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Classics in Mathematics

Charles B. Morrey, Jr. Multiple Integralsin the Calculus of Variations



Charles B. Morrey, Jr.

Multiple Integralsin the Calculus

of Variations

Reprint of the 1966 Edition

123



Originally published as Vol. 130 of the series Grundlehren der mathematischenWissenschaften

ISBN 978-3-540-69915-6 e-ISBN 978-3-540-69952-1

DOI 10.1007/978-3-540-69952-1

Classics in Mathematics ISSN 1431-0821

Library of Congress Control Number: 2008932928

Mathematics Subject Classification (2000): 49-xx, 58Exx, 35N15, 46E35, 46E39

© 2008, 1966 Springer-Verlag Berlin Heidelberg

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Die Grundlehren der

mathematischen Wissenschaften

in Ein2;eldarstellungen

mit besonderer Beriicksichtigung

der Anwendungsgebiete

Band 130

Herausgegehen von

J. L. Do o b • E . H einz • F . Hir^iebruch • E . H opf

H . Hopf • W . Maak • S. M acLane

W . M agnu s • D . M um ford • F . K . Schmidt • K . Stein

Geschdjtsjuhrende Heramgeher

B. Eck m ann un d B . L. van der W aerden



Multiple Integrals in the Calculus

of Variations

Charles B. Morrey, Jr .

Professor of Mathematics

University of California, Berkeley

Springer-Verlag Berlin Heidelberg New York 1966



Geschaftsfiihrende Herausgeber:

Prof. Dr. B. E c k m a n nEidgenossische Technische Hochschule Zurich

Prof. Dr. B. L. van der W aerdenMathematisches Institut der Universitat Zurich

All right reserved, especially that of translation into foreign languagesi It is alsoforbidden to reproduce this book, either whole or in part , by photomechanicalmeans (photostat, microfilm and/or microcard or any other means) without written

permission from the Publishers

(c) by Springer-Verlag, Berlin • Heidelberg 1966

Library of Congress Catalog Card Number 66-24 365

Printed in Germany

Title No. 5113



Preface

T h e p r i n c i p a l t h e m e of th i s book is '' the exis tence and dif fe ren t iab i l i ty

of the s o lu t ions of v a r i a t i o n a l p r o b l e m s i n v o l v i n g m u l t i p l e i n t e g r a l s / '

We sha l l d iscuss the co r re s pond ing ques t ions for s ing le in tegra ls on lyvery br ie f ly s ince these have been d iscussed adequa te ly in eve ry o the r

b o o k on the ca lcu lus of va r ia t ions . Moreove r , app l ica t ions to engineer

ing , phys ics , etc., are not discussed at all; h o w e v e r , we do discuss

mathematical app l ica t ions to such subjec ts as the t h e o r y of h a r m o n i c

in teg ra l s and the s o -ca l l ed ' ' ^^Neumann" p rob lem (see C h a p t e r s 7 and 8).

Since the p l a n of the b o o k is descr ibed in Sect ion 1.2 below we shall

m e r e l y m a k e a few obs e rva t ions he re .

In o rde r to s t u d y the ques t ions men t ioned above it is neces s a ry to

us e s ome ve ry e lemen ta ry theo rems abou t convex func t ions and o p e r a

t o r s on B a n a c h and Hilbe r t s paces and some spec ia l func t ion spaces ,

n o w k n o w n as ' ' S O B O L E V s p a c e s " . H o w e v e r , m o s t of the fac ts which we

use concern ing these spaces were known before the war w h e n a different

t e r m i n o l o g y was u s e d (see CALKIN and M O R R E Y [5]); but we h a v e in

c luded some powerfu l new re s u l t s due to CA LD ERON in our expos i t ion

i n C h a p t e r 3. The def in i t ions of these spaces and s o m e of the proofs

have been made s imp le r by us ing the mos t e lemen ta ry ideas of d is t r ibu

t ion theo ry ; howeve r , a lmos t no o t h e r use has b e e n m a d e of t h a t t h e o r ya n d no knowledge of t h a t t h e o r y is r e q u i r e d in orde r to read th i s book .

Of course we h a v e f o u n d it neces s a ry to deve lop the t h e o r y of l inear

e l l ip t ic sys tems at s ome leng th in orde r to p r e s e n t our desired differenti

ab i l i ty re s u l t s . We found it pa r t i cu la r ly e s s en t ia l to cons ide r ' 'weak

s o l u t i o n s ' ' of s uch s ys tems in w h i c h we were often forced to allow dis

cont inuous coeff ic ien ts ; in th is connec t ion , we inc lude an expos i t ion of

t h e D E G IO R G I— N AS H— M O S E R re s u l t s . And we inc lude in C h a p t e r 6

a proof of the a n a l y t i c i t y of the s o lu t ions (on the in te r io r and at theb o u n d a r y ) of the mos t gene ra l non - l inea r ana ly t i c e l l ip t i c s y s tem wi th

general regular (as in A G M O N , D O U G L I S , and N I R E N B E R G ) boundary con

d i t ions . But we confine ourselves to func t ions wh ich are a n a l y t i c , of

class C" , of class C^ or C^ (see § 1.2), or in some Sobolev space H^ w i t h

m an in tege r > 0 (except in C h a p t e r 9). Thes e l a t t e r s paces have been



vi Preface

defined for all real w in a domain (or manifold) or on its boundary andhave been used by many authors in their studies of linear systems. Wehave not included a study of these spaces since (i) this book is alreadysufficiently long, (ii) we took no part in this development, and (iii) thesespaces are adequately discussed in other hooks (see A. F R I E D M A N [2]»HoRMANDER [1], LiO NS [2]) as wcU as in m an y pa pe rs (see § 1.8 an dpapers by L I O N S and MAGE NE S) .

The research of the author which is reported on in this book hasbeen partia lly supp orted for several years by the O ffice of N ava l R esearchunder contract Nonr 222(62) and was partially supported during theyear 1961— 62, while the au thor was in France , by the N ational ScienceFoun dation under the grant G— 19782.

Berke ley , Augus t I966

C HA R L ES B . M O R R E Y, J R .



Contents

Chapter 1

I n t r o d u c t i o n1 .1 . I n t r o d u c t o r y r e m a r k s 1

1 .2 . T h e p l a n of t h e b o o k : n o t a t i o n 2

1 .3 . V e r y b r i e f h i s t o r i c a l r e m a r k s 5

1 .4 . T h e E u L E R e q u a t i o n s 7

1 .5 . O t h e r c l a s s i c a l n e c e s s a r y c o n d i t i o n s 1 0

1 .6 . C l a s s i c a l s u f f i c i e n t c o n d i t i o n s 1 2

1 .7 . T h e d i r e c t m e t h o d s 1 5

1 .8 . L o w e r s e m i c o n t i n u i t y 1 9

1 .9 . E x i s t e n c e 2 3

1 .1 0. T h e d i f f e r e n t i a b i l i t y t h e o r y . I n t r o d u c t i o n 2 61 . 11 . D i f f e r e n t i a b i l i t y ; r e d u c t i o n t o l i n e a r e q u a t i o n s 3 4

C h a p t e r 2

S e m i - c l a s s i c a l r e s u l t s

2 . 1 . I n t r o d u c t i o n 3 9

2.2. E l e m e n t a r y p r o p e r t i e s of h a r m o n i c f u n c t i o n s 4 0

2 . 3 . W E Y L ' S l e m m a 4 1

2 . 4 . P O I S S O N ' S i n t e g r a l f o r m u l a ; e l e m e n t a r y f u n c t i o n s ; G R E E N ' S f u n c t i o n s 4 32.5 . P o t e n t i a l s 4 7

2.6. G e n e r a l i z e d p o t e n t i a l t h e o r y ; s i n g u l a r i n t e g r a l s 4 8

2.7. T h e C A L D E R O N - Z Y G M U N D i n e q u a l i t i e s 5 5

2.8. T h e m a x i m u m p r i n c i p l e f o r a l i n e a r e l l i p t i c e q u a t i o n o f t h e s e c o n d o r d e r 6 I

C h a p t e r 3

T h e s p a c e s H"^ a n d H'^Q

3.1 . D e f i n i t i o n s a n d f i r s t t h e o r e m s 6 23 .2 . G e n e r a l b o u n d a r y v a l u e s ; t h e s p a c e s ii/^Q ( G ) ; w e a k c o n v e r g e n c e . . . 6 8

3.3. T h e D iR i CH L ET p r o b l e m 7 0

3 .4 . B o u n d a r y v a l u e s 7 2

3.5. E x a m p l e s ; c o n t i n u i t y ; s o m e SoB O L EV l e m m a s 78

3 .6 . M i s c e l l a n e o u s a d d i t i o n a l r e s u l t s 8 1

3 .7 . P o t e n t i a l s a n d q u a s i - p o t e n t i a l s ; g e n e r a l i z a t i o n s 8 6

vi i i Contents

Chapter 4

E x i s t e n c e t h e o r e m s

4.1. The lower-semicont inu i ty theorems of S E R R I N 904.2. Var i a t i o n a l p ro b l ems w i t h / = f{p); th e eq ua tion s (1.10.13) w ith N = \,

Bi = 0, A^" = A<^{p) 98

4.3. The borderl ine cases k = v 1054.4. The genera l quasi-re gular inte gra l 112

Chapter 5

D i f f e r e n t i a b i l i t y o f w e a k s o l u t i o n s

5.1. In t roduct ion 126

5.2. G eneral t he or y; r > 2 . . 1285.3. Extensions of the DE GIORGI-NASH-MOSER res ul ts ; v > 2 134

5.4. T he case I' = 2 1435.5. Lp a n d SCHAUDER estimates 14 9

5.6. The equ at io n a- V ^w + ^ *Vw + c w — Aw = / 1575.7. An alyt ic i ty of the solut ions of ana lyt ic l inear equ at ion s i645.8. An alyt ic i ty of th e solut ions of ana lyt ic , non-l inear , el l ip t ic equ at ion s 1705.9. Pro pert ie s of th e ex trem als ; regu lar cases 1865.10. T h e e x tr e m a ls i n t h e c as e 1 < ^ < 2 1915.11. The theory of L A D Y Z E N S K A Y A a n d U R A L ' T S E V A 19 4

5.12. A class of non -line ar eq ua tio ns 203

Chapter 6

R e g u l a r i t y t h e o r e m s f or t h e s o l u t i o n s of g e n e r a le l l i p t i c s y s t e m s a n d b o u n d a r y v a l u e p r o b l e m s

6.1. In t ro d u c t i o n 2 0 96.2. Inte r ior es t im ates for general el l ip t ic system s 2156.3. Est im ates near the bo und ary ; coerciveness 2256.4. W ea k solutio ns 2426.5. The exis tence theory for the DIRICHLET problem for s t rongly el l ipt ic

sys tems 2516.6. The an aly t ici ty of th e solut ions of ana lyt ic system s of l inear el l ipt ic

equat ions 2586.7. The an alyt ic i ty of the solut ions of an aly t ic nonl ine ar el l ipt ic system s 2666.8. The different iabi l i ty of th e solut ions of non-l inear el l ipt ic sys tem s;

weak so lu t ions ; a per tu rbat io n theorem 277

Chapter 7

A v a r i a t i o n a l m e t h o d in t h e t h e o r y of h a r m o n i c i n t e g r a l s

7A. In t ro d u c t i o n 2 8 67 .2 . Fu n d am en t a l s ; t h e GAFFNEY-GARDING inequal i ty 288

7.3. The var ia t ion al me thod 2937.4. Th e deco m posi t ion theore m . Fi na l resu l ts for com pac t manifolds wi th

o u t b o u n d ary 2 957.5. Manifolds w ith bo un da ry 3007.6. Different iabi l i ty at the bo un da ry 305y .y . Poten t ia l s , the decom pos i t ion theore m 3097 .S . Bo und ary va lue p rob lems 314



Conten t s ix

Chapter 8

T h e ^ - N E U M A N N p r o b l e m on s t r o n g l y p s e u d o - c o n v e x m a n i f o l d s

8.1. In t ro d u c t i o n 3 1 68 .2 . R esu l t s . Exa m ples . The analy t i c em bedding theore m 3208.3. Some im po r ta n t fo rmulas 3288.4. T he HiLBER T spa ce re su lts 3338.5. Th e local ana lysis 3378.6. Th e sm ooth ness resu l ts 341

Chapter 9

I n t r o d u c t i o n t o p a r a m e t r i c I n t e g r a l s ; t w o d i m e n s i o n a l p r o b l e m s

9.1. In t ro d u c t i o n . Pa ra m et r i c i n t eg ra l s 3 499.2. A lower sem i-con t inui ty theo rem 354

9.3. Two d imens ional p rob lem s; in t rod uct io n ; th e conformal map ping ofsurfaces 362

9.4. The problem of PLATEAU 37 4

9-5- The genera l two-d imen s ional par am et r ic p rob lem 390

Chapter 10

T h e h i g h e r d i m e n s i o n a l PLATEAU p r o b l e m s

10.1. In t ro d u c t i o n 4 0 010.2. V surfaces , their boundaries , and their HAUS DORF F m easures . . . . 407

10.3- The topological resul ts of ADAMS 41410.4. Th e min imizing seque nce; th e minim izing set 42110.5- The local topological disc pro pe rty 43910.6. T h e R E I F E N B E R G cone inequ al i ty 45910.7. Th e local differe ntiabil i ty 47410.8. Addit ional resul ts of F E D E R E R concern ing LEBESGUE v-area 48O

B i b l i o g r a p h y 4 9 4

I n d e x 5 0 4



Multiple Integrals in the Calculus

of Variations

Chap te r 1

Introduction

1 .1 . In trod u ctory remark s

Th e pr inc ip a l th em e of these lec tures is ' ' t h e ex is tence an d d i f fe ren ti

ab i l i ty of the so lu t ions of var ia t iona l p roblems involv ing mul t ip le in

tegra ls / ' I sha l l d iscuss the corresponding ques t ions for s ing le in tegra ls

only very br ie f ly s ince these have been adequa te ly d iscussed in everybook on the calculus of varia t ions (see , for ins tance, A K H IEZER [1],

B L I S S [ 1] , B O L Z A [ 1] , C A R A T H E O D O R Y [2 ], F U N K [1 ], P A R S [ 1] . Moreover,

I sha l l no t d iscuss app l ica t ion s to engineer ing , phys ics , e tc . , a t a l l ,

a l though I s ha l l men t ion s ome mathematical a p p l i c a t i o n s .

In general , I shall consider integrals of the form

(1.1.1) I [^,G-) = ff[^,z{x), V z{x)]dxG

wh ere G is a dom ain ,(1.1.2) X = ( A ; 1 , . . ., x"), z = (^1, . . ., z^), dx = dx^ . . . dx\

z{x) is a vector function, V^ denotes i ts gradient which is the set of func

t ions {zioc}, w h e r e z^^ d e n o t e s dz'^ldx°', a n d f{x, z, p) [p = {pi}) is

gene ra l ly a s s umed con t inuous in a l l i t s a rgumen ts . The in teg ra l s

b

f l/l + {dzldx)^ dx and j y [ ( ^ ) ' + ( ^ , ) ' ] dxidx^

a G

are familiar ex am ple s of inte gra ls of th e form (1.1.1 .) in which iV = 1 inb o t h ca ses, i ' = 1 in th e first case , i = 2 in th e sec on d case a nd th e

corresp ondin g func t ions / a re def ined respec t ive ly by

f{x,z,p) = yi +p^, f{^,z,p) =pi+piwhere we have omi t t ed the s upe rs c r ip t s on z a n d p s ince N =^ \, T h e

second integral is a special case of the Dirichlet integral which is defined

in genera l by

(1.1.3) D{z,G) = j\\Jz\^dx,f{x,z,p) ^\p\^ = 2[pi)'^.G i,(x

Anothe r example i s the area integral

<• < * ' i'-<^^ =Mimj+[m,j+m^p-''"G

M o r r e y , M u l t i p l e I n t e g r a l s



2 In t ro d u c t i o n

which gives the area of the surface

( 1 . 1 . 5 ) z^ == z^x'^.x^), {x^,x^)^G, / = 1 , 2 , 3.

I t i s to be no t iced tha t the a rea in teg ra l ha s the s pec ia l p rope r ty tha t i t

i s inva r ian t unde r d i f feomorph is ms (1 — 1 dif fe ren t iab le mappings , e tc . )

of the domain G onto o ther domains . This i s the f i rs t example of an in

tegral in parametric form. I shall discuss such integrals la ter ( in Chapters

9 and 10).

I shall a lso discuss briefly integrals l ike that in (1 .1 .1) but involving

der iva t ives of h igher order . And, o f course , the var ia t iona l method h a s

been used in problems which involve a ' ' func t iona l ' ' no t a t a l l l ike the

in teg ra l in (1 .1 .1 ) ; a s for examp le in p rov in g the R iem ann m app ing

theo rem where one min imizes s up \f{z) \ am ong a ll s ch l ich t func t io ns / (^ )def ined on the g iven s imply connec ted reg ion G for which f{zQ) = 0 and

f'(zo) = 1 a t some g iven poin t ZQ in G.

We sha l l cons ider on ly problems in which the domain G is f ixed;

va r ia t ions in G may be taken care of by t rans format ions of coord ina tes .

W e s ha l l u s ua l ly cons ide r p rob lems invo lv ing f ixed bounda ry va lues ;

we sha l l d iscuss o ther p roblems but wi l l no t der ive the transversality

condi t ions for such problems .

1.2 . The p lan of the book: notat ion

In th i s chap te r we a t t empt to p re s en t an ove ra l l v iew o f the p r inc ipa l

theme of the book as s ta ted a t the beginning of the preceding sec t ion .

However , we do not inc lude a d iscuss ion of in tegra ls in parametr ic form;

these a re d iscussed a t some length in Ch apte rs 9 an d 10. Th e mat er i a l in

this book is not presented in i ts logical order. A poss ible logical order

would be §1 .1 — 1.5, Chapter 2, Chapter 3, §§ 5.1 — 5-8, § 5.12, Chapter 6,

§§ 1.6—1.9, §§4.1 , 4.3> 4.4. Then the reader must skip back and forth as

required among the material of § 1.10, 1.11, 4.2, 5-9, 5-10 and 5-11. Thenthe rema inde r o f the book may be read s ubs tan t ia l ly in o rde r . Ac tua l ly ,

Ch apte rs 7 an d 8 could be rea d imm edi a te l y a f te r § 5-8 .

W e beg in by p re s en t ing background ma te r ia l inc lud ing de r iva t ions ,

unde r re s t r i c t ive hypo thes e s , o f Eu le r ' s equa t ions and the c la s s ica l

necessa ry condi t ion s of Leg endr e and W eiers t r ass . Ne xt , we inc lude a

br ie f a nd inco m ple te p res en ta t i on of th e c lassica l so-ca lled ' ' su ff ic iency "

cond i t ions , inc lud ing re fe rences to o the r works whe re a more comple te

p re s en ta t ion may be found .Th e second hal f o f th is chap te r p resen t s a reas ona bly comp le te ou t

l ine of the exis tence and differentiabil i ty theory for the solutions of

var ia t iona l p roblems . This beg ins wi th a br ie f d iscuss ion of the deve lop

ment of the d i rec t methods and of the success ive ly more genera l c lasses

of "admiss ib le" func t ions , cu lmina t ing in the so-ca l led "Sobolev spaces" .



1.2. The plan of the book: notation 3

Thes e are then def ined and discussed briefly after which two t h e o r e m s

on lower - s emicon t inu i ty are pre s en ted . Thes e are not the mos t gene ra l

theorems poss ib le but are selected for the s impl ic i ty of the i r p roofs

wh ich , howeve r , a s s ume tha t the r e a d e r is will ing to g r a n t the t r u t h of

s ome we l l -known theo rems on the Sobolev spaces . The r e l e v a n t t h e o r e m s

abou t the s e s paces are p r o v e d in C h a p t e r 3 and more genera l lower-

s e m i c o n t i n u i t y and ex i s tence theo rems are p r e s e n t e d in C h a p t e r 4.

In Sec t ion 1.10 the dif fe ren t iab i l i ty resu l ts are s t a t e d and some pre l i

m i n a r y r e s u l t s are p r o v e d . In Sect ion 1.11, an ou t l ine of the differenti

ab i l i ty theo ry is p r e s e n t e d . I t is f i rs t shown tha t the so lu t ions are ' ' w e a k

s o l u t i o n s " of the E u l e r e q u a t i o n s . The t h e o r y of the s e non - l inea r equa

t ions is r e d u c e d to t h a t of l inear equa t ions which , in i t ia l ly , may h a v e

discontinuous coeffic ients . The t h e o r y of these genera l l inear equa t ionsis discussed in de ta i l in C h a p t e r 5 • H o w e v e r , the higher order differenti

ab i l i ty for the so lu t ions of systems of Eule r equa t ions requ i red the s a m e

m e t h o d s as are us ed in s t u d y i n g s y s t e m s of e q u a t i o n s of h ighe r o rde r .

Acco rd ing ly , we p r e s e n t in C h a p t e r 6 m a n y of the re s u l t s in the two

recent papers of A G M O N , D O U G L IS , and N I R E N B E R G ([1], [2]) concerning

the so lu t ions and weak so lu t ions of s uch s ys tems . Bo th the L^^-estimates

a n d the ScHAUDER-type es t imates (concerning H O L D E R con t inu i ty ) are

p r e s e n t e d . We have inc luded sec t ions in b o t h C h a p t e r s 5 un d 6 p r o v i n g

the ana ly t i c i ty , inc lud ing ana ly t i c i ty at the b o u n d a r y , of the so lu t ions of

bo th l inea r and non-l inear ana ly t ic e l l ip t ic equa t ions and s y s t e m s ; the

mos t gene ra l ' ' p rope r ly e l l ip t i c " s ys tems wi th "complemen t ing bounda ry

c o n d i t i o n s " (see § 6.1) are t r e a t e d . The proof of a n a l y t i c i t y in th is genera

l i ty is new. In C h a p t e r 2 we pre s en t we l l -known fac t s abou t ha rmon ic

func t ions and genera l ized po ten t ia ls and conc lude wi th proofs of the

CALDERON-ZYGMUND inequa l i t ies and of the m a x i m u m p r i n c i p l e for the

s o lu t ions of second order equa t ions .

I n C h a p t e r s 7 and 8, we pre s en t app l ica t ions of the v a r i a t i o n a l m e t h o dto the H O D G E theory of harmonic integrals and to the so-called 5 - N E U -

MANN p ro bl em for exterior differentia l forms on s t rong ly p s eudo-convex

complex ana ly t i c man i fo lds wi th bounda ry . In C h a p t e r 9, we p r e s e n t a

brief discuss ion of ^^-dimensional parametric problems in gene ra l and

then d iscuss the two d imens iona l P la teau p rob lem in Euc l idean s pace

a n d on a R i e m a n n i a n m a n i fo l d. The cha p te r conc ludes wi th the a u t h o r ' s

simplified proof of the existence theorem of C E S A R I [4], D A N S K I N , and

SiGALO V [2] for the gene ra l two d imens iona l pa rame t r ic p rob lem ands ome incomple te re s u l t s conce rn ing the dif fe ren t iab i l i ty of the s o lu t ions

of such problems . In C h a p t e r 10, we p r e s e n t the author ' s s impl i f ica t ion

of the v e r y i m p o r t a n t r e s e n t w o r k of R E I F E N B E R G [1], [2], and [3] con

ce rn ing the h ighe r d imens iona l PLATEAU prob lem and the a u t h o r ' s ex

t ens ion of these resu l ts to var ie t ies on a R i e m a n n i a n m a n i fo l d .

4 Introduction

Notations. For the m o s t p a r t , we use s t a n d a r d n o t a t i o n s . G and D

wil l denote domains which are bounded unless o therwise spec if ied . We

d e n o t e the b o u n d a r y oiDby dD and its closure by D. We shall often use

t h e n o t a t i o n D C CG to m e a n t h a t D is c o m p a c t and D CG. B{xo, R)

d e n o t e s the bal l wi th cen te r at X Q and r a d i u s R. y^ and Fv d e n o t e the v-

m e a s u r e and [v — 1)-measure of5 ( 0 , 1 ) and ^ 5 ( 0 , 1 ) , r e s p e c t i v e l y . We

often denote dB{0,\) by Z. Mos t of the t ime (unless otherwise specified)

w e let Rq be ^ -d imens iona l number s pace wi th the us ua l me t r i c and ab

b r e v i a t e B{0,R) to BR, d e n o t e by a the {v — 1)-plane x" =0, and define

R^ ={x\x''>0}, R~={x\x'' <0}

(1.2.1) GR BnORi , ER ^dBnC .R , OR BROa

G^ BRHR-, Z =dBRnRv

If Sis a set inRq, \S\ deno te s its Lebes gue ^ -meas u re ; if ;t; is a p o i n t ,

d{x, S) d e n o t e s the d is tance of x from S. We define

[a^b] ={x\a'' < x°'<,b°', ^ = 1 , . . . , v, x^Rv}.

I n the case ofb o u n d a r y i n t e g r a l s , we often use dx'^ tod e n o t e nxdS

w h e r e dS is the surface area and nx is the -th c o m p o n e n t of the exterior

normal. We say t h a t a func t ion u C'^{G) iff (if and on ly if) u and its

p a r t i a l d e r i v a t i v e s oforde r < âre c o n t i n u o u s on G and u C'^{G) iff

u^ C'^{G) and each of its d e r i v a t i v e s of orde r < ^ can be e x t e n d e d to be

c o n t i n u o u s on G. If 0 < /i < 1, ^ Q(G) (or Q (G)) <^ (i.e. iff) u C^{G)

(or C^{G)) and all the d e r i v a t i v e s oforde r <n satis fy a H O L D E R ( L I P -

SCHITZ ii /J, = 1) cond i t ion on each compac t s ubs e t of G (or on the whole

of Gas e x t e n d e d ) . If u C^ (G), t h e n A {u, G) =sup \x2 — xi\-^.

\u(x2) — u(xi) I for xi and X2^G and Xi ^ x^. Ad o m a i n G is said to be

of class C^ (or Q , 0 < / < 1) iff G is b o u n d e d and each po in t PQ oi dG

is in an e i g h b o r h o o d n on G w h i c h can be m a p p e d by a 1 — 1 m a p p i n g

of class C (or CJJ), t o g e t h e r w i t h its inverse , on to GR U G R for s o m e Rin such a way t h a t P Q co r re s ponds to the origin and n U dG co r re s ponds

t o O R . li U^ ^ ' ^ { G ) > we d e n o t e its de r iva t ive s duldx^" by u,«. If ^ ^ C2(G),

t h e n Vû d e n o t e s the t ens o r u,oc^ w h e r e oc and /5 run independen t ly f rom

t o r . L i k e w i s e V ^ ^ = {^,a/Sy}, etc., and \\/û\^ — 2^\u,ocp\^, etc. IfG is

also of class C , then Green ' s theo rem becomes (in our no ta t ions )

f u ,oc(x) dx =J uUocdS = f udx'^.

G dG dG

S o m e t i m e s w h e n we wish to cons ider uas a func t ion of some s ingle x'^,

we wr i te x = (%*, x^) and u(x) =u(x^, x^) where x'^ d e n o t e s the r e m a i n

ing x^. One d imens iona l or{v — 1)-d imens iona l in tegra ls are t h e n in

d i c a t e d as m i g h t be expec ted . We often let o c d e n o t e a ' ' m u l t i - i n d e x ' \

i .e. a v e c t o r [ o c i , . . .,av) in which each o c t is a non-nega t ive in tege r . We

1.3- Very brief historical remarks

then def ined l«i u

Us ing th i s no ta t ion

(^,!),c« = ^ , f - = ( f i r - . . ( i ^ j ^

|a|=m

W e s ha l l deno te cons tan t s by C or Z w i t h or withou t s ubs c r ip t s . Thes e

con s tan t s wi l l, pe rh aps depend on o t h e r c o n s t a n t s ; in th is case we may

w r i t e C = C{h, jbt) if C depends on ly on h and //, for e x a m p l e . H o w e v e r ,

e v e n t h o u g h we may dis t inguish be tween d i f fe ren t cons tan ts in s ome

discuss ion by in s e r t ing s ubs c r ip t s , the re is no g u a r a n t e e t h a t C2, forexample , w i l l a lways deno te the s a m e c o n s t a n t . We s ome t imes deno te

t h e s u p p o r t of u by spt w. We d e n o t e by C^iG), Q(G), and C , {G ) the

se ts of func t ions in C°°(G), C^(G), or CJJ(G), respectively, which have

s u p p o r t in G (i.e. which van i s h on and n e a r dG). But it is h a n d y to say

t h a t u has s u p p o r t in GR \J GR <^ U van is hes on and n e a r ER (see 1.2.1);

we allow u{x) to be 9^ 0 on GR.

1.3 . Very brief h is torical rem arks

P r o b l e m s in the ca lcu lus of var ia t ions which involve on ly s ing le

in teg ra l s (1; = 1) have been d iscussed at leas t s ince the t i m e of the B E R

NOULLI'S. A l t h o u g h t h e r e was some ear ly cons idera t ion of doub le in te

gra l s , it was RiEMANN who a rous ed g rea t in te re s t in t h e m by prov ing

many in te re s t ing re s u l t s in func t ion theo ry by a s s u m i n g DIRICHLET'S

principle w h i c h may be s t a t e d as follows: There is a unique function which

minimizes the D IRICH LET integral among all functions of class C^ on a

domain G and continuous on G which takes on given values on the boundary

dG and, moreover, that function is harmonic on G.

RIEMA N N 'S w o r k was cri t ic ized on the grounds tha t ju s t becaus e the

in teg ra l was b o u n d e d b e l o w a m o n g the compe t ing func t ions it d i d n ' t

fo l low tha t the grea te s t lower bound was taken on in the class of c o m p e t

ing func t ions . In fact an e x a m p l e was given of a (1-d imens iona l) in tegra l

of the type (1.1.1) for which the re is no minimiz ing func t ion and a n o t h e r

was g iven of c o n t i n u o u s b o u n d a r y v a l u e s on the uni t c i rc le such tha t

D[z, G) = + 0 0 for eve ry z as above hav ing thos e bounda ry va lues .

The f i rs t example is the i n t e g r a l (see COU RA N T [3])1

(I.3.I) I{z,G) = | [ l + ( g ) ' J " ^ ^ . G = (0,1),0

the admiss ib le func t ions z being those ^ C^ on [0,1] w i t h

^(0) = 0 and ^(1) =- 1.

6 Introduction

O b v io u sl y I[z, G) > 1 for eve ry s uch z, I{z,G) has no u p p e r b o u n d and

if we define

Zr{x) = | _ ^

+ [1 + 3 (A; — r)^l[\ — r)2]i/2, r<.x<.\

w e see t h a t I{zr, G) -> 1 as r -> 1-.

The s econd example is the following (see COU RA N T [3]): It is now

known tha t Dir ich le t^s pr inc ip le ho lds for a circle and tha t each func

t i o n h a r m o n i c on the unit c irc le has the form

oo

(1.3.2) w{r,0) = + 2 r'^{ancosn0 + bnSinnO), (Un, bn const),

in po la r coord ina tes and t h a t the Dir ich le t in teg ra l is

(1.3-3) D{w,G)=jzZn(al +bl)

n= l

prov ided th i s sum converges . But ifwe definea^ =^-2 if == 1 ^ an =bn = 0 o the rwis e ,

w e see t h a t the series in (1 .3 .2) converges uniformly but t h a t in(1.3-3)

reduces to

^ 2 / A4

which d ive rges .

D IRICH LET'S princ ip le was es tab l ished r igorous ly in c e r t a i n i m p o r t a n t

cases by H ILBERT, LEBESG U E [2] and oth ers shor ly a f te r I9O O . T h a t was

the beg inn ing of the s o -ca l l ed ' ' d i rec t me thods " of the ca lcu lus of va r ia

t i o n s of w h i c h we sha l l say more l a te r .

T h e r e was renewed in te re s t inone d imens iona l p rob lems wi th the

a d v e n t of the M O R S E t h e o r y of the cr i t ica l po in ts of f u n c t i o n a l in whichM . M O R S E genera l ized his t h e o r y of cr i t ica l po in ts of functions defined

on f in i te -d imens iona l manifo lds [1] to cer ta in func t iona ls def ined on in

f in i te -d imens iona l spaces [2], [3]. He was able to o b t a i n the M O R S E

inequa l i t i e s be tween the n u m b e r s of p o s s i b l y ' ' u n s t a b l e ' ' (i.e. cr i t ica l

b u t not minimiz ing) geodes ies (and uns tab le minma l s u r faces ) hav ing

var ious ind ices (see also M O R S E and TO MPKIN S, [1] — [ 4]) . E x c e p t for the

l a t t e r (wh ich cou ld be r e d u c e d tothe case of cu rves ) , M O R S E ' S t h e o r y

was app l ied ma in ly to one -d imens iona l p rob lems . Howeve r , w i th in thelast twoyears, S M A L E andP A L A I S andS M A L E have found a modification

of M O R S E ' S t h e o r y w h i c h isapp l icab le to a wide class ofmul t ip le in te

g r a l p r o b l e m s .

V a r i a t i o n a l m e t h o d s are beg inn ing to be us ed in dif fe ren t ia l geometry .

For example, the author and Eells (see M O R R E Y and E E L S , M O R R E Y ,



1.4. The EuLER equations 7

[11] and C h a p t e r 7) deve loped the H O D G E theory ( [1] , [2]) by v a r i a t i o n a l

methods ( H O D G E ' S original idea [1]). H O R M A N D E R [2], K O H N [1], S P E N

CE R (KoHN and S P E N C E R ) , and the author ( M O R R E Y [19], [20]) have

app l ied va r ia t iona l t e chn iques to th e s t u d y of the 5 - N e u m a n n p r o b l e m

for exterior differentia l forms on complex ana ly t i c man i fo lds (see C h a p

te r 8; the au tho r encoun te red th i s p rob lem in his w o r k on the a n a l y t i c

e m b e d d i n g of rea l -ana ly t i c man i fo lds ( M O R R E Y [13])- Very recently ,

E E L L S and SAMPSON h a v e p r o v e d the exis tence of " h a r m o n i c " m a p p i n g s

(i.e. m a p p i n g s w h i c h m i n i m i z e an in t r ins ic Dir ich le t in tegra l ) f rom one

compac t man i fo ld in to a mani fo ld hav ing nega t ive cu rva tu re . S ince the

inf. of th i s in teg ra l is zero if the d imens ion of the compac t man i fo ld

> 2 , they found it neces s a ry to use a grad ien t l ine me thod wh ich

led to a non- l inea r s y s tem of pa rabo l ic equa t ions wh ich they thens o lved ; the c u r v a t u r e r e s t r i c t i o n was essen t ia l in the i r work .

1.4. The Eu ler eq u at ion s

After a n u m b e r of spec ia l p roblems had been so lved , E U L E R deduced

in 1744 the f i rs t genera l necessary condi t ion , now k n o w n as E U L E R ' S

e q u a t i o n , w h i c h m u s t be satis f ied by a min imiz ing or m a x i m i z i n g arc.

His de r iva t ion , g iven for the case N = v = 1, proceeds as follows: Sup

p o s e t h a t the func t ion z is of class C^ on [a, b] (= G) min imizes (for

example ) the i n t e g r a l I{z, G) a m o n g all s imila r func t ions having the

s ame va lues at a and h. T h e n , if C is any func t ion of class C^ on [^, h]

which van i s hes at a and h, the function 2: + AC is, for e v e r y A, of class

C^ on [a, h] and has the s ame va lues as at a a nd h. T h u s , if we define

6(1.4.1) w[X)=I[z+Xl:,G)=jf[x,z{x) +Xl:{x)^z'[x)+XC[x)]dx

a

(p m u s t t a k e on its m i n i m u m for A = 0. If we ass um e t h a t / i s of class C^

in its a r g u m e n t s , we find by dif fe ren t ia t ing ( I .4 . I ) and s e t t i n g X = 0 t h a t

h(1 .4 .2) /{C ' (^) 'U[x,z{x),z'{x)] + (:{x)f,[x,z{x),z'{x)]]dx = 0

(fv =

T h e i n t e g r a l in (1.4.2) is called the first variation of the in teg ra l / ; it is

s u p p o s e d to v a n i s h for eve ry f of class C^ on [a, 6] which van i s hes at a

a n d h. If we now assume that f and z are of class C^ on [a, H] (EU LER

h a d no c o m p u n c t i o n s a b o u t t h i s ) we can in tegrate (1 .4 .2) by p a r t s to

o b t a i n

h(1.4.3) jC{x)-{fz-~h]dx = 0, U=U[x,z[x),\/z{x)],etc.



8 In t ro d u c t i o n

Since (1.4-3) ho lds for a l l C as abo ve , i t follows th a t t he eq ua tio n

( 1 . 4 . 4 ) y,=h

must ho ld . This i s Euler's equation for the in tegra l / in this s imple case .If we write out (1.4.4) in full, we obtain

(1 .4 .5 ) fvv " +vz ' +px=z

which shows tha t Eule r ' s equa t ion is non- l inear and of the second order .

I t is , however, l inear in z" ; eq ua t ion s wh ich a re l inear in th e der i va t iv es of

h ighes t o rder a re f requent ly ca l led q^iasi-linear. The equa t ion ev iden t ly

b e c o m e s s i n g u l a r w h e n e v e r / ^ ^ =0. Hence regular v a r i a t i o n a l p r o b l e m s

are those for which fpp neve r van i s hes ; in tha t c a s e , i t i s a s s umed tha t

fpp >0 w h i c h t u r n s o u t t o m a k e m i n i m u m p r o b l e m s m o r e n a t u r a l t h a n

m a x i m u m p r o b l e m s .

I t i s c lea r tha t th is der iva t ion genera l izes to the mos t genera l in tegra l

(1.1 .1 ) p rov id ed t h a t / a n d th e min im iz ing (o r max imiz ing , e tc .) func t ion

z is of class C2 on the closed domain G which has a suff ic ien t ly smooth

bounda ry . Then , i f z min imizes /among all (vector) functions of c lass C^

with the s ame bounda ry va lues and C i s any such vec tor which vanishes

o n t h e b o u n d a r y o r G, it follows that 2:+AC is a" c o m p e t i n g ' ' o r ' ' a d

miss ib le" func t ion for each A so that if 99 is defined by

(1.4.6) (p{^) =I{z + U,G)

t h e n (p'{0) = 0. This leads to the condi t ion tha t

(1-4.7) ll\ C:Jpi + C'fz dx=0Q i=-l l a = l J

for a l l C as indi ca ted . T he int eg ral in (1 .4 .7) is the first variation of the

genera l in tegra l (1 .1 .1) . In tegra t ing (1 .4-7) by par ts leads to

Since th is is zero for all ve cto rs C, it follows t h a t

( ^ • 4 - 8 ) 2i M=f '' '-'' .N

which is aquas i - l inear sys tem of par t ia l d i f fe ren t ia l equa t ions of the

second order. In the case A^ =1, i t reduces to

(1.4.9) "-iV V

1.4. Th e EuLER equ at ion s 9

The equa t ion (1 .4-9) i s ev ident ly s ingula r whenever the quadra t ic form

( 1 - 4 . 1 0 ) 2 ' ^ a 2 . , ( ^ , ^ > ^ ) ^ a ^ ^ *

in A i s degene ra te .

We notice from (1.4-5) that ii N = v = \ and / dep end s on ly on p

and th e p rob lem is r egu la r , then E u le r ' s equa t ion reduces to

/ ' = 0 .

In gen eral , if / dep end s only onp(== pi), Eule r ' s equa t ion has the fo rm

an d ever y l inear vec to r function is a solutio n. In p ar t ic ul ar , if A^ = 1

a n d / = 1^12, Eu le r ' s equ a t io n is ju s t Lap laces eq ua t i on

a

In c a s e / = (1 + |_/>|2)i/4 as in th e first ex am pl e in § 1.3, we see t h a t

4fvP = {2-p^){^ +^2) -7 /4

wh ich is no t a lways > 0 . O n th e o ther han d fpp > 0 if |^| < ]/2 so

class ical resu lts wh ich we shall discuss la t er (see § 1.6) show t h a t th e

l inear func t ion z{x) = x min imizes the in teg ra l among a l l a rc s hav ing

\z-(x)\<^.

W e now re ve rt t o eq ua tio n (1.4-9)- If we tak e, for ins ta nc e, A^ = 1,

V =^ 2,f = pl — pl, then (1.4-9) becomes

which is of hyperbolic type. Moreover, the integral (1 .1 .1) with this /

o b v i o u s l y h a s n o m i n i m u m o r m a x i m u m , w h a t e v e r b o u n d a r y v a l u e s a r e

given for z. An yhow , i t i s we ll know n th a t bo un da ry va lue p rob lems a re

no t na tu ra l fo r equa t ions o f hype rbo l ic type , li v "> 2 s, g r e a t e r v a r i e t yof types may occur , depending on the s igna ture of the quadra t ic form

(1.4.10) . A similar objection occurs in a l l cases except those in which the

form (1.4-10) is positive definite or nega t ive def in i te ; we sha l l res t r ic t

ourse lves t o th e case whe re i t is po sit ive definite . In th is case Eu le r 's

equation is of elliptic type. Th e choice of th is cond i t ion o n / is re -enforced

by ana logy wi th the case r = 1 ; in tha t case fpp > 0 implies the con

vexity (see § 1.8) of f as a function of p for each {x, z) a n d t h e n o n - n e g a t i v e

def in i teness of th e form (1.4-10) i s equ iva le n t to th e con vex i ty o f / a s afunc tion of ^ i , . . .,pv for each set [x'^, . . ., x"^, z). O ur choice is re-enforced

fur ther by the c lass ica l der iva t ion g iven in the next sec t ion .

* Greek indices are sum m ed from 1 to i; an d L at i n indices are sum m ed from1 t o N. Hereaf ter we shal l usual ly employ the summat ion convent ion in whichrepeated ind ices are summed and summat ion s igns omi t ted .

1 0 In t ro d u c t i o n

1.5. Other c lass ical necessary condit ions

Su ppo se th a t / is of c lass C^ in i t s a rgu m en ts , th a t A^ = 1 , th a t z is

of class Ci on the closure of G, a n d t h a t z min imizes I(z, G) among a l l

func t ions Z of the same class which coincide with z o n t h e b o u n d a r y a n dare sufficiently close to z in the C^ norm, i .e .

\Z[x) — z[x)\ < ( 3 , |VZ(%) - Vz[x)\ < ( 5 , xon G.

In c lass ica l te rminology , we say tha t z furnishes a leêak relative minimum

t o I[z, G). We sha l l show tha t th is impl ies the non-nega t ive def in i teness

of the form (1.4.10) when z = z{x) a n d p = \/ z(x). W e n o t e t h a t o u r

hypo thes e s imp ly tha t fo r each C of the type above , van ish ing on the

b o u n d ar y , t h e fu nct ion 99(A), defined b y (1.4-6) is of class C^ for |A| <

Ao(>0) and has a re la t ive min imum a t 2 = : 0 . Th i s imp l ie s tha t

^ " ( 0 ) = f \Z a-^{x) C,«C,^ + 2 2:b-{x) a , . + c{x) m dxÔ

(1 .5 .1 ) G ^^'^ ^ J

a^^[x)=^f^^j,^[x,z{x),Vz{x)], b°^=fj,^^, c=f^^,

for all ^ as described. The integral in (1.5.1) is called the second variation

of the integral (1 .1 .1). By approximations , i t fol lows that (1 .5-1) holds

for a l l LiPSCHiTZ functions C w hich va nis h on th e bo un da ry . N ow let us

select a point xo(^G and a un i t vec to r A, and le t us choose new coord inate s y re la ted to : b y a t ran s for m at io n

(1.5.2) yy = i;dy{x^-x^), x^-x^^^z^ly''^ ôc^dioc y

w h e r e d is a. c o n s t a n t o r t h o g o n a l m a t r i x s o t h a t A i s the un i t vec tor in

the y i d i rec t ion , and def ine

co{y)=-a^{y)], 'ay'{y)â^^[x(y)-]dld$,

^ ^ ^ ^ 'hy {y) = b^[x{y )]dy, 'c{y) =^-c[x{y)].

Then i f G' denotes the image of G,

(p"[0) = / [ ^ ^ V*5(y)cL>,y ca,5 + 2'hy co'(D,y + 'c(jo^]dy > 0 .

Now, choose 0 <C h <. H so sm all t h a t the su pp or t of a> G G\ w h e r e

X f ( A - | y i | ) ( 1 - r / i f ) , if | y i | < / ^ , 0 < r < / f \

1.5.4) co(y^,...,y'') =y '- ' ^ ' ^ J . r^ ^ ^ - - ^ lo , o therw ise J

r2 :== (^2)2 + . . . + (y»')2.Then if we divide 99 ' ' (0) by the measure of the support of co and then l e t

h a n d H -> 0 so t h a t hlH -> 0 , we conc lude tha t

'all (0) = â^ (^^) ^1 ^1 ^ a''^{xo) >a A^ > 0

which is the s ta ted resu l t . This i s ca l led the Legendre condition.

1.5. Other classical necessary conditions 11

If we repeat this derivation for the case of the general integral (1 .1 .1),

a s s u m i n g / ^ C^ of c o u r se , w e o b t a i n

^"(0) =I{ZI<CUC$ + 2 2-bf,C'C\ + CijC'en dx > 0

< f (^) == hi 4 [ > ^ (^), V ^ [x)], 6?,. = /g^ pi^, c J- = /î 2?.

Making the change of variables (1 .5-2) and (1.5-3) and sett ing

co^(yi, . . ., y") = fâ)(yi, . . ., y "), {i= i, . . .,N)

where f i s an a rb i t ra ry cons tan t vec to r and co is defined by (1.5-4), and

l e t t i n g h and ^ -> 0 as above , we obta in

(^•5.5) 2 fp^ paxo.z{xo),Vz(xo)]Ao:Apii&'^0 for a l U , f ,

which is known as the Legendre-Hadamard condition (HADAMARD [1]). In

th i s ca s e , we s ay tha t the in teg ra l (1 .1 .1 ) o r the in teg rand / i s regular if

the inequali ty holds in (1 .5 .5) for a l l A 9^ 0 an d f 9^ 0. I t tu rn s ou t t h a t

the system (1.4-8) of Euler 's equations is strongly elliptic in the sense

defined b y N I R E N B E R G [2].

L e t u s s up p os e, n o w , t h a t / ^ C^ eve rywhere and tha t ^^ C^ on G

and min imizes I[z, G), as given by (1.1 .1), among all such functions with

t h e s a m e b o u n d a r y v a l u e s . A s i m p l e a p p r o x i m a t i o n a r g u m e n t s h o w s

t h a t z m inimizes / am on g a l l LIPSCHITZ func t ions wi th the s ame boundary va lues . Le t us choose XQ^G and a un i t vec to r X, and le t us in t ro

d u c e t h e y coordinates as in (1 .5-2) and le t us define (us ing part of the

notation of (1 .5 .4)

I (y^ + h)(p(rh-^l^) , —h<y^<0 , 0<.r<h^l^,

/î/2(/ji/2 _ y i ) .(p{rh-^l^), 0 < y i < A i/2, 0 < ^ < /? i /2 ,0 , o therw ise

(1.5.6)

w he re 99$ C ^ o n [0,1] w ith (p{0) = 1 an d 99(^) = 0 for Q near 1 . Since

the f i rs t var ia t ion vanishes , we have/ [/(^ , ^ + C/., V ^ + V u ) - fix, z,Vz)~ Cifz^ - ClJpl] dx->0,G

(1.5.7) fz^ ^ fz^(x, ^, V ^), fpi = fpi{x, z,Vz).

We not ice f i rs t tha t the in tegrand is 0(h) (since f^ and \7 Ch a re bo th

small) for x^R^ w he re 0 < y i < ^1^2^ 0 < r < Ai/2, g y se tt in g y i

= hrj^,r = Ai/2 ^ in Rl{—h < y i < 0, 0 < r < h'^^^), dividing by ^(»'+i)/2

a n d l e t t i n g A -> 0 , we obta in

/ | [/(^0,^0,i5)â+'^âl*X^)] -f{X0,Z0,p0) "

— Z^-<^ ^^ (p{Q)fpi{ô,zo,po)\drj[ > 0 .i. a J

Q^l

12 Introduction

W e may now choose a s equence {cpn} so q^niQ) -> 1 bounded ly . Th is l e ads

t o

(1.5.8) f{xo,zo,pio+^-^') -f{^o,zo,po) -Ao. i^fpi{xo,zo,po) > 0

w h i c h is th e Weierstrass condition (see GRAVES). In case N = 1, (1.5-8)

y ie lds the following more familiar form of th i s cond i t ion :

E{x,z,Vz,P) == f(x,z,P) — f{x,z,Vz) — [Poc — z,^)f^^[x,z,\Jz)^0

(1.5.9)

fo r all P and all x. The func t ion E(x, z,p,P) here defined is k n o w n as

t h e Weierstrass E-function.

H ESTEN ES and MACSHANE s tud ied the s e gene ra l in teg ra l s in cases

where v — 2. H E S T E N E S and E. H O L D E R studied the second variation of

the s e in teg ra l s . D ED ECK ER s t u d i e d the f i rs t var ia t ion of ve ry gene ra l

p r o b l e m s on mani fo lds .

1.6. Class ical suff ic ient con dit ion s

A de ta i l ed accoun t of classical and recen t work in this field is g iven

in the recen t book by F U N K , pp. 410— 433) wh ere o th er re fe rences are

given . I shall give only a br ie f in t roduc t ion to th i s s ub jec t .

I t is c lea r tha t the pos i t iveness of the s econd va r ia t ion a long a func

t i o n z g u a r a n t e e s t h a t z furnishes a r e l a t i v e m i n i m u m to I[Z, G) a m o n g

all Z ( = z on dG) in any f in i te d imens iona l space . However , if TV = 1, a

grea t dea l more can be c o n c l u d e d , n a m e l y t h a t z furnishes a strong

relative minimum to / , i.e. min imizes I[Z, G) a m o n g all Z^ C^{G-) w i t h

Z = z on dG ior w h i c h \Z{x) — z(x)\ <. d for s o m e ^ > 0 regard less of

the va lues of the d e r i v a t i v e s . W E I E R S T R A S S was the first to prove s uch

a theo rem but his proof was grea t ly s impl i f ied by the use of H I L B E R T ' S

i n v a r i a n t i n t e g r a l . Of course , the original proof was for the case N = v= 1; we presen t b r ie f ly an ex tens ion to the case N = \, v a r b i t r a r y .

Suppos e G is of class Q , z^ C^ (G), a n d / and fp are of class C^ in

t h e i r a r g u m e n t s , 0 < ju 0 for each C$ Q(G) ( c o m p a c t s u p p o r t ) .

B y a s t r a i g h t f o r w a r d a p p r o x i m a t i o n , it fo l lows tha t the s econd va r ia

t ion is defined for all f Hl^ (G) (see § 1.8). If we call the in tegra l ( I .5 . I )

h{z\^]G) we see from the t h e o r e m s of § 1.8 below that 7-2 is lower-semi-

con t inuous wi th re s pec t to weak conve rgence in H\Q {G) . Moreover, fromthe assumed pos i t ive def in i teness of the form (1.4.10), it follows from the

c o n t i n u i t y of the a'^P {x ) ( they ^ Q {G ) in fac t ) tha t there ex is t wi > 0

a n d Ml s u c h t h a t

(1.6.1) ^«^ {x)?ioc?i^ > mi |A|2, J [^'"^ W]^ ^ ^ 1 •

1.6. Classical sufficient conditions 13

Then, f rom the SCHWARZ and CAUCHY i n e q u a h t i e s , we c o n c l u d e t h a t

t h e r e is a i^ s u c h t h a t

(1.6.2) h{z\l:\G)>' j\\/^\^dx ~ K j l: dx.G G

Since weak convergence in HIQ{G) imphes s t rong conve rgence in L2{G)

(RELLICH 'S the ore m , T heo rem 3-4 .4), it fo l lows tha t there is a fo in

H\Q{G) (ac tua lly C^ (G)) which minimizes 12 among all I^^H\Q[G) for

w h i c h f C^ dx = 1. Since we h a v e a s s u m e d /2 > 0 for e v e r y f 9 0, it

G

fo l lows tha t

(1.6.3) l2(z;C]G) ÂijC^dxÂi > 0 .

G

F r o m the t h e o r y of §§ 5-2— 5.6 , it fo l lows tha t there is a un ique s o lu t ion

f of Jacohi's equation

(1.6.4) ^^î^(«"^f -) + iK -o)C = 0

w i t h g i v e n s m o o t h b o u n d a r y v a l u e s . It is to be n o t e d t h a t JA COBI 'S

e q u a t i o n is j u s t the E u l e r e q u a t i o n (z f ixed) corresponding to 12. I t is

also the e q u a t i o n of v a r i a t i o n of the E u l e r e q u a t i o n for th e orig ina l / ,i .e.

(1.6.5) ^ f = ^ { ^Jpc. [^.^ + eC, Vz + QVCl-fz [ sa m e ] } ^ ^ ^ .

I t fo l lows f rom Theorems 6.8.5 and 6.S.6 t h a t t h e r e is a un ique s o lu

t ion of the E u l e r e q u a t i o n for all suffic iently near (in C^ (dG)) b o u n d a r y

v a l u e s , in p a r t i c u l a r for the b o u n d a r y v a l u e s z -{ - Q, and t h a t z = z (Q)

satisfies an ord ina ry d i f fe ren tia l equa t ion

(1.6.6) p ^Fiz)

i n t h e B a n a c h s p a c e (C ^ [G)), w h e r e F{z) d e n o t e s the so lu t ion f of J a c o b i ' s

equa t ion (1 .6 .4) wi th z = Z{Q) for which C = 1 on dG. We shall show

be low tha t th i s s o lu t ion f canno t van i s h on G for Q suff ic ien t ly smal l ; it

is sufficient to do th i s for = 0, w h e n Z{Q) = our so lu t ion z, on a c c o u n t

of the c o n t i n u i t y .

So, let fi be th is so lu t ion . If Ci(^) < 0 a n y w h e r e , t h e n th e set w h e r e

th i s ho lds is an open set D and fi = 0 on D( C G). Since Ci is a so lu t iono n D, I(Ci, D) ^0 s ince Ci is m i n i m i z i n g on D. {D may no t be s m o o t h ,

b u t see C h a p t e r s 3 — 5)- But if we set ^ = î on D and C = ^, o the rwis e ,

f ^ HIQ {G) SO (1.6.3) hold s and we m u s t h a v e C = 0. H e n c e Ci{^) > 0

e v e r y w h e r e . Now, suppose Ci(ô) = 0. From Theorem 6 .8 .7 , it follows

t h a t we may choose R so s m a l l t h a t B(xo, R) G G and t h e r e is a non-

1 4 Introduction

vanish ing so lu t ion co of (1.6.4) onB(xo,R). L e t t i n g fi = coy, we see

t h a t V sa t is f ies the equa t ion

a n d v(xo) = 0 . Bu t f rom the max imum p r inc ip le a s p roved by E. H O P F

[1] (see § 2.8 ), it fo l lows tha t v^ c a n n o t h a v e a m i n i m u m i n t e r i o r to G.

Accord ingly Ci ^ 0a n y w h e r e in G.

There fo re it isposs ib le to embed our so lu t ion zin s. field of extremals.

Tha t i s , the re is a1 p a r a m e t e r f a m i l y Z{X,Q) of so lu t ions of Eule r ' s equa

t ion whe re Z{x, 0) == z{x), our g iven so lu t ion . Z(X,Q)^ C^{GX [— Ô>Ô ])

a n d ^ C^ (G) as a func t ion ofx foreach Q with ^^ > 0 . Cons equen t ly

there a re func t ions Po: {x, z) on the se t F, w h e r e

(\.6.7) F: x G, Z{x, ~QO) < ^ < Z{x, go),

which ac t as slope-functions for the field, i.e.

(1.6.8) Z,oc{x, Q) = P a [x, Z{x, Q)].

By v ir tue of the fac ts tha t Z[x, q) sa t is f ies Eule r ' s equa t ion for each Q,

tha t (1 .6 .8) ho lds , and tha t if {x, z) $F, t h e n z = Z(x, Q) for a u n i q u e Q

on [— ^0, ô], we find t h a t

^ ' ' ^ f,=fz\x^ z, P{x, z)l etc. , (^, z) ^ F.Let us define

/*(^, G) = ff*{x,z,Vz)dx,

(1.6.10) G'

/ * {x. z,p) =f , [X, z, P [x, z)]' \_p. - P « [x, z)\ + / [ ^ , z, P {X, z)\ .

W e o b s e r v e t h a t

Sl = [P- - Poc {X, Z)] ' {fp + / p ^ p^ Ppz} - Poczfvu

^^''^^^ +fz+fv,Po. z; f;^=fvJ^>z,P(x,z)].T h u s , ii z Gi(G) and {x, z{x)) ^F for x G, we see tha t

(1.6.12) / : [X, z (x), Vz(x)] - j; [X, z {x ), Vz{x)] =0.

Accord ing ly the integral I*(z,G) has the same value for all such z which

have the same boundary values. Moreove r , if z{x) = Z[x, q) fors ome q,

t h e n

^ ' I*{z,G)=I{z.G).

This in teg ra l 7 * (^, G) isk n o w n a s Hilbert's invariant integral. There fo re ,

if z Ci(G) and [x, z{x)) ^F for all x G, a n d z{x) = zo[x) on dG, t h e n

(1.6.14) I{z,G) - I(zo, G) = I(z, G) - I*{zo, G) = I(z, G) - I*{z, G)

= f E[x, z{x), P{x, z{x)}, Vz{x)] dx

1.7. The d i r ec t me t h o d s 15

w h e r e E(x, z, P, p) is the Weiers trass £^-function defined in (1.5-9).

Thus ZQ minimizes I{z, G) among all such z, and hence furnishes a strong

relative minimum to / , provided that

(1.6.15) E[x, z, P{x, z),p] > 0, [x, z)^r,p a r b i t r a r y .Th is s ame p roo f s hows th a t if (1.6.15) holds for all {x, z, p) in some domain

Ry where all the (x, z) involved ^ F, then I(zo, G) < I{z, G) for all z for

which [x, z[x), \7 z{x)'] ^Rfor all x^G.

I n the cases v = \, the Jacobi condition is f requen t ly s t a ted in t e r m s

of ' ' con juga te po in t s " . A co r re s pond ing cond i t ion for ) > 1 is t h a t the

J A C O B I e q u a t i o n has no non-zero so lu t ion which vanishes on the b o u n d

a r y dD oi any s u b - d o m a i n D G G; D ma y coinc ide w i th G or may not be

s m o o t h , in which case we say u van is hes ondD <û^ H\Q [D). The m o s tin te re s t ing cond i t ion is tha t the re ex i s t a non-van i s h ing s o lu t ion a> on G;

we have s een above tha t th i s is imp l ied by the p o s i t i v i t y of the second

v a r i a t i o n . If we t h e n set ^ = cou, w h e r e u = 0 on dG, t h e n the r e a d e r

may ea s i ly ve r i fy tha t

I2{^',^',G) = J[wâ°'û,ocU,p — uâ)L(o]dx > 0

G

for all u(^ H\Q (G), s ince Leo = 0.

In cases where v > 1 and A^ > 1 it is s t i l l t rue (if we c o n t i n u e toa s s u m e the s ame d i f fe ren t iab ih ty for G,/, and z) tha t (1 .6 .2) ho lds wi th

I2 defined by (1.5.1 ') even in the general regular case where (1 .5-5) holds

w i t h the i n e q u a l i t y for A 7^ 0 and f 7^ 0. T h i s is p r o v e d in § 5.2. So it is

s t i l l t rue tha t if /2 > 0 for all f, the E u l e r e q u a t i o n s h a v e a u n i q u e

so lu t ion for s u f f i c ien t ly nea rby bounda ry va lues . It is more diff icult

(but poss ible) to s h o w t h a t t h e r e is an A^-parameter field of e x t r e m a l s

a n d t h e n it t u r n s out t h a t s u c h a field does not lead so easily to an in

v a r i a n t i n t e g r a l . By allowing s lope functions P* {x, z) w h i c h are not

" i n t e g r a b l e " (i.e. t h e r e may not be zs s u c h t h a t z = P^ {x, z)), W E Y L

[1] (see a lso D E B O N D E R ) deve loped a compara t ive ly s imp le f i e ld theo ry

and s howed the exis tence of his t y p e s of f ie lds under ce r ta in condi t ions .

H i s t h e o r y is s ucces fu l i f / i s convex in all the p'l. To t r e a t more gene ra l

cases, C A R A T H E O D O R Y [1], B O E R N E R , and L E P A G E have introduced more

genera l f ie ld theor ies . The l a t t e r two not ic ed th a t ex te r io r d i f fe ren t ia l

forms were a n a t u r a l t o o l to use in fo rming the ana log of H i l b e r t ' s

i n v a r i a n t i n t e g r a l . H o w e v e r , the suff ic ien t condi t ions deve loped so far

a r e r a t h e r far from the necessary condi t ions and m a n y q u e s t i o n s r e m a i nt o be ans wered .

1.7. The d irect meth od s

The neces s a ry and suff ic ien t condi t ions which we have jus t d is

cus s ed have p re s uppos ed the exis tence and dif fe ren t iab i l i ty of an ex-

16 Introduction

t r e m a l . In the cases v = i, t h i s was often proved us ing the exis tence

t h e o r e m s for ord ina ry d i f fe ren t ia l equa t ions . Howeve r , un t i l r e cen t ly ,

co r re s pond ing theo rems for par t ia l d i f fe ren t ia l equa t ions were not ava i l

ab le so th e d i rec t me thods were deve loped to hand le th i s p rob lem and

to ob ta in re s u l t s in the la rge for one d imens iona l p rob lems .

As has already been said, H I L B E R T [1] and L E B E S G U E [2] had solved

the Di r ich le t p rob lem by es s en t ia l ly d i rec t me thods . Thes e me thods

were exp lo i t ed and popu la r ized by T O N E L L I in a series of p a p e r s and a

b o o k ([1], [2], [3], [4], [5], [7], [8]), and h a v e b e e n and s t i l l are be ing

us ed by m a n y o t h e r s . The idea of the d i r e c t m e t h o d s is to show (i) t h a t

t h e i n t e g r a l to be m i n i m i z e d is lower - s emicon t inuous wi th re s pec t to

s ome k ind of conve rgence , (ii) t h a t it is b o u n d e d b e l o w in th e class of

' ' a dmis s ib le func t ions , " and (iii) t h a t t h e r e is a minimizing sequence (i.e.a sequence {zn] of admiss ib le func t ions for w h i c h I[zny G) t e n d s to its

i n f i m u m in the class) which converges in the sense required to s ome

admiss ib le func t ion .

Tone l l i app l ied the s e me thods to many s ing le in teg ra l p rob lems and

s ome doub le in teg ra l p rob lems . In do ing th i s he found it e x p e d i e n t to

use un iform convergence (at leas t on in te r io r doma ins ) an d to allow

absolu te ly cont inuous func t ions (sa t is fy ing the g i v e n b o u n d a r y c o n

dit ions) as admis s ib le for one d i m e n s io n a l p r o b l e m s ; and he de f ined wha the ca l led abso lu te ly cont inuous func t ions of two va riab les ([6]) to h a n d l e

ce r ta in doub le in teg ra l p rob lems (see the nex t s ec t ion ) . In the doub le

i n t e g r a l p r o b l e m s (iV == 1, r = 2), he found it e x p e d i e n t to r e q u i r e t h a t

f[x, z, p) sa t is fy condi t ions such as

(1.7.1) m\p\^ — K <f{x, z,p), k>2, m>0, w h e r e

(1.7.2) f{x, z,p)'>0 and f{x, , 0) = 0 if k = 2,

in order to ob ta in equ icon t inuous min imiz ing s equences . Howeve r ,

Tone l l i was no t able to get a gene ra l theo rem to cover the case where /satisfies (1.7-1) with 1 < ^ < 2. Moreove r , if one cons iders in tegra ls in

w h i c h r > 2, one soon f inds tha t one w o u l d h a v e to requ i re ^ > i in

orde r to e n s u r e t h a t the func t ions in any min imiz ing s equence wou ld be

e q u i c o n t i n u o u s , at leas t on in te r io r doma ins (see Theorem 3.5-2). To see

t h i s , one needs on ly to n o t i c e t h a t the func t ions

l o g l o g (1 + 1 /|:v|), 1/|A ;|^ 0 < | A ; | < 1

are l imi ts of C^ func t ions Z n in w h i c h

j\SJZnYdx djnd j \\/Zn\^ dx for k<vl{h+\)

a re un i fo rmly bounded ove r the un i t ba l l G. In the ' ' bo rde r l ine ca s e"

k = V, it is poss ib le , in case / satisfies the s u p p l e m e n t a r y c o n d i t i o n

(1.7.2), to rep lace an a rb i t ra ry min imiz ing s equence of con t inuous func

t ions by a min imiz ing s equence each member of which is monotone in the

1.7. The direct methods 17

sense of Lebesgue ( i .e . takes on its m a x . and min . va lues on the b o u n d a r y

o f each compac t s ub -doma in ) ; the new s equence is e q u i c o n t i n u o u s on

in te r ior domains (see e 4.3)-

However , even th is Lebesgue smooth ing process does not w o r k in

gene ra l for 1 < ^ < r. In orde r to get a more comple te ex i s tence theo ry ,

the writer and C A L K I N ( M O R R E Y [5], [6], [7]) found it expedient to allow

as adm iss ib le , func t ions wh ich are s t i l l more genera l than Tonel l i ' s

func t ions an d to a llow correspo nding ly m ore genera l typ es of conve rgence .

T h e new spaces of func t ions can now be iden t i f ied wi th the B a n a c h

spaces HJ (G ) (see the next sec t ion) (or the Sobolev spaces Wl{G)) w h i c h

have been and still are be ing us ed by m a n y w r i t e r s in many d if fe ren t

connec t ions (see § 1.8). In th i s way , the w r i t e r was able to ob ta in ve ry

gene ra l ex i s tence theo rems . Unfo r tuna te ly the so lu t ion shown to exis twas known on ly to be in one of these genera l spaces and h e n c e w a s n ' t

e v e n k n o w n to be c o n t i n u o u s , let a lone of class C^! So these ex is tence

t h e o r e m s in thems e lves have on ly mino r in te re s t . Howeve r , at the s a m e

t i m e , i the wri te r was ab le to show in the case v = 2 {N a r b i t r a r y ) t h a t

these very genera l so lu t ions were , in fact , of class C^ after all p r o v i d e d

t h a t / satis f ied the condit ions (1 .10.8) below with ^ = r = 2. A g r e a t l y

s impl i f ied presen ta t ion of th i s old w o r k is to be found in the a u t h o r ' s

pap e r [15 ] ; r ecen t deve lopm en ts hav e pe rm i t t ed fu r the r s imp l if i ca t ionsand ex tens ions wh ich we shall discuss la ter .

In genera l , it is s t i l l not k n o w n t h a t the so lu t ions are c o n t i n u o u s if /

satisfies (1-7.1) with 1 < ^ < r. I n the case v = 2, N — \ T O N E L L I [8]

s h o w e d t h a t the s o lu t ions in th is case are c o n t i n u o u s if / is s u c h t h a t

t h e r e is a un ique min imiz ing func t ion in the smal l . More recent ly ,

SiGALOV ([1], [3]) s h o w e d t h a t the solution surfaces a lways posess con-

formal maps (poss ib ly wi th ver t ica l segments ) in the case v = 2, N = i.

I n the case v == 2, N = \, f == f{p) it was p r o v e d a long t ime ago by

A . HAAR (see also R A D O [2]) t h a t t h e r e is a un iqu e min im iz ing func tion

z w h i c h is defined on a s t r i c t ly convex doma in G and which satis f ies a

Lips ch i tz cond i t ion wi th cons tan t L, p r o v i d e d t h a t any l inear func t ion

which co inc ides wi th the g iven bounda ry va lues at th r ee d i f fe ren t po in ts

on the bounda ry has s lope < Z . This resu l t has recent ly been genera l ized

(not quite completely) by G I L B A R G and STAMPACCHIA [3]. The author

has comple ted the ex tens ion of H A A R ' S re s u l t s and has e x t e n d e d t h o s e

1 This work was completed during the year 1937 — 38 and the author lectured

on it in the seminar of Marston Morse during the spring of 1938 and also reportedon this work in an invited lecture to the American Mathematical Society at its

meeting in Pasadena, California, on December 2, 1939 [6]. The necessary theorems

about the H^ spaces (called ^ ^ at that time) (see § 1.8) were published in the papers

referred to above by CALKIN and the author [5]. The remainder of the work was

first published in a paper ( M O R R E Y (7]) which was released in December 1943," the

manuscript had been approved for publication in 1939-

Morrey, Multiple Integrals 2

18 Introduction

of STAMPACCHIA a n d GILBARG. Thes e re s u l t s a re p re s en ted and p roved

in § 4-2. T he resu lts inc lude GILBARG'S exis tence theorem for equa t ions

of th e form (I . IO . I3) wi th N = i, Bi = 0, A"- = A"" {p). T h e a d v a n t a g e s

of these theorems are that one can res tr ic t one 's se lf to LIPSCHITZ func

t ions z and no a s s umpt ion has to be made abou t how f{x, z, p) b e h a v e s

as 1 1 - > o o . Th e conv exi ty assu m ptio n for a ll (x, z,p) i s sugges ted by

the condit ions in §§ 1.4—1.6.

In the ex i s tence theo rems men t ioned above , the au tho r cons ide red

integrals of the form (1.1.1) in which v a n d N a r e a r b i t r a r y b u t i n w h i c h /

is convex in a l l the pl^; with th is convexi ty assumption , no d i f f icu l t ies

were in t roduced in the proofs by a l lowing AT > 1. Th e resu l ts ha ve been

extended and the o ld proofs grea t ly s impl i f ied by S E R R I N ([1], [2]); we

shall pres en t ( in § 1.8) a s imple low er-s em ico ntin uity proof b ase d onsome of his ideas and on some ideas of T O N E L L I . H ow ev er, for iV > 1,

the proper condit ion would be to assume that (1 .5-8) and/or (1 .5-5) held

for all (x, z, p, A, f ) . Th e au th or has s t ud i ed these gen era l in tegra ls ([9])

an d foun d th a t if / sa t is f ies th e co nd it ion s

mV^ - K <.f{x, z,p) <M V^, \fp{x, z,p)\<. M F ^ - i , k > 1

\M,\M<MVK m>0, F = (1 + 1 12)1/2

then a necessary and suff ic ien t condi t ion tha t I[z, G) be lower semicon-

t inuous on the s pace H\ (G) with respec t to un iform convergence is tha t

/ be quas i -convex as a func t ion oi p. A f u n c t i o n / ( ^ ) , p = {pl^} is quasi-

convex if and only if i t is continuous and

jflPo + VC(«)] dx >f{po) • \G\, C€ Q ( G ) ;G

th a t i s, l inea r vec to rs g ive the abs o lu te m in im um to I{z, G) among a l l z

with such boundary va lues (no te tha t l inear func t ions a lways sa t is fy

E U L E R ' S equa t ion i f f^C^). A necessary condi t ion for quas i -convexi ty is

jus t (1 .5 .5)- The author showed that (1 .5-5) is sufficient for quasi-con

vexi ty i f f{p) is of one of the two following forms

Wm = <Ppf K f ) c o ns t.

{2)f{p)=F{D i,...,D^^,). N = v+i

w h e r e F is ho m oge neo us of degree 1 in the Di a n d Di i s t h e d e t e r m i n a n t

o f the s ubma t r ix ob ta ined by omi t t ing the ^ - th co lumn o f the pi_ m a t r i x ;

or if for each p, the re ex i s t a l t e rna t ing cons tan t t ens o rs Af, Aff, . . .,

s u c h t h a t

fiP + 71) > / ( / ) ) + AtK + ^IKA + ••• + AX\\\\7il\.. .7^:;;

for all 71 . Under s ome add i t iona l cond i t ions the in teg ra l i s lower - s emi-

con t inuous wi th re s pec t to weak conve rgence in H\ (G), We discuss these

in teg ra l s in § 4 .4 - R ecen t ly N orm an M E Y E R S h a s e x t e n d e d t h e a u t h o r ' s



1.8. Lower semicontinuity 19

resu l ts to higher order in tegra ls and has i m p r o v e d t h e m s o m e w h a t . The

proofs are very s imila r to t h o s e of the a u t h o r for the f irs t order inte

gra ls an d we sha l l not p r e s e n t t h e m .

F i n a l l y , it s hou ld be p o i n t e d out t h a t the r -d imens iona l a rea in teg ra lin the n o n - p a r a m e t r i c c a s e w i t h AT > 1 is not c o n v e x in all the ^ j , but

is a regu la r in teg rand in the genera l sense ; for v = N = 2, the i n t e g r a l is

I(z,G) = J J l / l + (4)' + (4)' + (4) + (4)' + (TlJ^J'i^dy.G

f{x,z,p) = [1 + iPl)^ + (Pl)^ + iPl)^ + (Pl)^ + iPlPl - PlPiW^.

T h e i n t e g r a n d of a p a r a m e t r i c p r o b l e m is never regula r s ince it is

neces s a r i ly degene ra te bu t a large c lass of s uch in teg rands do satis fy(1.5.5) and (1.5.8) with the e q u a l i t y a l l o w e d ( w h e n e v e r / ^ C^). We shall

s p e a k a b o u t p a r a m e t r i c p r o b l e m s in C h a p t e r 9-

1 .8 . Lower semicon t in u i ty

W e beg in wi th a brief discussion of the spaces of admiss ib le func t ions

w h i c h we sha l l use ; a more ex tended d i s cus s ion inc lud ing comple te

proofs is given in C h a p t e r 3. I t is conven ien t to call these spaces S O B O L E V

s p a c e s ; in a d d i t i o n to the b r e v i t y of th is des igna t ion , it is a p p r o p r i a t es ince he proved s ome impor tan t r e s u l t s conce rn ing the s e func t ions [1]

a n d p o p u l a r i z e d t h e m in his book [2]. H o w e v e r , he was by no m e a n s the

f irs t person to use these func t ions . Beppo LEV I was p r o b a b l y the first

to use (in ( I906) adm iss ib le func t ions wh ich requ ired th e use of th e

Lebes gue in teg ra l to expre s s I{z, G); his func t ions (of two var iab les )

were con t inuous , abs o lu te ly con t inuous in each var iab le for a l m o s t all

va lues of the o the r , and the i r f i rs t par t ia l der iva t ives were in L2. In

discuss ing the a rea of surfaces , T O N E L L I [6] i n t r o d u c e d his n o t i o n of

abs o lu te ly con t inuous func t ions (ACT) in 1926. His definit ion was

i d e n t i c a l w i t h t h a t of L e v i e x c e p t t h a t the pa r t i a l de r iva t ive s we re

requ i red on ly to be in Li; he used these func t ions to discuss the doub le

i n t e g r a l p r o b le m s m e n t i o n e d a b o v e . But m e a n w h i l e in I92O , G. C. EVANS

{W > [2]) had encountered more genera l func t ions , essen t ia l ly those we

n o w use, in his s t u d y of p o t e n t i a l t h e o r y . R E L L I C H p r o v e d the c o m p a c t

ness in L2 of b o u n d e d s e t s in HI. SOBOLEV p r o v e d his well known resu l ts

o n t h e L p - p r o p e r t i e s of these func t ions in 1938 [1]. In 1940, J. W. CALKIN

a n d the a u t h o r ([5]) p r o v e d m a n y of the f u n d a m e n t a l p r o p e r t i e s of the s efunc t ions s ta ted be low. S ince the war A RON SZA JN and SMITH h a v e s t u d i e d

these func t ions in grea t de ta i l [1]. No d o u b t m a n y o t h e r s h a v e s t u d i e d

thes e func tions . R ecen t ly the s e func t ions h ave been used by m a n y p e o p l e

in ma ny different connect ions (see, for ins tan ce, D E N Y , F R I E D R I C H S [1],

[2], [3], FuBiNi, J O H N [2], LAX, M O R R E Y [1], [10], M O R R E Y and E E L L S ,

2*



2 0 In t ro d u c t i o n

NiKODYM, SHIFFMAN, SIGALOV ([2]); their use is now s t a n d a r d in p a r t i a l

differential equations (see F R I E D M A N [2], H O R M A N D E R [1], L I O N S [1]).

Definition 1.8.1. We say t h a t a func t ion z is of class Hl[G), r > 1, if

and on ly if z is of class Lr[G) and there ex is t func t ions pi, . . ., pv, also

of class Lr{G), s u c h t h a t

(1.8.1) jg[x)Pa,[x)dx= -lg,oc{x)z{x)dx, g^Cl{G),oc= \,... ,v.G G

Remarks. It is c lea r tha t the func t ions px are u n i q u e l y d e t e r m i n e d

u p to nul l func t ions and t h a t if z is of class Hj{G) and z'^{x) = z{x)

a lmos t eve rywhere , then ^* is of class Hl{G) and the same func t ions po^

will do for 2:*.

Definition 1.8.2. As in the case of the L^-spaces , the e l e m e n t s of the

space Hl[G) are classes of equ iva len t func t ions of class Hj(G). We d e n o t e

the c lasses of equ iva len t func t ions px by z^« and ca l l them the distribu

tion derivatives of the e l e m e n t z. An e lemen t z^ space H'^[G) if and only

if z and its d is t r ibu t ion de r iva t ive s up to orde r m — 1 are success ively

seen to Hj (G).

Remark. N a t u r a l l y we may rega rd an e l e m e n t z in H^ (G) as a d is t r i

b u t i o n a n d t h e n the d is t r ibu t ion co r re s pond ing to z^« would be the der iva

t i v e of z in the d is t r ibu t ion s ens e .

Definition 1.8.3. 99 is a Friedrichs mollifier if and only if 99$ ^^{Rv),q)(x) > 0, spt 99 (i.e. the s u p p o r t of 99) C ^ ( 0 , 1 ) , and

f (p(x) dx = \.B{0,1)

If ^ is loca l ly s ummab le on an open set G, we define its cp-mollified func

tion UQ by

u,{x)= I u{i)(pf{^-x)di, X^G, = {X^G\B(X,Q)CG},

B(X,Q)

The fo l lowing theorems a re a lmos t ev ident and a re proved in C h a p t e r 3

( in fact , Theorem B is e v i d e n t ) :

Theorem A. The space H^ (G) is a Banach space binder the norm

lUlli _ _ IMlo 4- V l l ^ !lo\\v,G — \r\\v,G^ ^ \r '°'\\v,G

a = l

2 ' l k ,

Theorem B. (a) If u^Lp{G) and UQ denotes its (p-mollified

then UQ ^ C'^(Gg) and UQ ->u in Lp(D) for each D CCG.(b) If u^ ^KQy ^^^^ ê ,a(^) = {u ,OC)Q(X) for X^GQ SO that UQ ->u

and UQ ôc ->u ôc in Lp(D) for each D C C G.

(c) The convergence in {a ) and {b) holds for almost all x.

The lower s emicon t inu i ty theo rems in th is sec t ion depend on s ome

we l l -known theo rems on convex func t ions which we now define:

1.8. Lower semicontinuity 21

Definit ion 1.8.4. A set S in a l inear space is said to be convex ^ the

s e g m e n t P1P2C. S w h e n e v e r Pi and P2^S. A function 99 is sa id to be

convex on the convex set S <=>

^[ (1 - A) fi + U2] < (1 - > ) (fih) + M^2), fi, f2€ 5 , 0 < A < 1.

R e m a r k . E v i d e n t l y cp is c o n v e x on the c o n v e x set 5 <^ the set of

p o i n t s (f, f) for w h i c h f ^ S and C > 99(1) is convex .

W e now s t a t e the c h a r a c t e r i s t i c p r o p e r t y of convex func t ions :

Lemma 1.8.1. A necessary and sufficient condition that cp he convex on

the open convex set S G Rp is that for each ^ in S there exists a linear func

tion ap ^ -\- b such that

(1.8.2) cp{0=apC^ + b, (p(^)^apS^ + b, ^^S.

If (p is of class C^ on S, this condition is equivalent to

(1.8.3) £ ( ^ , l ) - 9 5 ( | ) - 9 , ( a - ( | 2 > _ f j . ) 9 , ^ ( ^ ) > 0 , | , f 6 5 .

If Cp is of class C2 on S, this condition is equivalent to

for all ^ and all rj.

R e m a r k . The func t ion E(^, f) in (1.8.3) is seen to be the W E I E R S T R A S S

£^-f un ct io n .O u r f ir st lower - s em icon t inu i ty theo rem depen ds on J e n s e n ' s in

equa l i ty wh ich we now s t a t e :

L e m m a 1.8.2 (Jensens's i n e q u a l i t y ) . Suppose cp is convex on Rp, Sis

a set, pL is a non-negative bounded measure on S, and the functions ^^ ^

Li(S,pi),p = 1,. . . , P . Then

cp{i\ . . . ,F) < [f^{S)]-^f(p[^Hx), .. .,iP{x)]dii,

(1.8.4) , ^ r

i:p = [i,(S)r^f^Pdpt.s

R e m a r k . I.e. 99 (a ver age ) < a v e r a g e of 99.

Proof. Choose ap ( L e m m a 1.8.1) so t h a t^(C) + cipii^ - C ) < ^(f) for all f

and then ave rage ove r 5.

W e can now s t a t e and p r o v e our f ir st lower s em icon t inu i ty the o re m ;

in th is genera l fo rm, it is due to S E R R I N [2]:

Theorem 1.8.1. Suppose thatf = f[p) is non-negative and convex for

all p = {py] and suppose that z and each Zn^ ^\[^) ^'^^ t^^t z^—^z (i.e.

tends weakly to) z in Li{D) for each DCGG. Then I{z, G) and I[ZnG) are

each finite or + 00 and

I{z, G) < lim inf I{zn, G)

22 Introduction

Proof. The firs t conclusion is obvious . Le t Z) C C (^; we m ay s uppos e

t h a t D C Ga for s o m e a^ 0. Let 99 be a moUifier and, iox 0 <i Q <Z a,

le t ZQ and Z nq be the 99-mollified functions (of class C^ on D). F r o m

F a t o u ' s l e m m a , T h e o r e m B, and equa t ion (1 .8 .4) , we conc lude tha t( s i n c e / [ V - ^ e ( : v ) ] - > / [ V ^ ( ^ ) ] a.e.)

(1.8.5) I{z,D) < lim inf I[ZQ,D),

e->oL e t us now define

F{X) = / [ V Z{X)] , Fn{x) = / [ V Zn{x)] .

Then , f rom J ens en ' s inequa l i ty wi th d/j, = q)*(^ — x) d^ ( s ee Equa t ion

(1.8.4)), we c o n c l u d e t h a t

/ [ V ^ e W ] < ^ e W > f[VZne{x)]^FnQ{x), x^D, 0<Q<a.

Us ing th i s and Theorem B (se t t ing F^ {x) = 0 ou ts ide G) we conc lude tha t

(1.8.6) I{znQ,D) ^JFnQ ix) dx<I{zn, G).D

The weak conve rgence in Li{D') for each D' C. C. G imp l ie s tha t VZnQ

converges un iformly on Z) to V ZQ for each Q, 0 <. Q <. a. T h u s

(1.8.7) I{ze, D) = lim I{znQ, D) < lim inf I{zn, G).

Combining (1.8 .5) and [\ .^ .7)y we c o n c l u d e t h a t

I{z,D) < l i m i n f / ( ^ ^ , G)TO-* 0 0

from which the result follows eas ily us ing the a rb i t ra r ine s s of D.

W e now g ive a s imple proof of lower s emicon t inu i ty for a wide class

of in tegra ls but us ing a m o r e r e s t r i c t e d t y p e of convergence which is,

however , suff ic ien t ly genera l for the ex i s tence theo ry . The h y p o t h e s i s

th a t the /^^ be c o n t i n u o u s can be removed ra ther eas i ly . More genera ltheo rems have been p roved by S E R R I N (see Chapter 4) .

Theorem 1.8.2. Suppose f == f(x, z,p) and the f>pi are continuous with

f{x, z, p) ^ 0 for all {x , z, p), suppose f is convex in p for each {x, z), and

suppose Zn—rz in Hl{D) for each D (ZC.G, Then the conclusions of

Theorem 1.8.1 hold.

Proof. Choose D C G G. The weak conve rgence in Hl{D') for each

D' d a G impl ies the s t rong convergence of Zn to z in Li(D) (see Theorem

3.4.4). By choos ing a subsequence, s t i l l cal led Z n, in which I{zn, G) - ^ i t s

fo rmer lim inf, we may a s s u m e t h a t Zn{x) -^z{x) a.e. on D. We now

s uppos e I{z, D) < + 0 0 . T h e n , for each e > 0, t h e r e is a c o m p a c t s u b s e t

S G D on w h i c h z and V^ ( i. e. repres en ta t iv es ) are c o n t i n u o u s , on w h i c h

Z n converges un iformly to z, and w h i c h is s u c h t h a t

{\.8.8) I{z,S) >I\z,D) -8

1.9. Existence 23

(if I(z, D) = + c>o, we m a y ta k e I{z, S) > M, a rb i t ra ry ) . Then , f rom

Lemma 1 .8 .1 , we conc lude tha t

f[x, Z n{x), VZ n(x)] > / [ ^ , Z n{x), V z{x)] + fp[x, z(x), V z(x)] ' [V Zn{x) -

- \7 z(x)] +{fp[x,Zn(x),Vz{x)] -fj,[x,z{x),\7 z{x)]} • [VZn{x) - Vz{x)].(1.8.9)

The weak convergence imphes (see Theorem 3-2.4 (a)) the weak conver

gence oi VZn toVz in L\{D) which imphes , in tu rn , tha t

(1.8.10) Hm(fp[x, z{x), \/z[x)] • [\/Zn[x) — \/z{x)] dx = 0.

The un i fo rm conve rgence of ^ ^ to on S, toge the r wi th the un i fo rm

boundedness of the Li norms of V Z n and V z, i m p h e s t h a t

(1 .8 .11 ) hmf {fj)[x,Zn , Vz] —fp[x,z, Vz]} • [V^^^ — \/z] dx = 0.

He nce , from (1 .8 .8— 1.8 .11) , we conc lude th a t

I{z, D) — s<, lim (f[x,Zn{x), \/z{x)] dx< lim inf I{zn, G).

The theorem follows eas ily .

1.9 . ExistenceI f / sa tis fies th e cond i t ion (\.7A) w i t h ^ > 1, an exis tence theorem

can eas i ly be deduced f rom the lower-semicont inu i ty theorems of § 1.8

and Theorems 3-4.4 and 3-4-5- The following s imple lemma enables us

to prove eas i ly am ore gen eral exis te nce the or em (Th eorem 1.9-1 below)

Lemma 1.9.1. Suppose that fQ{p) is continuous and that

( 1 . 9 . 1 ) l i m | ^ i - i / o ( i ^ ) = + 0 0 .3 9 ^ 0 0

Then, for each M, there exists a function cp such that cplg) ' 0 for ^ >- 0

and (P{Q) - 0 as Q - 0 such that

(1.9.2) l\p{x)\dx<cp{\e\)*e

for every measurable e C. G and every vector p [x) satisfying

(1.9.3) p [p[x)]dx M.G

Proof. We def ine (p{Q) as the sup of the left member of (1.9.2) for all

^such that 1^1 < ^ and_^ satis fying (1.9-3)- If 9^(^) does no t - > 0 a s ^ - ^ 0 ,H an £0 > 0 and sequences {cn} a n d {pn} s u c h t h a t \en\ - > 0 a n d

r \pn{x) \ dx >£Q. We define

ip{G) =in f \p\~^fo{p): fo r \p \ >a,

* We often use 1^1 to denote the measure of the set e.

2 4 Introduction

Clearly ^(cr) -> + oo as c - ^ + c>o. For each n, le t gn be the subset of

en w h e r e \pn{oc)\ > cr = eol2\en\. O b v io u sl y

j \pn(x)dx <an-\en\ = £ol2^

en—gn

Consequently , s ince/o(_^) > | ^ | • 'y^(o') for | ^ | > (X, we ob ta in

(1.9.4) / \pn{x)\dx> £0/2, f Mpni )] dx >tp{an)'£ol2.

gn gn

Since or^ -> + 00 and ^'(o'^) -> + "^^^^ (1.9.4) contradicts (1.9.3)-

W e now s ta te and p rove ou r p r inc ipa l ex i s tence theo rem.

Theorem 1 .9 .1 . We suppose that (i) / and the fpi are continuous in

their arguments; (ii) / is convex in pfor each {x, z); (iii) there isa function

/ o satisyfing the conditions ofLemma 1.9.1 such that f{x, z, p) >fo(p) forall (x, z, p); (iv) F* is a(non -empty ) family of vector functions which is

compact with respect toweak convergence in H\[G)\ (v) F is a family,

closed under weak convergence inH\[G), such that each zinFcoincides on

dG with a fimction z"^ in F"^ (i.e. z ~ z'^^B.\^{G), see§ 3 - 2 ) ; (vi)

(-2^0, G) < + ^>o for some Z Q^ F; (vii) Gis bounded. Then I{z, G) takes

on its minimum for some z in F.

Remarks. Since we have no t made any a s s umpt ions on G o t h e r t h a n

boundedness and s ince the admiss ib le func t ions a re no t cont inuous , them os t conven ien t w ay to s peci fy th e boun da ry va lues ofa functio n is to

s t a t e t h a t z ~ z"^^ •^io (Q ^^^ som e given ,^*. T hu s th e family i^* defines ,

so to speak , the c lass o f boundary va lues be ing a l lowed. Ofc o u rs e , F *

could consist of a s ingle function ^*.

Proof ofthe theorem. L e t {zn} b e a min imiz ing s equence ; we may

a s s u m e t h a t I{zny G) < / ( 2 : o , G) =M. Using (i i i) and Lemma 1.9.1 , we

conc lude tha t the se t func t ions f\\/Zn{x)\ dx a re un i fo rmly b oun dede

and uniformly abso lu te ly cont inuous . From (v) , we conc lude for each nt h a t t h e r e is a 2:* ^F * su c h t h a t w^ =" Z n — z*^HIQ(G). From (iv) we

conc lude t h a t the re i s a subsequen ce , s t il l ca l led {n }, s u c h t h a t z* -v s ome

^* in-F* . From Theorem 3 .2 .4 (a ) wi th ^ = 1, it follows that the set

func t ions { \ z \ dx a re un i fo rmly bou nde d and un i fo rmly abs o lu te ly

econt inuous so tha t the same is t rue for {wn}. From Theorem 3 .2 .4 and

the co ro l la ry to The o rem 3-2.1, it fo l lows tha t there is a fu r the r s ub

sequence, s t i l l cal led {^}, such that Wn—7W^R\^{G). T h e n ,of

courseZn—7Z=:^z'^ + wvi\K\((G) and the theorem fo l lows f rom Theorem 1.8.2.

Actua l ly ou r hypo thes e s do no t imp ly tha t f{x, z,p) > 0 ; howeve r , it is

c lea r tha t /Q t akes on i t s min imum, s o tha t / i s bou nde d be low and the re

is no loss in generality (since G i s bounded) in a s s uming tha t / > 0.

Remark 1. I f / s a t i s f i e s (1.7 .1 ) wi th ^ > 1, it obviously satisfies (iii)

a b o v e .

1.9. Existence 2 5

Remark 2. In the case v = i, one conc ludes immed ia te ly the un i fo rm

absolu te cont inu i ty of the (un ique in th is case) abso lu te ly cont inuous

r e p r e s e n t a t i v e s Z n and hence the i r equ icon t inu i ty . The weak conve rgence

impl ies un iform convergence , the l imi t func t ion be ing obvious ly abso lu

te ly con t inuous .

Remark 3. In Theorem 1 .9 .1 , F could be, for example, the set of a l l

z s u c h t h a t z — z"^^ HIQ { G ) , for some 2:* in -F* (i^* ass um ed give n), an d

z satis f ies a sys tem of equations of the form

(1.9.5) <• [x, z {x)] zi,{x) + bij [x, z{x)] z3 {x) + Ci [X, z {x)-\ = 0,

i= i,. . . , P ,

whe re the af^, bij, a n d Ci a re eve rywhere bounded and con t inuous . Fo r ,

s u p p o s e t h a t {zn} is an y se que nce in i^ ^ ^ ^ -^ ^ in Hl{G) and le t {zr} b ean a rb i t ra ry s ub -s equence o f {zn}. From Theorem 3-4-4 (appl ied to each

sm oo th dom ain Z) C C G), i t follows th a t Zs(x) -^z{x) a lmos t eve rywhere

for some subsequence {zs} of {zr}. liv^,i = 1, . . ., P , a re a rb i t r a r y bo un

ded func t ions , then

<,• [x, Z s{x)]vi(x), bij [x, Z s(x)]v^x), Ci[x, Z s{x)] v^{x)

conve rge a lmos t eve rywhere and bounded ly to the i r l imi t s . By rep lac ing

2: in (1.9.5) by Z s, m ult ip ly in g by '* an d sum m ing , in te gra t in g the resu l t ,

and pass ing to the l imi t us ing the weak convergence , we f ind tha t z^ F.Remark 4. If 2 is th e vec tor fun ction defined b y

zi{x) = Z)°= w^{x), 0 < |(%| < Mi — 1, " = 1, . . ., N,

the in teg ra l I{z, G) i s equa l to an in tegra l J{w, G) of a var ia t iona l p ro

b lem involv ing der iva t ives of o rder '<mi of wK T h e n T h e o r e m I. 9 .I f o r i

impl ies a corresp ond ing theo rem for / . In fac t th e ex is tence the ore m of / .

GEL'MAN can be deduced immed ia te ly in th i s way .

In o rde r to ob ta in more mean ing fu l bounda ry va lue p rob lems , i t i s

convenien t to res t r ic t ourse lves to Lipschi tz domains G. Us ing the

genera l com pac tness the ore m s of § 3-4 we can pro ve th e fo llowing

b o u n d e d n e s s l e m m a ( b y c o n t r a d i c t i o n ) :

L e m m a 1.9.2 (cf. Theorem 3-6.4). Suppose G is Lipschitz, a is an

open subset of G, and r is an open subset of dG. Then there are constants

Ci{G , a,r,v) and C2{G,r,r,v) (depending only on the quant i t ies in

d ica ted) such that

II z| |J,« < Cx• {\\Vz\\U + \\ z\\\J-\ f ^ € H I ( G ) ,

Remarks. This l emma imp l ie s tha t i f F is any family of functions z

in Hl[G) for which the Lr{G) norms | |V2: | | J^^ of the f irs t derivatives are

b o u n d e d , t h e n t h e Lr norms of the func t ions wi l l be bounded i f the i r

Li -n or m s eit he r ov er a f ixed open sub set or of G or a f ixed op en su bse t T

26 Introduction

of ^G are b o u n d e d . The family F* cou ld then be rep laced by a c o m p a c t

(in Lr(dG)) family of functions really defined only on dG. As an e x a m p l e

of the use of our genera l ex is tence theorem in c o n n e c t i o n w i t h L e m m a

1.9.2, we m e n t i o n the p r o b l e m of P l a t e a u for a surface of leas t a rea

w h i c h has a g iven arc F, w i t h end p o i n t s on a mani fo ld M, as p a r t of its

b o u n d a r y , the re s t of its b o u n d a r y b e i n g r e q u i r e d to lie on M, On

a c c o u n t of c o n fo r m a l m a p p i n g (see C h a p t e r 9), it is sufficient to min imize

the Di r ich le t in teg ra l among all z^ Hl[B{0,i)] w h o s e b o u n d a r y v a l u e s

a long d'^B(0,\) (the closed upper semicirc le) are c o n t i n u o u s and give a

1 — 1 c o n t i n u o u s p a r a m e t r i c r e p r e s e n t a t i o n of F, in w h i c h the p o i n t

(0,1) corresponds to a f ixed po in t of F, and whose (possib ly d iscont inu ous)

va lues on d''B{0,\) lie on ikf (a lmos t eve rywhere ) . It t u r n s out (see §9-3)

t h a t the mapp ings f rom d+B(0 ,\) to F are e q u i c o n t i n u o u s for any m i n i m iz ing sequence . S ince th ey are u n i f or m l y b o u n d e d || z^ \\1^B s e un i fo rmly

b o u n d e d and a subsequence, s t i l l cal led {zn} conve rges weak ly in

Hl[B(0,\)] and uniformly a long d+B{0,\) to a min imiz ing func t ion z.

H o w e v e r , z (ha rmon ic and conformal ins ide 5(0 ,1) may not be c o n t i n u o u s

a long d-B(0,\) if M is a l lowed to have edges , as has been s hown in an

e x a m p l e of C o u r a n t ([2], [3]. H o w e v e r see, L E W Y [2]).

Remarks. The t h e o r e m s of th is sec t ion and the preced ing one can

be carr ied over to in teg ra l s invo lv ing de r iva t ive s of higher order of theform

I{z, G) = Jf[x, L[z), M{zy\ dx

G

w h e r e M[z) d e n o t e s all the d e r i v a t i v e s D°' z^ of highes t o rder where |^|

== nii and L[z) deno te s all thos e de r iva t ive s whe re 0 < |^| < w^-. It is

a s s u m e d t h a t / i s c o n t i n u o u s in its a r g u m e n t s and convex and differenti-

ab le in the set of a r g u m e n t s M{z). T h e r e are essen t ia l ly no differences in

the proofs . Of course each vec tor z in F (Theorem 1.9-1) w o u l d be s uch

t h a t z^ - ^* '*$ ^foH Q foi" s o m e ^* in i^*, /o = : / o [ M ( ^ ) ] , etc. F I C H E R A

h a s . o b s e r v e d t h a t lower-semicontinuity theorems for such in tegra ls can

be ob ta ined when M{z) consis ts only of some combinations of the de r iva

t ive s of highes t o rder and L(z) consis ts only of some of thos e of lower

o rde r .

1.10. The dif ferent iabi l i ty theory. Introduct ion

Of pe rhaps g rea te r in te re s t than the ex i s tence theo ry is the t h e o r y of

the d i f fe ren t iab i l i ty of the so lu t ions . In th i s chap te r , we shall confineourse lves to the n o n - p a r a m e t r i c c a s e ; we shall discuss the p a r a m e t r i c

case in C h a p t e r s 9 and 10.

The f i rs t resu l t about d i f fe ren t iab ih ty was t h a t of LICHTENSTEIN

w h o p r o v e d in 1912 [1] t h a t a so lu t ion z of class C" of a regu la r doub le

in teg ra l p rob lem {v = 2, N = 1) in w h i c h / is a n a l y t i c is of class C "

1.10. The di f feren t iab i l i ty theoy . In t roduct ion 27

and hence ana ly t i c by the f a m o u s t h e o r e m of S. BERN STEIN ([1]). The

same conc lus ion was s h o w n to ho ld (a) if z is of class C^ w i t h H o l d e r

con t inuous de r iva t ive s by E. H o p f in 1929 ([2]) and (b) if z is Lips ch i tz

b y t h e w r i t e r in 1938 ( M O R R E Y [4]). Us ing the l a t t e r r e s u l t , it follows

t h a t the Lipschi tz so lu t ions (ment ioned above) ob ta ined by A. H a a r for

t h e c a s e / = / ( ^ ) , r = 2, N ^= 1, G s t r i c t ly convex , are a n a l y t i c i f / i s .

B u t e x c e p t for prob lems whos e Eu le r equa t ions are l inea r and ce r ta in

in teg ra l s whe re / is of the form

f{x, z, p) = a(x, z) \p\^ + 2Z h^{^, ^)pcc + c {%, z){v = 2,N= 1)

t r e a t e d by H IRSCH FELD the re we re no o the r re s u l t s in which the so lu t ions

wh ich were s hown to exis t were shown to be dif fe ren t iab le un t i l thew o r k of the w r i t e r ([6], [7] see foo tno te on p. 17) in which the e n t r e

ex is tence and d i f fe ren t iab i l ity p rog ram w as ca r r i ed th ro ug h for essen t ia l ly

the c lass of p r o b l e m s in w h i c h / s a t i s f i e s the condit ions (1 .10.8) below

w i t h V = 2, N a r b i t r a r y , and k = 2; a l t h o u g h N was a l lowed to be > 1,

the convex i ty hypo thes i s on / w a s r e t a i n e d , t h a t is

< M i

Later ([8]) , the w ri te r app l ied these resu l ts to p r o v e the dif fe ren t iab i l i tyof his s o lu t ions of the p r o b l e m of P l a t e a u on a R i e m a n n m a n i fo l d.

A grea t ly s impl i f ied vers ion of t h i s old w o r k is to be found in [15],

recen t deve lopmen ts have pe rmi t t ed s t i l l fu r the r s imp l i f i c a t ions and

s ome ex tens ions .

T h e m e t h o d s u s e d in th is d i f fe ren t iab i l i ty theory wi th i = 2 would

not genera l ize to la rger va lues of v and it was not u n t i l the recent resu l ts

of D E G I O R G I [1] and N A S H [3], as simplified still more by M O S E R that

the ex is tence and dif fe ren t iab i l i ty program could be c a r r i e d t h r o u g h for

p r o b l e m s in w h i c h v '> 2, The m e t h o d s of NASH and D E G I O R G I were

ent i re ly d i f fe ren t ; NASH o b t a i n e d his re s u l t s as a b y - p r o d u c t of his w o r k

on pa rabo l ic equa t ions and confined himself to bounded s o lu t ions

w h e r e a s D E G I O R G I dea l t on ly wi th e l l ip t ic equa t ions but a l lowed

s o lu t ions in L^. Us ing M O S E R ' S s implif ication of D E G I O R G I ' S w o r k , a

s t u d e n t E. R. B U L E Y was able in the s p r ing of I960 to o b t a i n the re s u l t s

s t a t e d in T h e o r e m 1.10.4 (ii) for the cases where / satisfies (1.10.7) or

(1.10.7') and the a u t h o r was able to ob ta in thos e in (ii) (the (1.10.8)

cases) , (iii), and (iv). At a b o u t the s a m e t i m e , LADYZENSKAYA and U R A L ' -TSEVA ([1], [2], [3]) ob ta ined the i r r e s u l t s s t a ted in T h e o r e m 1.10.4 (v)

which inc lude the au tho r ' s r e s u l t s . Thes e re s u l t s are discussed fur ther

be low and m o s t of the deta i ls are p r o v e d in C h a p t e r 5. An u n f o r t u n a t e

fea tu re of the D E G I O R G I - N A S H re s u l t s is t h a t t h e y h a v e b e e n p r o v e d

onl y f or iV = 1.

2 8 I n t r o d u c t i o n

We now show how to prove d i f fe ren t iab ih ty in the case v = 1, A^

a rb i t ra ry . W e do no t a t t empt to p rove the mos t gene ra l theo rem.

Theorem 1.10.1. Suppose that v= \, that f^C^ everyw here, that f

satisfies (1.7-1) with ^ >> 1, and that there is a positive function Mi{R)

such that

(1.10.1) | ^ ( ^ , ^ , / > ) | , | / , ( ^ , ^ , ^ ) | < M i ( i ^ ) ( | + | :^ |^ )fo r

Suppose z minimizes I{z, G) (G = {a, b))\ among all A.C . functions with

the same boundary values. Then z^ C\a, hi and satisfies Euler's equations.

Proof. L e t f be any L ips ch i tz ve c to r wh ich v an i s hes a t a a n d h. Since

/ satisfies (1.7.1), it follows t h a t z is A.C. and \\z'\\l < oo . Since C is

b o u n d e d a n d z and f are absolutely continuous, we see from (1.10.1) that

'C*W -h'ix. z{x) + A f W , z-{x) + <C 'W ] + C«W -fAx, ^ + A f, / + A f ]

is domina ted for | A | < 1 by th e f ixed sum m abl e func t ion

2 ^ . ^ 1 ( 7 ^ ) [ 1 + 2 ^ - 1 ( g ^ + | / ( ^ ) | ^ ) ]

wh ere ^ is a bo un d for | I^'(x) \ a n d | C(^) | a n d R is one for A; _|_ | ^ _|_ ^^|2

for IAI < 1; he re we use d the ine qu ali ty

[a + h)^ < 2^- i (a^ + h^), ^ > 1.

Thus the func t ion (p{X) defined in (1.4.6) is of class C for \X \ < | and(p'ip) = 0 so th a t (1 .4 .7) holds .

Now (1.4.7) is of the form

(1 .1 0.2 ) / C C M ^ + I:iBi)dx = 0, C -(a) = f^(6) = 0a

w h e r e Ai a n d Bi a r e s u m m a b l e . L e t 7 1 ^ b e a n a r b i t r a r y b o u n d e d m e a s u r

ab le vec to r s uch tha t6

( 1 .10 .3 ) j 7i^{x)dx = 0a

Any Lipschi tz f van ish ing a t a a n d h is of the formy h

(1.10.4) f^y) = jn^ {x)dx = —J7 i^{x)dxa y

Breaking the in tegra l (1 .10 .2) in to the in tegra ls o f i t s two te rms , rep lac

in g xhy y m the second , then subs t i tu t ing (1 .10 .4) for C, in te rchanging

orders of in tegra t ion , and recombin ing , we conc lude f rom (1 .10 .2) tha t

( 1 . 1 0 . 5 ) 17t^x)\Ai(x) - .lBi{y)dy\ dx = 0.

Since this holds for all TT satisfying (1.10.3), it follows from a well-known

l e m m a t h a t

X(1.10.6) fpi [x, z{x), z'{x)] = Ai{x) == J Bi{y) dy -f- Q (Q = const.) a.e..

1.10. The differentiability theory. Introduction 29

in which the r igh t member is A.C. S ince / i s regula r , the equa t ions

(1.10.6) can be solved for the 'z^ in terms of x, z{x), and the in teg ra l .

T h u s 'z^ i s equiva len t to an A.C. func t ion . Thus 'z^ is easily seen to be

A.C. H en ce bo th s ides of (1 .10.6) are A.C and t h e r ig ht s ide ^ C^. Con

s e q u e n t l y z^ C^ so z^ C^ a n d E U L E R ' S equa t ions fo l low as usua l .

Fo r r > 1, th e the ory is no t so s imple of course . R a t he r tha n t ry in g

to presen t the mos t genera l condi t ions under which each par t o f the

demons t ra t ion can be ca r r i ed th rough , we s ta te now two s e t s o f con

d i t ions on the in tegra l func t ion / u nd er which d i f fe ren t iab i l ity resu l ts

have been ob ta ined .

Common Condi t ion , f ^ CI in its arguments or f and fp ^ CJJ~^ for

some n > } and some fji with 0 <i ju <, 1.

The use of Holder condi t ions in connec t ion wi th e l l ip t ic d i f fe ren t ia lequa t ions is very common. The des i rab i l i ty of the i r use a r ises in po ten t ia l

theory (see Chapte r 2 ) .

Bes ides th e com m on c ondi t ion abov e , we require / to sa t is fy one of

the following sets of condit ions for a l l {x, z, p):

imV^ — K <f{x,z,p) ^MV^,

\\fv\' + Ifvccl' + \fz\' + \U\^ < M f F2A:-2

(\A0.7)l\fpz\' + \fzz\'<MlV^^-^

\m i F^-2 I TT |2 < Zfplp'p {^' ^> P)<^h^^l V^~^\7t\^,

[o < M < M , k> \, F == (1 + | ^ | 2 + | j^ |2)i /2 , 0 <mi <M i,

\fpp\'=I{fpipir> \fpz\'=I{fpiz^)'> H' = I{<)'^ ec t .

( 1.1 0.7 ') S a m e a s ( 1.1 0.7 ) b u t / = / ( : v , ^ ) , F = (1 + \p\^)^l^.

imV^ — K <f{x, z, p) < MV^,

(1.10.8)

| / . | 2 + |/ . ^ |2 + | ^ | 2 < M ( i ? ) F 2 ^

\\fp? + \fpz\^ + \fpx\''^M[R)V^^-^

U i ( i ? ) F ^ - 2 l 7 r | 2 < 2 ' A j ^ 2 , ^ : 7 r j , 4 < M i ( J ^ ) F A ^ - 2 | 7 T | 2 ,

k>v, 0<m<M, 0 < mi{R) < Mi{R), F =- (1 + \p\^)^l^

\x\^ + \z\^ <R^

R em ar ks . W e no tice th a t (1 .10.8) redu ces to (1 .10.7 ') in case / does

no t depend on z (except for the i^ -condi t ion which is somewhat meaning

less if z i s no t p resen t s ince our domains G are b ou nd ed) . To see th e d if fe r

ence be twe en (1 .10 .7) an d (1 .10 .8) , we notc e th a t th e fu nc t io n / de f in ed by

f(x,z,p) = [\ + affix, z)pip^fl^

satisfies the conditions (1.10.8) but not (1.10.7) if the aff^ CI o r Q - ^

if ^ > 3 an d th e qua dr a t i c form is pos i t ive defin i te an d bo un de d abo ve

and be low in the obvious way.

3 0 Introduction

The f i rs t s tep in the d i f fe ren t iab ih ty theory is to no te cont inu i ty pro

pert ies of the minimizing functions as follows:

Theorem 1. 10. 2. In all cases if k'>v the minim izing functions are

Holder continuous on interior domains and at any boundary points in the

neighborhood of which d G is Lipschitz. If k — v, the minimizing functions

are Holder continuous on interior domains and

(1 .10.9) I / V^dxY'^^C'dlVWD^irlRy iiB(xo,r)c:B{xo,R)(ZG.

If k =^ V = 2, and f satisfies

(1.10.8*) m\p\^ ^f{x,z,p) <M \p\^

G is bounded by a finite number of disjoint J ordan curves, z*^Hl(G), and2:* is continuous on G, then any minimizing function with z — ^ * $ H\Q[G)

is continuous on G. li k == v, G is Lipschitz, the boundary values are con

tinuous, and f satisfies the supplementary conditions

(1.10.10) f{oc,z,p) > 0 , f{x,z,p^, . . . , ^ ^ - 1 , 0 , ^ ^ + 1 , . . . , ^ ^ ) = 0 ,

r = 1, . . ., N{p^ = {Pi}, i fixed)

then any m inimizing function with those boundary values is continuous on G.

Proof. If ^ > r an d I{z, G) is f ini te , i t fol lows from the f irs t assump

tions in (1.10.7) and (1.10.8) that the H\[G) norm o f the min imiz ing

func t ion is f in i te . Tha t z is continuous on the interior follows from the

co ro l la ry to Theo rem 3-5.1. The con t inu i ty on the bounda ry i s ob ta ined

by f i rs t ' ' f la t ten ing out ' ' a p iece of the boundary by a b i -Lipschi tz map

(see th e definit ion in § 1.2) a nd th en reflecting th e func tion.

In the case v = 2 = k wh ere / sa t isf ies (1 .10.8*), th e m inim izing

v e c t o r z satisfies

mD[z, B{xoR)] < I[z, B{xo, R)] < I[H, B[xo, R)] <MD[H, B[xo, R)]

(1.10.11)

w h e r e H i s the harmonic func t ion co inc id ing wi th z on dB(xo, R). If we

let (p{r) = D[z, B(xo, r)], t ake po la r coo rd ina te s abou t XQ, e x p a n d z in

a Fourier series

(1 .10.12) z = ' ^ + Z [an[r) cosnO + bn{r) smnB],^ n

and use the formula (1 .3 .3) for D[H, BR], we f ind th a t

R

<K 7 c2J»[aliR) +bl{R)]^K-R-q,'{R), K = Mlmn

from which (1.10.9) follows with// = \jK.

1.10. Th e different iabi l ity the ory . In t ro du ct io n 31

The condit ion (1.10.11) holds , obviously, with B{xo,R) rep laced by

any o the r s ub -doma in o f G, the re s u l t ing cond i t ion be ing inva r ian t unde r

con fo rma l mapp ings . The two-d imens iona l r e s u l t s on con t inu i ty a t the

bo un da ry invo lve th is idea and a re carr ied ou t in § 4-3- Th e proof abo ve

can be generalized to the general case k = v; the ha rmon ic func t ion H

i s rep laced by th e func t ion

H{r, d) =z + {rlRY[z{R, Q) ~ z], 0 < r < 7?,

w h e r e [r, 6) are spher ica l coord ina tes , 6 denotes a po in t on dB{0,i), a n d

z is the average of z(r, 0) ove r dB{0,\). The rema in ing re s u l t s a re p roved

in § 4 .3 .

W e mus t p rove fo l lowing in te re s t ing theo rem:

Theorem 1 .10 . 3 . Iffis of class C^ and satisfies (1.10.7) or (1 .10 . / ' ) for

some ^ > 2, then I(z) = I{z, G) is of class C^ over H\[G). If , instead, f

satisfies (1.10.8) for some ^ > 2, then I(z) is of class C^ over the space

*Hl(G) = H\[G) n C^{G), the norm in this space being

m ax 1 ( )1 + | |V^| |? .G-xeG

If we merely have 1 < ^ < 2, then I(z) is of class C^ over the corresponding

space. In either caes, if zo minimizes I{z) among all z with given boundary

values, then the first variation, i.e. the first differential Ii(zo,C), is zero,

where Ii{z, C) is defined 5y (1 .47) .

Proof. W e s hal l p rov e on ly the f ir s t s t a t em en t ; th e o the rs a re p rove d

s imila r ly . For a lmos t a l l x, w e m a y w r i t e

fix, z + l:,p + n)= f{x, z, p) + C 'fz^ + <f^^ + \ \fz^ z^ ^'^' +

1

+ £«;• C« IK fff = / (1 - ) {fvlT>\l^, z + tl:,p + tn-\- fp^pi^ix, z, p)} dt

0

and £fj- and Sij are given by s imilar formulas . I t is c lear that if Zn - ^2 : in

Hl{G), then/j,[Zn] ->fpM and/^[Zn\ -^fz W s t rong ly mLr^,r = kl{k~ \),

fvvi^n\ ->fpp[z]> fpz[zn] -^fpz[z], a n d fzz[Zn] -^fzz[^] s t rongly in Ls,

s = kj{k — 2), a n d t h e efj^(zs), e tc . , - ^ 0 s t rong ly in Ls. Thus we s ee tha t

i{z + 0= l(z) + ii(z,C) + Ihiz, 0 + R(z, C)

1 To see this, let {q } be any subsequence of {n}. Then there i s a subsequence {s}

of {q } s u ch t h a t Z s{x) -> z{x) and V ^s{^) -> V ^(^) a lmos t everywh ere so th a t

/ p M - > / 3 ? M for a lm o st all x. We hav e [/^[-^s]!^ and | / z M | ^ dom inated by

C ' Ml V^{x) and the s t rong convergence of jSg t o z in H'^{G) impl ies th e uniform

absolute cont inui ty of the set funct ions j V\{x) dx. A s imi lar a rgument ho lds fo r

fpp> £ij> e t c . «

32 Introduction

w h e r e / , / i , and I^ are c o n t i n u o u s in bo th va r iab le s and

\R{z,t:)\<s{z.i:)-m\^

w h e r e e[z\ t) ->0 un i fo rmly as ||C|| - ^ 0 for ||2:' — ^|| < s o m e h > 0.Definition. Any v e c t o r z ^ Hl[G) for w h i c h Ii[z, C) = 0 for all f as in

T h e o r e m 1.10.3 is called an extremal.

W e can now s t a t e our pr inc ipa l r e s u l t s :

Theorem 1.10.4. We assume that f is regular and satisfies the "common

condition.'' Then:

(i) If N = \ and z is Lipschitz and is an extremal, then z^ C^ [D) for

each D C C G. ( R e c al l the e x i s t e n c e t h e o r e m w i t h / = f{p) and G s t r i c t ly

c o n v e x m e n t i o n e d at the end of § 1.7).

(ii) / / / satisfies (1.10.7) or (1.10.7') with N = \, v arbitrary, and

^ > 2 or if f satisfies (1.10.8) with N == \ and k ^ v, then every extremal

€ Q {D) for all Dec G.

(iii) If f satisfies (1.10.8) with iV = 1, r arbitrary, and k = v or with N

arbitrary and k == v = 2, then any extremal which ^ C^^ [D ) for D G G G

and which satisfies the Dirichlet growth condition (1.10.9) ^•CJJ(Z)) for

DCCG.

(iv) / / / satisfies (1.10.7) or (1.10.7') with N = i, v arbitrary, and1 < ^ < 2, and if z"^^ HI [G), then there exists an extremal z^ C^ {D) for

all D G G G and which minimizes I{Z, G) among all Z such that Z — ^*

€ f f* o ( G ).

(v ) If f satisfies (1.10.8) with N = \,v arbitrary, and ^ > 1, then any

boimded extremal ^ CJJ [D) for all D G G G.

(vi) In all cases above, the extremal ^ C°° (D) for D G G G if / ^ C°^

and is analytic on G iff is analytic.

Remark 1. The re s t r i c t ion N = \ l imi t s the app l icab i l i ty of the re s u l t s

very se r ious ly . The r e m o v a l of th is res t r ic t ion would not only increase

the app l icab i l i ty of our re s u l t s in such fields as Dif fe ren t ia l Geome t ry

and Topo logy bu t is essen t ia l for the proof of the dif fe ren t iab i l i ty of the

so lu t ions of v a r i a t i o n a l p r o b l e m s of higher order . The r e m o v a l of th i s

res t r ic t ion involves the ex tens ion of the De Giorg i-Nash resu l ts to

s y s t e m s of e q u a t i o n s .

Remark 2. If f{x, z, p) is q u a d r a t i c in z and p, the E u l e r e q u a t i o n s

are l inear and more de ta i l ed re s u l t s are ava i lab le in th is case .

The re s u l t s in (iii) w i t h N a r b i t r a r y and k = v = 2 are thos e ob ta i ned before the war (1937— 38) by the a u t h o r and referred to earl ier .

The re s u l t s in ( ii)— (iv ) o the r th an thos e ju s t m en t ione d , we re ob ta ine d

ea r ly in I960 in con junc t ion wi th a s t u d e n t (E. R. B u l e y ) . The r e m a r

kable resu l ts in (v) were ob ta ined at a b o u t the s ame t ime by

L A D Y Z E N S K A Y A and U R A L ' T S E V A ([1], [2], [3]). Their results include

1.10. The differentiability theory. Introduction 33

thos e of ours in ( ii)— (iv) abov e which refer to the cases (1.10.8) with

N = \ an d > v since we have seen in T h e o r e m 1.10.2 a b o v e t h a t the

so lu t ions are c o n t i n u o u s in these cases . The t h e o r y of LADYZENSKAYA

a n d URAL'TSEVA is discussed briefly in § 5 • 'I 'I ; the y have t rea ted the s e

and o the r de ta i l s in the i r rece nt book [3] . Ho we ver , the re s u l t s in (ii)

and (iv) in the cases (1.10.7) and (1.10.7') are of in te res t because the con

t i n u i t y of the solutions doesn ' t fol low from the ex i s tence theo ry in the

cases where 1 < ^ < r .

R e m a rk s . The func t ions z w h i c h we are ca l l ing ex t rema ls are also

called weak solutions of the Eu le r equa t ions

It was noticed by L A D Y Z E N S K A Y A and U R A L ' T S E V A that identical proofs

could be us ed to p r o v e the dif fe ren t iab i l i ty of weak s o lu t ions of e q u a

t ions of the form

(1.10.13) -^^A^ = Bi, A^ = Anx,z,p), Bi = Bi{x,z,p);

tha t i s , so lu t ions z of the e q u a t i o n s

(1.10.14) f(CiA^ + C'Bi)dx = 0G

for th e f in Theorem 1.10.3. The equations (1 .10.14) are ob ta ined f rom

(1.10.13) formally by multiplying (1.10.13) by f, in teg ra t ing , and then

i n t e g r a t i n g by p a r t s ; a n d , of course , the equa t ions (1 .10 .13) a re ob ta ined

from (1.10.14) formally by following the de r iva t ion of E u l e r ' s e q u a t i o n s

g iven above . Th e a s s um pt ions on the Af a n d Bf corresponding to (1 .10.7),

(1.10.7 ' ) , and (1.10.8) are:

\A\^ + \A^\^ + | 5 | 2 + \B^\2 < Mf F2*-2 ,

\A,\^ + M P | 2 + \B,\^ + | 5 p | 2 < Mf F2^-4 ,

mlV^-^\n\^^A^{x,z,p)7li7r$,

w i > 0, ^ > 1, F = (1 + |^ |2 + |^ |2)i /2

(1.10.7 '") The s a m e as (1 .10.7") with A = A{x,p), B = 0,

V = {\ + 1: 12)1/2, k>\,

\B\^ + \B,\^ + \B^\^<Ml{R)V^^

(1.10.7")

(1.10.8")mi{R)V^-^\7i\^<A^pi 71^71^^ , 1^2,1 < M l ( i ? ) F ^ - 2

mi{R) > 0 , F = (1 + | ^ p ) i / 2 .

I n all cases , it is a s s u m e d t h a t the A^ an d 5 ^ ^ CJ if n = 2 or the

^ ? € ^^""^ a n d Bi^ CJJ~^ iin > } . A n d , of course , it is n o t a s s u m e d t h a t

M o r r e y , M u l t i p l e I n t e g r a l s 3

3 4 Introduction

Theorem 1.10.4. With these assumptions, the results (ii), (iii), (v), and

(vi) hold if the assumption that z be an extremal he replaced hy that that it

he a weak solution in H\{G) of (1.10.13) cind satisfy the additional condi

tions mentioned in the cases (iii) and (v).

L A D Y Z E N S K A Y A and U R A L ' T S E V A ([1], [2], [3]) and G I L B A R G and

o the rs have ob ta ined ex i s tence theo rems for e q u a t i o n s of the t y p e

(1.10.13). But in the last tw o or thr ee year s, V I S I K , M I N T Y , B R O W D E R ([3])

a n d L E R A Y - L I O N S {[6^)'\ have deve loped an ex i s tence theo ry for n o n

l inea r equa t ions in Banach s paces wh ich cove rs a wide class of equations

i n c l u d i n g m a n y of higher order and all equations of the type (1.10.14) in

which the Af and Bi satisfy (1.10.7") or (1 .10 .7 '" ) - The ir theory y ie lds

so lu t ions in H\{G) (or co r re s pond ing s paces in the higher order cases)

ra ther than c lass ica l so lu t ions and there fore takes the place of the ex is t ence theo ry ra the r th an th e d i f fe ren t iab i l ity theo ry . How eve r , comb ined

wi th ou r regu la r i ty re s u l t s we now have an existence theorem for any equa

tion of the form (1.10.13) which satisfies the (1.10.7") or (1 .10.7" ') condi

tions with N = \. T h e r e l e v a n t a b s t r a c t t h e o r e m of L e r a y and Lions is

s t a t e d and p r o v e d in § 5.12 where th is ex is tence theorem is p r o v e d .

1.11. Dif ferent iabi l i ty; reduct ion to l in ear eq u at ion s

In th is sec t ion we wish to give some ind ica t ion as to how one goesabout p roving the resu l ts on d i f fe ren t iab i l i ty which we s ta ted in Theorem

1.10.4. W e b e g i n by a p p l y i n g a dif fe rence quot ien t p rocess to the e q u a

tions (1.10.14) in which we r e g a r d the so lu t ion as k n o w n . We sha l l

assume f i rs t tha t the A^ and Bi satis fy (1 .10.7") with ^ > 2; we shall

ind ica te the modif ica t ions for the case (1 .10.8").

T o do th i s we choose an in tege r y, 1 < y < i , let ^y be the u n i t

v e c t o r in the x^ direction and define

^l (x) = h~^ \p {x-h Cy) - C* {x)], zi {x) = h~^ [z^ [x + h ey ) - z^ {x)],

( 1 .11 .1 ) 0 < \h\< a

where C bas s u p p o r t in a d o m a i n D' (Z C Ga. If we rep lace f by ^h in

(1 .10.14), make a change of coord ina te s x in the t e rms con ta in ing

C{x — hCy) or V C(^ — ^ ^v)y we o b t a i n the e q u a t i o n

( 1 .11 .2 ) fh-HCU^^f + C'ABi]dx = 0,

AAl =A'^[x-{- hey, z(x + hcy), \7z(x + he-y)] — Af[x, z(x), S7z(x)]

a n d A Bi i s g iven b y a s imilar formula. Now for each fixed h,0 <\h\ <, a,

t h e two t e r m s in AAf a n d ABi are defined and m e a s u r a b l e for a l m o s t

all X an d for s uch x can be expre s s ed in t e r m s of th e d i ffe rence q uo t ien ts

A z^jh and the i r der iva t ives us ing the in tegra l fo rm of t h e T h e o r e m of t h e

1.11. Differentiability; reduction to linear equations 3 5

M e a n . If t h i s is done (1.11.2) becomes

(1.11.3) = 0w h e r e

1

An(x) = | [ 1 + l^(^) + tAz\^ + \p(x) + tAp\^]-^+^^^dt

01

'^h(x)a^f^{x)=lA^pi^[x + they, z(x)+tAz, p(x) + tAp]dt0

1 1

^hh?i,^^lAf,idt, AneiyPn=JAt,ydt0 0

1

(1.11.4) AnPh = l [ \ + \z{x) + tAz\^ + \p(x) + M ^ | 2 ] ( A : - I ) / 2 ^ ^

01 1

^hO^ij = J Bipi^dt, Ahdjiij = J Bizjdt0 0

1

Anfl^Pn^ ^Bi^vdt, p{x)=\/z(x)

0Az = z{x + h Cy) — z{x), Ap = p(x -{- h ey) — p(x).

In th is case {k > 2), we see f rom (1 .10 .7") tha t the func t ions A^ etc.,

a re all m e a s u r a b l e , the coefficients aji, hn, cn, dji, en, and fji are all u n i

f o r m l y b o u n d e d ( i n d e p e n d e n t l y of h) and

( 1 . 1 1 . 5 ) ^ i | ^ | 2 < < f , - ( ^ ) 7 i t 4 ' Zi^hiA^W^^l' An[x)^\

for all h w i t h 0 <\h\<a.

N e x t , we let C^ = rj w^, w •= rj zl

w h e r e rj = i on D w h i c h we a s s u m e c. D'^, r] = \ — 2a~^ d(x, D) for

0 ^d{x, D) < ajl, a n d r] == 0 o the rwis e . Then we h a v eCU = rj(ze;% + n,a4), iqzy = w\^ - YJJ Z {.

If these C* are s ubs t i tu ted in to (1 .11 .3 ) , one o b t a i n s

JAn{{w\, + 77,a4) KIM^ - Vji) + ^tii^' + rieUPu\ +

(1.11.6) + wi[cl,^(w{, - ri,.z{) + dmow^ + rifl,Pn\)dx = 0.

Using (1.11.5), the Schwarz inequa l i ty , and the Cauchy inequa l i ty

( 2 | ^ 6 | < £ a 2 + £ - i & ^ e>0), (1.11.6) yields

(1.11.7) ( An\Vw\^dx<^C j Ah[\Vri\^zn\^ + \wf + YJ^P^dx.

F r o m the definit ion oi rj , we then conc lude tha t

(1.11.8) jAn\VZh\^dx^ClAh[{a-^+ i)\zh\^ + Pl]dx.D D'

3*

36 Introduction

B u t , f r o m L e m m a 3.4.2 and Theorem 3 .6 .8 , it fo l lows tha t A^ ->V^-^

in Lr(D'), Zfi ->z^y in L]c[D') and P;^ -^ F in L]c{D') so t h a t the r igh t s ide

of (1.11.8) is b o u n d e d i n d e p e n d e n t l y of h. A s t r a i g h t f o r w a r d a r g u m e n t ,

g iven in § 5-9, s h o w s t h a t we may lei h ^0 ( th rough a subsequence) to

o b t a i n the fo l lowing theo rem:

Theorem 1 .11 .1 . Suppose that z ^ Hl{G) and is a weak solution of

(1.10.14), where the Af and Bi satisfy ( I .IO .7" ) or (1 .10.7 '") with ^ > 2.

Then z and U = V^l^^ Hl(D) for each D (Z d G and the vectors py satisfy

DI

(1

(1

.11.

.11.

where

(1 .11.

9)

10)

11)

/D

Vfc-

v^-Vk-

-^vp

-^[x)al

-2 b' . =.

\^dx

fw• ^tzi

<.Ca

Ao c

•, v^-

-VD

p^[^'

lefy

V^dx

z{x),

, DCD„

/ -W].= Af,y, F*-2

. C C G , C

cfi = Bij^

Vfc-^dij = Bui, V^-^fr = Bi^y.

I n the case (1 .10.8") with k>:v, we can p r o v e in essen t ia l ly the

s ame manne r Theo rem 1 .11 .1 ' which is the same as Theorem 1.11.1 except

that h'^j, cf- and f\ must he replaced respectively by bf^ * V, cf^ V, and fl V

and dij must be replaced by V^dfj and we must assume that z satisfies

1.10.9). Moreover we conclude also that

(1.11.12) IVf^+^dx<Ca-^ fVf^dx.D D'

The proof in th is case is more diff icult but is simplified by us ing a l e m m a

of L A D Y Z E N S K A Y A and U R A L ' T S E V A [2] (see Lemma 5.9.1). There is no

re s u l t co r re s pond ing to (iv) of T h e o r e m 1.10.4 for the v a r i a t i o n a l p r o b l e m .

I n the case of the va r ia t iona l p rob lem we h a v e the following result :

Theorem 1 .11 .1 ' ' . Suppose that f satisfies (1.10.7) or (1 .10.7 ') , with

1 < ^ < 2, and z"^ ^ Hl{G). Then there is a z which minimizes I(Z, G)

among all Z such that Z -- z"^^ ^K^) <^^^ which is such that its derivatives

^ HI (D ) and satisfy (1.11.9) and (1.11.10) on each D d G Ga and the func

tion U = F^/2^ HI [D) for each such D.

The diff iculty in the proof of th is arises from the fac t tha t Ah a l w a y s

< 1 and th is causes t rouble in the dif fe rence-quot ien t p rocedure . To get

a r o u n d t h i s we min imize I[z, G) w i t h D{z, G) < K. For each K, we o b t a i n

c e r t a i n b o u n d s i n d e p e n d e n t of h and so we may let /^ -^ 0. Then we mayallow i^ ->c>o. The s pec ia l a rgumen t u s ed in th is proof is p r e s e n t e d in

§ 5 . 1 0 .

I n all cases , the n e x t t h e o r e m to be p r o v e d is the following:

T h e o r e m 1.11.2. Suppose that N = \ and that f or the A'^ and B and z

satisfy the hypotheses of Theorems 1.11.1, 1.11.1', or 1.11.1". Then the

1.11. Differentiability; reduction to linear equations 37

function W = U^ satisfies an inequality of the form

(1.11.13) / K , a K ^ l ^ , ^ H- B^PW) + ^{C^^PWô. + DP^W)]dx ^ 0

ifC^Lip,{G), C{x)ÔonG,

for some Ay 1 < A < 2, where the a* satisfy (1.11.5) (^ = ^), the coeffici

ents B, C and D are hounded and measurable, and P^Lp (G) and satisfies

(1.11.14) j f PUxY" <.ZrQ, DCCG

where Z depends on A, G, and \\P\\^^Q.

T h i s is p r o v e d in § 5.9. The idea is to set

^ =.yjV-'py, y)^Lipc(G), xp[x) > 0 on G,

in equa t ions (1 .11 .9) ( technica l lemmas a l low th is ) . I t t u r n s out to be

possible to choose e s ma l l enough so t h a t W = V^~^ = U^ satisfies

(1.11.13). In the cases (1.10.7), etc. , P = 1 w h e r e a s P = F in th e cases

(1.10.8) . The inequali ty (1 .11.14) follows in the cases (1.10.8), etc. , from

t h e H o l d e r i n e q u a l i t y in case k ^ v and f rom Theorem 1.10.2 in the case

k = V. The idea of s t u d y i n g the s o lu t ions of inequali t ies l ike (1 .11.13)

is basic to M O S E R ' S simplification [1] of the D E G I O R G I - N A S H theory. This

i s p re s en ted in §§ 5-3 and 5-4.B u t by c o m b i n i n g the re s u l t s of Theorems 1.11.1, 1 .11.1 ' , 1 .11.1",

a n d 1.11.2, we conc lude tha t the func t ion U satisfies the h y p o t h e s e s of

T h e o r e m 5.3-'I and hence U and there fore z and all the py are b o u n d e d

on each doma in D C. G G. T h e n the equa t ions (1 .11 .9) take the form

(5.3.21). I t follows from the t h e o r e m s in § 5.3 t h a t the py are H o l d e r -

c o n t i n u o u s on in te r io r doma ins . Then the equa t ions (1 .11 .9) take the

form (5.2.2) with Holder-continuous coeffic ients and it follows from

T h e o r e m 5.5.3 t h a t the de r iva t ive s of the py are H o l d e r - c o n t i n u o u s on

in te r io r doma ins f rom wh ich it fo l lows tha t z^ Cl(G).The re s u l t s unde r (v) of L a d y z e n s k a y a and U r a l ' t s e v a for b o u n d e d

weak s o lu t ions , when 1 < ^ < r, are somewhat more d i f f icu l t and are

p r e s e n t e d in §5- 11 . But as soon as the f i rs t der iva t ives are seen to be

b o u n d e d on in te r io r doma ins , the proo f tha t z^ C^ in the case ; = 2 is

t h e s a m e as t h a t s k e t c h e d a b o v e .

O n c e the so lu t ion {N — 1) z is k n o w n to $ Cl(D) for each D G G G,

i t then fo l lows tha t z satisfies the equa t ion (1 .10 .13) which becomes

a'^^x, z, Vz)zôc^ = g{^> ^> V^)(1.11-15) ,

a^^ [Xy z,p)=l (A;^ + AD, g = B-A^p.~ A%.

w h e r e in th e case of a va r ia t iona l p rob lem, th i s is E u l e r ' s e q u a t i o n and

3 8 Introduction

I f / s a t i s f i e s t h e " c o m m o n c o n d it i on " w i t h ^ > 3 , i.e . i f / a n d / ^ ^ Cl~^,

in th e var ia ti on al case , o r if ^ ^ CJJ"^ an d B ^ C'^~^ in general , i t fol lows

t h a t t h e a'^^ a n d g^ C^^ and the d i f fe ren t iab i l i ty of z s t a t e d i n T h e o r e m

1.10.4 follows from a repeated application of Theorem 5-6.3 as follows:

Since z^ Cl{D) for eac h Z) C C G, it follows t h a t a""^ a n d g^ Cl(D)

for such D s o t h a t z^ Cl(D) for such D b y Th eor em 5-6 .3 . T he n a^"^

a n d g^ Cl{D) for such D and hence z^ C"*(Z)) for such D. The re s u l t

fol lows b y indu c t io n . The C°^ resu l t fol lows b u t th e ana ly t ic i t y require s

a s e p a r a t e proof. The ana ly t ic i ty proof for th is case {N = 1) is pr ese nte d

in § 5 .8 whe re references are g iven .

I t i s to be no t iced tha t the f i rs t theorem above which does no t ho ld

for sys tems is Theorem 1.11.2 (see th e re m ar k a f te r th e proof of T heo rem

1.11.2 in § 5 .9)- Th is enable s us to show th a t solu tions ar e Lips chit z onin te r io r dom a ins . Bu t even if th i s cou ld be p rov ed , the D E G I O R G I - N A S H -

MosER theory (presen ted in §§5 .3 and 5-4) has no t been ex tended to

sys tems so we would s t i l l be unable to prove the Holder cont inu i ty of

t h e py , or even the i r con t inu i ty .

In ca s e a pa r t o f ^G i s s moo th and the bounda ry va lues a re s moo th

a long tha t par t , there i s a gap in the d i f fe ren t iab i l i ty resu l ts . By making

a correspondingly smooth (CJJ , n^ 1) change of var iab les and sub

t rac t ing o f f a s moo th func t ion hav ing the g iven bounda ry va lues , onereduces the equation (1.10.14) (or (1 .10.13)) to one of the same type in

which one works in a hemisphere and the so lu t ion is supposed to vanish

(in some Hl^ sense) on the f la t part GR {X^ = 0) of the hemisphere Gn

(see § 1 .2). The t ran s for m ed equ a t io n or sys t em h as the sam e p rop er t ies

as th e or ig ina l exc ept poss ib ly for th e bo un ds . O ne th en carr ies ou t t he

d if fe rence quot ien t p rocedure of Theorem 1.11.1 in the tangential direc

tions and s hows tha t the py ^ HI (Gr) foTr<.R and s a t i s fy the equa t ions

(1.11.9), y = "i, . . . ,v — \. Since i t i s t rue tha t pv^y = py,v for y < .v , all

the de r iva t ive s pv,y w i t h y <.v^ L2{Gr). Since we a l ready have in te r iord i f fe ren t iab i l i ty we m a y so lve th e equa t ion s (1 .10 .13) for the z^^^ = pl^,

in te rms of the o thers . Thus a l l the py ,y,y = 'i, . . . ,v, satisfy (1.11.9) and

(1.11.10) follows. However, Theorem 1.11.2 has no t been shown, in a l l

cases, to ho ld in s uch a bounda ry ne ighborhood . Bu t the fo l lowing

theo rem s cove r w ha t i s know n to th e au t ho r abo u t the bo un da ry behav io r

of so lu t ions :

Theorem 1 .11 .3 . Suppose a portion y of dG, containing a point PQ, is

of class C'^, n > \y suppose that the Af and Bf satisfy the comm on condition and any one of the sets (1.10.7' ')> (1-10.7'") ' ^^ (1.10.8") for some

r > 2 and ^ > 1, and suppose that z^Hl {G) and is a solution o/(1 .10 .14) .

/ / , in addition, N = \, z satisfies a Lipschitz condition in a neighborhood

ru y {rG G) of PQ on G and vanishes along y , then z is of class C^ in any

smaller neighborhood^ of PQ on G. If N ^ 1, the same conclusion holds if

2.1. I n t r o d u c t i o n 3 9

we know that z^ C^{r[J y ). Ify,A, and B^ C^ or analy tic, respectively,

so is z on Syi.

T h e o r e m 1.11.4. Suppose G is bounded, (p ^ Hl{G), N = i, B = 0,

and the A^" = A°'{p) ^ C^ and satisfy the inequalities (1 .10.7 ' ' ' ) with k — 2.

Then there exists a unique solution z of Equa tion (1.10.14) such that

z — (p^ HIQ (G) . / / G is of class C^ and cp ^ C^(G) then z^H^ {G) for any r.

If the A'^^Cl, G is of class C^ and (p^ C^iG), then z^ Q ( S ) . / / we

merely know that the A°'(p)^ C^ and satisfy the condition

^p^{P)^ock>0 if ; . # = 0 , p^Rv

and if G is of class C ^ and cp^ C' (^) then there exists a unique solution

z^ € ^(0) of Equation (1.10.13) such that z — (p on dG.

This i s jus t a combina t ion of Lemma 4 .2 .5 and Theorem 4.2.3 .Addit iona l boundary va lue resu l ts a re found in Theorems 4-2 .1 and 4 .2 .2 .

F o r N = \ a n d t h e py a l l bo un ded , th e equ a t io ns (1.11 .9) ta ke the

form (5.3.21) with G rep laced by GR, a s s u m i n g t h a t t h e a b o v e - m e n t i o n e d

t rans fo rma t ions have been made . S ince each py w ith 7 < r van ish es

a long OR, i t fol lows from Lemma 5.3.5 that these py ^ C^(Gr) for each

r <. R and, in fact these py each sa t is fy the ' 'Dir ich le t g rowth* ' condi

t ion (5.3-27). Since the derivatives of pv a re a l l de te rmined a s above

from those of the o ther py , i t fol lows that pp also satisfies such a condit ion and hence ^ C^ {Gr) for each r <CR (Theorem 3-5 .2). Th e C ^ resu l ts

a re ev id ent an d th e an a ly t ic i ty i s p ro ved in § 5 .8 . Th e resu l ts for A > 1

follow from th os e in §§ 6.8, 6.4, an d 6 .7 .

C h a p t e r 2

Semi-class ica l resul ts

2.1 . I n t r o d u c t i o nIn th is chapte r , we begin by proving some of the e lementary proper

t ies of harmonic functions . A proof of W E Y L ' S lemma (Weyl [2]) is

ins erte d in § 2 .3 for la t er referen ce; th e proof is incl ud ed a t th a t po int

s ince it i s close ly re la te d t o th e m ea n va lue pro pe r ty . The n in § 2 .4 th e

class ical notions of G R E E N ' S func t ions and e lementary func t ions a re in

t roduced and the s e no t ions and POISSON 'S in tegral formula for the c irc le

an d ha l f p lan e a re c a rr ied over t o th e r -d im ens io na l case . In § 2 .5 , the

s t ud y of po ten t ia l func t ions is beg un and the formulas for an d co nt in u i typroper t ies o f the i r f i rs t der iva t ives a re der ived . In §2 .6 , the formulas

for an d cont i nu i ty pr ope r t ies o f th e f i rs t der iva t ives of ce r t a in "gener a l ized

po ten t ia l func t ions " a re s tud ied and the H O L D E R con t inu i ty o f the

second der iva t ives of o rd inary po ten t ia ls o f H O L D E R con t inuous den

s i ty func t ions fo l low from the genera l resu l ts ; an example of a con-

4 0 Semi-class ical resul ts

t inuous dens i ty func t ion whose po ten t ia l i s no t everywhere of c lass C^

is given . In § 2 .7 we pr ese nt a proof of the now famous inequali t ies of

CA LD ERON a n d ZYGMUND ([!]) and ([2]) for s ingular integrals ; we con

fine ourselves to rea s onab ly s moo th func t ions inorde r to r e t a i n t h e

essen t ia l s impl ic i ty of the i r p roofs . The chapte r conc ludes wi th the proof

of the maximum pr inc ip le for ce r ta in second order e l l ip t ic equa t ions

which was g iven by E. H O P F ([1]).

2.2. Elemen tary p rop ert ies o f h armon ic fu n ct ion s

Definition 2.2 .1 . uis harmonic on G <— > u ^ C^{G) a n d

V

A ii{x) ^ 'ûôôi[x) =0, x G,

x=l

T h e o r e m 2.2 .1 . If u isharmonic on G, D (ZC G, D isclass C^, and

B{xo, R) CG, then

(2.2.1) Juôcdx^=^ 0dD

(2.2.2) u{xo)=r- lu{x)d2:

(2.2.3) u{xo) = \B{xo,R)\-^fu(x)dx.B{xo,R)

Proof. The first follows from G R E E N ' S t h e o r e m . If D = B{xo,r),

0 <r <R, (2.2.1) takes the form

(2.2.4) furdS =0dB(xo,R)

w h e r e Ur denotes the rad ia l der iva t ive . Le t ( r , p) be pola r coord ina tes

with po le atXQ, r = \X — XO\ a n d p on U =dB{0,\). Then (2.2.4) is

equ iva len t to

(2.2.5) j Vr[r,p)dZ =0, v(r,p)=u(xo +rCp), Cp = qp,

In tegra t ing (2 .2 .5) f rom 0 to jR yield s (2.2.2). U sin g (2.2.2) wi th R

rep laced by r, we obtain (2 .2 .3) by multiplying both s ides of (2 .2 .2) by

Fv r^-^ and in teg ra t ing wi th re s pec t to rfrom 0 to jR.

Theorem 2 .2 .2 . Suppose u is harmonic on the domain G, XQ^G , U

takes on its maximum value at XQ. Then u[x) = U{XQ).

The proof follows eas ily from the mean value Theorem 2.2.1 and theconnec tedness of G. The fo l lowing importan t un iqueness theorem fo l lows

i m m e d i a t e l y f r o m t h e m a x i m u m p r i n c i p l e :

T h e o r e m 2 .2 .3 . There is at most one function uwhich is continuous on

G and harmonic inG which takes on given continuous boundary values on

dG (G be ing ab o u n d e d d o m a i n ) .

2 .3 . W E Y L ' S lemma 41

Theorem 2 .2 .4 . If u is continuous and {2.2.}) holds for every B{XQ,R)

G G G, then u is harmonic. A harmonic function has derivatives of all

orders which are harmonic.

Proof.Since (2.2.3) holds, we see t h a t

u^ Ci(GO) with

uôi{x) = \B(xo,R)\-^fu(x)dx^ if B{xo,R)C.CG, (x= \, . . .,v.

d B (ajo. R)

B u t t h e n by G R E E N ' S t h e o r e m , we see t h a t u^^ also satisfies (2.2.3)- By

i n d u c t i o n , we see t h a t all d e r i v a t i v e s are c o n t i n u o u s and satisfy (2.2.3).

So, s uppos e XQ ^ G. E x p a n d i n g in T A Y L O R ' S series , we obtain

u{x) — u(xo) = {x°' — xfj uôc{xo) +

^^•^'^^ +-^{^"- ô) (^^ - ^ g ) ^,cc^M + R{XO, X)

\R(x,xo)\ <M ' \ x - xo\^ if \x - xo\ <R, B{xo,R)CCG.

It follows, by in tegrating (2.2 .6) over B(xo, r), d iv id ing by r^ \B{xo, r)\,

using (2.2 .3), and l e t t i n g r ->0, t h a t Au{xo) = 0.

F r o m t h i s one eas i ly ob ta ins the following corollary:

Corollary. If u^ G^{G) and satisfies (2.2.4) (i.e. (2.2.1)) for each sphere

B{xo, r) C Gy then u is harmonic on G.

F o r , by r e p e a t i n g p a r t of the proof of T h e o r e m 2.2 .1 , one conc ludes

that (2 .2 .3) holds for each such B{x{), r).Theorem 2.2.5. Suppose each Un 6 C^{G) and is harmonic in G, and

suppose that the sequence {un} converges uniformly on G to a function u.

Then u^ C^(G) and is harmonic on G.

For each Un satisfies (2.2.3) so the l imit function does a lso.

T h e r e a d e r can p r o v e the fo l lowing inequa l i t ies :

Theorem 2.2.6. Ifu is harmonic and in Lp{G), then

\u(x)\^ ^\B{x,r)\-^ f\u(y)\Pdy if B(x,r)c:G.

B(x,r)

Theorem 2.2.7. If u is harmonic and u^LîG), then \S7 u[x)\

< C{v) \\u\\\ d~'^~^''^ where dx denotes the distance from x to dG.

Theorem 2.2.8. / / u is harmonic onG, \u{x)\ < Af there, dx denotes the

distance of x from dG for x^G, there exists a constant C, depending only

on V, such that

\\/û(x)\^k\eJ^-^C^MS-^, k>\.

2.3. W e y l ' s l e m m a

W e beg in wi th a gene ra l i za t ion of the L e m m a of du B O I S -R A YM O N D :

Lemma 2.3 .1 . Suppose f <^ Li on the interval [a, b] of Ri and suppose

that

h(2.3.1) lf(x)g(x)dx = 0

4 2 Semi-classical results

for all g^ C^ [a, b]. Then f(x) = 0 almost everywhere. If (2.3-1) holds

only for all g ^ C^ [a , b] for which

b

(2.3.2) jg(x) dx = 0,a

then f{x) = a const, almost everywhere.

Proof. In the first case, it follows by approx ima t ions tha t (2 .3 .1 )

h o l d s for all g w h i c h are b o u n d e d and m e a s u r a b l e on [a, b] f rom which

the f i rs t resu l t fo l lows immedia te ly by s e t t ing g{x) = sgnf{x). In the

second case, let ^1 ^ C ^ [a, b] w i t h

b

f gi(x) dx = \.

aT h e n if g is any func t ion ^ C ^ [ a , b], we h a v e

b b

g{x) = g*{x) + gi{x)Jg{y) dy, f g'^ix) dx = 0

a a

a n d , of course , g * ^ C^ [a, b]. The second result follows.

Theorem 2.3.1 ( W E Y L ' S lemma, W E Y L [2]). Suppose that u ^ Li(D)

for each D G G G and satisfies

(2.3.3) f u{x) Av{x) dx = 0G

for all v^ C'^(G). Then u is equivalent to a harmonic function.

Proof. Suppos e B{XQ, a) C G, 0 <C s <, a, (p(r) ^ C"^ for r > 0, (p(r)

= 0 ioT r > a, (p{r) = (p{e) if 0 < r < £, and

(2.3.4) v{x) =(p{\x- xo\), X[r) = fu(x)d2:.dBixo.r)

T h e n v^ C ^ ( G ) and X^Li[s, a]. Then (2.3 .3) becomes

a a

fx(r) [I ( . -1 <p ')] dr=jX(r)y, {r) dr = 0,

e e

w h e r e ip may be any func t ion ^ C^[s, a] for which (2.3.2) holds (with

[a, b] = [s, a]). It fo l lows tha t X{r) = a c o n s t a n t a.e. on [e, a]. Since e

a n d a are a r b i t r a r y t h i s is t r u e for 0 < r < d(xo, dG); let us ca l l tha t

c o n s t a n t Fv • u(xo). U s i n g the definit ion of X{r) in (2 .3 .4) , mul t ip ly ing

by y*'-i and i n t e g r a t i n g , we o b t a i n

(2.3.5) u{xo) = \B(xQ,r)\-^ Ju{x) dx, B(xo,r) CG.B{Xo,T)

By ho ld ing r f ixed, we see that u is c o n t i n u o u s on G a n d by l e t t ing r ^0

w e see t h a t u(x) = u{x) a.e. so that (2 .3 .5) holds with u rep laced by u

on the r i g h t . T h u s u is h a r m o n i c on G.

2.4. POISSON'S integral formula; elementary functions; GREEN'S functions 43

2 . 4 . P oisson ' s in tegra l formu la; e l emen tary fu n ct ion s ; Green ' s

f u n c t i o n s

Sup pose G is a bo un de d do m ain of class C^ a n d u a n d v are of class

C^ onG = GU dG. Then, f rom G R E E N ' S theo rem, we ob ta in the fo rmula

/ {uAv — vAu)dx= / f^T ^T ~) dS = {uv,oc — vu,oi) dx'^

G dG dG

(2.4.1)

n be ing the ex te r io r no rma l .

Next , i t i s a wel l -known and eas i ly ver i f ied fac t tha t

i s h a r m o n i c ii y ^ 0, Moreove r

( ^ d S -J dn ~~ ] d B ( o , Q )

^ ^ < o . e ) [271, iip=2

So, let us define

- {v -2) fg^-'^-dS = - {v -2)rr, iiv > 2

(2.4.2) Ko{y)=\^^ w i I I . ,[ {27i)-^log\y\ if )^ =

22

Now, s uppos e G is bounded and of c lass C^, u^ C^(G), Au(x) = f{x)

on G, a n d XQ^G. Suppos e B{X{),Q)(ZG and we apply (2 .4-1) to the

d o m a i n G — B{x{i, Q) w i t h v{x) = Ko(x — XQ). T h e n w e o b t a i n

dB{xo,Q) dG G-Bixo.Q)

(2.4-3)

Let t ing ^ ->0 in (2 .4-3) , we obta in

(2.4-4) u{xo) =J[^J^ - ^ ^ ) ^S + f^oi^ - ^o)f(pc) dx

SG G[v(x) = Ko{x — xo)).

Definit ion 2.4.1 . The func t ion KQ, defined in (2.4-2) is called the

elementary function for Laplace's equation Au = 0.

If f[x) = 0, then (2.4.4) expresses u{xo) in t e rms o f i t s bounda ry

va lues and thos e of i t s no rm a l de r iva t ive . How eve r , from t he m ax im um

pr inc ip le , a ha rmon ic func t ion i s comple te ly de te rmined by i t s bounda ry

values alone. If, in (2.4-4), we h av e / (A; ) = 0 and cou ld t ake

(2.4-5) ^(^) = Ko(x — xo) + H[x, XQ) -= GO{XQ, X)



4 4 Semi-classical resu l ts

w h e r e H i s harmonic in x for each XQ an d is so chosen t h a t ? ; = 0 on 3G ,

then (2.4 .4) would express U{XQ) in te rms of i t s boundary va lues . Such a

func t ion v, if it exists , is called a G R E E N ' S function for G with po le a t XQ.

B y t h e m a x i m u m p r i n c i p l e , t h e G R E E N ' S function is unique if i t exis ts

at all.

From the discuss ion so far given, i t fol lows that {a ) if a G R E E N ' S

func t ion v exis ts for a given domain G and po in t XQ a n d [h) if u is of

class C^ on G a n d h a r m o n i c o n G, then (2.4-4) expresses U{XQ) i n t e r m s

of i t s boundary va lues . I f a G R E E N ' S function could be found and if i t

could be shown to be harmonic in xo for each x m G, then the func t ion

u def ined by (2 .4-4) w i t h / = 0 and u in the bounda ry in teg ra l r ep laced

by a func t ion u * would be ha rmon ic ; i t wou ld then rema in to s how tha t

u{x) ->^**(xo) d.s X -^XQ for each x^^ on dG. And, of course , p roving theexis tence of the G R E E N ' S func t ion requires proving the ex is tence of

ha rmon ic func t ions hav ing g iven bounda ry va lues . Th i s p rob lem i s

ca l led the Dirichlet problem.

Because of a l l the problems ment ioned in the d iscuss ion above , the

D IRICH LET problem is no t usua l ly so lved by proving the ex is tence of

th e G R E E N ' S func t ion . Ho wev er , the re a re two cases wh ere th is i s poss ib le,

nam ely th e case wh en G is a sphere which , obvious ly , ma y be assum ed to

have cen te r a t the or ig in and the o ther i s a ha l f -space which we mayas s ume to be tha t whe re x"^ > 0. W e now derive ' th e G R E E N ' S func t ion

for such a sphere .

L e t x^ BR — B(0, R) and le t x' be the po in t inverse to x w i t h

respec t to BR, tha t i s , the po in t where

'x^^ = R^x^'llx]^.

Using the spher ica l symmetry , i t i s easy to ver i fy tha t the ra t io \^ — x\l

If — x' \ is the same for a l l ^ on OGR SO t h a t

(2.4.6)H-x'

Thus we note tha t i f we def ine

( ^ [ l o g | f - x \ - log\i - x'\ - log( | ;^ | / i^ ) ] , V = 2

(2.4-7) G(x, I) - _ (^ _ 2 ) - i p-i [ | | _ ;^|2-v ._ | |: _„ x'\^-''{\x\IR)^-']

[ v> 2

t h e n G(x, f) is of the form (2.4-5)- Moreover, by using the formulas for

x' an d f', we see th a t

(2.4-8) G{1 x) = G{x, I )

s o th a t G i s harmonic in x for each f and vanishes for f in terior to BR a n d

x^dBR. Thus the func t ion u def ined b y (2 .4-4) w i th / = 0 an d v{^)

2.4. P OI S S ON ' S integral formula; elementary functions; G R E E N ' S functions 45

= G{x, f) is h a r m o n i c on BR, Fina l ly , s ince the function w = 1 an d G

satis fy the h y p o t h e s e s of the a r g u m e n t in the p a r a g r a p h c o n t a i n i n g

equations (2 .4-3) and (2.4-4), we conclude that

(2.4.9) | ^ , S ( | ) = 1, .^B,.

dB R

By computa t ion f rom (2 .4 .7) , we see tha t

dn{i)R-^^^Gôc = i ^ - i r ^ - i f ^ p l - x\-'{^°^ - A;«)

; | : ^ | / i ^ ) 2 - r | | _ ^ ' | - v ( | a _ ^ ^ a )

F o r I on dBn, we may use (2.4-6) to o b t a i n

( 2 - 4 - 1 0 ) ^ ^ = ( / ; 7 ? ) - i if - x \ - r { m - \ x \ ^ ) > 0

a n d t h u s o b t a i n Poisson's integral formula

(2.4.11) u{x) = {ryR)-' j\^-x\-''{R'^- \x\ )u*( )di

for the harmonic func t ion u which t akes on given va lues ^ * on SBR.

T o see t h a t u{x) - > w * ( f o ) as x->io if u* is c o n t i n u o u s , X^BR,

f o € ^^ i2 , we no te from (2.4.9), (2 .4-10), an d (2.4-11) th a t

u(x) - M*(|o) = ( r . i ^ ) - i / | | - x\-^{R^ - |^|2)[^^*(f) - u*(ô)]dS.

(2.4.12) ^ «

To show t h a t th is d i ffe rence -> 0 , we br eak the i n t e g r a l on the r igh t in

(2.4-12) into integrals / i ove r SBR H^ ( I O , Q) a n d I2 ove r ^^ i ? — B{ô, Q)

w h e r e we ma y choose g so t h a t | ^ * ( | ) — w * (f o) | < e/2 for | ^ ^jBi? Pi

B(Ô,Q), S being g iven . The r e a d e r may c o m p l e t e the proof. T h u s we

have the fo l lowing theo rem:

T h e o r e m 2.4 .1 . There is a unique function u which is continuous on

B(0, R), coincides with a given continuous function u* on dB(0, R), and

is harmonic in B(0, R). It is given in B(0, R) by Poisson's integral formula

(2.4.11).

W e can n o w p r o v e the fo l lowing importan t re f lec t ion pr inc ip le for

h a r m o n i c f u n c t i o n s :

Th eorem 2.4.2 (R ef lect ion pr inc ip le ) . Suppose u is continuous on G

and harmonic on the {possibly unbounded) domain G which lies in a half-

space bounded by the hy per plane 77, suppose Q n U = S, and supposeu{x) = Oforx^ S. Letr = G\J G' \J S^^\ where G' is the domain obtained

by reflecting G in U and S^^) is the non-empty set of interior [with respect

to IT) points ofS. Define U(x) = u{x)forx^ G and define U{x) = —u{x')

fo r x^r — G, where x' is the point obtained by reflecting x in II. Then U

is continuous on P and harmonic in P.

4 6 Semi-classical resu lts

Proof. That Uiscontinuous on T and harmonic near each XQ in

r — S<0) isobvious. So suppose XQ^S^^^ and choose 2 so small thatB{xo, R) c r and let G(xo, R) =B(xo, R) n G- Let Hbe that harmonicfunction inB{xo, R) which coincides with Uon dB(xo, R). From the

symmetry in Poisson's integral formula, itfollows that H(x) =0along5(0) nB(xo, R) and hence H{x) =u(x) on dG(xo, R). Thus H(x) =u{x)

on G(xo, R) and hence, using the symmetry in Poisson's integral formula,we conclude that H{x) = U(x) on B(xo, R).

The following Liouville-type theorem is another immediate consequence of Poisson's integral formula:

Theorem 2.4.3. A function which is harmonic and hounded on Rv is a

constant.

Proof. For, by differentiating (2.4.11) with u*=u andletting7 ^ o o , one finds easily that each u^x(x) =0.

Suppose wenow consider thehalf-space i?+ : J " > 0 and define

G{x, I) =Ko{x — i) — Ko('x —f) where V = —x\ '< = <,

i.e. 'A; is the reflection of xin IT : x^=0. We notice that

G(f, x) =G(x, ^) , G(x, f) = 0 for f ^ 7 7 .

Thus G is harmonic in | for each xand in xfor each | and thus wouldappear to be a candidate for a G R E E N ' S function for R . Clearly ^G(A;, f)/

dn{i) is just —dG(x, f)/«9|''. Along I*' = 0, we have

- dG{x, f ) / a | - - KoM^ - f) - ^ o , . ( ^ ' - f)2.4.13) ,

A straightforward computation shows that L is harmonic in x for eachil, is positive, and

(2.4.14) lL(x,QdC= 1.Rv~l

Thus if^*(fi,.) is continuous and bounded and we define

(2.4.15) u{x)=lL{x,Qu*{Qdil,

Rp-l

we see, as in the case of Poisson's integral formula for B{0, R), tha t u(x)

is harmonic in R and hounded there hy the sup of ^*( |^) , is continuous

on R , and coincides with «/* along 77. We now prove:

Theorem 2.4.4. There is a unique function u which is harmonic inR^ and continuous and hounded in R and which coincides on x^ =0

with a given hounded continuous function u*; uis given hy (2.4.15) (^f^d

(2.4.13).Proof. It remains only to prove the uniqueness. Ifui and U2 both

satisfy all the conditions of the theorem, then u =ui — U2 is bounded

2.5. Po t en t i a l s 47

and continuous on R^ and vanishes on 77. If we then extend u by reflection as in Theorem 2.4.2, then U is harmonic and bounded everywhereand is therefore a constant which must be zero.

2.5. Potentials

In formula (2.4.4) with v(x) = Ko(x — XQ) = Ko(xo — x), we seethat if u is of class C^ on G with

(2.5.1) Au{x) =f{x)

where G is of class C^, then the boundary integrals are harmonic so thatthe function U defined by

(2.5.2) U{x)=lKo{x i)mdi

G

would differ from ^by aharmonic function and hence would also be a

solution of (2.5.1).Definitions 2.5.1. The equation (2.5.I) is known as Poisson's equation

and the function C7 in (2.5-2) is called the potential of f.Unfortunately, if / is merely continuous, itdoes not follow that U

is necessarily of class C^; we shall give an example of this in § 2.6.Since there is no difference in the proofs, we shall consider the more

general integrals(2.5.3) U{x) JK{x- )md .G

The following definition is useful in this section and in the study 0/higher order differential equations:

Definition 2.5.2. A function (p{y) is essentially homogeneous of degree pin 3; if 99 is positively homogeneous in case p<C0 or, ii p is an integer> 0, 99 has the form

(2.5.4) q^{y) = M y) logbl = niy)where cpo is ahomogeneous polynomial of degree pand 991 is positivelyhomogeneous of degree p.

Theorem 2.5.1. Suppose f is bounded and measurable with \f{x) \ <Mand has compact support G G, K C^{Rv — {o}) and Kis essentiallyhomogeneous of degree 2 — v, and U is defined 6y (2.5.3)•

Then U^ C^ everywhere with

(2.5.5) U,4x)==fK^4x-i)f(i)di, | V C / ( ^ ) | < M | | V i ^ | | ? . 2 : - ^

rM,[3 | ivi^ | | ; ,^ +(2.5.6) \'VU{x2)~VU{xi)\^l+\\^^K\\l,:\og{\+AlQ)], if ^ > ^

|M, | | v2 i^ | | ; . 2 ; log (1 +zJ /^ ) , if Q<d,

where q =\xi — X2\, d is the distance of G from the segment x\ x^ and A isthe diameter of G.

4 8 Semi-classical results

R e m a r k . T h i s t h e o r e m h o l d s ifUa n d /are vec tor func t ions and Kis

a ma t r ix func t ion . The proof is the same with ind ices inser ted .

Proof. We first as su m e t h a t / ^ ^ c ( ^ ) ' "the proof inthe general case

follows by a p p r o x i m a t i o n s . We define

(2.5.7) U,{x)=JK{x- )f{ )d .G-B{x.Q)

I t is easy tosee t h a t UQ converges un iformly toUon any b o u n d e d set

a n d t h a t

(2.5.8) U,,4x) =-fK(x - |)/(f) di:+lK,4x - f)/(|) di;dB{x,Q) G-B{X,Q)

th is ho lds even ifB (x , Q) Cf G, s ince we may f irs t replace G bya la rge

reg ion G' Z ) GUB{x, Q). Since VK ish o m o g e n e o u s of degree 1 — r, itis easy tos ee tha t the v e c t o r V UQ t ends un i fo rmly on any b o u n d e d set

t o the vec to r Vdefined by

(2.5.9) V4x) = UM^) ==IK,4X -i)md^G

from which (2.5.5) follows easily.

Next , def ine

(2.5.10) V.Q[x)=JKAx- )f[ )d .

G-B{X,Q)T h e n

(2.5.11) \VQ[X)-V{X)\<.\ " "• ., , , ^

Dif fe ren t ia t ing as in (2.5.8), we o b t a i n

Fa,,^(^) =-fK,4x -f)/(f)^|^ + IK,. {X - i)fi^)di,dBix.Q) G-B(x,Q)

T h u s , ifwe d e n o t e the dis tance f rom :r toG by ^ 1 , we see t h a t

f M [ | l V i ^ l | ; . ^ + | l V 2 X | | ? . 2 : l o g ( 1 + z J / ^ ) ,

( 2 . 5 . 1 2 ) | V F e ( : ^ ) i < | if B{x, Q)nG4^0

l i k r | | v 2 i ^ l | S . 2 : l o g ( 1 + / J / ^ i ) , if B(x,Q)nG^0.

The result (2.5.6) follows by se tt in g |:V2 — ^ i | =Q, obs e rv ing tha t

\V(X2) - V{xi)\ <\V{X2) - VQ(X2)\ +\V{Xi) - VQ(XI)\ + \VQ(X2)-VQ(XI)\

and using (2.5.12) toe s t i m a t e | VQ(X2) — VQ{XI)\.

2.6 . G e n e r a l i z e d p o t e n t i a l t h e o r y ; s i n g u l a r i n t e g r a l s

F r o m T h e o r e m 2 . 5 . I , it fo l lows tha t the f i rs t der iva t ives of po ten t ia lfunc t ions are given by fo rmula s ofthe form

(2.6.1) V(x)=r{x~i)mdS



2.6. Genera l i zed po t en t i a l the ory ; s ingu lar in tegra l s 4 9

where F satis f ies the condit ions

(2.6.2) (a ) / ' i s pos i t iv e ly hom ogen eous of degree \ — v, a n d

( b ) r $ c i ( i ? . - { 0 } ) .

We shall begin with a s tudy of such integrals . All the results hold if V

a n d / a re vec to r func t ions a nd Z ' i s a m a t r ix func t ion .

By repea t ing the second paragraph of the proof of Theorem 2 .5 .1

w i t h K^oi rep laced by F, we conc lude the fo l lowing theorem:

Theorem 2.6.1. / / F satisfies (2.6.2), / satisfies the conditions of

Theorem 2.5-1, and V is given by (2.6.1), then V is continuous.

Theorem 2.6.2. Suppose F satisfies (2.6.2) and F^ C^[Rv — {O }), /

is hounded and measurable on the bounded domain G and f^ Cf^(D) for

every D C G G, and V is defined by (2.6.1). Then V^ C^+f" on eachD C C G and

V,o.{x) = C4{x)+Wo.{x).

(2.6.3) Wo. {x) = Urn fF,4x - f ) / ( | ) ^ | , x^Ge-*0 G-B(x,Q)

C.= -.jr(-y)dy'^,55(0,1)

the convergence being uniform on any DC . C G.Proof. We def ine VQ[X) by (2.5 .10), with \/K rep laced by F, a n d

obta in the inequa l i ty corresponding to (2 .5 .11) and the formula

(2.6.4) F,,«(%) ^ -JFix - f)/(|)^f; + W.,{x)SBix.Q)

w h e r e W^Q i s def ined by the in tegra l over G — B{x, Q) in (2.6.3). It is

c lear that the f irs t term on the r ight in (2 .6 .4) tends to Cocf{x), un i fo rmly

on any D G G G, Next , choos e a doma in D' of c lass C^ suc h t h a t

D GGD' GGG and b reak up the in teg ra l fo r W^Q in (2.6.4) into anin teg ra l ove r G — D' plus one over D' — B{x, Q ) . In the l a t t e r in teg ra l

an d in th e f i rs t te rm on th e r igh t in (2 .6.4), wr i te / ( | ) = f(x) +

+ [ /( f) - / W J - U s i ng t h e f ac t t h a t F^cc(x - 1 ) = -dF{x - f ) /^ f« and

in teg ra t ing ove r D' — B{x, Q ) , we ob ta in

V,,4x) = C. [x, U)'f(x)-^\F,.{x-l)f[i)dl-G-B'

(2.6.5) - jr{x ~ i) [/(I) -f{x)]d$', +frM^ - i) [/(I) -/(^)]^fBB(x,g) B'-Blx.Q)

C4x. D') = -lr{x-i)dr^, x^D.dD'

Goi{x, D') and the in teg ra l ove r G — D' ^ C^{D') and the l a s t two te rms

in (2 .6 .5) tend uniformly to the i r l imi ts . Thus the convergence of WXQ

to Woe in (2.6.3) is uniform on D,

Mo ney, Multiple Integ rals 4

5 0 Sem i-class ical resul ts

Diffe ren t ia t ing WOCQ and us ing the ideas employed to ob ta in (2 .6 .5) ,

we ob ta in

W.,,p(x) =. C^,«(^, D')f(x) +fr,.p{x - f ) / ( f ) d^ -G-D'

(2.6.6) - / r , . (A; - 1 ) [ /(f ) -fix)] di^ +Jr , «^ (X - i) Ua) - /{x)] dl8B{X,Q) B'-B{X,Q)

The f i r s t two te rms a re bounded and a re independen t o f Q ; the l a s t two

are 0(^'^""i). Thus, if ^ = |A;I — :V2| a n d B[x, Q) C D' for every x on the

s e g m e n t xi X2, we s ee tha t

\W,(X2) -W,(xi)\ ^CQ ^.

Using (2.6 .5) and the fact that

v,A^) - F ,(^) = -jr{x - i) [/(f) -f{x)]di'^ + w,4x) - w^x)dB{x,Q)

we ob ta inI Wg{x) - W{x) I < C / if B{x, Q) C D'.

T h e H O L D E R cont inu i ty fo l lows .

T h e o r e m 2.6.3 . Suppose F satisfies the conditions (2.6.2). Then

(2.6.7) jr^o.[y )dZ(y ) = 0, oc==\,. . . ,v .aj5(o. 1)

Proof. L e t Act de no te the left s ide of (2 .6 .7). T he n, from t h e ho m ogene i ty , we conc lude tha t

2^ , l o g 2 = / [ j r,oc[y)dS{y )\dR= j r,o.{y )dy

1 [dB{Q,R) \ £ (0 .2 ) - ^ (0,1)

= jr(y )dy '^-jr{y )dy '^ = 0.SB {0,2) dB{0,l)

From Theorems 2 .6 .2 and 2.6.3, we see that the f irs t derivatives of

func t ions V defined by (2.6 .1), and hence the second derivatives of the

potentia ls (2 .5 .2) (or (2 .5 .3)) are given by the formulas (2 .6 .3), wheret h e Woe are g iven by the singular (o r CAUCHY principal value) integrals

of the form

(2.6.8) W{x) = lim fA [x ~ ^)f[^)d^ = \imW^{x)

w h e r e A satis f ies the condit ions

(a) A i s pos i t ive ly homogeneous of degree — v

(2.6.9) (b) A^C^R,- {0})

(c) JA[y)dS{y)=0.E

A proof l ike that of Theorem 2.6.2 yie lds the following theorem:

Theorem 2 .6 .4 . / / G and f satisfy the conditions of Theorem 2.6.2,

A satisfies (2.6.9), and W is given by (2.6.8), then W ^ Cf*{D) for every

D G G G. If f^ C^{Rv) and has compact support, then W^ Cf^{Rv).

2.6 . Generalized potential theory; singular integrals 5 1

Proof. The las t follows s ince G can be a rb i t ra r i ly la rge . Us ing (2 .6 .9)

(c), we s ee tha t ii 0 <. a < Q, t h e n

Wa(x) - W,(x) =fA(x - I ) [ / ( f ) -fix)]di.

Clear ly we ma y a l low a -> 0 to ob ta in

I W(x) - Wg(x) I < C i Qf', B{x, Q)CD'CGG.

Proceeding as in the proof of Theorem 2 .6 .2 , we obta in

W,,4x) =JAA^ - f ) / ( | ) ^ | -fix) IA {X - ^)d^: ~G-D' dB '

6B{X,Q)

+ / z l , a ( ^ - | ) [ / ( f ) - / W ] i f , X^D,D'-B{X,Q)

(DCCD'CCG).

Thus if |:î — :V2| =Q a n d B{x, Q) C D' for every x on the s egmen t xi X2,

we find, as before , that

\W,{xi)-W,(X2)\<C2Q^.

The result now follows.

We now prove some theorems which wi l l be usefu l la te r and whichy ie ld add i t iona l in fo rma t ion in the ca s e tha t A also satisfies

(2.6.10) A(-y)=A(y).

Theorem 2.6 .5 . Suppose A satisfies (2.6.9) (^nd (2.6.10). Then

(2.6.11) jA(x-i)d^ = 0, x^B[xi, Q)CCB{XO, a)B{xo,a)—B{x^,Q)

(2.6.12) lA{x-i)di'^ = 0, x^B[xo ,a).d B ixo, a)

Proof. For each rj ^dB(0,\), l e t the ray th rough x in the d i rec t ion rjin te r s ec t dB[xi, q) in the po in t f i and dB{xoy a) in the point fo and define

(2.6.13) ô (^) = | l o - : ^ | , n{v) = \î-x\>

F r o m p l a n e g e o m e t r y , w e c o n c l u d e t h a t

(2.6.14) ô(^) 'Wo{~v) = a^ — \x — xo\^,

Wi(v) 'fi(—v) = Q^ — \x — xi\^.

Tak ing po la r coo rd ina te s wi th po le a t x, we conc lude tha t

f A{x-i)di = jAirj)[logy joi?]) - logtpi{ri)] dZ in)

= 2J^(v)\}^Sn{v) + l o g i / ; o ( — ^ ) — lo gv î (^ ) -~logy)i( — rj)']d2{v)

= 0

on account of (2.6.10), (2.6.9) (c) and (2.6.14).

4*

5 2 Sem i-class ical resu l ts

F r o m (2.6Ai), we conc lude tha t

(l ft AZ \ B{xo,a)—B{xuQ)

dBixo.a) dBixuQ)

Using (2.6.15) with xi = x, we conc lude tha t

lA(x-i)d^', = fA(x - I ; ^ 1 ; = ^ -1 lfjÂ{rj)dS {rj) = 0dBixo.a) dBix.o) dBiO.l)

on account of (2.6.10).

Theorem 2.6.6 . (a) Any two points xi and x^ in BR can he joined bya path X = x(s), 0 ^ s ^ I, in BR such that

I

J {R - \x(s)\)f'-^ds< (2iu-^ + \)'\xi — ^ 2 | ^ 0 < ^ < 1 .0

(b) There is a constant C{fji) such that any two points of GR can be

joined by a path x =^ x{s),0 < s < /, in GR such that

j{d[x[s)]}f-^ds < C ( / / ) - | : v i ~ x^^, 0 < / i < 1 ,0

where d[x) denotes the distance of x from OG R = Z RU OR.Proof, (a) Is easily verified as follows: Let Q = | :î — : 21- li R <. Q <.

2 R, choos e the po lygona l pa th ;%:i 0 A;2. l iO <.Q < R a n d | A;I | < i^ — ^ ,

\x2\ <,R — Q, choose the segment %i :\:2. I f | ^ i | > R — Q, \x2\ <: R — Q,

choos e the po lygona l pa th xi xz X2 w h e r e xs is on 0 Xi w i t h | ;^;3| = i^ — ^ .

If [î| > -?^ — ^, 1^2! > ^ ~ ^ , choose th e poly gon al p a th xi xs X4 X2

where xs is on 0 xi, x^ is on 0 X2, a n d \xz\ =:^ \x/^\ = R — Q.

(b) I t is also sufficient to pr ov e thi s f or 0 < | A;I — ;V21 == Q = kR w h e r e

0 < y < 1/3, say . Fo r such Q, the se t SQ of x s u c h t h a t d{x) > Q is the

part of the ball \x\ ^R — Q w h e r e x^{=y) > ^ . F o r s uc h Q, we chooset h e p a t h s xi X2 if xi a n d X2 ^ SQ, xi xs X2 if xi ^ SQ a n d X2 ^ SQ, a n d

xi Xs X4 X2 if xi ^ SQ a n d X2i SQ; he re xs is the neares t point of SQ t o xi

a n d X4 i s tha t nea re s t X2. A s t ra igh tforward ana lys is ver i f ies the resu l t

in all cases.

Theo rem 2 .6 .7 . Suppose that A satisfies (2.6.9) cind (2.6.10). Suppose

that f^ C^{BR), and suppose W is given by (2.6.8). Then W^ Cf'(BR) and

\W{x)\<[2i^-l^-^\A\\'iS^K{f)R''

^•- ' K{W )<C{v,ix)WA\\l^+\\vA\\ls]K(f).

Proof. F ro m Th eore m 2 .6 .5 , i t fol lows t h a t

W{x) = \A{x-m{^)-f[x)]d^BR

from which the f irs t inequali ty in (2 .6 .16) follows immediately .

2.6. Genera l i zed po ten t ia l theory ; s ingu lar in tegra l s 53

To prove the second inequality, we extend / t o Bû so tha t m ax /and hii{f) are unaltered and write

W =Wi- W2, W^{x) =JA{x- i)f{i) di

( 2 . 6 . 1 7 ) , 'W2{x)=fA(x-i)mdi.

Differentiating lî^ as in (2.6.6), we obtain

Wu,Jx) = - f(x) lA{x- )di:-lA(x-i) im - fix)] rfi; +dB2R dB{X,Q)

(2.6.18) + lA 4x- )[m-f{x)]di, X BR, 0<Q<R,

in which the first term vanishes. From (2.6.18), we obtain

\WlQ(X2) - WiQ(Xi)\ < 2h4f)[\\A\\l^ + (1 - /^)-l||VZl||?. J \X2 - ^ i h

(2.6.19) (1^:2-^li = 2^) .

From Theorem 2.6.5 and equation (2.6.17), it follows that

I Wuix) - Wi{x)\ = 11A (X - i) [/(I) - fix)] di\ < h,if) \\A\\l^-fi-^ Q".

(2.6.20) ^'^-^^

The result for Wi follows from (2.6.19) and (2.6.20).To prove the second inequality for W2, we first note that

VW2ix)=I Aix-i)fi )di

= JvAix-$)[fi )-fix)]di, X^BRBZR-BR

using Theorem 2.6.5 (or (2.6.15)). From (2.6.21), we conclude that

I vPF2(^)! < /1 vzi (^ - f) I • m - f(x) I diRv—B{x,d)

(^•^•^^) <(1 - / . ) - ! Kif)' II Vzl 11?. . (i? - iA;|)^-i

(^ = 7 ^ _ | : ^ | ) .

From (2.6.22), Theorem 2.6.6 (a), and the fact that \W2{x2) — Wîxi)]

is dominated by the integral of j V TF21 over any path from Xi to X2, theresult for W2 follows.

Corollary 1. Suppose that f ^ C^^(BR) and U is its potential. Then

U ^ C^'^^{BR) and there is a C(ILI, V) such that

(2.6.23) AU (x) =f{x), X^BR, h^S/Û) <C h^(f),

Proof. From Theorem 2.5.1, it follows that U ^ C^ with

u,4x)=fKo,4^-i)mdi.BR

Then , f rom Theorem 2.6.2 w i t h r= KQ^CC, we see t h a t VU^ Cl{Br)

for each r <. R, w i t h

C/,a^ W = Ca^/W + Hm / KoM^ ~ f ) / ( f ) d^

(2.6.24) '"^^ ^ . - ^ ( - . . )

5 5(0,1)

as one f inds , us ing the formulas (2.4.2) for KQ. Since each iô,a)8 satisfies

(2.6.9) and (2.6.10), the second resu l t in (2.6.23) follows. The f irs t result

follows from (2.6.24) and the fac t tha t AKo(y) = 0 ii y z^ 0.

By combin ing th i s r e s u l t w i th POISSON 'S in teg ra l fo rmula , we o b t a i n

the fo l lowing coro l la ry :

Corollary 2. Suppose/^ C^{BR) and u* is continuous on dBn, Then

there is a unique function u which is continuous on BR, coincides with u *

on dBn, ^ C^(Br) for each r <, R, and satisfies P O I S S O N ' S equation (2.5.1)

on BR,

W e s ha l l now give an e x a m p l e of a f u n c t i o n / ^ C^{Ba) the p o t e n t i a l

of which is not in C^(Ba). To do t h i s , we f ir st n o t e t h a t i f / i s b o u n d e d

and measurable on Ba and / ^ C^[B(XO, R)] {E{XO, R) C Ba), then its

p o t e n t i a l V^ Cj^[B(xo, R)] s ince one can w r i t e

V{x) = fKo{x~ i)f(S) di + fKo{x - f ) / ( ^ ) diB{xo,B) Ba-B(xo,R)

in which the f irs t term $ C^[^(ô, R)] and the second is h a r m o n i c . T h u s

w e h a v e the fo rmula s

F , cc^{x) = Coa^fix) + I Ko^ ^^(^ _ I) [/(^) _ f(^x)] dS {x^B{XQ, R)) .

Ba

So now we define /(O ) = 0 and

f(x) = [log\x\]-^\x\-^x^x^, x=jÔ, 0<a< \.

T h e n its p o t e n t i a l ^ C^{Ba — {O}) and the formulas (2 .6 .22) hold if

X ^ 0; but F,i2(:v) -> + ^^ as ;\; ->0.

Remarks. H o w e v e r , if / satisfies a D I N I condition on a d o m a i n Q,

t h e n F ^ C^(G). I n d e e d if, in T h e o r e m 2 . 6 . 2 , / s a t i s f i e s a D I N I cond i t ion

on each D C G, t h e n V F is c o n t i n u o u s and given by (2.6.3), the con

vergence be ing uniform on any D G G G. Simila r ly if, in Theorem 2 .6 .4 ,

/ s a t i s f i e s a D I N I cond i t ion on each D C G, t h e n W^ C^{G). H o w e v e r W

does not necessari ly sat is fy a D I N I cond i t ion . / is said to satis fy a D I N I

cond i t ion on a c o m p a c t set S if t h e r e is a pos i t ive , non-decreas ing , cont inuous func t ion (p{Q) s u c h t h a t

l/(î) — / W l < 9^{\xi — X2\), xi, X2^S, and

a

jQ-^<P(Q) dg <oa^ a > 0.

I.'], The C A L D E R O N - Z Y G M U N D inequalities 55

2.7. The Calderon-Zygmund inequalities

These very useful and now well-known inequalities are based on the

following well-known (see ZYGMUND) inequality for the H I L B E R T t rans

form w hich is actually the one dimensional case of the general inequalities.As was stated in § 2.1, we shall restrict ourselves to HoLDER -continuousfunctions. The proofs are greatly simplified; moreover these results are

sufficient for our later purposes.Theorem 2.7.1. Suppose f ^ Cl,{R^) (0 < < 1). Then the H I L B E R T

transform g, defined hy

( 2 . 7 . 1 ) W =l-i/f '

\X—^\>Q

^ C2 {R^) and, for each ^ > 1, ^€ Lp{Ri) and

(2.7.2) ||^||»<C(;^).||/||«.

Proof. The integral (2.7.1) is the one dimensional case of the integral(2.6.8). Accordingly g^ C^^{Ri) by Theorem 2.6.4.

I t is sufficient to prove the result (2.7.2) for f{x) > 0 since we may

always write / = /i —/2 where each/^^ C^^(Ri) and is non-negative.So we define

( 2 7 . 3 ) « ( ^ ' > ' ) = i f ( 7 y/ms

—oo

Since (2.7.3) is just P O I S S O N ' S integral formula for the upper half-plane(see § 2.4) we see th a t u is harm onic for y > 0 and is continuous for y > 0

with

u{x,0) =f(x).

Since for some A > 0, f^ C^onlx] < A + \ snd f{x) = 0 for \x\ > A,

we see, by writ ing/(f) =f(x) + [/(f) — f(x)] in (2.7-4) that v{x, y) tendsuniformly in A; as y ->0+ to g{x) for |A;| < ^ + 1/2, the convergence for

larger x being obvious. Moreover ?; is a conjugate harmonic function to .

If we let F{z) = u + i v, thenoo

— ooNow, consider the function [F{z)]^ for y > 0, where w^ denotes

that branch which is real and positive when w is (remember z* > 0 since/ > 0). From (2.7-5), we see tha t

\F(z)\P <M ' \z\-P for 1 1 > ^ + 1,

http://z/-P

http://z/-P

http://z/-P

56 Semi-class ical resul ts

say. A simple limit argument using CAUCHY'S theorem on hemispheresGR shows that

(2.7.6) j [F{x)]v dx = I [f{x) + ig{x)]Pdx = 0.

By integrating along the segment of length u = f{x) in the w plane fromi V to {u + i v), we obtain for each x

(2.77)

(2.7.8)

( / + ^* ) — (^'^)^ = p J wP-^ dw

segment

I p f WP-'^ dw\-l)/2

<^-z(^)(!/|p+ 1/1-1^1^-1).

Integrating (2.7.7) from — 00 to 00 and using (2.7.6) and (2.7.8), weobtain

(2.7-9) l[ig{x)]Pdx < C i ( ^ ) [ ( | | / | | « ) ^ + | | / | | » - ( | U | | « ) ^ - i ] .

Since [ig{x)]P = \g{x) l ^ e x p [ ± i | ^ ] ^^^ I ^ W I ^ k | . we see tha t

k D ^ < C i ( ^ ) [ ( | | / | | o ) ^ + | |/ ||o -( |l ^| |» )^ -i ]7 T

from which the result follows for any p for which cos(^ 7r/2) ^ 0.We take care of those cases as follows: We note that cos(^ 7r/2) ^ 0

if 1 < ^ < 2. So suppose p > 2 and let p' be the conjugate exponent(p-i +p'-i = 1 ) ; then 1 < ^ ' < 2. Let h^ Cf^iRi) and let k be itsHiLBERT transform. Then

(2.7.10) \\kr^ < c{p') \\h\\i,.

Moreover, it is easy to see that

(2.7.11) fh(x)g(x)dx = / - k{x)f{x)dx < C{p') ||/1|»- \\h\\l,

— OO —OO

by applying the H O L D E R inequality to the right side of (2.7.11) andusing (2.7.10). Since the functions h are everywhere dense in Lp,, thetheorem follows in this case also with C[p') — C(p),

Theorem 2.7.2. Suppose/^ C^^d^v), A satisfies (2.6.9) end

(2.7.12) A(-y) = ~A{y),

and suppose

(2.7-13) ^ W = 1™ / ^ ( ^ - ^ ) /( f )^ f •^^ RV—B{X,Q)

Then g^ C^i^v) Pi Lp{Fv) for each p > 1 and

(2.7.14) \\gt<C{P)\\A\\l^-\\f\\l

2.7. The CA L D E R O N - Z Y G M U N D inequalities 57

The hypothesis that A ^ C'^{Rv — {O}) is not necessary; it is sufficient for

/ I ^ C 0 ( i ? , - { 0 } ) .

Proof. T h a t g ^ C^ (Rv) follows from Theorem 2.6.4 and t h a t g ^ each

Lp{Rv) follows since g(x) = 0(|^ I"**) at c>o.

N o w , if we w r i t e ^ = x -\- rrj where r > 0 and rjÎ! and use (2 . / . 12),

we conc lude tha t

g(x)=l lA{f])h(x, ri)d^[ri),

( 2 7 . 1 5 ) i

h{x, f]) = lim r r-^f(x J^rr])dr.

\T\>Q

For each rj, let us w r i t e x = XQ + SYJ w h e r e XQ- rj = 0, T h e n

h{x, + sri,n) = \im r ^ l ^ i i ^ ^ ,(2 .7 .16) ^ -^^s -* i> . '

(p{t;xo,rj) =f(xo + trj)

F r o m T h e o r e m 2.7.1, we conc lude tha t h(xo -\- srj,r))^Lp in s for each

{xo, rj) w i t h

(2.7.17) ||v^||« < Cp • Ml. yj(s; xo,rj) - /^(ô + srj;r]).

By raising (2.7.17) to the p - t h p o w e r and i n t e g r a t i n g , we o b t a i n

(2.7.18) / \h{x, fj) \Pdx < Cl{\\f\\l)P [dx = dx^ds).

The result (2.7.I4) follows from (2.7.15) and (2.7.18) by f i rs t apply ing the

H o l d e r i n e q u a l i t y w i t h m e a s u r e -\A{y)\d^{y) to (2.7.15) to e s t i m a t e

|^(:v)|2? and then in teg ra t ing wi th re s pec t to x.

Before we can p r o v e the theo rem co r re s pond ing to T h e o r e m 2.7.2

w h e n A satisfies (2.6.10) instead of (2.7.12), we need the following useful

l e m m a s :

Theorem 2.7.3 . If u ^ Cl,(G), then

u(x) = f Koôc{x ~ i)uôc(S)di

(2.7.19) d

=r-^ I \x - i\-^ {x- - t)u,4i)ds.G

Proof. We may a s s u m e G C B{x, A), u = 0 in B(x, A) — G. T a k i n g

po la r coo rd ina te s at x, the in teg ra l is j u s t

]dr.f -Ur(r,rj)d2^(ri)

0 Li:

L e m m a 2.7 .1 . If xi and X2 $ B^,

(2 .7 .20) / | | - : ^ i | l - ' ' * | f - ^ 2 p - ' ' ^ l =BR

— ClKo{x2 — xi) + e{xi,X2,R),

if V > 2

-{27z)^Ko{x2-xi) + Ci +

+ 27tlogR'ê(xi,X2, R), v = 2

where Co and Ci are constants and e{x\, X2, R) converges uniformly on

BA X BAto 0 as R ^ o o .

Proof. If r > 2, i t fol lows t h a t a s J? -> oo, the in teg ra l ten ds t o a

func t ion oi \x2 — xi\ only which is homogeneous of degree 2 — r and so

mus t be a nega t ive mu l t ip le o f KQ. Clearly a lso

8{Xi, X2, R) = -f\S-Xi\^-'''\i-~X2\^-''dSBy-BR

converges as s ta ted .

In the case v — 2,\etx = {xi + X2)l2 a n d Q = \x2 — xi\ and s uppos e

R> A +Q. T h e n

f \^' - Xi\-^'\S - X2\-^dS < f \S - Xi\-^'\^ - X2\-^d^

B(x,R-\x\) BR

Bix,R+\x\)

Eva lua t ing the in teg ra l on the l e f t , we ob ta in

C2 + 2 7rlog[(i? - \X\)IQ] + cp[{R - \x\)lQ]w h e r e

C2 = l\i-Xi\-^'\i-X2\-^

(2.7.21)

Bix,Q)

« - / | c - I ei

B(O .l)

1 + ^ . 1

( « i = ( 1 . 0 ) )

<p{S) = J {\^ - Xl\-^-\^ - X2\-^ -\^ - x\-2)d^

B{x,R-\'x\)-Bix,Q)

= / (I f - eil2 |- l • i I + «l/2 r^ - II r^) '^ l.B(0,S)-B(0.1)

5 = (i? - \x\)lQ.

In l ike manner, the r ight integral in (2 .7 .21) is given by

C2 + 2 7r log[(2^ + 1 1)/ ] + (p[(R + 1^1)/^].Clearly (p{S) te n d s t o a lim it 990 as S - > o o . If w e ta k e Ci = C2 + 9^0,

t h e n

27r log(1 - l^l/i? ) + (p[(R - \X\)IQ] -(po<s

< 27rlog(1 + \x\IR) + 99[(i? + 1^1)/ ] - 990.

Theorem 2 .7 .4 . Theorem 2.7.2 holds, with \\A\\^^ replaced by MQ + Mi

in (2.7.14), if A satisfies (2.6.9) and (2.6.10) instead of (2.6.9) and {2.7 A2).

Here MQ and M, are the respective maxima of \A{x)\ and \\/A{x)\ for xondB{0,\).

Proof. W e s up po se f irs t t h a t / ^ C'^(Rv), the general case follows by

approx ima t ions . Le t u s de f ine

(2.7.22) h{x)==fR{x- i)f(i) di, R(y ) = Co-i I y | i -

http://x/IR

http://x/IR

http://x/IR

2-7. TheC A L DE R O N -Z Y G M UN D inequalities 59

Co =2 7Z ifV =2and o the rwis e isthe c o n s t a n t of L e m m a 2.7 .1 . T h e n ,

by Theorem 2 .6 .4 , A€ Q[Rv) w i t h

(2.7.23) AaW= Hm fR,4 - S)f(i)di ==fR{x-i)f,4i)di

f rom wh ich it fo l lows tha t h^ ^^(Rp). S i n c e / has c o m p a c t s u p p o r t , it

fo l lows tha t

(2.7.24) Vh(x) =0(\x\-^), Ah{x) =0{\x\-''-^) at CXD.

N e x t , we define

(2.7.25) k(x)= -fR{x- )Ah{i)di=-fR{x- )Ah(i)di +8R(x}

Ry BR

w h e r e SR converges un iformly to0onany c o m p a c t set onaccoun t of(2.7.24) . Since thes u p p o r t of/ is ins ome BA, wemaywrite (s ince

jAf(r))drj = 0)

BA

k{x) = en{x) - I \ J R ( X - i)R( - v)dS]Af{rj) drj

BA [BR J

= £IR{X) + fKo{x — ri)Af{rj)dri

BA

= £1R+ f KG,cc{x~r])f,a{r])d7] = SIR(X) +f(x),

BA

£iit(x) = SR{X) — jei{x, rjy R)Af(rj)dr)BA

for all r > 2. Since kand / are i n d e p e n d e n t ofR, SIR ^ 0. H e n c e , s u b

s t i t u t i n g the r igh t s ide of(2.7.25) for/(:^) in(2.7.13), we o b t a i n

g{x) = lim JH m j A(x-^)\ J--R(i - r])Ah(r])dr]Q-*0 \a^0R^-B(x.Q) [R^-Bix.o)

( 2 . 7 . 2 6 ) = : l i m | H m jAh(7i)\ J - A(x - S) R{i - rj)di]g^O ya-*0 B^-B(x,a) [R^-B{X,Q) \

If we now set f = ;t: — CO on the r i g h t in(2.7.26), we o b t a i n

d^

di].

(2.7.27)l im j l i m fS(x— r]; Q)Ah[rj) drX

^{y ' y Q) = fA(co)R{y — a))da).Rv-^Q

(2.7.28)

F r o m the h o m o g e n e i t y , we c o n c l u d e t h a tS{y\Q) = \y\^-'S(rj;Ql\y\), rj = \y\-^y, \rj\ = 1

^i-yiQ) = s{y>9)*

N o w , if \rj\ =1 and\co\ > 3/2, t h e n \rj — co\ > \co\l}. If \7]\ = 1 and

1/2 < \(o\ < 3/2, t h e n \A{co)\ < 2"Mo if h — co] < 1/2and R(rj — co )

http://co/l%7D

http://co/l%7D

http://co/l%7D

6 0 Sem i-class ical resu l ts

< Z" -!- Co if \r) -a>\> 1/2, MQ being the max. of |Zl(co)| for |co| =1.

F in al ly , if 0 < ^ < 1 / 2 ,

5(7-/; Q) = lA(co)R{f] - co)dco + JA(a))[R{rj - co) - R(7])]dco

Bv-BiOA/2) B{0,li2)-B(0,Q)(2.7.29)

a n d as Q -^0, S{r}; Q) converges un iformly for y on i7 to S{rj). F r o m t h e

ana lys is above , we see tha t

(2.7.30) \S(y; Q ) \ <C{v) M Q | y | i - l i m S ( y ; Q) = S(y)

s o tha t we may le t o* - > 0 and th en ^ - > 0 in (2 .7 .27) to y ie ld

(2.7.31) g{^) = I S(x-rj) Ah(rj) drj.

Now, in (2 .7 .29) , we assume tha t \r)\ is nea r 1 an d 0 < <1/2. By

defining

Sl{V'> Qy'^) ^J ( ^ ) ^{'^ ~~ <^)^<^>Rv-B(Q,l/2)-B{rj;r)

dif fe ren t ia t ing wi th respec t to rj^" and le t t ing r- > 0 , w e f i n d t h a t 5 ( ^ ; Q)

is of class C^ in r] a n d t h a t

Ry-B{0,l/2)-B{r],l/S) B {r],l/Z)

(2.7.32) +lA{a})[R,47]-co) - R,4rj)]dco; (0 < <1/2)B{0,l/2)-B{0,Q)

in this development, we used (2.6 .9) (c) for R^^^ In (2 .7-32), we may le t

^ - > 0 a nd conc lude from the un i fo rm conve rgence in rj for \r]\ n e a r 1

t h a t S C^{Rr — {0}) and tha t

(2 .7 .33 ) |V5(y ) | <[Ci(v) Mo +C2{v) Mi] \y \-^

w h e r e Mi i s a bound for |V^(co) | fo r | co | =1. Thus, from (2.7.31), we

d e d u c e t h a t

~g(x) = lim JS,oc(x — fj)h^oi(r])dr].

From (2.7.28), e tc . , we see that V 5 and Vi^ satis fy (2 .7 .12). From (2.7.23)

and Theorem 2 .7 .2 we conc lude tha t VA^ Lp(Rp) w i t h

(2.7.?4) \\ m<C{p,v)lft.

From (2.7-34), we see that V h can be app rox ima ted s t rong ly in Lp b y

func t ions kn^ C'^(Rv) satis fying (2.7-34) uniformly so that , by Theorem

2.7.2 again, g i s the s t rong l imi t (un iform on any bounded se t ) in Lp offunc t ions gn s o t h a t

\\gt<C*{p,v)\\VS\\i^-\\Vh\\l

from wh ich t he resu lt follows, s ince || V S| | ? _^ <C{v) • {MQ +Mi) from

(2.7.33)-From Theorems 2 .7 .2 and 2 .7 .4 , we obta in the coro l la ry .

2.8. Th e m ax im um principle for a l inear ell ipt ic equ at ion of th e second order 61

Theorem 2.7.5. If A satisfies ( 2 . 6 . 9 ) , / € C^^c{Rv)y and g is defined by

(2.7.13), then g^ Cl(Rv) O Lp(Rr) for every p > 1 and

| | g | | o < C ( r , ^ ) ( M o + M i )| |/ | | o

where MQ and M\ are defined in the statement of Theorem 2.7 A-

2.8 . The maximum princip le for a l inear e l l ip t ic equat ion of the

second order

We conc lude th is chapte r wi th the wel l known proof of th is p r inc ip le

d u e t o E . H O P F ([ ! ] ) .

Theorem 2.8.1. Suppose that u ^ C^{G), that ^^^ 6«, c and f ^ C^[G ),

c and f satisfy

(2.8.1) c[x) = 0, / ( ^ ) > 0 , x^G

and tha t u is a solution of the equation

(2.8.2) a'='^{x) uô:^{x) + b 'ix) u,a(x) + c(x) u{x) =f{x), x^ G,

assumed to he elliptic on G. Suppose that x^^G and that u takes on its

maximum value at XQ. Then u[x) = u{xo) for all x on the domain G.

Proof. S u p p o s e u[x) ^ U[XQ) = M. Then the s e t whe re u{x) < M is

open . There is a ba l l B[x2, R)ClG s u c h t h a t u{x) <.M for x^B{x2,R) —

— {î}> w h e r e xi^dB{x2, R) a n d u{xi) = M. Fina l ly , there i s a ba l lB[xi,R[)CG w i t h Ri < R. L e t Si = B{x2,R) ndB{xi,Ri) a n d 5 ,

= dB{xi, Ri) - B(X2,R), s o t h a t StU Se = dB{xi, Ri). T h e n

u{x) <, M — s on Si a n d u{x) < M on Sg

for som e £ > 0.

Now, l e t

h(x) = e~y r^ — e-y^^, r = \x — X2\.

L e t t i n g L 99 stand for a*^ 99 a/5 + ^" (p,oc, we see tha t

evr^Lh = 4y2 a'^îx^' ~ xfi{x^ — :vf) — 2y[a°^^6a^ + h''[x'' - % | ) ] .

We can choose y so la rge tha t Lh[x) > 0 in B[xi, Ri). Fina l ly

(2.8.3) h{x) < 0 on Se, h(xi) = 0 .L e t

(2.8.4) v(x) = u{x) + dh(x), (3 > 0 ,

w h e r e 6 i s smal l enough so tha t v(x) < Af on 5^. From (2.8.3), we see

t h a t v(x) < M on dB(xi, Ri), v{xi) = M so t h a t v h a s a m a x i m u m a t a

p o i n t xs in B{xi, Ri) while L(v) > 0 the re .But th is would imply tha t (s ince a l l vôc{xs) = 0)

(2.8.5) a^'îxs) vôc,3{xs) > 0 bu t vûpixs) r)°'rj^ ^0 for all rj.

Now, we may def ine new var iab les y a n d w by the ro ta t ion

yv = cl[x^ — xl), C^ = clr)°', w(y ) = v{x)

62 The spaces H'^ and H^^

w h e r e the m a t r i x c is chosen so t h a t c"^ a(xs) c = d(0) is d iagona l . Then

(2.8.5) is e q u i v a l e n t to

(2 .8 .6 ) 2 ' ^^^ (0 ) ' ^ \yy (0 ) > 0 but w^y6{0) C'^ C^ < 0 for all Cy = i

w h e r e all the d^y > 0. But the f i rs t inequa l i ty in (2 .8 .6) implies that some

one w^yy(0) > 0 which con t rad ic t s the second .

Coro l la ry . / / u and the coefficients satisfy the conditions of Theorem

2.8.1 except that we require c[x) < 0, then u cannot have a positive maximum.

If, also, f{x) -Ê 0, then u has neither a positive maximum nor a negative

minimum.

C h a p t e r 3

The spaces H"; and ilô

3 .1 . Def in i t ion s and f i r s t th eorems

In th i s chap te r we co l lec t s t a temen ts and proofs of the t h e o r e m s

w h i c h we need concern ing these func t ions . As we s t a t e d in C h a p t e r 1,

the s e and s imila r spaces have been d iscussed at grea t l eng th by m a n y

a u t h o r s and have ce r ta in ly p roved the i r wor th in connec t ion wi th thes t u d y of dif fe ren t ia l equa t ions .

Definition 3.1 .1 . A func t ion u is sa id to be of class H^ on G iff u is

of class Lp on G and there ex is t func t ions r^ of class Lp on G, a = oci,

. . ., ^v , 0 < 1 1 < w, s u c h t h a t

(3.1.1) fg(x)r4x)dx = {-iy-\fD-g{x)u{x)dx, g^C-(G).G G

The func t ions ra (or r a t h e r the classes of equ iva len t func t ions de te rmined

b y t h e m ) are called the distribution derivatives of u and r^ is hereaf te rd e n o t e d by D^" uovu,« ; we m ak e the c o n v e n t i o n t h a t 0°" u = uii\(x\ = 0 .

Remarks. I t is c lea r tha t if u is of class H^ on G, its d is t r ibu t ion de r iva

t ive s are de te rmined on ly up to add i t ive nu l l func t ions ; moreove r \i u*

differs from w by a nul l func t ion , then ^ * is also of class H'^ on G and

h a s the s ame d i s t r ibu t ion de r iva t ive s . The def in i t ions above ex tend to

vec tor func t ions and to complex -va lued func t ions . The LEBESG U E

derivatives ( S A K S p. 106) of the set functions Tâ(A;) dx are called the

egeneralized derivatives of u at po in t s whe re they ex i s t .

T h e o r e m 3.1 .1 . The space H'^{G) of classes of equivalent vector func

tions of class H^ on G with norm defined by

(3.1.2) ||«i|» = { /pl2 ] lip

dx\ (u = u^, . . ., u^)

3.1 . Definitions and first theorems 6 3

is a B A N A C H space. 7 / ^ = 2, the space is a H I L B E R T space if we define

(3.1.3) ( ^ , ^ ) ? = f Z Z Co^D'û^Wvidx.

In (3-1-2) and (3.1-3), Co, denotes the multinom ial coefficient \(x\\j(Xi\ . . .ocpl

Proof. The on ly p rope r ty requ i r ing p roo f is the comple tenes s . So

s uppos e {un} is a CAUCHY s equence in H^. Then each of the sequences

D°^ Ufi, w i t h 0 < 1 1 < w, is a CAUCHY s equence in Lp(G) and so con

verges in Lp{G) to some vec tor func t ion roc(=u if | ^| = 0 ) . But , for

each oc w i t h 0 < | ^ | < w, it follows that (3.1.1) holds in the l imi t so

t h a t the r^ are the co r re s pond ing d i s t r ibu t ion de r iva t ive s of u.

R e m a r k 1. I t will be c o n v e n i e n t to ca l l these e lements of H^ func

t ions and to say t h a t u is c o n t i n u o u s , h a r m o n i c , etc., iff s ome rep re s en ta t i v e of the c lass forming the e l e m e n t has the s e p rope r t i e s . Na tu ra l ly ,

also, t h e r e are many d i f fe ren t topo log ica l ly equ iva len t no rms wh ich

could be us ed .

R e m a r k 2. The spaces H^{G) have been def ined for all rea l m (see,

for example L I O N S [1]) and in te re s t ing re s u l t s in the t h e o r y of differentia l

equa t ions have been ob ta ined us ing the s e and other spaces such as t h o s e

introduced by A R O N S Z A J N and S M I T H [1], [2], C A L D E R O N and others.

A discuss ion of the s e ma t te r s , though in te re s t ing , wou ld l ead us r a t h e rfar afie ld into functional analys is an d is not rea l ly re levan t to our dis

cuss ion . Accord ingly , we sha l l not def ine them here . However , a spec ia l

class is us ed in C h a p t e r 8.

Theorem 3 .1 .2 . (a) Ifu^ H^(Q and D C G, then U\B ^ H^(D).

(b) Suppose u is defined on the whole of G and each point of G is in a

domain D such that U\D^ H^{D). Then there are functions r^ defined on

G such that {0°^ U\D) (X ) = ro:{x) for almost all x on D, and any D C C G.

(c) / / , in (b), the roc^Lp{G), then u^H^(G ).(d) Ifu^ H^(G), then Vû^ H^-^(G), 1 < ^ < m.

(e) Ifu^H^{G) and V'û^Hl{G), then u H^+'{G).

(f) Ifu^HîG) andg^ CZ '^G) (i. e. g^ CrHG)

and all its derivatives of order < m — 1 are LIPSCHITZ, then (3.1-1) holds

with roc = D'^ u ^ uôc.

(g) If u^Lp(G), u is absolutely continuous in each variable (on seg

ments in G) for almost all values of the other variables, and if its first

partial derivatives {which consequently exist a.e. and are measurable)^Lp(G), then u^Hl{G) and its partial and generalized derivatives coincide

almost everywhere.

Proof. P a r t s (a) — (e) are obv ious and p a r t (f) follows by a s t r a i g h t

f o r w a r d a p p r o x i m a t i o n of the func t ion g. To p r o v e (g), we n o t i c e t h a t if

g^ C ^ ( G ) , t h e n g[x) • u{x) has the abs o lu te con t inu i ty p rope r t i e s of u

6 4 T h e spaces H ^ a n d H^^^

and has compac t s uppor t in G. F r o m F U B I N I ' S theorem, i t fo l lows tha t

we may take the ra a s the pa r t i a l de r iva t ive s dujdx^" in (3.1-1)-

We recall the definitions of a moUifier 99 and the 99-mollified functions

UQ wh ich were g iven in Chap te r 1 (Def in it ion 1.8.3).

Theorem 3.1.3. Suppose u^H^(G), cp is a mollifier, and UQ is the

(p-mo llified function of u. Then UQ ->U in H'^{D) for each D C. d G and

(3.1.4) IP(XQ(X) = D°'Ue(x) if y)a = u^oc, 1 < | ^ | < m .

If m ~ \ and u^oc(x) = 0 a.e. on G, oc = \, . . .,v, and G is a domain,

then u{x) = const, a.e. on G.

Proof. If B {x, Q) C G, the func t ion gx Q defined on G b y

gx,(^) = cpta - ^) « ( y ) == Q-^cp{Q-^y))

^ C^(G). Hence , us ing Theorem B, § 1 .8 and the def in i t ions , we see tha t

D-u,(x) = f ( - 1 ) '^ ' ^ , , a ( f ) u{^) di = fgxQd) D-u(^) dS = ^/ '« ,(^) .G G

Th e rem ain in g s ta tem en ts fol low from Th eore m B , § 1.8 and t he con

nec tedness of G.

R em ar k . I t i s no t know n (and the w ri te r be l ieves i t i s no t t rue) tha t

a func t ion u ^ H'^ on a doma in G with suff ic ien t ly wi ld boundary can beapprox ima ted ove r the who le o f G by functions of c lass C^ on a domain

con ta in ing GU dG. We sha l l p rove th is , however , fo r a surpr is ing ly wide

class of domains in § 3-4.

The fo l lowing theorem is an immedia te consequence of Theorem

3.1.1, 3-1-2 b and c, and 3-1-3 and the details of its proof are left to the

reade r .

Theorem 3.1.4. Ifu^ H^{G) ^ ^ ^ C ^ Cf~'^(G) and C and its derivatives

are bounded on G, then C u^ H'^(G) and its derivatives are obtained from

those of t, and u by their usual formulas.

Proof. For if 99 is a mollifier an d Ce a n d UQ are th e 99-mollified fun ctio ns

we conc lude tha t D^ ^Q conve rges a lmos t eve rywhere and bounded ly to

D " C a n d D^" UQ converges s t rongly in Lp t o 0°" u on each D (Z G G,

0 < |(%| < m.

Defin i t ion 3 .1 .2 . A mapping T: x = x(y) of class C^ or Cf~^ of a

d o m a i n H on to a doma in G is said to be regular iff it is 1 — 1 an d all t h e

derivatives of order < m of the mapping func t ions for T a n d T - i a r e

u n i f o r m l y b o u n d e d .Theorem 3.1.5. Suppose that x ^= x[y ) is a regular mapping of class C^

of a domain H onto G, suppose u^ H^{G), and suppose v(y ) = u[x{y )].

Then v^ H'^{H) <^f^d, if all of the generalized derivatives D'^ U{XQ) exist at

XQ = x{;y o), then all of the generalized derivatives D * v(yo) exist and are

connected with those of u at XQ by the usual rules of the calculus.

3.1. Defin i t ions and f i r s t theorems 65

Proof. F r o m T h e o r e m '}A.'}, it fo l lows tha t we may a p p r o x i m a t e

s t rong ly to u in H'^'{A) for eachZl C C G. If Z) is the c o u n t e r - i m a g e of s uch

a A, the t r ans fo rmed func t ions conve rge s t rong ly in H'^(D) to v. The

las t s ta tement follows eas ily s ince a regula r family of se ts (see SAKS,

p . 106) a b o u t yo co r re s ponds to one a b o u t XQ.

W e now digress to give a s imple proof of the famous theo rem of

R A D E M A C H E R concerning L I P S C H I T Z functions.

Theorem 3.1.6. Suppose u satisfies a LIPSCHITZ condition on G. Then

u possesses a total differential almost every where on G.

Proof. Let ZQ be the set of po in t s whe re one of the gene rahzed de r iva

t ive s of u fails to exis t . Let E be an eve rywhere dens e denumerab le set

of po in ts C on Z. W ith each s uch f, let x = xz[y) be a r o t a t i o n w h i c h

carr ies the v e c t o r ei in the y-space in to f and let v^[y) = u[xc(y)]. The

generalized (f irs t) derivatives of v^ exis t at all points y for which xc(y) ^ Z Q.

Also, for each C, t h e r e is a set Z^ of m e a s u r e 0 s u c h t h a t the partial

d e r i v a t i v e dv^dy^ = Dyi v if xc(y) i Z^ (Theorem 3.1.2 g). If AJQ ^ ZQ U

(U Z^), t h e n all the directional derivatives in the d i rec t ions C exis t and

a re connec ted wi th the gene ra l i zed de r iva t ive s D"^ u{xo) by the u s u a l

fo rmula s . At any s uch po in t , it is easy to see, us ing the LIPSCHITZ con

d i t i o n t h a t u has a to ta l d i f fe ren t ia l .

F r o m t h i s , we conc lude the following generalization of T h e o r e m 3- L 5 *Theorem 3.1.7. Suppose x = x{y) is a regular mapping of class

C 7 - i ( m > 1 ) of H onto G, u^H^(G), and v{y) ^ u[x(y)]. T h e n v

^ H^(H) and if all the generalized derivatives 0°^ U[XQ) exist and all those

of the xy at yo exist, and if XQ = x(yo), then all the generalized derivatives

D'^ v{yo) exist and are given by their usual formulas.

Proof. Since x = x{y) is of class C^-^, v $ H^-^[D) on each D CCG.

If m > 1, we conc lude f rom tha t theo rem tha t any d e r i v a t i v e D^^ v of

o r d e r \(x\ < m — 1 is g iven by a fo rmula of the form

(3.1.5) D^v{y) = j:A^^{y)u,p[x{y)]

w h e r e 1 < /5 < m — 1 and the AQ are p o l y n o m i a l s in the de r iva t ive s of

t h e xy of orde r | ^| + 1 — | ^ | < m — 1 and hence are LIPSCHITZ. Since

t h e u^^ all ^ H^{G) at leas t , the theorem wil l fo l low from Theorem

3.1.2(g) and the special case m — \.

If UQ is a mollif ied function of u, defined on G , if Z) C C the c o u n t e r

i m a g e of GQ, and v*{y) = Ue[x{y)], t h e n v* is LIPSCHITZ and

(3.1.6) vl4y) = u,,p[x(y)] ^x^Jy) (a.e.)

I t is c lea r tha t we may let ^ -> 0 and conc lude v^ Hl[D) w i t h v^OL g iven

b y the l imi t of the r igh t s ide of (3.1.6). If, n o w the genera l ized der iva t ives

u ,^(XQ) and x^o,{yo) all exis t and e r u n s t h r o u g h a regula r family of s e t s

Morrey, Multiple Integrals c

6 6 The spaces H^ and H^Q

a t yo a n d E is th e i m a g e of e, these forming a regula r family a t X Q , we

c o n c l u d e :

11 H " V [ A y ) - ^ ,i8 (^ o ) ^" fa (yo ) ] ^ y II e I

( 3 . 1 . 7 ) <\e\-^ f lÂîy)] - ^,p(ô)\'Kiy)\dy +

+ I H ~ ^ ^. )8 ( A T O ) ' f \ x ^ ^ ( y ) - : ^ f « ( y o ) dy .

The s econd te rm on the r igh t c lea r ly t ends to zero and the f irs t does a lso

s ince it is

<,C\E\-^ f \u^p{x) — u^^{XQ) I dxE

w h e r e C d e p e n d s on t h e L I P S C H I T Z c o n s t a n t s .W e now p rove two theo rems conce rned wi th the abs o lu te con t inu i ty

p rope r t i e s of func t ions ^ H\[G).

Lemma 3.1.1. Each class of functions u in H\[G) contains a represen

tative uo which for each o c is absolutely continuous in x " - on each segment in

G with endpoints in GU dG for almost all values of x^ and U Q tends to

limits as [x^, x^ tends to the end points of any such segment.

Proof. If i^ = [a, b] is an y ra t iona l ce l l in G, t h e r e is a s equence of

va lues of ^ -^ 0 s u c h t h a t

lim { [|UQ{X ", X^) — u{x°', x^\'P + \UQ OC{X°', X^) — U X(X°' X^)|^] dx = 0

(3.1.8)

for a lmos t sIL x^, o c = \, . , .,v. For an y x^ for which (3.1 .8) holds , the

U Q a re un i fo rmly abs o lu te ly con t inuous and hence t end un i fo rmly to an

absolu te ly con t inuo us func t ion . The resu l t fo llows by orde r ing th e ra t i on a l

cells in G and choos ing success ive subsequences .

Theorem 3.1.8. Each class of functions in H\{G) (C H.\{G)) contains arepresentative u which has the absolute continuity properties of U Q in the

lemma and which retains these properties under a regular L I P S C H I T Z mapp

ing. In fact we may define U { X Q ) as the L E B E S G U E derivative of f u(x) dx

for each X Q for which that derivative exists. ^

Proof. Since regular families of s e t s co r re s pond unde r regu la r L I P

S C H I T Z m a p s , we see that ^[^^(y)] = u[x{y)] and so it suffices to s how

t h a t u has the s e p rope r t i e s in s o m e one coord ina te s ys tem. Als o , it is

c lea r tha t u ( X Q ) is defined if t h e r e is an A s u c h t h a txo+h

(3.1.9) \im{2h)-'' f\u{x) — A\dx = 0

in which case u ( X Q ) = A.

N o w , let Uo be a r e p r e s e n t a t i v e as in Lemma 3 .1 .1 and let [a, b] C

{A,B) C[A,B]C G. Using F U B I N I ' S theorem and L E B E S G U E ' S theorem,

3.1. Defin i t ions and f i rs t theore m s 67

i t fol lows that for each (x, there is a set Za of x^ of measure 0 such tha t i f

x'^^ is not in Za then U{s[x'^, x\^ is A.C. in [A'', B^^] and

(3.1.10) lim ( 2 ^ ) 1 - ' ' / / l%,a ( l ) I ^f = / 1^0,«(!«, ^Lo)! ^ 1 "

for a g iven dense denumerable se t o f x"^ on [ '^, 5^^], in cl u di ng A°^ a n d

J5« andxo+h

(3 .1 .11) l im (2 A )- ^ / lô(f) - uo{xo)\di = 0 (;i:o = K , < o ) )

fo r almost every XQ on [^^, J S ' ^ ] . Since the functions of x"^ in (3-1.10) are

a l l monotone and cont inuous , the convergence in (3 .1 .10) is uniform.

Now, l e t XQ = (XQ, X^Q) b e any poin t fo r which X'^Q is not iii Z ^ a n d" < ^S 0. Usin g th e un iform con verg ence in (3 .1 .10)

and the abs o lu te con t inu i ty o f uo{x'^, A;^Q), we see that we can choose an

xf > x^ on {A"", 3°") such that (3 .1 .11) holds a t (xl, x^^) a n d s u c h t h a t

/ | ^ 0 , a ( | ^ < o ) | ^ f " < l / 3

(2h)-Î dx°'fd^'='f\u,cc(i'', 0 | ^ < < £ / 3 . 0<h<ho.

Then ( s e t t ing 6 = x'^ — x^)

I a;o+/i xfî+h x'xo+h [

( 2 A ) - / |wo(|) - uo{xQ)\di - {2h)-^J / |MO(I) - « o ( 4 , < o ) | '^f I

I t c o — ^ xî—h x'xO~h I

< 1^0(^,40) - w o W ! + ( a ^ ^ ' / l M o d + d)~ Uo{i)\dS < 2 e / 3 ,

0<h<ho,

so that (3 .1 .11) holds and u{x°', X'^Q) is defined a nd = Uo(x°', X'^Q) for all

x"" on [^°', 6«].

Definit ion 3.1 .3 . A function u is absolutely continuous in the sense of

ToNELLi (A.C.T.) on G if and only if u^H\{G) a n d u i s cont inuous on G,

Corollary. If u is A.C.T . on G, it has the absolute continuity properties

described in Lemma }AA and if x = x{y ) is a regular LIPSCHITZ map of

a domain H onto G and v(y) = u[x{y )], then v is A.C.T . on G.Theorem 3 .1 .9 . Suppose F ^ C^(Rp), suppose each U'P^H\ {G) for

some A3? > 1, /> = 1, . . ., P , supposeU{x)=F[uHx),...,uP(x)],

(3.1.12) P

6 8 The spaces H!^ a n d H ^ ^

and suppose U and the Vu^ Lx(G) for some A < 1. Then U ^ H\(G) and

U^a{x) = Voc[x) a.e.

/ / the u^ are A.C.T., so is U.

Proof. For each ^, let ^2> be a representative of the class U'P whichhas the absolute continuity properties described in Theorem 3.1.8 andsuppose£7(:\;) =^F[u [x)]. Then tJ has these absolute continuity properties,is equivalent to U, and its partial derivatives dU/dx'^ = Fa almosteverywhere. The result follows from the h ypothese s a nd Theorem 3• 1.2 (g).

We conclude with two inequalities which will prove useful later (the

inequality f v(x) d x\ <i f \v(x)\d x for vector functions v is useful);

Theorem 3.1.10. Suppose z^H'^(G), D(ZG GQ, and (p is any non-

negative mollifier. Then

(a) / 1 / l ^ i V - ^(1) - V^z{x)\^f' <p*{i - X) dAdx < {Q \\z\\Z^)P

(b) \\^,\D-z\D\\ll,'<Q\\z\\Za iP>^).

ZQ denoting the mollified functions.Proof. It is easy to see that (b) follows from (a) and (3.1.2). By

regarding the derivatives D"^ z"^ with |ix | < m — 1 as components of a

vec tor, (a) is reduced to th e special case where m = 1. There is a domainD' CC.G such that D CD '^, Using the mollified functions, we mayapproximate to z\i) by fun ctions of class C^, so we ma y assume z^C^{D').Then

1

\z{S)-z{xy f<^Qvj\\/z[x + t{^~x)]\vdt, ^^B(X,Q), X^D.0

Thus, we obtain

/ I j\z{^)~z{x)\vcp^^[^-x)dAdx

^ Qv] dt (pliOll j \V z{x + K)\J>dx\d!:

= gpfdtl^*{0\ f\Vz(y )\Pdy ]di:<.(e\\z\\l^)P0 BQ [D\^ J

if D§ denotes, for each fixed vector |, the set oi sXl y = x -\- i where

x^D\ clearly Dt^ CD' CG for each [t, f) considered.

3.2. General boundary values; the spaces H^Q{G); weak convergence

We begin by defining the spaces H'^^ (G):

Definition 3.2.1. The space H^Q(G) is the closure in H^{G) of the setof functions C'^{G).

3.2. General boundary v a l u e s ; the spaces H'^Q{G); weak convergence 6 9

Theorems 3.2.1 ( P O I N C A R E ' S inequality). Suppose G(ZB(XQ,R) and

U^H^Q(G). Then

(3.2.1) j \Vû[x)\i>dx <,p - R rn-k)pj \jmu{x)'fdx, 0 < ^ < m.

G G

Proof. We sha l l p rove th is for w = 1, ^ = 0; the gene ra l r e s u l t

follows easily by induction, since if U^H^Q(G), S/^ U^H^^^[G)

( = Lp(G) if ^ = m). From Def in i t ion 2.3.1, it suffices to p r o v e the

t h e o r e m in the cas e tha t u^ Cl[B{xo, R)]. Tak ing po la r coo rd ina te s

(;', C) (C on dB{0,\)) with po le at XQ and s e t t i n g v(r, C) = '^(ô + ^ f), we

o b t a i n

(3.2.2) R^[R-r)v-ij j\v,r[sX)\^dsd^{^)

T E

us ing the H O L D E R inequali ty . The result (3-2.1) follows for m = 1 ,^ = 0

from (3.2.2) and the fac t tha t

B

j\u(x)\Pdx = Jr''-^J\v(r,C)\^drd2J-B{Xo,B) 0 E

Corollary. The space H'^Q{G) is a closed linear subspace of H^(G) and,

if G is hounded, the norm || u |^Q defined by

(3 .2 .?) M7o = [j\'7û(x)\vdxY''

is topologically equivalent to the norm \u\^ for u on H^f^{G). If p = 2,

the inner product may be taken on H'^Q to be

- N

(3.2.4) {^,'v)2o = j Z 2 ' ^ « ^ , « W ^ , = ' W ^ ^ (u = u^,...,u^, etc.).

For func t ions in H^ on a rb i t ra ry reg ions , we h a v e the followingt h e o r e m s :

Theorem 3 .2 .2 . (a) Suppose u ^ H'!^Q(G) and V(x) = u[x) for x ^G

and V(x) = 0 elsewhere. Then V^ H^[Rv) and V^ Hpo0) for any open

set A DG. Moreover 0°" V[x) = 0°" u(x) on G and D"" V{x) = 0 for

x^Rp — G if 0 <.\(x\ < w [almost everywhere).

[h) Suppose u^HîG), DcG, v^H^(D), v - U\D^ Hôi^),

U(x) = v{x) on D, and U{x) = u(x) onG~D. Then U^ H^{G), U - u^

H%{G), and Z)^ U{x) == Z)« v[x) on D and Z)« U[x) = D^ u(x) on G — DifO^\oc\ < w (a.e.).

(c) If u^ Hl{G), E is measurable, E G G, and u(x) — const, a.e. on

E, then Vu(x) = 0 a.e. on E (see M O R R E Y [16], p. 254).

Proof, (a) fol lows im m ed ia te ly f rom D efin i tion 3 .2 .1 and t he the ore m s ,

of Section 3-1-I- If we define V{x) = v(x) — u(x) on D and V(x) = 0

7 0 T h e spaces i f^ and H^^

e lsewhere , then the conc lus ions of par t (a ) ho ld wi th G rep laced by D.

B u t t h e n U(x) = u(x) + V{x) on G so the results in (b ) follow. Part (c)

fo l lows by choos ing an abso lu te ly cont inuous represen ta t ive u of u.

Theorem 3.2 .3 . If p > 1, the most general linear functional in H'^{G)

has the form

(3.2.5) m = J S Z At{x)u:,{x)dx

where the A^ ^ Lq(G) where p-^ + ^~^ = 1- ^ / :^ = 1, ^^^^ linear functional

has the form (3-2.5) i'^ which the A^ are hounded and measurable.

P roof .I t is c lear that any express ion (3.2 .5) defines a Hnear functional

on H^\ Converse ly , le t Bp be t h e sp ace of all ten so rs {99*} w he re eac h

(pi^ Lp{G) w i t h n o r m

lki= / 2" Sc^W Um dx\ .KQ Li= l 0^ | a | : ^ m J J

Then , f rom Theorem 3.1.1, i t fo l lows tha t the subspace M of all tensors

(p w h e r e (pi = u^^ a n d u^ H^(G) is a c losed l inear submanifold of Bp.

If we define Fi{(p) = f{u) for such tensors (p , we have | |iî|| = | | / | | and

i^ i c an be ex tended (Hahn-Banach Theorem) to a l inea r func t iona l F

ove r Bp with the s ame no rm. Bu t any l inea r func t iona l F on Bp h a s

the form

F{<p) = fS Z Atix)<pUx)dxQ i = l 0 < | a l ^ w .

where the ^* have the s ta ted p rope r t i e s .

F r om T heore m 3 -2 .3 , we im m ed ia te ly ob ta in :

Theorem 3.2.4. (a) A necessary and sufficient condition that u^ con

verges weakly to u(un — 7 u) in H^{G ) is that each component u^^^ of u^

converges weakly in Lp{G) to u\^.

(b) / / Un-ru in H^'{G)^ then Un-7 u in H^(D) for DcG (i.e.Un\ D —7 "l^l D )'

(c) If Un—7 u in H'^{G), x = x{y) is a regular transformation ^ C f ~^

of H onto G, Vn(y ) = Un[x(y )], and v{y ) = u[x(y )], then Vn—7 v in H'^(H).

(d) Ifun -V u in H^(G),C^ C f - i ( G ) , and all the D'^ C -^îth 0 < |^ | < w

are uniformly bounded on G, then l^ Un—7 iû in H^{G).

(e) If p'> \, bounded sets in H^{G) are conditionally compact with

respect to weak convergence in H^{G).

3 . 3 . Th e Dir ich le t p rob lem

In th is sec t ion , we in te rrup t our s tudy of the spaces H^ a n d K^Q in

order to i l lus t ra te the var ia t iona l method for p roving the ex is tence of

the so lu t ions of ce r ta in d i f fe ren t ia l equa t ions . We a lso es tab l ish D I R I C H -

L E T ' S principle.

3.3. The Di r ich le t p rob lem 7 1

Definition 3.3.1. If u and ^* ^ Hl(G) and u — u* ^ HIQ(G), we sayt h a t u and u* coincide on dG or u and u* have the same boundary valueson dG. li u^ HIQ{G), we say that u vanishes on dG.

R em ark. The con tent of the definition de pends onp

(but see Theorem3.6.2).Theorem 3.3.1. (a) Suppose that u^H\{D) for each DCC G and

satisfies

(3.3-1) / C ,a ^ , a ^ ^ = 0, ^^Cr{G),G

Then u is harmonic.

(b) If u^ H\[^) ^^^ satisfies (3.3-1), then u minimizes the D I R I C H L E T

integral

(3.3.2) D{u,G) = f\Vu\^dxG

among all functions with the same boundary values.(c) / / G is bounded and u* ^ Hl{G), there exists a unique function

u = u'^ on dG which minimizes D{u, G) among all such functions. Thatfunction is harmonic.

Proof, (a) The hypotheses imply that u is locally summable on G.

Since the C,a€ <^r{^) i f ^o^s, we conclude from (3.3-1) an d (3.1.1) t ha t

juAI:dx = 0, C^C '^'iG).G

The result follows from W E Y L ' S lemma (§ 2.3).(b) If u^HKG) it follows from Definition 3.2.I that (3.3.1) holds

fora l lC^ ^ lo {Q - Soif z^*^ H\{G),u'^ = uondG, a n d we se t C = u* ~ u

so u* = u + C and C€ - 20 {Q> we conclude that

D{u*, G ) = D(u, G) + 2fC,ocU,ccdx + ^(f , G) > D{u, G)G

unless C = 0.(c) Let {un} be a minimizing sequence and let f/j = ^* — Un. Each

Cn^HlQ(G) and D(Cn, G) is uniformly bounded. By POINCARE'S in eq ua lity IICwIII, ? an d hence l wfLc? is uniform ly bo un de d. T hu s a su bsequence, still called Un—yu . Since the norm of an element in a BANACH

space {L2{G) in this case) is lower semicontinuous with respect to weakconvergence, it follows that the D I R I C H L E T integral (=^(|i^,a| |2.^)^)has this property so th at D{u, G) < lim inf D{uny G). B ut u^H\ [G) and

f%-7f = ^* — ^so tha t C € Hloi^) ^^^ ^ = ' ** on dG. Thus u minimizes the D I R I C H L E T integra l. So if f ^ HloiQ, u + ^C = u* on dG forall Z and

D{u + ZC,G) =D(u,G) + 2A /C,a^ ,a^A; + A2 7)(C,G).G

Since A = 0 gives th e minim um , (3.3.1) holds and the results follow.

7 2 The spaces Hi' a n d H^^

3 .4 . B o u n d a r y v a l u e s

In th is sec t ion we in t roduce the c lass o f s t rongly LIPSCHITZ d o m a i n s

and p rov e th a t if G i s s t ron gly LIPSCHITZ a n d u ^ H^ (G) for some arbitrary

w > 1, th en u can be app rox ima ted in H^ (G) by functions each of c lass^^{r) for some Fz^G. W e t h e n p r o v e a v a r i a n t CALDERON'S ex tens ion

theo rem fo r s uch doma ins and d i s cus s bunda ry va lues on the bounda r ie s

of smoother reg ions . For the case m = \, some of the resu l ts a re ex tended

t o o r d i n a r y LIPSCHITZ domains i .e . domains of c lass C\. An example i s

given of a LIPSCHITZ doma in wh ich i s no t s t rong ly LIPSCHITZ.

Definition 3.4 .1 . A d o m a i n G is said to be strongly LIPSCHITZ iff G is

bounded and each po in t XQOI dG i s in a ne ighb orho od 3? wh ich is th e

i m a g e under a rotation and translation of axes of a domain \y[,\ <.R,

I> '* '!< 2L 2 in which XQ corresponds to the or ig in , "St H dG co r re s pondsto the locus of y ^ = f (y^) w he re / satisf ies a LIPSCHITZ cond i t ion wi th

c o n s t a n t L , a n d 3 1 0 G corresponds to the se t o f y w h e r e \y!^\ <, R a n d

f(yl)<r<2LR.Remark. A bounded doma in i s s t rong ly LIPSCHITZ iff it has a regular

boundary in the sense of CA LD ERON (p . 45).

Lemma 3.4.1. Any bounded open convex domain is strongly hiFSCunz.

Sketch of proof. L e t B (XQ, R) be the la rges t sphere C G, the g iven

d o m a i n . L e t xi^dG. From the convex i ty , i t fo l lows tha t any po in tin te r io r to any s egmen t xi x w i t h x ^ B(xo, R) lies in G. If we choose axes

so the y ^ axis runs a long the ray xi XQ, i t is easy to see that a part of

dG n e a r XQ can be represen ted in the des i red form.

W e need the fo l lowing we l l -known lemma:

L e m m a 3.4.2. Suppose f ^Lp (Rp), e is a unit vector, and fh is defined

hfh{x) = f{x + he). Thenfh ->f in Lp(Rv).

Proof. L et £ > 0. T he re is ag^C^c(Rv) s u c h t h a t \\gh — fh\\

= l k ~ / l l < ^ / 3 - S i n c e g i s a l so un i fo rmly con t inu ous , the re is a ^ > 0s u c h t h a t \\gh — g\\< si} for \h\ <d.

Theorem 3.4 .1 . Suppose G is strongly LIPSCHITZ and ti^H^{G). Then

there is a sequence {un}y each^ ^^{Gn) where G (Z C. Gn, such that Un -^u

in H^{G),

Proof. GUdG can be covered by a f ini te number of open sets 9^^,

each of which is e i ther a cell with 3 ^ C G o r a bo un da ry ne ighborho od

of the type descr ibed in the def in i t ion . There is a f in i te par t i t ion of un i ty

{Cj}, each of which has suppor t in some one 3 - and is of c lass C^ every

whe re ; i t c an be a r ranged tha t Ci{^) + * * * + CJ(^ ) = 1 on a do m ainFZ) GU dG. For each j , le t Uj(x) = Cj{^) u{x). For each j for which Cj

has s uppor t in G, Uj can be approximated by mol l i f ied func t ions which

are 0 near dG and can be ex tended .

So, le t us consider a j w h e r e Cj has s uppor t in a bounda ry ne ighbor

h o o d 3 t, and le t Vj(y) = Uj(x) and for \yl\ < R, l e t u s ex tend Vj{y) = 0

3 .4 . Boundary va lues 7 3

for y^>:2LR. Clearly, the function Wjn defined by Wjn{y) = Vj{y + n~^ Cp)€ H'î^^) where 9^+ is the part of 9^^ where >»' > / ( y ; ) - n-^ and also'^jn ->^j- in H^{^+) where 3^+ is where y ^ >/ (y i ) . There is a sequenceQn -> 0 such tha t Q^ <, (2L n)~^ and < distance of the support of Cj fromd 3 1 , and also so small that if we define Vjn = "Wongny then Vjn ->Vj on 9^+and each Vjn^ C'^ on Slg^. If we define Ujn{x) = 0 for x not in 9^ andequal to the transform of Vjn(y) for x in 9^, we see that Ujn^C°° in adomain "Z) G\J dG. We define Un =2unj on their common domain.

Lemma 3.4.3. Suppose 0 < s < r and S is a measurable set of finitemeasure. Then

j \ x - y\-^dy^rv{v ~ s)-ic»'-^ yvC'= | S | = \B{x, o)\,s

Proof. Since \x — y\~^ <: G~^ ii y i B(x, G) and \x — 3^1"^ > a~^ ify^B(x, a) and since | 5 (A; , cr)| = \S\, it follows that

f\x — y\~^dy <, f \x — y\~^dy = Fviv — s)~ô''~^.S B(x,a)

We next prove the following important theorem:Theorem 3.4.2. Suppose K is essentially homogeneous (see Def. 2.5.2)

of degree m — v {m'> 0) and ^ C^+i \Rv — {O}] and suppose f is definedon Rp and u is defined by

(3.4.1) u[x)=JK{x~y)f{y)dy.

Then

(a.) IffeQ{R.),ueC^{Rv),and

(3.4-2) Dûix) = j K,^{x— y)f{y)dy, 0 < | « | < w — 1

(3.4.3) D''u[x) = Co^f{x) + Yim I K,o:(x-y)f(y)dy, \oc\=m,

^ \y\ == ^•

( b ) / / / $ Lp(Rv) and has compact support C G C C Rv, then u ^ H^{D)for any bounded D, the formulas (3-4.2) hold almost everywhere, and

\\u\\l^<C{v,N,m,p,D,K)-\\f\\l^.

Remark. Formula (3•4-3) holds in (b) also, almost everywhere, but

we shall not prove this. Further properties of the functions in (34-1) incase (b) will follow most easily from certain S O B O L E V type theorems tobe given in §§ 3.5 and 3.7.

Proof. Part (a) follows from the methods and results of §§ 2.5 and 2.6.To prove (b), let us assume that f^Lp. Clearly K^cc is essentially

homogeneous of degree m — \(x\ — v.li this is > 0 , then Kâ is continuou s

7 4 The spaces H^ and H^^

a n d it fo l lows eas i ly by approximat ing to / t h a t D^" u i s con t inuous and

given by (3.4 .2). It m•— \oc\ — v =0, t h e n K^oc{x, y ) = Clog | ; \ ; — y\ +

-\- Kioc(x — y ) where C is a c o n s t a n t a n d Kixi s pos i t ive ly homo geneous

of degree 0. In that case, if _> > 1, the right side of (3.4.2) is bounded by

a co ns ta n t t im es || / | | J ; Up = 1, it fo l lows tha t the r igh t s ide ^Lr for

eve ry rw i t h rn o r m b o u n d e d b y a c o n s t a n t t i m e s ]| / | | ^ .

So , let us a s s u m e t h a t m— \oc\ — v = —s, 0 < s < r . Let Ux

denote the r igh t s ide of (3 .4 .2). Then, for a lmost a l l x,

\Uo:{x)\<M^,J\x-y\-^\f(y)\dx

(34 .4 ) \U4x)\p<Mf,^ f\x-~y {"^dyY ^ / I ^ — y i"^ \f{y) \^dy

IG

< M f ^ i • CP-^{V, S) • 1 G 1(2^-1) (i-^/»')Jj x — y\-' \f{y) \PdyG

by Lemma 3 .4 .3- From (3 .44) and the preceding d iscuss ion , it follows

t h a t II Uoc ||p,2) < C'll/JlJ for 0 < |(:x| < w— 1. So , s uppos e D is a fixed

b o u n d e d d o m a i n a n d wea p p r o x i m a t e / s t r o n g l y inLpby func t ions

fn^ C^ (G ) . Then a l l the Unoc->Ux inLp(D). Moreover , f rom the CAL-

DERON-ZYGMUND ineq ua l i t ies , i t fo llows th a t th e D^" Un converge s t rongly

in Lp{R) to some l imi ts f/aw h e n | ^ | =m. The results follow.

Theorem 3 .4 .3 . (Var ian t o f CALDERON's E x t e n s i o n T h e o r e m ) Suppose

G is strongly LIPSCHITZ and G (Z G D. There is a linear hounded extension

operator @ which carries H^{G) into H^Q{D) and which has the property

that if V = ( u, then v(x) =u(x) for x G.

Proof. W e beg in as in th e proof of Th eo re m 3.4.1 assu m ing , asw e

m a y , t h a t e a c h 3^ CZ). For each j for which Cj h a s s u p p o r t CG, we

define (Bj u as the ex tens ion of Uj{x) toD o b t a i n e d by s e t t i n g Uj(x)

= Oior x D — G.

F o r aj for which jh a s s u p p o r t in a b o u n d a r y n e i g h b o r h o o d ?ft, w edefine ©^ u first for u C [G) a n d s h o w t h a t it isb o u n d e d . T h e n @ u

= 2J®J '^ ' To define ©j u for such j a n d u, we define Vj{y) — Uj\_x[y)'] for

y $ 9 t + ( n o t a t i o n of the proof ofTheorem 3 .4-1) , ex tend the func t ion /

to the whole space toh a v e LIPSCHITZ c o n s t a n t L a n d e x t e n d Vj{y) =0

fo r y * >- fiy'p) w h e r e it i s no t a l ready def ined; c lear ly Vj{y) ^ C ^ i n t h a t

d o m a i n U. Now, le t y ^ C7 a n d let f b e aun i t vec to r wi th | f | <: L- C"

and le t w{r, f) = Vj{y + r C)- Then

0

(3.4.5) t^dO) =f'0 D'-w {r. C}dr =V](y).

4: LB

Let us choose a func t ion a)j{z) which is homogeneous of degree 0, of c lass

C^lRv — {0}] , wh ich has i t s s up por t in the cone 2:* > L | 2: j and is such

tha t i t s in teg ra l ove r ^ is 1. Multiplying (3.4.5) by co(C) and integrating

3.4. Boundary values 7 5

over 2!

(3.4.6)

we obtain

v^{y)

vf{z)

K4z)

la |=m

^ J(m

z \-^

ziU

2'C<xZ'â}j\

)d z

(2) (Coc = | a | ! / ^ i ! . . ,avl)V" w ' \x\=m

where X« ^ C^ [Rv — {O}] and is posit ively homogeneous of degree

m — V.

To define &j u, we beg in by ex tend ing Vj to the whole space by (3.4-6)

a n d l e t t i n g uf{x) = ^^{y). T h e n uf{x) = Uj{x) for x^G a n d uf ^ H^(A)fo r any bounded doma in A, by Theorem 3-4 .2 wi th

\\ur\\:_ ,<C{v. N. m. p, A)- \\u\\Z^.

Fina l ly we def ine uf^ = (S; u b y

uf'^{x) = C{x) uf(x)

wh ere C€ C '^(Z)) an d C{x) = 'I on G. The result follows.

A spec ia l case of the fo l lowing Theorem was proved by R E L L I C H .

Theorem 3.4.4. Suppose G is strongly LIPSCHITZ and w > 1. Thenbounded subsets of H^'{G) are conditionally (sequentially) compact as sub

sets of H^^-^G). If Un-7 U in H^{G), then Un->u in H^'^{G). The

theorem is true for any bounded dom ain G if we replace the spaces H'^(G)

and H^-HG) by H^îQ ^^^ Hô^{G), respectively .

Proof .J Su pp os e GdC D CdRvy Gc DQQ, (£ is the ex tens ion opera

tor of Theorem 3-4.3» and Vn — &Un. Then , f rom Theorem ')AAO w e

c o n c l u d e t h a t

(3 -4 7 ) II Vn, - Vn WZ a' <Q\\vn \\ID ^MQ.Fir s t , le t us suppo se t h a t ^ > 1. Th en , accord in g to Th eore m 3 .2.4(e ) ,

g a subsequence, s t i l l cal led {vn], s u c h t h a t Vn—zvm H^{D). It follows

from the formula in Definition 1.8.3 for VnQ a n d VQ and the weak conve rg

ence tha t D°'VnQ converges un iformly to 0°" VQ on G, for 0 < | ^ | <

< w — 1. Th en , t ak ing no rm s on H^~'^{G), we ob ta in

^ Un — '^ II ^ II '^n — '^TiQ II + II ^n Q — ^ e II + || ^ e — '^ ||

<2MQ -\-\\UnQ — Ue\\(u = V\G, Un = Vn\G)y

(UnQ = VnQ\G> '^Q = VQ\G)

us ing (3.4.7). T he r es ul t follows easily. If ^ = 1, all th e D^^ Vnq a re equ i -

con t in uou s and un i fo rmly bou nd ed s o th a t a s ubs equence , s ti ll c a ll ed

{un} ex i s t s s o tha t the Z)* Unq converge un iformly to some func t ions cpocQ

for each of a sequ enc e of ^ - > 0 . Since (3 .4-7) s t i ll ho lds th e sequ ence s

7 6 The spaces H^ a n d H^^

<âe (|^| <= ^ — ^) form a CAUCHY sequence in Lp{G) and so tend to l imi t

func t ions (fa in Lp(G). If we set u = cpx w he n |<%| = 0 th en w e see t h a t

u^H^~^(G), (foe = î,ocy q^xQ = UQÔC aud (3.4.7) holds for UQ a n d u. T h e

resu l t aga in fo l lows , bu t now, we do not know tha t u ^ H^. However , i fUn -7 u in H^, then th i s u mus t be the s ame a s the p reced ing one . The

la s t s t a temen t i s now ev iden t s ince we may a s s ume G C. D where D is

strongly L I P S C H I T Z .

Remarks. I t is c lear from Theorem 3.1-5 and 3-1-7 (change of variable

theorems) how to def ine the e lements of the spaces H^{Tl) for a compact

man i fo ld Tl, w ith or w i th ou t bo un da ry , of c lass Cf ~^. Fo r a g iven f in i te

cover ing U of 9Jl by coo rd in a te pa t che s ri w i t h d o m a i n s Gi in Rv a n d

ra ng es 3^^ in 9D one can define a norm by

(3.4.8) (II u \\;\)p = 2 ' ( l l î Wla,)^- ««W = <Mx) ]i

for ex am ple an d an y two such no rm s a re topologica l ly equ iva le n t . In t he

case p = 2, the co r re s pond ing inne r p roduc t wou ld be

(3.4.9) {u, v)^==2:f Z(^-J^"î^"ndx,i Q 0: 1 a l^ m

The definit ions can be carried over for tensors on Wl, bu t then , u s ua l ly ,

Wl must be of a h igher c lass than C^~^ s ince the re la t ions be tween thecomponents in d i f fe ren t coord ina te sys tems usua l ly involve a t leas t the

f i rs t der iva t iv es of th e coo rd ina te t ran s fo rm at io ns (see Ch apte rs 7 an d

8) . Als o , an ex tens ion can be made to non -compac t man i fo lds wh ich

possess coverings in which no point of W i s in more than K of the 3t^, K

being independent o f the po in t ; bu t then the topologica l equiva lence of

tw o nor m s (3-4.8) is no t usua lly tr ue . If G is a bo un de d d om ain of c lass

Cf ~^, th en we m ay cons ider ^G as a man ifo ld of c lass Cf "^ , th e s t r uc tur e

be in g def ined by th e map pin gs a l lowed in § 1.2, N ot a t i on s .

Theorem 3 .4 .5 . / / G is bounded and of class C ^~'^ the functions u ^C f ~^(G) are dense in any space H^{G ) with ^ > 1 and there is a bounded

operator B from H^{G) into H^'^dG) such that B u = U\QQ whenever

u^ C f - i ( G ) . If Un-û in H^{G), then Un -û in H';-^{dG). If p > 1,

the mapping B is compact.

Proof. To prove the f irs t s ta tement, we select a f ini te covering of G

by nei gh bo rho od s 9^^, each of whic h is a cell w ith c losure inter ior t o G

or is a b ou nd ar y ne igh bo rho od in th e sense of th e definit ion in § 1 .2 ,

N ot a t io ns ; in th e la t te r case we suppose t h a t 9^^ is m ap pe d on to / \ U 0*1by a regu la r map x = Xi(y) of class Cf"^, A be in g th e set 0 < y" < 1,

|y^| < 1. W e select a pa rt it io n of un it y {fs} , s = 1, . . . , 5 , of class

Q - ^ ( G ) , each Cs ha vin g su pp or t in some 9^^. Now , suppo se u^H^(G),

le t Cs ^ = i^s, and le t Vs{y) == Us[Xi{y)] for y^Fim case 3^^ i s a b o u n d a r y

ne ighborhood. For those s for which 3^ C G, w e c a n a p p r o x i m a t e t o Ug,

3 .4 . Bo und ary va lues 77

b y {uns}, in which each Uns^ ^TiQ- ^ ^ r t h e o t h e r s, w e c a n a p p r o x i m a t e

t o Vs by s imila r Vns o n / i U cfi, us ing Theorem 3-4.1; clearly each v^s n i a y

be chosen to vanish near G D dFi s ince v does . T he fi rst s ta tem en t fol lows

b y s e t t i n g Un ==Ûns where, of course Uns[xi{y)] = Vs(y) w h e n y ^ A

a n d Uns{^) = 0 elsewhere.

Nex t , s uppos e u^ Cf~^(G) and le t % and Vs be def ined as above . We

define BsU = Us\eQ. Then , c lea r ly Bu = u\Qg. F or each s for wh ich

îG G, BsU = 0. So le t s be one of the remain ing ind ices ; ev ident ly

Vs = Us u, Us being bounded . S ince Vg^ C f ~ ^ ( A ) , w e h a v e

l\l\'7^vs(y\ yl) - V^'Vs(0 , y l) \^\ ' dy l

(3.4.10) < {y n^-^fflz\'^'^s(v'> y l) \^Y'^dy ldri-<e(y -)

l i m £ ( ^ ) = : 0 , 0<^8(r)< 8(\) = \\vs\\:^,r.Q-*'0+

SO t h a t | |^s(0, y^) llr.î^ ^^ ev id en tl y bo un de d, since y*' is a rb it ra ry in

(3.4.10). F o r u^H^(G), i t is easy to see that BgU = Ws w h e r e Wg = 0

on dG — î a n d Ws i s the t rans form of Vs{0, yj,) on dG O^t w h e r e Vs is

an A.C. represen ta t ive of Vs (Theorem 3.1.8).

Now, s uppos e Un-yu in H'^(G). Then each Vns—^'^s and (3.4.10)holds wi th a func t ion 8[Q) which i s inde pen den t of n, s ince in case p = \,

the se t func t ions f \V^Vns\'^dy a re un i fo rmly abs o lu te ly con t inuous .e

From the fac t (Theorem 3-4 .4) tha t Vns-^'^^^s in H^~^{î) (since A is

s t r o n g l y LIPSCHITZ) i t fol lows eas ily from (3.4.10) that the l imiting values

^ns(0, y l) -^Vs(0, y l) in H^~'^(ai), The las t s ta tement now fo l lows f rom

Theorem 3 .2 .4(e ) .

The case m = \ in Theorem 3 .4 .4 can be genera l ized to LIPSCHITZ

domains , i .e . those of c lass CJ.Theorem 3 .4 .6 . If m= \, Theorem 3.4.4 holds for LIPSCHITZ domains,

i.e. those of class C J.

Proof. Using the no ta t ion of the proof of the preceding theorem, we

con clud e, s ince th e int eri or 3i% are s t ro ng ly LIPSCHITZ as i s / \ , th a t if

{un} i s any bounded s equence in H^(G), then a s ubs equence {q} of {n}

ex i s t s s uch tha t a l l the Uqs ->Us in H^~^{G) i f % C G and Vqs - > ^ s i n

H^~Hri) a n d h e n c e Uqs-Ûs in H^-HG). li Un-ru in / f^ (G) , then

Uns->Us a n d Vns ->Vs- The theorem fo l lows .An example . Tha t no t eve ry LIPSCHITZ doma in i s s t rong ly LIPSCHITZ

is seen by the following example of a bi-LiPSCHiTZ map in the case v = 2.

L e t {r, 6) a n d {R, 0) be pola r coord ina tes in R^. W e de f ine the mapp ing

T:R = r, 0 == 0 — lo g r , O < 0 < 7 7 : , 0 < y < 1 ,

T (origin) = origin

7 8 The spaces H^ a n d H^Q

B y c o m p u t i n g t h e d e r i v a t i v e s Xx, e tc . , where

X = R COS0, Y = R s i n © , x = r cosO, y = r sinO

one easily verifies that the map is bi-LiPSCHiTZ.

Remarks. W e have no t p roved an ex tens ion theo rem hke Theorem

3.4.3 ioT m = i w i t h LIPSCHITZ d o m a i n s .

3.5. E x a m p l e s ; c o n t i n u i t y ; s o m e S o b o l e v l e m m a s

As was po in ted ou t in the in t roduc t ion (§ 1 .8) , the func t ions in H^{G)

have been s tud ied f rom many d i f fe ren t po in t s o f v iew by many wr i t e r s .

We sha l l no t go deeply in to rea l var iab le proper t ies o f these func t ions

here bu t wi l l mere ly g ive a few examples to ind ica te the genera l i ty of the

func t ions a l lowed and th en sha l l p ro ve some SoBO LEV-type lem m as w hich

wil l ind ica te when such func t ions a re cont inuous or have severa l con

t i n u o u s d e r i v a t i v e s .

I t is easy to verify that the function / defined by

(3.5.1) f(x) = \x\-f^, x^B(0,i), (h + m)'P<v,

^ H^. F ro m th is , i t can b e eas i ly show n t h a t any func t ion / def ined b y

f[x) = J \x — S\^^djLi{e§), {h + m)'p <vG

w h e r e G is bo un de d a nd / is a f ini te m ea su re on G, ^ H'^ o n a n y b o u n d e d

domain D. (3 .5-1) shows also that , for a given m a n d p, the wildness of

functions of class H'^ increases wi th the d imens ion v. W e now p rove a

SoBO LEV-type l em m a gua r an tee ing con t inu i ty .

T h e o r e m 3.5 .1 . Suppose u ^ H^{G) where G is hounded and strongly

LIPSCHITZ and m > v\p. Then u is continuous on G and there is a constant

C{v, m, p, G) such that

\u(x)\^ C-\G\-^IPI^^\\V}U\\O + (m-vlp'n^^,\\ V ™ ^ ! " }

(3 .5 .2 ) ^ -^(A = d iam G).

/ / G is also (see Lemma 3-4.1) convex, we may take C = 1; u may he a

vector function.

Proof. If G i s s t rongly LIPSCHITZ, U may be ex tended to ^ H'^Q{D)

where Z) is a given open set D G w i t h || u |j^j) < Ci(v, m, p, G , D) - \u\^(}

and then may be ex tended to be 0 ou t s ide D. So, by al lowing a factor C,we may rep lace G by i t s convex cover . S ince u m a y b e a p p r o x i m a t e d i n

H'^[GQ) b y func t ions ^ C ^, for each ^ > 0 , we m ay assu m e G = GQ a n d

u^ C^(G) and then le t ^ ->0 in (3 .5 .2) .

By expand ing in a T A Y L O R ' S s e r ie s wi th rema inde r abou t each y in

G and then in teg ra t ing wi th re s pec t to y ove r G, we obta in ( for no ta t ion .

3.5- Examples; continuity; some SOBOLEV lemmas

see § 5.7 belo w )m.-i

(3.5.3) \ G \ ' u { x ) = ^ z u ^ r H - ^ y I ' ^ ^ < y ) ' { y - ^ y d y +1

_ | - y ( _ > l ) m [ ( ^ — 1)!]-î^^~M fvû[x+ t(y — x)]'(y — x)^dy

79

dt.

The inequa l i ty fo l lows by apply ing the H O L D E R inequa l i ty to each t e rm,

firs t se t t ing z = x + t(y — x) to hand le the l a s t in teg ra l ; when th i s i s

done the l a s t in teg ra l i s domina ted by

1

[A^l(m— \)\]'Jt^-^-'^l J\V'û(z)\dz\dt0 [Oixj) J

w h e r e G(x, t) consists of all z = x + t(y — x) for y ^G a n d \G(x, t)\= f\G\.

Corollary. If u ^ H\[G) with p '> v, then u is continuous.

W e now p re s en t a " D I R I C H L E T g r o w t h ' ' t h e o r e m g u a r a n t e e i n g c o n

t inu i ty (MoRREY [4] and [7]) :

Theorem 3 .5 .2 . Suppose u ^ Hl[B(xo, R)], 1 < ^ < r, and suppose

(?-5-4) f \\/ u\p dx < LPirldy-p+Pf", 0 <.r <d = R ~ \x ~- xo\,B{x,r)

0 < ju < \

for every x^B(xo, R). Then u^ C^[B{xo, r)] for each r <, R and

(3.5.5) \u{^)-u{x)\<CLd^-^'P{\i-x\ld)^

for \^-x\<dl2, C = C(v,p,fi).

Proof. B y a p p r o x i m a t i o n s , w e m a y a s s u m e u ^ C^{BR). L e t x and f

be giv en, le t ^ = | f — x\l2, a n d x = {^ + x)l2. For each 7 ]^ B{x, Q), w e

o b s e r v e t h a t

1

u(r)) - u(^) = (rj- - |« ) / ^ , a K + Hv - f ) ] ^ ^0

1

\u{7]) —u(^)\ < 2 ^ | | Vu[i + t(fi — ^)]\dt.0

Averag ing ove r B{x, Q) we ob ta in

(3.5.6) \B{x,Q)\-if\u{rj)-u{^)\d7i

B ( a . Q)

:.2Q\B{X,Q)\-Î \j\s7u[i + t{7i-^)]\dt

B(X,Q) LO

drj.

In te rch ang ing the o rde r o f in teg ra t ion , s e t t ing y — I + ^ (^ — I ) an d

n o t i n g t h a t y ranges over B{xt, tg) w h e r e xt = {i — t) i -\- t x which is

a t a d is tance dt >: S — 2Q -\- tQ '>dl2 from dB{xQ, R) and then us ing

80 The spaces H^ and H^^

(3.5-4) and the H O L D E R i n e q u a l i t y we o b t a i n

(3 .57 ) l\Vu(y)\dy < CiL{dl2)^-^-^lP{t Qy -^+^ Ci = C[v, p).

Bixutg)

T h u s we see t h a t the r igh t s ide of (3.5.6)1

(3.5.8) < 2C2 Q^-''L{dl2)^-'f'-''lPQ''-^+^ftf'-^dju, C2 = C2(v, p).

0

U s i n g the s ame re s u l t for the ave rage of \U[YJ) — U[X)\, we o b t a i n the

re s u l t .

We now prove anot her S O B O L E V Lemma ( SO B O L E V [1 ] , N I R E N B E R G [ 3 ] ) :

Theorem 3.5 .3 . Suppose u ^ Hl{Rv) with \ <= p < v, and has compact

siippOYt. Then u^ Lr{Rv), where r = v pl{v — p) and

(3.5.9) II «r < tri{\\».4i?''^v-"^t\\^M\ia = l

i \ can be p r o v e d by a p p l y i n g the inequali ty (3-5.9) for ^ = 1 to the

func t ion v def ined by

v[x) = \u(x)\*.

F o r , if th i s is d o n e , we o b t a i n

(3.5.10) f\u{x)\rdx= f \v{x)\^dx < fj {Wv^ocfi)'^"-> '' a = l

<i^(\\u\^yHP-i)lp]J{l\u 40)^1", (s = vl{v - 1))a = l

s ince r = p[t — \)j{p — \), The f irs t result in (3.5.9) follows easily since

r — [r s(p — \)lp] = s

a n d the second result follows from an e l e m e n t a r y i n e q u a l i t y .

W e s ha l l p rove the i n e q u a l i t y for = 1 and v = } , the proof for

r = 2 or r > 3 is s imilar . Clearly

0 0 0 0

\u(x,y, z)\ < / | ^ , i ( f , y, z)\dS, / I ^ , 2 ( ^ , ^ 7 , z)\d7],

00J\u^s(x, y , OI^C-

— 0 0

T h u sM/2

u{x,y,z)\si^^[ f\u,i{i,y,z)\di\

~ \l/2 / ~ \1^2

j\u,2(x.ri,z)\drj] I f \u,3{x, y, 0\dU •

3.6. M iscellaneous add i t ion al resu l ts 8 1

In teg ra t ing f i r s t w i th re s pec t to x then y , and us ing the SCHWARZ in -

e q u a h t y , w e o b t a i n

J \u{x, y, z)\^l^dx <l I \uî[S, y , z)\d§\ I j j\u^2{x> ' J, !^)\dxdr]— OO \ — C O / \ — O O

J J \u,2{x, y , C) IdxdCj

OO / OO \ i ; 2

ff \u(x, y , z) l^l^dxdy < j ff \ uî{S, y , z) \didy \— OO \ CO /

j J \^l',2(^,'r], z)\dxdrj\ [j J J \^,z{x,yX)\dx dy d^\OO / \ —O O /

f rom with the resu l t fo l lows by in tegra t ing wi th respec t to z.

Theorem 3.5.4. There is a constant C[v) such that if u ^ Hl[B{xo, R)],

there exists a function U^HIQ[B{XO, 2 R)] such that U{x) = u(x) for

x^ B(xo, R) and

\\Û\\l,j,^C-'\\u\\l^, where {'\\UIIJ,)P =J{\V u\^ + R-û^)Pl2dx.BR

Proof. From cons idera t ions of homogene i ty , i t fo l lows tha t i t i s

suffic ient to prove this for the unit ball B(0,i). But then the re s u l tfo l lows form Theorem 3 .4-3 ( the ex tens ion theorem).

Theorem 3.5.5. IfG is LIPSCHITZ and u {a vector) ^Hl{G), \<p <v,

then u^ Lr{G), where r = v pl{v — p) and

( 3 . 5 . 1 1 ) \\u\\?<C(v.p,G)-\\u\\l.

If u^ Hl{G), then (3.5-11) holds for each p <C v. If G = B{xo, R), we may

replace (3.5.11) by

( 3 . 5 . 1 2 ) \\u\\?<C{v,p)-'\\u\\l.

Proof. The las t s ta tement fo l lows f rom Theorem 3-4-6 ( the Extens ion

Theorem fo r m= \), If u^Hl(G) then, of course u^Hl{G) for each

^ < 1 . In order to prove the f i rs t s ta tement , we cover G by ne ighborhoods

3 li and choose a par t i t ion of un i ty {Cs} as in the proof of Theorem 3-4-5.

F or e ach 5 for wh ich the s up po r t of ^sClG, (3.5-11) holds for Ug, F o r t h e

rema in ing s , we may ex tend Vs(y) by posit ive reflection {vs{y^, yl)

•= Vs{ — y'^, y'v)) a n d t h e n Vs h a s c o m p a c t s u p p o r t i n \y^\ < 1, \y'^\ < 1.

So that (3 .5 .11) hold for such t ;^ and hence also for the corresponding Ug.

3 .6 . Misce l lan eou s ad d i t ion a l resu l t s

In this section, we include a selection of theorems which are useful

in d iscuss ing boundary va lue problems in the ca lcu lus of var ia t ions and

a few results useful in the case of differentia l equations of higher order.

8 2 The spaces H'^ a n d H^^

T h e o r e m 3.6 .1 . Suppose u ^Hl(G) and u is the absolutely continuous

representative of Theorem 3.1.8. Then

(a) if R = [a, b] is any cell in G, U\QJ}, is a representative of BR U, the

boundary value operator of Theorem3-4.5

for R\

(b) if x = x[y) is a &^'-LIPSCHITZ map of a part '31 of G onto FR U OR in

which OR corresponds to 3 1 (1 dG, v{y ) = u[x(y )], and (p{yl,) = -^[^(0, y^)]

for y l^OR, where y j = B u, then ^(y^ y^) ^ ^2?tei?) fo^ ^^^^ y^> varies

continuously with y ^ as an element of LJ,{GR) and, for almos t every y^,

^ (y^ y'v) - ^ ^ ( 0 . y'v) ^^ y" ^ O " ^ andv(0, y^) is a representative of(p(y l).

(c) / / B{xo, R) CZ G, then U\QS{X^^R) is a representative of B u, B

being the boundary operator for u; u\QB {xo,B)^Hl[dB(xo, R)]- for almost

all R for which B(XQ , R) C G.

Proof. This follows from Theorem 3.1-8 combined with the proof of

Theorem 3-4 .5 .

Theorem 3 .6 .2 . Suppose G is LIPSCHTTZ. Then U^HIQ(G) ifu^Hl(G)

and Bu = 0 a.e. on dG. Consequently, if U^HIQ{G) 0 Hl{G), then

Proof. The second s ta tement fo l lows f rom the f i rs t . To prove the

first, cover G b y ne ighb orho ods 9^^, choose a par t i t i on of un i t y I^s, a n d

define Us and?:;^ as in th e proof of T he or em 3.4-5- B y usi ng th e mo llified

func t ions USQ, we see th a t tho se Ug for w hich Cs ha s sup po r t i n te r io r to Gca ,n be app rox ima ted s t rong ly in H\{G) by functions ^ C^(G). So le t s

be one of the o ther ind ices . From Theorem 3.6.1, i t fol lows that the func

t ion Vs{y) ext en de d to be 0 for y" < 0 $ 77^(71 U J ' f ) . Fr om L em m a

3.4.2, i t fol lows that Wn -^Vs in Hl{ri) if ^£;^(y^ y ^) = Vsiy" — n-^ h, yl)

and the suppor t o f each Wn ^Fiii h i s smal l enough. By mol l i fy ing the

Wn, one ob ta in s a s equence % § , each ^C^(ri) s u c h t h a t Vns->^s in

HKFi), and hence one ob ta ins a sequence Uns^Lip^{G), e tc .

Theorem 3.6.3 . (a) Suppose G is LIPSCHITZ, u^Hl(G), and u is continuous on G. Then U\QQ is a representative of B u.

(b) Suppose G is LIPSCHITZ, Un—ru in H\[G), B Un has a continuous

representative (pn for each n, and xpn -^cp uniformly on dG. Then (p is a

representative of B u.

Proof, (a) follows from Theorem 3.6.1. (b) follows from Theorem 3.4-5-

Theorem 3 .6 .4 . Suppose G is LIPSCHITZ. Then there are constants

Ci(v, p, <T G) and C2(v, p, r, G) such that

f \u\p dx < Ci\ f \V u\v dx + i f \ B u\dSY]G [G \a ) \

J\u\Pdx <C 2\f\Vu\Pdx + if\u\dxY]G [G I T J J

for every u^ Hl{G); here a and r are subsets of positive measure of dG and

G, respectively.

3.6. Miscellaneous addit ional results 8 3

Proof. We prove the f irs t , the proof of the second is s imilar . If the

s ta temen t we re no t t rue , the re wou ld ex i s t a s equence {un}, each Un

^Hl{G) s u c h t h a t

(3.6.1) j\un\Pdx> n\j\Vun\^dx + \j \Bun\dSYy G [G \O J J

O n ac co un t of th e ho m og en eity , we m a y assu m e t h a t || w|| ^(? = 1 for

each n. Thus there is a subsequence , s t i l l ca l led { 71}, s u c h t h a t Un -> UQ

in Lp{G) a n d Bun->0 in Lp{a) (Theorem 3-4.6). Thus (see (3.6.1))

Vun-^0 in Lp{G), s o t h a t Un ~ÛQ in Hl(G) s o t h a t B Un-^B UQ in

Li(or), on ac co un t of Th eo re m 3.4.5 . B u t, from Th eo rem 3.1-3, i t follows

t h a t uo=^ cons t , w hich m us t b e 0 s ince BuoÔona. B u t t h i s c o n t r a d i c t s

t h e f a c t t h a t || Un ||J = 1 for ea ch n.

Theorem 3.6.5. Suppose G is LIPSCHITZ. Then there are constants

Ci{v, p, G) and C2{v, p, G, c) such that

{ \u\'P dx <, Ci ( \\J u\P dx if u = \G\~'^ f udx = 0

Q G G

j\u\'Pdx<,C2J\Vu\'Pdx if | 5 | > c | G | (c > 0)

G Gfor every u^ Hl{G); here S is the set where u(x) = 0. In case G = B (xo, R),

we may writeCi( r , p, G) = Ci(v, p) RP, C2(r, p, G, c) = C2{v, p, c) RP.

Proof. Again we prove only the second, the proof of the f irs t being

s imi la r. The l a s t s t a tem en t fo llows by hom ogene i ty . I f the s t a t em en t

were no t t ru e , the re wou ld ex is t a sequence {un} s u c h t h a t | | ^ ^ | | J = : 1

a n d Un->uo in Lp{G) a n d

f \un\Pdx > n J \\/Un\Pdx, \Sn\>i c\G\> 0

G Gfor each n. T h u s Vun - > 0 i n Lp{G) so Un ->uo in H\{G) an d wo = con st ,

on account of Theorem 3.1.3 ^^^ ^0 ^ 0 since V ^ o = 0 b u t ||«ô||J = 1-

But s ince Un ^= 0 on Sn, we conc lude f rom the conve rgence tha t

0 = lim \ \un — UQ^P dx =^ l im j \un — % \P dx = lim | /o |^ * | 5,^ |

wh ich con t rad ic t s the fac t tha t \Sn\ >: c\G\ > 0 for all n.

Theorem 3.6.6. Suppose G is a LIPSCHITZ domain and u^Hl(G)

where p '> v. Then u ^ C]^(G) and

(3.6.2) h,{u,G)^C[v,p,G)'\\ull, fz=\~vlp.

The corresponding result holds if G is replaced by a compact LIPSCHITZ

manifold {i.e. one of class CJ) with or without boundary y the norm being any

one of the norms discussed in the remarks after Theorem 3.4.4-

6*

http://bun/dSYy

http://un/Pdx

http://un/Pdx

http://un/Pdx

http://bun/dSYy

84 T h e spaces H^ a n d H ^ ^

Proof. We prove the f i rs t , tha t o f the second be ing s imila r . The

theorem fo l lows for any in te r ior domain D f rom Theorems 3 .5 .1 and

3.5.2 and the H O L D E R inequa l i ty . The theorem fo l lows for G by cover ing

G w ith ne igh bo rho od s 9 -, choosing a pa rt i t i on of un it y {Cs}> and con

s ider ing the func t ions Us a n d Vs of Theorem 3-4.5- Since Fi i s convex and

ea ch in teri or 3^^ is a lso, th e th eo re m follows for eac h Us on G.

Theorem 3 .6 .7 . Suppose, for each n, that Un ^ Hl(G) with p '> \ and

is continuous on G. Suppose also that 1|V 7 1 ,6? ' uniformly bounded and

Un converges uniformly to u on G. Then u^—ru in Hl{G). If G is LIPSCHITZ,

the same result follows if each Un is continuous on G and U n converges uni

formly to u on each D G G G .

Proof. We prove the f i rs t s ta tement ; the proof of the second is s imi la r

if one m ak es use of Th eor em 3-6.4 . Th e hy po thes es im ply tha t \\un\\l ga re un i fo rmly bounded . Le t {q} be any subsequence of {n}. There is a

fu r the r s ubs equence {r} of {q} s u c h t h a t Ur —7 som e ^0 in ii\(G) (Theorem

3.2 .4(e )) . B u t th en w e conc lude f rom Th eore m 3 .4 .6 th a t Ur-^uo in

Lp{D) for each LIPSCHITZ D G GG, SO t h a t u = UQ. Hence the who le

s equence Un—r u in H\[G).

Theorem 3.6.8 . (a) Suppose (p ^ Lp{G), D G G a, e is a unit vector, and

ipji is defined on D hy1

ipfi{x) = \ (p[x -\- t h e) dt, I h\ < a.0

Then fh^ ^P{^) fo^ ^^^^ ^^^^ ^ ^'^^ '^h-^(p i'^ I^pi^)-

(b) Suppose u^H\{G), D GG a, Cy is the unit vector in the %y direc

tion and vji is defined on D hy

i^hi^) = h~'^[u(x -\- h Cy) — u{x)].

Then vu in Lp{D).

Proof, (a) Let F(x, u) = (p{x -{- u e), \u\ <i a. T h e n F ^ Lp{D) foreach u and is measurab le in {x, u) over D X {—a, a). Also

a a

f \ f \F{x, u) \Pdx]du = f \ f \ (p(y) \Pdy] du<2a f \ q)(y) \Pdy

-a [D I -a iDiu) J <?

s ince D(u) G G for each u on {—a, a), D{u) be ing the set oi y == x + u

for u^(—a, a). H e n c e F^Lp{F), T = (— a, a) X D. N o w

' f P h [x) = J F{x, th)dt = h-^ f F(x, u) du

so y)h(x) is defined a.e. on D a n d $ L i ( D ) b y FU BIN I 'S theo rem. Als o

\Wh(x)\^<h-'^lb

f \F[x,u)\'P du

3.6. Miscellaneous additional results 8 5

Cons equen t ly ipn^Lp for each h] if h === 0, iph{x) = (p{x). Now, f rom

L em m a 3-4.2, i t follows t h a t

lim f \F(x, u) — (p(x) \Pdx = 0.

u-t-O J)

Accord ingly , we see tha th

Wh{^) — (p(x) = h~^ J [F(x, u) — (p(x)]du0

h

f \ fh{x) — (p{x) l^dx <h~^ J i f \F(^> ^) — ^{^) \^dx]duD 0 ID I

from which the result follows.

(b) Let u be the rep re s en ta t ive o f Theo rem 3 .1 .8 . Then vji is givena lmos t eve rywhere by1

'^h(^) = f u^y (x + they)dt.0

The result follows from part (a) .

We conc lude th is sec t ion wi th th ree theorems which a re usefu l in the

s tudy of d i f fe ren t ia l equa t ions of h igher order . There a re a g rea t var ie ty

of addi t iona l theorems l ike those above and those be low which can be

p roved by the me thods o f th i s s ec t ion .Theorem 3.6.9. Suppose G is bounded and strongly LIPSCHITZ, m > 1

and p '> \. Then, for each j , 1 <,j<,m— 1, and each £ > 0, there is a

constant C {v, m, p, G, j , s) such that

(? .6 .3) \\Vû\\l<e\\V"û\\l + C\\u\\l u^H^{G).

Proof. For o therwise there is a sequence {un} w i t h Un^ H'îQ a n d

I] Un 11 = 1 s uch th a t

(3.6.4) II VHin t > 4 ^Ûn \\l + n\\ Un i | ^

F ro m Th eor em 3 .4-4 , i t fo llows th a t we m ay as sum e th a t Un —> u

in H^-^(G). Since f V^'^^[J < i|^^!|^-i < ||^^||^^ = 1, it follows t h a t

Un->u in Lp(G), so u = 0. But then | | \7Ûn\\^ - ^ 0 so t h a t jj V ^ Un\\l

and hence | | Un \\^ -^ 0.

Theorem 3.6.10. / / u^H^(G ), there is a unique poly nomial P of degree

< w — 1 (o r — 0 ) such that the average over G of each D"-{u — P) is 0 if

0 < 1^1 < M — 1.

This i s eas i ly proved by induc t ion on m.

Theorem 3,6.11. Suppose G is a strongly LIPSCHITZ domain andG C B[XQ , R). Then there is a constant C{v, m, p, G) such that

\\V^ u\l <. C R'^-qSl'û\l, 0 < y < w - 1

for every u^H^^{G ) such that the average over G of each D°^ u is 0 for

0 < k l < w — 1.

8 6 The spaces IT^ and H:^^

Proof. W e f i r s t a s s ume tha t G CB[0,\). Suppos e the theo rem is n o t

t r u e . Then the re i s a ; , 0 < y < m — 1, a n d a s equence [un} of func t ions

of the type descr ibed such tha t

( 3 . 6 . 5 ) l | V ^ - ^ ^ K > ^ | | V ^ ^ ^ | | « , \\unt=^\.A subsequence, s t i l l cal led {uri) conve rges in i7^~^ {G) to s o m e u. F r o m

(3.6.5) it fo l lows tha t VÛn-Ô in Lj,{G) so t h a t Un->u in H'^{G)

and V*^ ^ = 0. By i n d u c t i o n on m and th e use of moUifiers, it can be

s h o w n t h a t ^ is a p o l y n o m i a l of degree < w — 1. Clear ly th e averag es

ove r G of each i ) ° ' ^ is 0 if 0 < | ^ | < m — 1. From Theorem 3 .6 .10 , it

fo l lows tha t u == 0. Since ^^^ - > ^ in H'^{G), th is contrad ic ts (3 .6 .5) .

3 .7 . P o ten t ia l s an d q u a s i - p oten t ia l s ; ge n era l i zat ion sIn th is sec t ion , we i n t r o d u c e the n o t i o n of ' 'quasipotentia ls** and

ded uce some of the i r d i f fe ren t iab i l i ty pro per t ies as wel l as some ad di t io na l

s uch p rope r t i e s of p o t e n t i a l s .

Definition 3.7 .1 . S u p p o s e ^ is a v e c t o r ^Lp(G), p > 1. We def ine its

quasi-potential u by

G G

(37.1)T h e o r e m 3.7 .1 . Suppose e^Lp(G), ^ > 1, u is its quasi-potential,

and V°^ is the potential of e°'. Then

(a) Each F« ^ HI [D) and u^H^ (D) with

II V 2 F | | o . p < C{v,p) Wet^Cr. \\VU\IJ, ^ C(v,p) \\e\\la

\\VV\\% < C{v,p)-A'\\e\\la, \\u\\% < C ( . , ^ ) - z l • | kK. ( .

for any bounded domain D, A being the diameter of D;

(b) u(x)= - F ^ « ( A ; ) , X^G (a.e.),

(c) fv^cc(u,<x-i-e°')dx = 0, v^Lipc(G)G

(d) if G — BR and e^ C^(BR), then u^ CJ^{ER) and

h^(Vu, BR) < C{v, fj) h^{e, BR) (O < ^ < 1).

Proof, (a) Fol lows immedia te ly f rom Theorem 3-4 .2 and i t s proof, (b)

follows from (3-7.1), Theorem 3.4-2, and the defin it ions . T o pr ov e (c),

le t {en}^ ^ r ( ^ ) ^ ^ ^ su p p o s e î - ^ ^ in Lp(G). For each n, we s ee tha t

u,(x) = l im fei{i)±^Ko(x - | ) | = - / i ? o ( * - I) < . . ( ! ) d^^^^G-B{x,Q) G

s o t h a tAun==~el^

and (c) holds for each n. The result follows by pas s ing to the l imi t , (d)

follows from Theorems 2.6.2 and 2.6 .7 .

http://vu/Ij

http://vu/Ij

http://vu/Ij

3 .7- Po ten t ia l s and qua s i -po ten t ia l s ; genera l i za t ions 87

Theorem 3.7.2. If u^ H\Q(G), then u(x) is given a. e. on G by

G G

(2.7.19)Proof. We have seen in Theorem 2.7.3 that this holds if ^ ^ C\^{G).

So, le t U{x) denote the right side of (2.7.19) and let Un^ ÎdQ ^^^ ^^^hn and Un ->u in HIQ{G). We note that

\ u { x ) \ ^ r-^ I \S - x \ ^ - ' '\v u{i)\d^G

1 1 U(x)\dx < f \Vu{i)\- \r-^f\i - x\^--dx] di <g'\\Vu\\laG G [ G J

(3.7.2) y,g- = \G\.Using (3.7-2) for the difference Un — U, we see that Un ->U in Li(G),

Theorem 3.7.3. (a) Suppose thatf^L\[G) and satisfies

(3.7.3) /1/(1) \ d ^ < . L ^ f-2+;., o < A < 1

for each hall B{x{),r)y and suppose V is its potential. Then V^H\{D)

for any hounded domain D and satisfies

(3.7.4) f\vV\^dx<C^(v,iu)-L^r''-^+^^, XQ^G, 0 < r < i^B{Xo,r)

where R is the diameter of G. Moreover

(3.7.5) fv(x)f(x)dx==-fvAx)VAx)dx, v^Hl^{G).G G

(b) If v>2 and f^ Us'{G), s' = vl{v + 2), then V^Hl(D) for anybounded D and

( 3 .7 .6 ) / l VF | 2 ^ ^ <C 2 ( r ) - ( | | / | | » , . ) 2 .

D/ / , also, f satisfies

f \f{x) \dx<Li i^^-i (rlRy-^+f", 0<r<^R, B(xo,R)cG

0 < ^ < 1 , i : i > | | / | | « , „ r = vl2

for all xo in G, then

(3.7.8) / |vF|2Â;<C(i^,/^)Lf(r/i?) ' '-2+2A^.-B(a;o.r)

(c) If f^ L2(BR) and satisfies a condition

f\f{x)\^dx'<L^r''-^+^f'

BiXo,r)(\BR

for every B{xo, r), then V^ CJ,(BR) and

hîf.Bn) <C{v,iu)L.

88 Th e spaces Hand iJ^^

Proof, (a) Let Wa(x) be the right side of the equation

(37.9) v,4x) = -rr^/ II- i-'d"'-x^)M) di.

o

By proceeding as in the proof of the preceding theorem, we easily con

clude that V and Wct^Li{D) for any bounded D. By approximating to

/ in L\{G) hy fn in C^ci^), we see as in the preceding theorem that F$

Hl(D) for any bounded D and that (37.9) holds a.e.

Now, we select xo^G and write

fii) =M^) +m) where /i(f) = m in B{xo,2r)

and/i(f) =0 elsewhere and let F& be the potential oi fjc. Let

(37.10) <P2{Q\x) = j\m\d^.-B(a;,e)n[(?-B(!Bo,2r)]

Then, from (37-9) for V^ we haveR

\ v V 2 ( x ) \ < r - ^ f \ i -x \ ^ ' ^ \ f { ^ ) \ d C ' = r-' f Q ^ ~ ^ < P ' ^ ( Q ; x ) d Q

r ^ 1

< Lr,-! R^-^ + {V-\)I Q^-^ dQ\

L r \

< ( ^ - 1)r7i(1 -X)-^Lr^-\ x^B{xo;r)

since obviously

0<.Q <r(:v$5(^o;^)).

Accordingly

/ I V F2 (A;) |2 i:^ <C {V, ;i) L2 ;^ -2 + 2A ^

B{xQ,r)

From the SCHWARZ inequality, we obtain

|VFi(A;)|2<r ;-i/i/2, where 0 < a < A and

(37.11) h=r- j\ -x\ - \md,

B(XO,2T)

B{Xo,2r)

since/i (I) =0 outside B{xo, 2r). In order to evaluate I2 define

B{x,Q)nBiXo,2r)nG

Then we see that

0 < < r ,

3.7- Potentials and quasi-potentials; generalizations 89

P r o c e e d i n g as with 992, we see t h a t

( 3 7 . 1 2 ) l2<C{v,A){A-o)-^Lr^-\ x^B(xo,r).

In tegra t ing (3 .7 .11) over B(XQ, r) and using (3.7 .12), we see t h a t

I \VVi{x)\^ dx < o-^(2r)'' L{2ry-^-^^ C{VJ,G) Lr^-\

Since th is ho lds for any a w i t h 0 < cr < A, we see t h a t

f\VVi(x)\^dx<C{v, A) Z2 r^-^+2^.

Using (3.7 .9), we see t h a t

u s ing Theorem 3-7-2 (wi th x and | i n t e r c h a n g e d ) .

T o p r o v e (b) we no te f i r s t tha t it follows from Theorem 3.4.2 t h a t

V^HIAD) for any b o u n d e d D w i t h ||V^ F | , , . ^ ^ < C(r) | | / | | L . . If

w e t a k e any A, we f ind, as in the proof of T h e o r e m 3-4.2 t h a t

\\/v{x)\^^' <g^«'-i. r-^ j \^ - x\^-^ 1/(1) |2 s' d^

So let Wx — rj V ^x w h e r e rj = i on BA on rj = 0 on and n e a r dB2A- T h e n

W^ Hlg,(Rv) and has c o m p a c t s u p p o r t , and

Wo:,^(x) =rjV,oc^ + r]jV,oc

II VTF||L,.2^ < II V^FIIL,.., + C'{glA)i-y2s' ll/IIL,.

H e n c e , for each A, we f ind tha t V F ^ L2{BA), us ing Theorem 3-5-3 . By

l e t t i n g yl -> + oo, we f ind tha t

J\vv\^dx<CHp){\\f\\o„)2.

T o p r o v e the s econd s ta temen t , choos e a p o i n t XQ^G and let i^ be the

d i s t a n c e of XQ from ^G. W r i t e

f{x)=Mx)+Mx)

where /2(A;) = 0 in B{xo, Rjl) and/i(A;) = 0 in G — 5 (A;O , i^/2), and letF fc be the p o t e n t i a l of fjc. C l e a r l y / g ^ L2 ., (G) w it h | |/2||L^(? < 11/11.'.^,

so F2 ^ Hl(D) on any b o u n d e d D . But s ince F2 is h a r m o n i c on B{xo, i^ /2),

it satisfies a condit ion (3.7-8) with // = 1 and Li = I j / IL ' - F ina l ly , i t is

eas y to see t h a t / i satisfies the cond i t ion of p a r t (a) w i t h L rep laced by

Li i^ -^ -^+ i . The result follows.

9 0 E x i s t en ce t h eo rem s

(c) Choose xo ^ BR and r > 0. Write f = fi + f^ where fi{x) = f[x)on B [ X Q , 2 r) and/i(A;) = 0 elsewhere; let V j c be the potential oifjc. Then,from Theorem 3.7-1 or Theorem 3-4.2, we conclude that

11V2 F i |2 £ A; < y IV2 F i | 2 ^ A ; < Z i L^ y»'-2+2/ .

Moreover, we conclude that V2 is harmonic on B(xo, 2 r) and

| V 2 F 2 W | < Z 2 / | | - ^ | - ^ | / ( | ) | ^ | , x^B(xo,r).Bii—B{Xo,2r)

If we define

B{x,Q)n[BR-B{Xo,2r)]

we find that0 0 0 0

I v 2 F 2 w I < ^ 2 / r'(P2{e> ^) ^ Q < ^ ^ 2 / r""^ 9^2( ; v) ^o < Z S L r ' ^ - i

s inequality^ince from the SCHWARZ inequality, it follows that

'LR'-^+i', g>RP2(Q>".-^, .J ,

Then V F satisfies the g row th condition for V Vi so the result followsfrom Theorem 3.5-2.

Chapter 4

Existence theorems

4 . 1 . The lower-semicontinuity theorems of Serrin

In C hapter 1. we proved two theorem s concerning the lower-semicontinuity of integrals I(z, G) (Theorems 1.8.1 and 1.8.2). Althougheach of the functions considered was required to be in the space H\{D)for each D < Z C . G, it was necessary in the first theorem only to requiretha t the functions Z n converged weakly in Li(D) to z for each D (Z <Z G,nothing being required concerning their derivatives. In the second,however, we required 2:^ —7 ^ in H\ ( D ) for each D G G G which impliesstrong convergence in Li and the uniform absolute continuity of the setfunctions f \VZn\ dx on such domains D. Th e writer, in his work referred

e

to in Chapter 1, required essentially this latter type of convergence inhis lower-semicontinuity theorems. In this section we present some ofS E R R I N ' S recent generalizations ([1], [2]) of these theorem s and someof those of ToNELLi in which S E R R I N merely requires that Z n ^-z(strongly) in Li{D) for each D C G G. O ur general hypotheses in th is

4 .1 . T h e lower-semicontinuity theorems of S E R R I N 9 1

s ec t ion conce rn ing / a re a s fo l lows :

(i) f{x, z, p) is continuous for all [x, z, p) (N a r b i t r a r y ) ,

(4.1.1) (ii) f{x, z,p) ->0 for all [x, z, p),

(iii) f{x, z, p) is convex in p for each (x, z).For much of th is sec t ion , we sha l l abbrev ia te (x, z) t o t.

We need the fo l lowing wel l known lemma which is an immedia te con

sequence of the definit ion given in § 1.

Lemma 4 . 1 . 1 . Suppose {/} is a family of convex functions defined on

a convex set withf(p) < / o ( ^ ) < + oa for each p in S. Then sup f(p) is a

convex function <.fo{p) on S.

S u p p o s e t h a t f(typ) satisfies (4.1.1) (t — (x, z)) and tha t i t s de r iva

t ive s fp are also co nti nu ou s. F or each L > 0, we define(4.1.2) fL{t, p) = max fO , sup [F{t, p, q)]\

w h e r e F{t, p, q) =f{t, q) + {p - q) 'fp{t, q).

L e m m a 4.1.2. The function fL{t, p) defined above has the following

properties:

(i) / L satisfies the conditions (4 .1 .1);

(ii) fL(t, p) ^f{t, p) for all (t, p), the equality holding if \p\ < ^ ;

(iii) for each compact subset ^ of the t-space, there is a constant A and afunction A((T), > 0 for a > 0 with X{a) ^^0 as a -^0, such that

fi^[ty P)^A{\ + \p\)

\fL[s,P) -fL{tyP)\<.X{\s -t\)'[\ +\P\), Sj^Z

\fL{t,P2)-fL[t,Pl)\<.A\P2-Pl\.

Proof. O bv ious ly / L > 0 . Us ing Le m m a 1 .8 .1 , we see th a t 0 an d

^{^y p>^) ^fi^'P) ^^^ each t a n d q, so t h a t / / , i s c o n v ex a n d < / . S in ce

F{t,p, q) i s un i fo rmly co n t inuo us on any com pac t s e t , i t fol lows t h a t / / ,

i s con t in uous . O bv ious ly fhif,p) = / ( ^ , p ) f or \P\ ^L. Now, c lear lyuniform inequali t ies of the form (i i i) hold for each function F(t,p,q)

w h e n \q\ <:L and s and t are in ^ ; also th e su p in (4.1-2) is ta k e n on for

s o m e q, depend ing on t a n d p. Clearly the f irs t inequali ty in ( i i i ) fol lows.

To prove the second def ine

foL{t,p) =m ^xF(t,p, q).

Then fL{t, p) = m a x { 0 , / O L ( ^ , P) ],

\fL{s,P) -fL(t,p)\<\foL(s.P) -foL{t,p)\.

AlsofoL(s,P)=F[s,p,q{SyP)l

foL{t, P) = F[ty p, q{t, p)] > F[t, p, q(s, p)]

foLis.p) -foL{t,p)<F[s,p,q{SyP)]-F[t,p,q{s,p)]

< m a x {|/(s, q) - fit, q)\ + \p - q\ ^ | / ^ ( s , q) - fp{t, q)\}.

9 2 Existence theorems

O n e o b t a i n s the s ame re s u l t w i th s and t i n t e r c h a n g e d . The th i rd re s u l t

i s p roved s imila r ly .

Definit ion. An in teg rand func t ion / is normal if and on ly if

(4.1.}) f{Xy z,p) -> -\ - oo as \p\ ->+ c>o for each {x , z).L e m m a 4 .1 .3 . If f is normal and ^ is any compact subset of the {x, z)

space, there is a constant a such that

f{x,z,p)â\p\- i, a>0, {x,z)^2^.

Proof. Suppos e th i s is no t t r u e . D e n o t e (x, z) by t. T h e n / s equences

{tn} and {pn} s u c h t h a t

f{in,pn)<'-\pn\-^^

Since f{t, p) > 0, we see t h a t R^ = \pn\ ->c>o. Let jtn = Rn^pn', wem a y , by t a k i n g a s ubs equence , s uppos e tha t tn —>ô and jtn -^no- Since

/ i s convex in p for each t, we h a v e

(4.1.4) f{tn, Rnn) < ( l - §-Îifn. «) + ^ £ ^ ^ " 0 ' 0 < i ^ < i ^ ^ .

H o l d i n g R f ixed and l e t t ing ; - ^ o o in (4.1.4) we conc lude tha t

/(ô, R n^) < f{k, 0), 0 < 2^ < CO

which contrad ic ts (4 .1 .3) ,

L e m m a 4.1.4. Suppose g(x,z,p) satisfies (4.1.1), vanishes outside a

compact set 2^= SQ X TQ in the {x , z) space, and has continuous partial

derivatives gp which are Lipschitz continuous in x, z, and p and satisfy

\gp\ < M for some M. Suppose Zn and z^ Hi(D) ^'^d Z n -^z in Li{D)

for each D CC.G. Then

Ig {z, G) < lim inf Ig {zn, G).

Proof. Let e > 0 and choose Q S O s m a l l t h a t 5o C GQ and

(4.1.5) / I V^ — V ^ e l^ :^ < e-So

ZQ be ing the 99 mo llified fun ctio n for some moUifier cp . N o w the W E I E R -

STRASS ^- fu nc tio n asso ciated w ith g is defined by

E{x, z, p, q) = g[x, z, q) - g[x, z, p) - {q - p) ' gp{x, z, p).

Since g is c o n v e x and \gp\ < M, we conc lude tha t

0 < E{x, z,p,q) <^2M\p — q\.

M a k i n g the a b b r e v i a t i o n s \J Zn = pn, V ^ = PQ> \l Z = p, we o b t a i n

g{x, Zn, pn) — g{x, Z , p)

(4.1.6) > g{x, Zn, PQ) + [pn — PQ) ' gv{^> Zn, PQ) — g{x, Z, pg)

- ip-po) 'gp{^> ^>PG) - 2 M \ P - P,\.

w h e r e 2M \p - PQ\ ^ g{x, z, p) - g{x, z, pg) ~ (p - pg) • gp(x, z, Pg).

4.1. T h e lower-semicontinuity theorems of S E R R I N 9 3

Now, f rom the domina ted conve rgence theo rem and FA TOU 'S l emma i t

fo l lows tha t

lim inf f [g {x, Zn> PQ) — g{x, z, pg)] dx>0 ,

(4.17)h m / PQ • [gj,{x, Zn, PQ) — gp{x, z, pe)] dx = 0.

Now, le t us wri teQ(X,Z) =gp[x,Z,Pg{x)].

From our hypo thes e s , i t fo l lows tha t Q i s con t inuous and has s uppor t in

5o X To a n dI Q{Xi, Z) - Q{X2, Z ) \<M'\Xi-X2\.

W e m a y a p p r o x i m a t e u n i f o r m l y t o Q by similar functions of class C^,a l l w i th th e s ame M ' ; s o we m ay a s s um e Q^ C^. Define

Znix)

zlx)

Then, f rom Theorem 3 .1 .9 , i t fo l lows tha t P^Hl{G) a n d h a s c o m p a c ts uppor t in G, and so

J VP{x)dx = f {Q[x,Z n(x)']Vzn — Q{x,z)'7z}dx +G

i Z n

G [z

G

+ j{lQcc(x, C)dC\dx = 0,

From th is i t fo l lows tha t

(4.1.8) I f[gp{x,Z n,pe)pn — gp{x, z,pg) p] dx\ <M' f\zn — z\ dx.So So

In tegra t ing (4-1 .6) over So and using (4.1-5), (4-1-7) and (4-1-8), we obtain

I(Zn, G) — I(Z, G) > —En — 2M S — M' J\Zn — z\ dx,

So

Ihnsn = l i m M ' f \zn — z\ dx = 0.

The lemma fo l lows eas i ly f rom th is and the a rb i t ra r iness of e .

Lemma 4.1 .5 . If q? is convex on an open convex set S, it satisfies a

uniform LIPSCHITZ condition on aity compact subset of S.

This i s wel l known.

Lemma 4.1 .6 . Suppose f(t, p) is strictly convex in p for each i (and

satisfies (4.1.1)), suppose ^ is a compact set in the t-space, and suppose

K, L, and R are positive numbers. Then there exists a positive constant K

such that

(4.1.9) Rt, P) -f(t. q)-A-[p-q)>K

when ever t^^, \p\ <,K , \q\ < L , \p — q\ '>:R, and A is a vector such

thatfif, r) —f{t, q) — A ' [r — q) ^0 for all r ( L e m m a 1.2.2).

94 Existence theorems

Proof. Let {tn}, {pn}, {^n}, {An} he chosen to satis fy the cond i t ions

a n d so t h a t the left s ide of (4 .1 .9) ten ds to its inf. Since ^ is c o m p a c t and

/ satisfies a un i fo rm L ips ch i tz cond i t ion in p for p on any b o u n d e d p a r t

of space, it fo l lows tha t the An are u n i f o r m l y b o u n d e d .

W e now p r o v e our f i r s t lower - s emicon t inu i ty theo rem ( S E R R I N [2],

T h e o r e m 12):

Theorem 4 .1 .1 . Suppose that f{x, z, p) satisfies one of the following

conditions (in a d d i t i o n to (4.1.1)):

A . / is normal.

B . f is strictly convex.

C. The derivatives fx, fp, andfpx are continuous.

Then if Zn and z^ Hl{D) and Zn -^z in Li{D) for each D G CG, it followsthat

I{z,G) < l i m i n f J ( ^ ^ , G).

Proof. We sha l l p rove the t h e o r e m in case A first. Let 5 an d T be

c o m p a c t s u b s e t s of the x and z spaces , respec t ive ly , le t L > 0, let SQ

a n d TQ be bounded open s e t s con ta in ing 5 an d T, re s pec t ive ly , and let

£ > 0. F r o m L e m m a 4.1-3. it fo l lows tha t there is a c o n s t a n t ^ > 0

s u c h t h a t

(4.1.10) f{x,z,p) •>a\p\- 1, X^SQ,Z^TO,\P\ < . L + \ .

Let 99 be a moUifier in [x, z, p) s pace and let /^ be th e 99-mollified func

t ion of / w h e r e Q is chosen so s m a l l t h a t

(4.1.11) \f{x,z,p)~f,[x,z,p)\<.e for {x, Z)^SQ x TQ^P\ <.L + \.

N e x t , we let oc{x, z) be of class C^ eve rywhere , have s uppor t in So X To

and equa l 1 on S X T; then def ine

(4.1.12) g'{x,z,p) =oc(x,z)feL{x,z,p).

I t is c lea r tha t g' satisfies the conc lus ions (i) a nd (iii) of L e m m a 4.1.2

and van i s hes for (x, z) ou ts ide So x T Q .

Now, l e t t ing {x, z) = t, ^^o = So X To, Z = ^ X T

(4.1.13) g'it.p) ^f,L{t,p) <.fS>P) ^f{t>P) + e

if ^ € 2 ^ 0 , \P\^L+ 1.

So now, s uppos e \p\ > L + 1. T h e n

fQL(t,p) =fQ{t, qp) + {p — qp) 'fep{t, qp) for s ome \qp\ <. L

= fS> qv)+S'(rp - qp) 'fMi> qv) [S = r f ^ >

(4.1.14) = S[Mt. qp) + [rp - qp) • f,p{t, qp)] + (1 - S)fS. qp)

<Sf,{t,rp) + {\-S)fS.qp)

<S[f{t, rp) + e] + (1 - S) [f(t. qp) - s]

<{2S-1)s+f{t,p)

4 .1 . T h e lower-semicontinuity theorems of S E R R I N 95

us ing the convex i ty and the fac t tha t S > 1. Here , rp i s the in te rsec t ion

of the segment qpp w i t h dB(0, L+ 1) . But now le t tp be the in te r s ec

t ion of the ray Op w i t h dB{0, L -\- \) and le t Sp be the po in t on the

line 0 p s u c h t h a t qp Sp \\ fp tp. T h e n a s imp le geom e t r ic a rgum en ts h o w s t h a t Sp^ B(0, L) s o t h a t

(4.1.15) S = i f ^ < | ^ | - I .

If, finally, we let g{t,p) =g^[t,p) (mollified) with a suffic iently small

we see , us ing the bound \g'(t,p)\ < (1 + \p\), t h a t

g{i>P) <f9L{t.P) +e( i + 1: 1) <f{t,p) +s{3 + 1^1)

(4.1.16) ^mp) + C8[\+f(t,p)]

k[t,p)-f{t,p)\^Ce for t^Z>\P\<^Lusing (4.1.10) for t^^Jo and the fac t tha t g(t,p)=0 otherwise . Us ing

the conclusions (iii) of Lemma 4-1 .2 for g', we s ee tha t g satis f ies the

hypo thes e s o f Lemma 4 .1 .4 .

N ow , su pp os e 7(2:, G) < + oo a n d 7 > 0. There a re compact se ts

5 C <^ in the xs pace , T in the z s pace , and a number L so large t h a t if 5 *

is the subset of S where z^T a n d \p \ < 7 , t h e n

(4.1.17) I(z,S*)>I(z,G) r)

(if I(z, G) =+ 0 0 , l e t Mbe a rb i t ra r i ly l a rge and S, T, a n d L can befound s o tha t I{z, S*) > M). Then, us ing (4.1 .16)

I{z, G) <I{z, S*) +rj ^ Ig(z, S*) +7)+ Ce\S*\

^Ig{z,G) +r} +C e | S * | < l im inf/^(^^, G) + ly + C e | S * |

< (1 + C e) l im inf/(^^, G)-\-r] + C e{\ + \G\).

Since r] a n d e are a rb i t ra ry , the resu l t fo l lows in th is case .

In case B,we proceed in the same way. But , in th is case we f ind ,

us ing Lemma 4 .1-6 for /^ tha t (4 .1 .14) can be rep laced by

(4.1.14') f,L{t, p) <f{t, P) + ( 2 S - \ ) 8 - K S

w h e r e x is for fg w h e r e ^ $ ^ o > \q \ < 7 , | r | < Z, + 1, and | ^ — ^ | > 1since, (4.1.14)

fQ{i> ^p ) + i^P - qp ) 'fQp{i> (1P) ^fQ[t> ^p ) - ^ •

Since a convex func t ion isLips ch i tz , it fo l lows tha t the suppor t ing

v e c t o r s A in Lemma 4.1.6 are jus t /27(^, q) for a lmost a l l q; cons equen t ly

one can mollify both sides of (4-1.9) to see that the K a b o v e i s t h a t f o r /

w i t h L rep laced by L + ^ and R = i rep laced by 1 — 2Q. Thus, if we

choose e < x/l, we see that (4 .1 .16) can be replaced by

g{t.p)<f{t.p) + Ce

\g{t,P)-nt.P)\<Ce for t^Z-\P\<I'.

The res t of the proof is the same.

In case C, we proceed by defining

g'[x, Z, p) = 0C{X, Z) fL{x, Z, p)

w h e r e (x[Xy z) was discussed above . By in s pec t ing the proof of L e m m a

4.1.2, we see t h a t g' satisfies the conc lus ions of t h a t l e m m a w i t h A((T)< ^ • cr if ^ is not va r ied and

g'[XyZyp) ^f(x, z, p), equa l i ty ho ld ing if {x,z)^^ and \p\ < L.

If we now mollify with respect top only we obtain (s ince 1 + |_/>| satisfies

a LiPSCHiTZ condition for all p) t h a t g{x, z, p) = ga{x, z,p) satisfies all

the cond i t ions of L e m m a 4.1-4 and

\g{x, z,p) — g'[Xy z,p)\ '<AG for all {x , z,p).

The re s t of the proof is th e s a m e .W e now r e m o v e the re s t r i c t ion in T h e o r e m 1.8.2 t h a t fp be con

t i n u o u s and gene ra l i ze tha t r e s u l t s omewha t (see S E R R I N [2] , Th eor em I3

and MoRREY [7], C h a p t e r III , T h e o r e m 4.I):

Theorem 4.1 .2 . If f satisfies (4-1.1) only and Zn and z^Hl(D) with

\\zn\\\^D ^'^^ ll-^lli.D 'i^'yiiformly hounded and Zn->z in L\[D), all for each

D CGG, thenI{z,G) < l i m i n f / ( ^ ^ , G).

Proof. Choose D CC G. If lim inf I{Zn, G) = + co, t h e r e is n o t h i n g

t o p r o v e . O t h e rw i s e , let £ > 0 and M be an u p p e r b o u n d for ||>?W||I.D

a n d ||-2:||i jr>. Define

g(x,z,p) =f(x,z,p) + e\p\.

T h e n g is n o r m a l . U s i n g T h e o r e m 4 .1 .1 , we o b t a i n

I(z, D) < Ig{z, D) < lim inf7^(^^, D)

< £M + lim inf I{zn, D) <sM + lim mil{zn, G).

The theorem fo l lows .

If, in Theorem 4-1 .2 , we requ i re on ly tha t Z n -^z in Li{D) (wi thou t

the un i fo rm boundednes s of ||'2:?^||i z))» t h e n the conclusion of the

theorem doesn ' t necessar i ly ho ld . An e x a m p l e , due to ARONSZAJN, is to

be found in the book by P A U C .

A t h i r d and perhaps less in te res t ing theorem, s ta ted be low, requires

the following definit ions:

Definit ions . An i n t e g r a n d / is of type I if and only if t h e r e are twopos i t ive func t ions X{a) and ^(c) defined for cr > 0 w i t h

limA(<7) = lim//(o') = 0, //(a) <, B a for a la rge ,

s u c h t h a t

\f(Xy z,p) ~f[y, w,p)\ ^X[\x - y\) • [1 +/ ( ^ , z^p)] + ii[\z - w\)

4 . 1 . The lower-semicontinuity theorems of SERRIJSL 97

for all X, y, z, w, p. An i n t e g r a n d is of type II if and on ly if t h e r e is a

func t ion X w i t h the prope r t i e s above s uch tha t

\f{x, Z,P) -f[y, W,P)\ < A(|^ -y\+\u~ V\) [1 + / ( ^ , Z,P)],

R e m a r k s . I m p o r t a n t e x a m p l es of i n t e g r a n d s of t y p e I and I I are

/ = / ( ^ , P)=A{X) F(p). f = f(x. z,p)=A (X, z) F{p),

re s pec t ive ly , whe re A [%) and A {%, z) are con t inuous func t ions wi th s ome

pos i t ive lower bound .

T h e o r e m 4 .1 .3 . (a) Suppose f is of type I. Then the conclusion of

Theorem 4.1.1 holds.

(b) Suppose f is of type II, z is continuous on G, Zn cind z^ HUD) for

each D G G G, and Zn converges in measure to z on G. Then I(z, G)

< l i m i n f / ( ^ ^ , G).

We refer to r e a d e r to S E R R I N ' S p a p e r ( S E R R I N [2]) for the proof. In

t h a t p a p e r S E R R I N defines the func t iona l I{z, G) in m u c h the s a m e way

a s LEBESGUE defined the a r e a of a surface :

^{z, G) = inf f l im inf / (^^ , Gn)\

for all s equences {zn} in which each Zn^ C^{Gn), GnG G, U Gn = G, and

Z n -^z in Li{D) for each D CCG. The scope of the discuss ion is en la rged

by a l lowing ' 'weakly d i f fe ren t iab le ' ' func t ions z in the discuss ion. A

vec tor func t ion z is weakly differentiahle on D if and on ly if it Li(D) and

the re ex i s t measures fii s u c h t h a t

I zicoi^^dx= —f wi djui for all oj ^ C ^ {D );

in case z is c o n t i n u o u s , it is of b o u n d e d v a r i a t i o n in the sense of T O N E L L I

( b o u n d e d v a r i a t i o n in the ord ina ry s ens e iiv — 1). S E R R I N then inves t i

g a t e s in s ome de ta i l th e re la t ion be twe en the in teg ra l s I{z, G) and I(z, G).

In many ca s e s it is s h o w n t h a t I{z, G) < I{z, G), the equa l i ty ho ld ing

w h e n z is ' ' s t rongly d i f fe ren t iab le* ' , i.e. z^H\ (D) for each D CC.G.

As an e x a m p l e , if we a l low func t ions of b o u n d e d v a r i a t i o n and define

hI[z, G)= f yr+V~^ dx {G = {a, b)),

a

t h e n I(z, G) is j u s t the l e n g t h of the arc^: = z{x), if z is c o n t i n u o u s . In

th is case , one can say t h a t I{z, G) < ^{z, G) and the equa l i ty ho lds if

and on ly if z i s abso lu te ly cont inuous . For de ta i ls of the resu l ts , the reader

is referred to the p a p e r of S E R R I N a lready re fe rred to.

Morrey, Multiple Integrals y

9 8 Existence theorems

4.2. V a r ia t io n a l p r ob le m s w i t h / = / ( p ) ; t h e e q u a t i o n s

( 1 . 1 0 . 1 3 ) w i t h N = I, Bi = 0, A'^ =A<^(p)

In th is sec t ion , we prove the theorems s ta ted be low. The f i rs t repre

s e n t s a comple te ex tens ion of the re s u l t s of A, HAAR discussed inChap te r 1 . STAMPACCHIA [3]I requ i red s ome add i t iona l s moo thnes s o f the

d o m a i n G a n d ofthe bounda ry va lues a s d id GILBARG.

Definition 4 .2 .1 . Let G be abounded s t r i c t ly convex doma in . A func

t ion (p defined ondG is sa id to satis fy the bounded slope condition with

bound M{BSM condition) iff with each xo^dG, there ex is t two l inear

func t ionsh{x) = q^{xo) + afc^x"^ — %g), | % | < M , k= 1,2

s u c h t h a t li{x) <,(p[x) < l2{x) for all :\; on ^ G .

Theorem 4 .2 .1 . Suppose G is bounded and strictly convex, cp satisfies

the BSM condition ondG, f =f{p) is convex and I{z, G) is given by

(1.1.1). Then there is a function ZQ which coincides with cp ondG, which

satisfies aLipschitz condition with constant Mon G, and which minimizes

I(z, G ) among all Lipschitz functions z =q? on dG. If f is ofclass C^ with

(4.2.1) fv,vSP)^^^^>^ ^ ^ '

then ZQ is unique and minimizes I{z,G) among all z^Hl{G) such that

z — ZQ^H\Q{G). If (p^HliG) with p'> 2, and G is of class CJ, thenZQ^ Hl{G). If f Q, C ^ , or analy tic, respectively, then so isZQ on G. If,

also, Gand 99 ^ C^, C"^, or analy tic, then so iszo on G.

Theorem 4.2.2. Suppose Gand cp satisfy the hypotheses of Theorem

4.2.1 and suppose that the A'^(p) ^ Cj^, 0<C /J' 0 if A 9 ^ 0 .

Then there is aunique Lipschitz solution ZQ of the equation

(4.2.3) lc,.A' (Vz)dx = 0, C Lip.G

G

which coincides with 99 ondG; ZQ satisfies aLipschitz with coefficient M

on G. If the A°' C^ with w > 1, then z CJ5+^(G). If G and the boundary

data also ^ CJJ'^\ z CJ5+^(G). The C ^ and analy tic cases also hold.

Theorem 4 .2 .3 . Suppose G is of class C^ and cp ^ C%{G) and suppose

the A°^^ CJ and satisfy

mi{p)\X\^<.Al^{p)Xo.h<^Mi{p)\X\^, mi(p)>0, Mi(p) <ymi(p).

(4.2.4)

Then there exists aunique solution ZQ 0/(4 .2 .3) ^ C^iQ- T^^ higher differentiability results ofTheorem 4.2.2 hold.

O ur me t ho d i s f ir st tosolve the equations (4 .2 .3) inthe cases where

G, cp, a n d t h e A°^ a re rea s onab ly s moo th ( the s e inc lude the Eu le r equa

t ions for the var ia t iona l p roblem) and sa t is fy (4 .2 .4) wi th m{p) and

1 He has recently improved his results

4.2. Varia t ional p rob lems wi th f = f (p ) 9 9

M{p) cons tan t s . W e then ob ta in bounds fo r the f i r s t de r iva t ive s wh ich

enab le u s to hand le the gene ra l c a s e s by app rox ima t ions .

The proofs of the f i rs t s ta tements in Theorem 4-2 .1 depend on the

fo l lowing two lemmas :

Lemma 4 .2 .1 . Suppose f(p) is convex for all p. Then there exists a

sequence {fn} of convex functions such that each ^ ^^{Rv), fn[p) ^f{p)

for all n andp, andfn{p) converges uniformly tof{p) on any hounded set in Rp.

Proof. S i n c e / h a s a s up por t ing p lane a t 0 (Lem ma 1 .8.1 (a)) we m ay

s ub t rac t tha t o f f and a s s ume tha t f(p) > 0 an d /(O ) = 0. Define gn{p)

as / L O was defined in the proof of Lemma 4-1.2 with L = n. It follows

(since /(O ) = 0, f(p) > 0) th a t / L O = / L a n d s o s at is fi es t h e c o nc lu s io n s

of tha t lemma. From the Lipschi tz condi t ion in ( i i i ) , i t fo l lows tha t , i f

gna deno te s the cp — cr-mollified function for some mollifier 99, thengna^C"" and

gna{p)<gn{p)+Aa<f(p) + Aa, \p\<. n.

W e may c lea r ly t ake

fn{P) = gnan(P) — ^Gn, On = ^jnA .

L e m m a 4.2 .2 . Suppose f and fn are convexy fn(p) ^fiP) for each n

and p, and fn - > / uniformly on any hounded set. Suppose that Z n and z all

satisfy a Lipschitz condition with coefficient M on the hounded region G and

that Z n ->z uniformly on G. Suppose Z n = z on dG and suppose, for eachn, that Z n minimizes In{^y G) among all Lipschitz Z =^ z on dG. Then z

minimizes I{Z , G) among all such Z .

Proof. L e t ^ == inf / (Z , G) am o ng all Z = 2: on dG. Le t £ > 0 a nd

choose di Z — z on dG s u c h t h a t I{Z,G) <C d + e. T h e n

In[Zn, G) < In[Zy G) < / ( Z , G)< d + 8.

Then l im sup/ (;2f , G) ^d.

O n the o the r han d (u sing the un i fo rm L ips ch i tz cond i t ion an d the un i form convergence of fn t o / ) ,

I{z, G) < lim inf7(^^, G) < lim mi[In(zn, G) + En] ^ d

(lim en = 0 ) .

L e m m a 4 .2 .3 . Suppose G is hounded and strictly convex and suppose (p

satisfies the BSM condition on dG. Let r= {(x, z)\x^dG, z = (p{x)}

and let E he the convex cover of F. Then there are convex functions q)~ and

— (p^ on G such that

(4.2.5) E= {{x,z)\x^G,cp-{x) ^z<,(p+[x)}.

(p~ and (p+ satisfy Lipschitz conditions with coefficient M on G and

(4-2.6) (p-{y ) = (p{y) = (p+(y ) for y ^dG.

If (p~{xo) — (p+{xo) < 0 for some XQ^ G, then (p~(x) — 99+(A;) < 0 for all

x^G.

1 0 0 Existence theorems

Proof. The h y p o t h e s e s on G and cp i m p l y t h a t Z h a s the form (4.2.5)

w h e r e 9 ? " and — 99+ are c o n v e x and (4 .2 .6) holds . Suppose Xi and X2^G

and s uppos e xs and X4, are the in te r s ec t ions of the l ine xi X2 w i t h dG. At

xs we have l inear functions /g" and l^ of s lopes < M s u c h t h a t

I-(x) < 99-{x) < 99+ {x) <l}(x), x^G, /g-(^3) = (p(X3) = It (^3)

w i t h the s a m e at x^^. I t follows easily from the e l e m e n t a r y p r o p e r t i e s of

convex func t ions of one var iab le (a long Xi, x^) t h a t

\(p-{xi) -(p-[x2)\ <.M '\xi-X2\, \(p+{xi) -(p+{x2)\ <M \xi -X2\.

T h e l a s t s t a t e m e n t is e v i d e n t .

Lemma 4.2.4. Suppose the A^" ^Cl{or C ) , n > 1, and satisfy (4-2.2).

Then, for each R, there exists a vector A%^ CJJ {or C^) which satisfies

(4.2.4) with Ml and Mi constants and also satisfies

\ARJ,{P)\^MI{R) for all p,A%{p) = A^[p) if \p\<.R^

Proof. From (4.2.2) and the c o n t i n u i t y we h a v e

(4.2.7) Al^{p)K.Xp^mi{r), \A^[p)\<.Mi{r) for \p\<^r.

L e t c o ^ C ( i ? i ) w ith cw (s) = O f or s < l ,co(s) = 1 f o r s > 2, and co '(s) > 0

for all s. Clear ly mi{r) is non- inc rea s ing and Mi( r ) is non-dec rea s ing . Let

us define

(4.2.8) A%{p) = i \ - coirjR)] A-[p) + c[R) w{rlR) ^ . , \P\ = r,

where c == c{R) is to be chosen . By c o m p u t a t i o n we o b t a i n

^Rp^(P)^ocPi^ ={\ - (o) Al^XAp + c m |A|2 + 5 (o'{c H ^ - f-1 HA-Aoc)

s = rlR, H = \p\-^{p'X).

Since for la rge r, \A{p)\ < 2 Mi(r) • r, we find using (4-2.7) that

Al^^AccA^^ [2k{\ - M) + ca ) ] | ; . | 2 + SO)'{cm - 2 M i | ^ | | A | ) ,

(4.2.9) 2k ^ mi{2R).

If we t a k e c >: 2 k, we see t h a t the r igh t s ide of (4.2.9) > ^ |AP for all

r on \R, 2 R] if

4kc{sa)') > 4 M 2 ( s c o ' ) 2 (Mi = Mi{2R))

which ho lds if ^ c > Mf • max(s a>'(5)).

L e m m a 4.2 .5 . Suppose G is bounded, cp ^ H\{G), the A^{p) ^ C^ and

satisfy (4.2.4) with mi and Mi constant and the condition

(4.2.10) \A^(p)\<M .

Then there exists a unique solution of Equation (4-2.3) such that z — (p^

HIQ{G). If G is of class C^ and 9? ^ C^{G), then z^ H^{G) for every r. If G

is of class CI, the A"^ are of class C\, and cp^ Gl^{G), then z^ Cl{G).

Additiona l theorem s concerning differentiability on the interior and at the

boundary hold as stated in Theorem 1.10.4 and 1.1 1.3.

4.2 . Variational problems with f = f [p ] 101

Proof. We use T h e o r e m 5.12.2 to p r o v e the exis tence . We set

(4.2.11) z = (p + u, A°'{x,p) = A'^[\/(p{x) + p],

and def ine A{u) and 51 (w, v) by

(4.2.12) '^l{u,v) = A(v), {A(v),w) = J w^aA^'iVq) + Vv) dx.G

We conclude, from (4.2.4) and (4-2.10) that

A-{q)=A-(0) + q,plA-^{tq)dt, | ^ (^)| < M i ( 1 + | ^ | ) ,

01

pocA^lVcpix) +p] =Po,A<'[V(p(x)] -hpocp^f^P^l-^^i^) + ^Pl^^

0

> m i | ^ | 2 - M i | ^ | - ( 1 + | V (^ ( : v ) | ) .

(4.2.13) {A^V^(p[x)-^p2-\-A-[\J<p{x)+p^-\}{P2.-pia)1

0

+ {1 ~t)Pi + tp2]dt>mi\p2-pi\^.

T h u s the A'ix.p) satis fy the condi t ions (5 . I2 .13) and (5.12.14). Also

{A(v),v) = Iv,,A-(x,Vv)dx>mi(\\vr2,)^ - Mi\\v\\l{\G\ + | | V 9 ^ i )

G

SO t h a t A satisfies the coerc iveness condi t ion (5 .12 .3) . Thus A satisfies

all the cond i t ions of T h e o r e m 5-12.2 so the re ex i s t s a s o lu t ion u^ HIQ(G)

of Equa t ion (4 .2-3) . The so lu t ion is easily seen, us ing (4.2 .13), to be u n i

q u e . The in terior differentiabil i ty results follow from the re s u l t s of

§ 1 . 1 1 and C h a p t e r 5.

W e m u s t now p r o v e the dif fe ren t iab i l i ty at the b o u n d a r y a s s u m i n g

G of class C^ and 99^ C^{G), To t h a t e n d , let xo^dG and map a ne igh

borhood 31 of XQ on G o n t o GR U OR (see §§1.2, 5.I, etc.) us ing a diffeo-

m o r p h i s m of class C^ so t h a t XQ co r re s ponds to 0 a n d '^ 0 dG co r re s ponds

to OR. Then Equat ion (4 .2 .3) assumes the form

(4.2.14) lv,ocA°'{x,Vu)dx = 0, V^G\{GR)

w h e r e the new A°'^ C^ and still satisfy the s ta ted cond i t ion , pos s ib ly

wi th d i f fe ren t bounds . We may rewri te (4 .2 .14) in the form1

fv,cc[af{x)u,^ + A^(x,0)]dx = 0, af{x) = JA^ [xJVu{x)]dtGR 0

a n d the a j ^ are b o u n d e d and m e a s u r a b l e and satis fy

(4.2.15 ) ^ ? ^ ( ^ ) ^ A ^ > w i | A | 2 , \a{x)\ < M i

a n d the A''{x,0)^ C^. F r o m L e m m a 5 . 3 . 5 , it follow s t h a t ^ ^ Cl{Gr)

for each r < R and satis f ies a Dir ich le t g row th cond i t ion a s in t h a t l e m m a

1 0 2 E x i s t en ce t h eo rems

Since this is t rue for each XQ, we conc lude th a t

(4.2.16) flVul^dx^L^r-^+^f", 0 < / ^ < l

for all B(xo, r).W e now reve r t to Eq ua t io ns (4 .2.14). A pply ing the d if ference q uo

t ie n t p ro ced ure of § 1.11 in a tan ge nt ia l d i rec t ion , we see th a t

Ufi = h-^[u{x + hey ) — u(x)] satisfies

fv,oc{K^'^h,^ + et)dx = 0 ,

w h e r e a^^ a n d e^ are given by formulas l ike (1 .11.4), the a^^ satis fy

(4.2.15) uniformly for small h, \\uji\\l^G{r) i^ un ifor m ly bo un de d for each

r <C R, a n d e^ satis f ies a condit ion(4.2.17) J\eh\^ dx < JJr^-^+' f^, 0 < r < i ? .

-S(i/o,r)n(?j5

I t fol lows, again from Lemma 5-3-5 that the u^ a r e u n i f o r m ^ H o l d e r

con t inuous on each Gr a n d \/ uji satisfies (4.2.17) with R rep laced by r

with an L a n d [ A , independen t o f h. T h u s w e m a y l e t A -> 0 and f ind tha t

t h e SIpy with 7 < v sa t is fy these condi t ions . S ince pv,y = py^v a n d w e

can solve for pv^p in te rms of the o thers , we see tha t Vpv also satisfies

(4.2.17) as do the py , and there fore py ^ C^^(G-r) for each r < R. T h e

higher differentiabil i ty results follow from Theorem 1.10.3.

Lemma 4.2 .6 . Suppose G is of class C^, z ^ ^ J ( ^ ) ' ^^^ ^ satisfies an

equation of the form

fv,oca°^^(x)z,^dx = 0, v^H\^{G), a^^^Cl{G).G

Then z takes on its maximum and minimum values on dG .

Proof. A p p r o x i m a t e t o t h e a""^ b y a^^ ^ Q ( ^ ) a n d l et Z n be the so lu

t ion coinciding on ^ G w i t h z of the equa t ions

T h e n Zn^ ^li(^)> ^n^ ^^(-^) for each D (Z G G, each Z n takes on i t s

m a x i m u m a n d m i n i m u m v a l u e s on dG a n d Z n -^z in C^{G) (see §§ 5-2,

5.5, and 5.6).

Proof of Theorems 4.2.1 and 4.2 .2 . In cas e 99 co inc ide s on dG with a

l inear func t ion l(x), t h e n l{x) is the des ired solution in e i ther case . So we

assume tha t th is i s no t the case .

W e f i rs t p rov e Th eor em 4 .2 .2 . W e f i rs t assum e th a t th e A^^C^and sa t is fy the condi t ions in Lemma 4 .2 .5 . Le t ^ be a moUif ie r . For each

n, we define

'(p^{x)^ (p±{x)^ Mn-^Tn~^\^-M^^ ^^G, xo^G,

4.2. Varia t ional p rob lems wi th f = f {p) 1 0 3

T h e n (p~ a n d — 99+ ^ C°°(GeJ and a re convex . Le t

T h e n Gn is convex and of c lass C^, Gn C Gii+i, GT - > < , a n d 99 conve rge

nicely to cp- on an y D C C G. F o r each w, le t Zn be the s o lu t ion o f Lemma4.2.5 with (p rep laced by cp~, s ay . Each Zn<^ ^^(Gn). By a difference

qu ot ie nt de vice like t h a t in § 1 .11, s t ar t i ng from (4-2.3) we find th a t eac h

^n,y == : wy sa ti sf ie s

lv,.[a-^{VZ n)pny ,(i]dx = 0, V^Hl,{Gn) a-^{q) =^ A^ iq)

and so each l inear combina t ion of the pny t a k e s o n i t s m a x i m u m o n dGn

s o t h a t \ ' 7Zn\ t a k e s o n i t s m a x i m u m on dGn.

Now, s ince (p~ i s convex , we conc lude tha t

a s i s s een by ro ta t ing axes . Acco rd ing ly

1

'al'{x) = JAl^[{\ -t)V<p^ + t^z„]dt,

0s o t h a t Z n — q)~ ^0 by t h e m a x i m u m p r i n c i p l e . I n l i k e m a n n e r Zn < <p^

on G,j. Co nseq uen t ly \ \7Zn\ < M ^ on dGn and hence on Gn a nd M ^ - > M .

Then a subsequence , s t i l l ca l led {zn}, is suc h t h a t ^ i con verge s u ni

formly on each D C. G G to 3 . Lipschi tz func t ion z = (p on dG a n d z

satis f ies a Lip sch itz con dit i on w ith coeffic ient M on G. Moreover , on

a n y D G G G, it follows from th e th eo re m s of §§ 5-2, S-S, and 5-6 tha t

the der iva t ives of a l l o rders converge un iformly on D to those of z. T h u s

^ is a solutio n. If th e A°'^ C^ but do not sa t is fy the condit ions (4-2.4)

wi th mi and Mi cons tan t s , we may f ind func t ions A%{p) which = A°'(p)

for \p\ <R,R> M, which do satis fy (4-2.4). For R> M, the s o lu t ion

z obta ined above for ^ i ? i s jus t tha t fo r A. F i n a l l y w e c a n a p p r o x i m a t e

a n y A'^^ C^~^ b y th os e ^ C^. This p roves Th eorem 4 .2 .2 . Th e h ighe r

d if fe ren t iab i l i ty resu l ts a long dG follow from Theorem 1.11.3.

In c a s e / i s regula r an d of c lass C^, we conc lude f rom T heo rem 4-2 .2

jus t p roved tha t there i s a so lu t ion ZQ of Eule r ' s equa t ion on G w h e r e z

satis f ies a Li psc hitz co nd it ion wi th coeffic ient M on G an d ZQ ^ H^ {D)

for each D GG G an d each r > 2. I t is im m ed iat ely e vid en t th a t . o isthe un ique func t ion min imiz ing I[Z, G) am on g a l l L ips ch i tz func t ions

Z = ZQ = q) on dG. The h igher d i f fe ren t iab i l i ty resu l ts fo l low as above .

I f / i s no t s m oo th , we m ay app rox im a te i t f rom be low a s in L em m a 4 -2 .1

and comple te the proof of Theorem 4 .2 .1 by us ing Lemma 4-2 .2 and

the un i fo rm L ips ch i tz bound M.

Proof of Theorem 4 .2 .3 . Let us f i r s t a s s ume tha t the A^" ^ CJ and

satis fy the cond i t ions of Lemma 4 -2 .5 . Then the re is a un ique s o lu t ion

z^ Cl(G) w i t h z = acp on G oi E q u a t i o n 4.2.3 for e a c h a, 0 < cr < 1.

As above we see t h a t V z t a k e s on its m a x i m u m on dG (for each a). We

n o t e t h a t z{x, 0) ^ 0 and, by dif fe rence quot ien t ing wi th respec t to c,

w e see t h a t z and V z v a r y c o n t i n u o u s l y w i t h a. In fac t Za^ C"J(G) and

satisfies

(4.2.18) fv,aAl{aV(p+Vu)Za,^dx = 0, v^Cl{G), z = a(p + u.

N o w , let us s u p p o s e t h a t the A^" satis fy the condit ions (4 .2 .4) with

mi(p) and Mi(p) v a r i a b l e . Let us choose R > m a x | V 9 ! ? ( ^ ) | and define

t h e A% as in (4.2.8) so Lemma 4 .2 .4 ho lds . Then , as long as | c V 9? + V |

= \Vz{x, a)\ •< R, u is 3 L so lu t ion of

a°'^(x,G)uôcp(x,G) = — aa'^^ (p^x^, Z{X,G) = a(p{x) + U{X,G)

a-^(x,G) = {A-^^[VZ{X,G)] + A^^^[Vz{x,G)]}{mi{\7z) + M i ( V ^ ) } .

(4.2.19)

T h e n we n o t i c e t h a t

(4 .2 .2 0) W 2 ( 7 ) | A | 2 < t 2 « ^ ( ^ , a ) / l a A ) 8 < 7 ^ 2 ( 7 ) , Mîy) = 2j[\ +y)

for such {x, a). From (4-2.19) and (4.2.20), it fo l lows tha t

w h e r e l\ip\\\o = ma x |^(JV;) |.

N o w , let xo^dG. Since G is of class C , the re is a ^ > 0, i n d e p e n d e n t

of xo y s u c h t h a t t h e r e is a un ique ba l l B{xi, Q) s u c h t h a t B(xi, Q) D G

= {XQ]. If we define

w(x) = K-P(Q-t -r-i), r=\x~-xi\, P = |||9^|!|g,

w e see t h a t

Lw ==KPtr-i'^ IZ^""" - {^ + 2) r-^ ^"'^(^"^ — ^1) (^^ — ^f)]

< K Ptm2{y) r-i-^[vy — (t + 2)]

{^/^(x)]^ KPtQ-^-^, W{XQ)=0, W{X)>:0 for x^G.

Clearly we may choose K{v, y, D) and t{v, y, D) so t h a t

Lw{x) < —C{v,y)P, x^G.

T h e n we see t h a t

L{a w ^ u) ^0 inG, a w :^ u >:0 on dG.

T h e m a x i m u m p r i n c i p l e i m p l i e s t h a t awû>0 in G and so, s ince

(T te ± ^ = 0 at ^0 , it fo l lows tha t

du ,(4.2.21) \Vu(xo)\ =: '-(xo) :.G\\7W{XO)\ <GCi(v,y,D)*P,

4.3- The borderl ine cases k =v 105

provided tha t max (c r | V 951 + | V ^ | ) < i^ . But ifw e t a k e

i?>liv<pK +Ciii|v2^|||S

the bound (4.2 .21) wil l hold forall ;^ on G and a l l c, 0 < o- < 1. This

p r o v e s t h e t h e o r e m .

4 . 3 . The borderl ine c a s e s k —v

In th is sec t ion , we comple te the proof of the cont inu i ty proper t ies o f

the s o lu t ions o f min imum p rob lems wh ich were s ta ted in Theo rem 1.10.2.

W e a s s u m e t h a t

(i) f{x, Z y p ) satisfies (4-1.1); and

( i i ) there ex is t numbers w > 0a n d Ks u c h t h a t

(4 .3 .1) / (^ , z,p)>mV'' — K for all {x , z,p), F = (1 + \p\^)^l^.

Lemma 4 .3 .1 . Suppose ^^H\{^), k'>\, and is itsaverage over ^ .

Then there is aconstant C = C{y, k) such that

/ { i f ( a ) - ? | 2 + | V a C ( ( T ) | 2 p / 2 ^ 2 ^ ( ^ ) < C / ' | V a C ( o r ) | ^ ^ 2 ' ( ( 7 ) .

Proof [k a n d vf ixed) . If th is is not so, 3{^'n] € ^ f c ( ^ ) s u c h t h a t

u z

O n ac c o un t of the homogene i ty , we may assume the le f t s ide = 1 for

e v e r y n. Since the H\ n o r m s of the func t ions f — Cw ar e all 1, a s u b

s equence —y some fo in Hl{^). B u t t h e n I^n — Cn t ends s t rong ly in HK )

to Co s ince VaZn-^Oin Ljc{y^).Thus f 0 i s a con s tan t , which m us t be 0 s ince

i ts in tegra l over ^ is0 . Bu t th i s con t rad ic t s the fac t tha t \\Cn ~ CnWl^u

= 1.

Theorem 4 . 3 . 1 . If f satisfies (i) and (ii) above and also

(4.3.2) f[x,z,p)^MV-

and if z minimizes I{z, G) among all vectors in Hl[G) having the same

boundary values, then z is Holder continuous on interior domains and

satisfies

(1.10.9) { SV^dx\^l'^C^[\Vn'{rlRY if ^ < ,

where C and [JL depend only on m, M , K, and v.

Proof. The hypotheses (4-3.1) and (4.3-2) imply that

(4.3.3) ^ Ipl" — K <f(x,z,p) <Mi\p\ + Ml, Ml =2"-! .

Let us define

(4.3.4) (p[r) = flpl'dx, Br== B{xo, r).

Using (4.3 .3) and the f a c t t h a t z is m i n i m i z i n g we f ind tha t

(4.3.5) m(p{r) < I(z, Br) + K\Br\^ I{Z, Br) + K\Br\

< ( K + Mi) \Br\ + MiJ \\7Z\'dxBr

w h e r e Z is any func t ion = ^ on dBr.

N o w , for a lmos t eve ry r, z^Hl{dBr) and is there fore essen t ia l ly

c o n t i n u o u s . For s uch r, let

Z{s,a) = z + (s/r) [z{r, a) — z]

(s, a) be ing "s phe r ica l coo rd ina te s " , a be ing on dB(0,i) and z be ing the

a v e r a g e of z{r, a) ove r dB{0,\), Since

1 V 2 |2 = Zf + 5-21 VaZ |2 = r-^{iz(r, o) - z]'^ + \ Vaz{r, o) |2},

we f ind th a t

J\\/Z\''dx==v-^f{[z{r, G) - ^]2 + I "Vazir, o) l^Yl^d^^Br dBir,a)

(4.3.6) < C{v) f I Vaz{r, o) \'d2; < Crcp'[r)dB (r, a)

u s i n g L e m m a 4.3-1 • From (4.3-5) we c o n c l u d e t h a t

(4 .37 )(p(r) ^Ar (p'{r) + B r\ A = Mi/w, B = yv[K + Mi)lm.

From (4.3.7) and the a b s o l u t e c o n t i n u i t y of (p it fo l lows tha t

q^(r) < (rlR)^iA(p{R) + j ^ - {T^'ARV-HA „ v)

(4.3.8) fV'dx^ 2 " - ! [yvT'' + ^(r)]

f rom which the result (1.10.9) easily follows. The H O L D E R c o n t i n u i t y

follows from this and Theorem 3-5 .2 .

Theorem 4 .3 .2 . Suppose k = v = 2 and f satisfies

(1 .10.8*) m\p\^ <f{x,z,p) <M\p\^, 0 < w < M" ,

instead of (4-3-1) ^'^^ (4.3-2). Then if G is bounded hy k> \ disjoint

J O R D A N curves andz*^H\(G) and is continuous on G, then any minimizing

function z with z — 2:* ^ H\Q[G) is continuous on G.

Proof. Since (1.10.8*) implies (4-3.1) and (4.3.2), the c o n t i n u i t y on

the in te r ior fo l lows f rom Theorem 4.3-1. If, in the a r g u m e n t in (4.3.5) we

rep lace Br by an a r b i t r a r y d o m a i n gC.G, we conclude, us ing (1.10.8*)

t h a t

(4.3-9) D{z,g)<.{Mlm)D{Hg,g)

Hg d e n o t i n g the ha rmon ic func t ion co inc id ing wi th z on dg.

N o w , let Po^dG; s uppos e it is on the b o u n d i n g c u r v e F, liFis the

o u t e r b o u n d a r y map the in te r io r of F conformal ly (us ing the R I E M A N N

m a p p i n g t h e o r e m , see for exam ple C hap te r 9) on to a d o m a i n G' in the

4.3- The borderl ine cases k = v 107

Upper ha l f -p lane so t h a t PQ c o r r e s p o n d s to the origin, a p a r t of dG

c o r r e s p o n d s to aa and a p a r t of G c o r r e s p o n d s to Ga. If JT is an i n n e r

b o u n d a r y , the s ame re s u l t may be o b t a i n e d by f i rs t performing an in

ve rs ion wi th re s pec t to a po in t in te r io r toF. The t r ans fo rmed func t ion

'z* is c o n t i n u o u s on 'G a n d ^ / f | ( ' G ) and the t r ans fo rmed func t ion 'z is

c o n t i n u o u s in 'G, ^H\{'G), satisfies (4.3-9) for s u b d o m a i n s '^ C 'G, and

'z — 'Z*^HIQ('G); all these things follow from Theorem 3-1-5, the in-

v a r i a n c e of the D I R I C H L E T in teg ra l unde r con fo rma l mapp ings , and an

a p p r o x i m a t i o n .

L e t us e x t e n d '^*, 'z and 'w = 'z — 'z* by the fo rmula s

'z*(x^,x^) = 'z*{x'^, —x^)y 'w{x^,x^) = —'w(x^, —x^),

(4.3.10) 'z{x^, x^) = ' ^ * ( A ; 1 , X^) Tf- 'w(x^, x^)

w h e r e 'G~ is the re f lec ted reg ion . From Theorems 3- l -2g , 3-1.8 and its

co ro l la ry , it follows that '2:*, 'z, and 'w^H\i^G{i) and '2:* is c o n t i n u o u s

on 'Go = 'G U 'G- ('Go = 'G\J 'G~ U p a r t of x^ axis on d'G). Let (p be

a mollifier and let ' ZQ, etc., be the mollif ied functions . Now, for each R,

Q <,R <C a, t h e r e is a set of r of pos i t ive meas u re be tween Rje and R

s u c h t h a t i^z is a l r e a d y A.C. in a for a l m o s t all r, see the coro l la ry to

Theorem 3 .1 .8) 'z[r, a) is A.C. in c w i t h 'za^, a) ^ L2 and

2 jr R 2 7 1

f\'zo(r,G)\^dG<2 f fr(\'zr\^ + r-^'za\^)drdo(4.3.11) 0 ' " BH ^ ^ ' '

<.2D{'z, BR)<e^(R)l27z.

U s i n g the m e t h o d of proof of Theorem 3-1-8 , we conc lude tha t we may

choose an r satisfying (4.3-11) and also the c o n d i t i o n t h a t %(r, a) con

ve rges un i fo rmly to 'z{r, a) and %,ty(^, o) -^ '2,a{r, a) in L2 for a s equence

of Q ->0. And, on a^, 'ZQ(X^, 0) = 'ZQ{X^, 0) and so converges uniformly

in x^ to ' 2 * ( A ; 1 , 0).

N o w , let us choose an a r b i t r a r y p o i n t P i in G^ and m a p Gy conf orma l ly

o n t o 5 ( 0 , 1 ) so t h a t P i co r re s ponds to the orig in . Then the t r a n s f o r m e d

func t ion "Z Q is c o n t i n u o u s on 5 ( 0 , 1 ) , ^ H\{Bi), "Z Q -^"Z in Hl{Bi) and

' ^ ( 1 , 9 9 ) c o n v e r g e s u n i f o r m l y on dB{0,\) to a func t ion "C{^) w h i c h is

the transform of';2:(r, a) on the pa r t o f^J 5 (0 ,1 ) co r re s pond ing t o ^ V ^^

'z*{x^, 0) on the p a r t of dB{0,\) c o r r e s p o n d i n g to Cf. It follows from

T h e o r e m 3,4.5 t h a t "^ has the con t inuous bounda ry va lues "C . Moreove r2n

(4.3.12) Osc "C < oj{B) + / I 'za{r, a)\da<. co{R) + s(R)

0us ing the SCHWARZ ine qu a l i ty ; he re co (P ) is the osc i la t ion of -2:* a long OR.

F i n a l l y , "z still satisfies (4.3-9) for s ub reg ions " g C 5(0 ,1 ) wh ich we

shall now take as c irc les Br = B{0, r). R e p l ac i n g Hg b y Z as in (4.3-5) we

o b t a i n

^(r ) = D{"z, Br) <. C' (MIm ) r (p'{r) (C = C(2))

f rom wh ich we conc lude tha t

cp{r) < ^(1) y i /^ , ^ (1) < e^(R)l27z.

S e t t i n gr 2n

ip[r) = J J s i / 2 | "zs{s, (p)\dsd(p,0 0

r 2j r

^{r) = J J\ "zs (s, G) I ds do (L == Mjm)

0 0

we find, success ivelyy )^{r)<27 ir(p(r) <^ e^(R)r^+^l^, \p{r) < e{R)r(^+^)J^i^

r

y(r) = js-^l^ y)'{s)ds = r-'^j^xp{r) + ^ / s-^l^y){s) ds

0 6<e(R)'{L + 1)y i /2L .

Thus we see tha t (s ince "z i s cont inuous ins ide)

<X{^)<iL+^)e(R).

Using (4.3 .12) and(4.3-13) we see that \"C{(p) — "z(0)\ < e if i^ is sin al l

enough. S ince Pi w a s a r b i t r a r y , t h e c o n t i n u i t y a t PQ follows.

Before deve loping the no t ion of a cont inuous func t ion monotone in

the sense of LEBESG U E, which was def ined in the in t roduc t ion , we in

t roduce the fo l lowing nota t ions and def in i t ions :

Nota t ions . For sca la r func t ions u continuous on G, we define

M(u, S) = m2ixu{x), m{u, S) = m i n ^ ( ^ ) ,xes xss

S c o m p a c t C G

(o{u, S) = M (u, S) — m{u, S)

ju+{u, D) = M[u, D) - M(u, dD) ]

^ a domain C G.,dD) 1

fji~{u, D) == m{u, dD) — m{u, D)

Definit ion. For a scalar function u con t inuous on G and a rea l numberw, we define T+{u, w) as tha t func t ion U s u c h t h a t U{x) =^ w w h e n e v e r

%^ a dom a in D for which u[y) '> w for all y in D and u{y) = w on

dD a n d U(x) = u{x) otherwise . We def ine T-(u, w) as tha t func t ion U

s u c h t h a t U{x) ~ w w h e n e v e r x^ D on wh ich u{y) < w, e t c .

The reader can eas i ly ver i fy the fac ts s ta ted in the fo l lowing lemma:

4.3 . The borderl ine cases k= v 1 0 9

Lemma 4 .3 .2 . Suppose u is continuous on G, w is a real number, and

U = T+ {u, w). Then

(i) U{x) < u[x) for x^ G and U {x) = u(x) for xf^ dG;

(ii) oj(U, S) < (jo{u, S) for every continuum S G G;(iii) fi+iU, D) < iA+{u, D) and fj.-{U, D) < ^~{u, D)

for every D G G;

(iv) ifM{U, dD) < w, then M(U, D) < w.

Corresponding results hold if U == T~ {u, w).

For ( i) is obvious and the others follow easily from the fact that the

set of X w h e r e U(x) < u(x) is an open set Q G G which is the union of

d o m a i n s D on each of which ^i(x) > w w i t h u{x) = w on dD] U{x) — w

on Q.

T h e o r e m 4 .3 .3 . Suppose f satisfies (4.1.1), (4.3-1), ^^^ the supplemen

tary conditions

fix, z, p) > 0, f{x, z, p^, . . ., pr-^, 0, ^^+1, . . ., p^^) = 0,

(1.3.10) r = 1 , . . . . A', ^* :^ { j }, 1* f ixed.

Suppose z is continuoiis on G and ^Hl(G). Then there is a function

ZQ^ Hl(G) which coincides w ith z on dG, for which I(zo, G ) < I{z, G), and

of which each component is monotone in the sense of LEBESGUE.Proof. W e s ha l l s how tha t we may rep lace each componen t z'^ in

tu rn b y a func t ion 4 wh ich is m on oto ne in th e sense of LEBESG U E, r e d u c

ing the in teg ra l a t e ach s tep . W e beg in wi th z^ .

For each n and each i, 1 < ^ < 2 ^ - i, w e defin e z£' , +z\^y and +2:^ b}^

induc t ion as fo l lows :

wl^ = M — 2-'^'i-{M — m), M == M(z^, G ), m =-m{z^,G)

Z i = -L (Z , ^ i i ) , Z ^^ = Z^^2n-i*

Cl ear ly e ac h +.'r is co n ti nu o u s on G. We note tha t i f D is a domain in

w h ic h M ( + 4 , a Z ) ) > < + i ^ „ t h e n + 4 +1 .1 W ^ •*"4W < ^ for x^D.

Also, if D is a dom ain in which 5^^+1,2 < -^ ( ^ 4 + i . i ' ^ ^ ) ^ '^ i - i . i '

t he n M (+ 4 + i . i ' ^ ) ^ ^w+ia (o therwis e H an ^0 in D where •^^+1,1(^0)

> ?£^i-i , i and hence a sub-domain A in wh ich '^4+1.1 (A:) > ^^+1,1 w ith

th e equa l i ty on th e bo un da ry ) . T hu s , if Z) i s a dom ain in wh ich

^ ( + 4 + 1 . 1 . dD) > <H, i .2 , th en / .+ (+ 4+ i . i ' ^ ) ^ 2-n-i{M - m).

N o w , s u p p o s e w e h a v e p r o v e d t h a t

/<+(+4+i.v ^ ) ^ 2 - ^-1 ' [M — m) w h e n e v e r

(4.3 .14) M ( + 4 + i .„ az )) > < + , . ,+ !( ^ < 2^+1 - 1 ) .

Then s uppos e M(+ zl+^ i+^, dD) > w++^ ^^^^ Th_en, from (4-3.14), we

c o n c l u d e t h a t M(+ zC,^,^,, D) <M {-^zl^,j, D) < M(+zl^,^,, dD) +

+ 2-n-i(M - m); _an(i also if w^^^^,^^ < M{+zl+,^,+,, dD) < < + i . , + i ,

t h e n M ( + 4 + i ^ ^ + p D) < ^^++1,^+1. T h u s we see that (4 .3-14) holds with i

rep laced by ' + 1.

T h u s it fo l lows tha t , for each n, +^i+i(^) < '^z\[x) for x^Gy the e q u a l i ty ho ld ing ii x^dG. I t is a l s o t rue tha t the -^z\ are e q u i c o n t i n u o u s on

Q ( p r o p e r t y (ii) of the l e m m a ) and, from (4.3.14) for i = 2^+ — 1, it

fo l lows tha t

/^+(+'ii, D) < 2-^(M — m) for e v e r y D C G.

H e n c e the +zl conve rge un i fo rmly to a func t ion +z^ for w h i c h

(4.3.15) f^+{+z^, D) = 0 for e v e r y D C G.

O b v io u s ly the i n t e g r a l is r e d u c e d at each s tep and from the lower-semi-

c o n t i n u i t y we c o n c l u d e t h a t it is r e d u c e d if z^ is rep laced b y +z'^. T h e n ,

s t a r t i n g w i t h +z^, we may form the func t ions ~zl^ and ~^^ in the ana logous

way us ing T" and s e t t i n g

w^j^ — m -\- 2~^ ' i • [M — m).

F r o m the l e m m a it fo l lows tha t iJi' {~z\, D) = 0 for each D and n so t h a t

t h e ~zl conve rge un i fo rmly to th e des i red func t ion z^ w h i c h is m o n o t o n e

in the sense of L E B E S G U E , since

^ - ( ^ 1 , D) = iu+(zl ,D) = 0 for eve ry D C G,Theorem 4 .3 .4 . Any family of functions z ^ Hl(G), each of which is

continuous on G and monotone in the sense of LEBESG U E, for which

IIV-^IIJ,^ is uniformly hounded ^ is equicontinuous on each D CC. G. If G

is Lipschitz, each z is continuous on G, and the functions are equicontinuous

along dG, then the functions are equicontinuous on G.

Proof. Since w[z,B{x{i,r)'] = aj[z, dB{xo,r)], we see t h a t the l a t t e r

func t ion is non-dec rea s ing in r. F r o m T h e o r e m 3.5.2 for ^ 5 ( 0 , 1 ) w i t h v

rep laced by v — 1, it fo l lows tha t

{co[z, B{xo, S)] Y < {co[z, dB{xo, r)] f

(4-3.16) ^Clj\Vaw(r,a)\'d2;,

6 <.r -< a, B(xo, a) C G, (w{r, a) = z{x))

for a lmos t r. Multiplying (4.3-16) by r-^ and in teg ra t ing f rom ^ to a yie lds

a

{o)[z, B{xo, d)]yiog(ald) < Cl J ar--^ f \Vaw{r, G)\''d2J dr

a

<Clf / / ' " - i p Wr{r, o) |2 + r-^\Vaw{r, G)\^]''I^ dr d^^0 dB{Xo,r)

{ 4 . 3 . 1 7 ) ^ ( G | | V 2 | | » . « ) "

wh ich p roves the f irs t result .

4.3- The borderline cases k = v 111

S u p p o s e , now, t h a t G is LIPSCHITZ, Z is c o n t i n u o u s on G, and z is m o n o

t o n e . Let xo^dG and m a p a n e i g h b o r h o o d 9^ of XQ o n t o Ga by a bi-LiPSCHiTZ

m a p X = x{y) so t h a t XQ is carried int® the orig in . Let yi be any p o i n t on

aal2, let R — ajl, le t e > 0, let Q{<R) be so s m a l l t h a t the oscil la t ion of

C(C(y) = ^[^{y)]) a long the p a r t of aa w i t h |y — y i | < ^ is < e/2, le t

(3 < ^, and suppose that a)[C, G-(yi, (3)] > e. T h e n

COLC, dG{yi,r)] > g, d <r ^ R ,

s o t h a t

(4.?.i8) co[C, 2'(n> ^)] > £/2, a < / < ^, (2'(yi, ) = G « n ^^(yi, r).

Then , p roceed ing as in (4.3-16) and (4.3.17), we f ind tha t

(4.3.19) {el2Ylog(QlS) < ( G l v C r r ™ ) ' < {K4^^UY

w h e r e Kv depends on ly on G and v. T h u s if the z are e q u i c o n t i n u o u s

a long dG, we may first choose Q s m a l l e n o u g h , i n d e p e n d e n t l y of z, and

then, us ing (4.3-19), choose d s ma l l enough so t h a t co[^, G{yi, 6)] < e.

R e m a r k . I t is eas y to see tha t Theorem 4 .3-4 . ho lds for vec tor func

t i o n s z if we i n t e r p r e t

aj{z, S) = sup|2:(:\:2) — -^(^i)!

and def ine a con t inuous vec to r func t ion to be m o n o t o n e in th e sense ofL E B E S G U E if and only if

a)(z, D) = co(z, dD) for e v e r y D C G.

F r o m T h e o r e m s 4.3-3 and 4-3-4, we eas i ly deduce the following

T h e o r e m :

Theorem 4 .3 .5 . Suppose G is Lipschitz, z* ^ Hl(G), z* is continuous

on G, and f satisfies (4.1.1), (4.3-1), ^'^d (1.10.10). Then there exists a func

tion z-^ Hl[G) which is continuous on G and coincides with < * on dG and

minimizes I(Z, G) among all such Z. If z"^^ Hl{G) and is continuous on

G, there is a similar function z which minimizes I{Z, G) among all similar

Z such that Z — z"^ ^ HIQ(G).

Proof. In each case , let {zn} be a min imiz ing s equence . In the first

case , each Zn may be rep laced by a zon, in w h i c h e a c h c o m p o n e n t is

m o n o t o n e , and the {zon} form an equ icon t inuous min imiz ing s equence .

I n the second case , we choose an e x p a n d i n g s e q u e n c e {Gn} (G n C Gn+i)

of LIPSCHITZ d o m a i n s s u c h t h a t U Gn =' G; and rep lace Z n b y a func t ion

zon s u c h t h a t zon{x) = Zn{x) ioTX ^ G ~ Gn and zo{x) is m o n o t o n e on Gn-The re s u l t ing min imiz ing s equence is e q u i c o n t i n u o u s on each D Cd G

by Theorem 4-3-4-

R e m a r k s . If is c lea r tha t theo rems l ike the main ex i s tence Theorem

1 .9 -1 , invo lv ing va r iab le bou nd a r y va lues can be p r o v e d . We l e ave the i r

f o r m u l a t i o n and proof to the r e a d e r .

4.4. The g e n e r a l q u a s i - r e g u l a r i n t e g r a l

In th is sec t ion , we s tudy in tegra ls in w h i c h / i s of class C^ and satis f ies

the general L E G E N D R E - H A D A M A R D condition

(^•5.5) fviv^^ {^> z> P) ^ - h ^' ^' > 0 (all X, z, p, A, I) .

W e s ha l l be i n t e r e s t e d in the way s uch func t ions va ry wi th p, so we sha l l

s u p p r e s s the a r g u m e n t s x a n d z for the p re s en t . W e beg in by genera l iz ing

such func t ions s l igh t ly as follows:

Definit ion 4 .4 .1 . We say t h a t a func t ion (p[p) is quasi-convex'^ if and

on ty if q){pi, + Aa I*) is convex in X for each p and | and c o n v e x in | for

each p and X.

Lemma 4 .4 .1 . Suppose (p is convex on an open convex set S. Then cp is

continuous on S.If\(p{^)\ <, M on S, then cp satisfies a uniform LIPSCHITZcondition with constant 2 Mjd on S^. If (pn s convex on S for each n and

<Pn{S) — ^9^(1) for each f, the convergence is uniform on each compact subset

of S. If cp is convex on the interval [a, h] in R\ and a <C c <C d <C b, then cp

is bounded on [a, b] by a number which depends only on a, b, c, d and its

values at these points.

The proof is left to the r e a d e r .

Theorem 4.4 .1 . / / / is quasi-convex everywhere, it satisfies a uniform

LIPSCHITZ condition on any bounded part of space. If p is given, there exisconstants A°- such that

(4.4.1) f{K + ^o.^^)>fm + A^X.^^ f o r a l U , f .

/ / / is also of class C^, then A'^ = fpo (p). Iffis of class C^, then f is quasi-

convex if and only ^/ (1.5.5) holds.

Proof. If is c lea r tha t if / i s quas i -convex , it is c o n v e x in each s ingle

Pi for f ixed va lues of the o t h e r s . T h u s it is sufficient to p r o v e t h a t a

func t ion (p{i^, . . ., i^) which is convex in each i^ satisfies a un i fo rm

LIPSCHITZ cond i t ion on e a c h h y p e r c u b e RQ : \^P\ < M. Let R d e n o t et h e h y p e r c u b e 112? | < M + 1. In the case P = 2,(p, being one d imens ion-

a l ly convex , is c o n t i n u o u s on dR and dRo and is b o u n d e d t h e r e by a

n u m b e r Q. H e n c e cp satisfies a un i fo rm LIPSCHITZ cond i t ion in f^ and i^

w i t h c o n s t a n t 2 Q on RQ. In th e case P = 3, we s im ilarly g et (p c o n t i n u o u s

on each face of dRo and dR. and b o u n d e d by a n u m b e r Q on the p a r t

w h e r e the v a r y i n g f 2> satis fy | f ^ | < M. Then aga in we o b t a i n the re s u l t

t h a t (p satisfies a LIPSCHITZ cond i t ion in each ^^ of b o u n d 2 Q on RQ.

T h e r e s u l t m a y be p r o v e d by i n d u c t i o n .

T h e l a s t s t a t e m e n t is obvio us (us ing L em m a (1 .8.1)) . It is clear (from

the def in i t ion of c o n v e x i t y ) t h a t any mollif ied function (pg is a lso quas i -

1 The terminology here is slightly different here than in the writer's paper

(MoRREY [9]) where functions (p satisfying this condition are called weakly quasi-

convex; the functions (p which we call strongly quasi-convex below were called

quasi-convex in the papers cited.

4.4. Th e genera l quasi-re gular integ ral 113

convex and ^ C°^ and so satis f ies (1 .5 .5)- From Lemma 1.8.1 , i t fol lows

t h a t

S ince 99 satis f ies a uniform LIPSCHITZ cond i t ion nea r p, we may choose as e q u e n c e of ^ - > 0 s o t h a t t h e n u m b e r s (fgp^ip) t end to l imi t s A'j, t h u s

verifying (4-4.1).

W e now ob ta in a necessary cond i t ion fo r lower - s emicon t inu i ty :

Def in i t ion 4 .4 .2 . We say tha t z^ couve rges Lipschitz to z ofi G if and

only if Z n conve rges un i fo rmly to z a n d t h e Z n a n d z satis fy a uniform

LIPSCHITZ cond i t ion (wh ich may depend on the s equence ) .

R em arks . I t is c lea r th a t th e LIPSCHITZ convergence of Z n to z on G

i m p l i e s t h a t Zn —7 zin H] (G) for any s > 1 b u t does no t im ply the s t ro ng

convergence of Z n t o z i n a n y Hj {G).

Theorem 4 .4 .2 . Suppose I{z, G) is lower-semicontinuous with respect

to this type of convergence at any z on any G and f is continuous. Then

(4.4.2) / / [ ^ o , -2:0, po + V C{x)] dx > / ( % o , - 0, pa) ' 'yniG)G

for any constant (xo, zo,po), any bounded domain G, and any LIPSCHITZ

vector C which vanishes on dG.

Proof. L e t XQ be any po in t , R be the cell XQ<,X°'^XQ-{- h, ZQ b e

an}^ vector of c lass C^ on RU dR, Q b e th e cell 0 < .v" < 1, a n d f b e

an y vec tor wh ich sa t isf ies a un iform LIPSCHITZ cond i t ion ove r the who le

spa ce an d is per iodi c of period 1 in eac h x°'.

For each n, define f7, [x) on R b y

CJ,(A;) = 7^-1 h CJ[n h-Hx ~ xo)].

T h e n t h e Ci tend to zero in our sense. Then, for each n, I(zo + Cn> R)

can be wri t ten as a sum of in tegra ls over the sub-hypercubes of R of side

n-'^ h. If r is one of these the integral over i t isn-'' h" Jf[xi + n-^ h | , Zn{xi + n-^ h | ) ,

Q

w h e r e

r ixl^x^" ^ x1 + n-^ h, x^ =• x^ + k^ n-^ h, 0 <k°' <n ~\

Zn{x) = Zo(x) + Cn{x), x^" = x'^ + n-^h^"", 0 < ^ < 1.

T h u s w e s e e t h a t

lim7(;?o + Cv, R) = / 1 / / [ ^ > M^)> Po{x) + V C(l)] d^\ dx > I{zo, R).^-^^ R [Q J

B y l e t t i n g ZQ a n d po be a rb i t ra ry cons tan t vec tors , se t t ing ^o(^) = ^0 +

+ poa' (x<x — XQ), d iv id ing by m(R) = h^ a n d l e t t i n g h -^0, we o b t a i n

(4.4.2) for G = Q an d C peri odic of per iod 1 in ea ch x'^. But i f G i s any

Morrey, Multiple Integrals g

bounded doma in and f van i s hes on dG, we may choos e a hype rcube Q'

of side t c o n t a i n i n g G a n d e x t e n d C{^) to be zero in Q' — G and of period

t in each x°'. Then a s imple change of var iab le ob ta ins the resu l t in

gene ra l .

Theorem 4 .4 .3 . / / / satisfies the conclusion of Theore m 4.4-2, then f is

quasi-convex (in p).

Proof. Clear ly we may suppress [XQ, ZQ). W e nex t no te tha t any mo l l i

f ied function fq aga in sa tis fies the sam e condi t ion s and ^ C^ . Th en i t

fo l lows f rom the der iva t ion in the in t roduc t ion tha t (1 .5 .5) ho lds for /^ .

I t t h e n fo llo w s f ro m L e m m a 4 -4 .1 t h a t / i s q u a s i - co n v e x .

Definit ion 4 .4 .3 . If / sa t isf ies th e conclusion of T he or em 4.4-2, we

s ay ihdit f is strongly quasi-conv ex in p\ ii f dep en ds o nly on />, we sa y

t h a t / i s s t r o n g l y q u a s i - c o n v e x .The fo l lowing theorem is in te res t ing . However , we sha l l no t p rove i t

here . The proof i s much l ike bu t s impler than tha t o f the more genera l

lower-semicont inu i ty Theorem 4-4 .5 be low. A proof i s to be found in the

wr i te r ' s pape r re fe r red to above ( M O R R E Y [9])-

Theorem 4.4.4. If f is continuous and strongly quasi convex in p, then

I(z, G) is lower-semicontinuous with respect to LIPSCHITZ convergence.

W e wis h now to p rove a lower s em i-con t inu i ty th eo re m w h ich invo lves

weak conve rgence . The theo rem i s p roved a f te r fou r l emmas .Definit ion 4.4 .4 . Suppose C ^Hj (G) and suppose i^ is a cell with

RG G. T h e n C is said to be strongly in Hl{dR) i f and only i f the boundary

va lues B C, as d efin ed in § 3.4, ^ Hl(dR) an d a lso there is a sequence {Cn}

of class Ci on R s u c h t h a t Cn-^C^ri Hl(R) and f^ - ^ J B f in Hl(dR).

R em ark . In th i s in s tance a rep re s e n ta t ive of th e bo un da ry va lues

BC is ob ta ined by ju s t t ak ing the re s t r i c t ion to ^ i ? o f an abs o lu te ly con

t i n u o u s r e p r e s e n t a t i v e £ , of f.

L e m m a 4.4.2. Suppose f ^ Hl(G). For each oc, i < oc <v, let {a°', h°')

he the open projection of G on the x"- axis. Then there exists sets Z °^ of

measure zero stich that if R : c'^ <^x'^ -< d°^ (oc = \, . . .,v) is any closed

cell in G with

C^f {a°', h"") — Z « a n d ^ « ^ [a^", h^") — Z « , ex = 1, . . ., r

then f is strongly in Hl{dR).

Proof. This is proved using the method of proof of Theorem 3-1-8.

L e m m a 4 .4 .3 . Suppose R is a cell with edges 2h^, . . ., 2 A" and

center XQ. Leth = m inh°', L = h-'^{h°' • /i«)i/2

Suppose also that 0 <C k < h, that f * * Hji^) for some D 3 R cind f * is

strongly in H][dR) with

U*tsR<k, | | V C * | | ? . < , B < M ( s > 1 ) .

- I °

'' ^^ ~ | \X*(X) -Xo\~K\x-:

4.4. Th e genera l quas i - regu lar in tegra l 11 5

Then there is a function C€ Hl{R) which coincides with C* on dR, iszero except on a set of measure

m[R) • [1 - (1 - h--^ ky],

and satisfies/ | V C | ^ ^ A ; < C - ^ - ( 1 +MS)

R

where C depends only on s and L.

Proof. For each x ^ R, x 4^ XQ, let x * (x) be the intersection of the

ray XQ X with dR, and for each x^ R define

(x = XQ)

Xo\ {x =?^ Xo)

Let U^ be the pyramid in R with vertex XQ and base the face F^ where

^« = x^ ± A«.

O n the pyram id J J ^ , introduce coordinates |^, . . ., 1""^, r by

x"" = xl -{- r h"", xy = xy + r ^y (0 <r <\, y = i, . , ,, V — i).

Then, if r and ^y are considered as functions of x, we have

r(x) - r, x*{x) = [|1(A;) + X], . , ., ^^-^x) + x^-^ h^ + x-].Similar coordinate systems may be set up on each of the other jfj ±.

Definer 0 ( 0 < r < 1 — ^ A-i) ,

^^^^ ^ I hk-^{r- 1 + ^ A - i ) (1 •~kh-^<r< 1).

Choose a sequence f* satisfying the conditions of Definition 4.4-4; andfor each n, define

^n{x)^<p[r(x)]-C:[x*{x)l

The each Cn(x) is LIPSCHITZ on R.B3' computing the derivatives of f« in JJi^, using the SCHWARZ in

equality and the fact that each | | ' ' | ^ hv, we find that

I V f „ W |2 < (1 + 2L2)^-2 ^^r) I VC:(f)|2 + 2(A-)-2 9,'2| C*(|)|2

Using the facts that <p(r) = (p'(r) = 0 for r < 1 — /j~i ^ and that

we obtain

y- i ^(r) < 1, (p'(r) = k-^h, 1 — A-i ^ < r < 1, / j ' ' > h.

l\VU\'dx<C i{s,L)kf[\vCS\' + k-s\C^\s]dS.

8*

Also

f\Cn\'dx:=fh-r^-^(p^(r)drf\CS\sdS<LkJ\Ct\sdS.n; i-fc/i-^ F; F;

Adding these results for all the J J ^ . we obtain th e result for each n; andalso ||C»||s,E is uniformly bounded. Thus, we may extract a subsequencewhich tends weakly in H]{R) to some function I^^H][R). Since eachCn = C* on dR, C* tends strongly in £§ to ^* on ^i^ and l^ — t,^ ondR.From the lower semicontinuity, the result follows.

Lemma 4.4.4. Suppose f is strong ly quasi-convex and suppose for allpi and p2, that

\fip2) -f{pl)\ ^K(\Pi\s-l + \P2\^-1 + 1) \P2-Pl\.

Then ifpo is any constant vector^ D is any hounded domain, andC^ HIQ(D),it follows that f[pQ + n[x)'] is summable over D and

fflPo + 7t{x)] dx > m{D) -fiPo).D

Proof. There exists a sequence of functions Cn, each of class C^ on Dand vanishing on and near dD, such tha t C^ - > f in ^ J (D). For each nand almost all x on D, we have

\f[pO + M^)]-f[Po + 7 l(x)]\

(4.4.3) <K[7 Z n(x) - n{x)] • [1 + \P o + nn{x)|^-i +\Po + n{x)Y-^ .

Using the H O L D E R inequality, and so on, and the strong convergence in/ / i p ) , we see that

lim (f[p{s + nn{x)] dx = / / [ ^ o + n[x)] dx.

Since / is quas i-convex, the result follows.Lemma 4.4.5. Suppose that f satisfies the hy potheses of Lemma AAA-

Suppose also that each Cn€ Hl[D) and is strongly in H l(dR) where R is acell with R G D, and that

C„^0 in LSR). ||C„[J.a«^M, ||C„IU^M.

Then for each po, f[po + 7Zn(x)] is sum mable for all siifficiently large n,and

liminf jf[pQ -I- Tinix)] dx > m{R) -fipo), < a W = Cix^^i^)-

Proof. Let K be the number in Lemma 4.4.4. For each n, let

kn = Un\\leE

and let h be the quantity of Lemma 4-4.3 for R- Since kn->0, wehave kn <.h for all n > some n\. For each such n, let r]n be the function of Lemma 4-4.3 which coincides on dR with f , an d let

Xn^ ^n — rin> 4 c . : = > ^ L ^ ' ^ 4 a = z l ^ ^ ' {^n =^ ^n + Mn) -

http://fip2/

http://fip2/

http://fip2/

4.4. The general quasi-regular integral 117

Then, s ince y ,^ — 0 on dR, we h a v e

f f[7to + con{x)] dx > m(R)f(po)-R

We a lso have (us ing Lemma 4-4 .3)

(4.4.4) IxnlU-^O. WnnWU^M, || o)« i|»B < M + || ;<„| |o ^.

A s in (4.4.3), we see t h a t , for each n, and a l m o s t all x on D,

\f[pO + COn(x) + >Cn{x)] — f[pO + COn(x)] \

< K ' \xn{x) I • {\Po + a)n(x) + >Cn(x)\^-^ + \Po + OJn{x)\'-^ + 1).

U s i n g the H O L D E R inequa l i ty (4 .4 .4) and so on, we see t h a t

lim f \f[po + 7tn(x)] —f[po + o)n{x)] \dx = 0,

f rom which the result follows.

Theorem 4.4 .5 . Suppose f is continuous in (x, z, p) and strongly

quasi-convex in p. Suppose also that there are numbers m and K, K y> 0,

such that

(i) f(x, z, p) > w

(ii) \f{x,Z,p2) -f{x,Z,Pi)\ <K[\ + [Pi\^-^ + | />2|^-l] • \P2 - Pl\

(iii) 1/(^2,^2,:^) -f{Xi,Zi,p)\ < i ^ ( l + | ^ | . ) [ | ^ 2 - ^ l | + \Z2-Zi\]

fo r all (x, z, p) with various subscripts.

Suppose also that Z n —7 ZQ {weakly) in Hl(G) and that either

(a ) each Zn and Zo are continuous on G and Zn converges uniformly to ZQ

on each closed set interior to G, or

(b) the set functions Ds(zn, e) are uniformly absolutely continuous on

each closed set interior to G, where

Ds(Zn> ^) = f \V Zn\^ dx.

e

ThenI{zo, G) < l i m i n f / ( ^ ^ , G).

Remark. If s = 1, weak conve rgence in G impl ies the h y p o t h e s i s (b).

T h e c o n d i t i o n s (ii) and (iii) are c lose ly re la ted to the co r re s pond ing ones

in (1.10.8).

Proof. We no te f i r s t tha t hypo thes i s (ii) impl ies

(4.4.5) \f{^,z,p) -f{x,z,0)<.K\p\{\ + \p\^-i).

Also , h y p o t h e s i s (iii) s imila r ly impl ies(4.4.6) \f(x, z, 0) - / ( O , 0, 0)1 < K{\x\ + | . ' | ) .

T h u s , for all [x, z, p), we h a v e

(4.4.7) \f{x, z,p)\ < 1/(0, 0, 0)1 + K{\x\ + 11 + 1 1 + 1 1).

There fo re I{ZQ, G) and the I{zn, G) are u n i f o r m l y b o u n d e d .

1 1 8 E x i s t en ce t h eo rem s

We sha l l se lec t subsequences {zrt} and fo rm g ra t ings &r as defined

be low^. We begin by se lec t ing a subsequence , s t i l l ca l led {zn} s u c h t h a t

I{zn, G) - ^ i t s f o r m e r liminf. For each oc, 1 < ^ < r , l e t (a°', h°') be the

open in te rva l p ro jec t ion of G on the %"• axis , we le t Z°^ be the un ion of

all the sets (for the Z n) in Lemma 4-4 .2 , and for each a, P, a n d {r, t)

(or n), we let ^^^p be the set of all x°'(i[a°'y 6*) — Z^" s u c h t h a t

ll^r«| |l ,c?(/) < - P , G{x'') being the set of a l l x'^ s u c h t h a t {x°',x'^)^G,

Su pp os e t h a t M is a unif or m b o u n d for || %||s,(? th er e be ing one on

accoun t o f the weak conve rgence . Ev iden t ly

|£^?«P I > ( ^ - a«) - M P - i .

No w we choose P i so th a t i lf P f ^ < (6i — a i ) /3 an d def ine

E v i d e n t l y \E\\^ 2{b^ — a^)l} so tha t there i s a po in t xl-^Êl w h i c h

i s in th e m idd le th i rd of the in te rva l {a^, b^). By def in i t ion , x\^ is in E^p^

for infinite ly many ;^; so we le t zn = z^ w h e r e {fit} are jus t these n

a r ranged in o rde r . Nex t , we choos e P2 s o tha t MP^^ < {b^ — ^ ^ )/3 a n d

define ^ ^

S=l n^ S

As before , there is an xl^ ( 7 t | an d in th e mid dle th ir d of (a^, b^) and wedefine Z 2t = înt where {%} are those in tegers for which ^ l i^^ iwtPa-

This p rocess i s cont inued unt i l oc = v. We def ine x'^^ = a"", x^^ = *»

a = 1, . . . ,v . N e x t, we defin e ^J+i^2i = î ,i ^^^ ' = 0, 1, 2, an d cho ose

Pv+i so large that MP~^^ < (1/5) of the length of the shorter interval

{XIQ, X\^) a n d {xl^, x\^). We def ine EIJ^^ = n U Îtp^- Then there is a

point :v;J+^ 1$ ^l+i and in the middle f i f th of the in te rva l [X\Q, X \ ^ . W e

c h o o s e 4 + i j = Zvn\ w h e r e n[ are those in tegers for which îiÊ]^'p^,

def ine '^ J+ i = 0 U £J ' p^, choose x]^^ 3 in the middle f if th of the inter-

v a l {x\^, x\^ and in TJ + j and de f ine Zvî,t =• 4+ i . r , whe re rt a re thos e

if su ch t h a t ^J+1 .3^ £ J, ' ,p , . T hi s pro ces s is re p ea te d for ^ = 2, . . ., i

u s ing the middle f i f th of each in te rva l (x'^^, x^^) a n d {x^^, x^^. T h e w e

define xl^^^^^^ = xl^^j, i = 0, . . ., 4, chooseP2V+1 so large that MP^}^^

< ( 1 / 9 ) th e len gt h of a n y of the i nt er va ls (:^J+ i ^_i, ^J+i,^) . Th is proc ess

is con t in ue d indef in i te ly tak in g th e m idd le 9 t h of each in t e rv a l fo r

^ = 2, . . . , r , th e n I /I 7 for oc — \, , . .,v, and in gene ra l the midd le1/(2^ + 1) of ea ch i n te rv a l. W e let @jt d en o te t h e 2*' cells w ho se faces

l i e a long hype rp lanes x"" = x"^^, ' = 0, . . ., 2^, r == (^ — 1) r + a . W e

now le t {zri] deno te the d iagona l s equence . Then i f R is any cell of any

1 This proc edu re is essent ial ly due to T ONE L L I (see T ONE L L I [8]). It is also similart o t h e e — d grat ing process used by L. C. Y O U N G ([1]).

4.4 . The general quasi-regular integral 119

(SA; for which R(Z G, we have each Z n s t rongly in Hl(dR) with ||^w||s.aiJ

un iform ly bo un de d and , of course (since th a t w as t r ue of th e or ig ina l

sequence) Zn->z in Ls(dR). M o r e o v e r , t h e q u a n t i t y L of Lemma 4-4 . ?

is uniformly bounded for a l l cel ls in any (^jc-

Now, we f i rs t cons ider the a l te rna t ive (a ) . Le t s be any pos i t ive

n u m b e r . F o r e a c h k, le t Djc be the union of all the cells of (^jc which a re

in te r io r to G. Since / i s bo un de d be low a nd I{zo, G) is finite, we first

choose ki s o l a rge tha t

(4.4.8) I{zn, G - Dkd > - els (w - 1, 2, . . .)

I{zo,DJc^)>I,{zoG)-e|5^

For this ^i , le t i^ i , , . ., Rqbe the cells of Djc^ and for each k>k\, le t

Rici[i= 1, . . . , ^ -2* ' (^-^ i ) )

be the cells of ^jc in Djc^. For each k, define x^{x), z^[x), p%{x) on Djc^

by def in ing them on each ce l l R of ©^ to be equal , respectively, to the

ave rage ove r R of x, ZQ[X), a n d PQ{X). Then, f rom hypotheses ( i i ) and

(iii), we conc lude tha t

\f[(x, z^{x), Po{x)] - f[x*(x), z* {x), p*{x)] I

(4.4.9) < K(\Po{x)\^ + 1) • (\x - xt{x)\ + \zo{x) - zf{x)\) +

+ K[]po(x) 1-1 + \pl{x) 1-1 + 1] • \Po(x) - ptix) I;

the method of proof is s imilar to that of (4 .4 .7)- If we le t

Cn = 2n — Z Q, jCn = pn — pO,

we see s imila r ly tha t

\f[x, Zo{x),po(x) + 7tn(x)] — f[x^ {x), Z^ (x), p* {x) + nn(x)]\

(4.4.10) <K(\Pn{x)\^ + 1) (1^ - xi(x)\ + \zo{x) - zl{x)\) +

+ K(\Pn{x) \s-i + \p^{x) + nn{x)\s'^ + 1) • \Po(x)~pn^)\>(4.4.11) \f[x, Zn{x),pn(x)] -f[x, Zo{x), pn{x)]\

<.K{\Pn(x)\^+ i)'\Zn(x) -Zo(x)\.

N o w , b y t h e H O L D E R inequa l i ty on each Rjd, we s ee tha t

(4.4.12) f \p^ {x)\s dx < I \po(x) \^ dx.

B y a p p l y i n g t h e MINKOWSKI inequa l i ty , we s ee tha t the in teg ra l s

(4.4.13) f\7ln(x)\'dx, f\pt{x) +7ln(x)\'dx

a re un i fo rmly boun ded . F ina l ly ,

(4.4.14) 1™ [\pQ{x)-pl{x)\^dx=:^0JJkl

H enc e , us ing (4 .4 .9) — (4 .4 .14) , we m ay choose a ^ so la rge th a t

(4.4.15) / l /[^ ,^oW , ^ o W ] - / K W , ^? M, A*W ]K^<^/5 ,

(4.4.16) f\f[x, Zo(x), Pn{x)] -f[x^{x), Z S(X), Pt(x) + 7tn(x)] \dx<8lS

(n = 1, 2, . . .) ,

and then choos e ni s o l a rge tha t

(4.4.17) / \f[x, Zn(x), Pn{x)] ~ f[x, ZQ[X) , Pn{x)]\ d X < e / 5 ,

vSince x^{x), z^[x), pf {x) a re con s tan t on each Rjdy i t fo l lows f rom Lemma

4.4 .5 tha tl i m i n f r / [ ^ J ( : ^ ) , zt(x),pl[x) + nn{x)']dx

(4.4.18) >:jf[4{^),zt{x),pt{x)]dx.

U sing (4.4 .8) and (4.4 .15)— (4-4.18), we see t h a t

l i m i n f / ( % , G) > I(zo, G) — s.

The result follows in this case .W e now cons ide r the a l t e rna t ive (b ) . Fo r each na tu ra l number q, w e

define

/ ^ ( A ; , Z , P) = [\ ~ aq{x, z)] f(x, z, p) + m • aq(x, z),

(0 (0 < i? < q),

^Q{X> ^) =\^{R~ q)^ ~2(R-q)^ (q<:R:Cq+ 1),

[ l ( i ^ > ^ + 1 ) , i ? = (|^|2 + |^|2)i/2.

I t i s easy to see th a t e a c h / ^ sa tis fies hypo the ses ( i)— (iii) w i th the sa m em and s ome Kq. Moreove r fq i s independent o f {x, z) for i? > ^ + 1,

and a lso

fq {x, z, p) < /g^.1 {x, z, p), lirnfq {x, z, p) = f(x, z, p).

Thus i t i s suff ic ien t to prove the lower semicont inu i ty for each q.

For a f ixed q, we no te tha t we may rep lace \zo{x) — z^(x) \ b y (pjc(x)

in (4.4.9) and (4.4.10) and \zn(x) — zo(x)\ by^^(%) in (4 .4-11), where

(pjc{x) == min( |^o(^) — z^{x)\, 2q + 2),

ipn{x) = mm {\zn(x) — zo{x)\, 2q + 2).

F rom the un i fo rm boundednes s o f the q)]c a n d ipn (q f ixed) , the un iform

absolu te cont inu i ty of the se t func t ion Ds(zn, e), and the fac t s tha t

\ivci(p]c{x) = 0, lim^^(% ) = 0

4-4. The general quasi-regular integral 1 2 1

a lmos t eve rywhere , i t fo l lows tha t the a rgumen t can be ca r r i ed th rough

as before for each fixed q.

Us ing s ubs tan t ia l ly the s ame proof, we can prove the following

theorem, the hypotheses of which a re c lose ly re la ted to the corresponding condit ions in (1 .10.7).

Theorem 4.4.6. Suppose that f satisfies the hy potheses of Theorem

4.4-5 wth conditions (ii) and (iii) replaced by

(ii') \f{x,z^,Pi)-f{x,Z2.p2)\

< K ( 1 + | ^ i| ^ -l + | ^ i | ^ - l + \Z2Y-^ + 1^2 -1)

{\Z2-Zl\ + \P2-P^\)

(iii') 1/(^2, z,p) -f(xi, z,p)\^ K{\ + \z\s + \p\s) . \X2 - xi\.

Then I(z, G ) is lower-semicontinuous with respect to weak convergence in

Hl(G). A corresponding theorem holds if f = f(x,p) and zi, Z2, and z are

omitted in (ii') and (iii ').

Proof. The right sides of (4.4-9), (4-4-10), and (4.4.11) can be replaced

re s pec t ive ly by

(4.4-9') K(\Po(x)\s + \zo{x)\^ +i). \x-x^(x)\ +

+ K ( \ + \ z o \ ^ - ^ + \ p o \ ' - ^ + i z ^ ' ' ^ + \ p n ' - ' )

' ( \ ^ o - z n + \ p o - p t \ ) >(4.4.10') K(\Pn(x)\s + | ô(^) |^ + \ ) ' \ x - x^(x)\ +

+ K{\ + \zo\^-^ + \Pn\'-^+ \zS \^-^ + \Pt+ 7tn\^'^){\z-Z ^\ + \Po-Pt\),

(4.4.11') K(\ + lôl^-^ + \Zn\'-^ + 2\Pn.\'-^) \Z n - ^ o | .

Inequali t ies (4-4.12), (4 .4 .13), and (4.4 .14) hold with s imilar inequali t ies

invo lv ing Z Q , Zny a n d z^. Also Z n -^ZQ in Ls(G). Cons equen t ly the rema in

der o f the proof may be carr ied over .

W e can now pro ve an ex is tence theor em s im ila r to Th eor em 1.9.1

f o r q u a s i - c o n v e x / .

Theorem 4.4.7. Suppose that f satisfies the hy potheses of Theorem Â-^

for some s > 1 or satisfies those of Theorem 4.4-5 "iîth s > v, condition (i)

being replaced in both cases by

(4.4.19) f[x,z,p)'>m\p\^--miy m > 0.

Suppose also that F* is a family of vector functions which is compact with

respect to weak convergence in H] [G) and that F is a non-empty family,

closed under weak convergence in Hj {G), such that each z in F coincides ondG with a z* in F*. Then I(z, G) takes on its minimum in F.

Proof. L e t {zn} be a min imiz ing sequence , suppose Z n — z^ = Wn

^ H\Q{G), Z ^ being in F * . Fro m (4.4 .19) and our hyp oth es is on F * , i t

follows t h a t | |V^?z ||? , l lV^*| |s , an d HV^^wi? are unif orm ly bo un de d.

F r o m P O I N C A R E ' S inequa l i ty , i t fo l lows tha t \wn^s i ^ un i fo rmly bounded .

Consequent ly , fo r a subsequence ot n, z^ —7 z* in F* a n d Wn—7W in

HIQ(G), SO tha t Z n z* + w in Hl{G) and z = z* on dG. If /

sa tis fies the h ypo the ses of Theo rem 4-4 .6, / i s low er-sem icont inuo us . I f

/ sa t is f ies those of Th eo rem 4-4.5, th en th eZ n

a re equ icon t inuous onin te r io r doma ins and we may a s s ume Z n converges un iformly to z on

such dom ains . Ag ain , / i s lowe r-semico nt inuou s and z i s min imiz ing .

Remarks. We have seen in Theorems 4-4-2 th rough 4-4-7 tha t i t i s

t h e strong quas i -convex i ty wh ich p lays the impor tan t ro le in the lower -

s emicon t inu i ty and ex i s tence theo ry . It is an unsolved problem to prove

or disprove the theorem that every quasi-convex function of p is strongly

quasi-convex. We now prove a genera l suff ic ien t condi t ion for s t rong

quas i -convex i ty and then p rove two theo rems g iv ing example s o f

spec ia l fo rms of / fo r wh ich quas i - con vex i ty imp l ies s t ron g quas i -con-vexi t}^

Lemma 4.4.6. Suppose v '>2 and ^ i , . . ., f""! ^ C'^{G), Then

Proof. This i s p roved by induc t ion on r . I f 1 = 2, (4-4.20) is ju st

wh ich is t ru e . So , suppose th e theor em t o be t r ue for 2 < r < X an d

th e n le t C^, . . . , C^ be fu nc tio ns of {x^, . . ., XK+^)^C^(G). Then, if we

call 5 the result ing sum on the left in (4 .4 .20), we obtain

5 = ^ ( - 1 ) «a = l

+ ( _ 1 ) K + 1

dx''2 ' ( - i ) ^ + ^^ = 1

aC c)(Ci, ^ - 1 ^ ^ ( S + l ^ rK \

- f 2 ; ( - i ^ ^ + ^ + ^

bg\.bxK+'^ b[x^.

bxK+1

C )

b{x^

, XK) =(-')•K+1 b 6i(Ci, ...,C^)

bxK+lb{x , XK ) +

ocp^i bx^bxK+1 b{x'^

^ = 1 bx^^^ a = l

., ;va-i, ;va+i, . . ., ;r^) +

C )d;i;* d( ;irl . A ^ a - l , A r « + 1 ,

the l a s t t e rm o f wh ich van i s hes by ou r induc t ion hypo thes i s . Bu t the

f i rs t two te rms a lso cance l as one sees by us ing the wel l -known method

of d i f fe ren t ia t ing a de te rminan t .

L e m m a 4.4 .7 . Suppose that the functions C , . . ., f , / / < r , satisfy a

uniform Lipschitz condition on G and that one of them vanishes on dG. Then

(4.4.21) CbJ^

J b{x^, . .r/')dx = 0.

Proof. S u p p o s e C^ = 0 on dG. Choose a large cell R c o n t a i n i n g G in

i ts in te r ior , ex te nd 4 to sa tis fy th e sam e LIPSCHITZ cond i t ion ove r the



4.4. Th e genera l quas i - regu lar in tegra l 123

whole space w ith C^ = 0 outside G, let 99 be a moUifier, and CQ he themollified functions. Clearly the integral in (4.4-21) formed with CQ converges to th at for f as ^ - > 0 , C = 0 on and near dR, and Ce€ C°°' So

we may as well assume f ^ C'^(R) and G = R. Then

B R a = l

dR a = l R a==l

where

^^d {x'^, . . . , ;i^ «-l , ; ^« +l , . . . , ;ir/ ) '

the last equality holding by G R E E N ' S theorem. But the boundary integral vanishes since C^ = 0 on dR, and the integrand in the second integral vanishes on R by Lemma 4-4.6.

Definition 4.4.5. A form

is said to be alternating if and only if the coefficient is zero unless all thea 's and all the i's are distinct and the interchange of two as or two i's

change its sign.Theorem 4.4.8. A sufficient condition that f he strongly quasi-convex

is that for each p there exist alternating forms

with constant coefficients such that for all n we have

f{p + n)>f{p) + Atni^--- + Aî::::-y 7^:; . . . TTI; .

Proof. This is an immediate consequence of the preceding lemma.Theorem 4.4.9. If the ajk are constants and

(4.4.22) f{p) = '^npipi

a necessary and sufficient condition that f be strongly quasi-convex is that

(4.4.23) aîL?i^m^Ô

for all X and | .Proof. If f — 0 on Z) *, we see from Lemma 4-4.7 with />« = 1 tha t

(4.4.24) jf[p + n{xy \ dx =f(p) m{G)+f afinU^) 7 ij(x) dx.G G

The condition (4.4.23) is just (1.5-5) and so is a necessary condition. Butif we introduce the F O U R I E R transforms (see VAN HO VE )

CHy) =^ {27i)-'"^fe-i^-yCHx)dx,

1 2 4 E x i s t en ce t h eo rem s

the integral on the r ight in (4 .4 .24) becomes

Re f a'î y^ y^ CH^ dy ^ 0— 0 0

if the condition (4.4-23) holds. Thus (4-4.23) is sufficient in this case.

Lemma 4.4 .8 . Suppose

m n

i = l ? = 1

for all X and y for which

Then there is a constant K such that

aij = Kbij (i = i,.. .,m; j = i, . . .,n).

Proof. W e m a y i n t r o d u c e n e w v a r i a b l e s f a n d rj b y

X = c ^, 3/ = drj,

c a n d d be ing nons ingu la r ma t r i ce s . Le t a a n d b be the matr ices of the

or ig ina l fo rms and A a n d B those of the t rans formed forms . Then

A = c' a d, B = c' b d (cl^ = Cji).

We sha l l show tha t there i s a sca la r K s u c h t h a t A — KB. W e m a ya s s u m e t h a t

Bii = \ {i =^ \, . . .,r)\ Bij = 0 o the rwis e , r ^m,n,

un le s s ^ = 0 in wh ich ca se . 4 = 0 a l so and the theo re m ho lds . B}^ t a k i n g

tjs = iîrjj z=z 0 (j z^ s, s = 1, . . ., n) in tu rn we s ee tha t

Ais = 0{i== i, . . .,m, s > r);Ais = 0(i^s,s=i,...,r,i= 1, . . . , m).

Th en , b y choos ing 1 < 5 < ^ < y an d se t t in g rj^ =.7 ]^ = \, nJ = 0,

j :^ s, j ^ t, w e h a v e(Ai, + Ait)î = 0 for all f w it h !« + f« = 0.

Thus the re ex i s t s a cons tan t K(s,t) s u c h t h a t

Ass + Ast = K(sJ), Ats + Att = K{sJ).H e n c e

Aii = A22 = ' " = Arr = K,

s o t h a t A = KB.

Theorem 4 .4 .10 . Suppose that N — v + \ and

(4.4.25) f{p)=F{Xi,...,Xrî),

where F is continuous in the Xi and

Thenf is strongly quasi-convex inp if and only i/Fis convex in theXt.

4.4. Th e genera l quas i - regu lar in tegra l 12 5

Proof. If F is convex in the Xi, it follows from Theorem 4-4.8 that/ is strongly quasi-convex in p.

Hence suppose / is given by (4.4-25) and is stronglv quasi-co nvexin p. If

AXjc = X^{pi + X.S^)-Xjc{pi),

then it is easily seen that

(4.4-26) AXjc^Xjcviâ^^'Also, since

p^XjcÔ (^ = i, . . . , r ) ,we have

(4.4.27) p^^X^j,i^=-d^fXi.

Now, choose a set of Xi not all zero and choose any p such that

Xt(p) = Xi.

Since / is strongly quasi-convex and hence quasi-convex, there are cons tan ts Ai such that

f{Pi + Li')^f{P) + At?i.i'.

Since / depend s only on the Xi, we must have

(4-4-28) AfXocS^ < 0 for all A, S with Xjcp^LS^ = 0(k=z \, ., ., V + 1).

O bviously, the n, t he equ ality m ust hold in (4.4.28). Using (4.4.26) and

(4-4-27), we see that(4-4-29) P^pAXjc=^ -MX iî), ^==\,.. . ,v).

XÂX^^DtX^î, Dt^XjcXjcvi-

Hence since the determinant of the coefficients of the A Xjc is not zero,

we must have(4.4-30) ^ -; ., f« = 0

for all X, I for which

(4-4-31) XiîÔ a n d Z ) ? A a f ^ - 0 .

Now, since not all the Xi are zero, assume Xjc 0. Then (solving for1* in terms of the |^ with i ^ k) we obtain

(4.4.32) Z {AtX,c-AtXt)l.î = 0

for all A, f for which

(4.4.33) Hm X^ - DlXi) Lî = 0 .i=|=fc

Fro m the preceding lemma , it follows th a t the re is a consta nt K such that

(4.4.?4) At X^ - At Xi = K(Dt X^ - Df^ Xi).

12 6 Differen t iabi l i ty of we ak solut ions

H e n c e

(4.4.35) A-,^KDt + L-Xi, L-== X^Â% - KD^).

From (4.4.29) and (4-4.35) it follows that

(4.4.36) AfX-î == KD^X^î + maXiî = CÂXj,,

C^={KXjc-L-pl).

Finally , i f we are given any values of the A Xjc, t h e q u a n t i t i e s

hi = p\AXic {i = \,, . .,v) a nd V i = X^cAXjc

a re de te rmined and the AXjc a re a l s o un ique ly de te rmined by the hi.

Us ing (4 .4 .29 ) , we ma y d e te rm ine th e Aa in te rm s of the hi = {i = 1, ...,v),

a n d s u b s t i t u t e t h e m i n t ohî = XjcAXjc = DtL^K

an d we m ere ly h av e to choose th e f* to sa t isfy th e equ a t io n

(Dtho: + hv+iXi)ii = 0 w i t h XiîÔ]

th is is a lways poss ible unless a l l the Df ha. = 0. Thus , un less these

l inear re la t ions in A Xi ho ld , we have

F{X + AX) = / ( ^ ^ + A . I O >_f{p) + AtXo.^^ = F{X) + CÂXu.

The resu l t fo l lows in genera l by cont inu i ty .

C h a p t e r 5

Differentiabi l ity of w ea k solu tion s

5 . 1 . In trod u ct ion

In th i s cha p te r we s upp ly the de ta i l s wh ich were om i t t ed in the s ke tch

of d i f fe ren t iab i l i ty theory which was presen ted in Chapte r 1 . I t was seenthere tha t th is theory involved a s tudy of the so lu t ions of genera l ized

l inear equa t ions of the form

/[C,a(«° '^^ , )8 + h°'u + e"") + C(c ' '^ ,a + du+f)]dx = 0, C^LipciG),G

(5.1.1)

in which the a*^ a re bounded and measurab le and the coeff ic ien ts b

a n d c^ Lv{G) a n d d^ Lî^{G)y and sa t is fy

(5.1 .2 ) m |A p <a«^ ( :^ ) ; , aA ^ , \a[x)\ < M f o r a .e . A;^ G a n d a l U ;

(5.1 .3) / ( l & P + |c |2 + \d\Yi^dx < {Corf^^yi^.

W e d o n o t a s s u m e t h a t a^'^ — a"-^ o r t h a t h"- — 0°". The proofs of the

theo rems in th i s gene ra l i ty a re new.

5.1 . Introduction 127

A l t h o u g h it is not s t r i c t ly neces s a ry for th e app l ica t ions , we p r e s e n t

in § 5.2 a gene ra l ex i s tence theo ry for s uch equa t ions and p r o v e an in

t e r io r boundednes s theo rem and an a p p r o x i m a t i o n t h e o r e m ; u s i n g t h e s e

t h e o r e m s a t h e o r e m on fur ther L2- type d i f fe ren t iab i l i ty is p r o v e d . If th e

coefficients ^ C^, the s o lu t ions ^ C^ on i n t e r i o r d o m a i n s . If a p o r t i o n

oi dG^ C^ a n d the b o u n d a r y v a l u e s ^ C^ a l o n g t h a t p o r t i o n , t h e n the

s o lu t ion ^ C°° a long th a t po r t ion . Thes e re s u l t s fol low by repea ted app l ica

t ion of the genera l theorems . This en t i re ana lys is ca rr ies over to s y s t e m s ,

even thos e whe re the a'lf m e r e l y s a t i s f y t h e g e n e r a l L e g e n d r e - H a d a m a r d

type cond i t ion

( 5 . 1 . 4 ) < / W A a A ^ | ^ f ^ ' > m l A | 2 | | | 2 , \a{x)\^M, m>0,

co r re s pond ing to (1 .5 .5) , p rov ide d in th i s c a s e tha t the a'^^ are continuousa t leas t .

In §§ 5.3 and 5-4, we present a much s impl i f ied genera l iza t ion of the

DE G I O R G I - N A S H - M O S E R resu l ts f rom which the i n t e r i o r b o u n d e d n e s s

a n d H o l d e r c o n t i n u i t y of the so lu t ions of (5-1.1) follow. In § 5-5 we

p r e s e n t Lp and S c h a u d e r e s t i m a t e s of the s o lu t ions of (5-1-1) under

va r ious hypo thes e s conce rn ing the coefficients. In § 5.6, we t r e a t the

e q u a t i o n

(5.1.5) Lu = a""^ u,o:^ + b " u,oc + c u = f;

in this case the coefficients a"^ m u s t be at l e a s t con t inuous . H ighe r

d if fe ren t iab i l i ty , bo th of L ^ - t y p e and Holde r con t inuous (Schaude r )

type a re ob ta ined in t e rms of the d i f fe ren t iab i l i ty o f / a n d th e coef fi cien t s.

I n all the s e s ec t ions , r egu la r i ty at the b o u n d a r y is p r o v e d w h e n e v e r

pos s ib le .

I n § 5.7 , we pro ve th a t th e so lu t ions of l inear ana ly t ic e ll ip t ic eq ua t ion s

are ana ly t ic on the in te r ior and a long an a n a l y t i c p o r t i o n of t h e b o u n d a r y

in case the b o u n d a r y v a l u e s are a n a l y t i c . The m e t h o d c o n s i s t s m e r e l yin f ind ing bounds for the d e r i v a t i v e s . T h i s m e t h o d has been us ed by

A. F R I E D M A N [1] for non-linear equations, using the device of G E V R E Y

(see a lso for sys te m s §6 .7) . B u t th e wri te r p res en ts h is p roof for non - l inea r

e q u a t i o n s in § 5.8 (see also § 6.7 for sys tems) s ince it uses a different

m e t h o d a n d the au tho r cons ide rs it m o r e i n t e r e s t i n g .

I n § 5 .9 , we s u p p l y the de ta i l s of the proofs of Theorems 1 .11 .1 ,

1.11.1' , and 1.11.2 w h i c h w e r e o m i t t e d in § 1.11. The s pec ia l a rgumen t

n e e d e d to prove Theorem 1 .11 .1" is p r e s e n t e d in § 5-10. The re s u l t s ofL A D Y Z E N S K A Y A and U R A L ' T S E V A ([1], [2], [3]) mentioned under Theorem

1.10.4(v), with k re s t r i c ted to be > 2 , a re p roved in § 5-11. In § 5-12, we

p r e s e n t the ve rs ion of L E R A Y - L I O N S of an a b s t r a c t e x i s t e n c e t h e o r e m ,

i n v o l v i n g ' ' m o n o t o n i c i t y " , for the s o lu t ions of non- l inea r func t iona l

e q u a t i o n s a n d a p p l y it to show the ex is tence of s o lu t ions of ce r ta in non -

1 2 8 Differentiability of weak solutions

l inear e l l ip t ic equa t ions of higher order . B R O W D E R , M I N T Y , and VisiK

h a v e e a c h w r i t t e n a g r e a t n u m b e r of p a p e r s on t h i s s u b j e c t ; we refer in

tha t s ec t ion to a r e c e n t p a p e r of each .

5 . 2 . G e n e r a l t h e o r y ; v > 2

W e beg in by def in ing i

B(u, v) == Bi{u, v) + B2{ii, v), L(v) = — j {e'^vôc + fv) dxG

(5.2.1) Bi(UyV) == f vôcd^"^ u^pdx, C(u,v) = f uv dx,G G

Bîu, v) ^= J (vôcb°' u -\- vc°^ uôc + du v) dxG

a n d we cons ider the e q u a t i o n(5.2.2) B(u, v) + I C{u, v) = L(v)

w h i c h is the s a m e as (5-1-^) except for the add i t iona l t e rm XC{u,v).

The equa t ion (5-2 .2) corresponds formal ly to the dif fe ren t ia l equa t ion

(5 .2 .3 ) -A(« ' ' ^^ , i8 + b'''^) — ^ ' ' ^ . a — du—lu =f— e'^^.

L e m m a 5.2 .1 . Suppose v > 2, q Lv/2(Q ^^^ satisfies

(5.2.4) J\q\''^^dx<^(Corf'iY^^ for all B{xo,r).

GnB(x„,r)Then, for each e, there is a Ci depending only an e, v, jui, CQ, and R, such

that r r r

j\q\\u\^dx<ej\S/ufdx + Cij\u\^dx,G G G

U^HIQ{G), if GcB(xi,R).

Moreover, there is a constant C2, depending only on v, such that

f \q\\u\^dx ^€ 260 Rî'{\\Vu\\l)^, u^Hl(G), GcB{x^,R).G

Proof. For each r, the re ex i s t s a f ini te sequence {C3?}, ^ = 1, . . ., P ,each C^(G) and h a v i n g s u p p o r t in s o m e B{xj,, r) s u c h t h a t

Then , u s ing the SOBOLEV lemma, Theorem 3-5.3> we o b t a i n

j \q\'\u\^ dx = j \q\'\u\^ ^l:ldx =^ ^ j \q\'\uj,\^ dxG G 33 = 1 P B{Xj„r)G

<(Corf'i)2:lf\up\^'dxY'^CoZirî2;f\'7up\^dx

<2ZiCorf'if 2:(Cl\Vu\^ + \VCp\^\u\^)dx {up=Cvu)G V

^2ZiCornJ l\vu\^ + [ 2 ' 1 V C 2 > P ) | ^ P W A ;

1 If complex functions are allowed, replace ?; by z;.

5.2. General theory; v > 2 129

f rom wh ich the first result follows. The second result follows from the

Holde r inequa l i ty , Theo rem 3 .5 -3 , and (5-2.4).

Theorem 5.2 .1 . There exist numbers M\ and Ao, which depend only on

i^(> 2), m, M, Co, /ui, and R such that

^^^^•^^ \B{u, u) I > (w/2) \\u}^ - Ao C{u, u), G C B(xi, R).

Moreover, there are constants C3 and C4, depending only on v, such that

\B{u, u)\^[m^ ei(R)] (\\Vu\\l)^ >[m- ei{R)] (1 + i?2/2)-i ( | |^ |1)2,

( l l ^ l l = l l ^ l l l )e^[R) ^ CsiCo R^^Y'^ + C4 Co R'^i.

Proof. This follows immediately from (5.1.2), (5 .1 .3), Lemma 5-2.1,

a n d the S c h w a r z and Poincare (Theorem 3 .2 .1) inequa l i t ies .

T h e o r e m 5.2.2 ( L e m m a of LA X and MILGRAM). Suppose in a real {or

complex) Hilbert space §, BQ (U, V) is linear in u for each v and (conjugate)

linear in v for each u, and suppose

(5.2.6) <^ \B^{u,v)\<M^\\u\\•\\vl

(ii) \BQ{u,u)\'>mi\u\^, mi>0.

Suppose To is defined by the condition

(5-2.7) Bo(u,v) = (Tou,v).

Then To and T^ are operators with bounds Mi and m , respectively.

Proof. It is c lea r tha t To is a l inea r ope ra to r wi th bound Mi. F r o m

(5-2.6) (ii) and (5-2.7), we see t h a t

w i II ^Ip < \Bo{u, u)\ = I (To ^> ^^)| < II' 11 * II Tou\\

s o t h a t

IITo /II > mi \\u\\.

I t fol lows eas ily that the r a n g e of To is closed. If the range were not the

whole space , there would be a t* s u c h t h a t BQ(U, v) = {To u,v) = 0 for

e v e r y u. But, by s e t t i n g u = v, if follows from (ii) t h a t v = 0. T h u s

TQ^ is a b o u n d e d o p e r a t o r w i t h n o r m < w^^.

Theorem 5.2.3. Suppose the transformation U is defined on HIQ{G)

by the condition that

(5-2.8) C{u,v) = {Uu,v)l„ veHl,{G).

Then U is a completely continuous operator.

Proof. T h a t U is an opera tor fo l lows f rom Poincare ' s inequa l i ty

(Theorem 3.2.1), s ince

| |C7^| | = s>uip{U u,v) = snp fuvdx ^2-^ R^\\u\\ if ||?;|| = 1.

G

Morrey, Multiple Integrals g

Next , suppose i% —7 u in HIQ(G). T h e n Un - > u in L2 (Theorem 3-4.4) and

\\U(un — u)\\ = sup J {Un — u)v dx <, 2'^^^ R \\v\\ • II ^ 4 — -^llo - ^ 0V Q

s o t h a t U i s compac t .Theorem 5.2.4. If X is not in a set b , which has no limit points [in the

plane), the equation (5-2.2) has a unique solution u in H\Q{G) for each

given e in L2{G ) and f in L2sf{G), where s' = vj{v + 2). If l^h, there are

solutions of (5.2.2) in w hi ch ^ 9^ 0 and e = f = 0, but the manifold of

these is finite dimensional. If Ao is defined as in Theorem 5-2.1, then

no real number Ai > Ao is in b .

Proof. Let us define AQ as in Theorem 5-2.1 and BQ b y

Bo(u, v) = B(u, v) + Ao C{u, v)an d define To b y (5.2 .7). T he n, eq ua tio n (5-2.2) is eq uiv ale nt to

(5 .2.9 ) To ^ + (A - Ao) Uu = w, w h e r e (w, v) = L {v),

L being a hnear func t iona l s ince v^L2s{G) an d (2s )~ i + (2s')~^= 1.

Moreove r , f rom Theorem 5.2.1, i t fol lows that BQ sa t is f ies the condi t ions

o f the Lemma o f Lax and Mi lg ram wi th mi = mj2. Accord ingly To has

a bounded inverse so (5-2.9) is equivalent to

u + [X — Ao) T^^ U u = TQ^ W .

Since T^^ U i s com pac t , the theo rem fo llows from t he R iesz th eo ry of

l inea r ope ra to r s .

Theorem 5.2.5. Suppose the coefficients, e, and f satisfy the conditions

of Theorem 5-2.4. There is a constant C, depending only on v, m, M , Co, i ,

and R, such that

||V^li2% < C[a- i l l^ i .^ + ll^i .^ . + | | / | | « , ,d ,DcGa, GcB(xi,R),

for any solution u 0/(5 .2 .2) with X = 0 in which u^ L2{G) and u^ HK D)for each D (Z C G.

Proof. We define r] as in the proof of Theorem 1.11.1 and def ine

V = rj U, U == rj u.

Making these subs t i tu t ions in (5 .2 .2) , we obta in

B{U,U) + ( [a^'^ri ocuU p — a^^^r] ocuU^^ — a'^^rj ocf] p\u\^ +

(5.2.10) o __ '

+ V ^"{^,0. + ^ ^ ,« ) + ^,aw(&°' ~ c«) C + rjfU] dx.Using (5.2.10), (5.2.5) for U, t h e CAUCHY inequa l i ty . Lemma 5 .2 .1 for U,

and the inequa l i t i e s

/ \fU\ dx < \fl%, • II Ull, < C 11/111,, • II U\l, <e{\ C7io)2 + C (| | / | |L ,)2G

we ob ta in the re s u l t .

5.2. General theory; v > 2 131

Theorem 5.2.6. Suppose G = GR, R ^ \, suppose the coefficients and

e and f satisfy the conditions of Theorem 5-2.5 on G, suppose u^L2(G),

u ^ HI (Gr) for each r <, R, suppose u = 0 along an, and s^lppose u is a

solution 0/(5.2.2) with A == 0. There is a constant C, depending only on v,

m, M, Co,and jui, such that

| | V « i . , < C[(R - r)-i |K||».^ + ||g||«^ + | | / | |» , , .B] .

Proof. The proof is identical with the preceding proof except that

we define Y$%) = \ if |A;| < r , r\[x) = \ — 2 [R — r)~^ {\%$ — r) if /

< 1 1 <.{r + R)I2, r]{x) =0 ii \x\->(R + r)j2. Then u, v, and U all

vanish on SGR.

Theorem 5.2.7. Suppose that the coefficients a'^/ b^, cJJ, and dn all

satisfy the conditions of Theorem 5.2.5 uniformly on G and converge almosteverywhere to a"-^, b°', c"-, and d respectively, and suppose that e'^ —7 e^ in

L%(P) and fn—^f in Lzs'iG). Suppose that Un-v u in H\{G) and that Un

is a solution of [S.2.2)n for each n. Then u is a solution 0/ (5.2.2) imth the

limiting functions.

Proof. For each fixed v ^ HIQ(G), we see that

<^ t ; , a ->a«^^ ,« , blv,oc->h°'v,oc, etc.

in L2(G), so that

Bn(Un, v) -^B{u, v), C[Uny v) -^ C(u, v), Ln(v) -^L[v).

Theorem 5.2.8. Suppose the coefficients and e and f satisfy the con

ditions of Theorem 5-2.4 dnd suppose that u^H\{G) and satisfies (5.2.2)

on G. Suppose also that a^"^^ C\{D), b^"^ Hl{D), ^«^ Hl{D), c^'is bounded,

d^ Lv{D), / ^ LîD), and V b and d satisfy

(5.2.11) fl\Vb\^ + \d\^f^dx<(Corf'iY'\ ^ i > 0B (a!o.r)n2)

for each D (Z G G. Then u^H\{D) for each such D where it satisfies

(almost everywhere) the differential equation

(5.2.12) — -^ (a ' ^^^ ,^ + b^'u + e^") + C'uôc + du + / = 0 .

/ / G is of class CJ, the coefficients and e and f satisfy the conditions above on

G, and u^ HIQ(G), then u^ Hl(G).

Proof. This is proved by the difference quotient procedure of Theorem

1.11.1. Let Z) C C G, choose D' and D" so D C G D' C C D" G C G

and Z)' C D'^, let vf^ C'^(G) with support in D\ let Cy be the unit vector

in the x^ direction and define1

Vji(x) = h-^[v{x — hey) — v(x)] = f — v y[x — th Cy) dt,

(5-2.13) 0

uji{x) = h-^[u(x -^ hcy) — u{x)], (p(x) = c'ûôi + du -\- f,

0<\h\ < H,

9 *

R e p la c in g v b y vn in (5 .2 .2) and making the ind ica ted changes of var iab le ,

we see tha t ^lfl sa t is f ies the equa t ion

(5.2.14) jv,^{a^ûn,p + E^j,) dx = 0 , v^LipcD'

w h e r e

^ y / i W = < ' ^ W Â^ + ^^r) + hlix) u(x + hey) + b°^(x) un[x) +

1

(5.2.15) + el{x) - A^y fcp{x + they) dt.

0

From our hypotheses and Theorem 3 .5 .2 , i t fo l lows tha t b i s con t inuouson D". F r o m o u r h y p o t h e s e s a n d SOBOLEV'S lemma (Theorems 3 .5 .3 and

3.5.5), it follows that (p^LîD"), T h u s (see § 3.6) for a su bs eq ue nc e of

A - > 0 , al^{x) converges a .e . and boundedly to a^y(x), u,p{x + hey)

converges in LîD') t o u^^{x), hi -^h'^y in Lv{D'), u(x + hey) û(x) in1

L2s(D'), Ufi{x) ->u^y in LîD'), el -ê'^^ in L2(D'), a n d f (p{x -{-they) dt

-^99 in L2(Z)'), so t h a t ®

It follows from Theorem 5-2.5 that || V^;^|| 2,D i s un iform ly bou nd ed , so

t h a t Ufi—yU^y in H\(P) for a further subsequence of A -> 0, since un - > u^y

i n L%{D). From Theorem 5 .2 .7 i t fo l lows tha t u^y satis f ies the l imiting

equa t ions on D. But then one may in tegra te by par ts in (5 .2 .2) to a r r ive

at (5.2.12).

In o rde r to p rove the l a s t s t a temen t , we p ick any po in t XQondG a n d

m a p a b o u n d a r y n e i g h b o r h o o d N of XQ in the p rope r way on to GR b y

a regular map of c lass C\. Then (5.2 .2) goes over into an equation of thesame form on GR. W e may repea t the a rgumen t o f the p reced ing pa ra

gr ap hs for each y <,v — 1, since e ach uji van is hes a long Cr. Us ing Theorem

5.2 .6 th is t im e , we conc lude th a t each u^y w i t h y <,v — 1 ^ Hl(Gr) for

each r < R. This impl ies tha t a l l u^yd w ith 7 < v — 1 an d 1 < ^ < r $

1,2(Gr). B u t n o w a x'' > e > 0, u^v is also in HI a n d u satisfies (5.2.12).

Since a^^ > 0 , we can so lve th a t e qu a t io n for u^vr and thus conc lude tha t

i t $ L2(Gr) a lso so tha t ti^v also H\{Gr) s o t h a t u^ Hl(Gr). Thus , s ince

each po in t XQ oi dG i s in a ne ighborhood on which u(^ HI, we conc ludet h a t ^ ^ i ^ K G ) .

Corollary 1 . / / , for some boundary neighborhood N of XQ on G \J dG,

the part N DdG is of class € { and if the coefficients and e and f satisfy the

conditions of the Theorem on N, and if u is the solution 0/ (5.2.2) in H\Q{G)

then u ^ H\{N') for each boundary neighborhood N' of XQ with N' C N.

5.2. General theory; v>2 133

In case the coeffic ients and e a n d / $ C^ a repe t i t ion of t h e a r g u m e n t

leads tothe fo l lowing resu l t :

Corollary 2. If the coefficients, e, and f C on the interior of G and u

is asolution o/(5-2.2),

then u C°° interior to G. If G is ofclass C^, the

coefficients, e, and f^C'^{G), and U^H\Q(G), then u^C'^(G). Local re

sults corresponding tothat in Corollary 1 also hold.

F u r t h e r r e s u l t s are p r o v e d in § 5.5.

The en t i re d iscuss ion of th is sec t ion carr ies over to s y s t e m s of e q u a

tions like (5-2.2) where (ifcomplex func t ions a re a l lowed)

B(u,v) =Bi(u,v) + B2(u,v), L(v) = - J {etv% + fv ) dx

B\ {u , ^) = / ^Ja if f/3 ^^> C(u,v) = f u^ v^ dx

B2{U, V)= f p;„ bfÛ^ + Vi{clûi^ +dij U%dx

(5.2-16)

u = {u^, .. ., u^), V = {v^,. . ,, v^).

T h e a s s u m p t i o n s on th e coeffic ients analogous to(5.1-2) and (5-1.3) are

tha t they a re a l l meas u rab le or in s o m e Lpand sa t is fy

(5-2.17) ni\7z\^âl^{x)7Ti7t^, SIITI, \a(x)\<,M, m>0

(5-2.18) /(l l + kl + IdlY^ dx icorf ifKBiXo,r)nG

In case the a'^f are continuous on G, {S.2A7 ) may be rep laced by t h e m o r e

genera l Hadamard type condi t ion (see (1 .5 .5))*

(5.2.19) m\A\^\^\^âf^{x)AocA^'^H^, \a{x)\<M, m> 0.

W h e n t h i s is d o n e , we confine ourselves to ^ H\Q[G).

The rea s on why t h a t can bedone isbecaus e we see byt a k i n g the

F o u r i e r t r a n s f o r m i of t h a t

/ îi Û ^!/s ^ ^ = ^ ^ / î^ y"^ y^ ^* (y) ^^ M y

G ' R

^m^J\y\'^'\U{y)\^dy = m J \\7u\^dx, U^HIQ{G)

Bv G

in case the a'^f a re constants satis fying (5-2.19). Incase the affare con

t i n u o u s on s o m e JT D G w e m a y , for each suffic iently small r > 0, cove r

G w i t h a f in i t e number of s phe re s B{xs,r), and can then choose Cs,

s = 1, . . .,5 , ^ Cl[B{xs, r)] sot h a t ff + • • • + f| = 1 on s ome doma in

13 4 Differe nt iabi l i ty of we ak solut io ns

/^ D G. Then, if E(r) i s a modulus of cont inu i ty of a, we f ind th a t

Bi (U, U) = j ^ < f {Ul^ - Cs. a Ui) « ^ - C,, ^ U^) dxG s

> I / < f <a </J ^^ - Ci MJlul^ dx8 G G

> I 2 / < f (^^) ^ l a < ^ dx-C2 8(r) ^ f \ V Us [2 dx

s G s G

- CiM (\u\^ dx >'^ y ] (\V Us\^ dx -- CiM j\u\^ dxG s G G

> ~ \V u\^ dx — Cs(r,m,M) \u\^dx, Us = CsU.

G G

5.3. Exten s ion s o f th e d e Giorg i - Nash - Moser resu l t s ; v > 2

L e m m a 5.3 .1 . Suppose that (i) co ^ HIQ(G), (ii) tp and xp m ^H\(G),

(iii) \p{x) > 1 on G, and (iv) ipcoôc and ipôcco^ L2(G). Then there exists

a sequence { C « } - > ^ * ^ ^\o{^)> *^ leÂ 'c/^ âcA Cn^Lipc[G), such that

y> Cw,a - > ^ co,a m Z2(<j ) « ^ ^ ^ Cn -^CO ip in L2s(G) where s = vj{v — 2).

If a)(x) > 0, Â^ Cn '^^y be chosen > 0 .

Proof. W e prove the l emma in the ca s e whe re co{x) > 0 , th e o thercase is eas i ly proved by wri t ing 0}{x) = a)+{x) — a)~(x) w h e r e a>^(x) > 0 .

By us ing the fac t tha t co^ ^2o(^) ^^^ ^^^^ ^Y choos ing abs o lu te ly con

t inuous rep re s en ta t ive s o f ip a n d ^ co tha t the l a t t e r func t ion ^ H\Q{r)

on any JT I) G if we define it = 0 ou tsi de G, T h u s ip oy^L^siG).

For each L, we de f ine the t runca ted func t ion OOL b y

{coix), \i x^G — EL,

; .r ^ p Ej, = {x\<o{x)>L}.

L, I I x^hLE v i d e n t l y

0 < ^ • (co — O O L ) <îpO), 0 < | ^ , a { c O — CO i:) | < | V ^ | 'C O ,

ip ' (C0,a — 0)L,oc) = ip0),o: ' X{EL)

%[EL) being the charac te r is t ic func t ion of EL. Cons equen t ly

(5 3 2) W'{^-(^L)->0 in UsiG)

ip,oc'(o) — (OL)-^ 0 an d ^(co ,a — co / ,, a )-> 0 in L2(G)

a s L ^ o o . M o r e ov e r, t h e r e e x i s ts a s e q u en c e {^n}, in which each^nÊipcG w i t h 0 < C w ( ^ ) < - ^ , s uc h t h a t CU-ÔJL in H\Q[G) a n d

Cw(^) -^ COL(^ ) and VC?i(^) -> VCOL(^) a.e. in G. Then i t is easy to see

that ( i) ip ' {COL — Cn) - ^ 0 i n L2s(G), (ii) (tp — ipK) • VCOL(%) ^ 0 in L2(G)

as X -> cxD, Z being fixed and y jK be ing the t runca ted func t ion o f ip, a n d

(iii) that ipK ' {V OJL — V Cn) ^ 0 as n -^ oo, K and L being fixed. Con-

5.3- Extensions of the de G I O R G I - N A S H - M O S E R results; v > 2 13 5

s e q u e n t l y

y){a)L — Q - > 0 in L2s{G) a n d !(5.3.3) } a s w - > c x 3 .

The lemma follows easily from (5.3-2) and (5.3-3) •Lemma 5.3.2. Suppose that U $ H\{D) for each D C C G, that U{x) > 1

and W = U^ for some A, 1 < 2 < 2, and that P^ Lv(G). Then for each

DCGG,

UÛ s{D), PUÛ iD), PÛ^L 2sr{D), s = vl(v-2),

s '= = i;/(^ + 2), V P F , PW, PVW a n d P^W^Li{D).

Proof. S u p p o s e t h a t D G Ga a n d rj is defin ed as usu al (i. e. as in th e

proof of T he or em 1.11.1, e tc . ) . T he n rj U^ H\Q {G) and i t fol lows from the

SoBOLEV le m m a (T heo rem 3.5.3) t h a t U^L^siD)- The rema in ing in

equali t ies follows from the H O L D E R i n e q u a l i t y .

L e m m a 5.3.3. Suppose (i) that U ^ H\{D) for each D (Z G G, (ii) that

U(x) ^ i in G, (iii) that w = V^^ L2(G) for some r > 1, and (iv) that

W = U^ satisfies {see L e m m a 5-3-2)

fCoc(a°^^W^ + b'^W) + C{c°'W oc + dW)]dx<0(5-3-4) J

for each C € ^ ipc(G) with C(^) > 0 ,

for some A with 1 < A < 2 ; ze ^ suppose also that a, b, c, and d are measur

able, a satisfies (5.1.2), and b, c, and d satisfy (5.1.3) for B(xo, r) C G,

Then w^H\ [D) for each D C . G G and

j\\7w\^dx<, C2Tâ-^ Jw^dx, B(xo,R + a)GG ,( 5 . 3 . 5 ) B(Xo,R) B{x^,R+a)

^ =: 1 + 4fî^, 0 <a <R ,

where € 2 depends only on m, M, v, A, jui, C i, and upper bounds for R and

the coefficients: C2 does not depend on r.Proof. I t fo l lows f rom the hypotheses . Lemma 5-3-2 , and Lemma

5.3.1 with y ) = t / ^ - i a n d w = rj^ U^'^ Uf-^ t h a t w e m a y s e t

(5-3.6) C = ^2 f72-A ^ 2 r - 2 {ri^LipcG)

in (5-3-4), UL being the truncated function (see (5-3-1)) of U. Since

UL,OC{X) =^ 0 a.e. on EL, w e c o n c l u d e t h a t

Co. = 7]^ U^-' Ul^-^[{2 - A) ?7,a + ( 2 T - 2) UL,a] + 2r]r],ocU^-' Ul'-\

(5 -37 )

The inequali ty (5-3-4) becomes (again us ing \/ UL = 0 on EL)

j {rf lPjf'^[X[2 - X)\/ U ' a-V U ^ (2 - A) U b-V U + AU cS/U +

(5.3.8) + dU^ + {2r - 2){A^ U L' a-^^ UL + ULb'VU L)] +

+ 2rjUUl''-^S7 r]-(Aa'\7 U + bU)}dx = 0.

13 6 Differen t iabi l i ty of wea k solut ions

Using (5.1 .2), the bounds for the coeffic ients , and the inequaHties of

ScHWARZ and CAUCHY, we conc lude tha t

(5.3.9) fv^ ^ r - ' [ | V t7 |2 + (T - 1) IV UL\^] dx

o< Z i / ( r ; 2 T P 2 + \\/ri'^)Ul'-'Û^dx. ( P 2 = , | 5 | 2 j ^ |c |2 - |_ |^ | ) .

G

Now, le t us define

(5.3.10) WL = rjUUl'K

Then, as was done in (5 .3-7), we find that

(5.3.11) VWL= UU l'^Vi] + riUl-^[V U + (r ~ 1 ) V C / L ] .

It follows from (5.3-9), (5.3-10), and (5.3.11) that

(5.3.12) f \\/ WL\^ dx < Z 2r^ f P^ wldx + Z stI \Vr]\^ U^ ' Ul""-^ dx.G G a

S u p p o s e t h a t rj i s def ined as usua l wi th a rep laced by a /2 , G b y B{XQ, a)

C G, a n d Dhy B {XQ, aj2). Then the inequa l i t i e s

I wldx ^ (a^l2) f \V WL\^ dx (wL^Hl^[B(xo,a)])

(5-3-13) J l\wL\^^dxY'<C3f[\VwLf-i-a-^wl]dx

<.{Cz+il2)f\VwL\^dx, s=vl{v-2)

BUo.a)

of PoiNCARE and SOBOLEV (Theorem 3 .5 .5) ho ld . Consequent ly

/ P 2 ^jdx < I / P^dx^'f f \wL\^^dxY' < Z4 a ^ i / jVze;z,|2 dx,

B(Xo,a) [Ba j \Ba J B{x„,a)

(5.3.14)

From th is and (5 .3 .12) we deduce tha t

j \VWL\^ dx <Z ^r a-Î m Ul'-^dx, 0 < a < oc,

( 5 . 3 . 1 5 ) JS(a;o.a) B {xo,a)

w her e 2 Z2 Z4 r^ ocî = \.

W e can now le t L -> 00 to conc lud e th a t

(5.3.16) J \S7 w\^ dx ^ Z ^r a-^ J w^ dx, 0 < a < oc.B{Xo,a/2) B{Xo,a)

Next , i f 0 < a ôc a n d B{xo, R + a) C G , we can cover B(xo, R)

by a f in i te number of ba l ls B{xi, a/2) s uch th a t e ach ba l l B{xi, a)C B(xo, R -\- a) and there a re no po in ts in B{XQ, i^ + a) w hich belo ng

t o m o r e t h a n K{v) of the B{xi, a). T he n from (5.3-16), we o bt ai n

f\\7w\^dx<2Jif\Vw\^dx<Z5ra-^2!fw^dx < K Z ^ r a - ^ fw^dx.B{Xo,R) B(Xi,a/2) Bix^â) B{Xo,R+a)

(5.3.17)

5-3. Extensions of the de G I O R G I -N A SH - MO S E R results; v > 2 137

For la rger va lues oi a, (5-3-17) holds with a rep laced by oc. T h u s we con

c l u d e t h a t

f\Vw\^dx< KZ^ra^ oc-^ a-^ f w dx,

B(x„,R) B(Xo,R+a)

a be ing an u p p e r b o u n d for a(<,R). The lemma fo l lows by us ing the

definit ion of/x us ed in (5.3.15) •

T h e o r e m 5.3.1. Suppose that the coefficients P, and U satisfy the hypo

theses of Lemma 5.3-3 ^^^^ T = 1. Then U is bounded on each domain

D C <Z G and

\U[x)\^ ^C a-* j \U{y)\^ dy, X^B{XQ,R)(5 .3 .18) B{x9,R+a)

0 < a^R, B{xo, R + a)cG

where C depends only on v, m, M, Co,and fx\.

Proof. Let us define

5 == vl{v - 2), Wo = C7, wn = U^"", Bn = B(xo, R + 2-^ a), Wn = jwldx.Bn

U s i n g L e m m a 5.3.3 we conc lude in t u r n t h a t wo^L2s{Bi), wi^ H\{Bi),

W2, = \wi_ \^^ L2[B2), W2^H l(Bs), etc. T h e n , u s i n g the inequa l i t i e s

(5.3.5) and (5.3.13), we o b t a i n

W}!^=if\wn.i\^^dxY'<Csl{\VWnî\^ + R-^wl_,)dx[Bn J Bn

(5.3.19) < 2C 3 C2 s^^-e 4« a-21wl_^ dx = Ko Kl Wn-1,

Bn-1

w h e r e Ko = 2C3 C2 s"? a-^, iî = 4 s^.

From th i s r ecu r rence re la t ion , we c o n c l u d e t h a t

W}!^"" < i^^ i -^" '^" ' • i ^ J i -^" ' ^ " ' 'Wo = C a-' WQ .

The theorem fo l lows by l e t t i n g ^ ->^^.

Corollary. Suppose the coefficients satisfy the conditions of Lemma5.3.3, that u^L2(G), and that u^ H\{D) for each D C G G and satisfies

f [v,cc{a°'û^p + b°'u) + v(c°'u,o: + du)] dx == 0, v^Cl(G).G

Then there is a constant C, depending only on v, m, M, Co (ind jui such that

1 +u^(x) <Ca-'l[\ +u^(y)]dy, x^B{xo,R),B{Xf^,R+a)

0<a^R, B{xo,R + a)cG.

Proof. Define V^{x) = 1 + u^{x) and set

v = ipV~û, if^LipdG), y )(x)ÔonG

in the e q u a t i o n a b o v e . An eas y app rox ima t ion s hows tha t th i s is legit i

ma te . S ince V V^x = u uôt so

a°'^V,ocV,^ = V-^ u^ a""^ u,ocU,^ <, a''^ u,ocU,p,

C€ LiPeG,


one sees tha t V satisfies

fy^,cc{a^^ F ,a + 'b^ V) + ipCC^ F a + 'dV) dx<0G

'h^ = (1 - F -2 ) 6«, 'c^ = c''+ F-2 6 ^ /^ = (1 - F-2) ^ .

Fo r the pu rpo se of p ro vin g the d i f fe ren t iab iHty of the so lu t io ns of

the va r ia t iona l p rob lems and the weak s o lu t ions o f the o the r d i f fe ren t ia l

equa t ions (1 .10 .13) . i t i s suff ic ien t to s tudy equa t ions of the forms

(5.3.20) / ( v C - ^ - V w + C - c - V ^ ) i ; ^ = 0G

(5.3.21) / [V C( « 'Vu + e) + C{c ' "û + / ) ] dx = 0G

in w hic h th e a«^ satisfy (5-1.2), c satisfies (5.1-3) {b = d = 0), e^ L2(G),

a n d f^L2sf(G), s ' = vl{v + 2), and e an d / sa t is fy cer ta in add i t ion a l

condi t ions ; we suppose , o f course , tha t u^H\[D) for each DGGG,

The more genera l equa t ions (5 .1 .1) a re in te res t ing in themse lves and

a re t r ea ted in a pape r by the au tho r ( M O R R E Y [14]).

Definit ion 5.3 .1 . A func t ion v ^ Hl{D) for each D C C G is a sub-

solution of (5.3.20) if and only if

f{V^ -a-Vv + Cc'Vv)dxÔ for each C^Lipc(G), C(^) > 0 .G

(5.3.22)

R em ark s . Th is cond i t ion i s fo rma l ly equ iva len t to th e cond i t ion

-^râ^'^VR — C'vô^ > 0 .

L e m m a 5.3.4. Suppose that (i) F is non-negative and convex on the

interval [0, 00), (ii) H = —e~ ^ is convex on that interval^ (iii) u is a non-

negative solution 0/(^.3 .20) on G (iv) V{x) =F(u{x)], and (v ) v^L2{G).

Then v is a sub-solution of (5.3.20) on G and

J\Vv\^dx<C a-^\G\ ii DcG a, GcB{xuR)D

where C depends only on v, m, M, fxi, Co, and R.

Proof. Fi r s t , we a s s ume tha t

i / e C 2[0 , CXD), -. \ ^H(U) <-e {s>0)

a n d t h a t H" i s bo un de d on [0 , 00) . T he n F^ C2[0, 00), and F, F', a n d

F" a r e b o u n d e d t h e r e w i t h F"[u) > [F'(u)]'^. L et u s set C = ^^ F'[u) inequa t ion (5 .3-20) , where rj is defined as usual . I t fol lows that

0 =^ j [ZrjSJrj' a-Vv -{- rj^F''{u) ^ u - a - S7 u -^ rj^ c ^ V v] dxG

^ f blîVv •a'S/v-\-C'S/v) + 2riy rj'a'SJv'] dx,

5.3- Extensions of the de G I O R G I - N A S H - M O S E R results; v > 2 13 9

s ince F" > {F')^, F i n a l l y

(5.3.23) jri^\S7 v\^dx^ Cf(rj^c^ + \Vrj\^)dxG G

f rom which the inequa l i ty fo l lows eas i ly , us ing Lemma 5-2.1.In the genera l case , H i s convex wi th — 1 ^ H(u) < 0 o n [0 , oco).

I t i s easy to see tha t H can be app rox ima ted f rom be low by func t ions

H hav ing the p rope r t i e s in the p reced ing pa rag raph . I t fo l lows tha t the

func t ions Vn(x) ->v(x) f rom be low and hence s t rongly in L2{G). Clear ly ,

also , Vn —7 V in H\[D) for each D (Z G G, on account o f the inequa l i ty

(5.3.23) which holds for each n. The inequa l i ty ho lds in the l imi t by lower-

s e m i c o n t i n u i t y .

Theorem 5.3.2 (HARNACK t y p e ) . Suppose that (i) u is a non-nega tive

solution 0/(5 .3 .20) on B^R^ B(xo, 2R) and (ii) the set S where u[x) ^ \

has measure > ci |5 2i ? | , c i > 0 . Then

u{x) > C2 > 0 for x^ BR

where c^ depends only on v, m, M, Co, /ui, and ci.

Proof. There i s a k, 1 < ^ < 2, suc h th a t \B2R — BJCR\

= (1/2) ci 1^2121. Th e n | 5 Pi BJCR\ > (1/2) ci \BJCR\. Let us define F(u)

= m ax[— log (w + s ) , 0 ] , wh ere 0 < e < 1 . I t i s easy to see t h a t F

sa t is f ies the hypotheses of Lemma 5•3-4 . Consequent lyf\Vv\^dx<Ci R"-^ w h e r e v{x) = F[u(x)'\.

Since v{x) = 0 on S and \S C\BJCR\ > (ci/2) |5A;i2|> if follow s fro m

Theorem 3 .6 .5 tha t

fv^dx^C2Rr

The theorem fo l lows f rom th is and Theorem 5-3-L

Theorem 5 .3 .3 . Suppose u is a solution of (5.3-20) on G. Thenu^ C^^(G) where 0 < //o < 1 ^'^d juo depends only on v, m, M, Co, and

/Lii, More precisely

\u(x) — u(xo) I < C L a-^ (\x — xo\IRY\ x^ B{xo, R), where

L = \\U\\IE-,S, B{xo,R + d)GG, r = vl2, d<R,

and C depends only on v, m, M , Co, and jui.

Proof. I t is suffic ient t o pr ov e th e in eq ua li ty . I t fol lows from th e

coro l la ry to Theorem 5 .3-1 tha t

\u{x)\ < CiLd-\ X^BR-^B(XO,R).

Le t us def ine w * an d M * as th e essen t ia l in f. a nd su p . of u{x) on

BR and le t us choose m (un ique) so th a t 1 5± | ^\BR\I2, S+ and S" be ing

the se ts o f po in ts x^ BR for which u(x) > m and u{x) < m, re s pec t ive ly .

If m* <m <M*, the func t ions [M * — u{x)]l{M'^ — fn ) a n d

140 Differentiability of weak solutions

[u(x) — m'^]l{fn — m*) satis fy the h y p o t h e s e s of T h e o r e m 5.3-2 on BR

w i t h ci = 1/2. I t fo l lows tha t wi < u{x) < Mi for x^ BR/2, w h e r e

fn^ =: m — h(m — m*), Mi = m + h{M* — m), A = 1 — C2 < 1,

C2 b ei ng the c o n s t a n t of T h e o r e m 5.3-2 w i t h ci = 1/2. The s ame re s u l t sho ld if w = m* or w = M* or b o t h .

N o w , let us define

q)(r) = [ess sup ^(A:)] — [essmiu{x)] for x^Br, r •<R.

We conc lude f rom the p r e c e d in g p a r a g r a p h t h a t

(p{2-^R) <^h^S, S = 2CiLd-\ n= \,2,,. .

T h u s

log(p(r) < l o g S — log A -\- {n + i) log A< l o g ( 5 / A ) + [(log/^)/(log2)]log{2?/r), if

n log2 < log{Rlr) < (^ + 1) l o g 2 .

F r o m t h i s it follows that

 0 and C which depend only

on V, m, M, Co, and jui, such thst

\\Vu\\l,<CL(rlRy-''+"\ 0 < r < R , L = \\VU\\IK, r = vl2,

for each R,0 <iR <,Ri, and each solution 0/(5-3-20) with || V ^HH.B < + >< -

Proof. E v i d e n t l y we may s u p p o s e t h a t the ave rage va lue of z* = 0.

From Theorem 3 -6 .5 , we conc lude tha t

\\U\\IJ,<CLR.

F r o m T h e o r e m 5-3-3 we t h e n o b t a i n

I u(x) - u(xo) I < Z . h | | o ^ ^ . ( i ^ / 2 ) - ( 1 ^ - xolIRr

<.Z2L'R^---f'^'\x-xo\''\ \X-XQ\<.RI2.

We define Y} as us ua l wi th a, G, D rep laced by ;', B{xo, 2r), and B{xo, r)

re s pec t ive ly , and put

(5-3-25) C{x) =r]^ '[u(x) -u{xo)], x^B(xo,2r), 0 < r < i?/4

in (5-3-20). We o b t a i n

0 = J^rj^[\/u ' a ' \7u + c{u — UQ) ' Vu] -\- 2rj{u — UQ) S77] ' a - Vu}dx,

The theorem fo l lows eas i ly by using (5-3-24) and the inequa l i t ies ofC A U C H Y and S C H W A R Z .

T h e o r e m 5.3.5 (Ex is tence and un iquenes s in the small)- There is a

constant JR2 > 0 which depends only on v, m, M, Co, and jbti such that there

exists a unique solution C/ o/(5- 1 -1) which^ H\Q \B (XO, R)] for each R < R^,

e^Lz B(xo, R), / ^ L9,sf [B(xo, R)], such that B(xo, R) G G; this solution

B y v i r t u e of Theorem 3-5-2, it is sufficient to prove that (5 .3-27)

ho lds for s ome K. Since R < R2, we conc lude , us ing Theorem 5 .3 .5 tha t

u ^ U + H on BR w h e r e U is the so lu t ion of (5.3-21) on BR w h i c h

^ HIQ(BR) and H is the solution of (5.3-20) such that H — u^ H\^{BR),

a n d t h a t

(5.3.28) l l V f / | l i ? < Z i l ^ l | ^ < Z i | ^ | | ^ , | | V i / | | / ? < Z 2 | | V ^ | | i ? < C 2 | | V ^ | | ( ? .

T h e n it follows from Theorem 5-3-4 that

N o w , let us define (p{s) = L"i sup | |V t /f^s for all e which sa t is fy

(5.3.26) with Li rep laced by L, i^ rep laced by 5 < i? , U be ing the solu

t ion of (5.3.21) ^HIQ{BS). Next , choos e an a r b i r t a r y e which satis f ies

(5.3.26) (Li rep laced by L). We m a y w r i t e U = Us + Hs on Bs w h e r eUs is the so lu t ion of (5.3.21) ^ HIQ(BS). O b v io u sl y e satisfies

l\e\^dx< [L^iSjRy -^^^'']' (rlSY-^+^^, 0<r< S.

T h u s , us ing the ideas of (5.3-28) and the definition of 99, we conc lude tha t

IIV UsWs ^ZiL^ ( S / i ^ ) - l + ^ I I V HsWs < Z 2 IIV t7||^ < Z2 L • (p{SIR).

N o w , s u p p o s e t h a t 0 <r < S <: R. T h e n

| | V ^ | | r < | | V t 7 ^ | | . + | |Viy5 | | r

^L{SIRy -^+^(p(rjS) +Z sL <p(SIR) • {rlSy-^+f^-.

Since e is a r b i t r a r y , we conc lude (se t t ing s = rjR, t = SjR) t h a t

(5.3.29) (p{s)^t^-^+f'q)(slt)+Zsq^{t)'{slty-''+f'\

O b v io u sl y cp i s m ono to ne an d 99(1) < Z i. So let us choo se a, 0 < cr < 1,

Then , obv ious ly

^ ( s ) < So s^-i+^, cT < s < 1, So < Z i a - ^ + i " " .

Using (5.3.29) with cr^ < s < a a n d t = (r~i s, we o b t a i n

(5.3-30) (p(s) < Si 5^-1+'^, where Si = So(1 + Zsco), co = a^^'''.

Since Si > So, (5-3-30) holds for o ' 2 < s < 1 . U sin g (5-3-29) w i th

cr < s < 0*2 a n d t = a~^ s, we conc lude tha t

^(5) < S2 5^-1+'*, 0r4 < S < 1 , S2 = So(1 + Z s (O) (1 + Z s 0)2) .

B y r e p e a t i n g t h e a r g u m e n t , we o b t a i n

(p {s ) < S s ^ - i + ^ , 0 <s < 1,

S= So(1

+Z3C0)(1 +Z3 a>2 )( l +Z 3ft>4 ). . .f rom which the theo rem fo l lows immed ia te ly .

The fo l lowing theorem concern ing the b o u n d a r y b e h a v i o r of solu

t ions of e q u a t i o n s of the form (5-3-21) is very eas i ly ob ta ined . More

gene ra l theo rems conce rn ing equa t ions of the form (5.1.1) are p r e s e n t e d

in the au tho r ' s pape r re fe r red to a b o v e ( M O R R E Y [14]).

5.4. The case v = 2 143

L e m m a 5.3.5. Suppose u is asolution 0/(5 .3 .21) which ^H\[Ga) and

which vanishes along Oa- We suppose that eand f satisfy the hy potheses of

Theorem 5.3.6 with B{xo, r) replaced by B{xo, r) 0 Ga for all XQ such that

B(xo, r) CB{0, a). Then, if we extend u to B{0, a) by

(5.3.31) u(x^,xl) = -u{-x\xl)

it follows that u C^{Ba) and satisfies (5.3.27) '^ith' G =Ba.

Proof. For, s uppos e C € Cc(Ba), an d we e x t e n d u, a ^a"^, e"", c'^,

a n d /, for oc <v, by (5-3.31) and e x t e n d a^"^, a"", e", and c^, for a and

/? < 1 , by the fo rmula

(5.3.32) (p(x\xl)=(p(-x-,x:).

T h e n

(5.3.33)

/ [Vf • (a-S/u + e) +C{c' Vu + / ) ] dx

= J[V C*'(ci-S/u + e) + C*{c'Vu+f)]dxGa

w h e r e C* (^^ K) =C (^', < ) - C ( -x\ < ) ,

^ * = 0 on dGa.

Accord ing ly the ex tended func t ion u satisfies (5.3.21) as e x t e n d e d to Ba

a n d e and /s a t i s fy cond i t ions l ike (5 .3 .26 ) the re . The l emma fo l lows .Theorem 5.3.7. Suppose G isLipschitz, u is the solution 0/ (5-3.21) in

HIQ(G), and e Z,2 (G ) , /^ L2s ,(G), and e and f satisfy

(5.3.34) J\e\^dx<Llr'-^+^f'

(5.3.35) f\f\dx<L2r^-^^^B(x.f,,r)nG

Then u C^{G) and satisfies a condition

( 5 . 3 . 3 6 ) / | V ^ | 2 ^ A ; < i ^ 2 ; . . - 2 + 2 ^ ^ K'^<^C\L\ + CILIB{Xo,r)f\G

where Ci and C2 depend only on v, m, M, jui, Co, JLC, and G.

Proof. The cond i t ion for in te r ior spheres is jus t Theorem 5 .3 .6- Each

p o i n t of 5G is in a set A^ C ^ w h i c h can be m a p p e d on Ga by a b i -L iP -

SCHITZ m a p as in § 1.2, Nota t ions . The form (5 .3 .2I) is pre s e rved by the

m a p p i n g and the t r a n s f o r m e d ea n d / satis fy the cond i t ions of L e m m a

5.3.5. The result follows.

5 .4 . Th e case v •= 2

T h e t h e o r y of §§5-2 and 5-3 carr ies over to case v = 2with on ly

ra the r mino r mod i f ica t ions . In fact the t h e o r y can begrea t ly s imp l i

fied in th i s c a s e . Howeve r , the re is a s l igh t complica t ion caused by the

fac t tha t the e x p o n e n t s' — v j{v + 2) = 1/2, so the e x p o n e n t 2s' = \

144 Different iabi l i ty of weak solut ions

and we cannot conclude from the CALDERON-ZYGMUND inequalities thatthe second gradient ] V^ F] of the potential F of a fu nc tio n/i n L\ is againin L\ and so cannot immediately conclude that V F ^ L2 as in Theorems

3.7.1 and 37.3.Lemma 5.4.1. Suppose u ^ H\Q{G) and q ^ Li{G) and satisfies

(5-2.4) j\q{x)\dx<.CQrf', for all B{xQ,r).

Then q • u and \q\ • \u\^^ -î(^) ^'^^

(5.4.1) j\q{x)'u[x)\dx^C^'C^'\\Vu\la'g"^'r-^-"\

(5.4.2) / I q{x) I • I u{x) \^dx < C2 • Co(|| Viî.«)2g'/V'^--^/2^

0 < / < /^,

q and u may he tensors; Ci and C2 depend only on X and fzi.Proof. It is sufficient to prove this for vectors u^ Q(^)- Then

(5.4.3) Ui(x) = - (27r)-l / I I - : |-2 (|a _ ^«) î^ (l) ^1G

using Theorem 2.7.3. Hence

J l q{x) • u{x) I ^:\; < (27i)-^f j\q[x) \'\S — x\-^'\\/u{^)\ d^dx.B(Xo,r)nG B{xo,r)nG G

(5.4.4)

Applying the SCHWARZ inequality to (5-4.4), we obtain

/ I ^(^) • ^ ( ) I ^^ ^ (2^)-^ f / ^^ / I ^ - P~ * I ^(^) I ^^-B{«o.r)n^ lGGf]B{Xo,r)

1/2X

(5.4.5) X [ / ^ f / I q(x) I • I ^ - ^ !" • I V w(|) |2^:vl^^^ 0 < ; . < / / ] .[G GnB{Xo,r) J

Using Lemma 3.4-3, we see that(5.4.6) / ^ f / I f - ^ M • k ( ^ ) N^ < 27r A-ig^ • Co ' ^

G GnB(xo,r)

Next, we define, for each |,

5(l.e)n-B(a;o.r)nG^

From our assumption on ^, we see that

(P^{Q) < Co Qfî and Co ^^ 1.

Accordingly

B{x^,r)nG

(5.4.7)

/ 11 — ^ r^ • i $'(^) \dx<, j Q-^ (P'^(Q) dQ.r)r\G 0

0 0

5.4. The case v = 2 1 4 5

The first inequality follows easily from (5.4.5), (5-4.6), and (5.4-7). The

second fo l lows f rom two appl ica t ions of the f i rs t wi th numbers A ' and

A " w i t h A' + A " = 1

Lemma 5.4.2. Suppose q satisfies the condition of Lemm a 5.4-1. Then,

for each e > 0, there are numbers C3 and C4 depending only on e, v, Co, / / i ,

and R such that

f\q\'\u\^dx<C^CoRf'^J\Vu\^dx u^ ^1^(0),G G

J\q\'\u\^dx<ef\Vu\^dx + CsJ\u\^dx G<ZB[XQ, R).G G G

Proof. The firs t fol lows by sett ing g = r = R in (5-4.2). To prove the

seco nd, le t / > 0 be given . Th ere exis ts a set of ^p as in the proof ofL e m m a 5-2.1. Then, as in the proof of tha t lemma,

Jl ^ I • I ^ |2 :v < 2 ^ / I ^ I • I 3? |2 ^^ < C Co rî^' / I V % P Â;

G 'P G 'P G

<, 2C Corfî f \\/u\^ dx + 2C Corfî f 2^\ S/Cp M ^ P dxG G

us ing the f i rs t resu l t and the fac t tha t each ^p has s uppor t in s ome

B (xp,r).

Us ing the s e l emmas in s tead o f Lemma 5.2 .1 , we eas i ly prove Theorem

5.2.1. In Th eorem s 5 -2 .4 an d 5-2.5 i t i s neces s a ry to requ i re t h a t / s a t i s f y

a condit ion of the form (5.2.4) in which case we find (us ing(5.4.1)) that

J\fv\dx^CiCo-\\Vv\\%'RîG

f\fU\dx<CiCo-\\VU ||».e • i? "! < £ (IIU l|.o)2 + C • £- 1 • R^"! • QG

in th e proofs of tho se theo re m s. In T he or em s 5-2.5 an d 5-2.6, th e re sul ts

are, re s pec t ive ly ,

II V u \\% < C [ a - i 11 ||o^^ + | | . Ijo ^ + Co Rî],

II V ^ i . , <.C[{R- r ) - i | | ^ | |o ,^ + | | . | | « ,^ + Coi^^ i ] ,

pr ov id ed th a t / a lw ays satis f ies (5-2.4). In Th eo rem 5-2.7, i t m us t be

as s um ed in add i t ion tha t th e /^ a nd / s a t i s fy (5-2 .4 ) un i fo rmly . Th e s ta t e

ment and proof of Theorem 5-2.8 carry over, us ing Lemmas 5-4-1 and

5-4-2 to show that d u and hence 99^ L^.

Th e theor em s in § 5-3 can be proved in m uc h th e same w ay as the y

were bu t r ep lac ing vjiv — 2) by an a rb i t ra r i ly la rge bu t f ixed f in i ten u m b e r 5 > 2 ; t h e SOBOLEV inequali ty (5 .3 .14) holds for any s in the

case r = 2 . H ow eve r , th e theo ry ca n be gre a t ly s impl i f ied in the case

y = 2 b y us in g th e original proof of the wr ite r as m odified to m ak e use of

recent s implif ications and to take care of the lack of self-adjointness in

th e equa t ion s (we hav e no t a s s um ed a^°' = a°'^ n o r b"^ = c*). We now

M o r r e y , M u l t i p le I n t e g r a l s | Q

present this simplified theory. All the theorems and their proofs generalize

immediately to vector functions except, of course, the last theorem which

concerns sub-solutions and corresponds to Theorem 5.3-'I-

Theorem 5 .4 .1 . Suppose u is a solution ^H\{G) o/(5.1.1) in which the

coefficients satisfy (5.1.2) and (5-1.3) wth v = 2. There are numbers C and

f ^ o , 0 < / o < /M i> which depend only on v, m, M, Co, and /ii such that if e

and f satisfy

(5.4.8) / I e\^dx< L\{rla)^f', j\f\ dx < L2(r/a)^

B{xii,r) Bixo.r)

0 < jbt < jLto, 0 < y < a, B (xo, a) C G,

then u^ CI ( G ) and satisfies

(5.4.9) J\Vu\^dx< C{L\ + Ll + L^ an) {rlafi",

L ^ - (11 ^^ W i a ) ^ ^ / (I V ^ |2 + a-^\u |2) dx,B{xo,a)

B(xo,r)cB{xo,a)cG.

Proof. Let us consider a ball B {XQ, a) C.G and let us redefine u in

B { X Q , 2 a ) — B(xo, a) so that the new u^ HlQ{B2a) and

\\'^\\U,Bix.,2ay<.Zl{y)L.

(Theorem 3-5.4)- Let us define

E ^ " = b'û + e ^ " , F = c'^'uôc + du + f, x^B(XQ, a).

E and F being 0 elsewhere. Then u, E, and F satisfy

(5-4.10) l[v,4a°^û,p-\-E°^) + vF]dx = 0, v^Hl^{Ba),B(Xo,a)

( 5 -4 . 1 1 ) | | i 7 | | o ^ ^ < i : 2 ( y / ^ ) ^ 4 . i : [ c i /V " ^ / 2 _|_ C iC o ( 2 a )^ /v ^ ^ - ^ / 2 ]^

as one sees from (5.1.3). (5-4.8), and Lemma 5-4.1 with G = B{xo, 2a).

Now, let us choose (i) polar coordinates with pole at X Q , (ii) a represen

tative u which is A.C. in d for almost all r and (iii) an i?, 0 <iR -Zi{L2 + Z2Lanl^) [Rja)^• | | V ? ;I ^ RBixo.B)

> -e{\\Vu\\lR)^ - 8(\\VH\\IR)^ - {Z^Ll+Zaân)e-HRIa)^^.

Using (5.4.11) and (5-4-12), we conclude from (5.4.10) that

(5.4.13) j\Vu\^dx<.Z^j\\/H\^dx+Z^{Ll + Ll + L'ân){Rla)'î^

B{xo,R) B{xo,R)

B{Xo,R)

2JT.

5.4. The case v = 2 1 47

where we have assumed that /j, < /î/2 and ju = (//i/2) — (A/4). If we nowchoose juo so that

/ o < /*i/2 an d juo < 1/2 Z5and if we set

(p(R) = J \Vu\^dx,B{Xo,R)

we see that

f\VH\^dx = I fr{Hf + r-^Hl)drdOo,B) 0 0

2JT.

= lf{ul{R. 6) + [M(i? , e) - « ] 2 } i 00

2J T

<ful{R, 0)d6<Rcp'(R).

0

Then from (5.4.13)> we conclude that

cp{R) < (2/.o)-i i^ 9 '(i ) + Z ,{Rla)^^, Z , = Z ^{L\ + Ll + L^ a^î)

from which we easily conclude (if 0 < ytt < //o), that

cf>[R) < [^(«) + Z 8( Z | + L | + Z2 A i) (2^/^)2^

On order to handle variational problems of degree other than 2(i.e. y :5 2 in (1.10.7) or (1.10.8)), it is necessary now to prove th e lemmascorresponding to Lemmas 5-3-1 — 5.3-3 and then prove the theoremcorresponding to Theorem 5.3-1-

Lemma 5.4.3. Suppose (i) that co ^ HIQ(G), (ii) that yj and xp co ^H\[G),(iii) \p{x) '> \ on G, (iv) y)co^x and yjâOJ^ L2(G), (v) P^L^ {G) and satisfies(5.4.14) J P^{x)dx< Corf'i, P{x)^0.

B(Xf,,r)G

Then there exists a sequence {^n} -> o) in H\Q [G) , in which each f ^Lipc{G), such that ip Cn,oc -îp o),oc and PipCn-^Py ôj in L2{G). Ifoj{x) ^ 0, the Cn ^(^y be chosen > 0.

Proof. As we noted in the proof of Lemma 5.3-1, ipo)^H\^^{r) onany r' D G if we merely define ip a> =^ 0 on F — G. Thus it follows fromLemmas 5.4-1 and 5-4-2 that Pipoj^L^ (G). The remainder of the proofis essentially the same as that of Lemma 5-3-1-

Lemma5.4.4.

Suppose that U^

Hl{D) for each D CG G, that U{x)>: i on G, that W{x) = [U{x)]^ for some A, 1 < A < 2, and that P satisfieshypothesis (v) of Lemma 5.4-3 on each D (ZC.G with Co = Co [D). Then,for each such D,

VW, PW, PVW, and P^w^Li(D).

Proof. For PU, P U^-\ and V C/ ^ L2{D).

10*

1 4 8 Differentiability ofweak solutions

Theorem 5 .4 .2 (Exis tence and u n i q u e n e s s inthes ma l l ) . There is a

constant i? 2 > 0 which depends only on v, m,M, Co, and jui such that there

is aunique solution U of (SAA) which ^HIQ[B [XQ, R)] foreach R< R2,

each e^L2{Bji), andeach f^Li{BR) satisfying (5 .2 .4); this solution

satisfies

llV?7i.c<Ci||e|lB+C2Coi?''i,

where Ciand C2 depend only on the quantities above. If u"^^ Hl[B {xoy R)],

and R<. R2, there exists aunique solution Hof the homogeneous equation

such that H — u*^ HIQ[B (XQ, R)].

The proof is l ike tha t ofTheorem 5•3-5-

Theorem 5.4.3. Suppose (i) that U ^ Hl{D) for each DCCG, (ii)

that U(x) > 1 inG, (iii) that U^ L2 (G), (iv) that W = U^ satisfies (5.3.4)

for some A with 1 < A < 2, (v) that asatisfies (5.1.2) and that b, c,and d

satisfy (5.1-3) on each D CCG with Co= Co {D). Then W^H\{D) and

is bounded on each D GG G.

Proof. Since U ^ Hl{D) foreach JD C CC, itfollows that i7^ ^ L2(D)

for every T > 1 and e v e r y D GGG. We may t h e n r e p e a t the f i rs t par t

of the proof ofL e m m a 5-3-3» s e t t i n g

w h e r e rj h a s s u p p o r t in D' and D GGD'a. The re s u l t is equation (5.3-12)w i t h G rep laced by D' and T by A:

/1 \ 7WL Pdx< Z 2 A2 /P2wldx +Z XJ\ Vfj p U^ Uf' dx.D' Df D'

Using Lemma 5-4-2 wi th £Z2 ^< 1/2, we f ind tha t

(5.4.15) j\VwL\ dx<^ j [Z^rf +Z \\/rj\ )mUf- dx,

I t is now poss ib le tolet L -> 00 and th i s ob ta in W ^H\ {D).B ut now, p ick a hal\B{xo, R) G D'w h e r e R <. R2 (which now depends

on D') and let Hbethe s o lu t ion of the homogeneous equa t ion (5 .1 .1)

which co inc ides wi th Won dB{XQ, R). Let w = W— H. T h e n z£ =: 0 on

SBR and satisfies (5-3-4) there. Ifwe t h e n set C =^"^, we see t h a t

^^^ =w^oc and C = ?!£ if z£' > 0

C,a = f = 0 if ze; = 0 (a.e.)

Accordingly (5-3-4) with wand t h i s f b e c o m e s

/ [vC • ^ • vC + ( + c) c • vC + dC^] dx<oB{xo,R)

which imp l ie s tha t f = 0 s ince R R2. T h u s w <,0 or W<, H on

B{XQ, R). But f rom Theorem 5-4-1, itfo llows th a t His H o l d e r c o n t i n u o u s

in te r io r to B{xo, R).



149

5.5. Lp an d S ch au d er es t imates

W e ob ta in fu r the r e s t im a te s conce rn ing the s o lu t ions of e qua t io ns

(5.1.1) when the a°'^ are cont inuous and the coeff ic ien ts and e a n d /

s a t i s fy va r ious s upp lemen ta ry cond i t ions . In ca s e the a^"^, }f-, a n d e"^ ^ C^^and the c«, d, and / s a ti s fy ce r ta in s upp lem en ta ry cond i t ions , we s hal l

s h o w t h a t \l ufC^^.

We sha l l ob ta in corresponding resu l ts fo r weak so lu t ions of sys tems

in § 6 .4 . Th ese resu l ts were o b ta ine d in th e tw o d im ens ion a l case in

MoR R EY [7] , C hap te r V I , § 6 , wh ere a d i f feren t m et ho d w as deve lop ed

to t rea t the case where the coeff ic ien ts a'^f were co ns t an ts sa t is fy ing

(5.2.17) and b = c = d = 0. Let us se t

an = aij, alf ^ afl = bij, a^f = Cij.I t tu rns ou t tha t i f the u^^ H\{G) and satisfy (5.1.1), i .e.

JlKi^ij K + hj K + di) + oji(bji ul + Cij u\ + ei)] dxdy = o,G

then the re ex i s t con juga te func t ions vi s u c h t h a t

-Vix = — [hi K. + Cij u% + ei), Viy = atj 4 + btj u{ + di.

If we le t w be the 2Ai^-vector [u^, . . ., u^, v^, . . ., v^) a n d g be th e 2 A^-

v e c t o r [di, . . ., d^, ^1, • • ., iv^), the s e equa t ions becomeAw.x + Bwy + g=^0

where A and B a re 2N X 2N s qua re ma t r i ce s hav ing a pa r t i cu la r fo rm.

By us ing the e l l ip t ic i ty and cer ta in theorems on A-matr ices (see B O C H E R ,

§§ 91— 96, i t i s seen th a t we m a y f ind a ffine t ran s fo rm at io ns

w = D w' a n d g' =^ C g

s o th a t th e t r ans fo rm ed equ a t ion s hav e the s ame fo rm a s the con jug a te

e q u a t i o n s a b o v e b u t w i t h

w h e r e h is a r b i t r a r i l y s m a l l . M o r e o v e r, t h e m a x i m u m a n d m i n i m u m

mag n i f ica t ions of the t r ans fo rm a t ions an d the i r inve rs e s a re bou nde d by

a cons tan t depend ing on ly on m, M, a n d h. An in te re s t ing gene ra l i za t ion

o f t h i s m e t h o d w a s e m p l o y e d b y C . MIRANDA ( [1]) , who in t roduced ex

teri or differentia l form s (see C ha pt er 7) ins tea d of th e conju gat e func

t i o n s .

We begin by considering equations of the form (5.1-1) on spheres BRor hemis phe re s GR where the s ymmet r ic pa r t o f the a^^ m a t r i x i s j u s t d""^

an d prov e loca l d i f fe ren t iab i l i ty . A ne ig hbo rho od of each in te r ior po in t

can be m ap pe d on s uch a s phe re by an af fine m ap in wh ich the m ax im um

and min imum magn i f ica t ion and thos e o f i t s inve rs e a re un i fo rmly

bounded by a cons tan t depend ing on ly on m a n d M. In the case of a

b o u n d a r y p o i n t , one can f irs t perform the a f f ine t r ans fo rma t ion as

above , fo l low tha t by a r o t a t i o n to m a k e x"^ ~Q the t a n g e n t p l a n e , and

fo l low tha t wi th a map of th e form

(5.5.1) V = ^ ^ ^ = 1, . . ., 1; — 1, V = X'' —f{x^, . . ., A;' '-!)

which carr ies a p a r t of dG i n t o the p l a n e a : x^= 0. The a n t i s y m m e t r i c

p a r t of the c o n s t a n t m a t r i x UQ^ may as well be omi t ted s ince

(5.5.2) f v,o:a^û,pdx = f vôcd'^'û^pdx if V^HIQ{G)G G

a s is easily seen by approx ima t ions , u s ing Lemma 4 .5 .7 -

H a v i n g m a p p e d a n e i g h b o r h o o d of XQ o n t o B2R or G2R, we then a l t e r

the coeffic ients by defining

Ê^W = ^f + (p(B~^x I) [a-^{x) - af(x)],

^^'^'^^ h%[x)=^(p'h-, c%= (pc-, dn=(pd,

w h e r e 9?^ C°°(Ri), q){s) = \ for s < 5/4, (p{s) = 0 for s > 7/4 and cp is

non- inc rea s ing . If ^ is a s o lu t ion of (5 .1 .1) wi th suppor t on BR or GR U (TR,

i t is also a s o lu t ion of

(5.5.4) fv,4alû,^ + b%u+ e"") + v(c%u,y, + dRU + f)dx = 0,G

V^HIQ(G), G = B2R or G2R.

W e t h e n w r i t e

(5.5.5) u = UR + HR, UR = Q2R[(aR — 0) • Vu + bnu + e] —

— P2R[CR- Vu + dRU +f]

w h e r e , in th e c a s e 5 2 i ? , Q2R a n d P 2 i 2 are the re s pec t ive quas ipo ten t ia l s ,

as defined in § 3.7, and p o t e n t i a l s , e x c e p t t h a t in the case 1; = 2, we define

pR(f) as the o r d i n a r y p o t e n t i a l o f / o v e r BR m i n u s its ave rage va lue . The

reasonfor

th is def in i t ionof PR in the

caser = 2 is

t h a tif V = PR{f),

t h e n , in a d d i t i o n to the re s u l t s in T h e o r e m 3-7.1, we a lso have

\\Vr,<C{v,q)RHf\\o

in all cases. This follows from the t h e o r e m s of § 3.6.

I n all cases , we see t h a t Q2R-^0 as :V -> 00. This is t r u e of P2R also if

r > 2; bu t if r = 2, P2R migh t become loga r i thmica l ly in f in i t e . So if u

h a s s u p p o r t in BR, it fo l lows tha t HR = 0 if 1 > 2, or HR = cons t , if

r == 2. In tha t c a s e

(5.5-6) UR —TRUR=^VR= Q2R {e + bR HR) — P2R {dR HR + / )

TR UR = Q2R [{aR ~ ao) ' \7UR+ BR UR] — P2R (C R ' V UR + dR UR) .

I t is t h e n s h o w n t h a t if R is s ma l l enough , \TR\ < 1/2 in HI(B2R),

w h e r e u and UR are k n o w n to be, and also in a s pace H\{B2R) for s ome

^ > 2 or a space C\{B2R). We then conclude that u e HI{B2R) or C]^{B2R)

15 1

as the case m ay be . A s imi la r p rog ram is ca r r i ed out for hemis phe re s G2R.

T h e g e n e r a l b o u n d e d d o m a i n G is h a n d l e d by a p a r t i t i o n of u n i t y .

Definit ion 5 .5 .1. In the space CIQ{B2R), we defi ne ||| ||||J gij =

hAe. B2R)_+ R-qe\\\lj,, J!|^|||o^ being the n o r m in C^ {B2R). In thespace C]^[B2R), we define

I n the s pace H\{B2R), we define the n o r m

We define the s pace L2>f^(B2R) to consis t of Ql\f^Lp{B2R) s u c h t h a t

J \f{x) \P dx <L2> r '-'P+'P^ < LP (2 Rf-v+vi^

B{xo,r)r\B2R

for every B{xo, r) and we define ||/||p,^,2i? =" ^^^- - •

Then, f rom SOBOLEV'S lemma (Theorem 3 .5 .3) and T h e o r e m s 3 7 . 1 —

3.7.3, we o b t a i n the fo l lowing resu l ts :

T h e o r e m 5.5 .1 . QR is a hounded operator from Lq{BR) to H\[BR) for

each q'> \, and from C^^^Q{ER) to C^^BR) with hounds depending on q and

/Lt (0 <C ju < \), respectively, hut not on R. PR is a bounded operator from

LP(BR) to HI(BR) and to HI{BR) with hound independent of R if \ <

p<^v and q = p vl{v — p); PR is also a bounded operator from LP^BR),

1 ^p <.v, or LP(BR) with p = r/(1 — /^) into C^BR), with hound in

dependent of R. In fact if u — QR{e) and C^LQIBR) then U^HI(BA) for

any A and \\Vu\\ls^ ^ C | |^ | | J^; if e^ C%{BR), then u^ CI(BA) with

hf,{S7u, BA) < C hf,{e, BR) for every A] if v = PR(f) and f^ Lp (BR),

\<P<v, then V^HKBA) for any A and || Vt; ||,%^ < C | | / | |o^ if

q =pvl{v —p)\ iff^Lp,f^(BR), \ <p <v, or to Lp withp = r /(1 — ju),

then v^ C'^{Rv) with hfi(^v, Rv) < C | | / | | , the norm being in the proper

space. Finally, if v = 2, and f satisfies the condition

(5 .57 ) / \f{x) \dx^L rn, all B{XQ, r), pix > 0,-B(a!o.r)nBB

then Pnif) C HI (BA) for any A and ^ PR(f) ^ L2,II(BA) for any A and

\\PR{f) M,2P. < CLR^i'i, I V P « ( / ) ||§ .„ ..,« < CL.

Definition 5.5.2. We say t h a t the coefficients a, h, c, and d satis fy

the i^^-condi t ions on a d o m a i n / " i f and on ly if the a^^ are c o n t i n u o u s on

r and the b"-, c°', and d are m e a s u r a b l e t h e r e and

(i) b^ and c«$ Lv(r) and d^Lvl2(r) if vl(v — \) < q <v and v > 2(ii) / ( | & | * ' + | c | * ' + \d\''^^)dx<L''r''n, /^i > 0,

rf]B(xo,r)

for every B(xo, r), if ^ = r > 2,

(m)b^^Lq(r), c^^Lr(r), and d^ Lp(r), p = v ql(v + q), ii q > v.

W e say t h a t the coefficients satisfy the C^-conditions on F if and on ly if

t h e a""^ a n d b^^ Cl{r) and c^ and d^"^ Li{r) and sa t is fy

(5.5.8) J (\c{x)\ + \d(x)\)dx^Lr'-^+^ , xo^Rv , r > 0.-B(a'o.r)nr

I t i s des i rab le to prove the fo l lowing lemmas :Lemma 5.5.1. (a) Suppose F is a hounded strongly Lipschitz domain

and (p^ Li{r) and satisfies (5-5.7) 'iîth BR replaced by F. Then, for each

£ > 0, there is a C [s, jbti, v, F) such that

(5.5.9) / I ^ W I • i ^ W I ^ ^ < CR'L '\\ u \\l • rî-%

u^HliF), \r\=y,R^

(b) Suppose u^ HI{B2R) with q > r . Then

( 5 . 5 . 1 0 ) \u[x)\ <^C[v,q)R^-->^'\\u\\, x^B^R.

Proof, (a) O n acco un t of th e exten sion t he or em (Th eorem 3.4.3)> we

m a y a s su m e t h a t u^ HIQ(F). Then, f rom Theorem 3 .7-2 , we conc lude tha t

/ I (p{x)\\u{x)\dx<F-^fdxf\ (p(x)\'\i — x\^-''\Vu(i)\di,B{xo.r)r\r B(rco,r)nr r

(5.5.11)

Ap p ly ing the H o lde r inequ a l i ty to th e doub le in teg ra l on the r igh t in

(5.5.11), we see that our integral / on the left sa t is f ies

/ < r - i 4 ^ - 1 ) / * ' • 7 l / ^ ii = J d x f \ ^ - x \ - ^ + ^ \ i p { x ) \ d ^B(xQ,r)rr) r

(5 .5 .12) l2 = fdxfli -X l -^C-i) I (p{x) I • I V ^ ( l ) j " ^ ! .

B(xo,r)nr r

F ro m L em m a 3-4 .3 , i t fo llows th a t

( 5 . 5 . 1 3 ) Iiâ-^F,{\F\ly y )^'''Lrf'i = G-'^FrR ''Lrf'i.

In o rde r to e s t ima te I2 , we f i rs t no te tha t0 0

/ 11 — ^ [- ("-D \q)(x)\dx<^ f s-^(»'-i) (p'{s; ^)dsB{xo,r)nr 0

(5.5.14) ==a(v — l ) ! ^ - ! - ^ ^ - ! ) ^ ( 5 ; ^)ds0

where

( 5 . 5 . 1 5 ) (p{s; i)=l$p{^)\dx^lBi$.s)nB{xo,r)nr [^

where

(Lsî, 0 < s < r

[Lrî, s ^ r .

Fr om (5.5.14) an d (5-5.15), we find th a t(5.5-16) l2<Zi{v, /Lii, o)Lrî-^(''-^)('\\u\\lY, 0 < a <fiil{v - $.

The result (5-5-9) follows by taking s = a{v — ^)lv < / / i / r . From th is , i t

follows that (5-5-9) holds for values of s s u c h t h a t juijv < £ < / / i .

(b) follows from Theorem 3.5.I with p = q ^ v a n d m = \ and from

the definition of the ' | | u IJ gR-

5-5. Lp an d S C H A U D E R estimates 153

Theorem 5.5.2. (a) Suppose the coefficients a, b, c, and d satisfy the

H\-conditions on BA where q'>vl{v ~ 1) and the equality holds only if

V = 2. Then for each R <, Aj2, the operator TR is hounded on H\[B2R}

and has abound of the form e{R) where e{R) ->0 as R -^0\ e{R) depends

on the moduli of continuity of the a'^^ and on v, m, and M and on

sup J ( | 6 | ^ + |c|»' + l^l^'S) dx in case (i) (of def. 5.5-2),

L and fii in case (ii), and

sup J {\b\Q + |c| ^ + \d\P) dx in case (iii).B{xo,R)nBA

(b) / / the coefficients satisfy the C^-conditions on BA, then, for each

R <, Ajl, TR is bounded on C\[B2R) with a bound of the form CR^ whereC depends on v, m, M, L, /bt, and the C^-norms of the coefficients a and b.

Proof. We set

eR = {aR — ao) ' VUR -\- bR UR, fR = CR- \/UR + dR UR,

rj(R) = ma.x\aR{x) — ao] .

In the H^ case we see, using Lemma 5-5-'I (with g){x) = l&i?!" + \CR\^ +

+ IdR]""^) and Theorem 3-5.5, that

(i) \\eRt^[v{R) + C\\bR\\^]^\\uR\\l4fR\\l<Cq(ii) II en ||S < [rj{R) + C{LR^î)^'^] '\\ UR | |J, UR t% < C(LRî) '|| UR | |J,

(iii) 11 CR ||o < [rj (R) + C\\bR ||o Ri-^'a-i '\\ UR \\l,

\\fRt^[\\cRr. + C\\dRtRi-^'ay\\uR\\l

in the respective cases; here we set ^ = r ql(v + q) and omitted the sub

script 2 i? on all the norms. If ) = 2 in (ii), the e inequality still holds

and fR satisfies (5.5-7) with L replaced by CL'\\UR\\1{\ + LRî). Inthe C}, case, we note first that

I bR{x2) - bR{xi) I < I (PR{X2) — (PR{XI) I • I bR{x2) I + (PR{XI) • | bR{x2) - bR{xi) |

(5.5.17) <{CBo'R-f^ + Bi) \x2-x1\f', X^^X^^B^R,

(pR{x)^(p{R-^\x\), \bR{x)\^BQ, X^B^R, BQ = m2ix\b{x)\,\x\Â

Bi - h^{b, BA).

Using this idea again, we obtain

\aR{x2) — aR[xi)\ < 2Ai\x2 — ^ l | ^ Ai = h^{a, BA),

/ \fR{x) \dx<LR^{\+R)''\\\u IIIJ.2E • ^^-1+^Bixo,r)riB2R

The theorem follows from Theorem 5.5.1-

http://x2-x1/f'

http://x2-x1/f'

http://x2-x1/f'

W e can now p r o v e the fo l lowing in te r io r boundednes s theo rem:

Theorem 5 .5 .3 . (a) Suppose that (i) the coefficients satisfy the H\ and

H\, conditions on the domain G, where <.q <. q', (ii) suppose

e^Lq, (D) and/^ Lp, {D ) for each D G G G, wherep' =v q'l(v + q') > '^

and (iii) suppose u^ Hl{D) and satisfies (5-2.2) (with ^ = 0) on such D.

Then u ^ Hi, [D ) for such D.

(b ) Suppose that (i) the coefficients satisfy the H\ and CJ conditions on

G, q ~> vj(y — 1) and 0 < / < 1; (ii) suppose e^ C^^{D) and / ^ Li^(Z))

for each DCC .G, and (iii) suppose u^H\{D) and satisfies (5-2.2)

(A = 0) on such D. Then u^ Cl(D) for each such D.

Proof, (a) E a c h p o i n t XQ in G is the c e n t e r of an ell ipse which can be

m a p p e d o n t o B^R by an affine ma p as descr ibed above , where R is sosmall th at the boun d TR is < 1/2 on H\[B2R) and on H\,{B2R) where

we assume f i rs t tha t q' <v ql(v — q), q <v. Let C$ Cl{G) and h a v e

s u p p o r t on the i m a g e of BR and let U be the t r a n s f o r m of C ^- T h e n , by

replacing z; by C ^ in (5.2.2), we see t h a t U has s u p p o r t on BR and sa t is

fies the t rans formed equa t ions (5-2 .2) (A — 0) w i t h e a n d / r e p l a c e d r e s

pec t ive ly by

J^ l = f e- - al^C,û, FR = Cf-c%^,ccu + C,4^%û,^ J^blu + e-).

(5-5.18)

Since U has c o m p a c t s u p p o r t , we see t h a t

U=UR + HR, UR -TRUR=VR= Q2R (ER + bR HR) -

— P2R {F R + dR HR) (HR = CO USt.).

Since q' <,v ql(v — q), it fo l lows tha t p' <,q so t h a t E^ Lq,(B2R) and

F^ Lp, (B2R) SO t h a t U^ HI,(B2R). Since it is c lea r tha t the coefficients

satis fy the ^ ^ „ - c o n d i t i o n s if they sa t is fy the //"J ,-conditions and q" <iq'

or if they sa t is fy th e CJ cond i t ions we may use the re s u l t to p r o v e there s u l t for any '. If Z) C C (^, we may find a p a r t i t i o n of u n i t y on D each

m e m b e r of w h i c h has s m a l l s u p p o r t as a b o v e .

I n the CJ-case, we f irs t show th a t u ^ HI, (D ) for s o m e q' > r, for

each Z) C C G. T h e n we see t h a t E^ Cl(D) so t h a t U^ CI(B2R). T h u s ,

a s above , u^ Cj^(D) for any D CC.G.

W e need the fo l lowing lemma for la te r re fe rence :

Lemma 5 .5 .2 . Suppose the coefficients satisfy the H\-conditions on BA-

Then there is an RQ, Q <C RQ <. Ajl, such that if u has support on BR andsatisfies (5 -1 -1) there with e^Lq andf^ Lp where v>2 andp = qvl(q + v)

> 1, then

'MU<Cx{le\ln + \\f\lE), 0<R<Ro.

If V = 2 = q, so p = i, the same result holds with \\f\\l^B replaced by

CLRfî if f satisfies (5-5.7). If the coefficients satisfy the Cji-conditions

5.5- LpandSCHAUDER estimates 155

on BA, there is an Ri, 0< i ? i < ^ / 2 , such that if u^ ClQ(Bji) and satis

fies (5.1.'I) there with e C^(Bji) and f^Li^(^[Bii), then

^^'-<C^{'\\e\\U + \\ni,,.n).uR •

The constants RQ, RI, Ci, andC2 depend on thequantities specified in

Theorem 5.5.2.

Proof. Since uh a s s u p p o r t inBR, we have s een tha t

U =UR+ HR SO UR— TRUR = VR + WR, VR = Q2R [c] — P^R ( / ) ,

W R =- Q2R {bR HR) — P2R (dR HR)

w h e r e HR = 0 if i' > 2and is a c o n s t a n t if > == 2and TRhas b o u n d

< 1/2 if i? < i? o orRi. li V = 2,we see, s ince \/ u = VUR and uhascompact support, that '\\U\\IR < C'WURWIH SO '\\HR\\IJ^ < C'\\UR\\IR.

S e t t i n g p = qvl(q -\- v) wefind as inthe proof ofTheorem 5 .5 .2 tha t if

r / ( r — 1) < q <iv, t h e n

II bR Hn tn < C 'II UR ||»« j] hn fk, \\ du HR \l < C 'f UR ||I«\\ dR | |o,,.^.

Since || &i? IJjj and|| i? ||J/2,E ^ - 0 asR->0, the result follows in t h i s

case . The o ther cases are proved s imi la r ly .

Definition 5 .5 .1 ' . We define then o r m s and spaces of func t ions on

G2R as they were def ined on B2R but w i t h B2R rep laced by G2R and the

ftmctions u (in C1(G2R) or H\[G2R)) being required tovanish along a2R.

The func t ions e'^ are s uppos ed tov a n i s h a l o ng ^ 2 i ? bu t not neces s a r i ly

a long a2R. W e def ine P2i2( / ) as th e re s t r ic t io n to G2R of the fo rmer P 2 R ( / )

w h e r e / i s t he ex tens ion toB2R by ' ' n ega t ive re f l ec t ion"

We define Q2R [e) = U hy the fo rmula

(5.5.20) U[x) =^ -fKo,4 - I)eHi)di + 2S fKoA^ - ye [ )d^^2 R ""^^ G2R

w h e r e we a s s u m e t h a t ehas b e e n e x t e n d e d by ' ' pos i t ive re f lec t ion"

f elx"", x') if x'' > 0,

(5.5.21) l[x\ <) = ] ^ , -

KQ is the e lemen ta ry func t ion for L a p l a c e ' s e q u a t i o n .

U s i n g the t h e o r e m s a b o v e , we o b t a i n the following result :

Theorem 5 .5 .1 ' . Theorems 5.5.1 and 5-5.2. and Lemma 5.5.2 hold with

BR replaced byGR; the H\- andC\-conditions onthe coefficients being

defined as above.

Proof. I t remains on ly to prove tha t the second te rm in (5 .5-20)

€ C1{Â) for any A. Le t th i s be deno ted by V. F o r %" > 0, we ha ve

= 2j, f^2Ko,4x - i)'[e-{S) -e-(x)]di

s ince e vanis hes a long ^ ^ an d th e in te gra l of Vîô ,a overJf?" converges

abs o lu te ly to 0 . F rom th i s we conc lude tha t

I V2F{:^ ) | < C{v,iLt) K{e, G2R) M ' * - !

f rom which the resu l t fo l lows , us ing Theorem 2 .6 .6 .

B y the m et ho d of p roof of Th eor em 5 .5-3. us ing the resu l t of T heo rem5.5.2 for G2R, we can prove the fo l lowing theorem:

Theorem 5 .5 .4 ' . Suppose that G ^ C^, the coefficients sa tisfy the H\-

and H\,-condition on a domain F Z) G, where vj{v — 1) ^q <iq' and

p ' = vq'l(v + q') ( > 1 ) , suppose e^Lq,(G) and f^Lp,{G), and suppose

u^H\{G) and is a solution of (5-1.1) on G. Then u^H\,[G). If, also,

G^ CJ^, the coefficients satisfy the Cj^-conditions on F, e^ C^(G), and

feLi,^(G),thenu^Cl{G).

We now prove the fo l lowing a pr ior i bound:

Th eor em 5.5.5 ' . (a) Suppose that G ^ C^ and that the coefficients

satisfy the H\-conditions on F Z ) G with q ^vl(v — 1), the equality hold

ing only if V = 2. Suppose also that e^Lq{G) and that f^ Lp{G), where

p == qvl(v + q) {>\) and where f satisfies (5-5-7) if p = 1- Suppose that

u satisfies (5-1-1) on G. Then

(5.5.22) l k K . « < c ( | | . | | » + i / K + ||«||?)

where C depends only on v, m, M , q, G, the moduli of continuity of a, and

hounds for the coefficients) if v =^ 2 =^ q, so p =^ \, the term \\f\\ must be

replaced by L.

(b) If G^C^^, the coefficients satisfy the Cj^-conditions on Fz>G ,

e^ C^{G) and f^Li^[x{G), then

( 5.5 .2 3) l | | « | | | i . e < C ( i ^ I | | « . e + l / i ? , „ . « + ||M|!S).

Proof. W e pro ve (a) w it h ^ > 1, th e proof of (b) an d th e last case of

(a ) a re s imi la r . Each poin t X{i of G is in a ne igh borh ood ^ C a ne ig hbo rhoo d

I t which can be mapped as descr ibed a t the beginning of the sec t ion on to

B2ROT G2RUa2R so th a t 9^ is m ap pe d o nto BR or GRU(TR, w h e r e R is sosmal l tha t Lemma 5 .5 .2 ho lds , and we may choose a par t i t ion of un i ty

{Cs} of the proper c lass , eachf^. hav ing s upp or t in s ome one ne ighborhood

9^. Then, if Us a n d Cs are the t rans forms of Cs ^ and fs , respectively (on

BR or GR), we s ee th a t Us satisfies (5-1-1) with e a n d / r e p l a c e d b y Eg a n d

Fs as given in (5 .5 .18) with the obvious changes of notation.



5-6. The equation a'S7û-\-b-Vi^-\~ou~?.u=f 1 5 7

Now, suppose there is no such cons tan t . Then there a re sequences

{^n}, {en], a n d {fn} s u c h t h a t

(5.5.24) II Un Wio = 1 , II ^ ^ \\U + Wfn WIG + II Un ||S.c.-> 0 ,

a n d Un—7 ti in H\{G). Since Un-û in L^(G) and hence in Ll{G) i t

fo l lows tha t u = 0, Thus we conc lude tha t Ens ->0 in Lq[G), [G = B2R

or G2R) Fins-^0 in Lp{G) and i^2ws-7 0 in Lq{G), F2ns = Cs,oca'^jf Un,p,

Fins = Fns—F2ns' T h u s V ^ ^ 2/2 ^ i w s - > 0 i n Lp{G) a n d S/'^P^RFÛS

-V 0 in Lq(G), s o t h a t P2R{Fns) - ^ 0 i n Hl(G). B u t t h e n Uns->0 in

H\{G) for each s (Lemma 5-5.2) s o t h a t ^ ^ - > 0 in H\{G). But th is con

t rad ic ts (5 .5 .24) .

Remarks. From Theorems 5 .5 .4 and Theorem 5 .5 .5 and the proof of

T he or em 5-2.4 w e conc lude th e following: Suppose G, the coefficients, ande andf satisfy the H\-conditions and either the conditions in Theorem 5-5-5 (^)

or (b) and suppose u^ Hl(G) and is a solution 0/(5.2 .2) withX = Ao, where

Ao is defined in Theorem 5.2.1. Then u^ Hl{G) if q > 2 or C]^{G), respec

tively, and the hounds (5.5-22) or (5-5-23) hold without the term \U^^Q on

the right. For i f no t , sequences [un], {%}> and [fn] wo uld ex is t sa t is fy ing

(5.5.24) w it ho u t th e te r m || » ||i,6? on th e rig ht . B u t, from th e proof of

T he or em 5-2.4, i t fol lows th a t w^ ~ > 0 in H\Q{G) and so in L\ (G) a n d t h e

remainder of the proof proceeds as before .

5 .6 . Th e eq u at ion a* V ^ u -}- 6 • Vw -{- cu — Xu =f

Using t he g enera l the or y of § 5-2 toge t her w i th T heo rem 5-2 .8 , we

eas i ly deduce the fo l lowing pre l iminary ex is tence theorem.

T h e o r e m 5.6 .1 . Suppose G is of class C \, a ^ C^ on G, b is bounded on

G, c^ Lv{G) and satisfies (5.2.11) there, and a satisfies (5-1-2) there. Then,

if X is not in a set b without {finite) limit points, there is a unique solution of

(5.6.1) L u — X u ^ {a ' \7 '^ u -\- b ' \/ u -\- c u) — X u =^ fwhich ^ Hl(G) n H\Q[G) for each f^ L2{G). If X^h, there is a non-empty

hut finite-dimensiona l man ifold of solutions in HI (G) 0 HIQ{G) with / = 0.

Proof. Let us define C(u, v) as in § 5.2, an d define

B{u, v) = J{v,cca°'û,p + v[(a^^ — b'^)u,cc — cu]}dx,G

L{v) = — ffv dx.G

Then, from Theorem 5.2.8 i t fol lows that (a) if X is not in a set b, there isa unique solution ^ of (5-2.2) in H\Q[G), and (b) that solution (if it exists)

^ Hl{G) an d satisfies th e diffe renti al e qu at io n (5.2.12) (a.e.). B u t (5-2.12)

reduces to (5-6.1) in our case.

In the theo rem above , we were fo rced to a s s ume tha t a^ G\(G) in

order to be ab le to use the H I L B E R T space proof given above and in § 5 .2 .

15 8 Differe nt iabi l i ty of we ak solut ion s

I t has long been known in the case v = 2 (see the art ic le by L. LICHTEN-

STEIN in the Enzyklopadie der mathemat ischen Wissenschaf ten [3]) and

was proved in 1934 by J . SCHAUDER for any v tha t i f the a, h, c, a n d /

were mere ly H O L D E R con t inuous , then the s o lu t ions ^ (^%,(G) and to

C^ (G) if ^ = 0 on ^ G a n d G is of class C^. We shall first obtain an ex

is tence theorem for the case where G is of c lass CJ, a is m ere ly co nti nu ou s

a n d h a n d c sa t is fy th e condi t ions abov e . W e sha l l th en ob ta in SCHAUDER'S

re s u l t s . The re a re s ome a p r io r i bounds wh ich have been ob ta ined by

CoRDES ([1], [2], [3]) for the case that the a"^^ a r e m e r e l y b o u n d e d a n d

m e a s u r a b l e w i t h & = c = 0 , A l th oug h the s e a re im po r ta n t for th e s t ud y

of quas i - l inear equa t ions , the resu l ts a re no t comple te ly genera l and

require ce r ta in spec ia l techniques for the i r p roofs . Accord ingly we sha l l

no t cons ider these resu l ts .O ur me th od of p roof i s to reduce the pro blem to one on spheres or

h e m i s p h e r e s BR^ or GR^ w h e r e a"^^ = Zl"'^ as in th e pr ec ed in g se ct io n; he re

w e m a y t a k e a^^ =^ a^^. As in that section, we shall consider f irs t func

t ions wi th s uppor t in BR or GR. W ithou t a l t e r ing the fac t tha t u satisfies

(5-6.1) on BR (or GR), we may alter the coeffic ients as in (5 .5•3)- We then

w r i t e , a s s u m i n g t h a t A = 0 in (5-6.1),

(5-6.2) u == UR + HR, UR = PR [f — (UR — ao) ' V^ U — bR ' \/U — CRU]

where in the BR case, PR i s the po ten t ia l as def ined before , and in the

GR case, PR{f) deno te s the po ten t ia l o f /* w h e r e / * i s the ex tens ion of

/ t o t h e w h o le of BR by nega t ive re f lec t ion :

(5.6.3) f*{x', K) = - / ( - ^ ^ < ) , ^' < 0 ;

/* may no t be con t inuous ac ros s x^ = O.liu^ H^Q(BR) and s p t . u C BR,

then we eas i ly see tha t

(5-6.4) u{x)==fKo{x~^)Au(^)d^ so HR{X)=\ ^ / ^ / ^ ^

B^ (const, if 1 = 2.

If U^HI[GR) n HIQ{GR) and has sp t . in GR U CR, then th i s t ime

(5.6.5) u{x) - UR{X) =fKo(x - f ) / * ( f ) di, f{x) =Au{x),BR

a n d / * is def ined by (5 .6 .3) ; th is fo llows from o ur def in i t ion s ince th e

ave rag e of ^i? = 0 even in th e two dim ensio nal case . T hu s if r > 2

(5.6.6) UR — TRUR = VR = PR ( / ) ,

TR UR = PR [(aR — ao) ' V^ UR + bR ' \7 UR + CR UR].

W e shall assu m e r > 2; we ha ve seen ho w to ta ke care of HR.

Definit ion 5.6 .1 . O n t h e s p a ce H\Q{B2R) or HI{G2R) we define

1kEo.H=J^-^*llv^-^-^fc;

5.6. The e q u a t i o n a'Vû-\-b'Vu-\-cu~?,u=f 159

on the space C^{B2R) or C^{G2R), we define

A i j = ^2i? or G2R,

III III be ing the n o r m in C^ {r.2R).

Definition 5.6.2. We say t h a t the coefficients a"-^, &", and c satisfy

the h-p-conditions on a region F (h an in tege r ^0) ^ a is c o n t i n u o u s and

satisfies (5.1.2), b and c are m e a s u r a b l e t h e r e , and

(i) if A = 0, and p < v/2, t h e n b^ U{r) and c^Lp,2{r) and

(5.6.7) / ( l ^ | ' + \c\''l^)dx<KRf'i, jLCi > 0;B{x„,E)nr

(ii) if A-= 0, and vl2<p<v, t h e n bÛiF) and c^Lp{r) and16|^ and I c|^ satis fy a condit ion l ike (5 .6 .7);

(iii) if / > 0 and ^ is chosen so t h a t (k —- \) p <,v <Ckp, then V^' a,

V^*-i S, V^-^c^L(yfj) for y = 1, . . ., y — 1 w i t h

/ (I V ' <^ I + I V^'-i & I -1- I V^*-2 c |)W:/ dx<K RîB{xo,R)nr

where mean ing le s s t e rms are omi t ted (V~ ^ b, e t c . ) ; if y > k, then V^* ci ,

V^'-i b, and V^'-^ c^Lp w i t h

y"( |V^*a | + | V ^ ' - i & | + IV^'-^cD^dx^RRfî, j = k, ..., h + 2,

Bixo,R)r\r

where V'^^^ a, V^+2 ^, V^^^ b are to be o m i t t e d .

The coefficients satisfy the /^-/^-conditions onF ^ they a lso ^ C^ (F).

Definition 5.6.3. We let E d e n o t e the set of b o u n d s and m o d u l i of

c o n t i n u i t y of th e coeffic ients and the i r r e levan t de r iva t ive s ; it will usually

be clear from the c o n t e x t w h a t q u a n t i t i e s are inc luded .

Lemma 5 .6 .1 . (a) Suppose G = BRQ or GRQ, 1 < _> < ^ < CXD^ h'> 0,

the coefficients satisfy the 0-p- and the h-q-conditions on G, and a°'^{0)= d"-^. Then there exists an Ri > 0, depending only on v, h, p, q, and E

and a constant Ci, depending only on Vyh,p, and q such that ifu^H^^ {BR)

then U^H%^[BR) and if U^HI{GR) H HU[GR), then U^H^+^GR)

nHl^{GR),and

'\\u\\i^^C^'\\Lu\\n, if 0 0, dependingonly on v, h, p, ^, and E such that if u^ H ^ Q ( B R ) , then u^ C^'^^(BR) and

ifu^HKGR) n HI^{GR), then u^ CI-'^GR), and

lll^lll^^^ < Cs'l lJLÎII^^, if 0<R<.R2

and spt u C BR or GR\J GR, respectively.

Proof. We f i rs t show tha t PR is a b o u n d e d t r a n s f o r m a t i o n f ro m

HU{B2R) to HI^^B^R) and from CI,{B2R) to CI^^{B^R). I f / € C^B^R)

a n d u = PR{f), t h e n

(5.6.8) V " u{x) = I Ko{x - f) \/^m d^, 0 < ; < A.

The resu l t then fo l lows f rom Theorem 3-7.1 for the H\'^^ case and from

T h e o r e m 2.6.7 for the Holde r no rm cas e .

I f / ^ C ^ ( G 2 i ? ) and s p t / C G2«U cT2i? , an d if u = PR(f), t h e n u is

given by (5-6.5) so t h a t

'V^' u{x) = I Ko(x -^) ' V V * ( f ) ^l> 0 < y < > ,

w h e r e 'V deno te s the grad ien t wi th re s pec t to the {x^,.. ., x^-^) va r iab le s .T h i s may be w r i t t e n in the form

' V i u=Uj~ Fy, Uj(x) = I Ko{x - I) (pj{S)dl (pj = ' V V ^

(5.6.9) Vj{x) = 2 f Ko(x - ^)<pj{^)d^,

a-.w h e r e /+ is the ex tens ion of / by positive reflection:

(5 .6 .10) /+(* ' ,<_i)

= / ( - * ' ,x',_-,), x'KO.

For e a c h ; , cpj^ LJ,{B2R) PI CI{E^R) with

(5.6.11) II <f>i\U < (2i?)*-^ 11/11*^, 'III mtn < (2R)n-i'\\\f\\l^.

The des i red inequa l i t ies for Uj follow as before (with h = 0). The H^

re s u l t s for Vj fo l low from Theorem 3-7-1. For x"^ > 0, we h a v e

V ' V 2 F ; ( : v ) == 2 / V'V^K^{x - | ) ^ , - ( f ) ^ f«^

= 2 / V ' V 2 i C o ( ^ - ^) V<Pm - 9 ^ i W ] ^ f

22-s ince the in teg ra l of V' V^ i^o (^ — I) ove r R~ converges abso lu te ly to

ze ro . Thus , u s ing the fac t tha t Vj is h a r m o n i c , we f ind tha t

I V3 V^[x) I < C • A^((^y, B2R) • (^^)'^-i.

I t fol lows from Theorem 2.6.6 t h a t Vj^ CI{G2R) w i t h the des i red bound .

It fol lows eas ily that PR is b o u n d e d in the G2R cases also.

I t is now easy to see, using (5-5-3) and (5-6.6), that

1 TR^.i^ < 8i(R;v, h, q, E), '\\TRI%\^ < e2{R',v, h, fji, E)

\imej{R\ . . .) = 0, y = 1, 2.

T h u s , if i^ < J^i, the n o r m of r^ in HliP^R) and in H^+^(r2R) is < 1/2

(A/2 = B2ROV G2R), so t h e r e is a u n iq u e so lu ti on 7 ? of (5.6.6) in b o t h of

5.6. The equation a''V^u-\-b'Vu-{-cu—}.u=^f 161

these spaces . Since the second space C the f i rs t , these so lu t ions mus t

coincide. Since u =UR, the re s u l t (a) follows. The proof of (b) is s imila r .

W e can now p r o v e the following apr io r i e s t ima te :

Theorem 5 .6 .2 . Suppose that G isof class Cf+^ and that the coefficients

satisfy the h-p-conditions onG for some p \. Then ^ a constant Ci,

depending only onv, h, p, E, and G such that

(5.6.12) \\u\ll^<C^{\Lulla + \\ul\a), u^Hl*^{G) r\H\,{G).

If G is of class C " ^ and the coefficients satisfy the h-ju-conditions onG,

then there is aconstant C2, depending only on v, ^, p, E, and G, such that

( 5 .6 .13) MU'<G4\\\Lu\\\lo) + \\u\\U, «€Cr'(G)nffio(G)-

Proof. We sha l l p rove the s e c o n d s t a t e m e n t ; the proof ofthe first is

s imi la r . Ea ch po in t Pof G is in a ne ighborhood on G which can be m a p p e das in§ 5.5, by am a p p i n g ofclass C^'^^, on to e i the r BR or GR in such a

w a y t h a t the coefficients of the t rans formed opera tor , s t i l l ca l led L,

satis fy the /^-/^-conditions with a°'^{0) =6"-^, and R issmal l enough for

the conc lus ions ofLemma 5 .6 .1(b) toho ld . Let {C^}, s— 1, . . ., 5, be

a p a r t i t i o n of u n i t y in which each Cs has s u p p o r t in some such ne ighbor

h o o d . We let Us be the t r ans fo rm of C uand Ls the t r ans fo rmed ope ra to r .

Now, s uppos e the re we re noc o n s t a n t C2 satis fying (5.6 .13). Then,

the re wou ld be a s equence [un] C^^^^[G) O H\Q{G) s u c h t h a t(5-6.14) \lu4\l%^=\, ||JL«„||| ^^0, Wunlla O

a n d we may a s s u m e t h a t Un, SI Un, and V^^^^ conve rge un i fo rmly to w,

V ^ , and V^w, re s pec t ive ly , whe re um u s t be zero on a c c o u n t of (5.6.14).

L e t Uns be the t r a n s f o r m of f § Un- T h e n Uns satisfies

(5.6.15) U Uns = Cf Ls < + Ms <,

w h e r e u isthe t r a n s f o r m ofUn and Msinvolves on ly u and its first

der iva t ives . From (5 .6 .14) and (5.6.15), it follows easily that ' | | | i f LgU^ +

+ MsU^\\\^R->0, But t h e n 'If ^ w s | | m ^ - > 0 for each s. But t h e n

| | | ^w| | |^G^->0 contrad ic t ing (5 .6 . I4) .

W e now p r o v e the fo llowing d i f fe ren t iab i l i ty t he or em :

Theorem 5 .6 .3 . (a) Suppose that G isof class CJ" ^ and that the coeffi

cients satisfy theO -— pand h — q conditions on G, where A > 0, 1 < <$',

and either A > 0, q -py or both. Then if u^ H^[G) r\H^Q(G) and

Lu^H^G), it follows that u^H\-^^{G). If we know only that u^Hl(D)

for each D G GG, it follows that u H^'^^(D) for any such D.

(b) If G is of class C^'^^, the coefficients satisfy the 0—p and h— /nconditions onG,u^ Hl(G) 0 HIQ(G), andLu^ C^i^), then u C^^^(G).

/ / we know only that u^H^(D) for each D G GG, weconclude that

u^C^'^^{D) for such D. If G, thecoefficients, and Lu C°°(G) then

u^C-(G) if u^Hl(G)nHl,(G); if u^Hl(D) for D G GG,

u^C^{G(o)).

Proof. The C^ results in (b) follow from the CJ"^^ results. We prove

(a) first. We first assume q = p, h = 1. We may assume the results and

notations of the first paragraph of the proof of Theorem 5-6.2 where,

however, we assume that the neighborhoods are small enough for the

conclusions of Lemma 5-6.1 (a) to hold for h = 0 and for h = \. Then Us

satisfies

(5-6.16) Ls Us = Cf Lsu* + Ms u*

where Ms is of lower order as in (5-6.15). Thus LgUs^ H\{rR) so that

each Us H^^[FR) so that u^H\[G). A repetition of the argument proves

the result for any h and the same p.

The argument for raising the exponent from p to p' = v pl(v ~ p)

(the Sobolev exponent) if ^ < r is similar; since Ms i^* is of lower order,

Ms u*^H%(rR) if u^ H^+ G). Up^v, then Ms u*^H^(rR) for any

q if u*^ H^^^{rR). A finite number of repetitions raises the exponent

from p to any desired q. In part (b), we first raise the exponent to some

q^v, at which time we can conclude that Ms u*^C^(FR) iiu^ H^'^^(G).

The following special theorem and its method of proof are essentially

due to NiRENBERG (see A G M O N , D O U G L I S , and N I R E N B E R G [1] p. 693 and

also §.6.5 for a more general theorem).

Theorem 5.6.4. Suppose that G ^ C\ and the coefficients satisfy the

0 — 2-conditions G. Then there exist numbers Ao > 0 and C, which dependonly on v, G, and E such that

\\u\\lc^<C\\Lu-û\\la, Areal, A>Ao, u^Hl{G) 0 HIQ(G),

Proof. For any given > i > 0, each point XQ is in a neighborhood or

boundary neighborhood in which \a{x) — a{xo)\ <Crji. We choose a

sequence Cs, s — \, . . ., S, such that each Cs î(^) with spt Cs in some

one such neighborhood and f f __ . . . _|. 2 ^ ^ ^ ^ jg Ug = Cs^ and

note that

^s L u = L Us -\- Ms u

where each Ms is an operator of the first order. Then

~{Lu, u)l= -Zi^sLu, Csu)l= -Z{Lus, %)?+ {M'u, u)ls = l s

(5-6.17)

where M' is of the first order. But

— [Lus, Us)l = —Ja'^Ûs,ccpUsdx —j{a"-^ — a^^)us,xÛsdx —

G G

(5-6.18) — f(b°'Us,cc + cus)usdx, af^ = a'^^{xos)

G

where XQS is in the small support of f§. Since, for each s,

— jalÛs,aÛsdx = Ja'^Ûs,ocUs,^dx > 0,

5.6. The equation a -^^ u -\- b ' Vw -\- c it — 2.u = f 1 6 3

we see, by summing (5.6.18) with respect to s and using the facts thatUs — Cs u. etc., that

G I- ^where M is an operator of the first order. Thus

-{Lu, u)\ > - ^ l i l V2 w|| • | |« | | - Ci(î) (II V«|l + i|M||) • | | « | |

S : - 2 ni II V 2 « II • 1 1 « I I - C ( ^ i ) • II u f . (II I l l ) .

But, now for real A,

\Lu-XuY^ \L uf-^ Xûf - 2K(Lu ,u)

> II L i. 12 + A2II u IP - 2rii(\\ V2 ^ IP + A2II ^ IP) _

- 2A C(rji) \\u\\^ > IJL ÎP (1 - 2rji C2) +

+ [A2(l - 2rji) - 21 C(f]i) - 2rji C2] || u |P

where C2 is related to the constant of Theorem 5-6.2. If we first choosefj i so small th a t (1 — 2 i i) > 1/2 and (1 — 2 ^1 C2) > 1/2, we m ay th enchoose Ao so large th at X^l2 — 2X C (rji) — 2 71 C2 > 1/2 for A > ô- Then

| | i : « _ ^ « | | 2 > l [ | | L « | | 2 + | | ^ l | 2 ]

from which the theorem follows.Theorem 5.6.5. The conclusions of Theorem 5.6.1 hold under the hy po

theses of Theorem 5.6.4. In fact no real A > Ao is an eigenvalue. IfG is ofclass C | and the coefficients ^ C^{G) with c(x) < 0, we may take Ao == 0.

Proof. Firs t, suppose Ai > XQ. We may approximate to L by operators Ln whose coefficients satisfy our conditions uniformly with a°^^ converging uniformly to a^"^ on C, each a^^^ C\{G). Define Lin = Ln — h Ias an operator on L2{G) with domain all u^HKG) which vanish on ^G.

If, for some n, Ai were an eigenvalue forl,^, then Lin would carry somenon-zero element into 0 which would contradict Theorem 5.6.4. ThusXi is not an eigenvalue for any n. Hence, ii f^L2{G), L^^{f) = Un isdefined for each n and || Un 1|| < C| | / | | | for all n. Hence a subsequence,still called {un}, —7 um H\{G) and 1 = 0 on ^G. Then Lin Un -^ Li uin L2 (cf. the proof of Theorem 5-2.7), so that Li u = f.

Now, the equation Lu~Xu=f is equivalent to the equation

— LQU + {X — Xo) u =f (L Q = L — XQI)

which is, in turn, equivalent to

u — {X — Ao) LQ'^ U = —L^^f.

As an operator on H^0^20> ^0^^ i^ com pact, since weak convergencein H\ implies strong convergence in L2. The first part of the theoremfollows from the RiESZ theory of compact linear operators.

11*

T o p r o v e the l a s t s t a temen t , s uppos e A > 0. If A were an e igenva lue ,

t h e n L u — Xu = 0 for s ome u^ C^(G) which van i s hes on G but is no t

^ 0. But s ince the coefficient of u in L u — Xu is c{x) — A < 0, th i s

wou ld con t rad ic t the maximum p r inc ip le (Theo rem 2 .8 .1 ) .

5 .7 . A n a l y t i c i t y of the s o l u t i o n s of an alyt i c l in ear eq u at ion s

In th is sec t ion , we p r e s e n t a s imple proof of the a n a l y t i c i t y of the

s o lu t ions of (5.6.1) on the in te r io r when the c o e f f i c i e n t s a n d / a r e a n a l y t i c

and s how tha t s o lu t ions of s uch equa t ions wh ich van i s h a long an a n a l y t i c

p a r t of the b o u n d a r y can be con t inued ana ly t i ca l ly ac ros s the b o u n d a r y .

By us ing the mapp ings de s c r ibed in the proof of Theorem 5-6.2, which are

a n a l y t i c in these cases , the p r o b l e m of local differentiabil i ty is reduced tot h a t for so lu t ions on spheres or hem is phe re s wi th a'^^ — Zl*^ . The local diffe

rentiability theorems imply a global theorem in case the entire boundary d G is

analytic. The t e c h n i q u e of proof is tha t g iven in a recen t pape r by M O R R E Y

and NiRENBERG; s ince the so lu t ions are k n o w n a l r e a d y to be of class C^,

i t is sufficient to o b t a i n b o u n d s for the de r iva t ive s . In view of L e m m a

5.7.2 below, it is sufficient to o b t a i n b o u n d s for the i r L2-norms .

L e m m a 5.7 .1 . Suppose u ^ Hl(Bji), u ^ H\{Br) for r <i R, and

( 5 . 7 . 1 ) Au=^f on BR.Then there is a constant Ci depending only on v, such that

f\\7^u\^dx<Cii ff^dx + f[d-^\Vu\^ + ^ - 4 ^ 2 ] ^ ^ !

(5 .7 .2) ^»' \^r+6 ^r+d J

0<r <r + d <R, 0<d<r.

The lemma holds with BR, Br, and Br+d replaced by GR, Gr, and Gr+s,

respectively, and the spaces HI [BR) and HI {Br) replaced, respectively, by

Hf[GR) and Hf{GR) (u = 0 along OR).

Proof. The second s ta tement fo l lows f rom the first by e x t e n d i n g u

a n d / to BR by nega t ive re f lec t ion . To p r o v e the first, define r] as u s u a l

w i t h a rep laced by d, G by Br.^d, and D by Br, and let

U =^ ri u.

T h e n U has s u p p o r t in BR, U ^ H \ [ B R ) , and U satisfies

AU =^ rif + 2S/ 7 ] ' "7u + AT] ' u = F.

The resu l t fo l lows f rom Theorem } .7 .1 .

Lemma 5.7.2. Suppose U^C°^{BR) [or C°°{GR[J GR)] and suppose

that there are constants M and K such that

(5.7.3) \\\7Vull^<M'P\KvR-v^ p = 0,\,2,...

Then u is analytic on BR [or GR] and, in fact

(5.7.4) \S/Pu(x)\ < Ci{m,v) 'Mpl(i.\K)PR-^-^, {r==vl2).

5.7- Analyticity of the solutions of analytic linear equations 165

Proof. Applying the SO BO L E V lemma, Theorem 3.5.1 , with m = \ -\-

-f [T] (r =vl2) and p = 2, to V u, we obtain

|v«»*W| <ci?-4^<?^||v«+^«i,K + (»«-T)-i^?^!|v'»+*«.|!l|(5.7.5) ' ' '

where Cis y~^^ or 2^^^yy'^^^> depending on whether we are considering

BR or GR. Inserting the bounds (5.7-3) into (5.7-5), we obtain

{rn—l

^ ^-^ (P + ; ) J +

+ (m- T ) - I - ^ ^ ^ , (p +m)n<:C(m,v)'M' R-^-^K^+-^-pi-^^- '

^ ^ m — \\^^ ' \ ' / r p\m\from which (5-7-4) follows easily.

It is convenient to introduce the following notations:

Notations. F o r / a n d u^ ^^(BR), we define

MR,p{f) = ipl)-^ sup (R - r)2+J'||V^/||i.,, P^O-R/2<: r - 2R/2<r<R

[p\]=pl if ^ > 0 , [: !] = 1 if : ^ < 0 .

Remarks. From Lemma 5-7-2, it follows that the analyticity of u onBR will follow from an inequality of the form

(5-7-7) NR,P(U)^M'KP, P ^ - 2

for some constants M and K. Our method of proof will consist in

demonstrating (5-7.7).

Lemma 5.7.3. Suppose uand f ^^{BR) and itsatisfies (5.7-1) there.

We suppose also that MR,p{f) and NR^P{U) < oo for each p. Then there

is aconstant C^, depending only on v, siich that

(5-7.8) NR, P {U) < C2 [MR, P(/) + NR, ^ _ I {U) + NR, P_2 {U)-\ , > 0.

Proof. We apply Lemma 5-7-1 (for BR) to each component of V^ u,

square, and add to obtain

j\\/Pf\^dx +^R, V Ml ' < ^1 {P0-' sup (7^ - r)4+2 V

^^''''^^ +/((5-2|V2>+l^i2 + ^-4|V^^^|2)^%]

We now obtain the following results using the notations

(5-7-10) / I V^/|2 doc < (/)!)2 M\^[f) '(R~r- ^)-(4+2p)

| | V 3 ? + I ^ | 2 ^ A ; < { [ ( ^ - 1)!]}2Ar|^^_3^(^).(ie-r-^)-(2+2 2» ^ > 1

/1V^^|2 i;k; <{[( j ;^ - 2)!]}2lS\^_^(u) - (R - r - d)-^P, > 2.

S e t t i n g

(5.7.11) d^{R-r)l{p^ 1), p > \ ,

in (5.7.9) and (5-7.10), we obtain

+ (1 -\-p-^)^n[p + \)^IPHP - l ) 2 } A l . , - 2 ( ^ ) ] , ^ > 2 ,

w i th ana logous re s u l t s in the ca s e s ^ = 1 and j ^ = 0 ( t a k i n g ^ = {R — r)l2

in the la t te r case) . The resu l t fo l lows .

Now, in order to prove the local differentiabil i ty , we write (5-6.1) (we

abs o rb the t e rm A u i n t o c u) in the form

(5.7.12) Au = F ^f — (a — ao) • V^w — b • V ti — cu.

In order to use Lemma 5 .7 .3 . we sha l l ob ta in bounds for MR^P (F) in

te rms of MR^p[f) a n d t h e ^R^q{u). As an a id in th is work we s ta te the

following lemma, the proof of which is left to the reader:

Lemma 5 .7 .4 . Suppose V and W are tensors in C'^(D) . Then

(5.7.13) \'7^{V-W){x)\<.J:{^^)\\Iv-QV{x)\'\\/aW{x)\, x^D.

Suggestion for the proof. By arranging the se ts o f common ind ices and

the se ts o f remain ing ind ices in to s ing le sequences , one may assume tha t

fc=ii ^ \, . . ., n, j = \, . . ., p.

Then, if oc d e n o t e s t h e m u l t i - i n d e x (oci, . . . , ocv),

fc=l iS+y=a

The resu l t fo l lows wi thout much d i f f icu l ty i f one no tes tha t

Now, s ince a, h, a n d c are ana ly t ic , the re a re nu m be rs ^ > 2 , L , and

RQ < 1 s uch tha t

/ . Q . . ^ | V ^ / ( ^ ) | , | V ^ a ( ^ ) | , \S7Vh[x)\, \yvc{x)\^Lp\Av,(5.8.14) \ ^ T A \ \ ^ € ^ i? o -

T h e o r e m 5.7 .1 . Suppose a, h, c, and f are analy tic on BR^ and satisfy(5.7.14) there, and suppose that U^H\[BR^ and is a solution of (5-6.1)

there where A — 0. Then there is an R^ < ; RQ, which depends only on v, A,

L, and R Q, such that u is analy tic in B R for each R <, R7

Proof. Su pp ose 0 < i? < i^o- F ro m th e re su lts of §§ 5.5 a n d 5.6, it

fol lows that u^C^ {BR) so th a t NR, p {u) is define d for every p. O b v i o us l y,

5.7- Analyticity of the solutions of analyt ic linear equations 167

MR, P (/) is also defined for eve ry p and, in fac t ,

^R,vU)-- sup ( ^ ! ) - i ( 7 ? - ; ' ) 2 + p f ^ L 2 ( ^ ! ) 2 ^ 2 p ^ ; ^ l | -

(5.7.15) m^T<R \ i J

<.Zi[v)'LR^+'='{AR)v.

Using (5.7.12), (5.7.14), and Lemma 5-7 .4 , we o b t a i n

| V ^ ( i ^ - / ) | <.LAR'\\7^-^'û\ + L I V ^^ + i^ l + L | V ^ ^ | +

+ ^ ( ? ) i ^ - 4 Q ' g ' ! ( | V ^ - ^ + 2 ^ | + i ^ ^ - ^ + i ^ j ^ |V2^-«^|)3 = 1

(5.7.16) =LAR\S7v-^'û\ +L{\ +pA)\V^^'û\ +

2> + l

X[\ + [p+ \-q)A + {p+ \ - - ^ ) ( ^ _ - ^ ) ^ 2 ] . | v i ^ + i - « ^ | .

Us ing the definit ion of NR^P(U), (5-7.15), (5-7-16), the fac t tha t A > 2,

a n d the f a c t t h a t MR^ p a c t s hke a n o r m , we conc lude tha t

39+2

MR,p(F)^MR,p(f)-]-LARNR,p(u) + 2L2;(AR)^NR^p^q{u).

(5.7.17)

N o w if we use L e m m a 5-7-3 w i t h / r e p l a c e d hyF and choose 0 < R < JR? ,

w h e r e i^7 is chosen so t h a tC2LAR7< 1/2,

we conc lude tha tV+ 2

NR,p(u)<2C2ZiLR^+ÂR)^ + 4C2L2;{AR)^NR,p,g(u) +

(5.7.I8) + 2C2NR,P_I{U) + 2C2NR,P-2{U).

We sha l l choose M and K so tha t (5 .7 .7) ho lds . Evident ly , we can

choose them so (5.7.7) holds ior p = —2 and p = —\. Suppos e (SJJ)

ho lds up to _/) — 1, > 0. Then, from (5.7.18), we conc lude tha t

NR,P(U) <MKP'2C2

v+2+ 2L2{AR7lK)Q

M - i Z i L RI+Â R7lK)P + X-i + K- +

^MKP

for any j ^ ^ 0 if we first choose i^7 so s m a l l t h a t

2 C 2 M - i Z i L i ^ f + ^ < 1/2

( to take care of the case p = 0) and then choose K so l a rge tha t

2C2 < l / 2 .

In o rde r to p r o v e a n a l y t i c i t y of u a long OR w h e n u van is hes the re and

/ , a, b, and c are ana ly t i c nea r 0 , we proceed in m u c h the s a m e way. But

n o w , it is c o n v e n i e n t to cons ider f i rs t on ly der iva t ives wi th respec t to

^ k . (


t h e x " ^ w i t h (x <iv since these all vanish along OR SO th a t Le m m a 5-7.1

applies on GR. W e the re fo re deno te x " ^ by y , der iva t ives wi th respec t to

x '^ by Dy , an d deno te by Va; th e gra d ien t involv ing only der iv a t ives w i th

respec t to the x'^ w i t h o c <,v. As an a id in la te r es t imates we s ta te the

following generalization of Lemma 5-7.4:

Lemma 5 .7 .4 ' . Suppose V and W are tensors in ^^{D). Then

\ y n v - w ) \ < i { ^ ) \ v i v ^ - \ v r ' w \ ,

9=0

l ^ ? v j ( F - T y ) i < i ' i ' ( ^ ) ( « ) | z ) » v « F H z ) r ' " v r ' ' w ^ i .

and corresponding inequalities hold for\DIVHV'W)\ a n d | V ^ V ^ ( F • M ^) | , e tc .

W e now in t roduce the fo l lowing no ta t ions :Notations. F o r u a n d / ^ C'^iGR) w i t h u = 0 a long O R , we define

M'n.Af) = (^0-1 sup {R - r)^-Pmm .rR/2^r<R

f (^!)-i s u p {R -r)^+P\\\7^V?.u\\l^, ^ > 0

s u p (R — r j2+2>| |v2+^w| |g , , p = - 1 , - 2 .

Lemma 5 .7 .3 ' . Suppose u and f ^ C'^i'GR), u satisfies (5-7.1) on GR,

and u = 0 along O R . There is a constant Cg, depending only on v, such that

Proof. We apply Lemma 5-7 .1 for GR to each com pon ent of V^ ^ an d

add , to ob ta in

j \V'^\/lu\^ dx <^CiU \Vlf\^ dx + j [d-^\S7\/%u\^ +

-{~d-^\Vluf]dA, p->0.

W e no te th a t V , V^c, an d Dy a l l commute wi th one ano the r and

(57-19) i V ^ ^ | 2 < | v 2 v r ^ ^ | ^ : ^ > 2 ; | V ^ ^ | < | V ^ ^ | , p=OA.

Afte r t ak ing (57 .19 ) in to accoun t , the l emma i s p roved in exac t ly the

same way as was Lemma 5 .7 .3-In es t imat ing der iva t ives of the form D^^^ D ^ . u, i t is convenient to

in t roduce the fo l lowing no ta t ion :

Notation. N'^,,Ju) = sup [(p + g) ! ] - i ( i ? - /)2+3'+« \\Df^ VMir>

iS/2S:r0. q > - 2 .

5.7- Analyticity of the solutions of analytic linear equations 169

Theorem 5.7 .1 ' . Suppose a, b, c, and f are analytic on GRQ and satisfy

( 5 .7.14) there, and suppose u^ Hf*{GEQ) and is a solution of (5.6.1) there,

where X = 0. Then there is an Rg < Ro> which depends only on v, A, L,

RQ, and corresponding quantities for certain ratios of the coefficients andf,

such that u can be extended to be analytic in BR for each R <, R ^.

Proof. As before , we k n o w t h a t u ^ C'îG-R) for each R < RQ. The

proof proceeds as in t h a t of T h e o r e m 5-7.1, but es t ima t ing on ly the

||V^ V^^| |2 . r - The es t i m ate (5.7-15) ca rr ies over w i th ou t chan ge . In

s t e a d of (5.7.16) we o b t a i n

\Vl(F-f)\^LAR\V^V%u\+L\VV%u\+L\V%u\ +

By using (5-7.19) and proceed ing as before , we obta in f i rs t the e s t i m a t e

(5.7.17) with MR,P and NR^P rep laced by M'^,^ and Aî?,^,, respectively.

As before we t h e n o b t a i n

( 5 -7 .21 ) N'j,^^(u)<M ^Kf

which g ives e s t ima te s for all d e r i v a t i v e s Dl S/û w h e r e ^ < 2.

In o rde r to es t ima te the rema in ing de r iva t ive s ^ , we n o t e t h a t we can

solve 5.6.1 for Dû in t e r m s of th e o the r de r iva t ive s .

( 5 . 7 - 2 2 ) Dlu=g+ ' '^B-DyDo.u + '^C^^^D.Dpu

in which g, the 3°", and 0°"^ are a n a l y t i c on GRQ, We s ha l l a s s ume tha t g

a n d the 3 °" and 6°"^ sa t is fy es t imates of the form

(5.7:23) | ^ ? V S ^ | , \DIVIB\, \DlV%C\<piq\LAP^^ in GR,.

W e s ha l l s how tha t if 7^g is s ma l l enough , then

(5.7-24) N'ji,:p,a{u) <MKP+QOP, ^ > 0 , ^ > - 2 , 0<R<RS,

w h e r e M, K, 6 are f ixed cons tan t s wi th K > 1 and d < 1/2. Diffe ren t ia t

ing (5.7.22) and using the bounds (5 .7 .23) and Lemma 5-7-4 ' , we find that

(5.7.25) \Dl+ \/lu\<p\q\LAP+Q +

m-Qn=0

N o t i c i n g t h a t A^^,J,,Q(^) acts l ike a n o r m , we nex t conc lude tha t

(5.7.26) ^R,^,M <LZ2(v) R^-^ÂR)P+Q +

+ L 2: Z im) il) ^^' n\{p + q - m - n)\ {A R)^+^[(p + q) !]- i X

M l -

1 The remainder of this proof could be replaced by a dominating function argu

ment like that used to prove the CAUCHY-KO VALEVSKY Theorem. Or a theorem of

H OL M G R E N [1] may be employed.

170 Different iabi l i ty of weak so lu t ions

Now, (5.7.24) will hold for 5 = — 2 , — 1, and 0 and all ^ > 0 if i^ is

s ma l l enough for (5.7.21) to ho ld and

(5.7.27) M KP+^QP'>M^Kl+^, p>0, ^ = 0 , - 1 , — 2 ,

s ince it follows from the def in i t ions tha t

(5.7-28) A ^ k ^ . . M ^ ^ k p + a M if J ' > 0 , ^ ^ 0 .

L e t us suppose, then that (5-7.24) holds for all q less than some pos i t ive

in tege r , wh ich we aga in deno te by q, and all ^ > 0. Then, us ing (5.7 .26)

and (5.7.24) and the fac t tha t , for _ > 0 and > 0, the factorial coeffi

c ien t of each t e rm in the doub le sum is < 1, we conc lude tha t

m=0 n=0

The r igh t s ide of (5.7.29) will be < 1 a lways {R < Rs) and (5-7.27) will

ho ld if we choose Rs, 6, K, and M in t h a t o r d e r so t h a t

LRl+^^\l2, d = mm{\l2, I / I 6 L ) ,

KO = m a x ( i C i , 2ARs), M = m a x [ 1 , Mi ( ^ / i ^ i ) ^ ] .

5 .8 . A n a l y t i c i t y of the s o l u t i o n s of an alyt i c , n on - l in ear , e l l ip t i c

e q u a t i o n s

In o rde r to conc lude the a n a l y t i c i t y of the so lu t ions of v a r i a t i o n a l

p r o b l e m s and of the quas i - l inea r equa t ions men t ioned in the i n t r o d u c

t i o n and t r e a t e d in §§ 5.1 — 5-6, it i s necessary to give a separa te proof for

the non-linear case s ince the m e t h o d s of the preceeding sec t ion do not

gene ra l i ze immed ia te ly . The a n a l y t i c i t y of the so lu t ions of a single linear

ana ly t i c e l l ip t i c equa t ion in two va r iab le s was p r o v e d by HADAMARD in

1890. The a n a l y t i c i t y of any so lu t ion u [x , y) of class C^ of a single non

linear ana ly t i c e l l ip t i c equa t ion wi th v = 2 was f i rs t p roved in a famous

m e m o i r of S. B E R N S T E I N [1] in 1904. O ther p roofs of B e r n s t e i n ' s t h e o r e m

were given by M. G E V R E Y , T . R A D O [1], H. L E W Y [1], and by B E R N

STEIN himself [2]. In 1932, E. H O P E [3] p roved the corresponding theorem

for a s ing le equa t ion but w i t h v a r b i t r a r y . In 1939, P E T R O W S K Y p r o v e d

t h e t h e o r e m for s ys tems wh ich are a l m o s t as gene ra l as the mos t gene ra l

e l l ip t ic sys tems yet cons idered (see §§6 .2 , 6.3). In 1958, M O R R E Y [12]

a n d A. FRIED MA N [1] p r o v e d the theo rem a lmos t s imu l taneous lyb u t by dif fe ren t methods for the mo s t gene ra l e l lip ti c s y s te m s ; ana ly t i c i ty

a t the b o u n d a r y was p r o v e d for the so lu t ions of s t rongly e l l ip t ic (see

§ 6.5) sy s te m s.

The methods of B E R N S T E I N , G E V R E Y , R A D O , and F R I E D M A N all

invo lve e s t ima t ing the m a g n i t u d e s of the success ive der iva t ives ; by

5.8. Analyticity for non-linear elliptic equations 171

t h a t m e t h o d , FRIED MA N was able to ob ta in co r re s pond ing re s u l t s for

quas i -ana ly t i c s y s tems invo lv ing the quas iana ly t ic c lasses of MANDEL-

BROjT. Those of L E W Y , H O P E , P E T R O W S K Y , and M O R R E Y all involved

ex tens ion to the c o m p l e x d o m a i n . L E W Y us ed his t h e o r y of hype rbo l ic

e q u a t i o n s to c a r r y out the ex tens ion and H O P E ex tended ce r ta in gene ra l i

zed po ten t ia l s to the c o m p l e x d o m a i n , a procedure wh ich had been used

b y E. E. L E V I . The proof of the wri te r combines th is idea wi th a s imp le

t y p e of impl ic i t func t ion theorem in a B a n a c h s p a c e and the idea of

§ 5-5. We presen t th is p roof in th is sec t ion . The proofs for more gene ra l

sys tems wi l l be s k e t c h e d in ^6.7.

We cons ider a so lu t ion u of an e q u a t i o n of the form

(5.8.1) (p{x,D°'u) = 0, 0<\(x\<2, oc = (oci, . . .,ocv)

in which 99 is a n a l y t i c in its a r g u m e n t s (x, Vx) for x n e a r XQ and r^ n e a r

D°' u (xo) an d is rea l for rea l va lues , and the e q u a t i o n is ell ipt ic at th i s

p o i n t ; t h a t is, the l inea r equa t ion

(5.8.2) 2<PrAô,ro)'D-v = 0a

is e l l ipt ic . By a t r a n s l a t i o n of axes , we may a s s u m e t h a t XQ = 0. N e x t ,

le t Q{x) be t h a t q u a d r a t i c p o l y n o m i a l s u c h t h a t

(5.8.3) V*<3(0) = V ^ w (O ) , ^ = 0 , 1 , 2 .

M a k i n g the s u b s t i t u t i o n

(5.8.4) u=Q + v

in (5.8.1), we o b t a i n a new e q u a t i o n of the s a m e t y p e in w h i c h V ^^ (O )

== 0, = 0, 1, 2. If we e x p a n d the new <p a b o u t the origin, we o b t a i n

a new e q u a t i o n w h i c h we can w r i t e in th e form

a'^fi vâ^ = b^vôc + c^v + ipix, D v) {0°" , h\, q c o n s t . ) .

F i n a l l y , by m a k i n g an a f f ine t r ans fo rma t ion of the x s pace , we o b t a i n

(writ ing u aga in)

(5.8.5) Au=^ M u + ip{x,Du), M u = b^'uôci- cu.

The Tay lo r expans ion of ip s t a r t s w i t h a homogeneous l inea r func t ion of

X w h i c h is followed by t e r m s in the x'^ and TX of second and h ighe r o rde rs .

We f i rs t suppose u defined and ^ C^ (^RQ) and i n t r o d u c e the idea of

§ 5.5 in w h i c h we w r i t e

(5.8.6) u = UR + HR, UR = PR[M U + y){x,D U)], HR = U — UR

w h e r e we m a k e the following definit ions:Definit ion 5.8 .1 . The space *C^(BR) cons is ts of a l l / ^ C«(5i j ) for

w h i c h /(O) = 0, the norm | | / | |o be ing hfx{f, BR). The space *C^(BJR) con

s is ts of all U^CI(BR) for which u{0) = Vu{0) = Vû{0) = 0, the

n o r m \\u\\2 be ing h^(Vû,BR). For / $ *C0(5i? ) , we define PR{f) as

t h e p o t e n t i a l o f / m i n u s the quadra t ic po lynomia l sa t is fy ing (5-8 .3) .

From the results in § 2.6, we obtain the following theorem:

Theorem 5 .8 .1 . PR is a hounded operator from *C2(5/j) to *C^^{BB)

with hound independent of R. If F = PR(f), then

(5.87) AF=f.

Thus, in (5.8.6), HR is harmonic.

Proof. The only thing not yet proved is (5.8.7). But, from its defini

tion, AF [x) = f{x) -\- C, C Si constant. Substituting x = 0 gives C = 0.

Now, as in § 5.5, we regard HR as known and shall try to find an

equation for UR. To that end, we define

.5 g g. VR = PR[M HR + yj(x,D HR)]

TR(uR;HR)=^PR[MuR + rp{x,DuR + DHR)-y){x,DHR)].

We easily obtain the following theorem:

Theorem 5.8.2. Suppose u is a solution ^ *CI{BRQ) of (S.S.S) and

suppose UR, VR, HR, and TR are defined as ahove. Then \\UR\\2, \\VR\\2, OLnd

II HR |] 2 are uniformly hounded hy some numher M2 for 0 <C R ^ RQ,

TR(0; HR) = 0, lim \\VR\\2 = lim \\UR\\2 = 0

(5.8.9) IITR(UIR, HR) - TR(U2R, HR)II2 < e{R) \\ UIR - U^RU,

lims(R) = 0 if ||/ /«||, ||C/i ij| |,| |C/2is| |<M2.R-*0

Thus, if 0 <i R ^ Ri, UR is the only solution with \\UR\\2 < M2 of the

equation

(5.8.10) UR-TR{UR;HR)=VR, | | //i j | |2<M2, HR given.

Proof. Obviously ||^||2 < L for 0 < 7 < RQ. L e t / = M u + ^p(x, D u).

Then, for x, xi, and X2 on BR, we have

(5.8.11) \Vû{x)\<LRf', \Vu{x)\^LR^+f', \u{x)\<LR^+f' ( A « < ^ )

+

2' |V^^(:v2) - V * ^ ( A ; I )

fc=0

1/( 2) - f(xi) I < IÎ • IVu(x2) - Vu{xi)I + IcI • \u(x2) - u{xi)I

(5.8.12) + \\7xf\ • \X2 - xi\ + IVr^l

<,ZiRf' '\x2~ .ri|^

where |Va;V^| and jVrV^j are bounds for the indicated gradients for

x^ BR and the r^ in the range defined by (5.8.11); we know that j Va; V j

is bounded and \Vrip\ ^C R^ in that range. Thus ||/||i ^ 0 like Rf

and it follows that || i?||2 ->0. Thus ||î?||2 < i^ + \\UR\\2 SO \\VR\\2 -^0 .

Obviously TR (0; HR) = 0. To get the other bound on TR, we note that

TR{UIR,HR)-TR{U2R,HR)

= PR[M(UIR - U2R) + y^{x, D UiR + DHR) - yj{x, D U2R + DHR)]

and estimate the bracket as in (5.8.11) and (5.8.12) above. If we assume

that Ri is chosen so small that s{R) < 1/2 and \\VR\\2 < M2/2, then it

fo l lows tha t the equation (5.8 .10) has on ly one so lu t ion UR w i t h \UR\2

< M 2 .

Now, a f te r the manne r of § 5-5, we wish to in t roduce ce r ta in s paces of

analytic func t ions , which we sha l l denote by *C|^(5/jij) and *Cf^{BhR)

and which are subspaces of *C^(BR) and '^C^(BR), respectively, such

tha t eve ry ha rmon ic func t ion in *Cf^(BR) is also in '^Cf^{BhR) and PR

is a bounded ope ra to r f rom *C^^(BfiR) to ^CKB^R) as a b o v e . The func

t i o n s in these new spaces can be e x t e n d e d to be a n a l y t i c on t h e d o m a i n s

BfiR in the complex %-space which are defined below. Given / defined in

s ome doma in , we h a v e to s how how to e x t e n d its p o t e n t i a l as given by

(2.5.2) to t h a t d o m a i n and verify the o the r p rope r t i e s .

N ow KQ {y) is a func t ion of | y | on ly and , for rea l y,

If, now, we cons ider complex vec tors y = yi + i y^ (yi and yo rea l

v e c t o r s in R^) , we see t h a t

(5.8.13) 1^(^)2 = |yi|2 - |y2|2 + 2iyi'y2.a = l

If we choose h, 0 <. h <c 1, and a s s u m e t h a t

( 5 .8.14) | y 2 | < A | y i | ( y i ^ o )

t h e n

(5.8.15)a = l

> Re2^(y<^)^ = |yi|2 - [ysp > (1 -^i^)\yi 2

Cons equen t ly , it is c lea r th a t w e can ex ten d KQ (y) u n i q u e l y to be ana ly t i c

in any reg ion of the type (5-8.14); one encoun te r s b ranch po in t t roub le

for h->\.

I n e x t e n d i n gthe

formula (2.5.2)to

complex va luesof x, it

e v i d e n t l ywil l no t do s imp ly to le t xbe complex , keep ing f rea l , becaus e iix = xi-\-

-\- i X2 and A;2 9^ 0, we n o t e t h a t

X — i = (Xi— i) + iX2

a n d \x2\ is not <. h - \xi — i\ for f n e a r xi. T h u s we must a l low f to

range ove r s ome r -d imens iona l ' ' s u r f ace" w h ich ev iden t ly mu s t pa s s

t h r o u g h X, there fore depending on x, a n d m u s t h a v e OBR as i t s b o u n d a r y .

Cons equen t ly , in orde r for (x — f) to be in a domain (5.8 .14) for | on

s uch a surface , x m u s t be re s t r i c ted to a set of xi + i ^2 s u c h t h a t(5.8.16) \X2\ <h\xi- f I for all | on OBR.

Since the m i n i m u m of the r igh t s ide of (5.8.16) is j u s t R — \xi\, we see

t h a t X = xi -{- i X2 must sa t is fy

(5.8.17) \x2\<h{R-\xi\).

http://ur/2

http://ur/2

N e x t , any surface S{x) over which we i n t e g r a t e m u s t h a v e the p r o p e r t y

t h a t if S = h +ih^S{x), t h e n |^2 — f2| < ^ |A;I — f i | . We may

descr ibe cer ta in such surfaces by

S(x) : | = | ( s ; A;), \h{s, X) — X2\ <h\li{s,x) — xi\, S^^BR,

Fina l ly , the in tegra l over such a surface is to be i n t e r p r e t e d p r o p e r t y .

W e beg in by defining the d o m a i n s BfiR and the func t ion spaces we sha l l

use:

Definition 5.8.2. The d o m a i n BhR in complex r -space cons is ts of all

c o m p l e x X =xi + iX2 which satis fy (5 .8 .17). The space *C^(BJIR) con

s is ts of all / ^ *C^(BR) w h i c h can be e x t e n d e d to be (s ingle-valued)

^ C^{BhR) and a n a l y t i c in BJIR, the n o r m b e i n g hfi(f, BJIR) (i.e. the H o l d e r

constant onBhR). The space *C^(BJIR) consists of all u *CI{BR) which

have s imi la r ex tens ions to BhR, the n o r m b e i n g hf^(\/^ u, BJIR).

W e now define our surface in tegra ls .

Definit ion 5.8 .3 . Suppose ^ is defined, and c o n t i n u o u s , and c o m p l e x -

v a l u e d on a set Dincomplex i^-space, suppose S : f = f (s), w h e r e

f(s) is a Lipschi tz vec tor def ined fors G w h e r e G is a Lips ch i tz

d o m a i n in r - s pace , and s u p p o s e t h a t f (s) ^Z) for s G e x c e p t

poss ib ly ats =SQ^G and ^[f (s ) ] is con t inuous excep t pos s ib ly at SQ.

T h e n we say t h a t S is admissible with re s pec t to gand define the in teg ra l

(5.8.18) f (S)d^=fm(s)]J(s)ds, J =d(sl ,SV)

in case it is abs o lu te ly conve rgen t .

Theorem 5 .8 .3 . Suppose / ^*C^^{BfiR), XQ ^B^R, and S :f =| ( s ) ,

s ^ BR is admissible with respect to the function

(5.8.19) %[^)Xo)=Ko{xo-^)f{^)

and I satisfies the additional conditions

(5.8.20) f i ( s ) - s , 5 ^ 5 , ^ , 1 2 ( 5 ) = 0, s^dBR.

Then the integral (5.8.18), with g defined by (5.8.19), exists. If i and | *

are both admissible and satisfy (5.8.20), the integrals have the same value.

Finally, if for each x^BfiR, S{x) : ^ =^(s; x) isadmissible with respect

to ^(f ; x) as defined in (5.8.19) and | ( s ; x) satisfies (5.8.20) for each x,

then the function Fdefined by

(5.8.21) F{x)=JK {x-- )f{ )d^Six)

is analytic on BhR and

(5.8.22) F,4x)=fKo,4 -i)f(S)dS.

Remarks. W e n o t i c e t h a t in o r d e r for 5 : | = | (s) to be admis s ib le

wi th re s pec t to (f ; X Q ) , i must sa t is fy a lso

. g . 12(^10) =^20 , \h{s) -X2Q\<h\s -xio\

Proof. F r o m the definit ion of KQ and from (5.8.15), (5.8.20), and

(5.8.23), and the fac t tha t f is Lips ch i tz , it fo l lows tha t | / [ f (s)] | and

I / (s) I a r e b o u n d e d a n d KQ satisfies

\Ko[xo - | ( s ) ] | ^ 2 i • (1 - A2)i- /2 Is - ^ l o P - ^ ^ ^^(5.0.24) , , , , ( i ' > 2 )

^ | V X o [ ^ 0 - | ( 5 ) ] | < Z 2 - ( 1 - ^ 2 ) ( l - ^ ) / 2 . | 5 _ ; , ^ 0 ] l - r ^ ^

so that the integrals (5 .8 .21) and (5.8 .22) converge absolutely . There is an

obvious modif ic ia t ion iiv = 2.If f an d f * are two admis s ib le vec to rs ^ ^^(Bji — { 10}), let f (5, t)

= (1 — Q f (s) + ^ {* (s) and define

(5.8.25) (p(t, Q) =fKo[xo - i(s, t)]f[^(s, t)] • / ( s , t) ds,BR-B(XIO,Q)

Diffe ren t ia t ion wi th respec t to t y ie ld s

(Pt ^

(5.8.26) _ r ' 5(j i_ . . . , |v - i_ c j ^y ^ .+ i_ . . . , J,

~J ^ d(s\... .s' ') '^^'

as i s s een by expand ing the de te rminan t wi th re s pec t to the y - th co lumn

and us ing the re la t ions

d(% ^) lds ' = ir^f l, + ^ 32 fv/5s« dt.

O ne sees in t u r n by us ing Lemma 4 .5-6 and the proof of Lemma 4 .5-7(for com plex-v a lued func tions ) th a t

(5.8.27) Mt'Q)= - 2^-^^'""h'^*^'^'''-y . ( 5 = l 6B{xto,Q) ^

From (5.8.24), (5.8.25) and (5-8.27), we see t h a t (p{t, g) -^(p{t, 0) and

(ft {t, Q) ->0 = (f t [t, 0) un iformly , so th e in teg ra l s have the s ame va lue

for f a n d f *. If | a n d f * a re mere ly L ips ch i tz it is easy to see tha t one

c a n a p p r o x i m a t e to each by C*^ adm iss ible func tions .To see the ana ly t ic i ty , p ick XQ = ;tio + i :^2o€ BJIR. I t is easy to con

s t r u c t a family S{x) of admiss ib le surfaces i = i {s , x) which ^ C^ as

a b o v e in b o t h s a n d x for x n e a r XQ. L e t us define

F{x,Q)=^J%[i{s,x);x]J{s,x)ds,BR-B{XI,Q)

w h e r e g is defined in (5.8.26). Clearly for each suffic iently small o > 0,

F{x, Q) ^C^ in X near :vo and we o b t a i n

PX^Y{X, Q) = fKo^y[x- | ( s , x)]f[S{s, x)] ds -

( 5 . 8 . 2 8 ) ' ' " ^ ' • ^ ' ' ^ • " • " '•' '

PX^Y{X,Q) =ifKo,y[x — i(s,x)]f[^{s,x)]ds —

-BK--B(a!i.e)

b y m a k i n g use of the devices used in (5.8.26) and (5.8.27). Using the

bounds (5 .8 .24), we see t h a t F(x, Q), Fx-^y{x, Q), and Fx2v(^> Q) convergeun i fo rmly to F, Fx^y, and Fx^y and t h a t Fx^y =iFx^y so t h a t F is

ana ly t i c wi th de r iva t ive g iven by (5.8.22).

Definition 5.8.4. For / ^*C0(J5/^^), we define U = PR{f) by the

fo rmula

(5.8.29) U{x) = -Q(x)+lKo(x-i)md , X^B^RSix)

w h e r e S{x) is admis s ib le for each x and Q is the q u a d r a t i c p o l y n o m i a l

satisfying (5-8.3).

W e m u s t now e s t i m a t e h*{\/^ U, BIIR) w h e n U =PRI/) and

/ € *Cl(BfiR); c lear ly we may a s s u m e , for th i s pu rpos e , tha t U is given

b y the in teg ra l in (5.8.29). As in § 2.6, we e s t i m a t e V^U{x) and t h e n use

a l emma l ike Theorem 2.6.6 w h i c h is inc luded in the l e m m a we s t a t e

and p rove a f te r in t roduc ing the fo l lowing conven ien t no ta t ions :

Definition 5.8.5. For x =xi +i X2 ^BJIR, we let d{x) be the dis

t a n c e of X from dBfiR and r(x) be the d is tance in the p l a n e X2 = cons t ,

of X from the in te r s ec t ion of t h a t p l a n e w i t h dBjiR. We define Si{x) to

b e the surface defined by

h = l i i ( s , ^) = s , S^BR — B{xi,r), r= r(x)l2

Si{x):

h =h2{s,x) =

(R~ \xi\ - r - \s - xi\) X2l{R - \xi\ -2r)

r <.\s — xi\ <, R — \xi\ ~ r

0 , S^BR~B{XI,R-\XI\ - r ) .

L e m m a 5.8 .1 . (a) If x — xi + iX2 ^BJIR, then

r(x)=R-\x2\lh-\xi\, d{x) = hr(x)iyi + h^

(b) BfiR is convex

(c) Any two points zi and Z 2 of BUR can he joined hy a path x= x{s),

0 < .*? < /, in BfiR such that

I

f {^ [x (s)]Y-^ ds<C{v,fi,h)'\zi-Z2\f'.

5.8. Analyticity fo r non-linear elliptic equations 1 7 7

Proof, (b) i s ev ident . To prove the formula for r(x), w e n o t e t h a t

(fi, X2)^BnR ^\X2\< h{R - l l i l ) and \ h \ < R ^ \ h \ < R - \x2\lh.

To p rove tha t fo r d(x) we n ot e t h a t (f i , ^2) € dB^R ^

(5.8.30) \h\<R, \i2\^h{R-\h\)

a n d t h e m i n i m u m o f

for all (li , ^2) satisfying (5.8.3O) is just that of

(r - \xi\)^ + [h(R -r) - \x2\]^ iovO <r <R

which is jus t the square of the d is tance in R2 f rom the po in t (\xi\, \x2\)

to the l ine ^^ = h{R — C^), (C^, C^) being the coordinates in 7^2-

Using (a) , i t fol lows that the dis tance of Bfi,R-r from dBjiR is> hrl]/i + h^. Also if {xi, X2) ^ B^R, t h e n (txiJx2)^Bh^R^r for all

^ < 1 — rjR (r <. R). Accord ingly the des i red pa th could be the po lygon

. 1 2:3 2:4 ^2 where zs a n d 24, are the neares t po in ts o f B^^R^r t o z\ a n d Z2

re s pec t ive ly , r = \z\ ~ Z2\-

We need a lso the fo l lowing two lemmas :

Lemma 5.8.2. Suppose XQ^B^R, Si(xo) is the surface defined above

and V is defined near XQ by the formula

V{x)=lKo{x-^)d^.Siixo)

Then for all x near XQ,

V(x) = — v~ ^ XQ X " ^ + cons t .

Proof. I t is easy to see that for each x n e a r A;O, a surface S2{x) can be

def ined by ^ =^ ^2[^\ ^) w i t h 5 ^ B{xio, r), s o t h a t i2{s, x) = f i ( s , XQ)

fo r s^dB{xio,r), and the whole surface S{x) = Si{xo) + S2{x) is a

admiss ible for x n e a r XQ. T h u s

V{x) + V2(x) = Vo{x) = JKoix - I ) d^, V2{x) =fKo{x ~ f ) d | .Six) Szix)

But for X rea l , Vo{x) jus t reduces to the po ten t ia l o f 1 . Thus

Vo(x) = {2v)-^ 2 ' M ^ + c o n st ., a n d

V2(x) = {2v)-^ 2 ' i^^" - 4 ) ^ + c o n s t .a

for the same reason , us ing a t rans la t ion of axes . The lemma fo l lows .

L e m m a 5.8.3. Iff^ *Cl{BhR) and L = | | / | |o, then\Vf{xo)\ <Ci{v,f^,h)L'[d{xo)r~K

I V V W I ^C2{v,fi,h)-L'[d{xo)r-^, xo^BuR.

Proof. There i s a cons tan t x, which depends on ly on v a n d h, s uch

th a t the po ly cy l inde r \^^ — x°'\ <Cxd{x), oc •= 1, . . ., 1 , lies in BfiR.

Morrey, Multiple Integrals \2

http://x2/lh

http://x2/lh

http://x2/lh

Since / ^ C^^{BfiR), we have (using Cauchy's integral formula withy : II — :\;| = Kd{x))

f { x ) = { 2 7 1 i ) - ^ j . . . j m d i ^ . . . d ^ - m ^ - ^ 1 ) . . . (1 ^ - x ^ )V V

V V

for any complex t , " - . The firs t result follows from the fact that

| ( | . _ ^ . ) - l ^ a | < | | q | . [ ; , ^ ( ^ ) J - l .

Th e second resu l t i s p ro ved s imila r ly .

We now prove the ana log of Theorem 5 .8 .1 ior BUR.Theorem 5 .8 .4 . PR(f) is a bounded operator from *C^(Bfiji) into

*C^(^/i i j) with bound independent of R.

Proof. I t is suffic ient to show that V^F ^C^(Bjiji), which we do by

e s t i m a t i n g \/^ F{x) in terms of d(x), F being the po ten t ia l as def ined in

Th eo rem 5-8.2. So, le t X Q = :vio + i ^20 € ^hR and le t Si ( X Q ) be the surface

defined above. As was seen in the proof of Lemma 5.8.2 , this surface

may be comple ted to a surface S(x) by adjoining a surface S2(x)

with the same r im, i f x i s near enough to X Q . B u t t h e d e r i v a t i v e s m a y b ecomputed by d i f fe ren t ia t ing on ly in rea l d i rec t ions . So , we may choose

S2(x) as the set of f = s + i X 2 0 , w h e r e s^B [ X I Q , r), r = r(xo)l2. T h u s ,

w e h a v e

F(X) = Fi{X) + F2(X), Fi(X) ^fKo{x- | ) / ( | ) i |S i (xo)

(5.8.31) F2(x) = jKQ{XI - s)f{s + ix2o) ds,

X = Xi + i X 2 Q .

Clearly F\ is analytic for x n e a r X Q and we can d i f fe ren t ia te unde r the

integral s ign as often as is des ired. Thus

Fi,o.^y{xQ) = JKo,ocpy(xo - I) [ /( I) -f{xo)] di +f(xo)' V,ocPv{xo)

St (xo)

in which the second te rm vanishes , us ing Lemma 5 .8 .2 . Now, f rom the

form of our integrals , i t fol lows that \di\ < i^-dimensional element of

a rea . From our cons t ruc t ion i t fo l lows tha t

\ds\<{i + h^r^dh

w h e r e d^i i s the pro jec t ion of the e lement on the rea l Rv. T h u s

\V^Fi(xo)\^Z i'L'l\h - xio\^-^-^dii <Z 2'L'[d(xo)r-^ (L = | | / | | i).

(5.8.32) Iiv-B(xio,f)

5.8. A na lyt ic i ty for non-l ine ar el l ipt ic equ at ion s 17 9

N ow jp2 is jus t th e po te n t i a l of a com plex-v a lued fu nc t ion . F i r s t we

dif fe ren t ia te

F2,cc(x) = j — — Ko(xi — s)f(s + ix2o) ds

B ixio, r)

= — Ko(xi — s)f(s + i:r2o) ds^ +

+ Ko(xi — s)fô,(s -I- ix2o)ds.

B(xio,r)

Diffe ren t ia t ing twice m ore and us ing Th eore m 2 .6 .5 , we o bta in

F2,ocpy(xo) = — f Ko,^y {xio — s)[f(s + t X20) ~ f(xo)] ds^ +5J5(ccio.r)

+ C^vf,o^(ô) + f Ko,fiy(xio — s)[fôc(s + iX20) — f,oc{xo)] ds.B(xio,r}

T h u sI V3F2(^0 ) \<ZsLr^-i +Z4L[d(xoV/-^ +Z^Lr[d{XQ)^-^

usin g L em m as 5.8.1 an d 5.8 .3 . T he the or em follows from (5.8.32},

(5.8.33) , and Lemma 5 .8 .1(c ) .

Theorem 5.8.5. / / i J '5 harmonic and H $ *C^(B/?) , then H ^ *Cf^(BhR)

for each h < \ and\ \H \\2n^C{v,^.h)\\H \\2.

Proof. F ro m for mu la (2 .4 .4) , i t fo llows th a t

H{x) = -f [H{S) KoM ^ -i)+Ko(x- i) HR,4m d^'^.

I t i s c lea r tha t H i s ana ly t ic in any Bhu w ith A < 1. Dif feren tia t ing , we

o b t a i n [x — Xi -{- i X2)

H,^y6[x) == - l{Ko,.^y 3 {x - f) [/f (f) - H(xi) - sjH(x^)' (f - xi) -

- V2 H(xi)' (I - î)2 /2!] + Ko,py6(x - I ) [H,4^) - H,.{x{) -

- V / f , a ( ^ l ) - ( | - ^ ^ l ) ] X

s ince i t fol lows from Th eor em 2 .6 .5 , d i f fe ren t ia t ion , an d an a ly t ic con

t i n u a t i o n t h a t

j Ko,oc6y d(x - i) d^'^ = J Ko^ccfiyd(x - ^) - (^ - x) d^'^ = . . . = 0

dBR BB R

and also i -~ x = ^ — xi -]- xi — x, e tc . Thus , us ing the proper t ies o f H

fo r X rea l and the fact tha t | | — xi\ < | f — ^ | < (1 + h^)^\^ — xi\,w e o b t a i n

\^^H(x)\ ^ZiL j \ ^ - xi'f-^d^^Z2L{R - \xx\Y-^

^Z zL[d{x)r-K L = \\H\\2


12*

W e can now p r o v e our i n t e r i o r a n a l y t i c i t y t h e o r e m :

Theorem 5.8.6. Suppose u^ CKBRQ) with u{0) = \7u{0) = S7û(0)

= 0 and satisfies (5.8.5) there, where ip is analytic in the {x, r)-space near

(0, 0) and has the properties described there. Then u is analytic near x ^= 0.

Proof. Let us choose ^ < 1. F r o m T h e o r e m s 5-8.2 and 5.8.4 it follows

t h a t I|i5/ i2||2fe <.Mz un i fo rmly for R < iô- If we res t r ic t ourse lves to

UiR and U^R in *C^(i5/iij) with norms < M 3 , we see that (5 .8 .11) and

(5.8.12) hold for c o m p l e x x in B^R SO t h a t if we t a k e J?2, 0 < i?2 < ^ 1 ,

s ma l l enough TR satisfies (5.8.9) with e[R) = 1/2, cons ider ing it as an

ope ra to r ove r '^C^[B}IR). T h u s , t h e r e is a un ique s o lu t ion UR of (5.8.10)

w ith ||wi?||2/ < Mz and ||î2||2 < M^- These mus t co inc ide so t h a t UR also

€ ''CI[BUR). The result follows.

W e now wish to cons ider a so lu t ion 1 of (5.8.1) where X[i is a p o i n t ona n a n a l y t i c p a r t oi dG a long which u v a n i s h e s . A s imp le ana ly t i c t r ans

fo rma t ion of the X coord ina tes enables us to a s s u m e t h a t AJQ = 0 and a

p a r t of G co r re s ponds to GRQ w i t h a p a r t oi dG co r re s pond ing to ORQ.

The s ubs t i tu t ions made ea r l i e r l e ad to the equation before (5 .8 .5). Then

an a ff ine t rans format ion , fo l lowed by a r o t a t i o n of axes yields (5.8.5) on

GRQ w h e r e ^ = 0 on ORQ and | V ^(0) | = | V^ w(0) | = 0 . We may a s s u m e

t h a t ^ has its prev ious p rope r t i e s . We wri te (5 .8 .6) , where we m a k e the

following definit ions;the

extra d i f f icu l tyin

definingPR{f)

arises fromthe fac t tha t the h e m i s p h e r e G^ is not a s m o o t h d o m a i n .

Definit ion 5 .8 .1 ' . The space *C«(G/?) consists of a l l / ^ CI{GR) w i t h

/(O ) = 0, the norm ||/||o being A^*(/, GR). The space '^C^{GR) consis ts of

all u^ C^ (GR) which van i s h a long GR and for wh ich a lso V ^(0) == V ^ ^ (0)

= 0, the norm ||f^||2 being hfi{\J^ U,GR). For f^*C^^{GR) we define

U = PR(f) by the fo rmula

U(x) = - Q(x) +lKo(x - f ) / i ( f ) di - 2fKo(x - f ) / i ( f ) d^

(5.8.34) '" ' ^

w h e r e Q is tha t quad ra t i c po lynomia l chos en so t h a t V C/(0) = V^ ^(0)

= 0, a n d / i is t h a t e x t e n s i o n o f / t o B^R defined by the cond i t ions

, , , / i W = = / W , ^^GR, / l ( ^ ^ < ) = / l ( - A ^ < ) , x^B^R(5.8.35)

/ l ( ^ ^ ) = / i [ ( 2 ^ - ^ ) f l , 0 < r < 2 i ^ , f ^ a ^ ( 0 , 1 ) .

Theorem 5 .8 .1 ' . Theorem 5-8.1 holds with BR replaced by GR.Proof. As in th e proof of T h e o r e m 5.8.1, we f ind tha t AQ =^ Q. Also

since t/ = 0 a long ji'' = 0, it fo l lows tha t Q is also. We need only prove

t h e H O L D E R c o n t i n u i t y oi\/Û. Let F\[x) and F2[x) d e n o t e the first and

second in tegra ls , respec t ive ly , in (5 .8 .34). That h/^{V^ Fi, BR)

'<Zih(i(f,BR) follows from Th eo rem 2.6.7 , C orollary 1. O b v io u sl y ,

hfji{f, BR) =h^{fi, B2R). Dif fe ren t ia t ing F2, we o b t a i n

^ 2 , a ^ y W - - 2 / X o , a ^ y ( ^ - f) [/l (f) - fl(x)] d^ + fl(x) V,a^y{x)

V{x) = -2jKo{x-^)d^.(5.8.36) ^ ^

O b v io u sl y Vis h a r m o n i c , and if a, say , < r , t h e n

(5.8.37) V,4x) = 2lKo(x-i)d$:, V,. y(x) =2lKo, y(x-i)dS' ,

Since V is h a r m o n i c , V^vw can be found in t e r m s of the other F,a/3y, and

we find from (5.8.36) and (5-8.37) that

\S/^F2{x)\ <ZiL J \S- x\f^-'-^ di +Z2LRÎ \^ - x\-^dS(.^)

<,ZzL[d{x)r-^, X^GR,

d {x) be ing the d i s tance of x from d GR. The theorem fo l lows f rom Theorem

2.6.6(b).

Again we define VR a n d TR by the formulas (5 .8 .8) and conclude eas ily

as before tha t the fo l lowing theorem holds .

Theorem 5 .8 .2 ' . Theorem 5.8.2 holds with BR replaced by GR.

N o w it w o u l d be des i rab le to be able to in t roduce s paces of a n a l y t i c

func t ions as was done above and p r o v e the ana logs ofthe r e m a i n i n g

t h e o r e m s . H o w e v e r , it t u r n s out to be more conven ien t to in t roduce f i rs t

spaces of func t ions which are ana ly t i c on ly in (x^, . . ., x^-^), keep ing x*"

rea l . In th i s way , we sha l l be ab le to e x t e n d / t o / i a n d to generalize eas i ly

the formulas (5 .8 .21) . We first define our d o m a i n s GhR and our spaces of

func t ions .

Definit ion 5.8 .2 ' . We define BohR to b e t h a t p a r t of B^R for which x" is

rea l , GUR as the p a r t of BO^R w h e r e x^ > 0, and G a s t h e p a r t of BQ^R

w h e r e x'^ < 0. The space '^C^(GfiR) cons is ts of a l l / ^ '^C^{GR) w h i c h canb e e x t e n d e d t o ^ C^{GhR) and to be a n a l y t i c in (x^, . . ., x^-'^) for x GJIR,

t h e n o r m \\f\\oh be ing hfi(f, GhR). The space *C^{GfiR) cons is ts of all

^ € *C^(GR) which can beextended to CJ^{GJIR) and to be analytic in

{x^, . . ., x'^-'^) for x^ GhR w i t h u =0 for all x GhR w i t h x^=0, an d

V w(0) = \7û{0) =0, the norm ||^| |2ft being h^^^û, GhR).

Theorem 5 .8 .3 ' . Suppose (i) that / i C^(BQhR) and is analytic in

[x^, . . ., x"-'^) for x BohR] (ii) XQ^ GhR] (iii) Si : ^ =| i ( s ) , s ^ BR and

5 * : f = f f ( s ) , s G , are admissible with respect to 5 i ( f ; : r o )

= KQ {XQ — f) / i ( f ) ; (iv) that Si and S* satisfy also the conditions

(5.8.20) and also the conditions that

(5 .8 .38) ir , (s ) = illis) =0

on their respective domains. Then the integrals of ^ 1 over Siand 5* exist.

If S2 and S2 satisfy their respective conditions, the integrals of ^ i over Si

and S2 cire equal as are the integrals of ^ i over 5* and 5 | . Finally if, for

each X ^ GuRy S {%) and S * {x ) satisfy all the conditions then the functions

F and F * defined by

(5.8.39) F(x)=lKo(x-i)fi(^)d^, F^x)=fKo{x-S)fi(^)d^S(x) S*{x)

are analytic in {x^, . . ., x*'~^) for x on GhR and of class C^ in x and

F,4X) = JKO,4X-Six)

F^^ix) = 1 KoM^-

- f ) / i ( l ) ^ l ,

-f)/i(l)^f.(5.8 40) ^^""^ \<oc<v.

"^ = / ^S*{x)

Proof. The s t a t e m e n t s c o n c e r n i n g F(x) and the in teg ra l s ove r Si

a n d 52 follows from the proof of Theorem 5-8.3, s ince the cond i t ion(5.8.38) gu ar an te es t h a t |J ' (s , ) E ^ 0 so t h a t the d e r i v a t i v e of ^ i w i t h

re s pec t to ^, which does not necessar i ly ex is t , is no t i n v o l v e d a n y w a y .

An en t i re ly s imi la r ana lys is shows tha t the in teg ra l ove r 5* e q u a l s t h a t

ove r S^: If we set

| * ( s , t)=s + i[{\ - ^ ) | * (s) +tSUs)] (1*^(5, t) = s"),

(p{t)=fKo{xo-i)fi(i)d^,S*{s,t)

a repe t i t ion of the ana lys is in the proof of Theorem 5.8.3 yie lds

v—l v—1 V

OR V = 1 ER V = 1 < 5 = 1

The f i rs t in tegra l van ishes s ince |*» ' = s" and the second vanishes s ince

I * (s, ) = s on ^ . The a n a l y t i c i t y in the case F follows as before and

t h a t for F* fo l lows s imila r ly ; the formulas (5.8.40) are o b t a i n e d in the

course of the proof.

Definition 5 . 8. 4' . F o r / $ *C2(GA i? ), we define U = Pnif) by the

fo rmula

U{x) =-Q(x)+Fi{x)+F^{x)

(5.8.41) Fi(x) =fKo(x-i)/i{$)d$ + f Ko{X - I ) / i ( f ) dSBZR-BR S ix)

F2{x) =f-2Ko(x - | ) / i ( a dS - 2JKo[x - | ) / i ( | ) d^GZR-GR S* {X)

w h e r e /i is the ex tens ion of / to BohR (J {B2R — BR) w h i c h is u n i q u e l ydefined by the formulas in (5.8.35) and Q is as u s u a l .

Definition 5.8.5'. For x = Xi -{- i X2[x\ = 0) in BQJIR, we let ^o(^)

be the dis tance f rom dBouR (in the 2v — \ space) and r[x) be the dis

t a n c e in the i^-plane x^ = cons t , of x from the in te r s ec t ion of t h a t p l a n e

w i t h dBfiR. For x^GfiR, we define d(x) as the d is tance of x from dGjiR.

Lemma 5.8.1'. (a) BQ^R and GUR are convex.

(b) r[x) = R - \xi\ - |4i /A, do[x) =hr{x)j]l\ + h^

d{x) =min [do{x), x''], 4 =[x\, . . ., xl~^).

(c) Any two points zi and Z2 of GUR can he joined by a path in GUR as inLemma 5.8.1. The same is true of BQJIR.

The proof is like that of Lemma 5-8.1.

Lemma 5.8.2'. Let S*{x) satisfy the conditions in Theorem 5-8.3' and,

for X $ GfiR, define

(5.8.42) V*{x) = -2fKo{x-i)d -2lKo(x-S)d .S*{X) G2R-GR

Then F * is analytic in x^for X^GRU, is harmonic there, and

(5.8.43) I V3 F* ( ^ ) | < C(v,fA,h)R-KProof. Choose XQ ^GhR. For xsufficiently near XQ, V * (x) is given by

(5.8.42) with 5 * (x) replaced by S* (XQ) SO th at F * is analytic in all x'^

and its derivatives of any order can be found by differentiat ion under

the integral sign. Thus F * is harmonic . For real Xi

(5.8.44) V%(xi) == - 2/Xo,«(^i - i)d^ = 2fKo{xi - i)d^',, oc<v.

Since the right side of (5.8.44) is analytic, (5.8.44) must hold for x in

GfiR. HenceV%py(xo) =2fKojy(xo - i)dii

from which a bound (5.8.43) follows if {(x, p, y) ^ (v, v, v). But since F *

is harmonic, this derivative has asimilar bound.

Lemma 5.8.3'. Iffi ^ C^ (^0hn) ^^^/i ^^ analytic in x^for all X^BQnu,

then^ ^ ^ \fAx^)\<C\(v,ii,h)-[d(x,)-\>^\

Proof. The proof of Lemma 5.8.3 with vreplaced by 1 — 1 demon

strates (5.8.45) with (5(:vo) replaced by the distance d{xQ) in the plane

x^ = xl from the intersection ofthat plane with d G^R ; obviously d (XQ)

>^(^o ) , so the result follows.

Theorem 5.8.4', PR is ahounded operator from *C2(Gftjj) to "^CKGUR)

with hound independent of R,

Proof. From the fact that Fi, AFi, and / i are analytic in x^ for x

on BofiR and AFi = fi for real x, it follows that this holds on BohR- It is

also seen that A F2 =0 on G^R.

Choose xo^BohR. For x =xi +X<ZQ, | ^ I — ^ I O | < ^, we can write

(5.8.46) Fi{x) = Fii[x)+Fi2{x), Fn(x)=fKo(xi-s)fi{s + ix'^^)ds,B(xio.r)

Fi2(%) =^fKo{x - S)fi (S)dS +JKO{X - S)fi(^)di.Si(Xo) ^ZR-^R

A s in the proof of Theorem 5-8.4, F1 2 is a n a l y t i c in all x° ' and

(5.8.47) I V S F i a W I < ^ i | | / | | o • [^W?"!.

I n the case ofFn, we first choose oc < v and d i f fe ren t ia te to o b t a i n

Fii,cc{x) = ~jKo{xi~s)fi{s + ix2o)ds'^ +fKo{xi — s ) / i , « ( s + ix'ôjds.

W e can then e s t ima te V^-F i i , a by

\V^Fii,4ô)\ < Z2\\fh[d(xo)r-'-

as was done in the proof of Th eore m 5-8 .4 , us ing L em m a 5-8 .3 ' th is t im e .

T h u s all second derivatives iî ,a /3 with (oc, ^) ^ [v, v) satisfy the desired

Holde r cond i t ion . But t h e n Fi^w does also since AFi — / i .

I n the case of F2, we see t h a t

F2[xi + ixm) =f-2Ko(x~ I ) / i ( I ) dS - 2fKo(x - f ) / i ( f ) d iS*{Xo) G2R-GR

fo r X = xi -{- i X20 n e a r XQ. Diffe ren t ia t ion y ie lds (see L e m m a 5-8.2')

V 3 F 2 ( ^ o ) = / i ( ^ o ) • V 3 F * ( ^ o ) - ifv^Koixo - f) [/i(f) -fi{xo)]di -8*{xo)

(5.8.48) -_2fv^Ko(xo - I) [/i(|) -fi{xo)]d^.

GZR-GR

The las t two in tegra ls can be e s t i m a t e d as in the proof of T h e o r e m 5-8.1'

a n d the f irs t term on the r i g h t in (5.8.48) has the e s t i m a t e

1/1( 0) • V 3 F * ( ^ o ) I < ^ 3 | | / l | l • R^-^ < ^ 3 | | / i l • [d(Xo)r-^

u s i n g L e m m a s 5-8.1' and 5-8.2'. The theorem fo l lows .

Theorem 5 .8 .5 ' . If H is harmonic on GR and H^"^C^ÎGR) , then

• ^ € *C' (< /ii?) OL'y^d the inequality of Theorem 5.8.5 holds.

Proof. This follows from Theorem 5.8.5 and the reflection principle .

Theorem 5.8.6'. Suppose U^CI{GR^) with VU(0) ='7Û(0) = 0,

u satisfies (5-8.5) on GRQ, and vanishes along OR, where y) is analytic near

(0 , 0) and has the properties stated near that equation. Then u is analytic in

GR and can be continued analy tically across GR.

Proof. As in the proof of Theorem 5 .8 .6 , one deduces tha t there is an

J?2 which is s u c h t h a t u^ '^C^{GhR) for eve ry R, 0 <i R -< R^. F r o m

Theorem 5-8.6, we conc lude tha t u is a n a l y t i c in any s uch GR. N O W , let

V be rep laced by v + 1 and let x^+^ = y. T h e n u satisfies on GR an e q u a

t ion of the form(5.8.49) Uyy = (p(y, X,U,U,a,Uy,Uy,u,Uôcp) (oC, ^ = i , . . ., v)

in which 99 is ana ly t i c nea r the orig in ; th is is o b t a i n e d by solving (5.8.5)

for U y y . If we now set

w 2 4 a ^ , « , ^ = 1 , . . . , V

5.8. Analyticity fo r non-linear elliptic equations 1 8 5

we see from (5.8.49) that the vector w^, . . . , I L ^ + ^ satis f ies a sys tem of the

C a u c h y - K o v a l e v s k y t y p e

in which the /^ ' a re ana ly t ic near the or ig in . S ince the or ig in could corres

pond to any poin t on an ana ly t ic par t o f the boundary , we sha l l s imply

p r o v e t h a t t h e w3 a re ana ly t i c a t the o r ig in . W e a l ready know tha t the

w^ are ana ly t ic in (x, y) on GR and in x for {x, y ) on GhR. O u r m e t h o d is

to use the proof of the Cauchy-Kovalevsky theorem to show tha t a so lu

t ion can be ob ta ined wi th our g iven va lues q)^ ( x , y o) which has a rad ius

of convergence independent o f y o, if yo is small enough.

Fo r each rea l yo > 0 bu t suff ic ient ly smal l , we m ak e the usu a l sub

s t i tu t ions to p repa re to p rove the Cauchy-Kova levs ky theo rem as i t i s

proved in the c i ted book by G OU RSA T, p p . 2 — 6 :

(5.8.50) *a)^x,r]; yo) = = i2)^{x, y o + 7 ] ) — (pj{x; yo).

This l e ads to equa t ions

( 5 . 8 . 5 1 ) -^ = p [ ^ , y o + r i > *< ^ + 9 { ^ ' > y o ) , * c o ^ + ( p x { ^ , y o ) ] -

Sub t rac t ing o f f the cons tan t t e rm and mak ing the f ina l change

(5.8.52) coHx,7 ]; y o) = *w^x, T J ; yo) — a^yo) rj

^^(yo) - f^ [0, yo, (p (0, yo), (fx(0, yo)],

we ob ta in the equa t ions

(5.8.53) j ^ = gH^> V>(^^< ^x]yo) = -aHyo) +

+ f^ [^, y,co + (p [ x , yo) + a ( y o ) r}> ^x + (fx [ x , yo)].

N o w t h e f^ [x, y, w, p) a re ana ly t i c wi th

\p{x,y,w,p)\ < M i for \x\''<^r, \y \ <r, \w^ < ^ \pi\ <r.

Moreove r , we know tha t the <p^ are anah^tic in x a n d

|^^*(:^,yo)| < ^ , \(pUi^>yo)\<^ if 1^1 < ^ , 0 < y o < ( 5 .

T hu s we see th a t th ere i s a o" > 0 s uch th a t the g ^ a re ana l y t i c m(x,rj,a), n)

for each small yo with

\g^{x,y,a),n',yo)\<.Mi if \x°'\^a, | ? 7 | < o ' , | a > | < o r , |7r| < c r ,

0 < yo < (T .

C o n s e q u e n t l y t h e r e a r e n u m b e r s M, r > 0 an d ^ > 0 suc h th a t each g^

i s domina ted by the func t ion

C ^ ^ . M[1 ^ y-i(7? + Afi + 1- ;v»' 4- col H 1_ tt,P)]. u - Q'^Enr]

1 8 6 Differentiability ofweak solutions

T h e r e m a i n d e r of the proof as given inG O U R S A T s hows tha t the re is a

(3' > 0s uch tha t the co ^ {x, r); y o) a re ana ly t i c wi th

\co^x,rj;yo)\ <. [ A ^ for |:^«| < ^ ', 17y | < ^ ', 0 < yo < '

w h e r e d' a n d t h e [x ^ are independent o f JQ. T h u s t h e w^ ob ta ined a s aboveby so lv ing the d i f fe ren t ia l equa t ions have the same proper t ies and mus t

co inc ide wi th those ob ta ined prev ious ly , fo r y > 0 . The i r ana ly t i c i ty at

the origin follows.

5 .9 . Propert ies ofth e extremals ; regu lar cases

In th is sec t ion we comple te the proofs of t h e t h e o r e m s a n d r e m a r k s

m a d e inC hap ter 1 concern in g th e d i f fe ren t iab i l i ty of the so lu t ions of

va r ia t iona l p rob lems and of the weak so lu t ions of the quasi- l inear diffe

ren t ia l equa t ions men t ioned in the in t roduc t ion .

W e begin by com ple t ing th e proof of Th eore m 1.11.1 which s ta tes in the

case of the quasi- l inear equations (1 .10.13) and (1.10.14), where the A

a n d Bi satisfy (1.10.7") or (1 .10 .7 '" ) wi th ^ > 2 , tha t the de r iva t ive s ^ j ,

and the func t ion U = VÎ^^Hl{D) for each D C CG and the de r iva

t ives sa t is fy the d i f fe ren t ia ted equa t ions (1 .11 .9) there . InChapter 1 , we

in t roduced care fu l ly the d i f fe rence-quot ien t p rocedure and proved tha t

the d i ffe rence q uo t ie n ts

^h(^ ) = h~Âz, Az = z[x-\ - hey) ~ z[x)

satis f ied the equations (1 .11.3) and that for each D C DaC D' CG,

(5.9.1) fAf,\Vzh\ dx {Ci + C2a- ) j Ah{x)zl{x)dx

w here Ci an d C2 a re ind epe nd en t of h. Now^, from the exis tence theory, it

follows that each z^y^Ljc w i t h ^ > 2. Consequently , s ince

1

Zh{x) = J z^y(x + t h ey) dt,

0

i t fol lows from Theorem 3.6.8 that Zh ->z,y in Lfc{D') for each D' C(ZG.

Moreove r , by a p p r o x i m a t i n g s t r o n g l y toz inH\[D) hy Zn^^C^ a n d b y

us ing the techniques of Theorem 3 .6 .8 and Lemma 3-4 .2 we conc lude tha t

Afi~->A inL]cj{jc-2)(D') if > 2a n d A \ ii k = 2. Th us , f rom the

Holde r inequa l i ty we s ee tha t the r igh t s ide of (5.9-1) tends to i ts ex

pec ted l imi t as A -> 0 . N ex t , s ince Aji (x) > 1, it follows that zn —7 u, s ay ,

in H\{D) for a s ubs equence of h->0. B u t from T he or em 3-4.4 (s inceD a a s t rong ly L ips ch i tz doma in C D', e tc . ) and the fac t tha t zj i - py in

L]c(D), it fo l lows tha t u = z^y soz^y — py^H\(D). Also, wes ee tha t

An(x)-Â(x) = F^-2(jv) a.e. and the coefficients af^^îx), hl^^{x), eH

^hip ^hip ^ fhi ^^^^ ' ' ^ ^ ^ b o u u d c d l y to the i r expec ted l imi t s as

h - 0, Pji {x) -> V {x ) a.e. and in L^r {D'). F r o m t h e u n i f o r m b o u n d e d n e s s

5.9- Properties of the extremals; regular cases 1 8 7

1 1

of the left side of (5.9-1), it follows that Alaliaz{j, Alc^^^-zi^^ t e n d1 ' i 1*

weakly in L2 {D) to ce r ta in quan t i t i e s and A^hh ' zji, A\Ph ' el, A^^dn • zji,

a n d A^fl' Pfi t end s t rong ly in L2{D) to the i r expec ted l imi t s a.s h-> 0th rough a fu r the r s ubs equence . By mul t ip ly ing the f i r s t two by bounded

meas u rab le func t ions and in teg ra t ing , u s ing the weak and s t rong con

vergence a l ready es tab l ished we see tha t the f i rs t two te rms tend weakly1 1

to the i r expec ted l imi ts , li C^Lipc(D), A^S/C a n d A^C t e n d s t r o n g l y

to the i r expec ted l imi ts so tha t we may le t A ->0 to ob ta in (1 .11 .9) and

(1.11.10).

Before proceeding fur ther wi th the genera l theory , we now d iscuss

the ex tremals of / {z, G) in the case (1.10.8) with v = k = 2, N a r b i t r a r y .We have a l ready seen in Theorem 4 .3-1 tha t any min imiz ing func t ion z

satis f ies a condit ion

(5.9.2) f\Vz\^dx<L^(rlR)^f', 0<r<R , B(xo,R)cG , ju > 0B(xo,r)

from which i t fol lows that 2 ^ C^ (G ). But now, i f we cons ider a domain

D (Z D'ad D' C d G, we know tha t the d i f fe rence quot ien t Z h satisfies

th e equa t ions (1 .11.3 ) in wh ich ^ ^ = 1, the alf^ satisfy (5.2.17), the hhyCfi, a n d dfi satis fy (5 .2 .18), and Ch and// i sa t is fy (5 .4-8) uniformly in h a n d ,

as we have a l ready seen above , each Zh ^ HI [D) w i t h n o r m u n i f or m l y

bounded . I t fo l lows f rom Theorem 5-4.1, w hic h ha s been see n in § 5-4 to

gene ra l i ze to s ys tems , tha t the zji satisfy a condition like (5-9-2) uni

fo rmly wi th G rep laced by D, so that this is t rue of each z^y w h i c h t h e r e

fore ^ C^{D). T he w hole an aly s is carries ov er for a solutio n ^ of (1 .10.14)

if the A^ a n d Bf satis fy (1 .3 .8") with k = v = 2, N a r b i t r a r y . O n c e t h e

z^ a n d pi^ are seen to ^ C^{G), the equa t ions (1 .11 .9) assume the form

(5.1.1) (with B {u, v), C {u, v), a n d L {v) given in (5.2.16) with coefficients

e, an d / Ho lder con t inu ous . Th us , in th e case N = 1, we conclu de from

§ 5 - 5 t h a t z^C^{G) and hence sa t is f ies Eule r ' s equa t ion ; then the

th eo re m s of § 5-5 or § 5 .6 can be used to ob tai n th e resu lts s ta te d in

C h a p t e r 1 t h a t z^C^ii f^ 6% or A^ a n d Bi^ C^''^ in case # = 2 or if /

a n d fp ^ C^-i or Af^ C^'^ a n d Bt ^ C^'^ in case n > 2. The cases where

iV > 1 a re t rea ted in Ch apte r 6 . Clear ly ^ ^ C ^ i f /^ C ^ or A a n d B ^ C ^ .

T h a t z is an al yt ic if / of ^ an d B are ana ly t ic fo l lows f rom the theorems

in § 5-8 as g en er al iz ed in § 6.7-We now prove Theorem (1.11.1 ') for which proof we need the follow

in g lemma due to L A D Y Z E N S K A Y A a n d U R A L ' T S E V A [2] :

Lemma 5 .9 .1 . Suppose the A^ and Bf satisfy (1.10.8") with ^ > r

and that z^Hl{G) D Cl{G), 0 < / ^ < 1 and is a solution of ( I . IO.14)-

Then there are numbers Ri and Ci > 0 which depend only on v, m, M, mi,

18 8 Different iabi l i ty of we ak solut ions

Ml, k, K, fi, and A^ {z, G) such that

JV^ (x) |2 (x) dx < C2 ie^/F*-2 \vS\^dx,B{xo,R) B{xft,R)

0<R ^Ri, SeHl,[B{xo, R)-].Proof. Suppose, first that | ^Lipc[B(xo, R)], and define

C^X) = f 2 ( ^ ) [ ( ) -z^Xo)]

this choice being allowed in (1.10.14) as one sees by an approximation.Using that C* in (1.10.14), we obtain

/ { | 2 - [Pi^t + (^' - 4)Bi] + 2i{zi - zi)S,.At}dx = 0.Bixo,R)

The hyp otheses (1.3.8") (iV = 1) imply th a t1

PiAUx, z,p)=piAU x, z, 0) +pip}JAy (x, z, tp)dt0

1

> / ) i ^ J ( ^ . ^ , 0 ) + W i | ^ | 2 | ( 1 + ^ 2 | ^ |2 )- l+ A :/2 ^^

0

> W2 F*^ — i^ 2, ^ 2 , K2 =^ (p (v , W i , M l , i ^ , ^ )

M(;t:, ^,:?{))| < M i F * - i , | 5 ( : \ ; , ^ , ^ ) | < M i F*.

It follows that

/ F * P ^k; < | { Z i | 2 + Z2 Z i^'' F* ^2 + Z3 i: Z?' F^-21VI |2} dxB{xo,R) B{xo,R)

L = hf, [z, G), Z i = Z i {v, mi, M i, K, k)

from which the lemma follows easily, since

li^dx< 2-1 R^f\ V I |2 dx < 2-1 Rf^JVf^-^ I VI12 dxBixo.R) B{xo,R) B{xo,R)

by Poincare's inequality, k being > 2. Th e lemm a follows easily for an y1 $ HIQ[B (XQ, R)] by an approximation.

The proof of Theorem 1.11.1' starts with the difference quotient procedure which leads to equations (1.11.3') obtained from (1.11.3) byreplacing bh, cu, 4 , and/; , Pu by b Ph, cn Ph, 4 Qh, and 4 Qh, respectively,where Ph is defined in (1.11.4) along with Uh and Ch and Qh, bh, Ch, 4 , andfh are now defined by

1 1

AhPhbh=fAzdt, AnPhC h =fBpdt,0 0

1

(5.9.3) AhQh=j[\ +\z{x) + tAz\^ + \p(x) + tAp\^]J^'^dt,0

1 1

AhQhdh =fBzdt, AhQhfh =JBxdt.

5.9- Properties of the extremals; regular cases 1 8 9

F ro m our hypot hes is (ver if ied for an y ex trem al if ^ > r and by an y min i

miz ing ex tre m al if ^ = '^), i t fol lows th a t z satisfies (1.10.9) and is there

fore ^ C|J {D) for an y D C C G ; so we m ay a ssu m e th a t G i s a l rea dy a sub -

domain of the domain of z a n d z^ C'|J(S). So, let us pick B {XQ, R -{- a)C GjiQ, 0 < a < i? , an d define r] ^ L ipc[B {XQ, R + a)] w it h ? = 1 on

B (XQ, R) and defineC = 7] Z, Z = rj Zh .

Subs t i tu t ing th is f in the equa t ions (1 .11 .3 ' ) , we obta in

fAh{{VZ + Vrj'Z n) - [ah - (VZ -Vrj - z^) + b^PhZ + rjehPh] +

+ Z ' [cnPn- {V Z -V 7} ' Zh) + dnQh Z -^ U Qh]} dx = 0

f rom which we conc lude in the usua l manner (s ince Pf < Qh) t h a tj Ah\VZ\^dx^j {Z iAhQnZ ^+Z 2,\Vri\^Ahzl+Z ^AhQh)dx

B{xo.E+a) B(xo.R+a)

Z i = Z i{v,N,mi,Mi,k,K).

Now, we observe tha t the func t ion z {x -\- h ey ) (— z -+ A z) satisfies

the hypotheses of Lemma 5-9 .1 for the func t ions *^ and *5 def ined by

*A(x,z,p) = A (x -\- hey ,z,p), etc . , which obviously sat is fy (1 .10.8")

e tc . T h u s w e m a y a p p l y t h a t l e m m a w i t h V{x) rep laced by

V{x + hey) = 1 + \z + Az\^ + \p +Ap\^.

By us ing the l emma in con junc t ion wi th the inequa l i t i e s

(5.9.4) < / [ ( 1 + \pi\ +tAp\^]rdt {Ap=p2-pi, etc.)

0

< C [ ( 1 + | ^ i P ) r + (i +\p2m {r=-^ + kjl or kjl)

we find, ii R-\- a <^R i{z{x + h ey) also ^ CI[B{XQ,R + a)\), t h a tjZiAhQh\Z\^dx<^Z^[R A-ajf' fAhlVZl^dx.

B{xo.R+a) B{xo,R+a)

T h u s , a R -{- a < , R2, depend ing on the s ame quan t i t i e s a s Ri, we ob ta in

the inequa l i ty (1 .11 .8) wi th D = B (XQ, R), D' = B {XQ, R + a) and also

JAhQhzldx < Z 5 a - 2 R^jAhQhdx.B(xo,R) B{xo,R+a)

U sing t hi s las t in eq ua li ty as well as the o the rs w e see th a t th e res t of th e

proof can be carr ied th rough as before except tha t now

AlbhPh'Z h, A\ch'SIZh. AQ\^^

converge on ly weakly in L2 \B (^0, ^ ) ] to t he ir l im its an d, for each1 1 1 1

^^Lipc[B[xQ,R)], AlVC> AlPhC, ^nd AlQlC t end s t rong ly to the i r

l im i t s .

19 0 Differen t iabi l i ty of wea k solut ions

The proof of Theorem 1.11.1" is deferred to the next section s ince a

new idea is required .

W e proceed to the proof of Theorem 1.11.2. H e r e w e m u s t a s s u m e t h a t

N = \ (see R em ar k be low) . How eve r we need a s s ume on ly the hypo thes e s

and conc lus ions of Theorems 1 .11 .1 , 1 .11 .1 ' , and 1 .11 .1" . From Lemma

5.3 .1 and these resu l ts we conc lude tha t we may subs t i tu te

C = coV~'py , co^LipcD, a)(x)'>0

in the equations (1 .11.9) or (1 .11.9 ') and sum on y . Let us cons ider the1

case (1.10.7) in which V = {\ + z^ -\- \p\^)^, the o ther case is t rea teds imila r ly . Us ing the re la t ions

VV,oc==pypy,u + Z Pc., pypy = \ P \^ = V^ - \ - Z ^ ,

| V F | 2 < | V i ^ | 2 + | ^ | 2 , Coc =-V-'[co,o:py + co{Py,c. - eV-^PyV,a)]

we f ind tha t V satisfies

f V^-^-'{a),4a-^V,p +'B^V) + co('C^V,oc +'DV)}dx<0

D

w h e r e

'5« = V-^l-a'^^zp^ + ^«(F2 - \ — z^) + e'^PVpy ]

'D =^ -g + F - 2 [ - (&« + c'^)zPo. + d{V^ - 1 - ^2) 4_ Vfypy]

g an d £ > 0 be ing chosen so th a t

a''^py,ocpy,p + Ve^ypy^oc -sa^^V,o.V,^ + gV^ > 0 .

I t i s c lea r th a t g may be taken bounded so a l l the coeff ic ien ts a re bounded

and the result (5.3-4) follows with

W =y^-'=U\ ?i = 2(k-8)lk, P = 1 .

1

In the case (1 .10.8), w her e F = (1 + l^ p) ^ we ob tai n (5.3-4) w i t h P = Fand somewhat different coeffic ients B, C, a n d D.

Remark. The rea s on tha t we mus t a s s ume AT = 1 in th is proof is th a t

if we set C* = co V-^ p\, above we ob ta in t e rms l ike

w hic h giv e co,a F "^ a'^^{VV,p ~ zp^) in the case N = \ bu t y ie ld no th

ing t rac tab le ior N '> 1.

The resu l ts above apply on ly to in te r ior domains and , except for thecases k = V = 2, to cases wher e A^ = 1. Ho we ver , if we hav e a w eak

solutio n of (1 .10.14) for the gen eral iV which is kn ow n to ^ ^jiiD), a t leas t

for Z) C C G, then the equations (1 .11.9) have bounded coeffic ients ^ C^

and the theory of § 5.5, as generalize d in § 6 .4 a l lows us to conc lude t h a t

t h e p^ ^ CJ so the z^ $ Q . T hen repea ted app hca t io ns of the theo rem s of

5.10. The ex t r ema l s in the c a s e 1 < C A < < 2 191

§ 5.6 and 6.3 to Eu le r ' s equa t io ns a llows us to conc lude the higher differen

t i ab i l i ty , ana ly t i c i ty , etc. If A^ = 1, all the higher d i f fe ren t iab i l i ty resu l ts

follow if we h a v e a weak s o lu t ion known to be Lips ch i tz . Thus , in the

case of an i n t e g r a n d of the f o r m / ( ^ ) , we conc lude the dif fe ren t iab i l i ty of

th e Lips chi tz so lu t ions found in § 4- 2 and a n n o u n c e d in §§ 1.7 and 1.10. And,

as was s t a t e d in § 1.10, L a d y z e n s k a and U r a l ' t s e v a h a v e s h o w e d t h a t

a n y b o u n d e d w e a k s o l u t i o n of (1.10.14) in the case (1.10.8") is differenti-

ab le , in the case N = \. But, for the general sys tem (1.10.14) for N

a r b i t r a r y , it is poss ib le to conc lude all the higher d i f fe ren t iab i l i ty of a

weak s o lu t ion if it is k n o w n t o $ C^ (D), D C G. The w r i t e r ( M O R R E Y [10])

showed th is us ing the space L2,^ of § 5.5 (see also N I R E N B E R G [1] for the

two d imens iona l case) . But it is more in te re s t ing and the t e chn iques are

re s u l t s are m o r e i m p o r t a n t u s i n g the Lp theo ry wh ich has been h igh lydeveloped for higher order equations by A G M O N , A G M O N , D O U G L I S , and

N I R E N B E R G ([1], [2]), B R O W D E R ([1], [2]) and w h i c h has been p re s en ted

in §§ 5-5 and 5-6 for a s ing le second order equa t ion . We sha l l p resen t the

t h e o r y as it appl ies to h ighe r o rde r equa t ions in C h a p t e r 6.

5 . 1 0 . The e x t r e m a l s in the case 1 < A: < 2

In th is sec t ion , we prove Theorem 1 .11 .1 ' ' . The only diff iculty in th i s

is due to the fac t tha t it is no longe r t rue tha t Ah(x) > 1 and the placesw h e r e A his, near ze ro do not necessar i ly occur at the places where z^ is

l a rge . So to avoid this diff iculty , we i n t r o d u c e a new idea , name ly to

rep lace our m i n i m u m p r o b l e m by t h a t of min imiz ing / {z , G) a m o n g

those func t ions for w h i c h a ce r ta in o the r in teg ra l / (z, G) < s o m e n u m b e r

K. This same idea wi l l be used aga in in §9-5 in connec t ion wi th the

p a r a m e t r i c p r o b l e m . The wri te r fee ls tha t a fu r the r exp lo i t a t ion of t h i s

idea could lead to fruitful results .

Definit ion 5 .10.1. In th is sec t ion we define

(5-10.1) J(z, G) = (1/2) / I Vz\^dx.G

For each z*^Hl(G), we d e n o t e by ZQ the ha rmon ic func t ion in G s uch

t h a t ZQ ~ z*^ H\Q[G) and we define KQ = J(ZQ, G).

Theorem 5 .10.1 . Suppose z* ^Hl(G). Then for each K > KQ, there

exists a function Z K which minimizes I {z, G) in the family FK of all z such

that ^ — ^ * ^ H\Q (G) and J [z , G) < K. The vector Z K satisfies the equation

(5 .1 0 .2 ) / K : a ( /* i ! > i + / p ^ ) + c«/ .»]rf^ = o , c € ^ i o ( G ) .G

for some /u = /LC [K) > 0. There exists a sequence {Kfi) _> _[_ 00 such that

Kn ' [x (Kn) - ^ 0 . If ZK 1 ^ ^0, fJ' is unique.

Proof. The exis tence fo llows im m edi a te ly f rom Th eor em 1 .9 .1 . T h a t

fi is u n i q u e if Z K 9^ Z Q is e v i d e n t . If / {ZK, G) < K, it is o b v i o u s t h a t

(5 .10.2) holds with ju = 0. If K > KQ and / {ZK, G) = K, t h e n Z R ^ ZQ

a n d is not h a r m o n i c so tha t the re ex i s t s a func t ion Ci^ L ipc (G ) such

t h a t

(5.10.3) / C l a 4 . a ^ ^ = 1.G

I t fo l lows tha t J {ZK + Afi , G) is increas ing wi th L S u p p o s e t h a t C sa t is

fies

( 5 . 1 0 . 4 ) / C : a 4 . « ^ ^ = 0G

a n d let us define

Z{x, A) = ZK(X) + A C(:v) - C A2 Ci W

w h e r e c is a c o n s t a n t . We eas i ly conc lude tha t J[Z {', A), G] < iC for allA w i t h I A| < Ao if c is suffic iently large and posit ive . Since / (ZK, G) = K

a n d Z K min imizes / {z, G) as a b o v e , we d e d u c e t h a t

(5.10.5) l{C:Jpi + C'fzi)dx^OG

for each C^ L iPc {G) which satisfies (5-10.4).

N o w , let C be any v e c t o r ^ L i pc (G). We can w r i t e

(5.10.6) C = C* + ACi where ?^ = f C^z^^^dx.

G

T he n C* satis f ies (5 .10.4). Ac cord ingly (5 .10.5) holds ioi C* = C — Ci so

tha t (5 .10 .2) ho lds wi th

(5.10.7) f^= - l('^Ci'fp + Ci'fz)dxG

as one sees by s u b s t i t u t i n g the va lue of 2. from (5.10.6). Since I (z , G)

> I {ZK, G) w h e n e v e r / (z, G) <. K and since J [ZK + ^ Ci, G) is in

creas ing , we conc lude tha t

A -i [/ {Z K - A fi, G ) - / (ZK, G ) ] > 0 for 0 < > l < h,

(^•^^•^) l i m A - i [ / {ZK -?iCi,G)-I (ZK, G)] = fx

using (5.10.7). Finally , let cp [K) = I {ZK, G). E v i d e n t l y 99 i s non- increas ing

a n d

(p{K+AK) <I(zK + ?iCi>G), AK = J(zK + )iCi,G)-J{zK,G).

Since A KjX -> 1 as A -> 0, we f ind tha t

(p'{K) = —[JL[K) for almost all K.

T h e n , it follows that fx is s u m m a b l e on [KQ, 00) and

ff^{K)dK<I(zo,G) -I[zi,G)

z\ be ing a minimiz ing func t ion for / {z , G).

5-10. The ex t rem als in th e case 1 < A < 2 19 3

We now prove Theorem 1.11.1".

Proof of Theorem 1.11.1". Suppose first that ^*$Ci(r) for some T D G .

Then, if Z ~ z"^^H\Q[G), there exists a sequence {zn], each Zn being in

Ci{F) with[zn

—Z*)\G^C\ [G)

such tha t Z n->Z in HI(G). Consequently^{^Ky G) -^ inf I(Z, G) ioT Z — z* ^ HIQ(G) SO tha t Z K —r some z 2.^ Kruns through a subsequence of the sequence {Kn\ of Theorem 5•'10.1

and z minimizes I{Z, G) among sll Z y Z — z*^ HIQ (G).

Now for K fixed and equal to one of the Kn above, we replace C ^^the equation 5-10.2 by C/ W = "-" [C(^ — ^ ^y) — C W] and concludethat the difference quotient ZRU satisfies

D'(5.10.9) + C*^ft(6ri^4»,.a + Ci,-.4ft ^ni.Pn)]dx = 0.

By putt ing t,^ = rf z\^, we find in the usual way that

Since / (ZR, G) < K, we may let A -> 0 to obtain the limiting equationscorresponding to 5-10.9) and the inequality

l(f, + F|-2) I VPKP

dx^C a-^f(f, VI + V\) dxD D'

(5-10.10) = 2C3 r^K'iLi(K) +Ca-^JY %:dx {DcD'^).

Th e right side of (5.10.10) rem ains bou nde d as i^ -> 00 in our subsequence,since K • fJi{K) -> 0 and / {ZK, G) -> min. Also

/ I VpK 1^dx = / F | • V^Î VpK 1 ^:vD D

(5-10.11) ^lfV^^dxY-''^ÎV^£ ^\\/pK\^dxY ^ (h = k(2~k)l2).

From this it follows that pK-y p in Hl{D), tha t V^^+^^^pK -7 F-1+^/2^în H\[D), that f/ = V^^^^Hl(D), and that we can let X ->00 in thelimiting equations [h = 0) (5.10.9) to obtain the equation (1.11.9).

Finally, suppose merely th at z* ^ HI [G). Let ZQ be a function minimizing / (Z, G) among all Z ^Z — z*^ Hl{G). Let {Gn} be a sequence ofdomains such that Gn C Gn+i for each n and G = \J Gn- For each ^,

there exists a sequence of functions {z*^}, each 2:* , ^^(GTI+I) , such that^tp -^ ^0 in HI (G n) as ^ ->oo. For each n and ^, choose a function 2:^^= lim-s: ^^ as in th e first p art of thi s proof. For each n, we can choosea subsequence of the Znp which converges weakly in H\[Gn) to some Zn-

This function Zn is minimizing for / on Gn and Zn — ZQ(^ H\Q{Gn). Foreach n, the function Z n = Zn in G^ and — ZQ in G — Gn is minimizing

Morrey, Multiple Integrals \ 3

for / (z, G) a n d Zn — z"^^ HIQ{G). The bounds for the in tegra ls of \Vpn\^,

V^~^ \Vpn\^, etc . , on each D C G G are un iform, so tha t there ex is ts a

subsequence of the Z n which converges weakly toward a func t ion z of

the type s ough t .

5 .11. Th e th eory o f Lad yzen sk aya an d Ural ' t seva^

In t his section, we assum e th a t ^ is a bo un de d w eak solu tion of (1 .10.14)

w ith A^ = 1 in wh ich t h e A^" a n d B satis fy (1 .10.8") with k> \. Since

w e m a y w r i t e1

poc A^ [x, z,p) =pocA^ {x, z, 0) +pccp^ I Al^ {x, z, t p) dt

0i t fol lows eas ily that the hypotheses (1 .10.8") imply the exis tence of

n u m b e r s mi a n d M\ s u c h t h a t

pocA^{x,z,p) > mi \p\^. — Ml, 1^1 < M i ( l + 1 1* -1)

( 5 . 11 . 1 ) \B\<Mi(\ + \p\^). ^^^^^

The mos t d i f f icu l t par t o f th is ana lys is i s tha t o f showing tha t

z^ C^ (G) for some / / > 0 . Fo r th a t pu rpos e , i t i s conv enien t to in t ro du ce

the fol lowing D E G I O R G I type s pace :

Definit ion 5 .11.1. z ^"^djciG; L;y; 6) iff (i) z^Hl(G), (ii) \z{x)\ <Lon G, (iii) for each sphere B {XQ, r) C G, w e h a v e

( 5 . 1 1 . 2 ) j\\Jz\^dx^yr''-T^,B{xo,r/2)

(iv) for each hi for which z(x) — hi ^d in B (XQ, r), we have

( 5 . 1 1 . 3 ) / I Vz\^dx<y \A(xQ, hi, r)\l\ + {a r)-^^ msix [z (x ) — A j H ,

0<a< \,

(v) for each ^2 such that A2 — z {x) < (5 on B (XQ, r), we have

f\Vz\^dx<y\B(xo, A2, ^) | | l + (o -r) -^ m a x [; 2 — '2^(^)]^},

0 < a < 1.

H e r e A {XQ, h,r) = B (XQ, r) D A (h) a n d B (XQ, h,r) = B [XQ, r) Ci B [h)

w h e r e A (h) a n d B [h) are respec t ive ly the se ts where z [x) ^ h a n d

z [x) < h.

Theorem 5 .11.1 . Suppose z ^')8]c[G', L',y',d), XQ^G , and a is the

distance of XQ from dG, Then

osc. [z,B(xo, r)] < Co(rlaYo, 0 < r < a, /^o > 0

where CQ and jbio depend only on k, L, y, d, and G.

1 For a complete account of their work, see their n ew book (LADYZENSKAYA

a n d URAL'TSEVA [3]).

5.11. The theory of L A D Y Z E N S K A Y A a n d U R A L ' T S E V A 1 9 5

I t i s convenien t f i rs t to prove the fo l lowing lemmas :

Lemma 5 .11 .1 . Suppose w ^ H\\B[xQy r)]. Then

I j\w{x)-w |«'/(»'-i) dxy-^^'" < C{v)j\ Vw {%) I dx

where w denotes the average value of w.

Proof. Fr om the hom ogene i ty of the s i tua t ion , we m ay a s s ume th a t

y = 1 an d w = 0. F ro m T heo rem 3-6 .5 , i t fol lows th a t

[\w[x)\dx ^ Ci{v)(\ S/w{x) \dx.5 ( 0 .1 ) 5 ( 0 .1 )

From Theorem 3-5-4 , we conc lude tha t there i s a bounded opera tor @

from H\[B (0,1)] to H\Q \_B (0 ,2)] such that iiW=^w, t h e n

j\VW{x)\dx<^C2J\\Vw{x)\ + \w{x)\^dx-B(0.2) 5(0.1)

<.Czj\Vw{x)\dx, W{x) ^w{x), x^B(0, 1).5 ( 0 . 1 )

Then, f rom SOBOLEV'S lemma (Theorem 3 .5 .3) , we conc lude tha t

l\w{x) \^^(^-^)dx<^f\W{x) \^^(^-^)dx <(C4f\'7 W{x)\dx^^^''-^^5 ( 0 .1 ) 5 ( 0 ,2 ) \ 5 ( 0 ,2 ) /

f rom which the lemma fo l lows .

Lemma 5 .11.2 ( D E G I O R G I [1]). There is a constant ^ == P(v) such

that if u^H\ [G) and B {XQ, r) G G then

(A - h)r^-î^{K A; r) < ^ / | \/u(x)\dx( 5 . 1 1 . 4 ) ^(£eo.7i.r)-^(a!o.A.r)

h<X, r(h,X\r) = mm [\A(xo, K,r)\, y vr" — \A(xo, h,r)\].

Proof. Let us define

r 0, if A:^ JB (XO, r) — A [XQ, h, r)

w (x) = I u (x) — h, a xÂ (XQ, h,r) — A {XQ, A, r)

[k — h, if ^ ^ ^ (XQ, Xy r)

T h e n Vw (x) = 0 unless xÂ (XQ, h,r) •— A {XQ, A, r) when V w (x)

= Vu {x) (a lmos t eve rywhere ) . App ly ing Lemma 5.11.1 and us ing the

definit ion of r , we obtain

T-[(A — A — wy + wt] <iCJ\Vw(x)\dx}\ t = vl{v — 1)

from which the lemma fo l lows eas i ly .

Lemma 5 .11.3 . For each (T$ (0,1), there exists a 6 {a, y , k,v) "> 0 such

that if u^S8]c(G , L, y , d), B {XQ, Q) C G, and H = m a x [u(x) — h] ^d

(xÂ (XQ, h, Q)), then

I ^ (^0, h + H + a^ Q, Q — a Q)\ = 0 whenever \ A [x{), h, Q)\ <,d • Q^

The corresponding result ho lds for the sets B {XQ, h, g).

13*

Proof. Define

Qj = Q — (T Q + 2-3 a Q, hj = h + G H + a^Q — 2-^ {a H + a^ Q),

j = 0, 1, 2, . . .

W e s ha l l s how tha t | ^ ( ^ o , ^hQi)\ -^0- We a p p l y the estimate (5 .11.3)

w i t h Q = Qj and Q ~ a Q = QJÎ to o b t a i n

(5.11.5) / I Vu\^dx<y ' \A(xo, hj,Qj)\ • (1 + {QJ - Qui)-^H^y

W e now e s t i m a t e the left side of (5.11-5) using (5.11.4) and the H o l d e r

i n e q u a l i t y and a s s u m i n g t h a t d ^ Q^ <: yp (Q — a QYJI to o b t a i n

/ I Vu\^dx^\A \^-^( f\ Vu I dxY" > \A [1-^ / f\Vu\ dxY

A \A I \A-A' I

(5.11.6) > j 5 - ^ | ^ " | i - ^ ( V i - Ay)î .4'I*<!-!/»').

[A = A(xo, hj, Qj+i), A' = A{xo, hjî, QJ_Î), A" = A (XQ, hj, QJ);

fo r \ - k < 0 , | ^ ' ' | > | ^ | , and \A'\ <\B {XQ, QJ+I)\ ~ \A\ since

\A (xo, ho, Qo) I = \A {xo, h, Q)\<ê'^'' < \B {XQ, Q - a Q) | / 2 . F r o m

(5.11.6) and (5.11-5) and the fac t tha t \A'\ < \A''\, we conc lude tha t

l ^ ' l < i 5 y i / ^ i ^ " | i + i / ^ ( V i - h,)-HQJ - QHI)-'[H' 4- {^fj'-

(5.11.7)

W e now p r o v e by i n d u c t i o n t h a t

(5.11.8) \A(xo,h3',Qj)\ <d'2-^*'^Q\ y = 0, 1, 2, . . .

p r o v i d e d t h a t we t a k e

e = min | i Q-^' I B (XQ, Q-GQ)], a^^l{2^''+^ ^ yi/*)4.

F o r j = 0, (5.11.8) is j u s t the cond i t ion \A {XQ, h, Q)\ < 0 • ^^ If we

assume tha t (5 .11 .8) ho lds for s o m e y, it follows from (5.11.7) and thedefinit ion of 6 t h a t it ho lds for y + 1 •

Proof of the t h e o r e m . We cons ide r the bal ls

Bj ~ B {xo, Qj), Qj = 4~^' • ^, y = 0, 1, 2, . . .

W e d e n o t e by coj the oscil la t ion of u on Bj and set

Ci = max [2MIQO, 2^+1], ( o = a),

w h e r e the c o n s t a n t s will be chosen be low. Then we h a v e

coo < Ci Qo-

Suppos e for s ome j t h a t we h a v e the oppos i t e inequa l i ty

a)j > Ci Qj,

T h e n

( 5 . 1 1 . 9 ) a ) ; > 2 ^ + V ; .

5.11. Thetheory of L A D Y Z E N S K A Y A and U R A L ' T S E V A 197

W e s ha l l s how tha t 5 can be chosen so that (5 .11.9) implies

125+2

5.11.10) wj^rja)jî +Qj, ^ = i - - i - < i

T o t h a t end, we set

(O = CDj_l, Jl = (jLli + JL12)I2

jU i == m a x w (x), fji<z = mmu {x), x Bj_i

Dt = A (xo, jLCi ~ 2-* (JO, 2 Qj) — A (XQ, fii — 2 " * - ! co, 2QJ),

t = 1, . . ., s.

W e a s s u m e t h a t \A[XQ, ]l, 2QJ) \ < \B(xo, 2 Qj)\l2; if not, we m u s t h a v e

\B{XO,]2,2QJ)\ <\B(xo,2Qj)\l2.

W e now a p p l y L e m m a 5.11.2 and the H o l d e r i n e q u a l i t y to the i n t e

g ra l of I V I ove r Dt to o b t a i n

\A(Xo, fii - 2-^-1 CO, 2Qj) |Ml-l/»') • ( - ^ ) ^ <|SA;|i)^|A:-lJ| û\^dx,

(5.11.11) t=\,...,s. '

W e e s t i m a t e the in teg ra l on the r i g h t in (5.11.11) using (5.11.3), (5.11.2)

in case 2~*co > 6, and thef a c t s t h a t \A(xo,]l, 2QJ)\ < \B(xo, 2 ^ ; ) | / 2 ,

Jt = jui — 2-1 0), and u{x) ^ ju i on B(XQ, QJ-I). We obtain

l\Vu\^dx<:V ;î l r M ( ^ o , 1^1 - 2- CO, 2^;) K^ + (2^;-)"^(2-«co)^], 2-*co<^.

(5.11.12)

U s i n g the fac t tha t | A {XQ, JLII — 2"* o), 2^^) | < | ^ {XQ, ju i ~ 2-i co, 2^;)

^\B{xo, 2Qj)\j2, (5.11.12) becomes

J J ' ~[Ky ^ p ^ [QI + (2-^-1 co)*], 2-* CO < ^, K = 2"- ! y,.

(5.11.13)

B u t now

K[Q^ + (2 -^ -1 co) ] > 2" -! 7v • (dl2)^ if 2-* CO > a

s o t h a t we may write (5.11.13) in the form

(5.11.14) f\Vu\f^dx<KiQ^.-^[Q^ + (2-^-1 co)*], Ki^Ki(y,d, k,v).

Dt

T a k i n g i n t o a c c o u n t our assumption (5.11.9) and n o t i n g t h a t co = coy_i

> (Oj > 2^+1 Qj > 2^+1 Qj,we obtain, from (5.11.11) and (5.11.14),

I A (XQ, //I - 2-^-1 CO, 2Qj) |A:(i-i/*') (2 -^ -1 co)^

< / 5 ^ | / ) i i ^ - i • 2 i ^ i - ^ p ^ ( 2 - ^ - 1 co)^.

In add i t ion , | A [x^^, ju i — 2-^-i co , 2QJ) \ >\A (XQ, /LII — 2-«-i co, 2^ )

s o t h a t

( 5 . 1 1 . 1 5 ) \A{Xo, JU i - 2-«-l CO 2 ;) \Mv-l)/vik-l) ( 2Ki gv-k l/ik-l) . | / )^ | .

19 8 Different iabi l i ty of we ak solut ions

W e s um thes e inequa l i t i e s over t from 1 to s to ob ta in

S'\A [xo, iix - 2 - s - i CO, 2 q ) \ < ( 2 i ^ i ^^ ^p^)i/(*^-i) | B (x^, 2 QJ) \

(5.11.16) (r = k(v - \)lv{k- 1)) .

If s is so large that

[s - l • (2 i^ i |5^) )/. • 2". ^J+ (v-fc)/(&-i)]i/r ^ Q g j (2 ^ j)

( the Qj occur to the same power on bo th s ides ) then

I A (^0, /î - 2- ^- 1 a>, 2^,-) I < 0 ( i ) (2^J)\ 2 /

If, in ad di tio n, 2« > Mjd, th en 2-« -i co < 6 an d i t follows from L em m a

5-11.3 (wi th h = fj,i~ 2 - s - i (o, a= 1/2, Q = 2QJ, H = 2-« - i co) t h a t

I A {xo, ^ 1 — 2 - s -i CO + 2-«-2 CO + 2-1 ^;, ^;) I -= 0.

This means (s ince z (x) > jU2 on ^Byî) that

coj < co;_i — 2-^ -2 co;_i + ^j

which is the des i red inequa l i ty ( 5 .11 .10) .

Thus we have s hown tha t , fo r any j ,

( 5 .11.17) co j < C i Qj or (Oj < rj coj_i + QJ, 0 < ?y < 1.

Hence we ce r ta in ly ob ta in

co j < rj coy_i + (C i + 1) 4~^' ^0.

By induc t ion , we f ind tha t

CO,- < rj^{wo + (C i + 1) [( 4 ^ )- i + (4 ^) -2 + • • • + (4 77) W ] } < C2 T ',

C2 = coo + ( C i + \)l(4r)- 1)

s ince we mus t have rj > 3 /4 . T he the ore m the n fo llows f rom th e a rgu

ment in the las t part of the proof of Theorem 5-3-3-

Theorem 5.11.2. Suppose z is a solution o/(1 .10 .14) on G with N = i,

where the A °^ and B satisfy (5.11.1),^ '^ which z{x) <,L onG and z^ H^ {G).

Then z^ Bjc {G; L;y ; d) for some numbers y and d ^ 0 which depend only

on V, m, M, K, k, and L.

Proof. Firs t we se t^{x) = e^'^'^^rjîx)

w h e r e rj is defined as us ua l to equ al 1 on 5 (XQ, rfl) and to van i s h nea r

dB {XQ, r) a n d i n G — 5 {XQ, r) a n d A i s a pos i t ive cons tan t . We obta in

ê^^[XrjÂ^po, -\- krj^-Â°'rj,oc -^ Brj^]dx = 0.

Using (5 .11 .1) , we conc lude tha t

Je^^[Xmirj^\ S/z\^ — 2, Mirj^ — k Mi\S7 rj\ - rj^-^ {\ + \ \7 z\^-^)Bixo,r)

(5.11.18) — Mirj^(\ + I Vz\^)]dx < 0.

5.11. The theory of L A D Y Z E N S K A Y A an d U R A L ' T S E V A 1 9 9

U s i n g Y o u n g ' s i n e q u a l i t y ,

(5.11.19) a^-^b == k-^[k — \) sa^ + k-^s^-^b^,

we conc lude from (5.11.18) an d (5.11.19) w ith e = 1 th a t

fe^'{[Xmi — kMi](r]\'7 z\)^ - {X + k)Miri^ ~ 2Mx\\7 rj\^]dx ^0.B(xo,r)

By choos ing A = 2 Af i/wi, for in s ta nc e, we find t h a t

B{xo,rm

s ince e^'^ i s bou nde d abo ve and be low.

Nex t choos e h, a w ith 0 < a < 1, an d B [XQ, r) C G, and define

x) — h, if z(x) > h,

if z(x) < h.

(z{x)^{x) = rj^(x) co{x), a}(x) = i

w he re ^(:\^) — 1 on ^{XQ, r — ar) and vanishes on and outs ide dB(AÔ> >')•

Since a),a(^) = zô:{x) if z{x) > h, we ob ta in

B{oco,r)

>:f[rj^(mi\ Va>i^ —M i) — kMico\ V?y | ^*~ i ( | V- |^ - i + 1) —

A{xo,h,r}

(5.11.20) ~rj^coMi(\\/z\^ + \)]dx.

Now, we choose h s o t h a t co (x) <,d = mil4 Mi and use (5 .11.19) with

£ = mil4 Ml {k — \) and we conclude from (5.11.20) that

flVzl^dx^yJli+ia r)-^' co ^ {x)] dxA{xo,h,r—ra) A{xo,h,r)

f rom which proper ty ( iv ) fo l lows . Proper ty (v) i s p roved s imila r ly .

For what fo l lows i t i s convenien t to no te tha t the hypotheses (1 .10 .8")

imply (5 .11 .1) which , in tu rn , imply tha t

(5.11.21) pocA°'{x,z,p) ^m2V^ — M2, I ^ I < M 2 F ^ - i , \B\<M2V^

w h e r e m2 a n d M2 depend only on m, M, K, a n d k.

Lemma 5 .11 .4 . Suppose that the hy potheses of Theorem 5-11.2 hold

with k '>2. Then there are numbers x, 0 <. K <C i, a n d Ci which depend

only on m2, M2, Co, /no, and v, such that

I V^ f 2 (x) dx < Ci {rlaô I F^-2 \\/^\2dx,

Bixo^r) B{Xi,,r)

0 <r <K a, B (xo, a) C G, a < 1,

for any bounded ^^Hl^lB {XQ, r)].

Proof. B y appr ox im at io ns , i t i s easy to see th a t if r < ^ we m ay se t

C{x) = [z{x)^z{xo)]mx)

in (1.10.14) . F r o m t h i s s u b s t i t u t i o n and Theorems 5 .11 .1 and 5-11.2, it

fo l lows tha t

l{^^[A-p^ + {z- zo)B] + 2f | , a ( ^ - zo)A^}dx = 0

B(xo,T)

> / { f 2 [W2 V^ -M2-2C0 (rjay oM2 V^] - M2 Co (rja)f'oF*-21 y f | 2 ^ xJB(a;o.r)

f rom which the result follows easily, since

fPdx^(r^l2)f\Vi\^dxB{xo,r) B(Xo,r)

b y P o i n c a r e ' s I n e q u a l i t y and V^~^ > 1.

A s a cons equence of t h i s l e m m a and Theorems 5.11.1 and 5.11.2, we

o b t a i n the fo l lowing usefu l theorem:

Theorem 5 .11.3 . Suppose the hypotheses of Theorem 5-11.2 hold with

k >2. Then if D (Z C G, there are constants RQ > 0, C2 and C3, depend

ing only on v, m, M, K, k, L, and D such that z^ C^{D) with

hfîz, D) < C2, fV^^^dx < CsRf'oJv^-^ \V^\^dx R< RoB{xo,r) B{xo,R)

for each bounded ^^H\Q[B[XQ, R)], B[XQ, R) (Z D.

W e can n o w p r o v e the ana log of T h e o r e m 1.11.1.Theorem 5.11.4. Suppose that the hypotheses of Theorem 5.11.2 hold

with N = 1, k ^ 2, and suppose D (Z G G. Then the py and the function

U = V^'^^HliD) and they satisfy

(1 .11.90 IV^'-^{C,4^''^py,P + h^Vpy + e-yV) +

+ ^V{c<-py,a + dVPy+fyV)]dx = 0

(1.11,10) / F ^ - 2 | ^p\''-dx < C{v, m, M, K, k, L, D) f V^dx.

D O

In addition V^+^^Li{D).

Proof. We let Z)' be a d o m a i n s u c h t h a t D C C D' C C G and a p p l y

th e d if ference qu ot ie n t p r oced ure . This leads to the/^-equations (1 .11.3 ')

((1.11.3) modified as descr ibed in § 5.9 in the proof of Theorem 1.11.1 ') ;

for 0 <i \h\ <C ho, Zfi is H o l d e r c o n t i n u o u s .

N o w , let us s u p p o s e t h a t R + a < 2ô (• - -yD'), s uppos e B^â

^ B (xo, R + a) C D', let 7 = 1 on BR, e tc . , and set

C =^r)Z h, Zh = 7] Zh

in the equa t ions (1 .11 .3 ' ) . The re s u l t is (see the n o t a t i o n of § 5-9).

JAn{{VZn + V?? • zn) • [an • (VZ/, - V ^ zn) + hnPhZh + r} enPn] +Bn+a

+ Z„[cnP„ • [VZn - Vf]z„) + dnQuZu + / A Qh'\]dx = 0

5.11. The theory of LADYZENSKAYA an d U R A L ' T S E V A 2 0 1

from w hich we eas ily con clud e as us ua l (s ince P | < Qn)

(5.11.22) lAu\VZ h\^dx<lAh{ZiQnZ l + Z 27 ]^Qh + Z 3 \Vri\^zl)dx.

Now , by us ing the fac t th a t z (x -\- h ey ) is a solu tio n of (1.10.14) for t h efunc t ions A°'{x + h ey, z, p) a n d B [x + h ey , z, p) and us ing the in

eq ua liti es (5.9.4), we see as in § 5.9 t h a t

JAnQnZldx<^Z^j[V^x) + V^x + hey)-\Zl{x)dxBR-i-a BR+Q.

< Z4 Cs{R + aYoJlV^-^ix) + V^-^{x + hey)] \ VZ ^ \^dxBR+Q

<Z 5{R + ay ojAnl "^Z n fdx.BR+U

So, i f we assume Zi Zs(R + «)''o < 1/2, w e o b ta in

(5.11.23) fAn\vZh\^dx<,Z6jAh(ri^Qk+\Vr]\2z^)dxBR+U BR+U

( 5 . 1 1 . 2 4 ) lAnQkZ ldx<Z ^lA„{rj^Qh + \Vri\^zl)dx.BR+a BR+a

W e m a y n o w l e t A -> 0 in these inequali t ies and in (1 .11.3 ') to obtain the

re s u l t s .

Nex t we p rove a p repa ra to ry l emma l ike Lemma 5 .3 -3 -

L e m m a 5 . 1 L 2 . Suppose the hy potheses and conclusion of Theorem

5 .11.4 hold and suppose w := V = F ^ 7 ^ ^ L2{D) for some r > 1. Then

w^Hl{A) for each A CC D and if B {XQ, R + a) G D,

J\ Vw\^dx < C^r^a-^ fw^dx, 0 < a < R ,B{xo,R) B{xo,B+a)

Q= i + 2/^^ i, C4 = C4 (v, m, M, K, k, D);

C4does not depend on r.

Proof. I t fol lows from Theorem 5-11-4 and Lemma 5-3-1 with

ip = 17-1+^/2 a n d co = rj^ ^LPV ^^lat we may set

in the equations (1.11.9')> VL being the t runca ted func t ion for V (see

(5.3.1)). Since VL,OC = 0 when V > VL, we see tha t

C,.=-Vi(r]^Py,o. + r]^V-^Py VL,.+ 2rjrj,.Py )

yV,oc=pypy,cc | V F | < | V j ^ | •

Using these resu l ts we obta in

| F ^ - 2 ] 7 ^ ^ 2 ( | v F | 2 + ; . | V F L | 2 ) ^ : ^

BR+U

< / ( Z i F^+2 F i ? 2 _j_ 72 1 v ^ |2 F * Vi) dx.BR+U

N o w l e tCOL = > FFi /2 , WL^^n F*^/2 17A/2

V OJL = Vi^ (>y V F + F Vi^ + A ^ V F L / 2 )

Vz L = F-i+^'2 VJ!^[(kr]VVl2] + (A^ V F L / 2 ) + F Vry].

Then , s ince ^ r = (A + ^ ) / ^ , we conc lude tha t

1 ^2 vfc+2 v]^dx = jF* coi dx

BR+U BR+U

< C3(ie + «)''o J { Z 3 F * - 2 F i r ? H ( | V F j 2 + Aj V F i | 2 ) +BR+O.

< C 3 Z i Z 3 T ( i ^ + ay^ jjf V^^^Vidx +

+ C3{R + a)fol(Z2Z3r + Z4)\Vri\^V'cVidx.BR+a

L e t us now define a by

2C3 Z i Z3 T ^''o = 1, a < 2?o,

rep lace a b y a /2 , set if = a /2 , assume Bad G and s uppos e a < ^. T h e n

/ \VwL\^dx<,Z^x ^\ Vrjl^V^V^dx.

W e m a y n o w l e t L -^ 00 and use the def in i t ion of rj to conc lude

f I \/w |2dx <Z5Ta-^ J w^ dxB{xo,a/2) B{xo,a)

w h i c h is exac t ly (5 .3-16) . The remainder of the proof is t h e s a m e as t h a t

of the proof of Lemma 5 .3 -3 ; the different result for Q he re is d u e to the

different definit ion of oc.

Theorem 5 .11 .5 . Suppose the hypotheses and conclusions of Theorem

5.11.4 hold. Then U is bounded on each A G G D.Proof. The proof is exac t ly the s a m e as t h a t of Theorem 5-3-L

Remarks. The higher d i f fe ren t iab i l i ty resu l ts now follow as in th e

o the r ca s e s . Bu t we have a s s umed above tha t ^ > 2. LADYZENSKAYA a n d

URAL'TSEVA have ex tended Lemma 5 -11 .4 to the cases 1 < A < 2; it

follows that Theorem 5-11-3 extends to those cases a lso. So the proof of

T he or em 5-11-4 goes thro ug h as far as th e e qu ati on s (5 .11.2 3) and (5.11.24)-

Howeve r , the wr i t e r ha s no t found a w a y in these cases to s h o w t h a t

J\ Vrjl^AhzldxBR+a

i s un iform ly bou nd ed . S ince the a u t h o r s a b o v e {L and U) g a v e no h i n t

a s to w h y t h i s is so , they may have over looked th is po in t . This i s jus t the

d if f icu l ty which caused the w r i t e r to re s o r t to the a r g u m e n t in § 5-10

which does no t app ly in th is case s ince we have assumed z b o u n d e d a n d



5.12. Aclass ofnon-linear equations 203

w o u l d h a v e to s h o w t h a t t h e Z K were un i fo rmly bounded . Howeve r , once

the re s u l t s ofT h e o r e m 5.11-4 are s h o w n toho ld , then Theorem 5.11.5

will follow also.

5 . 1 2 . A c l a s s ofn o n - l i n e a r e q u a t i o n s

In th is sec t ion we presen t essen t ia l ly the t r e a t m e n t ofL E R A Y and

L I O N S of a theory of non-linear equations initiated byM I N T Y , B R O W D E R

[3], and Vis iK . Each of the s e l a t t e r th ree au tho rs has wri t t en s eve ra l

p a p e r s on th i s s ub jec t ; we h a v e q u o t e d arecent one of e a c h a u t h o r . The

equa t ions wh ich can be so lved by t h i s m e t h o d a r e in s ome re s pec t s more

genera l than those d iscussed in the preceding th ree sec t ions , s ince h igher

o rde r equa t ion s and s ys tem s can be han d led bu t no t a l l of thos e p rev ious ly

d iscussed , namely those sa t is fy ing the condi t ions (1 .10 .8") , can be

h a n d l e d .

L e m m a 5 .12 .1 . Suppose u=f{x) is a mapping from Rminto itself

such that

( 5 .12 .1 ) lim|; |-i -/( ) = + oo.

Then the range of f is the whole of Rm-

Proof. Let UQ ^Rm and define /*(:v) =f(x) -~ UQ. T h e n / * satisfies

(5.12.1) . Cons equen t ly itis sufficient to p r o v e t h a t the r a n g e of any m a p

satisfying (5.12.1) contains the origin. Using (5.12.1) we s ee tha t we m a y

choose R la rge enough so t h a t

(5.12.2) \x\-^X'f{x)^\ for \x\=R.

But from (5.12.2), it fo l lows tha t the m a p p i n g

w=f{RmfiR^)i ifi = ifrom dB{0,\) in to i tse lf is homotopic to t h e i d e n t i t y . I t fo l lows tha t there

is a so lu t ion x in B (0, R) of / {x) = 0.

W e beg in wi th the s t a t e m e n t and proof ofthe a b s t r a c t t h e o r e m .T h e o r e m 5 .12 .1 . Suppose that V is a separab le, reflexive, Banach

space, V isits dual ( the space of l inear func t iona ls ) , and A isan {possibly

non-linear) operator from the whole of V into V which iscontinuous in the

weak topology of V on finite dimens ional subspaces of V (i.e. if Xn ->xo in

a finite dimensional subspace of V, then AXn —r {weak convergence) A XQ

in V). We suppose further that (see notation below)

(A (w) v)

(5.12.3) I ,[ -> + 00 as II7; II ^ + CX3 {coerciveness).Finally we assume the existence of an operator % from F X V to V which

has the following properties:

(i) 51 (w, ^') isdefined everywhere and carries bounded sets into bounded

sets and

(5.12.4) %{u,u) = A{u).

2 0 4 Differen t iabi l i ty of we ak solut ions

(ii) 51 {u, v) has the continuity properties of A as a function of v and

(5.12.5) {^{u,u) —% {u,v),u — v) > 0 for all {u, v) (mono ton ic i ty ) .

(iii) If Un —7 u in V and

(5 .1 2 .6 ) ( ^ ( ^ 7 1 , Un) — ^ ( w « , u), Un — u) ->0

then 51 (un, v) —7 ^ {u , v) for each v^V.

(iv) If Un-7 u and %{un, v) — ^v"^, then l^{un, v), Un) -> (t^*, u).

Then the range of A is the whole of V\

Notation, li v ^V a n d v' ^ V, t h e n {v\ v) m e a n s v'{v).

Proof. L e t {wj\ be a basis for V and let F ^ be the s pace s pan ned b y

2^1, • • •> ^w- The equa t ion

(5.12.7) {A [u), v) = (/, v), u,v^Vmi s r educed to a mapp ing

fi = Ai[xi, . . ., Xm), t = i, . . .,m

and the condi t ion (5 . I2 .3) i s reduced to (5 .12 .1) by the subs t i tu t ion

m m mu=2^xj Wj, v=2yj ^'j> (/> ^) = 2P yjy

e tc . Thus , fo r each m, there is a solution Um oi (SA2.7). Since

(A [Um), Um) = ( / , Um) < | | / | 1 * ' || Um \\

it follows from (5-12.3) that \\uni\\<M and hence tha t 1^ (um) ||* < M *

(using (i)) for some M and M * and a ll m. W e may the re fo re ex t rac t a

su bs eq ue nc e {^7^} suc h t h a t

(5.12.8) ^^- ^-^ ^ in V, A[un) —7 %vciV\ ^(un, u) -rip in V.

Since (5-12.7) holds with u = Un for all v in Vn, i t fol lows that

(5-12.9) x= f

since (/, ?;) = {A (un), v) -^(x, ^) for ^ € ^ F „ i w h ich i s dens e in V.

W e n e x t n o t e t h a t

l i m ( 5 l (Un, Un) — 51 {Un, u),Un — u) = lim{A (Un), Un)

(5.12,10) —\im.[A{un), u) — l im(5 l(^w, u), Un) + Vnn{%[un, u), u)

= l i m ( / , Un) — (; , u) — {ip, u) + {ip, u) = 0

on account of (5-12.8), (5 .12.9), and hypothesis ( iv) .

Now, from this result and (i i i ) i t fol lows that

51 {un, t") —7 51 {u, v) for each v^V.Then as in (5 .12 .10) , we conc lude tha t

\im{^ {Un, Un) — "^{Un, v), Un — v) = lim(A (Un), Un) ~

— \im{A (un), v) — lim(^ (un, v), Un) + lim{'tK(un, v), v)

= { x > ^ ) — i x > ^ ) — (^ (^' ^ )> ^ ) + ( ^ K ^ )> ^ ) -

5.12. A class of non-linear equations 20*5

T h u s

(5.12.11) {x — %[u,v),u — v)>0 for all t;.

If, in (5.12.11), vêsetv = u — w, w^ V, i > 0, we o b t a i n

(X — ^(u, u — S '^), w) > 0 for all w^V,

By le t t ing | -^ 0+ and us ing the cont inu i ty a long l ines we o b t a i n

(X — 91 (^^ ^)> ^) > 0 for all ze ^ F

f rom which it follows at o n c e t h a t

A(u) = S^{u,u)=x-f'

W e now p r e s e n t the ex i s tence theo rem of Vis iK essen t ia l ly as ex

tended by L E R A Y and L IO N S . We have allowed ^ to be a vector. The

theo rem conce rns equa t ions of the form

(5.12.12) (Au, w) [ 2 Zî{^> <5 > D'û)D^w^dx = (/, w)

w h e r e u ^= [u^, . . ., u^) and w = {w'^, . . .y w^) range ove r a reflexive

B a n a c h s p a c e F — Fi x • • • X F^v in w h i c h

a n d , as in T h e o r e m 5-12.1, (/, w) m e a n s / (?x^), /being a l inear func t iona l

on F ; he re d u d e n o t e s the t e n s o r {D°^ u^} whe re 0 < | {x | <.nii — 1 and

D^ u d e n o t e s the t ens o r {D^" u^] w i t h | ^ | = mi. I t can be s hown , as in

§§1.3 and 1.10 t h a t if the v e c t o r u min imizes the i n t e g r a l

J F (x,d u, D^ u) dx

G

a m o n g all vec to rs ^ in a space such as F , t h e n u satisfies an e q u a t i o n of

the form (5.12.12) where

Af(x, d u, D^ u) == dFldpi (Pi = Z)« u^).

Th us the ex i s tence theo rem i s an a l t e r n a t i v e m e t h o d t o t h a t of the ca lcu lus

o f va r ia t ions .

We sha l l f i rs t p rove the theo rem unde r s omewha t s imp l i f i ed a s s ump

t ions conce rn ing the A^; we sha l l ind ic a te la te r how our s impl if ied a ssu m p

t ions may be r e l a x e d s o m e w h a t . We s ha l l a s s ume tha t G is s t rong ly

L ips ch i tz (see § 3.4) unless all the F^ = Hîîfi) in which case we shall

allow G to be mere ly bounded . Conce rn ing the A^ ê sha l l assume.

(a) Each Al is m e a s u r a b l e in [x, | , rj) and c o n t i n u o u s in (f, rj) fora l m o s t all x on G; he re f = {f^} w h e r e | ^ | <,mi — 1 and YJ = {rjl}

w h e r e |^| = mf.

(b) We a s s u m e t h a t for all {x , f, rj), we h a v e

(5.12.13) \At(x,^,rj)\<Cfi +2'f 2' | f^l^- '+2 ' I^M^" ' l l -

2 0 6 Di f fe rent i ab i l ity of weak so lu t ions

(c ) W e a s s ume a s t rong mono ton ic i ty cond i t ion :

i | a |=mi

(5.12.14) >Q[\ri2-riilR^S,x) if \^\^R,\rii\^S,x^Gw h e r e Q{r,R,S, x) > 0 wh enev er r > 0 an d is non-d ecreas ing in r.

Theorem 5.12.2. We assume that G and the Af satisfy the conditions

above and we assume that the operator A defined in (5.12.12) satisfies the

coerciveness condition (5 . I2 .3) . Then the equation (5.12.12) has a solution

u^V for each f in V\

Proof. Let us def ine the func t iona l '^{u, v) b y

(^(u, v), w) =[21 2Af[x, du(x), D^v(x)]D^wi(x) +

(5.2.15) +Z Ai[x, du{x), Dû{x)]D°'w^{x)\dx.\oc\<mi j

I t i s c lea r tha t the condi t ions on the A^ a n d G e n s u r e t h a t % {u, v) is

defined for all (u, v), 31 carr ies bo un de d se ts in F X F in to bou nd ed se ts

in V, a n d iiun-û a n d Vn ^v ina, manifold of f ini te dimensionali ty in

V X V, then ^ {un, %) -7 91 (u, v). Clearly 51 {u, u) =^ A {u). The con

dit ion (5.12.5) follows immediately from the condit ion (5.12.14)- We

nee d only to verify con dit io ns ( iii) and (iv) of Th eo rem 5 .12 .1 ; our theo rem

wil l then fo l low from tha t theorem.To prove (i i i ) , le t us suppose that Un—7 u and that (5 .12.6) holds . Let

V a n d w^V and le t {r} be any subsequence of {n}. Since D"^ u\ - > Z)** W

in Lp (G) if 1^1 < mi (R e l l ich 's the ore m , Th eor em 3 .4-4) , i t fo llows th a t

we may choose a subsequence {s} of [r] s u c h t h a t dus[x) ->du (x) for

a lmos t eve ry x. Thus , fo r such x, d Ug {x ) i s bounded wi th respec t to s .

T h u s , the condit ion (5-12.6) together with (5 .12.14) implies thatlim jQ[\D'^{US — u)\, R(x), \Dû{x)\, x] dx = 0

R (x) be ing a common bound fo r | d Us (x ) \ a n d | (5 ^ (:v) |. I t follow s t h a twe may ex t rac t a fu r the r s ubs equence {t} s uch tha t we a l s o have

D^^ ui(x) -^D°'u^{x) a lmos t everywhere i f \(x\ = mi. Thus, if we define

{At[x,dun{x),Dmv[x)l i f | ^ | = w ^

(5- . ) J^^^^^ -\Al[x,dun{x),DÛn{x)-], ii\oc\<mi

/ f be ing def ined s imila r ly , then we see tha t f^;^ {x) — / J (x) -^ 0 a lmos t

eve rywhere . F rom (5 . I2 . I3 ) we conc lude tha t

(5.12.17) \Af{xî,rj)\2>' < C{\ + | | | P + 1^1^}, f=pliP - 1).From (5 .12 .17) and the un iform boundedness of \\un\\, i t fol lows that

ll/wilU' ^^^ uni fo rm ly bo u nd ed . Als o, since t; is a fixed ve ct or an d

d ut{x) ->du{x) in Lp{G) we see tha t the se t func t ions

5.12. A class of non- l inear equ at io ns 2 0 7

are un iformly abso lu te ly cont inuous . Thus /^ '^ -^ ft ^^ ^v (^) if 1^1 — 'î-

Since |Z)* w' \ ^ Lq(G) for some q '> p ii \oc\ < mi, i t fol lows that

{SE(ut, v) — 5 1K v), w) ->0, w^V.

Since {r} was any subsequence the resu l t fo l lows .

To pr ove ( iv ) , we assum e th a t v^V, Un -i^ u in V, and ^ {un, v) —7 v^

in V. Ag ain , le t { r} be an y subsequ ence of \n]. Since #^ —7 ^ it follows

as above tha t the norms f /^^Hj? / a re un iformly bounded . Thus we may

ex tra c t a fur ther subseq uence {s} of {f} such th a t b u^-^b umL^(G ),

6 Us{x) -^d u {x) a lmos t eve ryw here , and / /^ —7 some functions /* in

Lp, {G). As in the preced ing pa ra gr ap h i t fol lows tha t /g° \ - ^ / f in Lp, (G)

if 1^1 = m^ w her e / f ha s i ts pre vio us significance if | ^ | = mf. T h u s v^

is given by

Fr om th e s t ron g converge nce of /*^ to / f an d th e weak co nvergence of

D"^ u\ to 0°" u^ ii $x\ = mi and f rom the weak convergence of//' t o / f a n d

the s t rong convergence of D°^ u^ to D"^ u^ w h e n | ^ ] < w^-, the desired result

in ( iv) holds for the subsequence [s]. But s ince {r) was a rb i t ra ry , the

desired result follows for the whole sequence {%].

Remark 1. The reader can eas i ly ver i fy tha t the theorem s t i l l ho lds i f

the condit ion (5.12.13) on the ^^ is replaced by

M ? ( ^ , f , ^ ) | < c f i ^ ( % ) + 2 1 2 ' | f ^ ^ h ^ ' ' ^ ^ + 2 ' h ^ M >

1^1 = mi,

(5.12.18) r-HJ, p)>{p- l ) - i [ 1 - v-Hmj - \^$pi

t{h P) <[P - '^)+ v-^p(mi ~\oc\)

^"(^' ) > . ( / - " i f ? ; 7 J - V i ) - ^(^'^)' '^^^^^' '^^'^^ ^ ' '

K^Lp,, L^Lqîôc), q{i, oc) > 1,

q~'^{h oi) > \ — p-'^ + v-'^[mi — I ^ I).

The proof i s essen t ia l ly the same but makes s t ronger use of Sobolev 's

l e m m a .

I t i s ins t ruc t ive to spec ia l ize to the case where a l l the nti == \ a n d t h e

^f a re d i f fe ren t iab le wi th respec t to a l l a rguments . Then the condi t ion

(5.12.13) is implied by the f irs t inequali ty in (1 .10.7")- The las t inequali ty

2 0 8 Differen t iabi l i ty of we ak solut ions

in (1 .10 .7") impl ies tha t

1

(5 .12.19) = 2 " (VL - rjU) ivy - VU) I Ay ^ [x, i, rn + t(rj2 - f]i)] dt0

1

0

If ^ > 2, th e con dit io n (5.12.14) hold s w i t h ^ = mi r^. In case 1 r, \i\<.R , \rji\<S,

1

0

If Q{r, R, S) = 0 for some R, S, a n d r > 0, H {f^}, {rjin}, and {s^} such

t h a t \in\ < ^ , l^iTil < 5 " , a n d \sn\ > r and / ( s^ , 1^ , " yi i) ->0. If , for

s ome s ubs eequence , \sn\ rema ins bounded , a compacnes s a rgumen t

s hows tha t / - T ^ O . T h u s w e m u s t h a v e | s ^ | - ^ o o . B u t

[1 + \Sn\^ + \rjln + tSn\^] < 1 + i^2 + 2 S 2 + 2 | 5 ^ |2 .

Consequent ly we see tha t / (%, ^n, ^iw) -> + oo , a con trad i c t ion . T hu s

(5.12.14) holds in this case a lso. Clearly this monotonicity condit ion, then,

is impl ied by the very s t rong e lhp t ic i ty of the sys tem as assumed in

(1 .10 .7") .

Using (5.12.19) with rj2 = rj a n d 7 1 = 0 an d us ing the fac ts th a t

tP-2 (1 + | | | 2 _!_ | ^ | 2 ) - l + p / 2 < (1 + | | | 2 + ^2 | ^ | 2 ) - l+2 > / 2

we conc lude tha t

i l a l = l

( 5 . 1 2 . 2 0 ) ^m2\r]\^VP-^ -Mi\rj\ • (1 + \^\2)ip-i)i2 _ Mi\i\ -VP-^

> m 3 F ^ - M 3 ( l + |f|2)2>/2, V =(\ + | | | 2 + |7^ |2 )l/2.

In case a l l the Vt = H'^'oiG) (Dir ich le t p roblem) then (5 .12 .20) guaran tees

the coerciveness condit ion (5.I2 .3) provided the diameter ofG is sufficiently

small (Poincare ' s inequa l i ty ) . The condi t ions (1 .10 .7 '" ) and (1 .10 .8") a resuff ic ien t fo r the theorem provided tha t p '> v and the coerc iveness con

d i t ion ho lds . But , in these la t te r cases the coerc iveness condi t ion is more

of a res tr ic t ion than i t is in the case of assumptions (1 .10.7")- The

coerc iveness condi t ion corresponds roughly to the condi t ion tha t

f{x, z,p) >mV P — K for all {x, z, p).

http://rjln/

http://rjln/

http://rjln/



6 .1 . Introduction 2 0 9

C h a p t e r 6

R e g u la r it y th e o r e m s for the s o l u t i o n s

of genera l e l l ip t ic sys tems andb o u n d a r y v a l u e p r o b l e m s

6.1 . I n t r o d u c t i o n

I n the preced ing chap te r , we deve loped ra the r comple te ly the regu la

r i ty p rope r t i e s of w e a k and s t rong s o lu t ions of cer ta in f i rs t o rder var ia

t iona l p rob lems and second order d i f fe ren t ia l equa t ions involv ing only

one unknown func t ion . Comple te re s u l t s conce rn ing the so lu t ions of the

co r re s pond ing p rob lems invo lv ing s eve ra l unknown func t ions we re

o b t a i n e d o n l y in th e case where v = 2; these resu l ts were ob ta ined by

the wri te r before the war and were descr ibed in C h a p t e r 1. In the preced

ing chapte r some f i rs t d i f fe ren t iab i l i ty resu l ts were ob ta ined for ce r ta in

s y s t e m s of e q u a t i o n s b ut these resu l ts did not i m p l y the c o n t i n u i t y of

the f i rs t der iva t ives . In 1952, the wri te r p re s en ted a p a p e r ( M O R R E Y [ 1 0 ] )

a t the Arden House Conference on Par t i a l D i f fe ren t ia l Equa t ions , in

w h i c h it was p r o v e d t h a t any vec tor so lu t ion of class C^ of a regu la r

v a r i a t i o n a l p r o b l e m of class C^, 7 i > 2, was also of clas s CJJ. T he se re su lt s

s t i l l leave a gap in the t h e o r y for s ys tems , wh ich can only be filled by anex tens ion of the D E G I O R G I - N A S H resu l ts , deve loped in § 5-3, to s y s t e m s

or some en t i re ly new device . B ut, in p r o v i n g the CJJ dif fe ren t iab i l i ty

re s u l t s for s y s t e m s of e q u a t i o n s , the w r i t e r was forced to use the v e r y

i m p o r t a n t f o r m u l a s of F. J O H N ([1], [3]) for fundamen ta l s o lu t ions and

to u s e me thods wh ich a re app rop r ia te for the discuss ion of e l l ip t ic sys tems

of h igher order . Hence the inc lus ion of t h i s c h a p t e r in th i s book .

O u r p r e s e n t a t i o n w i ll be a modif ica t ion of t h a t to be found in the two

important papers of A G M O N , D O U G L I S , and N I R E N B E R G ([1] and [2])

a l though some ideas essen t ia l ly due to B R O W D E R ([1], [2]) have been

us ed . A brief b ut r a t h e r c o m p r e h e n s i v e t r e a t m e n t of the top ic s in th i s

c h a p t e r is to be found in the l a s t chap te r of the recen t book by H O R M A N -

DER [1]; he inc ludes some top ics such as ex i s tence theo rems on the

' ' i n d e x ' ' of a prob lem, e tc . , not inc luded he re . Many peop le , pa r t i cu la r ly

a m o n g the R u s s i a n s, h a v e c o n t r i b u t e d to the d e v e l o p m e n t of th i s theo ry

a n d h a v e p r o v e d s u p p l e m e n t a r y r e s u l t s not s ta ted he re . Many add i t iona l

p a p e r s are re fe rred to in th e p a p e r s to w h i c h we refer in th i s chap te r .

T h e t h e o r e m s p r o v e d in th is chapte r wi l l enable us to conc lude thedif fe ren t iab i l i ty of the so lu t ions of the h ighe r d imens iona l PLATEAU

problem d iscussed in C h a p t e r 10. Th ey wil l sugges t d i f fe ren t iab i l i ty

h y p o t h e s e s on th e func t ions Af(x,^,rj) in § 5.12 which , toge the r wi th

a n a pr io r i a s s umpt ion tha t e ach u^ ^C^j, wil l im ply fur ther d i f fe ren ti

ab i l i ty p rope r t i e s of the uK


2 1 0 Regularity theorems for general boundary value problems

We sha l l beg in by s tudy ing l inea r s y s tems of the form

(6.1.1) Liu{x,D)u^{x)=fj[x), j=\,...,N, x^G,

and sha l l be concerned a lso wi th the so lu t ions of (6.1.1) subject to b o u n

da ry cond i t ions of the form

(6.1.2) Brjc(x, D) u^{x) = gr{x), r=\,...,m, xondG,

where m is de te rmined be low. I t is as s umed tha t the re a re in tege rs Si, . . .,

sjsf and h, . . .,t]s[ s uch tha t e ach ope ra to r Lj^ is of orde r Sj + t]c\ since

w e m a y add an in tege r r to all the tj if we s u b t r a c t it from all the Sj, we

m a y as well assume tha t maxsy = 0. We a l low com plex- va lu ed fun c t ions

throughout and a l low the coeff ic ien ts in the o p e r a t o r s to be c o m p l e x . We

requ i re the system (6.1.1) to be ell ipt ic in the sense of the following

def in i t ion :

Definition 6.1 .1 . The system (6.1.1) is said to be elliptic if and on ly

if the d e t e r m i n a n t L [x, X) of the cha rac te r i s t i c po lynomia l s L'^^ {x, A) is

not ze ro for any real non-zero A; here L'^j^i^' ^) ^^ ^^^ principal part of

t h e o p e r a t o r Ljj^, i.e. the p a r t of orde r exac t ly Sj + tjc.

I t is to be n o t e d t h a t L is a h o m o g e n e o u s p o l y n o m i a l of degree

(6.1.3) P-Z [ s^ + h) -

I t is a s s u m e d t h a t if 5; + A; < 0 for s o m e (;, k), t h e n Ljic ^ 0. F r o m the

e l l ip t ic i ty condi t ion , it follows that for each k one of the o p e r a t o r s

Zj.jj. ^ 0, so t h a t , for each k, max(sy + t]c) = fe > 0.

S y s t e m s of th i s s o r t we re in t roduced by D OU G LIS and N I R E N B E R G .

T h e y s h o w e d t h a t if P = 0 in (6.1.3) then the equations (6 .1 .1) may be

solved for t h e u^ in t e r m s of t h e / ; a n d t h e i r d e r i v a t i v e s . In fac t , wi th our

n o r m a l i z a t i o n , vndCKSj = 0, it t u r n s out t h a t if s eve ra l fe = 0, t h e n the

co r re s pond ing u^ m a y be found in t e r m s of t h e r e m a i n i n g u^, fj, and the i r

de r iva t ive s . Th i s is easy to see as follows: (1) By re labe l l ing the u^, wemay assume that if i < 2 < * * ' < JV- (2) By reo rde r ing the equa t ions , we

m a y a s s u m e si < S2 < * • • < Sisr = 0. If ti ^ t2 ^ ' ' ' =' tr = 0 <C

< tr+i < • • • < ^iv, we must have Siv_e+i = - - - = s^ = 0, w h e r e Q ^ r,

a n d the e l l ip t i c i ty gua ran tee s tha t the m a t r i x of func t ions Ljjc(x) w i t h

N — Q + 1 < y < i V and 1 < ^ < r has rank r. D O U G L I S and N I R E N

BERG t h e n s h o w e d t h a t if the s e u^ are e l imina ted f rom the r e m a i n i n g

e q u a t i o n s , the reduced s ys tem is s t i l l e l l ipt ic . So it is sufficient to con

s ider reduced sys tems and we res t r ic t ourse lves to s u c h s y s t e m s ; we

presen t such resu l ts in § 6.2. To s u m u p , we a s s u m e

( 6 . 1 . 4 ) m a x s y = = 0 , tjc'>i, k = i, . . .,N, P > 0.

In order to s t u d y b o u n d a r y v a l u e p r o b l e m s , we impos e the following

cond i t ions of proper e l l ip t ic i ty (somet imes ca l led the roo t condi t ion) and

uniform e l l ip t ic i ty :

6.1. Introduction 2 1 1

Definition 6.1.2. The system (6.1.1) is said to he properly elliptic or

to satisfy the root condition or to satisfy the supplementary condition if a n d

only if P is even, say P = 2 m, and, for each pair | and | ' of Hnearly

independen t I ' -vec to rs the equa t ion

(6.1.5) L{x,S + z n = 0

h a s m roo t s wi th pos i t ive imag ina ry pa r t and m w i t h n e g a t i v e i m a g i n a r y

pa r t . W e s ay tha t the s ys tem i s uniformly elliptic on a set 5 if and only if

the re i s a number M s u c h t h a t

\L(x,K)\->.M-^>0, | ; i | - 1 , x^S (A real)

an d the abso lu te va lues of a l l th e coeff ic ien ts in th e ope ra to rs a re < M

for X on 5 .Lemma 6.1 .1 . (a) The conditions o f ellipticity and proper ellipticity

are invariant under transform ations of the independen t variables.

(b) / / r > 2, any elliptic system (6.1.1) is properly elliptic.

Proof, (a) This can be eas i ly verif ied by the reader.

(b) Su pp os e, for som e ^o, lo , a n d f , t h a t 2:0 is a ro ot of (6.1.5) c or

responding to the pair (fo , f i) , i -e . L {XQ, fo + - O fo) = ^- Th en ob viou sly,

— 2:0 is a root corresponding to (fo , — fo)- Sin ce, if r > 2, th e pa ir (|o, IQ)

can be deformed cont inuous ly in to ( fo , — fo)> always keeping the vectors

l ine arly in de pe nd en t, an d s ince no ro ot of (6.1 .5) is real , an d s ince

the roots of (6.1.5) [X Q f ixed) va ry cont in uou s ly w i th (f, | ' ) , i t

fol lows that , for any fixed x^ , f, f, th e n u m b er of ro ot s of (6.1.5) w it h

pos i t ive imag ina ry pa r t equa l s the number wi th nega t ive imag ina ry pa r t .

T he re sult (b) is no t necessar i ly tr ue if r = 2 as is seen b y rep lacin g

the system (6.1.1) by a s ingle operator Z> where

ox by \hx by)

The reader can eas i ly ver i fy tha t L2U = 0 for any u of the form

u = {\ — x'^ — y2) f{^z), z = X -\- y i

wh ere / i s an y holom orph ic func t ion . T hu s the D IRICU LET problem for the

ope ra to rL 2 fa ils ver y se r ious ly to ha ve a un iqu e so lu t ion . A no the r exam ple

of this sort was given by BICADZE. T h e o p e r a t o r s L\ and L2 a re no t p ro

pe rly e l l ipt ic . An ex am ple is given in § 6 .5 of a pro pe rly e l l ipt ic op er ato r

L for which the Dir ich le t boundary condi t ion is complement ing (see

Defin i t ion 6 .1 .3 be low) bu t which is such tha t every complex 1 is ane igenva lue forL. The fo l lowing sys tem is seen to be proper ly e l l ip t ic and

is even s tro ng ly e l l ipt ic as defined in § 6 .5:

,^ , ^ v '^xx %2/ + ' xyyy = /

(6.1.6)

14*

2 1 2 R eg u l a r i t y t h eo rems for g en e ra l b o u n d a ry v a l u e p ro b l ems

- ( A 2 - / ^ 2 ) 4 _ | _ ^ 2 ^ 6 .

T h e d et e rm in an t L(A, //) (A^ = A, A^ = ^ ) is

I - X f J i ^ (^2-/^2)3 I

We now cons ider the boundary condi t ions (6 .1 .2) ; we sha l l assume

t h a t t h e p o r t i o n o f t h e b o u n d a r y dG on which we cons ider these condi

t ions is suffic iently smooth, the degree of smoothness being specified in

each case being considered. Firs t of a l l , the m in (6.1.2) will be precisely

P / 2 . We require a lso tha t there a re in tegers hi, . . ., hm, some of wh ich m ay

be nega t ive , such tha t the order o f Brjc is <:tjc —- hr', we le t ho be the

la rges t o f 0 and the —hr. If tjc — hr<0, w e a s s u m e t h a t Brk^ 0 ;

o therwise we le t B'^j^ deno te the p r inc ipa l pa r t o f Brk- At any po in t XQ of

^ G , we le t ^ deno te the un i t no rma l a t XQ an d | an y (rea l) vec t or tan ge ntto (9G at XQ. L e t z^ (XQ, | ) , S = \,.. .,m,he the roo ts o f L(XQ, ^ + zn) =0

wi th po s i t ive imag ina ry p a r t ; the s e ex i s t s ince we a s s ume tha t the s y s tem

(6.1.1) is properly e l l ipt ic . Define

vt

( 6 . 1 . 7 ) L ^ { x o . i ; A = n [ z - z ^ { x o , mand le t \\D^(xo, ^ + z n)\\ be the ma t r ix ad jo in t to \\Ljjc(xo, f + zn)\\.

Definition 6.1.3. F o r a n y XQ^G and any rea l vec to r | t angen t to

^ G a t XQ, le t us regard L^ (XQ, f ; z) and the e lemen ts o f the ma t r ix

II Qrk {^0, l i -r) II as po lynomia ls in z, w h e r e

Qrk(xo, I ; ^) = 2 ; Bl^{xo, ^ + zn) D^{xo. ^ + zn).

The sys tem (6 .1 .2) o f boundary opera tors i s sa id to sa t is fy the comple

menting condition {with respect to the sy stem (6.1.1)) if and only if the rows

of the Q m a t r i x a r e l i n e a r l y i n d e p e n d e n t m o d u l o L^ [XQ, S; Z), tha t i s ,

t h e p o l y n o m i a l2 * ^r Qrkixo, i;z)^0 (mod L+)r = l

only if the Cr are all 0.

In the sys tem (6 .1 .1) the maximum order of any der iva t ive of a

p a r t i c u l a r u^ which occurs i s tjc. I t is to be noticed that some (or a l l) of

t h e hr are a l lowed to be nega t ive , in which case der iva t ives of u^ of order

h ighe r tha n fe m ay occu r amon g the bo un da ry con d i t ions . As an ex am ple ,

w e m i g h t h a v e

Lu~Au= [Dlr + Dl. + D|s) u, I = (fi, h, 0 ) , ^ = (0, 0, 1),

Bu = aDlû + {hiDaî + h^DaP' + hz)Dl.u +

+ (Cl l ^ l i + 2Ci2 Daî D^2 + C22 Dl. + ICizDa^l + 2C23Da^2 + C33) X



6,1. Introduction 213

P3 being a cubic polynomial. Then

B{^ + zn)=az^-^{hh^ b h + 63) z^ + (cii If H )z +

and the complementing condition is merely that B{^ -\- z n) not be

divisible by L^ (considered as polynomials in z) for any f 9^ 0.

Notations. In this chapter, we shall obtain estimates for the solutions

of various problems which involve constants which do not depend on

the particular solution. We shall often denote such constants simply by

C but Cs occurring in different places may denote different constants.

On the other hand, we shall sometimes distinguish different constants

by using subscripts; however, we do not guarantee that C2, for example,always denotes the same constant. We shall frequently use Zi, Z2, . . .

to denote similar constants. Our notation here is not consistent, but the

situation is usually clear from the context. Estimates for solutions of

boundary value problems in a half-space where the operators Ljjc and

Brk have constant coefficients and coincide with their principal par ts will

involve constants which depend only on a common bound for these

coefficients, for M {Dei. 6.1.2), and for J-i where

/ = inflmaxig^'l}, l f | = = 1 , i ' n = 0,

the Q'^ being the various m x m determinants in the matrix (J of (6.3.19),

below, which is related to the ^-matrix above. Such constants will be

said to depend only on E; interior estimates will not depend on / . Con

stants which depend also on bounds and/or moduli of continuity of all

the coefficients and possibly on the domain G will be said to depend only

on E and E'. If a constant depends on other quantities such as h, q, /LC,

etc., we may write C = C{h, q, /bt), etc. In § 6.3, we shall replace v by

V -\- \, let X = (%i, . . ., x^), and y = x^+'^; we then often let X = {x, y).

We say that a vector function has support in GR U CTR if and only if it

vanishes outside GR and on and near ^R; it need not vanish on GR. If

a = {oci, . . ., {Xv) is a multi-index, we define

oc\ = (ail) (^2?) . • . io(v\), x^" = (x^)""! . . . (x^'^v, etc.

Definition 6.1.4. A function or vector function 99 is said to be essen

tially homogeneous of degree s if and only if (1) 99 is positively homogeneous

of degree s or (2) s is an integer > 0 and

(p{x) = (pi{x) log!A;] + (p2(x)where (pi is a homogeneous polynomial of degree s and 992 is positively

homogeneous of degree s.

Remark. If 99 ^ C (Rp — (O}) and is essentially homogeneous of

degree s, then any first partial derivative is essentially homogeneous of

degree s — 1.



2 1 4 R e g u l a r it y t h e o re m s for g en era l b o u n d ary v a l u e p ro b l ems

I n § 6.2, we in t roduce ce r ta in fundamen ta l s o lu t ions of e q u a t i o n s

(6.1.1) with constant coeffic ients and pre s en t in te r io r e s t ima te s for the

so lu t ions of systems (6.1 .1) with variable coeffic ients ; the pr inc ipa l

resu l ts are s t a t e d in T h e o r e m s 6.2.5 and 6.2.6. In § 6.3, we i n t r o d u c e the

"Poisson kernels" of A G MO N , D O U G L I S, and N I R E N B E R G ([1], [2]) and

VoLEViCH and in t roduce ce r ta in s o lu t ions of c e r t a i n b o u n d a r y v a l u e

p r o b l e m s for a half space in w h i c h the Lj]c and Bric have constant coeffi

c ien ts and coinc ide wi th the i r p r inc ipa l par ts . These resu l ts are us ed to

o b t a i n the princ ipa l resu l ts on higher d i f fe ren t iab i l i ty , g iven in T h e o r e m

6.3.7, and on ' ' c oe rc ivenes s " as given in Theorem 6 .3 .9 . In § 6.4, we first

ob ta in re s u l t s on local coerciveness and higher d i f fe ren t iab i l i ty on the

in te r io r for " w e a k s o l u t i o n s " of equations (6 .1 .1) and then ob ta in s imi la r

re s u l t s at the bou nd a ry fo r weak s o lu t ions of (6.1.1) w hic h satisfy the b o u n da ry condi t ion s in a cer ta in wea k sense . In § 6.5 we tr ea t the Dir ich le t p rob

lem for the "s t rong ly -e l l ip t i c " s ys tems in t roduced by N I R E N B E R G [2]. In

§ 6 .6 , we ex ten d th e ana ly t i c i ty re su l ts fo r l inear sys tem s g iven in th e recent

p a p e r by M O R R E Y and N I R E N B E R G to a p p l y to solutions sat is fying

ana ly t i c general bounda ry cond i t ions a long an a n a l y t i c p a r t of the b o u n d

a r y ; the d e v e l o p m e n t s are l ike those of § 5.7. In § 6.7, we e x t e n d the

re s u l t s of the a u t h o r ( M O R R E Y [12]) and A. FRIED MA N ([1]) concern ing

t h e a n a l y t i c i t y of the so lu t ions of non-linear s y s t e m s to the case of gene ra l

non - l inea r ana ly t i c bounda ry cond i t ions ; the d e v e l o p m e n t s are like

thos e of § 5.8. Fin ally , in § 6,S, we obta in resu l ts on higher d i f fe ren t iab i l i ty

of the so lu t ions and weak so lu t ions of ce r ta in non - l inea r p rob lems and

t h e n p r o v e a pe r tu rba t ion theo rem gene ra l i z ing tha t (Theo rem 12.6) in

the large paper of A G M O N , D O U G L I S , and N I R E N B E R G [1]. Our estimates

are of two t y p e s , the Lp type invo lv ing Lp e s t i m a t e s of the so lu t ions

and the i r de r iva t ive s and the Schaude r type invo lv ing e s t ima te s of

H o l d e r n o r m s .

W e h a v e not a t t e m p t e d to p r e s e n t the l a t e s t d e v e l o p m e n t s in the

t h e o r y of ell ipt ic differentia l equations nor h a v e we a t t e m p t e d to t r a ce

t h e d e v e l o p m e n t of the subjec t . But a grea t many re fe rences are to be

found in the recen t books and expos i to ry pape rs by H O R M A N D E R [1],

L I O N S [2], M I R A N D A [2], R O S E N B L O O M , B R O W D E R [1], [2], G A R D I N G [2],

STAMPACCHIA [1], N I R E N B E R G [3], and the author ( M O R R E Y [15], [18]).

W e h a v e not p r e s e n t e d an a c c o u n t of w h a t m i g h t be c a l l e d ' ' n a t u r a l "

o r ' V a r i a t i o n a l " b o u n d a r y c o n d i t i o n s . We present only special cases of

s uch p rob lems in C h a p t e r s 7 and 8. We h a v e in mind he re the regu la r i tya t the b o u n d a r y and the bounda ry cond i t ions s a t i s f i ed by the so lu t ions

u of e q u a t i o n s of the form (6.4.1) where we requ i re tha t the s e ho ld for all

V in some l inear manifo ld V and pe rhaps requ i re u to satis fy (poss ibly in

a weak sense) fewer than m b o u n d a r y c o n d i t i o n s . STAMPACCHIA [2] has

s h o w n t h a t if u{N = 1)^ H\[G) and satisfies an e q u a t i o n of the form

6.2. Interior estimates for general elliptic systems 215

(5-2.1) for aU v in s o m e V s u c h t h a t HIQ{G) C V C Hl(G) and if dG

satisfies a mild degree of s moo thnes s , then u^ C^(G). For h ighe r o rde r

e q u a t i o n s t h i s a p p r o a c h to b o u n d a r y v a l u e s was f o r m u l a t e d by VisiK

and SoBOLEV [1]. Quite a few resu l ts a long these l ines have been obta inedrecen t ly by SCHECHTER. NO d o u b t o t h e r w o r k e r s in the field, such as

A G M O N , B R O W D E R , N I R E N B E R G , etc., have contributed results but the

a u t h o r is no t fami l i a r w i th them.

6 .2 . In ter ior es t imates for gen era l e l l ip t i c sys tems

In th is sec t ion , we cons ider in te r ior es t imates for the so lu t ions of the

genera l e l l ip tic sys tem s of the form (6.1.1). We sha l l be able to inc lude a

s imple proof of the Lp e s t i m a t e s . Our m e t h o d is es s en t ia l ly tha t of §§ 5-5a n d 5-6. We begin by w r i t i n g the e q u a t i o n s in the form

(6.2.1) Lojjcu^- =fj - {Ljk - Lojk) u^,

w h e r e LQJ]C deno te s tha t ope ra to r wi th cons tan t coe f f i c ien t s ob ta ined

from the principal part of Ljjc by e v a l u a t i n g the coefficients at the p o i n t

XQ. Since these equa t ions wi th cons tan t coeff ic ien ts cannot , in genera l ,

be reduced to a s impler form by a f f ine t r ans fo rma t ions of the x va r iab le s ,

w e can on ly a s s ume XQ to be the origin and h a v e to cons ider the gene ra lcase . Moreover , for s uch s ys tems , the re is gene ra l ly no s imple law of

reflection, so th e b o u n d a r y n e i g h b o r h o o d s r e q u i r e the more compl ica ted

t r e a t m e n t g i v e n in the next sec t ion .

W e s ha l l a s s ume tha t the coefficients in Ljjc, and the func t ions u^

a n d / j s a t i s f y the fo l lowing min imum cond i t ions :

Minimum conditions. In case Sj = 0, we require the coefficients in the

principal part of each Ljjc to be continuous and the others to be bounded and

measurable. Ifsj < 0, we require merely that all the coefficients^ C^'^~^ i[G).

We shall assume that m > 0, m being the total order of the system. We

assume that fj ^ H~^^ (D ) and i Jfc {D) for some ^ > 1 and each D G G G.

We do not assume that the system is properly elliptic.

Remarks. The reader wi l l see t h a t s o m e of the re s u l t s can be genera l i

zed by r e l a x i n g s o m e w h a t the b o u n d e d n e s s r e q u i r e m e n t s as was done in

§§5-5 and 5-6. H o w e v e r , an a t t e m p t to s t a t e the mos t gene ra l theo rem

poss ib le would add g r e a t l y to the c o m p l i c a t i o n s w i t h o u t a d d i n g m u c h

in te res t . Af te r read ing those sec t ions and th i s one, the reader wi l l be able

to fo rmula te and p r o v e w h a t e v e r r e s u l t s he may need in th is l ine .A s in §§ 5-5, 5-6, we sha l l f i rs t assume tha t the v e c t o r u has s u p p o r t

in BR and satis f ies (6 .1 .1) there . Next, for each suffic iently small R, we

a l t e r the coefficients of the Lj]c ou ts ide BR t h u s o b t a i n i n g new o p e r a t o r s

LRJJC, the coefficients of which reduce to t h o s e of LQ^A; on, n e a r , and o u t

s ide of dB2R, the requ i s i t e con t in u i ty p rop e r t i e s be ing p re s e rved . Th en

2 1 6 Regularity theorems fo r general boundary value problems

we def ine P2i2( / ) to be a par t icu la r so lu t ion UR on B2R of the equa t ions

(6.2.2) LojicU%=fj.

N e x t , w e e x t e n d u to be zero on B2R — BR and wr i t e u =^ UR-\- HR

w h e r e UR = P2R[Lo{ti)] s o t h a t HR i s a so lu t ion of the homogeneous

equa t ions (6 .2 .2) . I t i s then shown tha t HR i s a po lynomia l o f bounded

degree wi th \\H\\^ < C \\u\\^ in a conv eni en t space . If we rega rd HR as

k n o w n , t h e n UR satis f ies an equation of the form

. UR— TR UR = VR , VR = P2R(f) — P2R [{LR — Lo) HR]^(0.2.3)

TR UR = P2R [— (L R — LQ) UR]^ [LQ U = f — (L R — LQ) U) .

If R i s smal l enough, | |TR\\ <, i/2 and i t fol lows that

Mt<C,\\f\\r+C2{R)\\u\\0,

w her e the nor m s are specified below. In case / an d the coefficients ha ve

add i t iona l s moo thnes s p rope r t i e s , i t i s s hown tha t u then possesses

addi t iona l smoothness proper t ies . These resu l ts a re suff ic ien t fo r in te r ior

boundednes s theo rems and , toge the r wi th the re s u l t s in the nex t s ec t ion ,

a re suff ic ien t fo r the es t imates in the la rge inc lud ing the boundary .

W e now s e t abou t de f in ing the ope ra to r P2R. Suppos e t ha t we le t

Lo(A) be the de te rminan t o f the ma t r ix LO;A;(^) of charac te r is t ic po lyno

m ia l s , le t Ll^(?i) be the cofactor of Loijc(k), and le t Lo(D) a n d Ll^(D) be

the corresponding opera tors . Then a so lu t ion of (6 .2 .2) can be ob ta ined

by s e t t ing

(6.2.4) U^ = Ll\D)Fi w h e r e LoFi=fi

and our problem is reduced to so lv ing the second equa t ion in (6 .2 .4) . We

now do tha t heu r i s t i c a l ly by Four ie r t r ans fo rms . In t roduc ing the t r ans

forms by the i r usua l fo rmulas

F(y) = (In)-'' f e-^^-yF (x) dx, (r ^ vjl),

we find formally

i'» Lo(y) P(y) = f(y), F{y ) = (-i)^ L^^ (y) fiy^-

Taking the inverse Four ie r t rans forms , we f ind formal ly

(6.2.5) ^'

F(x)=fK(x-i)mdi

K(x) = (In)-''( — i)"^ j e^^' y L^^(y ) dy .

1 O f course {L R — LQ) U denotes t h e vector F defined b y

Fj = LEJJCU^ — LQ j jc U^

2 Recall th at m i s the total order of the system an d m ay be odd if v = 2 .

6.2. Inte r io r es t im ates for general el l ip t ic syste m s 21 7

I f we eva lua te the in tegra l fo r K by in t roduc ing po la r coo rd ina te s in the

y space , we obta in

(6.2.6) K{x) = (27€ )-^(-i)^fL-^(G)lfr^-^-^eirix-o)^A^2 ;.

If Im (x ' a) were > 0 , the in tegra l in the brace would converge to

(6.2.7) Jv-i-m{^'(y )> JhM = hl{—iw)-^-^, A > 0 , m<v—i.

In case m '> v — 1, the in tegra l would d iverge . But we not ice tha t

(formally)

(6.2.8) A^'M -^'^Jh^vM -If we define

(6.2.9) j ^ ^ ) = ^ : ^ ^ - ^ ^ ^ ^ ^ ^ '

Co - : 0 , Ct= \ + 2-^ + " ' + t-^ if ^ > 1,

we see that (6.2.8) holds for all integers h siadp; for the log, we take the

pr inc ipa l log and cu t the w p lane a long the nega t ive imag ina ry ax i s .

Then , ins tead of def in ing K by (6 .2 .6) wi th the brace rep laced by

Jv-i-m, we choose P as a pos i t ive in teger such tha t

(6.2.10) V — \ — 2P <0

and defineKl(x) = (27i)-^{~-t)n^+^P fL-'(a)Jr-i-m.2p{a'x)dZ

(6.2.11) i:K (x) = AP K% [x] , M p {x) = Lo K* (x).

This func t ion K will be seen in Theorem 6.2.1 below to be the desired

func t ion . We f i rs t p rove the fo l lowing lemma:

Lemma 6.2 .1 . If yo is a real vector 9 ^ 0 , there are functions cl[y),

analytic for complex y near yo, ^^^^ l^^l t^^ transform ation

(6.2.12) Cr = cl{y)X^

is a rotation of axes for each real y near yo ^^'^^ such that

Proof. Le t 7 ° = y ° ' / | y | . M akin g a, fixed ro ta t ion o f axes , we may

a s s u m e 770 = (1 , 0, . . ., 0). If, in (6.2.12), ^ de no tes t h e row , we define

cl(y ) =7 ]°", (X = \, . . .y V

and the n no t ice th a t the re i s a un iq ue w ay to comp le te the m a t r ix c^ (y)

in s uch a way tha t the ma t r ix i s o r thogona l , cJJ (y) = 0 for ^ > )/ > 2,an d each de te r m ina nt in th e u pp er lef t ha nd corn er i s pos i t ive . Th e cJJ

a re ana ly t i c .

Theorem 6.2.1. (a) The functions K% , K, and Mp are analy tic for all

complex X = xi -j- i X2 1 ^ 0 with \x2\ <. hi\xi\, where h\ depends only on

the ellipticity hound E.

(b) Kp and K are essentially homogeneous of degree m -\- 2P — v and

m — V, respectively , and satisfy

(6.2.13) K*(-x) = (-\)^K*{x), K(-x) = {-\)^K{x).

(c) Zl^-i Mp{x) =Ko{x)^) if v>2; a v = 2, z1^- i Mp{x) = Ko{x)

+ const.

(d) If/^ CÎ(^R) ^^^ ^ 5 defined by

(6.2.14) F(x) =lK(x-^)f(i)d^,BR

then F^ C^(BR) with hf,{V^F,BR) < C hf,(f, BR) and if f^ CÂBR),

F^ Q + f e ( 5 ^ ) uîth A ^ ( V ^ + ^ F , BR) < C h^{V^f, BR), where C depends

only on E and on fi. Moreover LQ F = f.

(e ) If f^Lq(BR) for some q > i and F is defined by (6.2.14), then

F^H^BR) with | | V ^ i ^ | | , % < C - | | / | l o ^ and if / ^ / f g , ( 5 ^ ) then

F^H'^+îBR) ^^^'^^'| |V^î^||J,jj < C i | | V ^ / l i ^ B where C depends only

on q and E and C^ depends on q, E, and k.

Proof. We begin by using carefully the definit ion of Jh{w) a n d b r e a k

in g up the i n t e g r a l for Kp in to in teg ra l s ove r ^ + and ove r ^ - w h e r e

or • X > 0 and < 0, respec t ive ly . This y ie lds

K(^) = CplL^â)'{a'x)h[-log$y'x\ + Cn + inl2\d^ +

+ Cp j L^^{G) ' {G ' x)h[-\o^\a ' x\ + Cn~i7il2\dZ,

E-

Cp= \lh\(27 iiY {h = 2P + m — v).

Now, s ince LQ is of degree m, L^'^{a) - {a - x)^ is even or odd acco rd ing

as V is even or o d d . T h u s

2Cp f L^^{a)' {a' x)f^ [-log\a' x\ + Ch]dZ> v even

\i Ji Cp J LQ^ (o)' [G ' x)^ d2^ , V o d d .2;+

The p rope r t i e s of X J in (b) fo l low from th is represen ta t ion and the

a n a l y t i c i t y of Kp follows by m a k i n g the r o t a t i o n

(6.2.16) G''==c^{x)Ty

o f L e m m a 6.2 .1 , w h i c h is a n a l y t i c in :\; n e a r an :vo ¥^ 0. W h e n t h i s is d o n e ,

^+ is j u s t the s ubs e t of ^ w h e r e r^ > 0 and

G ~ G{X,T), G ' X ^ \x\r^.

T o p r o v e (c) we n o t e t h a t

Mp(x) = {27z)'-^{-$Pfj,-i-2p(o'x)d2;-

(6.2.15) K(x) =

^ KQ is the kernel for P OI S S ON ' S equation, as defined in (2.4.2).

6.2. Interior estimates for general elliptic systems 219

We shall prove (c) in the case v — 2k; the proof for v odd is similar. Then

we notice that

and C is a constant. If ^ = 1, so i = 2, this is the result, since Pr — 2n.

If ^ > 1, we find by taking zl^~i of this, that

zd^-l Mp[x) = - (27l)-Tr • 2 • 4^-^{k - 2)\ {k - 1)! ^2-"

which is seen to reduce to KQ (X) ; one uses the formula

Fv = vyv, JTV = yv-i • 2 J cos" 9 dd.

0I f / $ CI[BR), F is given by (6.2.14), and F% is given by

(6.2.17) F%[x) =JK%[x - I ) / ( f ) diBR

it follows that

LoFi(x) = fMp(x - f ) / ( | ) d^, APF'l.(x) = F{x)BR

LoAP-^F*(x) ^JKoix - ! ) / ( ! ) d^

BR

so that it follows from Theorem 2.6.7 that F^ C'^^{Bji), satisfies Lo F

= f, and h as the stated bound. I f / $ C^^(Bji), we see by integrating by

parts successively that V^' F is given by (6.2.14) in terms of V^/, j < k.

Thus the remaining results in (d) follow. The results in (e) follow from

these, the usual approximations, and Theorem 3-4.2.

We now define our spaces and norms.

Definition 6.2.1. For each integer A > 0, we define two pairs of

spaces, *H^Q{BR) and '^'^H^{BR) of vectors having components in

various J^*-spaces ( > 1), and *CjJo(5ij) and **C^(jBi?) (0 < ^ < 1) of

vectors having components in various Cj^{Bji) spaces with respective

norms defined as follows:

(6.2.18)j k

0**\MtB = I'\MCR''

where we define the norms

'll9'llU=i^-^l|V*-'9'||%

(6.2.19) 'MU = S[R-'h,{Vt-^<p) + R-'-^WW^-'CPWIR]

s-0\\\(p\\\R=^im,x\(p(x)\.

XSBR

W e now define our o p e r a t o r PRhoif) d e p e n d i n g on the in tege r ho

defined in § 6.1 in connec t ion wi th the b o u n d a r y v a l u e p r o b l e m .

Defin i t ion 6.2.2. I f / $ * H%{BR) for some A > Ao > 0 and > 1 or

/ ^ s ome *Cô (BR) for s ome h ^ ho and /^, 0 < // < 1, we define PRhoif)

= U w h e r e

m = Ll^[Fi - F*], Fi = fK{x- $)MS) d^

(6.2.20) ^«F* (x) = y — V Fi(0) • x\ Q = m-{-ho —si—v

( the sum be ing 0 if ^ < 0).

The following lemma s implif ies the h a n d l i n g of the de r iva t ive s of

low order .

Lemma 6.2.2. (a) If u ^ H\>{BR) p ^ q ^ 1, £ > 0, and s is an in

teger with 0 < 5 < ^, then

Ci = Ci{s,p,q,v, sj), Q = v{p — q)lpq.

Thus

1kl lU<C2( | |V^^l lS^+i? -^-^ | |ÎW, C, = C,(p,q,vJ).

(b) Ifu^ CJ,{BR), 0 < / / < 1, e > 0, and q > \ , then

hf,{\Ji-û, BR) <eR'h^{Vû, BR) + CsR^-t-^-f^M^, 0<s<t,

111 V ^'' u \\\R < e R^+^ h^Vû, BR) + C4 i^^"^"^ \\u\\%, 0<s<t;

Cs = Cs{s, qy fJiy Vy s, t), C4 = C4(£, q, JLC, V, s,t), a = vjq.

Thus

'M\R^Ci{q,li,v.t)[h,(ytu.BR) + R-*-'-^'\\u%j,].

(c) The same results hold with BR replaced hy GR.

Proof. In e a c h p a r t , the las t s ta tement fo l lows f rom the f i rs t . From

cons ide ra t ions of h o m o g e n e i t y , it fo l lows tha t it is sufficient to p r o v e the

f i r s t s t a temen t in e a c h p a r t for the case R = 1. If (a) were not t r u e on

Bi, there would ex is t a sequence {un} with ' | |^7i| |^ = 1 s uch th a t Un —7 u

mH%{Bi)hut

T h e n u = Q and (since s > 0) V^~^ ^î ^ 0 in Lp (Bi). B u t t h e n \7 * Un-^0

(s trongly) so tha t w^ -^ 0 in H\{Bi), c o n t r a d i c t i n g the fact that ' | |^w||î

= 1.

Suppose one of the f i rs t inequa l i t ies in (b) fails to ho ld on Bi for s o m e

s > 0. Then the re ex i s t s a s equence {un} su ch t h a t '|||^? i|||i === 1, V^~^ Un

converges un iformly to V*~^ u for s ome ^, 0 < s < ^, and

Ki'^^-'Un) > eh^VÛnyBi) +n\\un\\^,.

I t then fo l lows tha t uÔ and it is easy to s ee tha t hf^{\7i-s Un, Bi) ^ 0

if s > 0. But t h e n h^{\/tun,Bi) ->0 so ' | | | ^ ^ | | | * , i ->0 , a c o n t r a d i c t i o n .

6.2. Interior estimates for general elliptic systems 22 1

T h e o t h e r i n e q u a h t i e s in (b) are proved s imi la r ly . Pa r t (c) is p r o v e d in

t h e s a m e way.

Theorem 6 .2 .2 . PRUQ is abounded operator from '^H^Q{BR)to '^'^H^(BR)

and from *C^Q(BR) to **C^(5i2) with hound independent of R if h > ho

> 0. The bounds depend only on h, ho, p, or /u , and E. Moreover if U

= PRhoif), then U satisfies the equations

Loj,UHx)=f^{x)-f*(x)

where ff is the Maclaurin expansion of fj out to [and including) the terms

of degree ho — Sj — v.

Proof. By i n t e g r a t i n g by p a r t s we see t h a t

Fi{x) - Ft(x) =f ^ C.K4x,i)D''Mi) di, C„ = ^ ^ ^ T ^

BR \ix\=ho—si

K4x,^)=K„{x-^)- ^ l v « X . ( - f ) • ; c ^ Q = m + ho-si~v

(6.2.21) K^x) = D-K%^{x), ^ C,D-K^x) = K(x).\oc\=ho~si

Since (see Theorem 6.2.1) Ko, is es s en t ia l ly homogeneous of degree Q, we

h a v e

Kcc(y) =Ki4y)log\y\ + Kôîy), V^K^y) = V^ i ^ ia (y ) • lo g |y | +

+ K2scc(y)

where iCia is a h o m o g e n e o u s polynomial of degree q and K^cx. and K^so:

a re pos i t ive ly homogeneous of degrees q and Q — s re s pec t ive ly . Thus , if

w e SQi X =^ R y, ^ =^ R 7}, y, Tj ^ Bi, t h e n

K4Ry, Rfj) = [K,{y - f,) ~ ^ ±V' K4-f,)-y^]- R« +

(6.2.22) * e

+ R«logR-[Kicc{y -fj)- ^^y 'Ki4-'>j)-y ^] '

a n d the second term vanishes-. Thus

(6.2.23) f\Koc{x,S)\di^CRQ+^, j\Ko.{x,^)\dx<.CRQ^\ X.^^BR.

BR BR

The ana lys i s above a s s umes tha t ^ > 0 . I f ^ < 0 , the results (6.2.23)

follow from L em m a 3-4-3- U sin g the Holde r inequa l i ty , we f ind tha t

\Fi(x)-F*(x)\^

( 6 . 2 . 2 4 ) < ( C i ? e ^ - ) « - i / 2: C.\K4x,^)\\v'^-^'fi(^)\9dS

s o t h a t we can say t h a t

\\Fi - Fff, < Zi R^^^ \\ V'^--^/z | |? < Z2 R^^-^^-ô II v ' ^ - V ^ i ? .

The fac t tha t Fi ^ H'^+h-'' w i t h || v^^'^"^^ Fi||o < Z3 || V'^"'^ fi||^

follows from Theorem 6.2 .1 . U s i n g L e m m a 6.2.2 we o b t a i n the first

re s u l t . The Ho lde r re s u l t s a re ob ta ined s imi la r ly . The l a s t s t a temen t

follows from Theorem 6.2 .1 .

I t is con ven ien t to in t ro du ce the fo llowing te rm inolo gy:

Def in i t ion 6 .2 .3 . The opera tor L ( i. e . th e m at r i x [Lj]^) is said to

satisfy the h-conditions [h > 0) on a set G if an d on ly if

(i) ii Sj = h = 0, the coeffic ients of the highest order derivatives in

t h e o p e r a t o r s Ljjc are cont inuous on G, the o the rs be ing bounded and

m e a s u r a b l e t h e r e ; a n d

(ii) if ^ — Sj > 0, the coefficients in Ljjc^ C\~^^~'^{G). T h e o p e r a t o r

is sa id to sat is fy the h — fi condi t ions on G if and only if the coefficients

in the ope ra to rs Ljjc^ C^~^^{G).

Given the ope ra to r L defined on BA, we define LR b y

(6.2.25) LRJJC - Lojk + (p(R-^ \x\) {Ljjc - Lojic), 0 < i^ < Ajl,

w h e r e LQJJC is define d ne ar (6.2.1), an d 99 is a fixed fun ctio n ^ C°^{Ri)

w i t h (p(s) = \ for s < 1, (p{s) = 0 for s > 7/4, an d 99' (s) < 0 for

1 < s < 2.

Theorem 6.2.3 . (a) Suppose that L satisfies the h-conditions on B ^

and LR is defined as in (6.2.25). Then LR satisfies the h-conditions on B^R

and {LR — LQ) is a bounded operator from **i7^(52i?) ^f^io '^H^Q{B2R)

with bound e (R), which depends on E an d E', and which - > 0 as R ->0.

(b) 1/ L a lso satisfies the h-ju-conditions (0 < a < 1) on BA, then sodoes LR and LR — LQ is a bounded operator from **C^(B2R) into

*C^Q(52i?) with bound of the form C R^^ where C depends only on E and E'.

(c) / / in (a) or (b) above h> \, then the bound of the operator LR — L Q

is <,CR where C depends only on E and E'.

Proof. T he proof of (a) is s imila r to b u t s im pler t h an th a t of (b) a nd

the proof of (c) is similar. We sketch the proof of (b). Suppose u^

* * CI{B2R) a n d / = {LR - LQ) U. Then

« l , 'f c W - ^ o , - fc = 9 ^ ( ^ " M ^ I ) [ ^ ? f c W - < f c ( 0 ) ] if \oc\=sj + tjc

^hk(^) = (P{R~^ l^D^hi^) if I I < 5; + ht h e difj^ being the coefficients in Ljjc. If \^\ = h — s y ( > 0 ) , D^fj is a

sum of terms of the form

Dy (pR' D^ {a^j^ - a^^j,) • D^ u^ a n d D'' cpR • D^ afj^ - D^ u^,

(pR{x) = (p{R-^\x\),

w h e r e y, d, e tc . , a re mul t i - ind ices and

I r l + 1 1 + 1 1 =h + h, \y \ + \d\ <h-Sj,

1^1 + \M + \Q\<h + h, \>c\ + |Jl| <h-Sj.

The de r iva t ive s o f the afj^ and the i r Ho lde r cons tan t s a re bounded in

dependen t ly o f R while we eas ily see from th e definit ion of ' | w| | |J i j an d

6.2. Interior estimates fo r general elliptic systems 2 2 3

the form of (pR t h a t

i l l ^^ ^^ | | |2 i? < (2i^)*'^+ '^-|^ l+î^, hf,[Dû^,B2R] < [2Ry^^^-\^\'K

\lâ — a^W^R ^ C'R^, a = JLC ii h — Sj = 0; cr = 1 if A — s - > 0 ,

(6.2.26) ( ^ = ** |||« |||^ J.

I t fol lows eas ily that hfi(\/^~^^fj B2R) < , C K and the res t follows s ince

the fj vanish on B2R — B^^RJ^.

Definition 6.2.4. If L satis f ies the A-condit ions on Bj^ a n d R < A / 2 ,

we def ine the opera tor TR on the spaces **-fif^(52i?) and **C^(52i?) by

the cond i t ion tha t

TRU = — P2R [{LR — LQ) U] .

From Theorems 6 .2 .2 and 6 .2 .3 , we immedia te ly conc lude the fo l low

i n g t h e o r e m :

Theorem 6 .2 .4 . / / L satisfies the h-conditions on B^, then TR is a

hounded operator on '^*H^{B2R) for each p '> \ and R ^ Aj2 with hound

8(R,p) where s depends also on E and E' hut - ^ 0 as R -^0. If L also

satisfies the h— ju conditions, then TR is hounded on **C^(^2i2) ^^^^

hound e{R, fi) o f the form C(ILI, E, E') • R"^, a = /u or \ as in (6.2.26).

R e m a r k . TR depends on the in tege r ho which we may, o f course ,

regard as f ixed .

W e can now prove the fo llowing in te r io r d i f fe ren t iab i l i ty the or em :

Theorem 6 .2 .5 . Suppose that L satisfies the h-conditions on the bounde d

domain G, u^^Hf+^'(G) for some q > i, and the fj = Lj]cU^^H\~^j{D)

for each DCCG and some p^q and h > AQ- Then u^ ^ H^^'^'^ (D) for each

DGGG.If, also, L satisfies the h — JJL conditions on G and the fj^C^~^j(D) for

each DC CG , then u^^C*^+ ^{D) for each such D. If L and f^C^{G),

thenu^C^{G).Proof. The las t s ta tement fo l lows f rom the preceding ones . To prove

the f i r s t s t a temen t s uppos e D C C G and s uppos e D C C A C G G.

T h e n the fj^H^"^) {A). W e firs t assu m e A > 1 + Ao an d shall show th a t

w^ ^ H*^^^ (D ). W i th each po in t XQ of D i s assoc ia ted an i ? > 0 such t h a t

B{xo, 2 ^ ) c z l a n d t h e b o u n d s of TR on the spaces '^*H^[B{xo, 2R)]

an d * *i7J [B {XQ, 2i? )] are bo th < 1/2; we ta ke ho = 0 for this proof. W e

can find a C^ pa r t i t ion of un i t y f i , . . . , Cs def ined on a domain F Z^ D

an d such th a t each Cs has sup po r t in some one of th eB [xo, R) •

L e tti^ ~ ^g u^; we sha l l show tha t each ^lsÎI*q^^(G).

To do th is , we suppose tha t the suppor t o f Us is in BR .Then , c lea r ly

US^*'^H^,(B2R) ^n d

(6.2.27) Ljjc < = Csfj + Msjjc U^ = fu if3 = Ljk U^)

a n d fs^*Hl{B2R) s ince the opera tor Mgjjc is of order ^Sj + tjc — 1.

22 4 Regulari ty theorems for general boundary value problems

Now, write Ug= USR + HSR where USR = P^R [ ^O S ^ S ] . H US ^ C^ (B2R),

t h e n

mlK(x - ^)LosimK(i)d^ = LllFis(x)

= mLosimjK{x-i)uf{^)di=LosjK(x-i)u^($)d^ = u^(x).B2R B2R

T h u s , from the definit ion of P2R, we see t h a t H^j^ is a p o l y n o m i a l of degree

<,tk — V. T h u s UsR satisfies

UsR = P2R{fs) — P2R[{LRS — LQS)USR] — P2R[(LRS — LQS) HSR], Or

UsR — TsR UsR = VsR = P2R ifs) ~ P2R [{LRS — ^Q s) HSR] .

Since IT^ij |] < 1/2, it follows that USR and hence US^^'^H\{B2R).

Accord ing ly ^ * *^J [A] for s omez l Z^D. The a r g u m e n t may be r e p e a t e d

choos ing R s ma l le r if necessary . In a f in i t e number of s teps we f ind tha t

To ra ise the exponen t f rom q to p, we proceed s imila r ly but choos ing

R so t h a t | i r i j l | < 1 / 2 in **HI^{B2R) w h e r e qi = vql{v — q), the

Sobo lev exponen t co r re s pond ing to q. Then, from (6.2.27), we see t h a t

fs^ *H\[B2n). A f in i t e number of repe t i t ions accompl i shes our p u r p o s e .

In case L satisfies the h — fji cond i t ions and / ; $ C^~^j, we ma y t a k e p so

l a r g e t h a t u^^Cf^^^^^'^ and t h e n fg^ *C^ and we may conc lude tha t

u^ ^ Cfl''^^ by t a k i n g R s ma l l enough .Fo r l a te r use, we need the fo l lowing lemma and t h e o r e m :

L e m m a 6 . 2 . 3 . There exists constants Ci(v, s, t, p, q) and € 2(7 ] , s, t, ju , q)

such that

1 /f IIU < Cx R-t-^ IIH ||o^, ' IH11^ < C2 R-^-^-^ IIH ||o^,

Q=v{p-q)lpq, G = vlq, p>q

for all polynomials H of degree < s .

Proof. Let n be the l a rge r of s + 1 and t + \. T h e n , f ro m L e m m a

6.2.2 with £ = 1, ^ rep laced by n, and s rep laced hy r = n — t, WQ seetha t (s ince V H = 0)

II V^HWl^ < C[R-t-q H\\l h,[VtH) < C^R-t-o-^\\H\l^.

The lemma fo l lows f rom Lemma 6 .2 .2 .

Theorem 6 .2 .6 . Suppose L satisfies the h-conditions on B^. Then there

exist constants i?2> 0 and C2 depending only on v, h, q, E, and E' such

that

** i u |i,\ < C2 [* IIL «IIJ + *'\\u 11? ], *'\\u 11; =^R-h-'^-^ 11 «^ 1;

0 < i? ^ i?2, M* € H*g''{BR), Q = v\q- \)lq.

Also there exist constants Rz> 0 and Cz, depending on v, h, fi {0 •< ju < \)

E, and E' such that if L satisfies the (x — h conditions on BA, then

* * | | | « f c < C 3 [ * | L « ! l | ^ « + * ' | | | « | | ? ^ ] , *'\M\i^^R^'-"*'\\u\\u

0< R<R3, M* € CJf+''(5j;) .

(6.2.30)

6.3- E s t i m a t es n ea r t h e b o u n d a ry ; co e rc i ven es s 2 2 5

Proof. We prove the second, the proof of the f irs t is s imilar . We omit

the subscr ip ts /^ i? . We confine ourselves to R so small that | | TR || < 1/2

in the space **CjJ;(52i2). Suppose U^^O^;^^{BR), define LR, P2R, a n d

TR as above (with some fixed ho < A); we assum e u = 0 on B2R — BR.

L e t UR = P2R(LO U). Then, as was seen in the proof of Theorem 6.2.5 ,

u^ •=u\ + H\, w h e r e H\ is a polynomial of degree <,tjc — v, a n d

(6.2.28) UR-TRUR = PR(f)-PR[{LR-Lo)HR], f=LRU^Lu.

Using the definit ion of UR, we obta in (Z^ ind epe nd ent of R)

* * III ^ i ? I ll ^ < z f * III ^ III ^ * * III / i T / j III ^ < z f * IK ^ i p .

Using (6.2 .28), theorems 6.2.2 , 6 .2 .3 , and 6.2 .4 and the fact that | | TR\\

< 1/2, we find fur the r th a t

(6.2.29) * 1 l | ^ i ^ f c < ^ 3 l l | / | | | ^ E + ^ i ( ^ ) * * | | | ^ f c

From the def in i t ion of the norm, we conc lude tha t

'\\\uRr^n<2,Z*\\\f\\\lJ,+Z,el{R)**\\H\l^ (^4 = 77^)

.•.*'\\\HRr^I,<*'\\\ur^B + *'\\\uE\r^n.

From (6 .2 .30) , (6 .2 .29) , and Lemma 6.2.3, we conclude that

**|||/^flill^« < Z * ' | | | « ! | | ? B + Z*\\\f\\\lj, + e2(R) **\M tR* * \ \ H \ ' 'u E < z r tH \ ' i n + ^ n f \ & R + ^ s { R ) * n M iu


6.3. Est imates n ear th e b ou n d ary; coerc iven ess

W e now ca r ry ou t the bounda ry e s t ima te s fo r the bounda ry p rob lem

descr ibed in § 6 .1 , na m ely

(6.3.1) LjjcU^=fj in G, BricU^ =^ gr on dG,

We sha l l beg in by cons ider ing func t ions def ined on hemispheres GA

ins tead of full spheres BA bu t o the rwis e s ha l l t ry to imi ta te the deve lop

ments of the preceding sec t ion . For func t ions u hav ing s uppor t in GR U OR

we obta in es t imates for u in te rms of those for /and^ as def ined in (6 .3-1) .

The fac t tha t the coeff ic ien ts in the boundary condi t ions a re a lso

var iab l e causes an ex tra c om plica t ion w hich we tak e care of in th e fol low

ing way: We def ine the opera tor PR (/, g) for the pair (/, g) to be a ce r ta in

so lu t ion of the equa t ions

(6.3.2) LojkU^ = / y o n R+\ %"+! > 0, BQrTcU^ = gr on a

where the fj a n d gr have s uppor t on G2R U (y2R a n d a2R, respec t ive ly .

Then if u i s g iven wi th suppor t in GR U CTR, we alter the coefficients of L

and also B ou ts ide GR to ob ta in ope ra to rs LR a n d BRrk as in the preceding

Morrey, Multiple Integrals \ 5



22 6 Regularity theorems for general boundary value problems

sec t ion and t h e n w r i t e u =^ UR + HR w h e r e

(6.3.3) UR = PR[f- [LR - Lo) u, g-{BR~ Bo) u].

I t t u r n s out a g a i n t h a t HR is a p o l y n o m i a l of bounded deg ree . We define

(6.3.4) TR((P) = PR[~{LR - Lo)cp, - (BR - Bo)cp]

a n d the equation (6.3 .1) becomes

(6.3.5) UR -TRUR^VR = PR [f - (LR - Lo) HR, g - (BR - Bo) HRI

We def ine

PR(Lg) = UR+VR-WR, UR = PR(0,g), WR=PR(0,y),

(6.3.6) y, = BorkV%

a n d VR is defined in a way very s imila r to the Pho,2R (/) of the preced ing

sec t ion , / be ing a s moo th ex tens ion of / to B2R', PR(0, g) and PR (0, y)

are defined by m e a n s of the ' T o i s s o N k e r n e l s " i n t r o d u c e d by AGMON,

DouGLis , and N I R E N B E R G ([1], [2]). We wri te PR i n s t e a d of P2R in th i s

sec t ion , but s t i l l work on G2R. In o r d e r to o b t a i n the Lp re s u l t s , it is

s h o w n how to expre s s the de r iva t ive s of h ighes t o rde r by m e a n s of sin

gu la r in teg ra l s of GALDERON-ZYGMUND t y p e ; t h i s is an idea s imi la r to

one used by B R O W D E R [1], [2]. We shall replace r by 1 + 1 and x^+^ by

y and sha l l wri te the o p e r a t o r s LOJTC(L>) and Bork(L>) in the forms Lojjc

(Dx, Dy) and Bork (Dx, P>y) w h e r e x = (x^, . . ., x").

W e now deve lop formulas for the Poisson kerne ls by a t t e m p t i n g to

solve the e q u a t i o n s

(6.3.7) LojTc(Dx, Dy) u^ = 0 on R+,

Bork(P>x, Dy) u^(x, 0) = gr(x) on a,

formal ly by i n t r o d u c i n g the F o u r i e r t r a n s f o r m s u^ and gr with re s pec t

to X defined by(6.3.8) uk(X,y) = (27t)-'' f e-'^''^u^(x, y)dx, (r = vl2)

w i t h a s imila r formula for gr. This y ie lds the e q u a t i o n s

(6.3.9) Lojk(iK Dy) u^(X, y) = 0, y > 0

Bork(il Dy) u^(l 0) = gr(X), y = 0.

I t is c o n v e n i e n t to norma l ize the s e equa t ions by s e t t ing

(6.3.10) ^^(A, y) = (i \X\)-^k um |A| y)

gr(}i) = (im)-^rGr(X).

T h e n the equations (6 .3 .9) in w = |>l| y b e c o m e

(6.3.11) Lojk((r, -iDn)U^(lu) = 0 , u>0,

(6.3.12) Bork((r, ~iDu)Um 0) - Gr(?i) (a = | > l | - U , \a\ = 1).

6.3- Estimates near the boundary; coerciveness 227

W e beg in the so lu t ion of the s e equa t ions by s t a t i n g the following

w e l l k n o w n l e m m a :

Lemma 6.3 .1 . (a) Every solution U of a single ordinary differential

equation {with constant coefficients) of the form

L[-iD)U = 0, L[z) = aoftiz -rj)^j

is of the form ^

U(u) =2' ;M^*''' "^

where each Pj is a polynomial of degree <,mj ~ 1. Moreover, every function

U of this form is a solution.

(b) / / s ^ any rj, the equation

L{—iD)U=P(u)e^^'^ ( P a po lynomia l )has a unique solution U of the form Q [u] e^ ^ ^, w here Q (u) is a polynomial

of the same degree as P. If s = rj, the equation has a unique solution U of

the form u'^jQ {u) ^*' ?«* where Q is a polynomial of the same degree as P.

Definition 6.3 .1 . A func t ion of the form P{u)e^^'^, [P a po lynomia l )

w h e r e / w s > 0 is said to be exponentially decaying.

We now prove the following fundamental theorem of A G MO N , D O U G -

L i s , a n d NiRENBERG ( [ 1 ] , [2]):

Theorem 6.3 .1 . We suppose that the matrix of operators LQJJC is properlyelliptic. Then there are fixed rectangles R+ in the upper half-plane and R~

in the lower half-plane such that R^ contains all the zeros with positive imagi

nary part of LQ {a , z) for each a with | or | = 1 and R~ contains all those with

negative imaginary parts. The operators Bork satisfy the complementing

condition with respect to the LQJJC if and only if there are exponentially

decaying solutions Us of the homogeneous equations in (6.3.11) which satisfy

(6.3.13) BorJc{(^>-^Du)U^(a,0) =:drs, r,s=\,...,m.

If they exist, these functions are unique and are analytic near \a\ = 1. / /we introdtice the functions U^{o , z) by the formulas

(6.3.14) U^,{o, z) = {27z)-^fe~i^^U^{a, u)du

0

we see that the U^ are analytic in G near \G\ = 1, are rational functions of

z having denominator L^ [a , z), defined as in (6.1.6), and we have

(6.3.15) U'l{a, u) = jei^^U\{G, z)dz.dR +

Finally there are poly nomials Pj § {G, Z ) in z such that

(6.3.16) Lojjcia, z) U^{G, Z) = PJS{G, Z).

Moreover, the U^ satisfy the conditions.

(6.3.17) fBork{(r,z)U^{G,z)dz = drs> r,s= \, . . .,m..OR

15*



2 2 8 Regularity theorems fo r general boundary value problems-

Proof. Since the zeros of Lo(a, z) v a r y c o n t i n u o u s l y w i t h a, the se t

\a\ = 1 is co m pa ct, a nd no zero is real , the ex is tence of th e recta ngl es

R+ a n d R- follows.

Un t i l the l a s t pa rag raph o f th i s proof, we a s s ume tha t o* i s a cons tan tand suppress i t . Le t us cons ider the matr ix LQJ]C[Z) of charac te r is t ic

po lynomia l s . By app ly ing the e lemen ta ry ope ra t ions o f (a ) in te rchang ing

t w o rows and (b) adding to one row another row mul t ip l ied by a po ly

nomial , one can replace | | Lo;A; (-s) || b y a t r i a n g u la r m a t r i x || ZyA;('2 ) |i in

w h i c h Ljjc(z) ^ 0 if y > ^. Cle arly

L,{z)=-±nLjj{z).

If we factor Ljj{z) in to Lf^ (z) • L~' (z), then , c lea r lyN ^

^ ( ) = i t n^ti (^)' ^ = ^^i'

nij being the degree of L^^. Moreover , the sys tem

(6.3.18) Ljjc{-iD)U^ = 0

i s equiva len t to the homogeneous sys tem (6 .3-11) .

Next we see tha t every exponent ia l ly decaying so lu t ion of (6 .3 .18)

(and hence (6.3 .11)) can be obtained as follows: Firs t le t U^ be any

exponent ia l ly decaying so lu t ion of L^]:^{—iD)U^ = 0. There a re

Wiv l inear ly in dep end ent so lu t ions U^' (Le m m a 6 .3.1). W it h ea ch g iven

s uch U^, we assoc ia te th e pa r t ic u la r so lu t ions of Lem m a 6 .3 .1(b)

for the equations LN_I,]^T_IU^-^ = ~L]S^_I,N U^, LN-2,N-2U^~^

= — L N - 2 , N - 1 U^~^ — J^N-2, N U^, e tc . , in tu rn . Next , take U^ = 0,

le t U^-^ be any exponent ia l ly decaying so lu t ion of L^-i, N-1 U^~^ = 0,

and de te rmine the o the r U^ as the par t icu la r so lu t ions ob ta ined by so lv

ing the o the r equa t ions in tu rn . The re a re m]s^_i such so lu t ions . This

process m a y be co nti nu ed yield ing m == Wi + * * * + ^ iv l inear ly in dependent exponent ia l ly decaying vec tor so lu t ions of (6 .3 .11) .

Thus the to ta l i ty of these vec tor so lu t ions is an w-dimens iona l com

plex vec tor space V. If U^V, each express ion Borjc(—iD)U^(0) is a

l inear func t iona l Fr{U). T h u s t h e t r a n s f o r m a t i o n Xr = Fr{U) is SL \ — 1

t rans fo rma t ion f rom V t o Cm the range of the t rans format ion is Cm. T o

show th is , we se t

Lo+ (z) = z^ -i- ai z^-^ + •• ' + am

and def ineLt^{z) =z^ + ai z^-^ + '" + a^, ^ = 0, . . ., w - 1.

We not ice tha t i f R+ is a rectangle enclos ing the roots of LJ , then

6 .3 . Es t im a tes nea r t he bo un da ry ; coe rc iveness 229

F o r , ii y <: p, the degree of the numera tor i s < w — 2 a n d t h e r e s u lt

follows by replacing ^i^+ by a large c irc le ; if y = ^, the result follows

s im i la rl y; i f ^ < } / < w — 1, t h e n u m e r at o r differs fro m zy-^-^ L^ {z)

by a po lynomia l o f degree<

y — 1<

m — 2 . N ex t , we w r i t em— 1

(6.3.19) BorJc{z)Ll'{z)=Sq'/^^ {modL^);

we cons ide r the ma t r ix Q =(ql^) as hav ing m rows ( indexed wi th r) a n d

m N co lumns . The complemen t ing cond i t ion i s equ iva len t to the s t a te

m e n t t h a t Q is of ra n k m. T hu s , if Xr a re any numbers , the re a re cons tan t s

cip s u c h t h a t

(6.3.20) ci^ql^ = Xr.

W i t h t h e s e ci^, we define

(6.3.21) UHu) -l- fci ^L ., ,(z)ei- dz.

eR +

T h e n

Borici- ^ * i) ) ^ ^ ( 0 ) = - 1 . r C z ^ ^ O r ; f c ( ^ ) i : S ^ ( ^ ) ^ % ^ = ^ ^ ^

I t is c lear from (6.3.2I) that the vector U^ i s an exp one nt ia l ly decay ing

solu t ion of the homogeneous equa t ions . I t i s un ique in sp i te o f the fac t

t h a t t h e cip satis fying (6.3 .20) are not unique.

T h u s , for each a, the re i s aun ique s o lu t ion U^{a, u) of (6.3.11) and

(6.3.13)- Since Lo(or, z)'^ a n d t h e LOJA: an d L§* an d B^rk are analytic in or,

i t is easy t o see th a t w e m a y choose, for each s , functio ns Csi^ [a) which a re

ana ly t i c in a ne ar a ny give n co w hic h satis fy (6 .3 .20) w ith Xr = drs'

T h u s U^{a, z) i s ana ly t ic in a a lso . I f we in t roduce U^ by (6.3 .I4), i t is

c lear from (6.3.21) that

f7j(a, z) =(-i)csip(a) [ ^ ( a , z)]-^Ll\a, z)Li_,_^{a, z)

1 To see tha t LQ {a,z) i s ana lyt ic in a, w e n o t e t h a t

cnicf) =fz^lL^ic Z)]''^LO'{G, z)dz =-j z^ L^'^ L^dzdR+ dR+

so th a t each Cn i s ana lyt ic in a an d

(L^)-^ L ' = m z-^ 4 - f ^n (O ) z-n-ln= l

w h e r e t h e s er ie s c o n ve r ge s u n if o rm l y f or | ^ | > s o m e A, i n d e p e n d e n t ofa; of

course i?+ is a f ixed rec tangle in th e upp er ha l f -plane which co nta ins a ll roots of

LQ {a, z) for all a wi th | cr | = l.

2 3 0 R eg u l a r i t y t h eo rem s fo r g en e ra l b o u n d ar y v a l u e p ro b l ems

where , near any co , the Csip{a) satisfy (6.3.20) with Xr = drs- T h e s t a t e

m e n t s a b o u t t h e U^ follow.

T h u s , f rom th is theorem i t fo l lows tha t the equa t ions (6 .3 .11) and

(6 .3 .12) have a un ique exponent ia l ly decaying so lu t ion which is g iven byUm u) =Z GsWU^s{^> ^)> U^s = ieinzxjic{a, z)dz

w h e r e R+ is a f ixed rectangle in the upper half-plane which contains a l l

the roots of Lo+ (cr, z) for each o w i t h \a\ = 1. If we now use th e sub s t i t u

t ions (6.3 .10) and formal ly a pp ly th e inverse Fo ur ie r t rans f orm , we

o b t a i n

(6.3.22) u^{x, y) = J L^^x — f, y)gr{S)d^, w h e r e

(6.3.23) L^r(^x,y) = C^r j \X\-h+^rdXJe^^^-^-^\^'^y 'W^,[a, z)dzRV dR+

C^r ^ (2n)-''{-i)h-K.

T he in teg ra l in (6 .3 .23) i s d iver gen t , bu t if we in t r od uce p o la r coor d ina tes

(s , a) w ith s = |A[ an d \G\ = 1, we obtain

L^rf^x^ y ) _ C^^ jdj; jU^,{a, z) \js^-^-h-^K ^^^(^•«'+^^)^s dz.i: dR+ Lo J

By rep lac ing the b racke t by Jv-i-tk+hr {a • x + y z) where the Ju {w) aredefined in (6.2.7) and (6.2.9), we obtain

(6.3.24) L^^r^^^ y ) _ Ci^rjdZ JJv^i-t,^hr[o'X + y z)U!^{a, z)dz.

s eR+

I t i s convenien t a lso to have the fo l lowing in tegra ted kerne ls .

Definition 6.3.2. W e define, for larg e P ,

*L^ / (^ , y ) = {-\ra^r jd^jJ._ i_ t,^nr-2p[o'X + y zm{a, z)dz,E dR+

*L%{x, y ) = Dl^L%^,,{x, y), Ll^{x^ y) =A^^Lf,{x, y ) .

(6.3.25)

The fo l lowing lemma is very usefu l :

Lemma 6.3.2. Suppose (p is essentially homogeneous of degree n > 0

and (p ^ C^ (Rv+i — {0, 0 }) and suppose

(6.3.26) <p(X, S ) = f{X-S) -2lv^<p{-S)-X^.

Then

1 Z ) | ^ ( Z , S)\ < C ^ | . Y | ^ + i - l ^ l | S | - i if 0 < | X | < ^ | ^ 1 ,

\P\<n, A <\

\D^^(p{X, S)\ = \DP(p{X ~ S)\ < C^\X- S\n-\^\ if | i8 | > ^ ,

w h e r e CA, an d Q de pe nd also on bo un ds for the der iva tiv es of 99 on

^ B ( 0 , 1 ) .

6.3- Estimates near t h e boundary; coerciveness 2 3 1

Proof. The las t s ta tement i s ev ident . Moreover , i f \^\<,n, t h e n

D^(p{X, S) i s ob ta ined f rom D^cp{X — E) by the fo rmula co r re s pond ing

to (6 .3 .26). So i t is suffic ient to pr ov e th e f irst s ta te m en t only for | |^ | = 0 .

W e w r i t e

3 = S^, | l i = 1 , <p{X) = cp^{X)\og\X\+<p^{X)

w h e r e (px is a polynomial of degree n a n d 992 i s pos i t ive ly homogeneous of

degree n. T h e n w e h a v e

(6.3.27) D-<p(X) = D-<px{X) \0g\X\ + Cp2a:{X)

wh ere 992a is po sit iv ely ho m og ene ou s of de gree n — | « | . T h us

9)(Z, S I ) = S ' » 9 ) i ( S - i X - I) [logs + l o g | S - i Z - f |] + S « 9 P 2 ( S - iZ - |)

- S » 2 < ^ ^ " [ i 5 - 9 , i ( - f ) l o g S + 9 )2 « ( - l ) ]

= S » U S - 1 Z - | ) - ^ < ^ ^ Z ) X - | )

oc\ = {oci\)...{ocv\),

since, from (6.3.27) it follows that D°'(p{—^) =(p2„{—^). The re s u l t

follows since | S- i X | < ^ < 1.

Theorem 6.3.2. (a) The functions * L ^ ^ "^L^, L^\ and L\' are analy

tic for all real {%, y ) 9^ (0, 0) with y > 0 and are essentially homogeneous ofdegrees 2P + tjc — hr ~ v, \a\ + 2P + tjc — hr — v, tjc — hr — v, and

\oc\ + tjc — hr — V, respectively.

(b) For each r, P, and oc, the functions above sa tisfy

k Jc k k

(6.3.28) 2C,cD^^*Lf, = *L''/. ZC-DlLl' = Lkr, y > o| a l= t | a | = i

Z B,,,*m.,0)^6lM ,{x), C. = ^ ^ - J ^

where Mp is defined in {6,2.\\); it is assumed that P is large.

Proof. T h e an al yt ic i ty for y > 0 is ev ide nt from (6.3.25) an d th e

definit ions of the Jn given in (6 .2 .7) and (6.2 .9). The essentia l homogeneity

fo l lows f rom an a rgument s impler than tha t g iven in the f i rs t par t o f the

proof of Theorem 6.2 .1 . Since | <71 = 1, it follows im m ed ia te ly from (6.3.25)

(and (6.2 .8)) that

4 " ' * ^ p + i a i ( ^ , y ) = * - ^ ' / ( ^ , y ) . y > o .

Thus the first formula in the second line of (6,3.28) follows. From (6.3.25)

if follows that

Z Lojjc''L%'{^, y ) = C]pjdZ j J Q { O ' X + y z)Z U^jc{a, z)U^,(a, z)dzfc S dR+ ^

= 0 , Q = v - \ +Sj + hr-2P, y > 0, C]p = (27 t)-''(--\)Pi'j+K

http://0g/X/

http://0g/X/

http://0g/X/

on account of (6.3.16). Likewise

^Botic'^Lfix, y)-\-'Clpfd2;fMa'X + yz)ZBotk{a, z)W,{a, z)dz^ 2 dR+ k

'dp = (-\)P(27z)~*'iK-h, l==v - i -h hr- ht-2P.

If P is large, we may let y -^ 0+ and obtain

ZBotit^L^'ix, 0) = (-i)P(27t)-^dlfj,_i.2p{a'x)dZ

k £

from which the last line of (6.3.28) follows, using (6.2.11). The remaining

formulas in (6.3.28) follows easily.

To prove the analyticity of ^V^^ at a point (x^, 0), A:O =5 0, we make the

rotation

a = a{x,T) '. G"^ — d^i\x\~^ ' x) ly, G - x = \x\' T^

of Lemma 6.2.1 for x near X{^. Then we set

r i = COS99, T1+°' = of^ sin99, |co| = 1,

*iV5r(A;, cp, z) = sm''-^(pJU^[G{x, T),z]dco.| a > | = l

Then *N^/ is analytic in its arguments for 99 near the segment [0, jt] in

the 99-plane, X complex but near XQ, and z outside and on dR+. Also, for

a suitable integer n and constant C,

n

"^L^/ix, y) = C n fjn (\x\cos(p + yz) *N /{x, cp, z) dz\ dcp, y > 0.

0 UJ2 + J

We conclude the analyticity at {XQ, 0) by noticing that the segment

[0, 7r] may be replaced by a nearby arc from 0 to JT along which

/ m cos99 > 0 except at the end points.

Theorem 6.3.3. Suppose h > / o > 0 and ho + hr >: i for each r and

suppose that the gr satisfy the following conditions for some L:

(i) gr C^-^^^-(GA) with hf,(V^+Kgr,GA) <L, for every A.

(LRf +K+f'-K 0<.t h-{ hr, \X\^2R, y > 0,

(ii) I V^gr(X) I < \LRf^^K+f'-t(XJ2R)e-i[\ + log|Z/2i?|], 0 < ^ < ^,

[LRf +K+f'-*{Xl2R)6-t Q<t h + hr

where Q = ho -\- hr — v — 1 and \X\ ^ 2 R, y > 0 in the last two lines of

(ii). Suppose also that u is defined by

u^x, y)=fZ ^C.Ll^(x, y; S)D-gr{S) d^, qr =~-ho + h - \a r lai=ar

(6.3.29) Ll^{x^ y; I) == Llr{x - | , y) -^D^^Li^(~l o)|-f,|/5|<T

T ^ l ^ l + tjc — hr — V = tjc + ho — V — 1.

6.3- Estimates near the boundary; coerciveness 2 3 3

Then u^ ^ O^^^ ( G A ) for every A and

(6.3.30) 2'\\\^'\rt2M<cL.k

If the gr satisfy (ii) only for 0 <t <^ h + hr — \, then U^Ô^+^-'^{GA)for any A and (6.3.30) holds with h replaced by h — 1. In either case u is

a solution of the boundary value problem (6.3.7) with gr replaced by gr — gf

where gf is the expansion of gr about 0 out to and including the terms of

degree Q. Of course if Q <,0, the conditions for 0 < ^ < ^ are to be omitted.

Proof. We may extend the L^^{x,y) to be of class C^ on the whole

of Rvî preserving the essential homogeneity and the analyticity along

y = 0, k; 9 0. From Lemma 6.3.2 we conclude that

rc-|xK*+;io-»'-iî.|l|-i if \B\<:T, |x|<^|f | , A<\

Thus we see that u^ ^ C"^ (analytic in fact) for y > 0 and we ma y differen

tiate under the integral sign as often as desired to conclude that ^ is a

solution of the equations (6.3.1) with L;^; replaced by LQÂ; and t h e / j

being possible polynomials of degrees <, ho — Sj — v — 1; the convergence

of the integral (6.3.29) at CXD is guaranteed by the fact that \oc\ = r + v.

Next, we note that any derivative of the form D^^""^^ D'^^^ u^

{h ~ ho + \ oi the :\;-derivatives can be shifted onto gr) is of the form

(6.3.31) jCD^r{x~^,y)y[^)d^, y = D'gr, \d\=h + hr,

awhere F is positively homogeneous of degree —v. Since DxFipc — | , y)

is absolutely integrable over a with value 0 (if y > 0), we ma y replace

y ( ) by y (f) — r [x] in (6.3.31). Thus we find that

(6.3.32) < CLJ[\X - f 12 + y2]-(»'+l)/2 . I f _ ;t:|^^| < CL y^^"!.

aBy differentiating the j — th equation in (6.3.1) (as modified above)

ho — Sj times with respect to y and using the ellipticity, we see that we

can solve for the D* *+*« ^^ in terms of the other derivatives D^^'^û^.

By performing suitable differentiations we obtain the derivatives

j)h-h^+iDh+ho^k in terms of the i)^-'^''+2 2)|+/io-iÂ:^ and so on. Thus we

obtain a bound (6.3.32) for \/h+^+û^. From Theorem 2.6.6 it then

follows that

(6.3.33) h^{\/^k+û^, GA) < CL for all A,

We now wish to show that w is a solution of the boundary problem

(6.3.7) except for the polynomials ^f. Let (p^C'^{Ri), (p{t) = 1 for ^ < 1,

(p[t) = 0 for ^ > 2, 99' ( ) < 0 for 1 < ^ < 2. If, for large n, we define

23 4 R egularity theorems for general boundary value problems

w e see t h a t e a c h {gnr} and its de r iva t ive s of orde r <^h -\- hr are equ i -

con t inuous , conve rge un i fo rmly on any bal l to thos e of gr, and the gnr

sa t is fy condi t ions (ii), at l e a s t w i th the log t e r m a d d e d . T h u s the corre

s p o n d i n g u^ and all the i r de r iva t ive s of orde r <.t]c + h converge un i

fo rmly on any GA to u^ and its d e r i v a t i v e s . T h e n , for each fl, Hfi — Id'On —

~ Un w h e r e uon is defined by (6.3.29) with L^'^{x,y',^) rep laced by

L\^ [x — I , y) and Un is the Maclau r in expans ion of UQn out to t e r m s of

degree r. It follows from Theorem 6.3.2 that U{)n satisfies the h o m o g e n e o u s

equations (6 .3 .7). Since LQJJCU^ is j u s t the co r re s pond ing Mac lau r in

expans ion of LQJJCU^^, it fo l lows tha t u^ also satisfies (6.3.7). Hence u

satisfies (6.3.7).

L e t us define

u^^x, y) ==12 2C.L%^/{x, y; i)D-gr{^)d^r i a l =Sr

. V^.

For each n, we find by integrating by parts , etc., tha t

^ * f c = C 7 * ^ - F * ^ <^^\-y\. Alu-^^ = uK etc..

( • • " ^ Vt^{x^ y) = fZL%''(^ - I y)gnr{S)dâ r

w h e r e V^ and F * ^ are the Maclau r in expans ions of U^ and U^ ^ out to

t e r m s of degree tjc + ho — v — \ and tjc + ho — v — \ + 2P, respec

t ive ly , and U^ has the s ame fo rm ula w i th L^^^ rep laced by L^^. By us ing

Theorem 6 .3 .2(b) wi th P la rge , we f ind tha t

BorlcUt^X, 0) =fMp{x - i)gnr(^)dâ

Bor1cUl[x,0) =gnr{x).

Since F^ is a p o l y n o m i a l of degree ^ t + ho — v— \, BorkVl^ is a

p o l y n o m i a l of degree <,ho + hr — v ~ 1 = q.^ — v. H e n c e

(6.3.35) BorJcUi(x, 0)=gnr{x) - ^ ~DPgnr{0).

Since V*^^(0 , 0) = 0 for 0 <,t ^tjc-{-ho--v — \, the b o u n d

(6.3.30) will follow from (6.3.33) and b o u n d s w h i c h we now es tab l ish

for \7û^(x, y) w i t h t = tjc + ho — v. From (6.3.29) it fo l lows tha t

V « ^ ^ ( ^ , y ) = / 2 ' ZCccV'Ll^{x-ly)D-gr(^)d^, t = h + ho ~ v,

6.3- Estimates near t h e boundary; coerciveness 2 3 5

Using the bounds ( i i ) fo r D'^gr {\<x\ = ho + hr — 1, V*L\^ pos. horn

deg ree — 1) we ob ta in

I V * ^ ^ ( ^ ) | <.JC[\X~ f |2 + y 2 ] - l / 2 . i : . i ^ A - V / ^ + l ^ f +

R p - ^ aR

T he second resu l t on d i f fe ren t iab ih ty is p ro ve d s im ila r ly .

Corollary. Suppose A > Ao > 0, ho -{- hr^ i for each r, and suppose

that gr^Ct't^~^{(y 2R)' Suppose also that u is given by (6.3.29). Theî

(6.3.30) holds with h replaced hy h— \. If gr^Cl-l^'{a2R), then (6.3.30)

holds. In either case, u is a solution of the boundary value problem {6.} J)with gr replaced by gr — g* where g* is the poly nomial in (6.3.35)-

Proof. ¥oT u = U —V w h e r e U = AÛ*, U* is giv en b y (6.3.34)

a n d V^ i s th e Tay lor ex pans io n of U^ out to te rms of degree tjc -\- ho — v —

— 1.

We now deve lop a represen ta t ion of u^ in which the h ighes t o rder

d e r i v a t i v e s D^k^^ u^ are expressed as in tegr a ls of C a ldero n-Z ygm und

t y p e .

Theorem 6 .3 .4 . Suppose / > AQ > 0, q > i, and ho + hy ^ \ for

each r and suppose the gr satisfy the following conditions:(i) gr $ H^^h { G A ) with II V ^"^^r gr ilg ^ < -^ for every A ;

( i i ) 1 k r r 3 ' i < ^ ;(iii) if Q = ho -\- hr — V — \ > 0, then V ^ gr is continuous on any G A

with 0 < ^ < ^; and

(iv) for \X\ > '^R we have

'Rh+h,-{v+i)/q~n\^X\l2R)Q-t{\+log\Xl2R^^, if 0<t<ZQ

^Rh+h,-iv+l)fq--t^^X\l2R)6-t, if Q<t<^h + hr.

Suppose, fur ther that u is given by (6.3.29). Then u^^ H ^ ^ ' ^ ^ { G A ) for any

A and

Z 'W^^falR^^^' HW '^^^'^^^^tA^L for every A.

Proof. We begin by def in ing

r ^ ^ ^ 6 ^ ^ « " ( ^ ' ^ ) ^ ^ ^ ^ « + ^ ( ^ ' ^ ) (^, >') 4= (0, 0 ) , y > 0

K^-^'^^) Nl^[x,y) = ~DêLlU x,y) \e\ = \ {e=\,.,,,v)

and whe re we ex tend L^^ to be long to C^ on Rvjî — {O, 0} an d to be

essen tia l ly hom oge neo us of degree fe + ^0 — v. We then def ine

u^{x,y)=U ^[x,y)— V^[x,y ), vi^{x,y ) = Tu^{x,y), [qr =-ho + hr — \)

UH x, y ) =[2 ICJ 2 ^ M | - ( X - S)Dr^gr{l f]) +R^+i *• ' ° ' ' " ^ ' ' l i ^ i = i

+ mj{X - S)DID^gr{l rj)\ di drj, T^ D^^D l

(6.3.37) \^\==h-ho, t = tjc + ho-\.

|V*gr(X)|<{^^

2 3 6 R e g u l a r i ty t h e o r e m s for g en era l b o u n d ary v a l u e p ro b l ems

t/** being the Taylor expansion of U^ out to terms of degree fe + ô —-V - 1.

By using moUifiers and the m ethod s of the proof of Theorem 6.3.3. we

may approximate to gr by functions ^Cc{R^+-^. For such functions we

may integrate by parts in (6.3.37) if fe + ^0 > 'I and conclude thatuk(x^ y) = u^(x, y), since

Ni'(x, y) = -Ll'{x, y), y > 0, (x, y) 4= (0, O)

MZ'ix, y) + Nl%{x, y)=0, {x , y) =# (0, 0).

In case A; + ^0 = 1, we may obtain the same result by first removing a

small sphere B {x,y\ Q) from GR, integrating by parts, letting ^ 0, and

using the result

j [x'M)J'[x, y) + yNl'{x, y) ]^5 = 0;6^(0.1)

since this integral depends only on the values of the L^^^^ for {x, y ) near

dB[0, 1), its vanishing is seen to follow from (6.3.36) by altering the U s

near (0, 0) so that they ^ C^ [B{0, 1)]. Then v^ has the form

v^x,y)=f2 ZC.S Z D'j, M ^ ^ {X - S) Z )f ^+^ gr{S) +jl+ r | a | = 7 i o + 7 i r - l l l e | = l

4 D^^N^r^x - S)Dri>D^gr{S)j dS

which is of the form 2.6.1. The results concerning the derivativesD^-hoj)t^+ihuJ^ then follow from the Theorems of §§2.6 and 2.7. The

same results for the other derivatives of order h + h are found by dif

ferentiating the equations (6.3.7) as in the proof of Theorem 6.3.3.From (6.3.37) we see tha t u^ (= u^) is a sum of terms of the form

(D{X) =IM(X, E)D^gr{S)dE, 1/5 I = >o + Ar

M {X, E) = M{X - E) -^^DyM{-S), Q = tjc + ho-v-\y

and M is essentially homogeneous of degree h + h^ — v — 1. By the

method of proof of Theorem 6.2.2 we conclude that (recall that v has

been replaced by r + 1)

(6.3.38) j\M{X, E)\dX<CR h-^fô, j\M{X, E)\dE <CRh+ô.

Letting. CO = 0)1 + 0)2 where coi is the integral over GZR, we obtain

(6.3.39) I o)i[X) |ff < [CRh-^HY-^ f\^(X, E) I • \DPgr{E) l^dEG3 R

using the H O L D E R inequality. From (ii), we conclude that

(6.3.40) IID^gr IIS < CR^-h'Wgr t%^ < CLR^-fô.

6.3. Estimates near the boundary; coerciveness 237

By integrating (6.3.39) and using (6.3-40), we obtain

Using Lemma 6.3.2 and (iv), we see that if | ^ | < 2 i?,

I^v+i—Gzn

Thus, also

||C02|».2B^CL7?«.+*.

Corollary. Suppose A > o 0, Ao + ^r 1 /<5 <3cA ^, ^'^d suppose

that gr^H^'^ 'ifi^R) and has support in G^RÔ^R. Suppose also that u is

given hy (6.3.29). Then U^^H^^^^[GA) for any A and u satisfies the con

clusions of the theorem with L replaced hy 'Igr ' ^R- Moreover u is a

r

solution of the boundary value problem (6.3.7) with gr replaced by gr — g*

where gf is the polynomial in (6.3.35).

Theorem 6.3.5: Suppose A > ô > 0, ô + ^r > '1 for each r, u^

//<*+7ii Q " jgy ^^Q}I ^ ^^^ ^(IQ}I J ^ ^ ^(iQJi ^k^C"^ jgy | ^ | > i Q and y > 0

for some i?o > 0, and

(6.3.41) LojJcU^{x,y)=ft> y>0, BorkU^=gr 0^ y = 0,

(6.3.42) \Dlu^(X) I <C^(1 + I Z D r ô-^-i-^^+i/^

^ = 0 , 1 , 2 , . . . , \X\^Ro, y > 0 ,

where ff and gf are polynomials of degree <, ho — Sj — v — 1 and

ho -\- hr — V — 1, respectively. Then u^ is a polynomial of degree < fe +

+ ho —V — \.

Proof. Let 99 be a mollifier in the A;-variables and let w^ {x, y) be the

mollified functions:

^0(^» y) = J fPei^ ~ l )^^(f, y)d^ = J (p^{x — S)u^{L y)d^

y

= I f <(^ - f)<(f^ 7])didr].R B{X,Q)

It is easy to see tha t all derivatives of UQ of order <,t h — 1 are con

tinuous in [x, y) and C^ in ji: for 3/ > 0 and the u^ clearly satisfy equations

like (6.3.41) ani(6.3.12)if^ > 0 (the polynomials are altered but are of the

same degrees). By repeatedly using the equation (6.3.41) and its deriva

tives, we conclude that each u^ ^ C°° (R '+i) if ^ > 0. Since u^ converges

uniformly with all derivatives to w if \X\ <. Ro and y < 0, it is suffi

cient to prove the theorem for u ^ C"^ {Ri'+-^).

Let v^ — Dl u^ where t > max( ;fc + ho, ho — Sj, ho + hr). Then v^

satisfies (6.3.41) with/^^ and ^* ^ 0 and, from (6.3.42) we conclude that

\vHX)\<C{\ + \X\)-'"^'^ X^RUi-

238 Regularity theorems for general boundary value problems

N o w , if we let v^ (A, y) be the Four ie r t r ans fo rm of % as defined in (6.3.8)

a n d i n t r o d u c e F^(A, ^) by (6.3.10), i.e. V^{X,u) = {imykV^{X,ul\u\),

w e see t h a t | F* (A, w) | < C | A | *fc for all (A, u) and satisfies

Lojjc(a,-iDu)V^{ku) = 0, B^rkio,-iDu)V^[X,0) =0, r=\,.,.,m.Since V^ is b o u n d e d for w > 0 and A fixed, it m u s t be exponen t ia l ly

decay ing and hence ^ 0 by T h e o r e m 6.3 .1 . T h u s all th e Di u^^ 0 and

we conc lude f rom th is and (6.3-42) that

uJc{x, y) =Z4{y)^^> pic<.ric=^h + hQ — v - \.

B y a p p l y i n g the o p e r a t o r s D ^ L Q for T > TA ; — 2 W we see t h a t e a c h u^

satisfies Z)^ L Q U^ { % , y) = 0. Of course , if pjc < 0, ^^ ^ 0. O t h er w i se ,

s ince DILQU^ = = 0, it follows that Dl+^^ c^^(y) = 0 iov \^\ = pjc, th i sbe ing the coefficient of x^ in D^ LQ U^ { X , y). Thus, us ing (6.3 .42), we con

c l u d e t h a t the c^ {y ) are p o l y n o m i a l s of degree < TA ; ~- pjc- Next , s uppos e

t h a t 0 ^ p < pjc and t h a t all th e c^ (y) with \P\ '>p are po lynomia l s of

degree zjc — \P\. Then, s ince DlLoDl^-û^{x, y) = 0 and Dl^~û^(x, y)

con ta in s no t e r m s w i t h \p\ > _/) we see as a b o v e t h a t Dl'^'^^'^^'^~''^c^ (y)

= 0 ii \p\ = p. T h u s s u c h c^ sere po lynomia l s wh ich , by (6.3-42) must be

of degree ^r^ — p- The fac t tha t each u^ is a p o l y n o m i a l of degree

< T j c now follows by induc t ion .

Definit ion 6.3 .3 . For > 0 we define the spaces ' ^ H ^ [ G R ) , *C^{Gji),

* * H ^ { G R ) , and **C^(Gi?) jus t as they were def ined in Defin i t ion 6.2.1

w i t h BR rep laced by GE\ i f / € *^g or *C^ we requ i re V^fj to van is h on

^R bu t not necessar i ly a long OR, t — 0,. . .,h — SJ ~ 1.

Definition 6.3.4. li hoÔ, h "> ho — 1, and ho + hr '> 1 for eve ry r,

we define the space *Q(o ' i ? ) to cons is t of v e c t o r s g w i t h grKiC^^%^{GR)

w i t h n o r m* ' 1 1 1 . o r I I I T i V I I I a \\\l^+'hr

Ills |!Ui2 — 2 J l l lferlllÊ •

r

If / > ho, we define the space * 'H^ [ O R ) to consis t of all vec to rs g such

t h a t gr[x) = gr{x, 0) w h e r e gr^H^'^^'{GR) and V^gr = 0 a long 2!R for

0 < ^ < ^ + A,-— 1 w i t h n o r m* 'II ^ IIJK = inf 2 " 'II gr U \ gr {X , 0) = gr (x).

L e m m a 6.3.3. ( L I O N S ) There are bounded extension operators T from

*^J(Gij) into ^H^{BR) and from *C^^{GR) into ""CUBR).

Proof. Let t — max h — Sj, We may define xf=F, w h e r e F{x,y)

= f{x, y) if [x, y) ^ GR, F(x,y)=Oiiy^ 0, (x, y) i GR,t+i

F{x, y) =2!Qp(^> —^y)> y < 0

w h e r e the C* are unique ly def ined by the cond i t ionst+i

2 ; ( - s ) « ^ Q - 1, u = ,0 ..., t.

6.3- E s t i ma t es n ea r the boundary ; coerc iveness 239

Definit ion 6.3 .5 . Suppose h^ ho>0 and ho + K \ for eve ry r.

F o r / i n * C J ( G ^ ) and ^ in * 'C^(a i , ) or f o r / i n *H^,{GR) a n d ^ ^ * ' ^ J M

we define

PR{Lg)=Un+VR-WR = PiRig) + P2R(f) +P3R{f)

w h e r e UR is defined by (6.3.29) in t e r m s of g,

V%=Ll'(Fi~Ff), Fi{X)=JK{X-S)fi{E)dS, f = xf

(6-3.43) ^ x « ^ "

F^{X) =2ji\^^^'^^^' g = 2m + ho-si-v-i,

T be ing the ex tens ion ope ra to r of L e m m a 6 . 3 . 3 , and WR is defined by

(6 .3 .29) with g(x) rep laced by y(x, 0) w h e r e

yr{X) = BorJcV^^(X)^yro{X) -y:^{X), yf {X) =^ B^ricLl'Ff {X),

Theorem 6 .3 .6 . PR is a hounded operator from *H^(GR) X'^'HÎOR) to

**H^{GR) and from '^C^{GR) x^'C^{aR) to **C^(Gi?) with hound in

dependent of R. If u = PR (/, g), then

(6.3.44) Lojjc u^ = fj — / f in i?++i and B^rk u^ = gr — gf on a,

ff and gf being Maclaurin expansions of fj and gr out to the terms of degree

ho — Sj — V — \ and ho -\- hr — v — \ if these are > 0, heing 0 otherwise.

The second result holds if h = ho — 1 if ho — 1 > 0.Proof. F r o m the two corollaries it follows that PIR is bounded f rom

*'H^{aR) to **H^(GR) and from *'C|^(o-i?) to ''*CI{GR) and t h a t

UR = Pmig) is a so lu t ion of (6.3.44) w i t h / ; = / f = 0. Now V% coin

c ides wi th the func t ion U^ in (6.2.20). If f^*C^{GR), the n o r m of / in

'^C^'ÎGR) < C ' R^~K t i m e s its n o r m in *C^(Gi2). We see by i n t e g r a t i n g

b y p a r t s ho — si t i m e s t h a t

Fi{X) = f 2C.K4X - S)D^fi[S)d3

K4X)=DfK,^,(X)

(see (6.2.11)) so t h a t Fi ^CI'^+^'''{BA) for any A. T h u s , we see t h a t

yr^C^'^^^(BA) for any A and t h a t yô is a sum of t e r m s of the form

lr{X~S)f{S)dE, f = D'Ji, \a\^ho-si.

/ € Q " o ' ^° ( ^2 i ? ) , r ess. hom. degree ho + hr — v — 1.

W e see also from (6.3.43) that y* is the Maclau r in expans ion of yro outt o and inc lud ing the t e r m s in X*, t — ho + hr — v — 1. Accord ing ly the

Maclau r in expans ion of yr out to t e r m s of degree t van is hes . F rom th i s

r e p r e s e n t a t i o n it is eas i ly seen tha t the yr satis fy the cond i t ions on the gr

in Theorem 6 .3 .3- Thus in th is case P2R and PSR are seen to be b o u n d e d

ope ra to rs f rom *C^(G2R) to **C^(G2R) and f rom Th eor em 6 .3 .3 , it

fo l lows tha t VR = VR — WR satisfies (6.3.44) with^^. = gf = 0. The case

A — Ao — 1 ^ 0 follows b y in sp ect ion in t h e *CjJ; case.

In c a s e / ^ * iJj (G2ij ) , i t fo llows f rom T heo rem 6 .2 .2 an d i t s p roof

t h a t t h e y r satisfy condition (i) for the gr in Theorem 6 .3 .4 . Condi t ion

(iii) follows from condition (ii) and the relevant S O B O L E V lemmas (§ 3 .5).

For a lmos t a l l X in R^+^, we see tha t \7^y{X) is a sum of terms of the

form

co{X)=JM {X,E)ME)dE, MS)= D-M S), \oc\=ko-si,

M(X, E) ^ M(X - E) - V ^D^M(~E)

w h e r e M is essentia l ly homogeneous of degree Q — t a n d Q = ho -{- hr —

— V — \ . From th is i t fo l lows f rom Lemma 6 .3 .2 tha t

j\M[X, E) I dX < CRH+^-i, j\M[X,E)\dE<^ CRH^K'K

{ii ho + hr ~ t> 0).

Using the method of p roof of Theorem 6 .2 .2 and the fac ts tha t

(6.3 .46) ll/o i« ^ CR'^-'^oZ'W fi nW ^ CR> ^-H *\\f\l^nI

we obtain the result ( i i ) for the y r by s e t t ing t = 0, L =C* \\f\\^^2Eabov e and us ing Le m m a 6 .2.2. Now , if ^ — ^ < 0 , | Z | '> '}R,\E \ < 2R,

we s ee tha t

(6.3.47) \M{X, E) I <C \X\^-t, f \fo{E)\dE < CRi^^^ni-ifQ) ||/ol|o ,^B2 R

so th a t th e bou nd (iv) hold s in this case . If ^ — / > 0 , i t fol lows from th e

es s en t ia l homogene i ty tha t

\M(X,E)\ <C\X\^-t[i+log\Xl2R\] {\X\^}R,\E\ <2R)from which the bound (i i) again follows as in (6 .3 .47). The remaining

results follow easily from (6.3.45), (6.3.46), and (6.3.47)-

Definition 6.3.6. Suppose ho is th e smalles t inte ge r suc h th a t Ao > 0

a n d Ao + ^r > 1 for ev er y r. Ii h ^ ho, we s ay tha t the ope ra to r s Brjc

satis fy the A-condit ions in B(xo,A) if and only if the coefficients all

^ C^+ '-' [B {xo, A)] ; th ey sa t isfy the h — /u condi t ions there i f and only

if the coefficients all ^C^'^ '^ [B (xo, ^ ) ] ; w e a llo w t h e h — /u condit ions if

we mere ly hav e A > AQ — 1 p r o v i d e d t h a t A > 0 .W e can now imi ta te the deve lopmen ts a t the end o f the p reced ing

section. We firs t prove the following theorem like Theorem 6.2.5 .

Theorem 6.3.7. Suppose A > / o > 0 a n d Ao + ^r > 1 for every r.

Suppose G is of class C^'^'^"^, 0 = m ax ^^ , and suppose the operators

L = {Ljjc} and B = {Brjc} satisfy the h-conditions on a domain F Z) G.

6.3 . Estimates near t h e boundary; coerciveness 2 4 1

Suppose that u^ ^ H*^^^^ (G), fj^Hl~^^{G), and gr = gr on dG, where

gr^Hl+^(G) and

(6.3.48) p>:q , fj = Ljjc U^, gr = BrJc U^.

Then u^^W^'^^{G). If G is of class O^'^^, L and B satisfy the h — ix con

ditions on a domain J ' D G , and u^ ^H*^^^° (G ), fj ^ C^"^^ (G), and

gr^C^+^^{dG), then ^ ^ ^ Q + ^ / / G, fj, gr, L, and B^C"^, so does u^.

If h^ho, fj^Cl+'^{G), gr^Cl+^r^sG ), / ^ o > 1 , w ^ e Q + ' ^ « - i ( G ) , then

Proof. T he proof is s im ilar to th a t of Th eo rem 6.2.5 from w hich th e

interior results follow. But now, a lso, with each A;O on ^ G is associated an

i? > 0 an d a m ap pi ng of class C*{'^^~'^ (in the first case) of a neighbor

hood of XQ o n t o G2R', R can be t aken s o s ma l l tha t the t r ans fo rma t ion TRdefined in (6 .3 .4) ( in terms of the coordinates on G2R) ha s bo un d < 1/2

in a g iven des i red space . Supp ose C $ C ^ (G) an d has supp or t in th e ne igh

borhood of Xo co r re s pond ing to GR U (TR and le t U be the t rans form of

C u . Then, as in (6 .2 .27), we see that

LjjcU^ = Cfj + Mjjcil^=ff

BrJcUf^ = Cgr + NrJcil -=g?

where the t i lda s deno te the t r ans fo rmed func t ions and the ope ra to rs

Mjk a n d Njjc involve o n ly der i va t iv es of lower order . S ince these a re

smoother than those of h ighes t o rder , the proof proceeds as before . The

bas ic fac t i s tha t HR i s aga in a po lynomia l on account o f Theorem 6 .3-5 .

Next we prove a loca l theorem l ike Theorem 6 .2 .6 .

Theorem 6 .3 .8 . Suppose B and L satisfy the h-conditions {h > ho) on

B A . Then there are constants R2'> 0 and C2 depending only on v, h, q, E,

and E' such that

**\\ulln<C2[*\\Lutj, + *'\\Buf^j, + **'\\u\U, 0<R<R2,

ulc^Hf+^iGM), * *'i M i? B = 2 ' ^ - « * - ' ' - e l | « * ! | ? B , e = (v+l){q~ \)lqk

whenever u also has support in G R U OR. Also there exist constants i?3 > 0

and C3, depending only on v, h, /u , (0 <C jix <. '^) and the quantities above

such that if L and B satisfy the h — fi conditions on BA, then

0 0.

Proof. The proof i s exac t ly ana logous to tha t o f Theorem 6 .2 .6 . The

differences involve only PR: we le t u = UR -\- HR,

UR = PR [LO U, BO U), TR(P= —PR [{LR — Lo) (p, (BR — Bo) 99]

Morrey, Multiple Integrals 1 6

a n d t h e n w e h a v e

UR -TRUR = PR{f,g) - PR [{LR - Lo) HR, {BR - Bo) HR,]

f = LRU^ Lu, g = BR U^ B U.

We conc lude th is sec t ion wi th the fo l lowing wel l -known ' ' coerc ive-n e s s " i n e q u a l i t y .

Theorem 6 .3 .9 . Suppose h '> ho ^ 0, ho -{- hr ^ \ for every r. Suppose

G is of class O^'^^"'^, to = max^;^, and suppose L and B satisfy the h-con-

ditions on a domain F Z ) G. Then

1 11 u^ fto' <C \Z mia'^ +1 \\gr tâ'^ +2\\^' WIG] >(6.3.49) fc L ?• ^ k \

ufc^Hl^+^(G), gr^m. +^r^G), gr = gr on dG,

where C depends only on v, h, q, G, E and E', and fj and gr are given by(6.3.48). In case G is of class C^^'^^ and L and B satisfy the h — fi condi

tions on r, then a similar inequality holds for the various ||| \j^-norms. This

last result holds if h = ho — 1 if ho — 1 > 0.

Proof. We prove the f i rs t s ta tement , the proof of the second is s imi la r .

G can be covered by a f ini te number of neighborhoods 9^p each of which

is e i ther a sphere B (P, R) w i t h B(P,2R) C.G or is the image of GR U OR

under a mapping r o f c lass Cî^^"^ [to == max^^:) w hi ch m a p s BA on to a

ne ighborho od D9^p w i th A ~>2R. Since the t r ans fo rm ed ope ra to rs

under such mappings s t i l l sa t is fy the /^ -condi t ions , i t i s c lea r tha t we may

choose each R < t h e n u m b e r R^ in Theorems 6.2.6 or 6 .3 .8 . And there is

a p ar ti ti o n of u n it y f i, . . ., Cs of class C\^^~'^ (G) each Cs hav ing sup po r t

in some one such ne ighborhood.

No w suppose ther e is no such cons tan t . Th en H a sequence {un} s uch

that the left s ide of (6.3-49) is unity for each n, u\ —7 u^ in H*^'^^{G) a n d

the bracke t on the r igh t in (6 .3 .49) ->0 . Then we mus t have u = 0 a n d

h en ce < - > 0 in H*^+^~^{G). Now le t Uns = CsUn. T he n < s - > 0 in

Hf'^^~^{G) a n d Uns satisfies

(6.3.50) Ljjc Uns = ^sfnj + ^jks '^n, ^rk ^ns = Cs gnr + ^rks ^w

where the ope ra to rs Mjjcs a n d Nrjc s involve on ly der iva t ives of lower order .

Clearly , th en , th e r igh t s ides of (6.3.50) - > 0 s tro ng ly in th e resp ectiv e

spaces . Thus , from Theorems 6.2.6 and 6.3 .8 , i t fol lows that each u^g -> 0

s trongly in H^^'^^{G). But s ince u^ =2'^ts> ^^ i^ con t rad ic t s ou r a s s um p

tio n t h a t th e left s id e of (6.3-49) is 1 for each n.

6 . 4 . W e a k s o l u t i o n s

In this section, we consider solutions of equations of the form

( 6 . 4 . 1 ) G ? = 1 lal^-î fc=l \^\<^QiTc '

dx = 0 ,

v^^ CîG), Qjjc = tjc + Sj — mj

6.4. W eak solutions 2 4 3

where we s ha l l a s s ume tha t the Sj, tjc, a n d MJ satis fy

m in(s j + fe) > 1, inm(tjc + Sj — MJ) > 0, minw,- > 0 ,(6.4.2)

maxs^- = 0 ,and where we a lso assume tha t there ex is ts an in teger ho which may he

negative s u c h t h a t

(6.4.3) min(/A; + ô) > 0 , min(Ao — ^j + %•) > 0 .

I t fo l lows tha tN

(6.4.4) 2m = 2^ (sj + tj) >N, mj <sj + tj < 2m.9 = 1

W e no te tha t i f the u^, a^^, a n d t h e f^ are suffic iently differentiable ,

t h e n t h e u^ satis fy the equations (6 .1 ,1) where

LjkU^'= 2 2 (-iy^^^D -aîDûK fj= 2 {-\T^D-fi.

(6.4.5)

For each XQ we def ine the opera tors LO;;A: h y

(6.4.6) Loj'jcu^ =- I 2 ' ( - iy^âfi(xo)D-^û^\oc\=mj \^\=Q}jc

and then def ine the opera tors LQ a n d L^^ as in §§ 6.1 — 6 .3 . W e a s s u m e t h a t

the opera tor Lo is e l l ip t ic and sha l l denote the co l lec t ion of numbers Sj,

tjcy mj, ho, V, and a bound for the coefficients in (6.4.6) and for Lo(l)~^ for

A real wi th |A| = 1 b y £ . I n th e case of vari ab le coefficients, E' shall

s tand for the moduli of continuity of the coeffic ients in (6 .4 .6) and

bounds for the i r der iva t ives and for the o ther af^ and the i r de r iva t ive s a s

r e q u i r e d .

W e no t ice tha t the equa t ions

f[v,oc(a'='û,^ + hû + / « ) + v{c^ ^,/S + du + / ) ] dx = 0G

f [v,ocp(a'='û — f^^) + v,oc(b°'u ~f) + v{cu —f)qdx = 0G

are special cases of (6.4.1) (here oc a n d ^ are s ingle indices) .

As in § 6 .2 , we begin b y c onsid ering solutio ns u hav ing s uppor t in a

smal l ba l l BR and define the coeffic ients a|/ . b y

'^lU^) - <ii. = <p{R-^\A) [«rt^W - ag{x^)-\

' 0 , o th erw ise

w h e r e 99 is defined in (6.2.25).

Definit ion 6.4 .1 . If ^ > ho, w e s a y t h a t the a's satisfy the h-conditions

on a set F if and only if

16*

244 Regularity theorems for general boundary value problems

(i) ii h =ho and sj =0, the aj'^ w i t h \a\ =nij and \ \ =Qjk are

c o n t i n u o u s ; and

(ii) ii h— Sj + \oc\ > 0 t h e n the a^^^ c^-sj+icci-i f^py^ o the rwis e the

af^ are b o u n d e d and m e a s u r a b l e .The as satisfy the h— ju-conditions on F if and only if the coefficients

af^ w i t h \a\ =mj and | ^ | =QJJC^CKF) at leas t , those for w h i c h

h — Sj + \oc\>0 belonging to C^-'^+^'^^ir), the r e m a i n i n g a/s be long

ing to C«(r) at leas t .

Definition 6.4.2. For h> ho, we define the spaces **H^(B2R) and

**C^{B2R) of vec to rs uj u s t as they were def ined in §§ 6.2 and 6.3 e x e p t

t h a t now h may be < 0. We define the space '^H^{B2R) to consis t of t h o s e

vec to rs / = {ff} s u c h t h a t

(^m^-sj+\oc\^B2R), if h-Sj-^\oc\>0,

\^Lq(B2R) , if h~ Sj+\oc\<0,f

4

W e def ine the space *C*(52i j ) to consist of t ho se v e c t o r s / s u c h t h a t

j^C%^>+\''i(E2R), if h-sj+\a\>0

\ € C%{E2R) , if h~sj+\cc\<0

7i-s;+|al>0 ' h-Si + \a.\<Q

Definition 6.4.3. If / > AQ and / C*H^{B2R) or *C^(^2 i j ) , we

define PhoR (/) = UR w h e r e

I | a | < : m i

(6.4.8) .

a n d F\'^ is the Maclau r in expans ion of -Ff out to t e r m s of degree af

w h e r e af = 2m — v— \ if Ao — s + | ^ | < 0 anderf =2m +ho ~

— si + $x\ ~ V — \ \i ho — si + \(x\ > 0.

Us ing the m e t h o d s and re s u l t s of § 6.2 and a p p r o x i m a t i o n s to t h e / ?

by s moo th func t ions we conc lude the fo l lowing theorem:

Theorem 6.4 ,1 . PJIQR is a bounded operator from '^H^(B2R) into

**/fJ(jB2ij) and from *C^(E2R) into '^'^C^(B2R) with bound independent

of R if q :> \, 0 <ju < 1, and h> ho. If UR = PhoR{f), then UR satis

fies (6.4-1) with G =B2R and a'^^ replaced byag;\ and / ? replaced byft "" fj^" ^^^^^ fj°^ ^ i^^ expansion of f out to terms of degree ho — s^ +

-\- \(x\ — V — \ {if this is > 0). Moreover if u has support in B2R and

satisfies the same equations with the f*'^ ^ 0, then u^ — U\ 0or is a

polynomial of degree < max(^A; +ho —- v— \ytjc + si — v— $. It is

understood that the ^g^^ are defined in (6.4.7)-

http://ytjc/

http://ytjc/

http://ytjc/

6.4. Weak solutions 245

The last statement follows since the mollified functions u^ satisfy the

same equations as does u with the/? replaced by their mollified functions

fi^; of course Q is assumed so small that UQ has support in B2R (we may

work in B2R+d with ^ < ^). ThenTkl

^0IK(x - i)LoimU^{i)di]= Ll'Loimf K{x - ^)i,^(^)d^

J ^ ! a \^mj

J>2R

{6.4.9)

since UQ has compact support. Using (6.4-9), integrating by parts and

letting Q -^0, we obtain

(6.4.10) u^(x) = Lf 2 ' (-i)'°''i^^i^r(^)

where Ff is defined in (6.4.8). The result follows by comparing (6.4.10) and

(6.4.8) and calculating the degrees of the polynomials Lg^(— 1)l'^'Z)°'Fj\

Remark. We could have defined PhoR(f) by defining Fl^" as the

expansion of Ff out to terms of degree 2m — v — 1. This would give a

definition of PR which is independent of ho and, moreover U% would

satisfy the equations without polynomials / j * . However, the presence

of theho

and the polynomialsJ

is not a serious drawback and theirpresence is necessary for the treatment of boundary values.

Definition 6.4.4, We define the coefficients UR as in (6.4-7) and

define TR on the spaces **i7j(J52i?) and **C^(B2R) by

(6.4.11) T^UR = UR = PnoR(fR), fU = - « . - < . ) D-<-

Remark. TR obviously depends also on ho.

Then if u has support in BR and satisfies (6.4.1) and if we define

UR = PhoRifit) where/R is defined in (6.4-11) with URreplaced by u, we

see from Theorem 6.4.1 that

(6.4.12) U=:UR + HR

where H\ is a polynomial of degree < max(fe -}- ho — v — 1,fe — r— 1)

and the equations (6.4.1) are equivalent to

(6.4.13) UR-TRUR = VR = PUORif) - PkoR[« - 0 • D^HR].

The remaining developments of § 6.2 can be repeated to yield the follow

ing results:

Theorem 6.4.2, If h'> ho and the a's satisfy the h-conditions on B^,then TR is a hounded operator on '^*H^(B2R) for each p '> \ and each

R •< Ajl with bound e(R, p) where e also depends on h, ho, E, and E' hut

8(R, p) -^0 as R -> 0. / / the a' s also satisfy the h — ju-conditions, then

TR is hounded on **C^(B2R) with 6{R, ju) of the form C{ILI, h, ho, E, E') x

X i , a = ju or \ according as h = ho or h ^ ho.

Theorem 6 .4 .3 . Suppose the a' s satisfy the h-conditions on G, u^ ^

H\^+h^[G) for some q> \, the f^^Hl-'^+^^^'iD) if h - sj + \oc\ > 0 and

fj $ Lj)(D) if h — Sj + \(x\ < 0 for each D G O G and some h > ho and

p ^ q, and u satisfies (6.4.1) on G. Then u^^ H^^^^{D) for each DC C G.

If, also, the a' s satisfy the h — ju conditio7 ts on G and theff^ C'*~^^+I' I(Z))

ifh — Sj + \oc\>0 andff^ C^^(D) ifh-sj + \(x\^ 0, then u^^ Cy^{D)

for each D C G G.

The proof of th i s is j u s t Hke t h a t of Theorem 6 .2 .5 . The e q u a t i o n s

replacing (6.2 .27) for Us are of the form

k l/S |<ei/tdx = 0, v^CriB^R)

(6.4.14) g!s=Csf^ + k%,w h e r e if |^| = mj the kf^ invo lve on ly de r iva t ive s of u^ of orde r <^;A:-

T h e k'jg are not u n i q u e l y d e t e r m i n e d but a definite set can be o b t a i n e d

by rep lac ing v^ in (6.4.1) by Cs '^^ and w r i t i n g

D- {Cs v^) = Cs D- v^ + [D - {Cs v^) - Cs D« v^

and then sh i f t ing the Cs fac tor to the b r a c k e t in (6.4.1) to o b t a i n

L fe1/51 J

k 181

T h u s the equa t ions (6 .4 .14) mus t be e q u i v a l e n t to

X [D O (Cs u^) - & DO u'''\ + 2 2 C^" (fs " ) - fs D- vi] X

X \22atuD^ui'-mdx = 0, v^Cr(B^R).

k ]•

Theorem 6 .4 .4 . Suppose the a' s satisfy the h-conditions on B^. Then

there are constants R2'> 0 and C2, depending only on ho, h, q, E, and E'

such that ifuis a solution 0/(6.4 .1) withu^^H*l+^(Bii), a n d / ^ "^H^iBji),

then

**\\ u lU ^ C2 [ * | / | | J B + **'|| u WU. * * ' i » nn = 2 R-''-"-' II « * liJj.,

k0<R<R2, Q=v(q- \)lq.

Also there exist constants C3 and R^, depending only onho, h,jbt {0 <C JLI < 1),

E, and E' such that if u is a solution 0 / ( 6 . 4 . 1 ) , u^ '^*C^{BR) and has

support in BR and if f^ *Cg(5ij) , then

**NU^ 111^ <r Cc r*lll f lll' - u **'!IU^ l'i|o„i **'l!Uy No — J?Q-V-/^ . **'i| /110\\\U\\\^R S=, ^21 \\\j\\\txR\^ |,||^|||li?J' .!i!^|||lR — ^ l^lll-K-

Remarks. It is c lea r tha t we may allow thefj w i t h h — Sj -\- \oc\ < 0

to be d is t r ibu t ions in some proper ly def ined space H^s^+l^l-^ s uch

6-4. Weak solutions 247

s paces have been de f ined and us ed ex tens ive^ in t h e s t u d y of differentia l

equations (see H O R M A N D E R [1], A. F R I E D M A N [2], L I O N S [2], L I O N S and

MAGENES, and m a n y o t h e r s ; see Chapter 1 ) . We could say tha t 99^ H~*

if and on ly if cp =^ A* (p w h e r e (p^H\\ no doub t s paces C~* could be

def ined s imila r ly . Moreover the coefficients a'jl may be allowed to be

s o m e w h a t m o r e g e n e r a l as in §§ 5-5 and 5-6.

W e now wis h to cons ider weak so lu t ions sa t is fy ing genera l boundary

cond i t ions in some weak sense and would l ike to obta in g loba l resu l ts

l ike those in § 6.3. We s ha l l cons ide r bounda ry cond i t ions of the form

(6.4.15) (Z I DyCnBrJcyU -gry]dS==0, C ^ C {G )

w h e r e the o p e r a t o r s Bricy are of orde r ^tjc — hr — pr and w h e r e , if {x^ y)

a r e b o u n d a r y c o o r d i n a t e s the equations (6 .4-15) reduce to

j Z IDlC^ix ) [Bric y{x , Da,, Dy ) u x, 0) - gry{x)] dx =: 0

(6.4.16) C'€ C-{G2R).

We assume that the integers pr and ho satisfy

(6 .4 .17) min^r > 0, m i n ^ o +hr -\- pr > 1

as well as (6.4.3)- ^ ^ assume that Sj, tjc and mj continue to satisfy (6.4.2).

We define Borkv = 0 if \y\ <pr) if I7I = pr, we define B^rkv as t h a to p e r a t o r of orde r tjc — hr — pr o b t a i n e d by rep lac ing the coefficients of

the p r inc ipa l pa r t oi Brjcy by the i r va lues at the orig in . Then we define

(6.4.18) BorJc = 2(-' ) 'DyBorJcy\y\ = 'Pr

We suppose that for each point on the part of dG in which w e are interested,

that the operator B^rk above and the operator LQ satisfy the root and comple

menting conditions 0/ § 6 . 1 .

R e m a r k . IfBrk{x, y\B>xy Dy) isany o p e r a t o r oforde r fe— hr w i t hsuffic iently differentiable coeffic ients of degree <tjc — hr — pr in Dy,

i t can be w r i t t e n in the form

Z{-^)^' '^DlBrjcy{x,y;D^,Dy)\y \^Vr

in wh ich the o p e r a t o r s Brky are of orde r <.t]c — hr — pr- This jus t if ies

the re s t r i c t ion of our a t t e n t i o n to b o u n d a r y c o n d i t i o n s of the form

(6.4.16).

Definition 6.4.5. We define the space *'^J(c72i?) [h > Ao, 5 > 1) tocons is t of thos e vec to rs gry for which there ex is ts a gry^H^Q^^^'^'^{B2R)

if / + A r + l r | > 1 and HIQ{B2R) if h+hr +\y\< \ s u c h t h a t

grvi^, 0) = gry(^) on a2R, and define the n o r m by

*'l|g||J,2«=inf/ Z 'll rrir.V"" + i: R' '' ''-"\\grr\\l,A.[h-i-hr+lyl'^l h+hr+\y\<l J

We define the space * 'C^(a2ij) to consis t of those gry^ Cg+'''-+'^l{a2i2) in

case h +hr + \y\ \ a n d gry^ Cl^[a2R) o the rwis e , w i th no rm

We define the space *'C^(a27?) inthe same way as *'C^(o=2i?) but re

p lac ing 1 b y 0 .

Definit ion 6.4 .6 . For^ in one of the spaces above, we define PhoR{0, g)

= UR, w h e r e

u^l,{X) =fL^r(^x~ ly)gry (f ) di {r n o t s u m m e d ) ,

Uy'^ i s the Maclaur in expans ion of ^g j ou t to te rms of degree h — hr — v,

a n d t h e L^^ are defined in (6.3.24).

Theorem 6 .4 .5 . PhoR is independent of ho and is ahounded operator

from *'H^(a2R) into *'^H^{G2R) and from either *'C^(a2i?) or *'C^(a2R)

into **C^(ff2/2) ^ith bound independent of R. Inany case UR is a solution

of the homogeneous equations (6.3.7) for y > 0 and satisfies the boundary

condit ions (6 .4 .16) with Brky replaced by B^rky-

Proof. Using the methods of p roof and the resu l ts o f Theorem 6 .3-3we conc lude tha t Pn^R is abounded ope ra to r a s s t a ted . Fo r y > 0, we

see f rom Theorem 6 .3 .2 tha t

( 6 . 4 . 1 9 ) i:Ujicuii{X) = o, y = i,...,ivfc

for each fixed r and 7 . Now LoJA; is an operator of order tjc + s ; and so the

degree of ^ LQJJC U*^'^ is ^ — Sj — hr — v. I f th is i s nega t ive ,k

(6.4.20) ILojjcu; '(X) = 0.k

If i t is posit ive , then tjc — hr — v'> tjc +Sj s o t h a t

k

vanishes l ike X^ w h e r e g = \ — Sj — hr — v. In this case (6.4.20) follows

on account of (6 .4 .19).

Now, s uppos e the gry^ C^{O2R)' Then f rom Theorem 6 .3 .3 and i t s

proof we conc lude tha t

(6.4.21) ?^° -<( - '«) = {Lw; I T=r.Also ^ Bosjc ^J?*" (X ) is of degree hs — hr — v. I f th is i s nega t ive ,

k

(6.4.22) lBosku;^'(X)=^0.

6.4. Weak solutions 2 4 9

If i t is posit ive we conclude (6.4 .22) us ing (6.4 .21) and the fact that

[ul'!^(X) — 14*!''^ (X)] vanishes a t the or ig in to the power 1 + hg — hr — v.

We wish now to define PR{f, 0) . B y v i r tue of Le m m a 6 .3 .3 , we m ay

as wel l assume tha t / i s a l ready ex tended to B2R and we then us e the

norms defined in Definit ion 6.4 .2 .

Def in it ion 6 .4 .7 . F o r / i n th e space *C^(G2i2) or *HI{G2R), h > ho,

we def ine (we omit the subscr ip t R)

F]'{X)=dl'(~\y'^D'^l{X) , (/, e not summed)

%t{X)=fK(X-S)fl(S)dS, f = rf

W^^^[X)=Z li-^r^Dlf ZCpLY{x,y;i)Dlgry{lO)dS

g%{X) = Borlcy V^^^{X),

qri = m3ix{h + hr + pr — "^, hr + pr + Si — \e\ — "i)

w h e r e i J** is the Maclaurin expansion of-F |^ out to the terms of degree

2m — l^l — r — 1 (if th is i s > 0 , F p * ^ 0 o therw ise) .Theorem 6 .4 .6 . PR(f, 0) is a bounded operator from *C^(G2R) into

**C^(G2i?) and from *H^{G2R) into **^J(<^2i?) ''^ith bound depending on

E and not R. If Y = PR(f, 0), then Y satisfies the equations (6.4.1) "i^^ith

a replaced by a^ and satisfies the boundary conditions (6,4.16) with Br icy

replaced by Borky ^ ^ ^ gry ^y 0.

Proof. By v i r tue o f Lemma 6 .3 .3 , we may a s we l l a s s ume tha t

/ € *C"J^( 2i?) or '^Hq{B2R)' I t is easy to see , us ing the carefully construc

ted de f in i t ions o f the no rms tha t the mapp ing v ia PR fro m / t o F is

bounded a s s t a ted . Le t u s de f ine Fo , F* , go, a n d ^ * b y

F = F o - F * , g^go~g*

ylcle^ _ Lg*J^J^*, gly' = Borky V^^ ''.

Then we see eas i ly by checking the degrees of the po lynomia ls tha t the

ve cto r F^^* satisf ies th e hom oge neo us eq ua tio ns (6 .3 .7). If , for som e

(l> £)>fi^ C7{B2R) i t fol lows that V^' satis f ies the equations (6 .3 .1) with

a rep laced by ao a n d

// ^ = (— 1 )i«I D'fl' d\ (1,8 n o t s u m m e d ) .

By v i r tue of our hypotheses (6 .4 .17) , we see tha t

h + hr-\- pr— \>ho + hr-\- pr— \>0

hr + pr + Si-\e\-\=(h + hr + Pr-i)-(h-Si+ \8\)

a n d qri > 0 in all cases. Also , in all cases, gj ^ be ha ve s at oo like a fun ctio n

which is essen t ia l ly homogeneous of degree hr + pr + si — \e\ — v ~ \

a n d gj.y* is a poly no m ial of degree < th is . Since th is < ^r; — r in all

cases , t h e p o l y n o m i a l g^^ may be neglec ted in the express ion for W ifdes ired. Since Lf[x, y\ | ) is , for a fixed | , of the form G\(X) — G^*(X),

wh ere Go satisf ies the hom oge neo us eq ua tio ns (6 .3 .7) and G J is a M aclau rin

ex pa ns io n of Go, it follows t h a t W sa t is f ies the homogeneous equa t ions

(6.3.7) for 3/ > 0 w her e i t is an al yt ic ; th e conv ergen ce a t 00 of th e in te

grals follows since L^ has degree — 1 in 111 a n d D^ ^J^ ha s d egr ee < — 1 in

| | | a t CO. A proof l ike th a t in th e f irs t par t of Th eore m 6 .3 .3 shows t h a t

t h e m a p p i n g f r o m / t o W v ia Pnif, 0) is bounded as s ta ted ; one sees tha t

^ r r ^ î'^'^îÂ) for every A with the expec ted bounds .

Now for some (Z, e) , le t u s a ss um e t h a t / j ^ C^{B2R). T h e n t h e

gly€ C'^{BA) for every A a n d W^' can be seen to ^ C'îW+i)- Let us

ap pr ox im ate to ^J.^ by gly^^ ^Ti^v+i) using the device in the proof of

Th eorem 6 .3 .3 . Th en the de r iva t ive s D^W}^ converge as des ired in

Rv+i t o \/^W^^ and we have (omi t t ing [I, e))

Wl{X) = Wi,(X) - Wl*{X), grn ^'L{-\r^m'gry nlyl==2)r

Pfô(Z) = / L * ' - ( ^ - | , y ) ^ , „ ( f . O ) ^ f ,

aa n d TF^^^* i s the Maclaur in expans ion of W^^^ out to the te rms of degree

tiz — V — \ -{- m a x (A, sz — | £|) =^ tjc + qri — hr — pr — v- B y t h e m e

thod of p roof of Theorem 6 .3 .3 , we conc lude tha t

BorJcWUlO)=grn(lO).

T h u s , by pass ing to the l imi t , we see tha t

Boric W^'^(^, 0) = ^[^(1, 0) - ^[^* (I , 0)

w h e r e gl^ * (X ) i s the Maclaur in expans ion of gl"" {X ) out to the te rms of

degree qri — pr — v. Since the po lynomia ls gly^ could be neglec ted in the

formula for W, we mu st ha v e ^J.^* ^ 0. T he res ul ts follow.

Fr om here on th e deve lopm ents para l le l those of § 6 .3 . Th e theo rem

corresponding to Theorem 6 .3 .5 is :

Theorem 6.4.7. Suppose u^ ^ HI^"'^{GA) for each A and each k, u^ ^ C

for \X\ <, some RQ and y > 0, and suppose u satisfies (6.4.1) with a

replaced hy a^ and f^ replaced hy ff^ and satisfies (6.4.16) with Brky replaced

by Borkv ^ ^ ^ Sry replaced hy gf^ where ff"- andgf^ are poly nomials of degrees< Ao — S; + 1^1 — V — 1 and ho -\- hr + pr — v — 1, respectively. Sup

pose also that u satisfies

\Dlu^(X)\ < Cp(1 + iXlY'^+^^-^-^-^+^f^ ^ = 0, 1, . . ., \X \ > Ro.

Then each u^ is a poly nomial of degree ^tjc + ho — v — 1.

6.5- The Dirichlet problem fo r strongly elliptic systems 2 5 1

Remark. Since if UR = PR(f,g), t h e n UR satisfies (6.4-1) with a repla

ced by UQ an d satisfies (6.4-16) w it h Brky rep laced hyBorjcy with no res idua l

p o l y n o m i a ls / * a n d g*, we nee d Th eo rem 6.4-7 only for th e cases / *

= g* = 0.

By para l le l ing the prev ious deve lopments we may conc lude the fo l low

ing re s u l t s :

Theorem 6 .4 .8 . Suppose ho is the smallest integer satisfy ing all the con

ditions of this section, suppose h > AQ, suppose the a' s and the coefficients

in the Br icy satisfy the h-conditions on a domain F z^G, suppose that for

each point XQ of G the operator LQ is properly elliptic and that LQ and the

operators Bork satisfy the root and complementing condition for each XQo n

dG, suppose G is of class C l'^^^~^{to = maxfe ) , and suppose u satisfies

(6.4-1) and (6.4-15) "i^^here f^^H^{G), Q = m a x ( 0 , h — Sj + \oc\),gry^Hl(G), T = : w a x ( l , h + hr+ \y \), and u^^ Hl^+^{G ). Then

k [j.oc r,v k J

where C depends only on h, q, G, E, and E'. If the a' s and the h' s satisfy

the h — ju-conditions on rz:)G, G is of class C*^^^, u^^ C*^'^^{G),

ff^ Cf,{G), andgry ^ Cl(G), r = m a x ( 0 , h + hr + \y]. Then

I I I I ^ ' \ m ' ^(^\I U\\\U + I \kry\ ila + I\\u'II?.fc L ? , a r.y k

where C depends only on h, ju, G, E, and E'.

If G , ff, gry , and the a' s and h' s satisfy the conditions in the first

statement for some h' >h and q' >q and u^ ^ H^^^'^[G), then u^ ^ H*"^^^' (G ).

U ^y ffy gry> <^^^ i^^ ^' s and b' s satisfy the (Holder) conditions in the

second statement with some h' ^h and ifu^^ H^^^'^[G), then u^^ Cf^^'^^' (G).

If Gy fjy gry y ^^d the a's and b's^CîG) and u^^Hl'+îG), then

u^^ C^(C)

6.5 . The existence theory for the Dirich let problem for s trongly

el l ip t ic system

I t is c lear th a t th e resu lts of § 6 .3 im ply th a t for e ve ry A, th e set of

s o lu t ions o f the homogeneous bounda ry va lue p rob lem

(6.5-1) LjkU^ + {—\)'^û^ = 0 o n G , BrkU^=0 ondG,

form a manifo ld of f in i te d imens iona l i ty , p rovided the Ljk a n d Brk

s a t is fy th e s up p lem en ta ry an d comp lemen t ing cond i t ions of § 6 . 1 . I t i sev ident (s ince for one equa t ion we regard L^ ^ as the iden t i ty ) tha t the

Dir ich le t boundary condi t ions a re a lways complementary for a s ing le

equa t ion of o rder 2 m which s a t i s f i e s the s upp lemen ta ry cond i t ion .

However the fo l lowing example due to SEELEY a n d c o m m u n i c a te d orall} '

to the au thor (essen t ia l ly ) by F . B R O W D E R s hows tha t eve ry complex

n u m b e r A may be an e igenva lue : Le t t ing {r, 6) be pola r coord ina tes in the

p lane , we cons ider the opera tor

on the annu lus TT < r < 27r. Fo r each com plex A, th e func tion

U(r, d; X) = Jo{w) sinr, w = X^''^ e'^

is an e igen-function for the given eigenvalue A, Jo being the Besse l func

t ion . Tha t i s , U satisfiesLU -XU = 0.

In th is sec t ion we presen t the ex is tence theory for the Dir ich le t

problem for a s t rongly e l l ip t ic sys tem, as def ined by N I R E N B E R G [2] ; in

th is p roblem, the e igenva lues a re i so la ted . The deve lopments para l le lth os e in §§ 5-2 an d 5-6.

In d iscuss ing s t rongly e l l ip t ic sys tems , i t i s convenien t to change

the no ta t ion s o m ewh a t f rom th a t u s ed in §§ 6 .1— 6 .4 . W e a s s ume th a t

t h e o p e r a t o r Ljk is of order Sj + sjc where we a lso assume

(6.5.2) s y > 1 , y = 1 , . . . , TV; thus m = si + h SN ^ N.

In the p rev ious no ta t ion tjc was the max imum o rde r o f any o f the Ljic

for fixed k. Thus if we define s = m a x s ; , t h e n

(6.5.3) 5 = m a x s ; , tjc = s -\- sjc, old Sj = — s + new Sj,

w h e r e tjc re fe rs to the prev ious no ta t ion and the Sj are new.

Definition 6.5 .1 . If we let

LM , D)= z < . W D ", L;^ = z «?. ".

L'y^ d e n o t i n g t h e principal part of Lj^, then the system is sa id to be

strongly elliptic if and only if

(6.5.4) Ia|=sj+Sfc ? =1

x^G, A r e a l , ^ c o m p l e x .

A vec to r u will be said to satisfy 0 Dirichlet data on 5 G if and only if

(6.5.5) V^-iw*^ = 0 , r = \, .. ,,S]c on dG.

In order to f i t these spec ia l boundary va lue problems in to our genera l

f ramework , we rep lace the index r by the pa i r {j, r); then the ope ra to rs

Brk, in [x, y) coord ina tes , a refo , if y ^ ^

Since, in t h e old notation, Brt was of order tjc — hr, i t fol lows that

(6.5.7) hjr = s + Sj+\-r.

6.5- The Di r ich le t p rob lem for s t rongly e l l ip t i c sys tem s 2 5 3

In ord er to pro ve our ex is tenc e theo rem s, we begin a s in § 5 .2 b y

assuming , in addi t ion to (6 .5 .4) , tha t

(6.5.8) afjc^C[-^-''^-^{G), | a | > s j , + 1 ,

a'^j^ bounded and meas u rab le , a l l oc,j, k.

In tha t case , i t i s poss ib le to in tegra te by par ts to e l imina te der iva t ives

of u^ of order h igher than S]c and thus ob ta in

(6.5.9) ^ ^ *

T he in te gra t ion s b y p ar ts a re no t un iqu e b u t we sha l l assum e th a t the)^

are done in some fixed way, obtaining the definite formula (6 .5-9) forB (u, v). However these in tegra t ions a re performed, i t fo l lows eas i ly tha t

(6.5.10) z Z SA?i{x) ?„if>i^'' = z 2 ' « r * ^ I ^ J * -

W i t h t h e s e a s s u m p t i o n s , t h e e q u a t i o n s

(6.5.11) LjjcU^ + (—'^)'iU3 ' =fjy Bjcrjui = 0 ondG

may be so lved in the weak sense by so lv ing the equa t ions

B{u,v) + A C {u , v) = L (v)^ '^*^^^ C(u,v)= f 2Ju^v^dx, L{v)= j2;{~\Y^f^vJdx

G ^ G în which B (u, v) is defined in (6.5-9) and G need on ly be bounded .

As in § 5 .2 we first p ro ve th e following th eo re m :

Theorem 6.5 .1 (Gard ing 's inequa l i ty ) . We suppose that G is bounded

and the coefficients Af^ are bounded and measurable and that those w hith

\(x\ — s; and \^\ = sjc are continuous and satisfy

ReZ I IAî{x)Lk&'^^>M-^Z \M^'A&\^>( 6 . 5 . 1 3 ) ^ '^ |a |= sj i/5|=Sfc i

M > 0 , A rea l , f com plex .

Then there exist constants Mi and Ao, depending only on v, M, the Sj, G,

bounds for the coefficients, and the moduli of continuity of those Af^ for

which \oc\ = Sj and |/51 = sjc, such that

| 5 ( ^ , ^ ) | < M i . | | ^ l i . | | t ; | | , u,v^ô

^ ^ ReB(u,u)^ (mill) • || u [p - XQ C {u, u), mi = A f - i

where the space ^0 ^5 the space of vectors u in which u^^ H^^[G) and

(6.5 .15) ( ^ , ^ ) = / 2 ' ^CaD ^'u^D'^vUx, Ca = \(x\\l(x\.

Q J \<x\ = SjProof. I t is sufficient to prove this for u, v^ ^TiQ- The exis tence of

Ml fo l lows immedia te ly f rom Poincare ' s inequa l i ty (Theorem 3-2 .1) . To

2 5 4 R eg u l a r i t y t h eo rem s fo r g en e ral b o u n d ary v a l u e p ro b l ems

prove the second inequality, let e > 0. G can be covered by a finitenumber of spheres B {xi, ri) such that \A'^^{x) — A^^(xi) \ < e wheneverx^ B(xi, Yi) and \(x\ =sj and | j ^ | — sjc. Clearly there is asequence {Ct},

t = \, . . ., S, inwhich each f ^ C^{Bv) and has support insome oneB (xi, Vi) and Cf + * * * + f ^ 1 on G. Then, as in the proof of the corresponding in equ ality at the end of § 5.2, we see th at

ReB(u,u) =ReB'{u,u) +Re f Z I ^^Â^^DûWû^dxQ t | a | = s j |<8|=Sfc

= ReB''(u,v) + RejZZ 2 ^?i D^^D^ ^"l dx, ul^^Ctu^(6.5.16) ^ * '*' *^ ' ' '

where B"{u,v) is a form like B(u,v) inwhich, however, "Af^Ô

whenever \(x\ =Sj and |/9| =sjc. But now the last term in (6.5.16)-R^flS ZAt{^t)D-u\DPu\dx

+ Re Zi: I[Ag{x)-Afi(xt)-]D«uiDûidx.G t \oc\=sj l/Sl=Sfc

Clearly the absolute value of the second term in (6.5.17)

(6.5.18) < Zi(r, si, . . ., SN,N)'8' Z WM^-t

If we introduce the Fourier transforms ul of uf by their usual formulas(see between (6.2.4) and (6.2,5)), the PL ANCHE RE L theorem shows thatthe first term in (6.5.17)

(6.5.19) =flReZ ZA iy y^ ufdy mi:S\\M -

Now, using the fact that Vî w| = Ct VîU3 + Mlu^, where the Mf are

of order Sj — \, and using (6.5.16) — (6.5.19), we see th a t

Re B {u, u) > (mi — Zi e) |1 u f — Re B'" [u, u)

where B'" is aform like B".Alm ost e xac tly as in § 5-2, we conclude th e following tw o th eorem s:Theorem 6.5.2. Suppose the transformation U isdefined on §0 (see

Theorem 6.5.I) by the condition that

(6.5.20) C{u,v) ={Uu,v)^Q, v ^Q.

Then U is completely continuous.Theorem 6.5.3. If X is not in a set gwhich has no limit points in the

plane, the equation (6.5.12), with L {v) defined by the more general expression

(6.5.21) L[v) = [2: Zf^D-vJ'dx,

has aunique solution u in ^for each set of ff inL2{G). If X^î, themanifold of solutions uof the equation with L =0 has a finite non-zerodimension. It is assumed that the coefficients satisfy the conditions ofTheorem 6.5.1.




Now we not ice tha t the equa t ion (6 .5 .12) wi th L{v) given by (6.5-21)

is of th e form discu ssed in § 6.4 w it h

(6.5.22) h = —S, Mj = Sj, hjr = S -}- Sj -\- 1 — r, pjr — 0 .

Cons equen t ly we may conc lude tha t i f the afj^ satisfy (6.5.8) the/ , ? $ i7r«^+l«i(G) an d G is of c lass Cf«-i , then the u^^ Hl^'^(G) H - ^ | J ( Q

and the in tegra t ions by par ts y ie ld ing (6 .5-9) may rea l ly be performed

(in reverse) to yie ld the following theorem:

Theorem 6.5.4. Suppose the a"-^ satisfy (6.5-4) and (6.5.8) and G is of

class Cf *~^. Then if 1 is not in a set g which has no limit points in the plane,

the equations

(6.5.23) LjTcU^ + {—^Yiui =-fj o n G

have a unique solution u with uJ ^ H^^^^ [G) f\ H^^ (G) for each f for whicheach p^ H2~^^(G ). If X^^, the manifold of solutions of the homogeneous

equations has a finite nonzero dimension.

Of course , h igher d i f fe ren t iab i l i ty fo l lows f rom more assumptions

about the coeff ic ien ts and the fj . We shall not s ta te these s ince i t is

poss ib le to e l imina te many of the assumptions made in (6 .5 .8) about

th e coeffic ients , as w as don e in § 5-6. W e pro ve be low a th eo re m

(Theorem 6.5.6) corresponding to Theorem 5-6.4, but only for sys tems in

w hich 5i = 52 = • • • — sjv = w > 0; th e m ore gen eral sys tem s in tr oduce d i f f icu l t ies . We mus t f i rs t p rove the fo l lowing theorem which is

s t rongly sugges ted by the resu l ts above and has been proved in A G MON -

DOUGLIS-NIRENBERG [ 2 ] .

Theorem 6.5 .5 . / / the system (6.1.1) is strongly elliptic, it satisfies the

supplementary condition and the Dirichlet boundary conditions are comple

menting.

Proof. T h a t t h e m a t r i x Ljjc s a t i s f i e s the s upp lemen ta ry cond i t ions

follows from the fact that the set 5 of coefficients a"-^ satisfying (6.5-4) is

convex and con ta in s the s e t

a% = 0 if j ^ k or if j = k a n d oc i^ 26 w it h li^l = Sj(6.5.24 ?A - - t- \t^\ J

obvious ly the coeff ic ien ts above correspond to the sys tem

A^JU3 =fj, y = 1 , . . . ,N.

C l e a r l y t h e d e t e r m i n a n t L{X] x) of the ma t r ix || a^^. Xoc|| cannot van ish for

any rea l X or there would ex is t a complex f so tha t

con trad ic t in g (6 .5 .4) . T he res t of th e sup ple m en tar y condi t ion requ ires

proof only if 1 = 2. T he n if G and T a re o r thogona l un i t vec to rs then the

roo t s z oi L{a + zr\ x) = 0 a r e n e v e r r e a l a n d v a r y c o n t i n u o u sl y w i t h

the coeffic ients as they vary over S.

2 5 6 R egularity theorems for general bounda ry value problems

To s how tha t the Di r ich le t cond i t ions a re complemen t ing , it is

suff ic ien t, accord ing to Th eor em 6 .3 .1 to show th e ex is tence of ex pon ent i

a l ly decaying vec tor so lu t ions U ^ of

Lojjc{a,~iDu)Ul{a,u) = 0, Dr-i Uf,{a,0) = df dl, \a\ == \,

(6.5.25)

where , here , we a re us ing the {x, y ) no ta t ion of § 6 .3 . The se equ a t io ns

made non-homogeneous a re ord inary d i f fe ren t ia l equa t ions of the form

N Sj+sjc

(6 .5 .26) 2 " Ibjjcp(a)i-iDu)i'UHa,u) =fj{a.u)fc=l 29=0

where (6 .5 .4) implies that

Re 2](6.5.27) i.k

Sj + Sjc

2> = 0|^ *c ? ^ > W i 2 ' ( l + iW 2)s, | ^ i | 2 ,

i

[\o\ = 1), mi = M - i .

Let us now define H^ as the set of a l l vectors U (hencefor th we

s uppre s s a) s u c h t h a t U^^ C^ic-i with Z)«fc-i U^ abs o lu te ly con t inuous

and sa t is fy ing

(6.5.28) Dl m{0) = 0 , p= {),. . .,sic- \

and wi th no rm de f ined byo« S

(6.5.29) \\U=j' ' i '\\DvUyfdu<o<.

F o r U a n d V having f in i te norm, we def ine

B{U.V) =( ^ \ v i j:hicv{-iD )vm +

(6.5.30) 0' -^^ """

we also define Bl{U, V) s im ilarly w ith [0 , 00) rep lace d by [a, h].

Now, l e t u s in t roduce the Four ie r t r ans fo rms U and F of a n d V in

HQ. Then, for such U and F , we see tha t

(6.5.31) B{U,V)= Z lhicvZvU^[z)V^{z)dz.

From (6.5.27) and (6.5 .31), we see that B satis f ies the hypotheses of BQ

in the Lemma of Lax and Milgram (Theorem 5 .2 .2) . Consequent ly i f

fj $ C^~^5 [0, 00) an d v an ish es for i/ >1, say, there exis ts au n i q u e U

i n ^ 0 s u c h t h a t

(6.5.32) B{U, V)=I 2 ' / ; V^'du, V^Ho.

The usua l d i f fe rence quot ien t p rocedure shows tha t U^^ C + fc a n d

satisfies (6.5.28) and (6.5.26). So, to find U\^, le t a)\^ be an y func t ion

^ C^^^k [0, 00) an d vani shi ng for w > 1 an d satis fying

Then , l e t U be the so lu t ion above wi th

Sj + Skfl = 2 i:hicv{-iD)P(4^.

k 39 = 0

Th en we m ay def ine

Uft = o>fe - UK

The following theorem and i ts proof are essentia l ly due to N I R E N -

BERG (see A G M O N - D O U G L I S - N I R E N B E R G [1] , p . 693) :

Theorem 6 .5 .6 . Suppose that G ^ Cf ~ , that Sj = s for every j , thatthe a°-^ with \(x\ =^ 2s ^C^[G), the other coefficients being hounded and

measurable. Then the conclusions of Theorem 6,SA hold, In fact, there exist

real numbers Ao and C, which depend only on v, G, E, and E' such that

h J \ k J

u^^Hf{G)r\Hl^{G), A r e a l .

Proof. W e firs t pro ve th e las t s ta te m en t. T he proof of th e f irst is l ike

tha t o f Theorem 5 .6 .5 .

Fo r any g iven r]i > 0, each po int X{) i s in a ne ighborhood o r bounda ry

ne ighborhood in wh ich \a[x) —a (:Vo) | < iq i w h e r e a {x) = {ajic {x)}

w it h 1^1 = 2s . W e choo se a sequ en ce f , ^ = 1, . . . , T , suc h t h a t ea ch

Ci € ^"^ [G) w ith sp t f i in some one such nei gh bo rho od an d Cf + • • • + C |

= 1. W e let u^ = ^t u^ a n d n o t e t h a t

wh ere ZJj^ is th e p rin cip al p ar t of Ljjc a n d Mjict is of ord er < 2s . T he n,

using (/ , g) to deno te the L2 inne r p roduc t , we ob ta in

(6.5.33) ^ ''*= 2 ' [ ( - ^Y{L]k < uf) + {M ;,, UK ui)]

w h e r e M'^-^^^ is of ord er < 2s . B u t

{6.5.34) ' '

i

w he re Z - ^ is th e o pe ra to r w ith t h e co ns ta nt coeffic ients afj^{xt) w h e r e1^1 = 2s and Xt is in the small support of Ct- Since for each t {ci. (6.5-9),

(6.5.17), and (6.5.19))

ReZi- ^Y{L-kt< 4) =Rej Afi(xt) D- u{D^<dx(6.5.35) ^ G9,k.ocJ

iMorrey, Multiple Integrals \ n

w e see, us ing (6 .5 .33)— (6 .5 .35) th a t

\\2s.

(6.5.36)

C(»? i )2 " lh ' ^ IP« -M l« *^ l l ' ' > -22»? i i «FMl« | i oj , k

Z 3 ( i « r ) 2 .

(6.5.37)

But now, for real 1,

j \ k I j \ k J

From Theorem 6 .5 .5 , it fo l lows tha t the b o u n d a r y v a l u e p r o b l e m s a t is

fies the cond i t ions of § 6.3, so t h a t

(6.5.38) (1* 112 )2 < Ci f2'(ll^;fc«*IIT + (II^IITl-

Using (6.5.36) and (6.5.38), we see that (6.5.37) becomes

> ( i -z 2 ^ i C i ) 2 ' ( i i i ; f c «* ' n 2 +

+ [A2(l - Z2V1) - 2AZ3 - Z2V1 CIKIMH^

— 2 SiWLjicW'Wr + ^HMr if A > s ome Ai.

The result follows eas ily from this and (6.5.38).

H igh er differen tiabil i ty resu lts follow from § 6.3 .

6 .6 . The an lyt i c i ty of the s o l u t i o n s of a n a l y t i c s y s t e m s of l in ear

el l ip t ic equat ions

In th is sec t ion we ca r ry ove r the m e t h o d s and re s u l t s of § 5-7 to

a p p l y to s y s t e m s of the type (6.1.1) on the in te r io r and to thos e of thet y p e s t u d i e d in § 6.3 at the b o u n d a r y . The deve lopments fo l low those of

MoRREY-NiRENBERG except tha t we t r e a t gene ra l bounda ry cond i t ions .

F o r our re s u l t s on the in te r io r , we cons ider so lu t ions on a s phe re BR.

Since we a l r e a d y k n o w t h a t the s o lu t ions ^ C ^ , we s ha l l a s s ume tha t

U^C^(BR). We define

eAf,Br)

(6.6.1)

dj) (u , Br) =

1/2, p'>So = minSj (<0) ,

1/2

^ > —t, t = m a x fjc

/ ' 2 ' | V - ^ . - + ^ / , - ( : ^ ) | 2 ^ ^Br ?•

Br ^ ' J

where, in general, \/Q(p = 0 if q < 0. W i t h [p\] given by (5-7.6), we define

(6.6.2

MR, P if) = [P l ] - i sup (7^ - ry-^P e^ (/, 5,), p ^ s^ ( > - ^)i ? / 2 ^ r —t.R/2^r<B

6.6. Analyticity for linear systems 259

A s in § 5.7, we sha l l show tha t

(6.6.3) iVij,p(w) < M L ^ p ^ - t

fo r s ome cons tan t s Mand L.

W e beg in by s t u d y i n g e q u a t i o n s of the form

(6.6.4) L%u^=fj, y = 1,. . . , iV,

w h e r e the L^j^ are opera tors wi th cons tan t coeff ic ien ts and invo lv ing

de r iva t ive s on ly oforde r Sj + tjc. We use th e n o t a t i o n of§ 6.2 u n t i l we

s ta r t cons ide r ing bounda ry va lue p rob lems . S ince , for e a c h y , t h e r e m u s t

b e at leas t one non-zero opera tor Z^^ , we m u s t h a v e

(6.6.5) SO + J 5 > 0 .

Lemma 6.6 .1 . Suppose u^C^{BR) and satisfies (6.6.4) on BR. Thenthere is aconstant K\ {v , E) such that

do{u,BR) <.Kieo{f,BR),

Proof. Mult ip ly ing bo th s ides of (6.6.4) by K{x —f)(see (6.2.11)) and

in teg ra t ing wi th re s pec t to | we f ind tha t

(6.6.6) L%m[x ) =Fj(x), m[x) j K{x- | ) u^^(f ) f,

BR

Fj{x)^lK{x-^)fj(^)dlBR

O p e r at in g on bo th s ide s of (6.6.6) with L ' and s u m m i n g , we o b t a i n

u^{x) = V^'Fj{x)

f rom which the lemma fo l lows , us ing Theorem 6.2 .1 .

L e m m a 6.6.2. Suppose u C^ (BR) and satisfies (6.6.4) on BR. Then

there is aconstant K2 [v, E) such that, for 0<Cr<Cr-^d<.R, r>d,

do (U, Br) < i^2 U (/, Br^s) + Z ~ ^-Q (/> ^r+s) + Z ^~^ ^-Q (^ > r A '

(6.6.7)

Proof. Let 99 bebefined as in (6.2.25) et seq. and let

{6.6.S) C( ) = ^[^"Mkl - ^ +^)]. U^{^) = C(^) ^^(^).

T h e n U has s u p p o r t on Br+6 C BR and satisfies

Sj + «fc

(6.6.9) L%m: Fj =Cfj+SIV9C-L jcau''

fo r app rop r ia te ope ra to r s Ljjcq. Moreove r

(6.6.10) V - ^ / F ; - = C V - ^ . 7 ^ - + ^ V * : - > / Z M / ; + 2 ; ^V^C'MjTcqui^q=l k q=l

w h e r e Mjq and Mjjcq are app rop r ia te ope ra to r s w i th con s tan t coef fi cien ts

of orders —Sj — qand fe— , respec t ive ly , be ing zero if these in tegers

17*

are nega t ive . S ince

(6.6 .11) I V ^ C W I <.Z^[v,(p) • ^ - ^ ,

the lemma fo l lows by apply ing Lemma 6 .6 .1 to the vec tor U and us ing

(6.6.10) and (6.6.11).Lemma 6 .6 .3 . Suppose u ^C^ {BR) and satisfies (6.6.4) on B^. Then

there is a constant Kz [v, F) such that

\MR,j>{f) +IM n,j,_ ,{f) +iNR,^_ ^{u)\, p>0.[ Q = i a = i )

NR,P{U) <KS\

(6.6.12)

Proof. S u p p o s e r i s any number wi th Rjl ^ r < R and le t

(6.6.13) d^(R-r)l{p+ \) so R - r - d = [i -—^'l'{R - r).

Then, i f _> > 0 , V ^ ^ a ls o s at is fi es (6 .6 .4 ) w i t h / r ep la ce d by V ^ / , so tha t

Lemma 6 .6 .2 ho lds wi th a l l the ind ices increased by p. Mul t ip ly ing bo th

sides of this result by (pl)~^ • {R — r^+P and using (6. 6. 13), we obtain

(p\)-^R-ry+Pdp(u,Br)

< (p\)-^K, [e^f Br^s) {R~r- dY^p[\ - j ^ ^ " +

+ f^{P + ^)'e^_,(l Br^,) '(R-r- ^)^+^-<?(l - ^ ) " ' " ' ' ' ' ' +

f rom which the resu l t fo l lows by tak ing the sup .

W e tu rn now to the gene ra l ana ly t i c s y s tem

(6.6.14) Ljjcuf^==fj

which we wri te in the form(6.6.15) L%^''= fi + {Lîic - Lik) «*= = Fi

where L^^ is th e o pe ra tor who se coeffic ients are ju s t tho se of th e h igh est

order der iva t ives eva lua ted a t the or ig in . We se t

(6.6.16) V-'i-^^'Fj =- V-'j-^^fj + 2 ' 4 l ( ^ ) • Vh-^^-ûf^, 5o < A < 0 ,

whe re the aj^, to be abb rev ia ted by a^^, a re app rop r ia te ana ly t i c t ens o rs .

(W e r ec all t h a t V ^ 9? = 0 i f ^ < 0 . ) A c co rd in gl y, t h er e a re n u m b e r s^ > 2, L , an d RQ, s uch tha t

{^'^•"^y) \\7â^^<^p\LAv, pô

a O O ( o) =.0 , \a^^(x)\ <^LA '\x\,

in B^R o -

6.6. Analyt ici ty for l inear sys tems 261

Applying the vector inequality to (6.6.16) and using Lemma 5-7-4,we obtain

\2:1 v-'i-^pF^|2ii/2 ^ i^; I v-'^+^fj|2ii/2 +

\Z I v-^y+^7^i 1211/2 < [2 ; I v-^.-^^/ ; |2]i/2 +

10 J L ? J

+2^1 ^^^W I [2'V^fc+2?-^^^|2]i/2^ :/><o.a = i L fc J

Using (6.6.17) and separating out the term where q = 0 and }c = p, in

case ^ > 0, we obtain for 0 < ^ < i^ < -Ro (setting r = vj2),ep{F,Br)<plLAPyl^^r^ +

(6.6.18) t V ,^\

+ Z in'(p->c)\LAP--d,.,(u,Br), p^O,

ep(F,Br)^[p\]LAPyl^^r' + 2:L[(p->c)\]dp_,(u,Br),p<0.

Thus, from the definitions (6.6.2), we find that2 ) - l

Mn,p{F) ^y l'^L{AR)pRt^^ + ^L(AR)P--NR,4U) +

t p

(6.6.19) +LAR'NR,P(U)+^^^^^^^L(AR)P--R'NR,,,-,{U)

t+ pMR,^[F)^yl'^LAvRP^t-^' +ZLR'NR,P_,[U), so<p <0,

We now apply Lemma 6.6.3.If

we takeR so

small that(6.6.20) KsLAR< 1/2,

we see that we can solve for NR^ p {u) in terms of the non-homogeneousterm and the NR^q{u) with q <p. The result is, for p > 0,

NR,^{U) < 2KJyli^LRt^^Z{AR)v- — SQ (which is the general case), the double sum f o r p + \ <.q

< — So is missing and q ranges from 1 to — so in the triple sum. We note

2 6 2 R e g u l a r it y t h e o r e m s for ge ne ra l bounda ry va lue p rob le ms

< 2i-2> yl'^ LAô iR«+%+^ if ^ i^ < 1 / 2 .

Thus, if we assume that

(6.6.22) NR,q{u)<MRi+So- '''PQ, —t<q<p, P>~SQ

we see that

iVi?,pM < iK^R^+ôÂl^-Vyl' LA +^ZML2^-vp^ +

+Z ZÂ'^' 2^-^-^MP^-^ +2! Z "z"^^'^ 2Q+^-p-^MP'<-^ +

+ VMP2>-« + 2-iL2'P^"4 <^^- '' o+^P {AR< 1/2)(6.6.23)

if M and P are chosen so that (6.6.22) holds for — ^ < ^ < — S Q — 1;

it is easy to see that P may be chosen > 1 and large enough so that the

coefficient of M in the brace is <P ^/4i^3 and then M may be taken

so large that the remaining terms in the brace are ^M P^j^K'^. We

therefore conclude the following theorem:

Theorem 6 .6 .1 . If u is a solution o/(6.6.14) and fj and the coefficients

in the opefators Ljk are analytic at XQ, then u is analytic at XQ.We now consider the analyticity at the boundary. Using an analytic

change of independent variables, we may transform a boundary neigh

borhood along an analytic portion of the boundary into GRQ, the part of

the boundary corresponding to ORQ. We begin by considering solutions u

of (6.6.4) which satisfy the boundary conditions

(6.6.24) Borjc u^ = gr on an, R <Ro,

where Bork has its significance in § 6.3. We assume tha t

(6.6.25) ^0 > 0, Ao + ^r > 1 for each r,

ho and hr being the numbers in § 6.3. We assume that the boundary con

ditions and equations satisfy the conditions in § 6.3 and we use the (x, y)

and other notations of that section. Instead of (6.6.1) and (6.6.2), we

define (SlVô-îVl/jhr, P > 0.

ep (f. ^'•) = j 2 ' II V *o-«,+»/j hr, 0>p>so-ho,

f2' l|V%+*oV?M*|k, p>0,(6.6.26) ^J.(^.Gr) = [|.||^,^+^^+^^,.||^^^ 0 > p > - t - h o ,

|2'||vAo+^v^|s|k,i '>o,

cp(g. Gr) = j^-ii vô+Vi' ^s | |G„ 0>p>-ho-h,h^m3.xK,

6.6. Analyticity for linear systems 263

the no rms be ing the L2 norms . S ince all func t ions cons ide red ^C^(Cr) ,

w e m a y t a k e ^s € ^"^ (^r) • We then def ine

NR^P(U) =s u p [ p ! ] - ! { R - r)«+Ao+2>dp(u,Gr)^ p - t ~ ho

R/2<r<B(6.6.27) MR, P ( /) - s u p [p ! ] - i {R - ry+ô-^P e^(f, Gr), p so - ho

R/2^r<R

QR,p{g) =s u p [pirHR - r^'-'ô-^Pcpd Gr), p -ho- h

We shall f irs t es tablish the analog of (6.6.22).

L e m m a 6 .6 .1 ' . Suppose u, / , and g ^C"^ (GR) and have support on

GRUOR, suppose g^C^ {OR) , suppose gs (x, 0) = gs (x), suppose u

satisfies (6.6.4) on GR and (6.6.24) on OR. Then there is a constant K[{v, E)

such that(6.6.28) do (u, BR) <K[ [eo ( / , BR) + co (g, BR)] .

Proof. W e n o t e t h a t u, f, g, a n d g satisfy the h y p o t h e s e s on any GA

w i t h A R and van is h ou t s ide GR in R^+i. Let U = PR (/, g). T h e n ,

f rom Theorems 6 .3 .6 and 6.3-5, it follows that u^ — U^ is a. p o l y n o m i a l

of degree <,tjc +ho — v — 1 and hence that (6 .6 .28) holds , s ince only

the h ighes t o rde r de r iva t ive s of t h e u^ are involved .

Lemma 6 .6 .2 ' . Suppose u, f, and g ^ C ^ (GR) , suppose g ^ C^ (CR)

and g (x, 0) =g (x) on OR, suppose u satisfies (6.6.4) on GR, and suppose usatisfies (6.6.24) on CR. Then there is a constant K'^{v, E) such that

{ho—So

eo{f> Gr+d) +2d-Qe_q{f Gr+6) + Co(g,Gr+d) +

(6 6 2Q^ ^^ *^^ ^

a=i a=i J

0 <r <r+ d <R, r > d.

Proof. D efin e C an d C7^ by (6 .6 .8). Then U satisfies (6.6.9) and

(6.6.30) BoskU^ = Cgs+'i:'7^C'BsJcqU^ ys on GR.Q = l

If we define ys by (6.6.30) with gs rep laced by gs, we s ee tha t ys{x, 0)

= Ys (x) and, in fact U, F, y , a n d y sa t is fy the hypotheses of Lemma 6.6.V.

J u s t as in the proof of Lemma 6 .6 .2 we f ind tha t

2'V^^_g(/, Gr^6) +*i;Vî_^(^, Gr^s)a=i Q=i

ho+h t+ho

2:d-^c_ _ ,{g, Gr^s) +Id-^d_,(u, Gr^s)g = l g=l

The lemma then fo l lows f rom Lemma 6.6 .1 ' .

By d if fe ren t ia t ing in the t angen t ia l d i rec t ions on ly and proceed ing

as in the proof of Lemma 6 .6 .3 , we p rove the fo l lowing lemma:

eo (F, Br+6) < eo ( /, Gr+s) + Zi

oo(y, Br+d) <co{g, Gr+d) +Z2

L e m m a 6.6 .3 ' . Suppose u, / , g, and g satisfy the hy potheses of Lemm a

6.6.2' on GR. Then there is a constant K '^[v, E) such that

NR,^{U) ^KAMn,^{f) +Z MR,p.a{f) + QR,p{g) +I Q=l

ho+h t+ho ^

+SQR,P~Ai)+INR,p_,{u)\, p>o.q=l q=l )

Remark. In th i s proof, we use the fac ts tha t

dp{u,Gr) = do(Vlu,Gr) if pÔ,e_q{V%fGr)êp_q(fGr),p,q>0,

C_q(V%g, Gr)<Cp_q(g, Gr), p,qÔ.

To handle the genera l ana ly t ic sys tem, we wri te i t in the form

(6.6.31) L%u^=Fj, BoskuJ' = ys = gs+{Bosk-Bsic)u^ on OR,

w h e r e Fj is defined as before, in (6.6.15) and gs i s ana ly t ic on GR w i t h

gg [x, 0) = gs {x). As before , we have

t+ha

\/hQ-Sj+^F.^^hQ-Sj+^f. +â]%{x, y)Vh+h-^^-û^(x, y), 0 > A > so — / oq= 0

t+ho

V ^ o - V ^ Ys == V ^ o + V ^ gs + Z Hk ( > y ) V*A-+ô+^-û^ {x,y), 0 > A > — ô — -

aOO(o) = bOO(0) =0, \a^^(x,y )\ <LA ]/\x\^ +y ^,

I b^^{x, 0) I <LA ]/\x\^ + y^ ,

with bounds l ike those in (6 .6 .17) for V^fj, V^ ^^^ V%b^^, V^gs-

Using the prev ious ana lys is and the idea of the remark above , we obta in

the inequali t ies (6 .6 .18) and (6.6 .19) with t rep laced b y ^ + AQ and we

obta in the corresponding inequa l i t ies for QR,p(g). As before , i f we take

R(<Ro) s o s ma l l tha t IK^LAR <,\j2 th is t ime , we can ob ta in an

inequali ty l ike (6 .6 .21) with t rep laced b y ^ + ô an d te rm s on th e r ig h tcoming f rom the ana lys is o f y . An ana lys is l ike the prev ious one demon

s t ra te s the ex i s tence o f numbers R> 0, M, a n d P s u c h t h a t

(6.6.32) NR^P{U) <M Ri+ô+'^ PP, p^—t — ho, 0<R<. Ri.

Now, in o rde r to p rove ou r ana ly t i c i ty theo rem, we mus t ob ta in

bounds like (6.6.32) for all the der iva t ives of u. For th is purpose we def ine

dp,q{u,Gr) = 2 ; i | Z ) t ' ' " ' " " ' ' V S ^ * ^ k , p>:0, q ^ - t - h o

^^ NR,P,^{U) = snp[(p + q)\rHR - ry ^ô^P^^dp,,(u, Gr).

Using the idea of the remark above , we conc lude tha t

(6.6.34) NR,p^q(u) < NR,p^q{u) if g' < 0 a n d j ^ > 0.

W e s ha l l s how tha t the re ex i s t numbers R2, 0 < R2 < Ri, M, P ^ \,

6 .6 . An aly t ic i ty for l inear sys tem s 2 6 5

a n d 0 < 1/2, 7?2, P, a n d d depending only on the bounds for the coeffic i

e n t s , / , a n d g and the i r de r iva t ive s , s uch tha t

If we differentia te the j — t h e q u a t i o n —Sj-\-a t imes wi th re s pec t

to y , 0 < ( ; < A o , the eUip t ici ty imphes th a t w e can s olve fo r D!^''^'^ u^

in t e rms o f the o the r de r iva t ive s , ob ta in ing

(6.6.36) D^û^ =f-^ +2 ;iblfjx, y )D'i^^-^\/tu\ 0 < ( 7 < Ao

where /<^^ and 5^^^ a re su i tab le func t ions ana ly t ic ne ar (0 ,0 ) . W e m ay

a ss um e t h a t t h e / ' s a n d h' s satisfy

(6.6.37) ^

for each set {a,Q,co). Differentia t ing (6.6 .36) with a = ho and us ing

(6.6.37) an d L em m a 5-7-4, we ob tai n, f o r ^ > 0 an d q >0,

k

e=l co=0 A=0 n= 0

(6.6 .38 ) X y ^ l D*^^+' »+^- v ? + ^ ^ ^ |.

F ro m (6.6.38) an d (6.6 .33), we ob tai n, f o r ^ > 0, ^ > 0,

(6.6.39) +IBli,^LAP^^-'-^NR,a>^,,^_,(u)RP^^-'-f'+^-^,

Q,0},2.,f.l.

D 2 > Q ^ P^-q^' [(A -f // + CO - g) !]

N ow , from (6.6.34), it follows t h a t (6.6.35) ho lds for all §- < 0 a nd

a l l ^ > 0 , p ro vid ed m ere ly th a t

(6.6.40) 0<R<^Ri a n d MPP+Q < M (P(9)^+^(9-ff for all p>0,

0 >:q'> — t — ho.

Then, from (6.6.39), we see that if (6.6.35) holds for all {p\ q') w i t h ^ ' > 0

a n d —t — ho<.q'<q{>^0), then it will hold also for all {p', q') w i t hq''^q, if, for example, we choose

AR2=--, Pd^P, P l 9 > 1 , 6 > < 1 , M > M , 2 2 >+ ^ M > 2L - (2 /l )- ô + ô ,

P > 1, ^ P 0 > 1, an d 0 < 1 /1 6 1 .

Since A was s uppos ed > 2 , the s e cond i t ions m ay be s a t is f ied .

6.7 . The analyt ic i ty of the s o l u t i o n s of analyt ic nonl inear e l l ip t ic-

s y s t e m s

In th is section w e ext en d t h e resu lts of § 5 .8 to gener al e l l ipt ic

s ys t em s ; the ana ly t i c i ty a t the boun da ry is p rove d fo r the s ys tems a ndbo un da ry c ondi t io ns d iscussed in § 6 .3 .

As usua l , we cons ider ana ly t ic i ty on the in te r ior f i rs t . We cons ider

s o lu t ions u {= u^, . . ., u^) of sys tems of the form

(6.7-1) (pj{x,Du) = 0, t=\,...,N

in which the cpj are ana ly t ic for a l l va lues of the i r a rguments near the

va lues of those a rguments a long the so lu t ion a t XQ. The sys tem is e l l ip t ic

a long the so lu t ion in the sense tha t the l inear equa t ions of var ia t ion

(6.7.2) Ljjc (x, D)vi^^~ %' [x,Du-^XD v]^^Q - 0

form a sys tem of th e ty p e discussed in § 6 .2 . W e shall use th e no tat io ns

of tha t sec t ion .

In (6 .7 .1) , we may suppose tha t XQ i s the or ig in . In addi t ion , we make

t h e s u b s t i t u t i o n

(6.7.3) ui' =,vk j^pk^ k=^ \, . . .,N

w h e r e p^ i s tha t po lynomia l o f degree < fe such tha t

(6.7 .4 ) V ' "P* (0 ) = V ^^^ (O ) , 0<r<tk.

W e the re fo re a s s ume tha t XQ = 0 a n d 0°^ u^ (0) = 0 for | ^ | < fe in

(6.7.1). I f we expand the q)j about the origin, we may write (6 .7 .1) in

the form

(6.7.5) L%u^ = Mfjû^ + WJ(^> D^)

w her e the op era tor s L^^ ha ve co ns tan t coeffic ients an d are zero or of

o r d e r Sj + h* th e M^ j. h av e c o ns ta nt coefficients an d a re of low er ord er,

a n d t h e ip j a re the rema inde rs . S ince the de r iva t ive s D"^ u^ (0) = 0 if

\oc\ ^tjc, we see by differentia t ing the j — t h equation in (6 .7-5) up to

— Sj t imes , tha t the Tay lo r expans ion o f ip j beg ins wi th a homogeneous

po lynomia l in x of degree 1 — Sj, l inear te rms in the D'^ u^ with coeffici

en ts homogeneous and l inear in x, and quad ra t i c t e rms in the D"^ u^

w ith co ns t an t coeff icien ts .

Definit ion 6.7 .1 . W e de f ine the s paces *C^(5 i ? ) o f vec to rs / fo r wh ich

fj^^J^'î^R) w i t h \7 ''fj{0) = 0 for 0 < f < - - s ; a n d **C ^(5 ij) of

vec to rs u for which U^Ô^{BR) w i t h V^ u^ {0) = 0 fo r 0 <.r <.tjc',

we def ine the norms by

(6.7.6) *||/|Ui, =ZK[V-îf^, En), ** lk lk i ? -2 ' ^4V^fc^^ Bn),i k

D e fi ni ti on 6 .7 .2 . F o r / ^ * C ^ {BR), we define P /j( /) = U, w h e r e

(6.7.7) U^ = M^ - PK u^ = Ll^Fu Fi (x)=fK (x - i) fi (f) i fBR

6.7. Analyticity for non-linear systems 267

w h e r e P* is the p o l y n o m i a l in (6.7.4). We are he re s uppos ing tha t the

o r d e r m of the o p e r a t o r LQ i s >0; o the rwis e LQ is j u s t a c o n s t a n t and

w e set Fi = L^^ fh

In o rde r to s h o w t h a t PR is a bounded ope ra to r f rom *Cfi{BR) we

need some more re f ined resu l ts than those in Theorem 6 .2 .1(d) .

T h e o r e m 6.7 .1 . PR is a hounded operator from '^Cfjt{ER) to **C^{BR)

with bound independent of R {dependent only on v,ibi,E). Moreover, if

U = PR (/) and w > 0, then U satisfies

(6.7.8) L%U>'=fj.

Proof. We e x t e n d / y to ^C'l^ [B^R) (p rope r ly ; cf. Lemma 6 .3 .3) and

w r i t e

F^[x) = Fn[x) - Fn{x), Fjiix) =jK{x- | ) / y ( f ) ^ |

(6.7.9) Fj2{x)=fK{x- f ) / , . ( I ) dl

B2R-BR

F r o m T h e o r e m 6.2 .1 , we conc lude tha t

(6.7.10) FIJ^C^~^^(BR), h^^^-îFi^, BR)<Zih4V-'ifj, BR).

By d if fe ren t ia t ing Fj2 and us ing Theorem 2 .6 .5 , we o b t a i n

Dm î-s,Fj2{x) ==lD A(x-i) [D-s,fj{i) - D-s,fj(x)] di,B2R—BR

(6.7.11) A(y) = D^K(y), A(-y)==A{y), X^BR.

From (6 .7 .11) , we see eas i ly tha t

(6.7.12) I v ^ - ^ i - « / i ^ y 2 ( ^ ) | < ^ 2 ^ ^ ( v - ^ . 7 ; > BR)'{R - 1^1)^-1.

The result (6 .7 .10) for Fj2 follows as usua l f rom Theorem 2 .6 .6 . Thus , if

we define u by (6 .7 .7) , we see t h a t U^Ô^{BR) and u satisfies (6.7.8).

T h u s U^ satisfies

w h e r e Pj is a p o l y n o m i a l of degree < —Sj. But by dif fe ren t ia t ing and

s e t t i n g X = 0, we f ind tha t Pj ^ 0. The theorem fo l lows .

N o w , as in § 5.8, we a s s u m e t h a t w is a so lu t ion of (6.7.5) and t h a t

u^ **C^(î?o)> ^0 > 0. For 0 < i^ < Ro, we w r i t e

(6.7.13) u = UR + HR, UR = PR[M U + y)(x, Du)'\,

If we define

VR = PR[MHR + y j {x, D HR)](6.7.14) TR{UR, HR) = PR[M UR + ip{x, DuR + DHR) - yj(x, DHR)1

t h e n the e q u a t i o n for UR b e c o m e s

{6 .7 .15) UR — TR [UR, HR) = VR

and HR is seen to satisfy [6.7 -^) with fj = 0 on BR,

2 6 8 Regularity theorems forgeneral boundary value problems

N o w , if ^ isa so lu t ion onBRQ, t h e n it is clear that ** | |^ | | i ? <Mi

independen t ly o f R. The proof of the fo l lowing theorem isiden t ica l w i th

t h a t ofTheorem 5 .8 .2 :

Theorem 6 .7 .2 . Suppose u is a solution ^*'^C[x{Bii^ of (6.6.5) and

suppose UR, VR, HR, and TR are defined as above. Then **|| ^i? {{R, * * [ | VR\\R,

and * *|| HR \\R are uniformly bounded by some number M^ for 0<C R<. Ro,

HR satisfies (6.7.8) with f =0on BR, and

TR (0 ; HR) = 0, lim**\\VR \\R = lim* *| | ^ ^ | |^ =0,2?->0 R-*Q

* * | | Tn(UiR; HR) - TR{U2R; HR) \\R < e(R) * * | | UIR- U^R \\R,

l ime{R ) = 0 , if * *|i HR \\R, * *\\ UIR \\IR, * * 1 | U2R \\R < M2.

Thus, if0 <C R Ri, UR isthe only solution 0/ (6.7.15) for which * UR \\

< M2, provided * *|| HR \\R < M 2, HR being supposed given.

Now, a s in§ 5.8, we int ro du ce th e following spa ces of ana ly t ic func

t ions ; we us e the no ta t ions of tha t s ec t ion .

Definit ion 6.7 .3 . Thespace *C^h{^hR) consis ts of a l l / ^ *C ^(-S/j)

which can be e x t e n d e d toE^R SO t h a t e a c h / y $ C ~ ^ ^ ( 5 ^ i ? ) a n d is holo-

m o r p h i c inBJIR. T h e space '^*C[jtfi{BjiR) consis ts ofall z / ^ * * C ^ ( 5 i ? )

w h i c h can be e x t e n d e d toBUR sothat each ^^^CJf (5/^/?) and is holo-

morph ic on B^R. We def ine the norms by

(6.7.16) mUR^ZKiy-'ifhSuR), **||«iU=2'A^(v%w* 5;,^).j k

The fo l lowing theorem isproved by imi ta t ing the p roo f of T h e o r e m

5.8.3:

Theorem 6 .7 .3 . Suppose 0<, h<, hi, hi being the number in Theorem

6.2.1_(a), suppose / $ '^C^h{BhR), suppose XQ^B^R, suppose 5 : f = f (s),

5 C BR, isadm issible with respect to the function ^ ( | ; XQ) =K(xo~i)f {i),

and f satisfies (5.8.20). Then the integral (5.8.18) exists. If and f* are

both admissible andsatisfy (5-8.20), the integrals have the same value.

Finally, if for each x^BfiR, S{x) : f — ^{s; x) is admissible with respect

to g ( f ; x) and satisfies (5.8.20) for each x, then the function Fdefined by

(6.7.17) F[x)=JK[x~^)f[^)d^8{x)

is analy tic on B^R and

(6.7.18) LoF{x)=f{x), D<'F{x)==lD° K{x-S)f{ )dS, \(x\<m.Six)

Definition 6.7.4. For f^*Ch/z{BhR), we define PR{f) =U, w h e r e U

is defined by (6.7.7) with Fi rep laced by i t s ex tens ion

(6.7.19) Fi{x)=lK{x- )fi{ )dlSix)

6.7- Analyticity for non-linear systems 26 9

Theorem 6.7.4. PR is a hounded operator from "^Cjifx (ShR) to ^'^Chfx [BUR)

with hound depending only on ii, h, and E and not on R.

Proof. Let f^*Chf^ {BJIR) and suppose / is extended to BhR as in the

definition and is also extended so tha t es^ch fj^C~^^{B2R) and is zero

near 2^2R, the norm of / being increased at most by a factor C (E); the

possibiHty of this is proved as in Lemma 6.3.3 using r instead of x . We

then write

Fi{x) = Fii{x) -Fi2{x), Fi2[x)=JK{x - i)fi{^) dl

Clearly F12 is analytic on BfiR and

ym^l-s,Fi2{x) - j \7^-HA{X-^) / Z ( | ) -^^D-fi{x) • (^ - x)-\ d^,B 2 R - B R L | a j=o J

A{y) = \7^K(y), X^BUR

for, since A satisfies (2.6.9) and (2.6.10), we see by induction and differen

tiation using Theorem 2.6.5 that

(6.7.20) j\J'^-hA(x — S)'{^ — xYd^^ 0, 0 < | ^ | < - s z .

BZR—BR

Thus

I V^+i-îFz2 (^0) I < ^(^, h E) h^ {V-Hfi, BUR) • [^(^0)]^-^, ^0 € B^R.

(6.7.21)

Since the derivatives up to the order — s; of the /y join up continuously

across SBR, we find that

Drri-^-HFu{x) = ID^-^K(X ~ i)D-îfi(S)d^ +Six)

(6.7.22) + JDm-iK{x - ^)D-Hfi[^)d^ = (pi(x) + (p2(x),

B2R-~BR

(p2(x) = J D^-^K(xi — s)D-Hfi(s + ix2o)ds,

BiXio,r)

x=xi+i X2Q, xo=XiQ+i X2Q s BjiR, r=r (ô)/2.

where S {x) — Si [XQ) U ^2 (XQ), SI {XQ) is the surface in Definition 5-8.5,

and 82= {x = s -{- i X20 \ s ^ B (xio, r)}. In case m = \ and — s? > 0,

this has to be proved by removing a sphere B (xi, g) as is done for 992

below. For x near XQ, we may find D^ cpi [x) by differentiating under the

integral sign to get

Z ) 2 ^ ^ ( x ) = f D A { x - I) [D-Hfi(S) - D-hfi{x)-]d^ +

SiiXo)

+ lDA{x-i) [D-Hfid) - D-Hfi{x)-]di

BZR-BR

s ince, by rep lac ing Si (XQ) by a surface near it bu t con ta in ing a t h i n a n n u -

lu s of p o i n t s ^ =z s + i X20, s^B {xio,r + e) — B [xo, r), we can p r o v e

t h a t , ,

JDÂ (X -S)di + JDo:A{x - i)d^

= jA{x - D^f ; - jA{xi - s)ds', = 0, xi^B(xio, r).

dB-2R dBiXio.r)

Using Theorem 2 .6 .5 and a n a l y t i c c o n t i n u a t i o n . T h u s

(6.7.23) I V2 991 (^0) I < ^ (/^, h, E) ' [r (ô)]-i.

N o w , for p o i n t s x = xi + i:V20, X\^B {XIQ, r) we w r i t e

(6.7.24) 952e(^) = / ^ ( ^ i — ^ ) / ( ^ + ^ '^20)^5,

B{Xio,r)-B{Xi,Q)

r{y) = D ^ - 1 K (y), / ( I ) = D - i / z ( | ) .

Differentia t ing (6.7 .24) with respect to x°', i n t e g r a t i n g by p a r t s , and

l e t t i n g ^ -> 0, we o b t a i n

(6.7.25) Dccp'zix) = —jr{xi — s)f{s + ix2o)ds(x' +

+ f r{Xi — s ) / , a ( 5 + iX20)ds.

B ( % o , r )

Since these are rea l in tegra ls of a complex -va lued func t ion /, we maydifferentia te (6 .7 .25) us ing Theorems 2.6.2 and 2.6.5 to o b t a i n

D^Da(p2[x) = —Jr,p(xi — s) [f{s + 1x20) —f(xi — tX2o)]ds'^ +

dB(Xio,r)

+ Cpfâ(Xl + iX20)+J rj(xi — S) [ / , a ( s 4- ^ ' ^ 2 0 ) —f,o:(Xl + iX2o)]ds

Cp = —jr{—s)ds^5 5 ( 0 . 1 )

s ince F^p can be t a k e n as a zl in Theorem 2 .6 .5 . F rom th i s and L e m m a5.8.3 and the fac t tha t d {XQ) < r [ X Q ) , we conc lude tha t

(6.7.26) I V 2 (p2 {x)\^Z ill, h, E) . [d {xo)Y-^.

The result follows from (6.7.21), (6.7.22), (6.7.32), and (6.7.25).

Theorem 6 .7 .5 . Suppose H ^* '^Cfi {Bji) and satisfies (6.7.8) withf= 0.

Then H^ **Cjifi(BhR) and

**\\H\\nR<C{f^,h,E)-*^HU.

Proof. We may define V to be an ex tens ion of H to B2R to h a v esupport in B2R and so th at * * | F||i2 < Zi (^ , r) • **\\H\\R. AS in the

proof of L e m m a 6.6 .1 , we conc lude tha t

Hi(x) = Ll'Fi(x), Fi{x) = JK{x - f ) / ; ( f ) i f ,

(6.7.27) B^R-SB

Mx) = L%VHx), xfB2R.

6.7- An alyticity for non-linear systems 2 71

From (6 .7 .27 ) , we immed ia te ly conc lude tha t H i s ho lomorphic on BJIR.

Moreove r

S 2 R — B a

SO th a t

I S/rn+l-SjF(x) I ^ Z îfl. h, E) • h^(S7-îf], S^R) • [^(%)]"-!

< Z3(fi, h, E)Z K(V hH, BR) • [d{x)r-\ x^BuR.k

The result follows eas ily from this .

The proof of the fo l lowing in te r ior ana ly t ic i ty theorem is essen t ia l ly

l ike that of Theorem 5.8.6; one verif ies the conclusions of Theorem 6.7-2

for the spaces "^^ChfiiBhR).

Theorem 6 .7 .6 . Suppose u ^ **C/,(-Bij) for some JR > 0 and satisfies

(6.7.5) there where L^j^, Mjjc, and the ipj have the properties as described

above. Then u is analytic near x = 0.

We now wish to prove ana ly t ic i ty a long an ana ly t ic por t ion of the

boundary of the solutions of a sys tem of the form (6.7.1) which satis fy

non- l inea r ana ly t i c bounda ry cond i t ions o f the fo rm

(6.7 .28) ;^ r ( ^ , ^ ^ ) = 0 , r=\,...,m

wh ere we now use th e no ta t ion of § 6 .3 ; the or der of LQ is now 2 m. W e

a s s u m e t h a t t h e )^r conta in der iva t ives of u^ of order ^tjc — hr, w h e r e

some of the hr may be nega t ive , and we a s s ume tha t the l inea r ized boun

d a r y c o n d i t i o n s

(6.7.29) ^^Xr(X>D^+^D^ IA-0 = 0

satis fy th e cond it io ns of §§ 6.1 an d 6.3 . W e assum e th a t ho i s the smal les t

in tege r s uch tha t

(6.7.30) ho > 0, Ao + Ar > 1, ^ = 1, . . ., w ,

and we a s s ume tha t ou r s o lu t ion u i s s uch tha t u^ ^ O^'^^^ (9 ) in a neigh

bo rho od on G of a po in t Xo on dG, as s umed ana ly t i c nea r Xo.

W e beg in by mak ing an ana ly t i c change o f independen t va r iab le s

which carr ies xo in to th e orig in, 9^ o n to G/jo U Cijo, a n d dG C)^ o n t o

(^RQ] we then us e the {x, y ) no ta t ion of § 6 .3 . In th e resu l t ing equ a t io ns

(6.7.1) an d (6.7 .28), we m ak e t he su bs ti t ut io ns (6.7.3) w here P^ i s tha t

po lynomia l o f degree tfc + ho su ch t h a t (6.7.4) hold s for all r, 0 < r

'<tjc + ho. I f we then expand the new 99 a n d Xr about the or ig in in the

(x, y, p) space , we obta in the sys tem (6 .7 .5) and the boundary condi t ions

(6.7.31) Bork U ^ = Cork U^ + COr {x , 0, D u)

w h e r e Bork has cons tan t coeff ic ien ts and conta ins a l l and only der iva

t ive s D^û^ of order t]c — hr, Cork has constant coeffic ients but is of lower

order , and the Taylor expans ion of co r beg ins wi th a homogeneous

p o l y n o m i a l in (x, y) of degree ho -\ - hr + 1, l inea r t e rms in the 0°" u^

with coeff ic ien ts homogeneous in {x , y) of degree > 1, and second powers

of the d e r i v a t i v e s ; the expans ion of ip j h a s its prev ious fo rm excep t tha t

the homogeneous po lynomia l in {x , y) has degree /ô + 1 — ^j-

The fo l lowing lemma can be proved by t h e m e t h o d of proof of L e m m a

6.3 .3 :

L e m m a 6.7 .1 . There ex i s t bounded ex tens ion ope ra to rs ti JE from

Ci(Gjt) to0^{G2R), r^jR from CI[BR) to0^[B2R), T^JR from 0^(GR) to

0^{BR), and T^JR from CjîaR) to 0^(G2R), with hounds independent of R

if the norms are defined using A^, and which are such that ti JR (/) vanishes

near ^^ând r2jR(f) vanishes near dB2R. Thus

The cons t ruc t ion in Lemma 6 .3 .3 is to be rep laced by

(6.7.32) F(x,y)=^CiF[x,-j ), y<0S = l

t o yield T 3 ; i ? .

Definit ion 6.7 .1 ' . W e define the space *C^ {GR) as the space of vec to rs

/ s u c h t h a t fj^C^^-'i{GR) with V^/y (0 , 0) = 0 ior 0 < Q < ho — Sj, the

no rm * | |/ || i? be ing ^h^{\7ô~^jfj, GR). We define the space * 'C^(î?) as

J

the s pace of v e c t o r s g s u c h t h a t gr ^ C^^^^^" {GR) w i t h V ^ ^ r (O ) = 0 for

0 < ^ < Ao + ^r and n o r m *'\\g\\ = = X " ^ ^ ( W ' ^ r ^ r , OR). We define the_ r _

s pace **C^{GR) as the space of vec to rs u s u c h t h a t U^^C*/^'^^^{GR),

VQu^(0,0) = OioYO<^Q<tjc + ho, and no rm **|| i^l| =2 '^ i" (V *" +' '^^ ^ Gij).

P'or f^'^Cfi{GR) and g^*'C{GR), we define PR(f,g) as follows: Let

fj =T2j,2R[rsjRfj], gr =TÔrRgr, Or = ho + hy.

We then def ine PR ( /, g) •= u, w h e r eu^ = U^ ^ V^ ~W^ — P^

w h e r e U^ is given in t e r m s g by (6.3.29), V^ is given by (6.3-43)» ^ ^ is

def ined by (6.3.29) with gr rep laced by

yr{x) =BorkV^(x,0), [sll x) ]

we a s s ume , of cours e , tha t gr{x) = 0 ou ts ide a2R and , as us ua l , P^ is

chosen so t h a t \/^ u^ (0,0) = 0 for 0 < ^ < A; + ô-

Since PR ( /, g) as we have def ined it differs from the PR (/, g) ofDefin i t ion 6.3.5 by a p o l y n o m i a l ofdegree ^tjc + ho, the following

theorem fo l lows f rom our normal iza t ion at the orig in and Theorem 6 .3 .6 :

T h e o r e m 6 .7 .1 ' . PR is ahounded operator from *Cft(GR) X * 'C^(a i? )

into **C^(Si j ) with bound depending only on [fJiy E). Moreover, if U =

PRify g) then U satisfies [6.7.S) with f replaced by f everywhere on i?+^^

6.7- An alyt ic! ty for non-l ine ar syste m s 2 7 3

and also satisfies

(6.7.33) BQricU^{x,0)=gr[x), all X.

N ow w e assum e th a t w is a solution of (6.7.5) on GR and (6.7.31) on

OR a n d t h a t u^ **C^{GR^, RQ > 0. F o r 0 < i^ < jô, we wr i te

u = UR + HR, UR = PR [MQ U + ip (x,y, D U), CQU + CO {X, D U)].

If we define

VR = PR [MO HR + ^P {X , y, D HR), CO HR + W (X, D HR)]

(6.7.34) TR {UR, HR) = PR [MO UR + ip [x, y ,DuR + D HR) -

~ y ) {x, y , D HR), COUR + CO (X, D UR + D HR) — (JO{X, D HR)]

then the equa t ion fo r UR reduces to (6 .7 .15)- An analysis l ike that in the

proof of Theorem 5 .8 .2 demons tra tes the fo l lowing theorem:

Theorem 6 .7 .2 ' . The conclusions of Theorem 6.7.2 hold with BR replaced

by GR. In addition HR satisfies (6.7-33) ^^^^ ^r — 0 on GR.

As in § 5 .8 , we no w intr od uc e ce rta in space s of function s wh ich are

analytic in the ;^:-variables only.

Definition 6.7.3 ' . We define BohR (as in § 5.8) to be t h e p a rt oiBhR

w h e r e y is real , G^R as the pa r t oi BohR w here y > 0, an d G^^ the par t o f

BohR wh ere y < 0 . Th e space "^Chfz (GUR) cons is t s of a l l v e c t o r s / ^ *C^ {GR)which can be ex tended to GhR s o t h a t fj^C^^~^^(GjiR) a n d fj i s ana ly t ic

in X for all (x, y) in GJIR. T h e space *'C/i^(G/îj) co nsi sts of all ^ ^ *Ci^{aR)

s u c h t h a t gr can be ex tended to OJIR SO t h a t gr^C^^'^'^'[GhR) a n d gr is

ana ly t i c in GUR] he re GUR is the part of BQ^R w he re y ^ O . T h e space

'^*Chfi{GhR) consists of all u^ **Cf,{GR) which can be ex tended to GhR

s o t h a t u^ ^ C*^^^^ {GhR) a n d u^ is analytic in x fo r {x, y ) in GhR. We define

the no rms a s u s ua l :

*\\fUR =Z K[WVy. ÛR]. etc.We wish now to ex tend the def in i t ion of the opera tor PR (/, g) to the

s pace *Chfi{GhR) X '^'Chtx{GhR) and wis h to s how then tha t PR is a

bounded ope ra to r f rom tha t s pace to '^'^Chf^{GhR), We cons ider the

separa te te rms in the def in i t ion 6.7.1 ' -

Theorem 6 .7 .3 ' ( a ) . Suppose g ^'^'Chfji{GhR), gr = r4srRgr where

Sr = ho + ^r, (^'y^d we define

uTc = U^- pJc^ Ui^{x^ y ) = f^J Z C.L^^{x - I , y)L-gr{S, 0)dS

where Cx = \(x\\l(x\ and P^ is the usual poly nomial. Then u^**C hfz{GhR),

satisfies the homogeneous equations {6.7 .S) on Rj^+ i, satisfies the bounda ry

conditions (6.7-33) for all real x a n d all complex x on GhR, and finally

(6.7.35) '^*\\u\\h<C{iu,h,E)' *'\\g\\ provided 0 <h <h2{E) <hi{E).

Money, Multiple Integrals ^ g

Proof. From Theorem 6 .3 .2 , we conc lude tha t the L^'^(x,y) a re

analytic for a l l {x, y) ¥" 0 s u c h t h a t y is real an d > 0 an d x == Xi + i X2 is

c o m p l e x w i t h | ^ 2 | < / ^ ] / | y p + \xi\^ if 0 <C h <h2(E). For s uch h, w e

e x t e n d gr t o ajiR and then ex tend U^ t o GhR by the fo rmula

UHx, y) =f^Hx, y; f)^f + J ^Hx, y; f ) i | ,

^Hx, y; f) =Z ZC.Ll^[x - i, y)D''gr{i), D. = |«|!/«!r \«\=Sr

w h e r e

S(x,y) :i == s + i^2 (s), SÔR,i s adm iss ib le . W e hav e seen in § 5 .8 th a t such in tegra ls a re ind epe nd ent

of the pa th 5 {x, y ) as long as i t is admiss ible so that U^ i s ana ly t ic . Neara n y p o i n t [XQ, y^), we may choose a f ixed surface containing a disc

f == s + ^ X2Qy I s — XIQ\ < Q, independen t o f [x, y). W e may ca lcu la te the

der iva t ives of U^ by d i f fe ren t ia t ing in the rea l d i rec t ions , so tha t we

wou ld f ind tha t a de r iva t ive Dx Dh'^ô U^ {x, y ) is a sum of terms of the

form

C jD^rix - I y)y{i)d^ + JD^rix - | , y)y[^)d^S{X,y) O-GR

w h e r e y (f) — D^rgr, r{x, y ) — Dh'^h Ll^(x, y ) so t h a t P i s h o m o g e n e o u sof degree ~v. N ow , if y > 0, we find, as in th e proof of T he or em 6.3.3

and by us ing ana ly t i c con t inua t ion , tha t

fD^rix - I y )d^ + JD^rix - 1 y ) d^ = 0 , [x, y)^GnRS {X, y) a-aR

SO th a t we m ay rep lace y ( |) ab ov e by 7 ( |) — y {x) and thus ob ta in a

bound l ike (6 .3 .23) for a l l such derivatives . Using the differentia l equa

tio ns we get a simila r bo u n d for i ^fc+' o+i an d he nc e co ncl ud e th e in eq ua lit y

(6 .7 .35) and thus the theorem.Theorem 6 .7 .3 ' (b ) . Suppose f ^""Cnii {GR) and we define V^ by (6.3.43)

where fj = T2;.,-2i2 T3A,-2i? //, ^j = ho — Sj. Then V^ ^J^*^'^^^ {BA) for any

A, V^ is analy tic in x for (x,y)^BohR , V^^Cy ^^{BohR), hf,(Vh+ôV^,

BohR) < G {/u, h, E) *| | / | | /i, and yr{X) ^ Bork V^(X) satisfies the conditions

at 00 in Theorem 6.3.3 and the analy ticy conditions in Theorem 6.7.3 '(a) .

Thus W^^C*^'^^''{GA) for any A, W^ it is analy tic in x for (x,y)^GfiR,

Wfc^Cyô^GA), andhf,{Vh+hoW^, GQUR) <.G{fA, h, E) *||/||/,. Ifv^ = V^-

— TF^ — P ^, P^ being the usual normalizing poly nomial, then v satisfies{6.7 .K) with fj replaced by fj on R^'+i U G^R and satisfies (6.7 .}}) with

gr =" 0 on a U OhR-

Proof. The resu l ts fo r rea l [Xy y ) fo l low from Theorem 6 .3 .6 . We may

extend/y to i^v+i U BQUR by def in ing / , - as above on B2R, d e f i n i n g / ; {x,y)

= 0 o n Rv+i—B2R, e x t e n d i n g / - to GhR as in th e def in i t ion , and the n ex t end -

6.7- Analyticity for non-linear systems 2 7 5

ingfj to BohR us ing the formulas (6 .7 .32). I t is c lea r tha t the e x t e n d e d ^ -

is analytic in x for {x, y) on BQUR. We may then extend Fj to Rv+i U BQJIR

by defining

Fj{x,y) ^ l^j{x,y; | , fj)dS drj + J ^j(x, y; ^,rj)d^drj,S(X,y) B2R-BR

S i K y]Lr]) =K(x-^,y ~- rj)fj(^, rj), [x, y) ^ EQUR,

w h e r e

(6.7.36) S{x,y)\^^s + ih (s, t) X, y), r) = t (real), {s , t) ^ BR

i s admiss ib le for g; . An ana lys is l ike tha t in the proof of T h e o r e m 5.8.3

s h o w s t h a t Fj is a n a l y t i c in x for [x, y)^BojiR and t h a t we may differen

t i a t e Fj 2m + ho — Sj — i t imes , shif t ing ho — Sj of the dif fe ren t ia t ions

o n t o the ^•. The re s u l t is

(p{x, y) =.lr{x-ly - fj)f(l rj)didrj + J r(X ~ E)f[E)dE

(6.7.}7) s^^'^y) BZR-BR

w h e r e / $ C^^[B2R U BohR) and F is h o m o g e n e o u s of degree — r.

N e a r a p o i n t {xo, yo) in BohR, we may t a k e S {x, y) as the un ion of a

f ixed surface 5i, given by (6.7.36) with (5, t) ^ BR — B (jîo, yo' , y) and

the disc ^ = s -\- i X2,o, ri = t w i t h (s, t)^B (^10, yo; ^ ) . Then , for p o i n t s

[x, y) w i t h X = xi-^ i xô and [xi, y)^B (A;IO, yo',r), {6.7. '^7) becomes(p {x , y) -= (pi (x, y) + (p2 {x , y),

(6.7.38) ^ i (Z) = fr(X- E)f{E)dE + f r(X - E)f{E)dE,

q)2{x, y) ~ f F(xi — s, y — t)f{s-\- ixô, t)dsdt.B{Xio,yo'^r)

An ana lys is exac t ly l ike tha t appl ied to the cpi in the proof of T h e o r e m

6.7 .4 shows tha t

(6.7.39) I V2 (pi{X)! < Z(/^, K E) • h,{f, BOUR) • [d[X)r-^

d{X) be ing the dis tance f rom X to dBohR w h e r e BOUR^^ cons idered as a

set in i^2»+i. We may dif fe ren t ia te 992 with re s pec t to an x° ' o b t a i n i n g the

fo rmula ana logous to (6.7.28). This may t h e n be dif fe ren t ia ted wi th

r e s p e c t to y or an x^ so t h a t we o b t a i n a bound (6.7-39) for \\J x \/(pi[X)\.

I t then fo l lows tha t any de r iva t ive Dh'^ô~^r^ ^C^(BohR) w i t h the

des i red bound . The re s u l t s for V then follow from the differentia l equa

t i ons . The proof of th e re s u l t s for W differs from the proof of T h e o r e m6.7.3(a) only in the cons ide ra t ion of the conve rgence at 00, s ince y

does not h a v e c o m p a c t s u p p o r t .

T h u s we o b t a i n our f irs t des ired result .

T h e o r e m 6.7A'. PR is a hounded operator from *ChM{G-hR) X '^' [CuixOhR)

into **Chpi(GjiR) with bound C(ibt, h, E).

18*

Theorem 6.7 .5 ' . Suppose H^**CJ^I{GR), satisfies (6.7.8) with the

fj = 0 on GR, and satisfies (6.7.33) withgr = 0 on OR. ThenH^ * *QA* (^hR)

and

(6.7.40) * *|| H WnR < C (//, h, E) * *|] H \\R if 0 < /Z < A2 < Ai,

^2 heing the number in Theorem 6.7 .3 ' (a ) .

Proof. L e t H^ = ri,t^+ho,RH^ and define H^(x,y) = 0 elsewhere on

F^+i and define

(6.7.41) fj(x, y ) = Lo3 k HK gr(x) = Bork H^ {%, 0).

Th en | r = 0 on OR and on o* — G2R and / , - = o on GR and on R^^^ — G2R.

We now def ine f = r' f w h e r e (p is def ined as usu al ( = 1 for s < 1, == 0

for s > 7/4, etc.)

f,[X)=<p{R-^\X\)r,{X)

f.[x,y) =ZCJ^[x, - ( T , - + 1 ) - M y ]

wh ere the CJ a re the un ique con s tan t s s uch th a t

Tj + l

Th en T ' i s a bo un de d ex tens ion op era t or an d we see th a t fj (X) = 0 for

X ^ BR and in i?v^i — B^R. Then we define F by (6.3 .43) and then define

W as ind ica ted immedia te ly be low and def ine U by (6.3.29) in terms of g.

T h e n U, V, a n d W have the requis i te d i f fe ren t iab i l i ty bounds every

w h e r e , V i s ana ly t ic in BUR, U i s ana ly t ic in GRH U ORh, if h is small

enough , and the fr are ana ly t ic in BfiR. From a minor ex tens ion of

Theorem 6 .7 .3 ' (a ) , i t fo l lows tha t W is analytic in x on GRhUaRh.

Fina l ly , f rom Theorem 6 .3 .6 , i t fo l lows tha t u = U ^ V — W satisfies

(6 .3 .4 4) w i t h / * = ^ * = 0 s i n c e / a n d g are 0 near 0 {g f could be a poly

nomia l van i s h ing on a). From Theorem 6 .3 .5 , i t fo l lows tha t H ~ u

is a polynomial of low degree. The results follow.

Theorem 6 .7 .6 ' . Suppose w^**C^(Gi?Q) and satisfies (6.7-5) on GR

and (6.7.31) along GR, where the ipj and cor satisfy the analy ticity conditions

specified near the respective equations. Then u is analy tic at the origin.

Proof. Suppos e 0 <C h <, h2 (E) < hi {E). From Theorem 6 .7 .2 ' and

6.7 .5 ' , i t fol lows that **\\HR\\h,R < Ms for 0<R < R2. T h e t h e o r e m s

above show tha t the conc lus ions of Theorem 6 .7 .2 ' ho ld for the spaces**Ch,.(GhR). Thus if 0<R<R2<Ri, UR is the only solution of (6.7.15)

w ith **|lwi?| |;^ < M 3. T h u s th e pr ev iou s UR m u s t ^**C}if^{GhR). T h u s ,

this is t r ue of u a lso . Th e e l l ip t ic i ty shows th a t we m ay , a f ter d i f fe ren t ia t

ing the j ~ t h equation in (6.7-5) — Sj t imes , solve for D*^ u^ in

te rms o f the o the r de r iva t ive s . In t roduc ing the va r ious de r iva t ive s o f

6.8. Differentiability. Aperturbation theorem 277

t h e u as new var iab les w\ we see tha t the w^ [x, y ) are ana ly t ic in x for

[x, y ) on GfiR and satis fy asys tem of the form (5 .8 .49) . The ana ly t ic i ty at

(0 , 0) follows from our previous discuss ion of that sys tem in the proof of

Theorem 5 .8 .6 ' .

6.8. Th e differen tiabil ity of the s o l u t i o n s of n on - l in ear e l l ip t i c

s y s t e m s ; w e a k s o l u t i o n s ; ap ertu rb at ion th eorem

In th i s sec t ion we f i rs t ob ta in resu l ts concern ing the d i f fe ren t iab i l ity

of the solutions orweak so lu t ions of non-l inear e l l ip t ic sys tems which

s a t i s fy gene ra l non - l inea r bounda ry cond i t ions . Wea s s u m e t h a t the

e q u a t i o n s of v a r i a t i o n , asdefined in(6 .7 .2), are properly e l l ipt ic a long

the so lu t ion and the l inear ized boundary condi t ions , as def ined in (6 .7 .29)

satis fy the complemen t ing cond i t ion . We m a k e the co r re s pond ing

a s s u m p t i o n s r e g a r d i n g thenon- l inea r s y s tems ana logous to thos e in

§ 6.4. Weconc lude thesec t ion wi th a p e r t u r b a t i o n t h e o r e m for s uch

n o n - l i n e a r b o u n d a r y v a l u e p r o b l e m s .

We sha l l ob ta in our resu l ts on the in te r ior f i rs t and sha l l make use

of the spaces ^H^Q(BR), etc . , defined inDefin i t ion 6.2 .1 , onbal ls BR

in te r io r to ou r doma in .

Theorem 6.8 .1 . Suppose that u ^ Ch(G) and satisfies the equations

(6.8.1) (pj(x,Du) = 0, y = i - . . . , i V ,

on G, where (pj{Xyp)^ C^-^;(9^), h'> i, and is anopen set in [x,p)-

space containing all the points [x, Du); we assume that the equations of

variation are properly elliptic. Then u ^ H^''^^{D) for each p'> i and each

D C. C. G and thederivatives D^ u^, | ^ | < h, satisfy the corresponding

differentiated equations [almost every where if \d\ =h). If, also, the

^3 ^ C^^'^i^) then the u^ ^ Cl^+^{D) for each DC CG. If the cpj are of

class C (analytic) on J l , then the u^ are of class C (analytic) on G.Proof. We f i rs t p rove thef i r s t s t a temen t ior h = 1.L e t XQ b e a n y

p o i n t ofG and le t B(xo, A) CG; we he rea f te r a s s ume tha t XQ =0. Let

0*0 b e ap o si ti ve n u m b e r < ^ / 2 , le t y b e a pos i t ive in teger < i^ , le t Cy be

the un i t vec tor a long the x axis , and define

ul(x) =a-^[u^(x + acy) — u^(x)], \a\ < (TQ.

T h e n t h e u^ sa t is fy the equa t ions

N

U 2 <fc (^) D^ ^a (^) =fja (x), (or n o t su m m ed )1

(6.8.2) ci'ji(x)=f(pj i{x+atey,p(x)+ t[p{x+aey)-p(x)]}dt

01

fjo(x) = - f (fja^y {same} dt, {p= {p^}).

Clear ly a^-> Ujla"^ = {^"ji}' ^^^•) ^^^foj-^fj in C~^j, w h e r e

(6.8.3) (^^jci^) = (pjp^ [x, Du{x)] , fj(x) = - (pj:^v.

So th e eq ua tio ns (6 .8 .2) are pro per ly e l l ipt ic an d t he co ns tan ts C2 an d

R2 in Theorem 6 .2 .6 a re independent o f a. So, choose R < min( i^2 , ^ /2 )

and le t CR € ^ T (^R) w i t h C j == 1 on BRf2 and define 17^ = ^ â- Multiply

ing (6.8.2) by C, we see that Ua satisfies

(6.8.4) IL^, VI = Fi. = C«/y. + 2" ^"n <k k

where the operators Af^^ (which involve the derivatives of CR) a-re of

order < S j + fe — 1 and t he Z^^ hav e the a^f as coefficients.

I t fol lows that the Fja converge in C~î to ce r ta in Fj which a re ex

press ible in terms of x a n d t h e D^ u^, \^\ < tjc. From Theorem 6 .2 .6 i tfo l lows tha t the norms of Ua are un iformly bounded in **H^{BR) for

any given ^, i f JR is sm all eno ug h. T hu s a sub sequ enc e of values of or -> 0

ex i s t s s uch tha t Ua —7 in '^*H^{BE) to some U. By writ ing (6.8 .4) in the

form

(6.8. 5) Z Ulc Ul = Fja - Gja, Gja = y ; ( L ? , - Ljj,) Ulk k

a n d n o t i n g t h a t Gja->0 in H-^^(Bji), we see tha t Ua-Û in '^*H^{BR)

and tha t t / sa t is f ies the l imi t ing equa t ions . Thus U^^^^HI^{BRI2) a n d

sa t is f ies the d i f fe ren t ia ted equa t ions on BRJ^. Since XQ w a s a r b i t r a r y t h efirst result follows for h = \.

The remain ing resu l ts fo l low from repea ted apphca t ions of Theorem

6.2.5 s ince the #y satis fy.

{6.% .6) Ljjc u\ (x) = Z a^j^ {x) D^ u^^ = fjy {x) = — (pja^y (x, D u).k,P

Since u^^ H^^-^^{D) for eac h ^ > 1, it follows t h a t u^^ C^^,{D) for each

ju\ 0 < / ^ ' < 1. Th us , if the cpj^ CJ-^^(9^) for som e //, 0 < / / < 1, it

follows first the af^ and/y in (6.8.3) € C'^J, s o t h a t u^^ Cl^+^. T h e n t h ea s a n d / ' s $ C''^ so tha t ^^^ Cj*+i . I f , "now the (p^^ Q " '" ' ' i t will follow

t h a t t h e as a n d f s^C^-'^ s o t h a t u^^H*'^+^ for each p> \. If

(fj^ C^"*, it follows as before that u^^ C*^^^. T h e a r g u m e n t m a y b e

r e p e a t e d .

The fo l lowing theorem is p roved in a s imi la r way us ing Theorem

6.3 .8 for boundary ne ighborhoods .

Theorem 6.8.2. Suppose that ho is the smallest integer satisfying the

conditions . , , r ,

ho>0, ho -}- hr > \ for each r.

Suppose u^^ C*^+ô(G) and the u^ satisfy (6.8.1) on G and

(6.8.7) ;fr(^, ^ ^ ) = 0 ondG.

Suppose, for some h > ho, that G is hounded and of class C*"^' , 0 = max^^:,

(fj^ C^~^î^), and ir^ C^'^^''{W), where 9^ and W are appropriate

6.8 . Differentiability. A perturbation theorem 2 7 9

neighborhoods in the {x, p)-space containing all the points [x, D u{x)] for

x^G. We assume that the linearized equations are properly elliptic and the

linearized boundary conditions satisfy the corresponding complementing

conditions on dG. Then u^^ H^^'^^{G) for each _> > 1 and the D^ u^

satisfy the differentiated equations on G. If, also, the g)j ^ C^"^' (%) and

Xr^ C^'^î^), i^^'^ ^ ^ € C^^'^^{G). Corresponding results hold in the C"^

and analytic cases. / / ô > 1, G is of class Cjo+ô^ u^^ C^*^+'ô~i(C),

(fj^ Cl^-'^{^), and the Xr^ C^^^^^^Wj, then u^^ Cy ôf^Q).

Proof. We prove the f irs t conclusion for h = \. The in te r io r r e s u l t s

have been p roved in Theorem 6.8 .1 . So, let XQ be a bounda ry po in t and

map a ne ighborhood o f XQ o n t o GA in the usua l way . Le t Cy be the un i t

vector in the (new) xv direc t ion , assumed tangent ia l . I f we def ine the

u^ as before , we see tha t they sa t is fy (6 .8 .2) and the boundary condi t ions

where 6^^ and gj" are defined in terms of the Xrp'^ a n d ~Xrxy as in (6.8.2).

From Theorem 6 .3 .8 and the convergence in C^^'^^'' of b° a n d g'^ to &r and

gr, respec t iv e ly , we conc lude th a t we m a y f ind an i?2 > 0 such th a t th e

bound in (6 .3 .49) holds on GR, in de pe nd en tly of cr if 0 < i? < R2. T h e n

we defin e Ci? an d U^ as before , we may repea t the a rgument of the pre

ced ing proof t o show th a t U^ ~>u^y in H*^'^^^ {GR) . Usin g th e differentia lequa t ions , we f ind tha t the no rma l de r iva t ive u^^ also H * ' ' ' ^ ^ ^ ( G R ) . T h e

fur ther d i f fe ren t iab i l i ty resu l ts a re ob ta ined by repea ted appl ica t ion of

Th eorem 6 .3 .7 - Th e l a s t s t a tem en t i s p rov ed in the s ame wa y .

The equations analogous to (6 .4 .1) are of the form

{6.S.S) [ ^ 2 ' ^ ' ^ ^ ^ * - ^ ? ( ^ » ^ ^ ) ^ ^ = 0 , v^C^[G)G j= \ | « | ^ «

where we shall assume that S; , t)c, mj, ho, a n d QJJC satis fy the condit ions in(6.4.1), (6.4.2), (6.4.3), and (6.4.4) and 99 involves der iv a t ive s D^ w^ wh ere

\P\ ^Qjk' W e as s ume tha t e ach cp'^ w i t h \(x\ = w y$ C^ a t leas t and th a t

the system of operators in (6 .4 .6) is properly e l l ipt ic for each set of

n u m b e r s

(6.8.9) < f0 = 9^^ (^0, po). \(X I = W;, j /5 I = QJJC

for (xo, po) in a neig hb orh oo d 9^ of a l l po int s [x, D u(x)']. T h e t h e o r e m

analogous to Theorem 6 .8 .1 may be s ta ted as fo l lows :

Theorem 6.8.3. Suppose that Sj, tjc, mj, QJ^, ho, and the 99 satisfy theconditions stated above and suppose u^^ C*' +'*o(G) and satisfies equations

(6.S.8) on G. Suppose, for so m e A > Ao + 1, that the cpf^ C''('j!fl) where

T = m a x (0 , A — S; + 1^1) and ^t has its usual significance. Then

u^^H^^+ ^{D) for each D C C G. If the cp^^ Cl(^), 0 < / ^ < 1, then

uic^ Cl^+^^{D) for each D C C G.

2 8 0 R eg u l a r i t y t h eo rems fo r g en e ra l b o u n d ar y v a l u e p ro b l ems

Sketch of proof. W e prove the f i r s t s t a temen t ioi h = ho + i. T h e

second s t a t em en t follows from th e the or y of § 6 .4 . T he resu lts for larger

h fo l low by repea t ing the a rgumen t .

We begin by rep lac ing v^ in (6.8.8) by v^ = a~^ [v3 (x ~ a Cy) — v^x)],

Cy having i t s usua l s ign if icance ; we assume tha t v^ C'^(G) and tha t

10*1 is sufficien tly sm all. B y p ro cee din g as in § 1.11, w e see t h a t th e u^

satis fy equations of the form (6.4.1) where the oF^^^ and/^*^ are defined by

formulas like (6.8.2) for those {j, oc) for which 99^$ 0{^) for som e T > 1,

namely those for which HQ — Sj -\- \(x\ > 0 (for each j , there is a t leas t

one such a since HQ — Sj + nij > 0). For the remaining (; ' , a), for which

th e 99" are no t diff eren tiabl e, we w rit e

JD^'vKx)- (p'^[x,u(x)]dx = J (p'^[x,u(x)]'i — jD'^vi^ix — taey)dt\dxG (? 1 0 J

1

= jD^vi^[x)[-f^^{x)]dx, f^^[x) = f (p^[x + tGey ,u{x + taey)]dt.G 0

(6.8.10)

Since an y 99" w ith | ^ | = m ; is diffe renti able , we see t h a t t h e u^ satis fy

eq ua tio ns of th e form (6.4.1) wher e the a^^^ = 0 an d the/^" " are given in

(6.8.10) a ho — Sj + \(x\ < 0 . Th e us ua l a rgum en t can be ca r r ied th rou gh

us ing Theorem 6 .4 .4 .The boundary condi t ions ana logous to (6 .4 .15) a re :

(6.8.11) j Z ZDy C'xry[^ '^^i^)']^^ = ^> C^C'-iG)eo r \v\<Vr

where the ^ry invo lve de r iva t ive s D^ u^ with | /5 | <.tjc — hr — pr a n d

are such tha t i f C h a s s u p p o r t o n a b o u n d a r y p a t c h m a p p e d o n GR, t h e n

(6.8.11) reduces to

(6.8.12) [2^ Z Dl^'''Xry [^,0\Du[x,0)]dx = 0

if we use the (x, y) no ta t ion o f § 6.3 . We sha l l require tha t the ^ry w i t h

\y\ = pr sha l l be d i f fe ren t iab le and tha t the opera tors

BorJcU^^2: Z ( - ^ ) ' ^ ' & r f c r 0 ^ ^ ^ i ^ ^ , qr = h ~ hr - Pr(6 8 13 ) \v\=Pr \d\ = Qr

Kicvo = Xry p^, [^0, po], (^0, po) ^W, (X= {x, y )),

W being a ne ighborhood of a l l the [X, D u{X)], X^G, sa t is fy the com

p lemen t ing cond i t ion wi th re s pec t to the Lojk- W e a s s u m e t h a t Sj, tjc, mj,

Qjjc a n d ho sa t is fy the i r p rev ious condi t ions and tha t ho, hr, a n d pr

sa t is fy the condi t ions

pr>0, ho + hr+pjc'>\, tk — hr— pr'>0

a n d w e a s s u m e t h a t ho (probably nega t ive) i s the smal les t in teger

sa t is fy ing a l l these condi t ions .

6.8. Differentiability. A perturbation theorem 281

U s i n g t h e m e t h o d s of proof ou tHned above and in § 6.4, we can p rove

the fo l lowing theorem:

Theorem 6.8.4. Suppose that the Sj, tjc, QJJC, ho, hr, pr, (p'j, ^^^ X^v

satisfy all the conditions above, suppose that the u^ ^ Qic+hQ ^^^ ^^^ satisfy

the equations {6.S.% ) on G and (6.8.11) on dG, suppose that, for some

h^ho+ i, G is of class C'«+^ (p^^^ C^'St) and Xw^ C'îW), where

T = m a x {0 , h ~ Sj -^ \a\), co = m a x ( 1 , A + ^r + 171), and 9^ and W

are neighborhoods of the sets {[x, D u (x)], x^ G} in the appropriate [x,p)-

spaces. Then u^^ H^^^'îfi) for each p^X.lf, also, G is of class C^^"^^

^ y € C ^ ^ ) , and xry^ C^{^'). thenu^^ Cy ^{G). Uho > 1, u^^ C^^+'ô'i,

G is of class C*^^^^, 99^^ C^^^), and Xry^ Cîi{^')y where K =^ max (0 , ho

— Sj + $x$ andX = m a x ( l , ho + h + \y\), then u^^ C^*+'ô(G).

We now wish to prove the perturbation theorem of A G M O N , D O U G L I S ,

an d NiR ENBER G ([1], T he or em 12.6). We f irs t prove the following s tan

d a r d l e m m a :

Lemma 6.8 .1 . Suppose the Ljjc and Brjc satisfy the ho — JJL conditions^

on a domain G of class C^^'^^^ and suppose the system L is properly elliptic

and the boundary operators Brjc satisfy the complementing condition on dG.

Suppose also that the problem

(6.8.14) Ljjc u^ = 0 on G, Bric u^ = 0 on dG

has a unique solution. Then there is a constant C, independent of u, such

that if Ljjc u^ = fj on G and BricU^ = gr on dG, then

(6.8.15) 2 'lll^*iy '^^'«< ^fZ'lll/il!!^^^- + 2'lll^r|!|^+N

where gr is any function ^ C^^'^^'{G) such that gr = gr on dG and

(6.8.16) |||a^|!|^ = ^^(V^9)) + i ; m a x | v X ^ ) ! .

Proof. Suppos e th i s is not t rue . Then the re is a s equence u^ s uch

t h a t the left side of (6.8.15) equals 1 for each n and the r igh t s ide tends

to ze ro . F rom the e q u i - c o n t i n u i t y , we may a s s u m e t h e n t h a t Ljk u^~>0

a n d Brjc u^->0 un i fo rmly . But, f rom Theorem 6 .3 .9 and the nega t ion of

(6.8.15), we conc lude a lso tha t Un-û un i fo rmly and u ^ 0 and also

€ Cjl''^^^{G) and u satis f ies (6 .8 .14). This contradicts the h y p o t h e s i s .

L e t us n o w s u p p o s e t h a t the n u m b e r s Sj, tjc, mj, QJJC, ho, and hr satisfy

the cond i t ion in T h e o r e m s 6.8.1 and 6.8 .2 , tha t G is of class Cô+'^+^,t h a t u^^CI^'+ô+^{G), and t h a t (pf^ ^ 6^^+'-'^ and ;^f ^ C^«+'''-+^ in

t h e i r a r g u m e n t s , and t h a t the u^ satis fy the equations (6 .8 .1) on G and

{6.S.7) on dG. Thes e re s pec tive equa t ion s of v a r i a t i o n , as defined in (6.7.2)

1 See Definitions 6.2.3 and 6.3.6.

2 8 2 R eg u l a r i t y t h eo rem s for g en e ral b o u n d ary v a l u e p ro b l ems

and (6.7 .29), reduce to

LjTc v^ -=Q on G, Brk z^ = 0 on dG

Ljjc v ^ ^ 2 I < W D^^K r = sj + h

We wish to f ind a solution u = UQ -\- v oi t h e e q u a t i o n s

(6.8.18) (pf^ {x, Du) = fj(x) inG, Xr i^' Du) =gr(x) ondG,

fo r a l l / a n d g w i t h * | | | / | | 1 ^ a n d * ' || |^ | || |^ s u f f i c i e n t l y s m a l l , w h e r e w e

d e f i n e

C 6 8 1 Q ^ *lll/"lll^ — T III/'Jll'io-sj- *'III i? Iljo — in f y III ^^111^0+ '-^0.0, ly j Illy ilU — Z / Illy; IIU ' Ills IIU — "^^ Z ( Ill& llliWj J.

the norms on the r ight having been defined in (6 .8 .16). Suppose we define

m (^>P) =•• <P?' [ > ^ ^ 0 {X) + P] , Xr (^, P) = xT i^' D uo (X) + P] .

(6.8.20)

S ince we hav e a s s umed th a t UQ^ Cj^+ô+^(G), it follows that cpj^ cô+2-sj

a n d Xr^ cô+hr+2 ^^ j^g a rg um en ts and

(6.8.21) (pj(x,0)~0, ;fr(^, 0 ) ^ 0 .

Now, le t us define y jj a n d cor b y(Pj(^>P) - Z I! 9JvHx> 0)p} = ipj(x,p), T = Sj + tjc

(6.8.22) ^"^'^^Xr{x>P) — Z ZXrp^{x>0)P} =" COr{x,p), X = h — K.

T h e n t h e y jj a n d y)jp^ Qo+i-s^ and the Xr a n d Xrp^ d^+hr+i [^ t h e i r

a r g u m e n t s a n d

(6.8.23) WA^' 0) ^ Wjpi^> 0) = ^ ' COr(::t:, 0) = (JOrp{x, 0 ) .

The equa t ions (6 .8 .18) then become, in tu rn ,

(6 .8 .2 4 ) (pj{x, D v) =fj{x) i n G, Xr{x> -D v) = gr{x) on dG,

, ^ 3 2 5 ) ^^^ '^^ ^ ^^^^^ " " '^^ ^^' ^ ^ (^^^ ^ ^ ^BrJc V^ = g r (^) — C' r [^, -C (^ )] o n ^ G

where L^j^ and Brk are exac t ly the opera tors in (6 .8 .17) .

W e n o w s t a t e t h e t h e o r e m .

Theorem 6.8.5. Suppose (1) that all the numbers Sj, tjc, to = max^^t,

'^jy Qjk, ho, and hr satisfy the conditions in Theorem s 6.8.1 and 6.8.2; (2)

that G is of class Cô+ô+2^ (3) ff^^t u^^ Cy ^^[G), (4) that cpf^ Cô+2-s,and Xr^^ ^ cô+hr+2 ^^ ^j^^^y arguments in % and W, (5) that the u^ satisfy

(6.8.1) and (6.8.7) "îth (pj replaced by cpf^ and Xr by xf\ (6) that the L-

system in (6.8.17) is properly elliptic, (7) that the Brk satisfy the complement

ing condition on dG, and (8) that the homogeneous system (6.8.17) has only

the zero solution.

6.8. Different iabi l i ty . A pe rtu rb at io n theo rem 283

Then v^^ C^^'^^^^'^ [G) and there exist numbers A and J5 >0 such that

if the norm s of f and g, as defined in (6.8.19), are < ; A, then the equations

(6.8.18) have a unique solution v of norm <C B. Ifho> \,all the differenti

ability requirements may be reduced by 1 in terms of ho {G of class C^o+'io+i^

etc.) and we conclude that v^^ C^^^^^^^[G).

Proof. The addi t iona l d i f fe ren t iab i l i ty of vfollows from Theorem

6.8 .2 . Us ing th is d i f fe ren t iab i l i ty , we in t roduce the func t ions cpj, Xry Wh

a n d cor as in (6 .8 .20) and (6.8 .22) and we conclude (6.8 .23). From Lemma

6.8 .1 , i t fo l lows tha t the problem

(6.8.26) Ljic w^ =jon G, Brjc w^ = gr on dG

h a s aun ique s o lu t ion wh ich we deno te by P (/, ^) == ze' in which Pis a

bounded operator from *CjJ X * 'C2 (norms in (6 .8 .19)) into **C]J whichconsis ts of a l l vectors u^u^^ C^^'^^ (G) w i t h

T h e n t h e

(6.8.27)

* * | | 1

equations (6 .8 .18)

V -{- T V =w

W e wis h to s how tha t

**\\\Tvi - Tv2K<-**III ^ " | l ! / « 2

(6.8.28)

w h e r e L :

L e t

(6.8.29)

is sufficiently sm all

f*{x) =y)j[x,Dvi

and le t u s s uppos e tha t

^ \ \ \ n •

a n d

w =

\\\vi

:W],

k

III h+ho

(6 .8 .25) are equivalent to

-PiAg)^

- M \ \ '

^* W =

Tv =

if **

= (Or [x ,

= Pbp,oj].

l l l ^ ^ l i l S < ^

Dvi{x)],

i=\,2

1, 2,

(6.8.30) **lvill<L, **lvi~vtll= R

Then, from (6.8.23), we see that

(6.8.31) \frMfU)\<CL-R xG

Now, l e t t ing the p^ be a rranged in as ingle sequence p^, .. ., p^, we see

t h a t ^

/ f w - / A W -y ;*2 w = s ^ H i ^ ) • [ i w - ^ i w i ,1

(6.8.32) Aj'i(x)=fipjj,i[x,p{x,f)]df, p(xj')=p2(x) +

+ f[Pl{x)~P2{x)].

In case ho — Sj =0, so tha t ho = — Sj = 0, all we need to do is show

t h a t hf,{ff, G) ^CLf'R, From (6.8.32), (6 .8 .23), and the fact that y )j

a n d yjjp^ CJ, we eas ily f ind that

\Aji{x)\<CL, \Aji(x2) - Aj'i{xi)\<C 1/^1x2-xilf" if Z < 1 .

The desired result in this case follows.

More genera l ly , a der iva t ive of o rder r = ho — Sj (> 0) of /f^ — /fg

^ff is a sum of products of a derivative of order q of an ^^-^ and one

of order r — q oi SL (p\ — pl), 0 <, q <,r. Any der iva t ive of o rder q of an

Aji is the sum of terms of the form

1

Q =.ly,^^^j,[X,p{X, f)] • [Dr^P^l) . . . {DympJCn) dt'

0

(6.8.33) K=^[ki,. . .,km), W > 0 , \oc\ + \yi\ + \-\yra\=q,

7co > 0 , CO— \,. . . ,m\im>Oy ipjiaK = D^Dpici. . . Dpkmipjpi

where oc and the yo) are m ult i-in dic es. If m > 2 or ^ < r, it is easy to see

tha t eve ry s uch t e rm Q^ C^ a n d t h a t

h^(Q) < m a x | V ^ W | < C^> \Q(^)\ ^ ^ ^ >

XeGth e las t hold ing even if ^ = T. If 5' = r , th e only non -differ entiab le te rm s(ass um ing r > 0) ^ are of th e form

1

Q = J tpj pi j)ic[x, p {x, t)] D^ p^ {x, t) dt, \p\ =r.0

L e t Q be s uch a t e rm and abb rev ia te the in teg rand ip Q. T h e n1

^(^2) - ^(^1) - /v ^( ^2 , t') [Q{X2, n - Q(xi, f)] +

0+ Q{xi,r)\jp{x,^j')-^[x^,t')-]dt'.

Using the definit ions of p [x, t') a n d Q [x, t') = D^ p, and the fu r the r

differentiabil i ty of y)j, we ob ta in

\Q{^2) — ^ ( ^ I ) | <CL \X2 — xi\f.

T h e o t h e r ff and the correspondingly def ined g^ = g*^ — gf^, can be

handled in the same way. Thus i f ** | | | z; |||^ < L, we con clu de from th es e

re s u l t s and the boundednes s o f P t h a t

* * | | | r ? ; i - Tv2\\\^^< C L ' ' * * | | | ? ; i - ? ; 2 | | | ^ ,

if L is small enough. The result follows.

The following special case of a variant of Theorem 6.8.5 can be ex

t e n d e d t o s y s t e m s :

Theorem 6.8.6. Suppose G is of class Q , the A''{x, z,p) ^C%,

B{x, z,p)^ C^, zo^C^(G), zo satisfies the equations (1.10.13) (N = 1) ,

and the equation o f variation

/ . . ..X ^(^^^) ^ r i (^'^^'^ + ^'^) - ('"'^'^ + ^ 0 = 0 with Z = ZO .(6.8.34) ^

a°^P = Al^ [x, z{x),^z{x)-\, h^ = Al, c^ = Bp^, d =: B^

has only the zero solution if ^ = 0 on dG. Then there are constants C,

^0 !> 0, and CTQ > 0 such that the boundary problem

(6.8.35) L{z]l:) = 0, C = 1 o n ^ G

6.8. Differentiability. A perturbation theorem 2 8 5

has a unique solution C € ^ ^ [G) provided || 2: — ô || < <7o. If we denote

this solution by % [z), then we have

\\%{z2)-%[z^)\\<C\\z^-zr\\ if | | ^ i - ^ 2 | i < a o ( | ^ | | = l| |9 ' |l!I).

(6.8.36)

Thus there exists a unique function Z from [—^0,^0] i'^to CJ^[G) which

^C^[— Qo, Qo\ and satisfies

(6.8.37) % = ^^^)' ^ ( 0 ) = ^ o .

Proof. From Lemma 6 .8 .1 i t fo l lows tha t there i s a cons tan t C\ s uch

t h a t if / ^ C^^{G) a n d ^ ^ C | (^)> the re is a un iqu e s o lu tion of the p rob lem

(6.8.38) L(2:o;C) = / o n G , C ^ ^ o n ^ Ga n d t h a t

(6.8.39) l | C i < C i ( | | | / | | | o + | g | | ) ( | k | | = | | k | p .

So we set f = fo + o) in (6.8.35) w he re Co is th e solu tio n of th e pr ob le m

(6.8.35) with z = ZQ. Then we write (6 .8 .35) in the form

(6.8.40) L{ZO;CD) = - [L[Z] W)-L(^0; CO)] - [L{z; Co) - L{zo\ Co)].

From this , (6 .8 .38), and (6.3 .39), we find that

(6.8.41) l l c a | | < C 2 | | . - . o | | ( | | c . | | + ||C o||).

T h u s , if C2 11- — 2:0 II < 1/2, we see t h a t

(6.8.42) \\w\\ < 2C 2 | | . - ôll • IKoll, llCi < llôl • (1 + 2 C 2 I . - ôll).

Now, if we have CA; = ^^{^k), \^k — .ô|| <o 'o , ^ = 1, 2 , th en we h av e

L(^2, C2) - L{zi, Ci) = 0, C2 - Ci = 0 on ^G , and

L{z^, C2 - Ci) - - \.L{Z2) Ci) -L[zv, Ci)] .

A n anal ysis l ike th a t in (6 .3 .40)— (6.8.42) shows t h a t

IIC2 - fi ll ^ C3 1^2 - ^lll • llCli < C4 1^2 - ^ l l | .

Thus g (Z) satisfies a LIPSCHITZ condit ion so the theorem follows.

Theorem 6.8.7. Suppose that G, the A^, B , and ZQ satisfy all the hy po

theses of Theorem 6.S.6 except the last. If X{)^ G, there exists an R'> 0

such that the problem (6.8.35) "^Hh z ^= ZQ and G = B{XQ, R) has a unique

solution C (ind C(x) > 0 in l^{xo, R).

Proof. We notice from (6.8.34) thati:(ô;C) = a«^{x)C,ap + 'b-{x)C,. + 'c{x)C

where the a°^^ a n d 'c^ Cl(G) a n d 'b'^^ C^(G) and the coeffic ients depend

on zo , of course . Now for each R s u c h t h a t B {XQ, R) C G, we define

VRiy) =CR(xo + Ry ) = 1 +ojR(y), y^B(0,\).

2 8 6 A variational method in the theory of harmonic integrals

Then the desired COR satisfies the conditions

Al^WR^.p + {Al^ - Al^) (DR^o^p + RBla>^ , + R^ CRCOR + R^CR = 0,

(6.8.43) COR =0 on dB(0,\), Alf(y) =a'^^[XQ + Ry), etc.,

Af = A%^(0) = a-n^o).

From (6.8.43), we conclude that

IICORW <{ZiR + Z2R^+^ + ZsR^ +Z4R^) \\COR\\ + R^Z,.

The result follows.

Chapter 7

A variational method in the theory of harmonic integrals

7.1 . Introduction

In this chapter, we show how one can apply the variational method

to the study of the theory of harmonic integrals. In his first paper on th e

subject, W. V. D. H O D G E [1] used a variational method in this theory to

study certain boundary value problems for forms defined on domains in

Euclidean space (using Cartesian coordinates). But, in order to carry

his theory over to compact Riemannian manifolds, he and subsequent

writers found it expedient to employ methods involving integral equa

tions (see H O D G E [2], K O D A I R A , D E R H A M - K O D A I R A , and references

there in). More recently, M I L G R A M and R O S E N B L O O M ([1], [2]) and G A F F -

NEY ([1], [2]), have treated certain problems by their ''heat equation"

method involving parabolic equations. The variational method was

applied togeneral compact Riemannian manifolds by M O R R E Y and

E E L L S and to such manifolds with boundary by M O R R E Y [11]. A closely

related method was employed concurrently by F R I E D R I C H S [3] in both

cases. Certain boundary value problems had been discussed previouslyby D U F F and S P E N C E R and by C O N N O R ([1], [2]).

One of the principal results which was obtained early is now known

as the ''Kodaira decomposition" and states that

(7.1.1) S2-©ee:e® or&l= r@r@c^r

where Sj denotes the space of forms of degree Tg which are either all

even or all odd, which have components in S2, the inner product being

defined in § 7.2; §^ denotes the space of harmonic fields in S2, an d (S

and ^ are the respective closures of the spaces {da} and {d^} wherea and ^ are smooth (see § 7-2 for definitions). In case all th e forms in

volved are smooth (which is th e case if M and the given form in C are

smooth—see below) this decomposition leads to the principal theorem

of H O D G E which is that there is a unique harmonic field in each cohomology

class of closed forms (i.e. those co with dco — 0) on M. This statement is



7.1. Introduction 287

e q u i v a l e n t to the s t a t e m e n t : Suppose M is of class C°°, co^ C^ ( M ) , and

dco = 0; then there is a unique harmonic field H (^ ^^{M)) such that

(7.1.2) 0) = H + d(p for some ^ ^ C ( M ) .

The writers mentioned above, except for E E L L S , F R I E D R I C H S , and

MoRREY, were not espec ia l ly concerned e i ther wi th the dif fe ren t iab i l i ty

of M and of th e forms or w i t h the n a t u r e of the e l e m e n t s in the spaces

K and '^ a b o v e . Not only does the v a r i a t i o n a l m e t h o d y i e ld a very s imple

proof of the decompos i t ion in (7.1.1), the f in i te d imens iona l i ty of § (in

the case where M is w i t h o u t b o u n d a r y ) , and the fac t tha t in the gene ra l

case in (7.1-1) where c ^ ® and d^^, there ex is t fo rms a and ^ in the

SoBOLEV spaces HI (M) s u c h t h a t

(7.1.3) doc = c, dp = d, doc = 6^ = 0,

bu t a l s o s hows tha t the genera l resu l ts ho ld for mani fo lds M of class CJ

on wh ich the metr ic t ens o r is mere ly L ips ch i tz . The success of the m e t h o d

depends on (a) the use of the S O B O L E V spaces HI (M), ^ > 2, (b) the

recent resu l ts concern ing the dif fe ren t iab i l i ty of ' 'weak s o lu t ions " of

e l l ip t ic d i f fe ren t ia l equa t ions (see C h a p t e r 5), and (c) an i n e q u a l i t y due,

in the case of forms , to GAFFNEY [1] and in the case of a s ingle differentia l

e q u a t i o n of higher order to GARBING [1]. The resu l ts in (b) above lead

to ve ry exhaus t ive re s u l t s conce rn ing the dif fe ren t iab i l i ty of v a r i o u sso lu t ion forms . However , some addi t iona l d i f fe ren t iab i l i ty is o b t a i n e d

by us ing the spec ia l charac te r of the e q u a t i o n s (see Theorem 7 .4-1) .

I n the case of a c o m p a c t m a n i f o l d w i t h b o u n d a r y , it t u r n s out t h a t

t h e SOBOLEV space H\(M) sp l i t s in to the two closed l inear subspaces

H\+ [M) and ^ | ~{M) of forms a for w h i c h noy = 0 and to) = 0, respec

t ive ly , on the b o u n d a r y h M, n co and t co be ing the n o r m a l and t a n g e n t i a l

p a r t s of CO (see §7.5)- The GAFFNEY inequa l i ty ho lds , on each of the s e

s paces s epa ra te ly and th i s , toge the r wi th the bounda ry d i f fe ren t iab i l i ty

t h e o r y of C h a p t e r 5, m a k e s it possible to ca r ry ove r the preced ing theo ry

to each of these par t ia l spaces . By us ing these two spaces and ce r ta in po ten

t ia ls def ined there in , it is possible to prove firs t a decompos i t ion l ike

(7-1.1) in w h i c h oc^Hl+ and ^^H\-. In th i s decompos i t ion howeve r ,

| ) may have in f in i t e d imens iona l i ty (see ^7J)- After th is is s h o w n a

g r e a t v a r i e t y of b o u n d a r y v a l u e p r o b l e m s t r e a t e d by D U F F and SPENCER

and C O N N O R ([1], [2]) are treated.

I t was p o i n t e d out t h a t F R I E D R I C H S was w o r k i n g on the s e p rob lems

a t a b o u t the s ame t ime as were E E L L S and the au tho r . Af te r a n u m b e rof conferences, it s eemed as t h o u g h t h e r e was enough difference between

our resu l ts as well as our m e t h o d s to warran t pub l i s h ing bo th ve rs ions

and th i s was d o n e . The vers ion presen ted here is m a i n l y due to M O R R E Y

and E E L L S and M O R R E Y [11]. In §§7.2—7.4, we cons ider manifolds wi th ou t

b o u n d a r y and in §§ 7-5— 7.8 , we cons ide r man i fo lds wi th bounda ry .

2 8 8 A variational method i n t he theory of harmonic integrals

W e do no t a s s um e tha t the man i fo ld is o r ien tab le ; the re a re com ple te ly

para l le l theor ies involv ing only "even forms ' ' o r on ly ' 'odd forms '* (see

DE R H A M - K O D A I R A ) . We sha l l assume tha t a l l the forms cons idered

are even or e lse tha t they a re a l l odd ,

7 .2 . F u n d a m e n t a l s ; t h e G a f f n e y - G i r d i n g i n e q u a li t y

Definition 7.2 .1 . We adopt the usua l def in i t ion of a Riemannian

manifold of class C^ (wi th o r wi thou t bounda ry ) and d imens ion v a n d

define those of class C^, 0 < /* < 1, in th e obviou s w ay : an y two a dm is

s ib le coord ina te sys tems a re re la ted by a t rans format ion of c lass C^,

and if 6 i s one such wi th domain G CRv, t h e i n d u c e d c o m p o n e n t s

giAx)^Cl-^ on G,Assumption . We assume that any manifold M is compact and of class

at least C [ . *

We shall be concerned with exterior differentia l forms of degree

r {0 <,r '<v) onsi mani fo ld M ; we call the se s im ply r-forms. I n t h e d o m a i n

of any coord ina te sys tem, such a form may be represen ted as fo l lows :

(7.2.1) 0) =2a)u...irdxi' A . . . A dx^^

w h e r e a)u...ir ^^^ th e components of co in th a t coo rd ina te sys te m an d Adeno te s the ex te r io r p roduc t . I f two coo rd ina te s ys tems wi th coo rd ina te s

(x ) a n d {x ) ove r lap , the re la t ion

0Ji,...ir(^) = £ lcoj,„j,[x(x)\ —^ - ^ - ,

(7.2.2) f"t~ ^ ^^^ ev en fo rm s

/ / I / I for odd fo rms , / = ^ { ! / • • • ^ g | ,

holds be tween the components . The no ta t ion in (7-2 .1) i s o f ten abbrev ia

t e d t o

(7.2.3) oj='^co^dx^

w here / [or o th er cap i ta l le t te r , correspondingly] den otes a sequence in

w hi ch 1 < H < ^2 < • * • < V < V.

T h e re lat io n (7-2.2) allow s us to define (£2 as follows:

Definit ion 7.2 .2 . An r-form o) onM will be said to be in 2p {resp. HI)

^ i t s components a re in Lp (resp. HI). If oj a n d rj are ^ '-forms of the same

ki nd in S2, we define thei r inner product b y(7.2.4) (CO, rj) = JF{P; co, ri)dS[P)

M

* Actually t h e reader will observe that many of the results have proper gene

ralizations t o manifolds of class H^ where p > v. Such manifolds are of class Cj^

with ^ = 1 — vjp.

7.2. Fundamentals; the G A F FN E Y — G A R B I N G inequality 289

w h e r e in any admis s ib le coo rd ina te s ys tem

F{P) oy ,ri) ^i;gi^{x)coi(x)riK{x), dS{P) = r{x)\dxK .. dx'<,

(^•^•5) . . .

I,K

I P'H ^1 . . . ^ * l ^r

aij-fCir(x) = [g(x)] 1 12

a n d g is the d e t e r m i n a n t of the gij (and the g'^^' m a t r i x is the inverse of

t h e gij m a t r i x ) . It is easy to see t h a t the va lue at P of i^ is i n d e p e n d e n t

of the coord ina te s ys tem p rov ided tha t co and rj are of the s ame k ind .

The fo l lowing theorem is well -known and e v i d e n t :

T h e o r e m 7.2 .1 . For each r, 0 < r < i , the totality of even (resp. odd)

r-forms in Q^ ( ^^^ equivalent forms identified) forms a real Hilbert space

with inner product given hy (7.2.4).F o r m s w are often regarded as a l t e rna t ing t ens o rs and w r i t t e n in the

form

(7.2.6) co = Jy ZoH.. . . ir dxh A . . . A ^xh.

W h e n t h i s is done the c o m p o n e n t s are defined for all va lues of the

indices from 1 to v and are a n t i s y m m e t r i c in the ind ices , a c o m p o n e n t

be ing zero if two ind ices have the s ame va lue . In th is case , co t r a n s f o r m s

like an ord ina ry cova r ian t t ens o r , excep t for the fac tor e in (7-2.2) andF(P;co, rj) has the form

(7.2.7) F{P; w,ri)=y^ 2'^*^^^ • • -^^'^^^ Wi....*.^^....fc.H ir

w h e r e the i's and k's all run independen t ly f rom 1 to v. We shall not

often use th is form but it is very useful in c o m p u t a t i o n s (see the proof of

T h e o r e m 7-2.4 below and §§ 7-5, 8.3, e tc . ) .

W enow

wishto

i n t r o d u c ean

inne r p roduc t in tothe

spaceH\.

Definit ion 7.2 .3 . Let U = ( ^ 1 , - - -, UQ) be a f in i te open cover ing of

M by coord ina te pa tches Oq w i t h d o m a i n s Gq (LIPSCHITZ and in Rv) and

ranges Uq. If co and rj^Hl (see Def. 7-2.2), we define

(7.2.8) ({co, rj))vi = {co, rj) +Z f SIco^'^Vit-^^'Q=l fY^ I a = l

w h e r e the co ^ and tj^f^ are the c o m p o n e n t s of co and r] with re s pec t to d q.

T h e n we define the co r re s pond ing no rm

(7.2.9) ||coiu = ( K c o ) ) r .

Theorem 7 .2 .2 . For each coordinate cover U and each r, the space of

r-forms in HI forms a real Hilbert space HI''' with inner product given by

(7.2.8). Any two such inner products are topologically equivalent.

T h i s is ev iden t .

Morrey, Multiple Integrals \g

2 9 0 A var ia t io nal me thod in the theo ry of harm onic in tegra l s

Definition 7.2.4. If o) is an r-form ^ H\ w i t h r <^v, we define its

exterior derivative dm SiS t h e r + 1—form defined by

(7.2.10) dw =2^coja^dx'^Adxl.

If r = V, we define dco = 0 ; we have a s s umed co given by (7-2.1) or

(7.2.3), of course .

Remark. We note tha t i f we wish to wri te

do) = ^ {d(ji>)jdx^J

t h e n w e m u s t h a v e

(7 .2 .11) (dco) J =2;\-iy-^dcoy ldxJyy = - l ^

(Jy =Jl> •"> h-1 > Jv+ly • • •, jr+l) •This definit ion is se t up so that the following vers ion of Stokes '

theo rem ho lds .

Theorem 2.3 .7 (S tokes ' theorem). If b is an oriented manifold of class

C^ in M having dimension r + 1 <^^^ boundary b h of class C ^ and if co is

even and of class C^ on b , then

(7.2.12) Jdco=Ja)b 6 6

where bb

has its usual orientation induced by that of h.

We do not need th is theorem and there fore omit i t s proof.

Def inition 7.2.5. If co is an r- form ^H\ giv en b y (7.2.1) an d r > 1, we

define its co-differential ^ co by the cond i t ion tha t

(7 .2.13) [oj,d(p) = (da),(p)

for every 99^ H\'^~^. If co is a 0-for m , w e defin e ^ co = 0.

Theorem 7.2.4. If co ^ HI and is given by (7-2.5) with r > 1, then

{dco ) = r - ^ 2 :(dco )jc ,. .. jcr .^dxKA. . .Adx^r-. ,

y^ ~ ^)- ki...kr-i

-(day )k,„.jcr_,=g-^co^jc,...Jcr-^x- + (gl^ +P-^ g°'^r^-)copfc... .jcr-. +r- 1

(7.2.14) +2IgVg%g°'^C0^k^...Jc,_^tk^^^,.,Jc^_^.d=X I

If CO is even (resp. odd), then d co and 6 co are even {resp. odd).

Proof. W e may choos e cp in (7-2.13) to have support in a coordinate

p a t c h w i t h d o m a i n G; we a s s ume cp given in the form (7-2.6). Th en , from

(7.2.4), (7 .2 .7), and (7.2 .11), we obtain

(co, dcp) == ^ f^ghh . . . gVr coa) \^{~^)'-^ 9 (,.;)

G (*).0') y = l

rdx

a v = i («),(>•) '^

( 7 .2 .15 ) {i) = i^...i,. {j'y)=h.. .jy-ijy+i.--jr. e tc .

Essent ia l ly the fo l lowing lemma has been proved by GAFFNEY [1 ] :

Lemma 7.2 .1 . Take any £ > 0 and r such that 0 < / < r. Given any

point %{s^M there is a coordinate system 6 mapping the open hall B (0 , q)

of radius q in E v onto a neighborhood U of XQ and a constant I such that

(7.2.18) D(co) > ( ! - £ ) / Z (îxo^)^dx -l(oj, co)

for any r-form co ^ Hl'^ whose support is in U.

Proof. Let us set Bg = B{0, q). Then for any co whose suppor t i s in

U we can write

(7.2.19) D(co) = f Z W' '" îxo^ ^JxP + 2 ^^^^ ojixo^ CO J + c^^ coi coj] dxB, iJ

a n daiJ«^(x) = aJi^°^{x), ciJ[x) =^ cJi(x),

where the as are combina t ions of the gij only and the h's a n d c's are

s imila r combina t ions of the gij and the i r f i rs t par t ia l der iva t ives . Con

sequent ly i t fo l lows f rom our assumption tha t the as are Lipschi tz and

t h e Vs and c ' s bounded and measurab le a t leas t .

We begin by choos ing a f ixed coord ina te sys tem mapping 0 in to XQ

a n d BQ onto a ne ighborhood of XQ^ with ^^^(O) = dtj. Since the a's a re

LIPSCHITZ, we may choose q s o s ma l l tha t

(7.2.20) faiJ^'^x) o)ix<^cojx^dx > DQ[(O) — - I Z iîxo^)^^^,J 2 J J ^

BQ BQ

w h e r e DQ (co) is the Dirichlet integral in the euclidean case with cartes ian

coord ina tes . S ince the h's a n d c's are bounded and s ince for any ?7 > 0

we have

\2(X^\ <rjoc^ + rr^^^,

i t fol lows thatD(co) > Do(oj) — e f ^{o)ixo)^dx — l{o), OJ).

BQ

Now s uppos e tha t the MJ are of class C^ on BQ and van i s h on and

n e a r SQ. Then by us ing the formula (7 .2 .13) , we may wri te

B>o{o)) = [do dco + ddo co, co)o.

U sin g th e fo rm ula s (7.2.11), (7.2.14), (7.2.4), an d (7.2.5) w ith ^«^ = ^«^

we f ind th a t(do dco + ddoco)o = — 2jfîôoJidx= f ^{coix<^)^dx = Do{co).

(7.2.21) ^' "'

Using approximations , the las t equali ty in (7 .2 .21) holds for a l l forms co

in HI with the s ta ted p rope r t i e s .

7.3- The variational method 2 9 3

Theorem 7.2.6. For each r = 0, . . ., n and coordinate covering U of M^

there exist constants K^ > 0 and L]x such that

(7.2.22) D{o}) > i ^ u ( K ^ ) ) i : - ^ u K oj)

for every o)^H\^,

Proof. From Theorem 7.2.2 i t is suffic ient to prove this for some

pa rti cu lar I t . Le t IX = [U\, . . ., UQ) be an open covering of M by coord i

na te pa tches such tha t each poin t ;^^ M is in some Uu satisfying (7.2.18)

wi th 6 = - , s ay . L e t Gi , . . ., GQ be the doma ins in E'^ s u c h t h a t Ujc

= Ok{Gk) for all k. There exis ts a f ini te sequence (pi, . • ., cps of Lipschitz

functions on M, each of which has suppor t in te r ior to some Uq, and such

t h a t

(7.2.23) Z ( p s [ ^ ) = 1

for all x^M.

Now if (7 .2 .22) were fa lse for the U jus t described, there would exis t

a seq ue nce {co^} of r-fo rm s in H\^ s u c h t h a t (D cop) a n d {cop, cop) were

un ifo rm ly bo u n d ed b u t Ijco ) llu ^ ^ = ^- T he n, for som e s, q, and s ome

subsequence , s t i l l ca l led mp, we wou ld have

( ^{(psoyf)l-dx-^ oo ,

whe re cps has s uppor t in Uq, s ince

s\\(^Avi<^Z\\^sO}p\\n

a n dQ .

II 95 (0 11 = (995 (.Op, (fsojp) +2^ 2^ ((fs a>^^0|a dx.a==i4 I, a

But i t is easy to see that D{(psO)p) a n d {(psOOp, (psOJp) are un iformly

bounded . From our choice of ne ighborhoods we have reached a contrad ic t ion wi th the fac t tha t

D {(fs (Op)^^ Z ((Ps cof)l« dx.^ J I,oc

7 . 3 . Th e var ia t ion a l m eth o d

We begin wi th the fo l lowing lemma.

Lemma 7.3 .1 . Let 9)1 be any closed linear m anifold in the space £2 ^/

r-forms on M. Then either there is no form (joofW which is in H\'^ or thereis a form COQ ^^ Wr\HY with (coo, coo) = 1 which minimizes D{(o) among

all such forms.

Proof. If Wl conta ins no form in Hl'^, the re is no th ing to p rove . O the r

wise let {CJOJC} be a min imiz ing sequence , i . e . one such tha t {cojc, (ojc) = 1

a n d (Ojc^Wl C) Hl'^ ioT k = 1 ,2 , . . . , and such tha t D((ojc) approaches i t s

infimum for a l l co^ 501 H Hl'^. From Theorem 7 .2 .6 i t fo l lows tha t the

{{(jojc, a)jc))u are un iformly bounded . Accord ingly , a subsequence , s t i l l

cal led {cojc}, exis ts which converges weakly in Hl^ to some form COQ. B u t

from Theorem 7 .2 .$ . cojc te nd s s tron gly in S j to coo an d D{co) is lower-

semicont inuous wi th respec t to weak convergence in Hl'^. The proof of

the l emma i s now comple te .

Definit ion 7.3 .1 . A harmonic field co on M is a form in HI on M for

w h i c h dcD = d CO = 0 a lmos t everywhere . We wil l le t §^ denote the

linear manifold of harmonic f ie lds on M of degree r.

T h e o r e m 7.3 .1 . For each r = 0, . . . , r ( = di m M) the linear manifold

^^ is finite dimensional.

Proof. The ^ | - forms a re dense in Sg , s ince the Lipschi tz forms a re .

L e t Wli = 82- There is a form co i in Tli H Hl'^ which min imizes D (co)among a l l such forms wi th {co, co) = 1. L e t WI2 be the c losed l inear

manifold in S^ orthogonal to coi , and le t C02 be the co r re s pond ing m in imiz

ing form in 50^2• By cont inu ing th is p rocess , we may de te rmine success ive

minimizing forms coi , C02, C03, . . . , each satisfying {cok, ojjc) = 1 and be ing

or thogonal to a l l the preceding ones .

Now if D (coi) > 0, the re are no ha rm on ic fields ^ 0 since D (coi)

<Z)(co2) < • • •. O n th e o the r h an d , suppose D{co]c) = 0 for all values

of k. Th en b j^ Th eore m 7-2 .6 , ((COA;, COA;))U i s un i fo rmly bounded in k,whence a s ubs equence {cop} converges weakly in Hl^ and hence s t rong ly

in £ | to som e form coo in Hl'^. T hi s is im po ssib le s ince t h e co r form an

o r thonorma l s ys tem in S^ .

Theorem 7 .3 .2 . For each coordinate covering U of M there is a constant

Ao such that

(7.3.1) D(co) >Ao((co,co))n

for any oj in H^ which is orthogonal to §^'.

Proof. Fo r, le t coo be t h a t for m in H^^ ( there is one s ince each harmonic f ie ld is in ^ | ) which min im izes D{a)) among a l l 00 in HY w i t h (w , co)

= 1 an d CO orthogonal to §^ . Then c lear ly D (coo) > 0 and by homo gene i ty

(7.3.2) D{a)) >i ) (coo ) (co , co )

for all CO in i^l*" and orthogonal to §^. By Theorem 7.2.6 we see that

(7.3.3) /û((ct>, co))u < {1 + i: u/ ^( co o) } /) (c o ),

from which (7.3-1) follows.

Theorem 7 .3 .3 . Suppose coo is any form in S2 <^<^ orthogonal to § ^ .Then there is a unique form QQ in H^ and orthogonal to § ^ such that

(7.3.4) (î3o, iC) + (îô, aC) = (coo, C)

for every C in H^. Moreover, the transformation from coo to QQ is a bounded

linear transformation from S^ into H\'^.

7-4 . F ina l resu l t s for com pac t mani fo lds wi t ho ut bou nda ry 2 9 5

Proof. We see f rom Theorem 7-3 .2 tha t

I(aj) = D{w) — 2 {co, coo)

(7.3.5) > Ao {{CO , oj))n - {hl2) (CO, 0)) - (2/Ao) {coo, coo)

> ( V 2 ) {{co, co))n - (2/Ao) (coo, coo)

for all CO in Hl^ and or thogona l to §^ . Also , s ince {co, COQ) is a (continuous)

l inear func t iona l on Hl'^, i t fol lows that I{co) i s lowe r-semico nt inuous w i th

respec t to weak convergence in Hl'^. From (7.3.5) we see that {{cok, cojc))u

i s un iformly bounded in any min imiz ing sequence ; the ex is tence of the

minimiz ing form QQ is es tablished as usual . Since co = 0 is a l lowed we

h a v e

(7.3.6) I {Do) < 0, (Ao/2) ((i^o , Qo))n < (2/Ao) {coo, coo),f rom which the las t s ta tement wi l l fo l low when the o thers a re es tab l ished .

Now let f be any form in Hl'^ and o r thogona l to §^ . Then

(7.3.7) /( i^ o + A C) - / (i3o) + 2A [{dQo, dC) + {SQo, dC) -

- ( c o o , C ) ] + A 2 i ) ( C ) .

Since / {D o + ^ C) > - {^0) for all A, we see t h a t (7-3.4) m u st ho ld for

all f or tho go na l to §^ an d h ence for a l l C in Hl'^ s ince any such can be

wr i t t em un ique ly in the fo rm ^ = H -{- Co, w h e r e Co i s o r thogona l to^^ a n d dH = dH = 0. If Qi also sa tisfie d (7-3.4) for all C in Hl^, w e

w o u l d h a v e

(7.3.8) {dDo ~ dQi, dC) + {SQo - SQi, dC) = 0

for all C in Hl'^, in par t icu la r for C = Do — Qi, f rom which i t would

fo l low tha t Qo — Oi wa s har m on ic . B ut the n i^o — i^ i = 0 , s ince b o t h

a re o r thogona l to ^^ .

Definition 7.3.2. The form Do of Theorem 7.3-3 is called the potential

of coo-W e observ e th a t if a l l th e forms in (7-3.4) we re suffic iently differenti-

ab le , then (7 .3-4) would imply tha t

(7-3.9) ADo^ddDo + ddDo=^coo.

7 .4 . Th e d ecomp os i t ion th eorem. F in a l resu l t s for comp act man i

fo ld s w i th ou t b ou n d ary

The defining equation (7.3-4) for potentia ls is a special case of the

e q u a t i o n s

(7.4.1) {dco - <p, dC) + {doj - ip, dC) - {rj, 0 = 0

for all C in H\. We wil l have occas ion be low to use these more genera l

equ a t io ns . R efe rr ing to th e def in ing formulas for d a n d d, for any f in

H\ whose support is in a s ingle coordinate patch, we see that (7 .4 .1) is

2 9 6 A var ia t ion al m ethod in the theo ry of harm onic in tegra l s

of the form

/G IJ

(7.4.2) + t/[&^^« W cojaôc + ci'îx) CO J +f{x)]}dx = 0

for al l C in ^20^ ^^^ sy m m et ry p rop ert i es of the coeffic ients an d th eir

dependence on the gij and the i r der iva t ives have been noted in (7 .2 .19) .

T h e e' s and / ' s a re l inea r combina t ions o f the rp ' s, ip ' s, a n d rj' s w i t h t h e

coefficients of the 99' s an d ^ ' 5 in th e e' s and those of the rj' s in t h e / ' s

not involv ing the der iva t ives of the gij; tho se of the 99' s an d ip ' s in t h e / ' s

poss ib ly do involve such der iva t ives . Thus i f the manifo ld M is of class

C^ (at leas t CJ), then the gij and hence the a s and the coefficients in

t h e e' s and of the rj' s in t h e / ' s a re of c lass C^~^; the b' s an d c ' s an d th ecoeffic ients of th e 99' s an d y )' s in t h e / ' s a re of c la s s C^"^ (bounded and

meas u rab le i f k = /bt = \). M oreo ver, if G = ^ (0, R) and the gij (0)

= dij, t h e n t h e a' s reduce to what they a re in £^ .

By repea t ing the a rgument a t the end of the proof of Lemma 7 .2 .1

and a s s uming t ha t C van i s hes on and nea r dG, we s ee tha t we may add

ce r ta in cons tan t s to the a' s which do not affect the value of (7.4.2) for

any such C, whence

(7.4.3) a^«^«^(0) =d}l...did''^

T h u s th e eq ua tio ns (7-4.2) are of the ty p e discu ssed in §§ 5.2 and 5-5.

F ro m th e resu lts of §§ 5.2 an d 5-5 we m a y dr aw t he following co n

c lus ions ab ou t t he har m oni c f ields a nd po ten t ia ls d iscussed in § 7 .3 .

Corresponding resu l ts ho ld for the more genera l equa t ions (7-4 .1) :

(i) / / M is of class C\ and co is an r-form in £2 on M, then the potential

Qofo) is in HI on M if r > \;Q and its derivatives are in HI in any coordi

nate sy stem if r = 0. If oj^2p with p'>2, then Q^H \ and in C| if

p '> V, fx = \ — vjp.

(ii) / / M is of class C^, where 0 < /^ < 1 and co^C^^, then Q^ Cj^ if

r>\]Q^Clifr = 0.

(iii) / / M is of class C\ and 0 ) ^ 8 2 , then the components of Q and their

derivatives in any coordinate sy stem are in H\. If co^ S>p, then Q and its

derivatives are in H],', Q^ C]^ if p ^ v and ju = \ —- vjp.

( iv) / / M is of class C^, 0 < /^ < 1 and > 3 < < co $ C^""^, then

^€ C^-^;ifco^Cf^^ then Q^C îfr = 0.

(v) / / M and co are analy tic or C^, then Q is analy tic or C^, respectively.

(vi) / / 0) is a harmonic field and M is of class Q , then co is of class

C^~i (CJ for 0-forms); if M is analytic or C°°, then co is also.

We begin wi th a lemma which fo l lows a t once f rom the re la t ions

d d = d 6 — 0 w h e n e v e r M and the forms are of c lass C^.

7-4 . F ina l resu l t s for com pact mani fo lds w i thou t bou nda ry 2 9 7

Lemma 7.4 .1 . Suppose that (x and ^ are in H\ on M. Then

(doc, d^) = 0 a n d [da, ^) - [oc, d^).

Proof. S u p po se U = (C7i, . . ., UQ) is an open covering of M by co

ord ina te pa tches . There is a sequence (pi, . . ., cps of Lipschitz functionssu ch t h a t 995 > 0, 991 + • • • + 995 = 1 on M a n d su ch t h a t th e s u p p o rt

of cps + (ft i s conta ined in some Uq w hen eve r th e su pp or ts of 995 an d

(pt in te rsec t . I f we wri te as = cpsoc a n d ^t = ^t P, t h e n

[doc, d^) ^Z ^^ocs. d^t), [doc, P) =Z (^ocs, ^t),s,t s,t

the sums being over a l l ordered pairs (5 , t) s uch tha t the s uppor t s o f tps

a n d (pt in te rse c t (and hence bo th l ie in some Uq). Thus the proofs of bo th

pa r t s a re reduced to the ca s e whe re bo th have s uppor t in te r io r to s omeUq. But then in the doma in Gq, w e m a y a p p r o x i m a t e to t h e c o m p o n e n t s

of oc a n d ^ s t rong ly in ^2 ^y C^ f u n c t i o n s; w e m a y a p p r o x i m a t e t o t h e

gij by func t ions gptj of class C^ in s uch a way tha t the gpij satis fy the

s ame L ips ch i tz cond i t ion a s gij, and the respec t ive f i rs t der iva t ives

converge a lmos t everywhere to those of gij. The result follows eas ily .

Theorem 7.4.1. Suppose M is of class CJ, co is in S2 on M, and Q is

the potential of co . Then

(i) a = dQ and ^ = dQ are in HI and satisfy

(7.4.4) CO = doc + dp = 6 (dQ) + d(dQ),

da = dp = 0.

(ii) If co^ HI, then a and ^ are the potentials of dco and doj, respectively,

so that doc and d^ are also in HI.

(iii) If a)^Sp(p '> 2), then Q, dQ, and SQ are in HI and in C^ if

p '> V and /Lt = i — vjp.

( iv) / / M is of class C^ with 0 < /^ < 1 and co^ C^^~^, Ihen Q, dQ,

and SQ are in C^~^ with k '> 2.Proof. We shall prove that (7-4-4) holds in some neighborhood of

each point of M an d w ill also show t h a t e ach p oi nt J\: of M is in som e

n e i g h b o r h o o d U s u c h t h a t

(7.4.5) (doc, dC) + (6 oc, 60 - (dco, C) = 0

(dp,dC) + (dp,SC)-(dco,0 = o

for all C in HI with s uppor t in U if co is in HI, or

(7.4.6) (doc, dC) +(doc-co,dO =0

(dp - CO , dt) + (dp,dC) = 0

for such C if CO is merely in S2. Since M can be covered by a f in i te number

of such ne ighborhoods and any l^^H\ may be wri t ten as the sum of a

finite nu m be r of Cs, each of wh ich ha s sup po rt in som e one such neigh

bo rh oo d, (7.4.5) an d (7.4.6) ho ld for all C in H\ on M. T h e r e g u l a r i t y

res ult s follow from eq ua tio ns (7.4-5) an d (7.4 .6), th e resu lts for sy s tem s

of §§ 5.2 , 5-5 an d 6.4 , and th e rem ar ks a t t he be gin nin g of this section.

Choose PQ^M and an admiss ib le coord ina te sys tem 6 w i t h d o m a i n

G = B{0, R) s u c h t h a t gij(0) — dfj. Then the equa t ions (7 .3-4) , when

expre s s ed in t e rms o f the componen ts Qj (and cer ta in in tegra ls o f

jacobians which have cons tan t coeff ic ien ts and which vanish i f the com

po n en ts ^-^20 ^^^ ad d ed (cf. § 7.2)) ta k e th e form (5.2.2), w he re B{u, v),

Bi(u, v), a n d B2(u, v) are defined in (5.2.16) an d the coefficients satisfy

(5.2.17) an d (5.2.18) (wi th/ /I = 2 s ince our coeffic ients are bo un de d) . Le t

u s a p p r o x i m a t e t h e c o m p o n e n t s coj, as s umed in H\[G), a n d gij by func

t ions (Dpi a n d gpij^ C'^ (G) s o t h a t copi-~>a)i in HI (G) a n d the gpi j -^gtj

uniformly so tha t a l l the gpij sa t is fy the same Lipschi tz condi t ion as the

gij an d the i r der i va t iv es converge a lmos t eve ryw here to those of th e gij.I t i s c lea r f rom Theorem 5-2 .1 tha t we may assume tha t R is so small

t h a t

Bp (u, u) > m i (II u 111)2, u $ H\Q (G), mi = w / 2 ,

for every p (and degree r) and that (5-2.6) is sa t is f ied for a l l^ . Then, by

w r i t i n g Qpi = Oi + rjpj, rjpi^HlQ(G), we see us ing Theorems 5.2.1

and 5.2.2 that , for each p, there i s a un ique so lu t ion vec tor Dpi of the

eq ua tio ns co rresp on ding to (7-3.4) w ith coo repla ced b y cop and wh ich

coincides on dG w i t h Qi. F ro m t he in te r ior reg ula r i ty th eor em (§ 5 .2 ,

C^ case) i t fol lows that each Qpi^C'^(G). Also, from Th eo rem s 5.2.1

and 5.2.7 , we see that Qpi -^Qi in H\[G). I t fo l lows tha t the vec tors

^j>j = {^^p)j a n d ^pK = (dQp)K converge s t rongly in 22 {Q to ocj a n d

^K' For each p i t is easy to see that the vectors ocpj a n d ^pK satis fy the

equations corresponding to (7 .4 .5) and (7-4.6) with the co i rep laced by

t h e copi. F r o m T h e o r e m s 5.2.1, 5.2.2, 5.2.5, and 5.2.7, it follows that the

a^j ->(xj a n d ^PK-^PK s t rongly in H\[r) for each 7^ w i t h F cG. T h u s

a J a n d ^K € H\ [F) for such F and sa t is fy the equa t ions corresponding to(7.4.5) and (7-4.6). If the co i are merely in 82 {G), we can s t il l ap p ro x im a te

t o co i s t rongly in S2(^) by the copi. In this case we s t i l l have the s trong

convergence of apj to ocj a n d ^pK t o ^K in HI (F) ior F ^G s o t h a t oc a n d

P^H\ and satisfy (7.4.6). In case co^ H\y the equations (7-4-5) are the

de f in ing equa t ions wh ich s how tha t oc and ^ a re the respec t ive po ten t ia ls

of doy a n d d a>, s o t h a t d a, doc, d p, a n d d^ all ^ HI.

We next p rove a decompos i t ion theorem for r - forms in S2 on M which

s t rengthens the resu l t o f K. K O D A I R A m e n t i o n e d i n o u r I n t r o d u c t i o n .Theorem 7 .4 .2 . If oo ^21 on M, then there are H\ forms H, oc, and p,

where H is a harmonic field, such that

(7-4-7) o} = H + d<x + dp

doc = dp = 0, oc = dQ, p = dQ,

7 .4 . F ina l resu l t s for com pac t mani fo lds wi th out bo un da ry 2 9 9

where Q is the potential of co — H, The three forms in (7 -4-7 ) (ire mutually

orthogonal in fi^, and the totality of the elements in each is a closed linear

manifold. If Hi, oci, i are any forms in HI, w here Hi is a harmonic field

and

(7.4.8) a)=Hi-^d<xi-{-dPi,

then Hi = H, d oci =^ d a, and dî = d^.

Proof. Since §*" is f ini te dim ens ion al , th er e is a uni qu e H ^^^ which

is near es t co. T he n co — ^ is o r th ogo na l to §^ , whence i t s p o te n t ia l Q

ex i s t s and has the p rope r t i e s men t ioned in Theo rem 7 .4 .1 . Thus (7-4.7)

def ines a decompos i t ion which is eas i ly seen to have the des i red type ,

u s i n g L e m m a 7-4.1. Moreover , suppose {ocp} is a sequence of HI forms

s u c h t h a t dap = 0 for each p a n d docp->some TJ s trongly in £2, each

ap = dQp being or thogona l to § '* ; then we see tha t

D (ocp — ocq) = (d(Xp — docq, 6ocp — Sag) - > 0 ;

i t fol lows that the ap form a Cauchy sequence in HI, us ing Theorem

7-3-2. Hence the ocp - ^ s o m e oc s t rongly in HI w i t h da = 0, d oc = 7] .

A s imilar proof holds for the manifold oi d^' s.

N e x t , s u p p o s e t h a t Hi, oci a n d î are HI forms satisfying (7-4-8),

w h e r e Hi i s a harmonic f ie ld . From Lemma 7-4-1 we see tha t the th ree

componen ts a re mu tua l ly o r thogona l , f rom wh ich we immed ia te ly con c l u d e t h a t Hi = H a n d d (oc — ai) = d (î — P). Since these las t two are

or thogona l by Lemma 7-4-1 they mus t bo th be zero .

Theorem 7.4 .3 . If w^ H\ on M, then in the decomposition of Theorem

7 -A-2 we have d a and d^ in H\ also.

Proof. This follows at once from Theorem 7-4-1, s ince H is a lways in

H \.

Theorems 7-4-2 and 7-4-3 lead immedia te ly to the fo l lowing theorem:

Theorem 7.4.4. Each cohomology class of forms in HI contains a uni

que harmonic field.

Proof. For if dw = 0, t h e n doc must be c losed and hence zero in

(7.4-7)-

R em ark s . T he fur ther d i f fe ren t iab i l ity p rope r t ies of th e forms doc a n d

dp in (7-4-7) follow from Theorem 7-4-1 and the other regulari ty propert ies

s ta ted a t the beginning of the sec t ion .

In h i s book . Geom e t r ic In teg ra t io n Theo ry , H . W hi tn ey [2] cons idered

a class of differentia l forms (the "fla t" forms) analogous to those described

in the following definit ion.Definition 7.4 .1 . An y-form co in £2 on M is said to be S2-flat <^

th er e exis ts an S2 form rj and a sequence {cop} in HI s u c h t h a t cop-^co

a n d dcop -^r] s trongly in S2 on M.

Theorem 7.4.5. An r-form co in S2 on M is S2--fl^l tf ^'^d only if

6 a^H\ in the decomposition (7-4-7 )-

3 0 0 A var ia t ion al m ethod in the theo ry of harm onic in tegra l s

Proof. L e t co be S2— flat , and le t rj a n d {co ,} be as in Definit ion 7-4-1.

If we write (7-4.4) for each p, we have Hp, d ocjt, a n d d^p in H\, w h e r e ocp

i s the po ten t ia l o f dcop. I t fol lows that a i s the po ten t ia l o f rj, w h e n c e

^ /% is in H\.

Converse ly , suppose d aisin H\. W e m ay ap p ro x im a te to co s t rong ly

in £2 b y {a>^} w it h e ac h co^ in H\. Then if we write (7 .4-4) for each^, the

forms /^^ and d^'^ are in H\ a n d d^'^ tend s t rongly in S2 to d^. Now, if

we define ooj, b y

0}p = H + doc + d^'^

we see th a t {co^,} is a seq uen ce a s in De fin itio n 7-4.1; i t fol lows that

CO is S2-flat.

7 .5 . Man ifo ld s w i th b ou n d ary

Definitions 7.5.1. For a ^^-dimensional manifold M w i t h b o u n d a r y

6 M of c lass C^(0 < / < 1) (C ^, anal ytic ) we ado pt t h e s t an da rd defini

t ion : each point of M U & M is con ta ine d in some se t 3^, open onMyjhM,

which is e i ther the homeomorphic image of the un i t ba l l in Rv or of the

p ar t of i t for w hich jt^ < 0 in wh ich la t te r case , th e poin ts where x^ = 0

co r re s pond to ^t O b M; any two coo rd ina te s ys tems a re re la ted by a

t ran s for m at io n of c lass C^(C'^, ana ly t i c ) . Any coo rd ina te s ys tem wi th a

L ips ch i tz doma in G is admissible i f i t is re la ted in this way to one of the

prefe rred ones and 'St H b M corresponds to a par t o f d G. We sti l l a l low

even and odd forms and def ine the spaces 2p a n d HI and the inne r

products (co, rj) and ((co, rj))]x and the i r corresponding norms as before .

The d i f fe ren t ia l opera tors d a n d 6 are defined as before .

Assumption . We assume M compact and connected and of class at

least C\.

Theorem 7.5.1. The spaces S2 (^nd H\ of forms of a given kind and

degree are Hilbert spaces. The operators d and S are bounded operators fromHI to S2, preserving even-ness or odd-ness, and D{m) is lower-semicon-

tinuous with respect to weak convergence in H\. Finally, if con -^co in

H\, then con -^co in Qp.

In o rde r to de f ine bounda ry va lues and the no rma l and t angen t ia l

parts of a form on b M, i t i s necessary to def ine admiss ib le boundary

coord ina te s ys tems .

Definition 7.5.2. An admissible boundary coordinate system on a man i

fold M w i t h b o u n d a r y bM oi class C^ is a coordinate sys tem of c lassC^ wh ich m ap s i t s (Lipschi tz ) dom ain G\J G on to a bounda ry ne ighbor

h o o d SI in s uch a way tha t a, t h e p a r t oidG on x'^ =^ 0, i s no t em pt y an d

open and i s m ap ped on to Slr\bM, and such tha t the metr ic i s o f the form

v - l

(7.5-1) ds'^ =: '^gy6{x^,0)dxydx^ ^[dx^'Y on o*.

7.5- Manifo lds wi th bou nda ry 3 0 1

Lemma 7.5 .1 . (a ) / / M (U hM) is of class C^, each point PQ of bM is

in the range of an admissible boundary coordinate sy stem.

(b) / / ^ > 2, each point PQ of b M is in the image of a coordinate

system of class Cj^"^ which satisfies all the other conditions of Definition

7.5.1 (^'yid is such that the metric takes the form (7-5.1) i^ith [x^, 0) replaced

by any x in G. If M is of class C^ {analy tic) this coordinate sy stem m ay be

taken to be of class C °^ [analy tic) and hence admissible.

(c) / / [x) and [y ) are overlapping admissible boundary coordinate sy tems

[or two sy stems as in (b)) , then,

( 7 .5 .2 ) y« , ( - ; , o )=y :« (v ; ,o ) -o , oc<v, y:,( .<,o) = i .

Proof. L e t [x ) be any coo rd ina te sys tem of c lass C^ which m ap s

G\J a o n t o a b o u n d a r y n e i g h b o r h o o d ^ of PQ in the u s ua l manne r .

Suppose ( I ) i s another s imi la r coord ina te sys tem and le t 0^^ a n d g^^ b e

the metric tensors of the (f) and [x ) s ys tems , r e s pec t ive ly . T hen we wa n t

C'va = dva, at least if k^ — 0 , s o tha t we wan t

at leas t on ^^ = 0. If we multiply (7.5-3) by d^°'jdx^ and s um we ob ta in

(7-5.4) g^^-^ = ^=.r6l, r = ^^,

since , of cou rse, |^ = 0 w he n x^ = 0. Multiplying (7.5-4) by g^^ a n d

s u m m i n g , w e o b t a i n

(7-5.5) f j ; = ^ ^ '

where we f ind , by mul t ip ly ing by d^^jdx^, using (7.5-4) and summing on

X, t h a t

(7.5.6) r'^g^^= 1.

A coord ina te sys tem as in (b) may be ob ta ined for | in a 1-sided neigh

borhood of a by solving the differentia l equations (7 .5-5) with ini t ia l

cond i t ions

(7-5-7) ^M^;.o)=|\ x<v, ^ - ( i ; , o ) - o .

If the [x) system already satisfies (7-5-1) it is clear that

(7 .5 .8 ) T = l , | | = ^ . „ , g = ^J o n 4 ' = 0

which proves (b) .

In order to ob ta in an admissible coord ina te sys tem (f ) , i t i s necessary

only to define functions x^ of class C^ on a 1-sided neighborhood of a

which satisfy (7-5.5), (7-5-6), and (7-5-7) along a. This may be done by

3 0 2 A v a r i a t i o n a l m e t h o d i n t h e t h eo ry of h a rmo n i c i n t eg ra l s

defining

(7.5.9) x^(K, ^') = ^"f^^tWv - K)f{%) <^Vv>

for ins tance , where 5(f^ , —S*') deno te s the (v — 1)-d imens iona l ba l l wi th

cente r a t |^ and rad ius — I*' > 0, cp is a mollifier in Rv_i a n d (p* (y )

Definition 7.5.2. S up po se o) ^ HI on M. W e s a y t h a t t h e tangential

part tco or the normal part n co of co vanishes on h M \i and only if

(7.5.10) ft>n...«r«'0) = - 0 if all iy<v, or

(7.5.11) f^n .. . ir(^ v> 0) = 0 if som e iy = v,

re s pec t ive ly , x be ing an admis s ib le bounda ry coo rd ina te s ys tem.

Remark. These a re invar ian t ly def ined by v i r tue of (7 .5-2) .

L e m m a 7.5.2. (a) / / ojn —7 oj in H\ and t ajn(n cun) vanishes on bM for

each n, then ta)(na)) vanishes on b M.

(b) / / M is of class C* with ^ > 2 and co^ C^ and tco — 0 (nco = 0)

on b M, then t dw = 0{n d o) = 0) on b M.Proof, (a) follows from th e theo re m s in Ch ap ter 3 on HI func t ions .

Using a part i t ion of unity , one sees that i t is suffic ient to prove (b) for

forms CO having suppor t in the range of some admiss ib le boundary co

or di na te sy s tem . Fo r such an co, one sees im m ed iat ely from (7.5-10) an d

(7.2.11) that if (7.5.10) holds, then all the (dco)j^,, ,j-^+^ with a l l ;V< '»^

satisfy (7.5.10). Also, if (7-5.11) holds we see from (7-2.14) and the facts

t h a t gpoc = g*'°' = dvocin a,n admis s ib le bounda ry coo rd ina te s ys tem, tha t

(<5 co )^ l... ^y_i = 0 if a n y iy = v . This proves the resu l ts .

Definition 7.5.3. If 99 a n d y j are given by (7-2.1), we define

(7.5-12) {cp,^){P)=F{P;cp,xp)

w h e r e F{P;q), \p) is defined in (7-2.5).

L e m m a 7.5.3. Suppose oc and ^ are r and (r — \)-forms respectively,

in HI which are of the same kind. Then

(7-5.13) [oc,d§) = [doc,P)+ j{{-\y -^noc,t^)dS{P).bM

If either naort^ = OonbM, then {d oc, d^) = 0.Proof. We prove (7-5.13) f irs t . There is a part i t ion of unity Cs, s

= 1, . . . , 5 , such t h a t if the su pp or t s of ^s a n d f.in te rsec t , the i r un ion is

in the range of some one coord ina te pa tch . Thus , by se t t ing as = Cs oc,

Pt = Ct P> we see t h a t i t is suffic ient to pro ve th e the or em for th e case

t h a t oc an d ^ bo t h ha ve th e i r su pp or t s in the ran ge 9^ of some one coo rd i-

7 . 5 . M a n i fo l d s w i t h b o u n d a r y 3 0 3

na te pa tch . I f ^ C M^^\ (7 .5 .13) follows from §7.2 with no boundary

in teg ra l . So suppo se 5y?: i s a bo un da ry ne ighborhood a nd [x) i s an admiss ib le

b o u n d a r y c o o r d i n a te s y s t e m w i t h d o m a i n GR U OR ( in th is chapte r ^^ < 0

on GR). W e m a y a p p r o x i m a t e t h e gy d and and the componen ts o f oc a n d ^

by smooth functions (Theorems (3.1-3) and (3-4.1) on GR. Then by look

ing at (7.2.15) we obtain

^ ' ' ^ "^ ^ r\a^ yd) (?)

Now if for each y we se t j y = v and hence iy = v, sin ce ^''^ = 0 if a < r,

move the index iy (= r) to the f irs t subscript of oc, rep lace ^^ b y ^ 1 . . . y^r-i

a n d j! ^ by / i . . . /^- i , we ob tain

(oc,d6) = {doc,6) + - r - ^ I Ig^^^^"'g^'- ''^ '- '^^vki. . .Jcr~i X

from which (7.5-13) follows since all the h . . . / r- i are < v, s ince jy = v .

Clear ly dS(P) = r(x) dS(x) s ince gvx(x) = dv<x for x on (TR.

To p rove the s econd re s u l t we may app rox ima te a s above and , s ince

d^ (p = d^ (p = 0, we s ee tha t

{da,d^) = ±f(nSocJ^ )dS{P) = 0

bM

if (X, fiand the metr ic a re smooth , s ince i t fo l lows f rom Lemma 7-5-2 tha t

^ ^ <x ==0 i f^ ^ = 0 o n & M . T he re s u l t follows by pas s ing to the l imi t .

The following corollary will be useful la ter:

Corollary. If M is of class C^ with ^ > 3 and co ^ C^(M), then

{dco,dtp) + {dco,d7 p) = (Aoj,if) + J (-\y{(nda)JC) + (idoj,nC}}dS,bM

(7.5.14) A = dd + dd.

I t i s now importan t to deve lop a formula for DQ (CO) , w h e r e

(7.5.15) Do(o)) = (do), da))o + (SQCO, doco)o,

do and the inne r p roduc t s be ing fo rmed us ing the Euc l idean me t r i c , o)

hav ing i t s s uppor t in GR (pe rha ps 7^ 0 on OR) and be ing s moo th . F rom

(7.2.11) and (7.2.14), we obtainr + l

(dco)mi . . . mr+i = ^ ( — 1 ) ^ " ^ CO{my) x^y

(7.5.16) ^=1{do a))si . . . sr-l = — I CO asi . . • Sr-1 X^ -

aWe substi tute the express ions (7 .5 .16) into (7-5.15) and form the inner

produc ts us ing the form (7 .2 .7) and the Euc l idean metr ic in which g^^

= d^K e tc . W e ob ta in

(7.5.17) (dco,da))o = , , \ , j I I(-'^y'-^co{m'y)xmyCO{m'y)xm6dx.

3 0 4 A var ia t ional m ethod in the theo ry of harm onic in tegra l s

W e break the s um on y a n d d in to the s um where y = d, t h a t w h e r e

y <C d an d th a t w here y > ^ . In th e f i rs t sum , we rep lace m^ b y ii . . . ir

and ca l l My = oc. In the s um where y <i d, we move the index ms to the

first place in m^ a n d m o v e nty in to the f irs t place of w^, us ing the anti

s ymmet ry o f the ind ice s ; we then rep lace m^Q ( in which bo th niy a n d ms

ha ve b een om itted ) b y s i . . . Sr_i and call ms = ^ a n d my = oc. W e d o

th e correspo nding th in g for the su m wh ere 7 > (5. Then we combine

these two sums wi th the in tegra l fo r (^0 cjo,doO))o. The result is

^ 0 ( a > ) = ^ J ^ K..Ar'^°^^^ + Jy^Z ^ ^ ^ {(ôcsi. , ,Sr-ix-X

X CO^si. . . Sf-l xP — OJxsi... Sr-1 xP OJ^si. . . Sr--[ x^') ^^

for smooth forms , f rom which we obta in

(7.5-18) Do(a)) = (^(D%<^dx + f ^ {OJVTCD^TXP — COVTXÔJ^T) dS

T : \ < 51 < • • • < Sr_i < V

for any form co $ C^ on Gji which vanishes near the spherical surface of GR.

Lemma 7.5.4. (a) / / co (considered as a set of functions) ^ H\ on GR,

CO is zero on and near the spherical part of the surface of GR, and if

either t(jo = 0 or nco = 0, there exists a sequence cop of similar formsof class C^ on G R which converge strongly in H\ on GR to co and such that

t a)p = 0 or n cop = 0 (respectively) for each p.

(b) If o) satisfies the hy potheses o / ( a ) , then

(7.5.19) D^(M) = fS{îi)x^)^dx.

Proof. Clearly (b) follows from (a) and (7.5.18). Also, since the con

d i t ion t CO = 0 is ju s t the s ame a s s ay ing tha t the co' s w i t h ir < v van is h

on OR and w co = 0 is th e sam e as say ing t h a t thos e w i t h ir = v van is h

on OR, par t (a ) i s jus t reduced to proving the theorem for func t ions . I f

the func t ion co is not required to be zero on CR, w e e x t e n d co to the who le

of BR b y co(—x^,xl,) =co(x^,xl) a n d t h e n n o t e t h a t t h e h mollified

func t ions wi th respec t to a spher ica l ly symmetr ic moUif ie r o f co are of

class C°^ an d ha ve su pp or t C BR if h i s smal l enough; these tend s t rongly

in HI to CO on BR. If co is zero on OR, we beg in by ex tend ing co to BR b y

co(— x'',xl,) = •—c o(x^, xl) and t hen p roceed ing a s abo ve ; we no te th a t

the moll if ied functions vanish on OR.Lemma 7.5.5 (GAFFNEY [1]). With each point P of M and each

£ ^ 0 is associated an admissible coordinate sy stem ® with domain G and

range 11 and a constant I such that

D(a)) > (1 - e) / 2 ; (oJi:,o.)^ dx - l(ay , oy )

'].(i. Differentiability a t t he boundary 3 0 5

jor any jorm co^ HI with support on U and either t(o = 0ornco = 0on

Proof. This has been proved for in te r ior po in ts in Lemma 7.2 .1 . If P

i s a bo un da ry po in t , i t i s c lea r th a t w e m ay choose an admiss ib le bo un da ry

c o o r d i n a t e s y s t e m w i t h d o m a i n GR U GR which carr ies the or ig in in to P

a n d GR U OR in to a ne ighborhood of P in which gij{0) = dij. T h e n ,

exac t ly as in the proof for in te r ior po in ts , we conc lude tha t we may

choose R s o s ma l l tha t

D[cd) >DQ{O)) — e j 2^ {(Oixo^)^ dx — l(co, w)

for some / and al l oy ^ H\ w ith su pp or t in 9^, DQ {CO) having i ts s ignificance

i n L e m m a 7-5A- The resu l t fo l lows f rom tha t lemma.

The fo l lowing theorem fo l lows f rom the lemma above in exac t ly the

same way as in the case of Theorem 7-2.6.

T h e o r e m 7.5 .1 . For each finite sy stem 91 of admissible coordinate

sy stems whose ranges cover M, there are constants k and I such that

D{o)) '>k\\ojf-l{oy ,oy) (k > 0)

for any form in HI with either tco = 0 or nco = 0 on b M, the norm being

that corresponding to 9?:.

7.6 . Dif ferent iab i l i ty at the bound ary

In th is sec t ion , we d iscuss the regula r i ty a t the boundary of the

solu tion s of th e equ atio ns (7 .4 .1). T he res ul t in g system s (7.4 .2) for bou n

dary ne ighborhoods a re no t qu i te the same as those in Chapte rs 5 and

6 becaus e we a re requ i r ing on ly tha t some (no rma l pa r t o r t angen t ia l

pa r t ) o f the componen ts van i s h on b M. However , the genera l ideas a re

th e sam e as th ose in §§ 5.2, 5.5 an d 5.6. O n acc ou nt of equ at io n (7-5.18)

and Lemma 7 .5 .5 and i t s proof, i t fol lows that if Po€ bM, there is an

admis s ib le bounda ry coo rd ina te s ys tem wi th doma in GR in which the

or ig in correspon ds to P Q an d goi^{0) == docp, and if e i ther nco or t(p = 0

on bM, then the corresponding boundary in tegra l to tha t in (7 .5-18)

(which then vanishes ) may be subtrac ted off and the equa t ions (7-4 .1)

take the fo rm

/ [<a « f ^j + K ^' + 0 + ^' K < + ^ii^' + h)\ ^^=^^GR

(7.6.1) v^H\t, at^{0)=dijd-^ i,j=\,...,N,

where H'^ denotes the set of a l l vectors v ^ H\ s u c h t h a t

(7.6.2) all v^ = 0 on 2^E, v^=^0 on (Ti? for i=\,...,k

(0 <k <^n).

1 Here as elsewhere if (x> h as support in a boundary neighborhood 9^, it is not

required t o vanish on 6 M Pi 5^.

k be ing g iven . W e have a l ready no ted tha t the as are combina t ions of

t h e gap whereas the h's, c*s a n d d's involve a lso the der iva t ives of the

g's] t h e e^ are combina t ions of the g's a n d t h e n o n - h o m o g e n e o u s t e r m s

in equations (7 .4 .1) and the ft involve these and the der iva t ives of the g ' s .

We sha l l ob ta in resu l ts concern ing the d i f fe ren t iab i l i ty of u b y

obta in ing such resu l ts fo r U == C ^ w here f is a 0-form ^C'^(G/j) wh ich

has s uppor t on GR\J OR; resu l ts fo r in te r ior ne ighborhoods have a l ready

been o bta ine d in § 7 .4- Assu m ing th a t v has s uppor t in GR U CR a n d t h a t

v^ Hl^, we rep lace v^ b y f v^ in (7-6.1) and find that U satisfies (7.6.1)

w i t h e an d / rep laced by E a n d F, respec t ive ly , where

^ ^ Fi = Cfi + Â<f<p + K^' - ot,u^ + < ).So we begin by cons ider ing so lu t ions u of (7 .6 .1) which have support on

GR U OR a n d w h ic h $ ^ | J .

We begin by a l te r ing the coeff ic ien ts ou ts ide GR as in (5.5.3) to

obtain new coeffic ients a^fn, etc . Then, if u^W^, is a solution of (7.6.1),

and has s uppor t in GR U OR, it is also a solution of

/ [vU i<^R <P + KR^'' + < ) + ^' KR < + îJR^^ + /^)] dx = 0,

(7.6.4) v(^Hl*.

We then write , as in (5 .5-5),U=ÛR + HR,

(7.6.5) UR = Q2R [{aR — ao) - Vu + BR-u + e] +

+ P2R [CR'Û + dR'U +f]

where we now wish to def ine the vec tors U = Q2R{^) a n d V = P^nif)

to s a t i s fy the bounda ry cond i t ions

(7.6 .6) U^ « , 0) = F*' « , 0) = 0, i= \, . . ., k.

So we define Q2R a n d P2R as follows:

Definit ion 7.6 .1 . We def ine the vec tors U = Q2R(e) a n d V = P2R{f)

as follows: li i <, k we define U^ [x) and F* (x ) as in Definit ion 5.5 .1 '

and Equations (5.5-19), (5.5-20), and (5-5.21). If i>k, we define P2R ( /)

as the res t r ic t ion to G2R o f t h e p o t e n t i a l o f / o v e r B2R, w h e r e / i s d e f i n e d ,

b y e x t e n d i n g / t o B2R by pos i t ive re f lec t ion :

(7.6.7) fK-x' ')=f«.^n>

with the usua l modif ica t ion ii v = 2. li i "> k, we define e by pos i t ive

reflection and then define

Ui(x) JZ f-KoA^ - i)eni)d^ + W%{x),

(7 ,6.S) W^x) = ^1 [Ko{x - i) - Ko{x' - me'(S)dS>G2R

x' = (xl, — x^) if X = [x^, x"^).

7-6 . Di f feren t iab i l ity a t the bo und ary 3 0 7

Lemma 7.6.1. (a) / / U = Q2R(e) and V^P^RIJ), e^LziG^R),/ £ Lp{G2R), p > 2vl{v + 2), p > \, then U^ and F* vanish along OZ R

if i <.k and. U and F* are solutions of

^2R GzR

(7.6.9)

If i > k, U^ and F* are restrictions to G^R of functions which are definedon Rv and satisfy {7.6.7); they are solutions of (7-6.9) for all V^H\{G2B)

which vanish on ^^R-(b) If H^ HI{G2R), vanishes along G^R and sa tisfies

(7.6.10) fv,,d-^H,fidx = 0, V^HI^{G2R),

GzR

then H is harmonic on B2R if extended by negative reflection. IfH^H\ [G^R)

and sa tisfies (7.6.10) for all v ^ HKG^R) which vanish along^^^, then H isharmonic if extended to B^R by positive reflection.

Proof, (a) The results concerning U^ and V^ for i <,k were proved inTheorem 5-5-'i'- If the ^^^ C^ and have support in G^R U O^R, then the^? € G^^[B2R) and have support in B2R. Since K[s(x — I) — KQ[X' — | )is the Green's function for the lower half space, it follows that W^, as

defined in (7.6.S) ^ C^ on each closed half-space (but not necessarily onthe whole space), this can be proved by writing

Wi(x) = - jKoix - I) e^iS) di + 2jKo{x - | ) e^{i) diB2R Gtn

and using the method of proof in Theorem 5.5.1'. It follows thatU^^ CJ^(G2R). If we approximate uniformly (with the same Holder condition) to the ^" by sm ooth functions e^^ and (7-6.9) holds for each n, itwill hold in the limit. Then

/^,a((5«^ Ul, + el,) dx = fv{Ul, + el,) dS~-Jv{A Ui + <,.«) dx.GzR 02R G2.R

(7.6.11)

By integrating by parts, we obtain

Ui{x) =f-Ko(x-^) s'elU^) di + Wl^x).

From Chapter 2, we conclude that

AWi=- el, (x) , AUi(x) = - Z el ,, , (x)

Ui,Ax) = Wl,,{x)^AWi as ^^'-^0

since W^ = 0 along cr2/2- Thus the integral on the left in (7.6.11) vanishes

for each n. If ft^ CI{G2R), t h e n / ^ ^ CI{B2R) iii>k and so Zl F^ = fi

and F*y = 0 along G2R since F*' is even w ith respect to x^.

20*

308 A variational method in the theory of harmonic integrals

I n p a r t (b), it follows from (7^6.10) and the in te r ior d i f fe ren t iab i l i ty

t h e o r e m s of C h a p t e r 5 t h a t H is h a r m o n i c on G^R- ^i H vanishes a long

(72ij and H is the extens ion of HtoB2R by negative reflection (see (5.5.19)),

t h e n H is easily seen to ^H\{B2R) by us ing the abs o lu te con t inu i typ rope r t i e s of § ^A. If, in (7.6.10), we rep lace G^R by ^272 and let

V ^ C^ (B2R), we see that

( 7 . 6 . 1 2 ) B2R G2R^

v[x^,x^) =v[x,,x^) - v[x^, - X^)^H\^{G2R).

If we do not a s s u m e t h a t H vanishes a long a^R bu t do a s s u m e t h a t

(7.6.10) holds for all v w h i c h v a n i s h a l o n g ^ 2 i ? , we o b t a i n a result l ike

(7.6.12) if we let H be the ex tens ion by posit ive reflection, let v ^ C^ [B^R)and define v by

V {x[., A;*') — ?; [x^, x'^) + V [x^, — % * ') .

T h e d e v e l o p m e n t s in prov ing regu la r i ty at the b o u n d a r y now proceed

as in § 5.5- We define the n o r m s and spaces as in Definition S-S-"! except

t h a t the spaces C^^Q{G2R) will consist of thos e vectors in C|^(S2i?) which

v a n i s h a l o n g ^ ^ j j (the s phe r ica l pa r t oidG^R', reca l l tha t , in t h i s c h a p t e r

G2R is th e lower h e m i s p h e r e ) ; we do no t requ i re them to vanish a long

(72ij. H o w e v e r , the range spaces H\^ [G2R) and CJ* {G2R) of the o p e r a t o r sQ2R and P2R are spaces of vectors which satis fy the b o u n d a r y c o n d i t i o n s

i n L e m m a 7-6.1. W i t h t h a t u n d e r s t a n d i n g we o b t a i n the re s u l t .

T h e o r e m 7.6T. The results of Theorem 5.5-1 hold.

I n our case , the coefficients UR are Lips ch i tz and the hRy CR, and da

(in (7.6.4)) are b o u n d e d and m e a s u r a b l e if th e manifo ld M is of class C\

and , more genera l ly the UR^ C^~^ and the bn, CR, and dR^ C^~^ if M is

of class C^, > 2, and so in these cases sat is fy the H^ cond i t ions in

Defin i t ion 5.5-2 for any q. We note again that if M is of class H^ withp "> V, then the coefficients satisfy the H\-conditions of Definition 5-5.2

with q <.p- So if we w r i t e the equation (7-6.5) in the form

UR — TRUR=-VR + WR, VR - : Q2R{e) + P2,R[f)

TR UR = Q2R [{aR — ao) ' V UR + hR UR] + P2R [CR '\/UR + dR- UR]

WR = Q2R [(aR — ao) ' \7 HR + hR HR] + P2R [CR ' V HR + dR-UR],

(7-6.13)

we ob ta in the t h e o r e mTheorem 7 .6 .2 . The results of Theorem 5-5.2 carry over to our case.

N o w , we no t ice , if our g iven da ta e a n d / van is h ou t s ide G2R and i^~,

w e may carry over the ana lys is wi th R rep laced by any A ; we then no t ice

tha t Q2A{e) = Q2R{^) and p2R{e) = P2A{^) for any A and we may let

A -> 00. If ^ van is hes ou t s ide GR, we t h e n see t h a t the h a r m o n i c v e c t o r

'].']. Potentials; t h e decomposition theorem 3 0 9

H2A = H^R can be ex ten ded to th e whole space w here i t i s bo un de d so

t h a t w e h a v e .

Theore m 7 .6 .3 . / / U^H\{GR), vanishes on^^, and satisfies (7 .6.4),

and if HR is defined in (7-6.5), then HR = const, if v = 2 and HR= 0 if

r > 2. If V = 2 and i < k, H^^ = 0. Ifv = 2 and i > k, then 11% is the

average value over B2R of the ordinary logarithmic potential of ff, where ff

is the extension to B^R hy positive reflection of c%^^ u[^ + dRij u^ + ft.

R em ark . O ne mu s t reca l l th e def in i t ion g iven in § 5-5 of P2i? ( / )

w he n r = 2.

T h u s , if e a n d / ^ Lq{G2R) for some q > 2, for example , R is small

enough , depend ing on q, ^ vanishes ou ts ide G2R, U^H\{G2R), U^ = 0

a long (y2Rii i <= k, a n d u satisfies (7-6.1), then u^ H\[G2R) a n d

Corresponding resu l ts ho ld i f e a n d / ^ ^^0(^272)- B u t now, s uppos e th a t

t h e c o m p o n e n t s u are co m po ne nts of a form a> wh ich satis f ies eq ua tio ns

(7.4.1) in the large on M and s uppos e we a s s ume tha t the fo rms q),ip,

a n d Tj^^qior some ^ > 2 . W e assum e th a t n co ^=^ n ^ = 0 ior d^ forms

o) an d C cons idered or e lse th a t ^ co = ^ C = 0 . Then we know tha t

Q)^H\ a t any ra te and f rom our in te r ior resu l ts o)^H\ on any D w i t h

D d G M. I n a n y b o u n d a r y n e i g h b o r h o o d , t h e n U = C '^ satisfies(7.6.1) with e a n d / r ep la ce d b y E a n d F as given by (7.6 .3). Now the

te rm s ^ ^f, Cfi, a n d C,a f ^ &q but the second te rm in E a n d t h e t e r m s

V C • (^ — ^) ' uinF^ 2s, wh ere s = 2r / ( r — 2) an d the te rm V C * ^ * V ^

m ere ly ^ S2. If ^ < s , we conclu de from t he the or em s abov e th a t each

U^H\', if q'>s, we conc lude tha t U^H\. Since this is t rue in each

s u f f ic ien t ly s ma l l bounda ry pa tch , we conc lude tha t (jo^H\ \i q ^s ox

t o H] otherwise . In th is la t te r case , then we conc lude tha t u^ £§, where

s' = V sl(v — 5). W e ma y rep ea t th e a rg um en t (wi th s ma l le r pa tches if

necessary) to conc lude tha t 00 ^H\ if §' < s' o r in H], otherwise . Clearly

a n y q may be reached in a f in i te number of s teps . I f M is of class CJ,

0 < // < 1, a n d if 99, ip , a n d ty ^ C^, we firs t show that co^Hl for some

q '> V which imp l ie s tha t u^C^, so that -E and F^C^ s o tha t co^Cl.

Th e high er differe ntiabil i ty resu lts can be ob tai ne d as in §§ 5-5, 5-6.

7 . 7. P o t e n t i a l s ; t h e d e c o m p o s i t i o n t h e o r e m

In this and the next section, we shall assume that a l l of our forms areof the same k ind , comple te ly para l le l theor ies be ing obta ined for each

k i n d .

Definit ion 7.7 .1 . We def ine the closed linear manifolds ^ J a n d ^ ^

(see Th eo rem 7.5-1) of i / | as th e tot a l i t y of forms in HI for which nw = 0

a n d t (JO = 0 on b M re s pec t ive ly .

31 0 A var ia t iona l m ethod in the theo ry of harm onic in tegra l s

Ju s t as in Sec t ion 7-3, we obta in th e fo llowing res u l t :

Lemma 7.7 .1 . Let 9K he any closed linear m anifold of S2 such that

9)1 n ^ J ($ ^) is not empty . Then there exists a form co in ^ r\ ^t (^2^)

which minimizes D [w) among all such forms with (co, co) = 1 .

Theorem 7.7 .1 . The manifold ^ + ( § ~ ) of harmonic fields in $ J ( $ ^ )

is finite dimensional.

Theorem 7.7.2. If a> ^ $2^($2^) ^^^^ ^^ S>2-'Orthogonal to § + ( § " ) , then

there are positive constants A+ and %- for each 3^ such that

D{a)) > A + | | a ) l ! 2 ( A - | | a ) P ) .

Theorem 7.7.3. / / ^ C S2 ^'^d is 22-orthogonal to §"^(^~), there is a

unique form Q+ (Q-) in ^ J _L §"^(¥2" J- §~) ^^^^ ^^^^

( 7 7 . 1 ) (î3+ d c ) + { d D + , S O = (^ , c ) , c e ^ i m ) >

Definition 7.7.2. The functions i^+ and Q~ are ca l led the p lus -poten t ia l

and minus -po ten t ia l o f YJ, re s pec t ive ly .

The de f in ing equa t ions [JJ-^) for the potentia ls are a special case of

the more gene ra l equa t ions

(7.7.2) {dcD ~ cp , dC) + {dm - y j, dC) - {rj, C) = 0 , C^ î o r $ "

which were discussed in Section 7.4 . The differentiabil i ty results for suchequa t ions on the in te r ior o f M follow from the discuss ion there given.

But now, suppose we se lec t a po in t P on b M and choose an admiss ib le

b o u n d a r y c o o r d i n a t e s y s t e m w i t h d o m a i n GR a n d r a n g e a b o u n d a r y

ne ighborhood H o f P s uch tha t gij(0) = dij. In such a sys tem the con

d i t ions no) =^ n ^ = 0 for ^ ^ and toj = tC = 0 for ^ ^ co r re s pond

un de r a p rope r o rde r ing of the s e ts / and / t o the equa t ions (7-6.1 ).

Ac cordin gly we see th a t if the s up po rt of f is confined to IX, th e sys tem

(7.7.2) reduces to the sys tem (7.6.1) discussed in Section 7-6- F r o m t h e

theorems of Sec t ion 7 -^ we m ay conc lude th a t th e d i f fe ren t iab i l i ty

resu l ts fo r the p lus and minus po ten t ia ls s ta ted near the beginning of

Sec t ion 7 A fo r po ten t ia l s , ho ld r igh t up to the bounda ry . W e now ex tend

these results as in Section 7 A and summarize as fo l lows :

Theorem 7.7.4. Suppose co ^ S2 0 §"'"(22 0 §~) (^f^d Q is its plus

{minus) -potential

(i) / / M is of class C\, then Q, dQ, and dQ are in ^ ^ (^2^)-

(ii) / / M is of class C \ and oj is in S p with p '>v, then Q, dQ, and dQ

are in ^ ^ ( ^ ~) and C^ if ju = \ — vjp.(iii) / / M is of class C^ and w ^ C^'^ (^ > 2, 0 ândco^ C ^ - ^ then Q^ C^-^.

(iv) / / M and OJ are of class C ^ or analy tic, then so is Q

(v) If Q and co are 0-forms, then Q has an add itional degree of diffe

rentiability in all cases above except the second half of [m).

T."]. Po ten t ia l s ; the decomp os i t ion theorem 311

In all cases, if we set a = dQ and ^ = SQ,

(7.7 .}) da + dp = d[dQ) + d{dQ) =00, da = d^ = 0 ;

(da, dC) + (doc - CO, dC) = {dp - (D , dC) + {dp, SC) = 0 , C^^i {^2) •

(7.7.4)

Proof. The resu l ts fo r D follows directly from the discuss ion above

and Sec t ion 7-6 . The proof of the results for dQ a n d SQ is like that of

Theorem 7.4.1 where i t is a lready done for the interior of M. We choose

a b o u n d a r y p o i n t P and an admis s ib le bounda ry coo rd ina te s ys tem o f

the type des c r ibed in the p reced ing pa rag raph and app rox ima te ( i f

necessary) to co a n d t h e gij by smooth func t ions . For each of the approxi

ma t ing func t ions Q, we see from formula (7.5-14) that AQ = co a n d

(7.7.5) ndQ = 0 if i ^ ^ ^ 2 ^ a n d tdQ -=^Q ii Q^^^

s ince , in the in tegra l over h M, t^ is ar b i t ra ry if f ^ ^ ^ an d ^ f i s

a rb i t ra ry i f C € $2^. F r om Le m m a 7 -5 .2 we s ee th a t t d(p = 0 w h e n e v e r

t(p = 0 an d 99 a n d M are differentiable . From Lemma 7.5-2 i t fol lows

a l s o tha t

(7.7.6) n(p = 0 '->nd(p = 0.

Hence , f rom th is and (7-7 .5) , we see tha t bo th a an d /^ € $2^($2^ ) if co

a n d D ^ ^ J (^^ ) a t e ach s tage of the app ro x im a t ion , th a t a a n d p satis fy(7.7 .3) an d hence (7 .7-4) us ing Le m m a 7 .5 .3 . Th e appr ox im at io n m ay

then be carr ied th rough as before on each Gr w i t h r <C R. Since a finite

number o f the s ma l le r bounda ry ne ighborhoods cove r b M, the re s u l t s

(7.7.4) for all C in ^ J (^^ ) fo llow an d the d i f fe ren t iab i li ty of a a n d / ? n o w

follow from Section 7'6-

Remark. Except in the case of ze ro forms i2 , the ind iv idua l der iva

t ives of the ind iv idua l components o f Q do not, in gene ra l have the s ame

diffe ren t iab i l i ty proper t ies as do dQ a n d SQ ( the coo rd ina te t r ans fo r ma

tions wil l not a l low it) .

The fo l lowing two theo rems a re u s e fu l and impor tan t :

Theorem 7 .7 .5 . Suppose rj ^^2 ^ H I, i J + and H~ are its projections

in § + and § ~ and Q+ and Q- are the plus and minus potentials ofrj — H^

and rj — H~~, respectively, and a^ = dQ'^, /5± == dQ^. Then

(i) A:+ is the plus potential of drj and p- is the minus potential of drj

and dr)^22Q §"^ ^^ ^ ^^ € S2 0 §~ .(ii) Ifr}^%^, then p+ is the plus potential dr]^ &2Q $'^.

(iii) Ifrj^ '^2> ^^^^ ^~ ^^ ^^^ minus potential of drj^ S2Q § ~ .Proof. These results follow from Theorem 7.7-4, equation (7-7.4) and

Lemma 7 .5 -3 .

Theorem 7.7.6. (i) If rj'^ ^^t ^'^^ V~ ^ '^2> t^^^^ ^^^ unique forms a

and P where A : $ ^ J and is 22-orthogonal to § + and P^'^2 ^'^^ ^^ S2-

orthogonal to § ~ such that da — dr]+ da =^ 0, dp = drj", dp = 0.

312 A var ia t ional me thod in the theor y of harm onic in tegra l s

(ii) If 7 ]^ HI, there are unique forms y ^ ^ ^ O (S2 0 § ' ) ^^ ^ s in

$ 2 n (S2 0 $~) S'i^ch that dy = drj, dy = 0, ds = drj, de = 0.

Proof. Th e un iquenes s is ev iden t . To p rove ( i) , l e t i ? + an d i 2~ be the

resp ectiv e plu s an d m inu s po ten tia ls of 7 + — H+ a n d r]~ — H~ and le t

r = SD'^ a n d E = dQ~. From Theorem 7 .7-5 , we see tha t F is the plus

poten t ia l o f dr]+ a n d E i s the minus po ten t ia l o f dr]~. Then , f rom Theorem

7.7 .4 we conc lude tha t a — dP a n d ^ = SE have the des i red proper t ies .

To prove (ii), let Q+ a n d Q~ be the respec t ive p lus and minus po ten t ia ls

of rj — H+ a n d TJ — R- and le t

A=dQ+, B = dQ-, y = dA, e = dB

and (ii) follows from Theorem 7.7.5(1)-

Definition 7.7.3. We define the linear sets ® and 1) as the sets of allforms of the form doc a n d d^, w h e r e oc^'^^ ^ ^ ^ P^^2> re s pec t ive ly .

We now can prove an ana log for the Kodaira decompos i t ion theorem

for man i fo lds wi th bounda ry .

Theorem 7.7.7. The sets © and ® and the set ^ of all harm onic fields

in S2 on M are closed linear m anifolds in S2 <^^^

{7JJ) 22 = ee©§.

Moreover, if 00^ H\, its £2 projections y , e, and H on ^,%, and § belong

to ^ 2 ", ^ ^ , and H\, respectively, and dy = ds = 0.Proof. T h a t K an d ® are closed l inear manifo lds fo llows im m edi a te l y

f rom Theorem JJ-^ and 7.7.2 and that ^ is a lso follows from the interior

reg ula ri ty th eo rem s of §§ 5.5- U sing L em m a 7-5-3, we see th a t © an d ^

are or th ogo na l an d th a t ( an d % a re bo th o r thogona l to ^ 0 HI and , in

fact , ii H^Hln{22Q^G'^), t h e n / / $ $ ( ^ J a n d ^^ a re bo th eve ry

where dense in £2)-

Now, s uppos e rj^H\ and le t y a n d s be i t s p ro jec t ions on g and %,

re s pec t ive ly . Us ing Theorem 7-7-(> we conc lude the ex is tence of un ique

forms a an d /5 in $ J O (S2 0 §+) an d $ ^ 0 (S2 0 ©") resp ectiv ely,

s u c h t h a t

{7.7 .^) da = y , doc = 0, 3 ^ = 0, d^ = e .

Since y a n d e are the projections of 77 on g and %, we see from {7'7-^)

t h a t a and /9 satisfy

(da, dC+) + (da, dC+) - (v , dC+) = (da, dC+) + (da, dC+) -

- (dri, C+) = 0

^ • • ^ (d^, di-) + (d^, dc-) - (n. dc-) = (dp, dc-) ++ (dp,dC-)-(drj,C-)=0

for all C"*" i n ^ J a n d f- i n ^ ^ . T h u s a is the plus potentia l of drj a n d ^ is

the minus po ten t ia l o f drj. The results follow from Theorems 7-7-3 and

7-7.5-

"J.']. Po t e n t i a l s ; t h e d eco mp o s it io n t h eo rem 3 1 3

The fo l lowing theorem conta ins fur ther in format ion concern ing the

decompos i t ion [7J . 7 ) ,

Theorem 7.7.8. Suppose co ^ S2 <^^^ y , £ , cind H are its projections on

(£ , , and ^ , respectively, and suppose Q^ and Q~ are the plus and minus

potentials of o) — H, respectively. Then

(8.7.10) y^doc, e=^dp, oc = dD+, ^ = dQ-, doc = 6^ = 0.

If co^ HI, then <x and ^ are the plus and minus potentials of dco and dco,

respectively. We have the following differentiability results on the closure

ofM:(i) / / M is of class € { and co^ &p{p ^ 2), then y , e, and H^ Qp and

a, ^, Q+ , and Q-^H],] if also w^ HI, then y , e, and H^H \ with y , d

and H in C^ in case fx = \ •— vjp and p ^ v.(ii) / / M is of class C^ with ^ > 2 and 0 < /^ < 1 and if co^ C"^"^,

then H y y , and e ^ C^~^ and oc, ^, Q+, and Q- ^ C^~^; if also, co ^ C^~^,

then H, y , and s^ C^~^.

(iii) / / M and w are C^ or analy tic, so are oc, ^, y, e, Q^, and Q~.

(iv) In the case of zero forms, H is a constant, and £ = O.Ifw^Hl with

dco = 0 or if 0)^ 22 cif^d w = dt] where 7]^ HI, then y = 0', if o)^H\ and

dw = 0 or if 0)^ 22 ^nd m = drj where r}^ H\ then e = 0.

Proof. Suppose , f i rs t , tha t o) ^ H\. If we then define oc, ^, y, s, i2+,

a n d Q- by (7.7-10), the results follow from (JJ.S) and (7.7.9). In case co

is merely in £2, we use the left s ides of (7 .7-9) and approximate , us ing

th e Theo rem s of § 7 .6 and Th eor em 7-7-2 . T he regu la r i ty re su l ts and th e

las t s ta tement fo l low from the fac ts tha t oc a n d ^ are the respec t ive

poten t ia ls o f dco a n d dco, since dH = SH = 0, in case co^ HI. The la s t

results for co m erel y in S2 follow from L em m a 7.5.? .

W e may now p rove a s l igh t ly s t reng thened fo rm o f an inequa l i ty due

to Fr iedr ichs [3]-

Theorem 7.7.9. There ^'s a A > 0 such that if co ^H\ j ^ ^ , thenD{co)-> Kloof.

Proof. Fo r if a) ^ i / | J _ § , then

(7.7.11) w=y + 8, dy = de = 0, y ^ ^ J , £^"^2 > {y,s)=0.

Hence from (7.7-9) and Theorem 7-7-2, we see that

D{w) = D{y ) + D{8) ^2,-^\\y \\^ + ?i-\\8r ^X[2\\y\\^ + 2\\s\\^]

> A II CO IP, ;i = m in[A +/2, A - /2 ] .

The fo l lowing theorem comple tes the ana logy wi th the case for a

c o m p a c t m a n i f o l d w i t h o u t b o u n d a r y .

Theorem 7.7.10. If f] ^22 cc^d is 22-orthogonal to § , there is a unique

form Q in H\ and 22-orthogonal to § such that

(7-7.12) [dQ, dC) + (SQ, dC) - {v> C), C^Hi

7.8. Boundary value problems 315

re s u l t s on the b o u n d a r y d e p e n d on the g i v e n b o u n d a r y v a l u e s as well

as on the dif fe ren t iab i l i ty of M and are s ta ted be low.

The fo l lowing theorem is seen (from their proofs) to be e q u i v a l e n t

to Theo rems 3 an d 4 of the paper b y D U F F and S P E N C E R (pp. 150,15I):

T h e o r e m 7.8 .1 . (a) / / co is any closed form in H\, there is a unique

harmonic field H such that (r = degree of co).

co = H + d^, tH = tco,^,dp^'^^{tp = td^ = 0), 0 < r < w - l .

(b) / / CO is any co-closed form in HI, there is a unique harmonic field H

such that

cjo = H + d(x, nH = ncjo,oc,d^^} {noc = ndoc = 0), i < r ^ n .

In either case, the differentiability results are as follows:

(i) / / M is of class C} and co^Hl, p >2, then H is also and co andH^ C^ on M if p ^ V and // = 1 — v/p. Ifr = 0,co and H are constants.

(ii) / / M is of class CKC^, analytic) and co is of class C^- i (C"" , C"),

^ > 2 , 0 < ^ < 1 , then H is of class C^~^ (C ^ , analytic).

Proof. If ^co = 0, t h e n f r om T h e o r e m s 7-7-7 and 7.7-8, we see t h a t the

t e r m 6^ = 0 in th e decompos i t ion wh ich p roves (a); (b) is proved s imi

la r ly . The dif fe ren t iab i l i ty resu l ts fo l low from Theorem 7-7-S.

T h e n e x t t h e o r e m is a re f inemen t of T h e o r e m 2 of the p a p e r of D U F F

and S P E N C E R .Theorem 7 .8 .2 . Ifrj is any form in H\, there is a form co in HI such

that tco =^ trj and dco is a harmonic field. If, also, YJ = d%for some y ^ in H\,

then there is a unique co of the form co = d^ with | in HI which satisfies the

conditions above.

Proof. Let H- be the pro jec t ion of drj on §-, let oc be the m i n u s

p o t e n t i a l of drj — H~, and le t

(7.8.1) CO = rj — y, y = da, s = doc.

T h e ny

ands are

in ^and

from (7-7.3) (^= ^), we

h a v ety = 0 = t e, dy -\ - ds = drj — H~,

dco = dr) — dy = H~ -{-de.

From (7.8.2) and the l a s t s t a t e m e n t in T h e o r e m 7-7-^, we see t h a t

^ c w € ( § e ® ) n ( § e e ) = = § -

Now, s uppos e Tj = dx for s ome x in H^. T h e n co =^ d(x — oc) from

(7.8.1). Suppos e co i also satisfies all these condi t ions . Then

t(a) — coi) = 0, CO — co i == dv{v = X — oc — | i ) , d{co — coi) ^ § .

But then , f rom the definit ions of g and ^ and T h e o r e m 7-7-S, we o b t a i n

CO — COi^ '^2> ' ' • ^(^ ~ ^1) € ^ n § , - • . d{cO — COl) = 0, . ' . CO ~ COl

^ § © T ) , CO — CO = (5r § © g , . • .CO — coi$ §-, CO —coi^S2 —©~

(T he or em 7-7-5 (i)),

. • . CO — CO = 0.



3 1 6 The d-NEUMANN problem on strongly pseudo-convex manifolds

W e now cons ide r bounda ry va lue p rob lems for ha rmon ic fo rms as

dis t inc t f rom harmonic f ie lds . We begin by defining

//io = ^2"-n $2-, §0 = |)+n r = «»n i/|o-

T h e n HIQ consis ts of all HI forms co w i t h tco = n(o = 0.

Theorem 7 .8 .3 . ^0 consists of the element 0.

Proof. O b v io u sl y if H ^ ^0, t h e n H ^ HIQ and min imizes D(H)

a m o n g all forms in HIQ. We may clearly f ind a R i e m a n n i a n m a n i fo l d M'

of the same class as M s u c h t h a t M C M'. If we e x t e n d H to van is h on

M' — M, t h e n H satisfies the s ame cond i t ions on M' and hence satis f ies

(7.8.3) {dH,dC) +(dH,dO =0

for all f in H^ and hence in ^2^ or ^ and so has the in terior differenti

ab i l i ty p rope r t i e s on M' s t a t e d in §§ 5-5 and 5-6. The theorem fo l lowsfrom the u n i q u e c o n t i n u a t i o n t h e o r e m of A RON SZA JN et al.

Definit ion 7.8 .1 . A form K is harmonic on M if and only if Ky dK,

a n d dK^ HI on any doma in in te r io r to M w i t h d dK + d SK = 0 t h e r e .

Theorem 7 .8 .4 . If oj^ HI, there exists a unique harmonic form K in

H\ such that t K = too, n K = no). The differentiability results for K are

the same as those in Theorem 7.8.1 except that in the case of zero-forms,

K^Clifm^Cl.

Proof. W r i t e K = oj ~\- rj and min imizeD{oj + ^) = D{rj) + 2{drj, dw) + 2(drj, dco) + D (co)

among a l l rj in ^20 H (S2 © $0). T h e n D{rj) > A || 77 p so t h a t the m i n i m i z

ing function exis ts as us ua l . Then K is easily seen to satisfy (7.9.3) for all

f in HIQ SO t h a t K is h a r m o n i c on the in te r io r of M (using the differenti

ab i l i ty resu l ts of § 7.4). Since YJ^HIQ f) (S2 0 §0) , the equ iva len t equa t ion

(7.8.4) [df] + doj, dC) + (drj + dco, SC) = 0

fo r all C$ HIQ is of the form (7.6.1) on b o u n d a r y c o o r d i n a t e p a t c h e s . The

differentiabil i ty results follow from § 7-^- The uniqueness follows from

Theorem 7 .8 .3 .

More genera l theorems were proved in the a u t h o r s p a p e r [11]. But

t h e s t a t e m e n t s of the re s u l t s are too long to be rep roduced he re .

C h a p t e r 8

T h e 5 - N e u m a n n p r o b l e m on s t ro n g ly

p se u d o -c o n v e x m a n i fo ld s

8.1. I n t r o d u c t i o n

In th i s chap te r we pre s en t a s impl i f ica t ion of the recent so lu t ion due

t o J. J. KoHN ([1], [2]) of the so-called ^ - N E U M A N N p r o b l e m i n t r o d u c e d

by G A R A B E D I A N and S P E N C E R for complex exterior differential forms on

8.1. In t ro d u c t i o n 3 1 7

a compac t complex -ana ly t i c man i fo ld wi th s t rong ly p s eudo-convex

bo un da ry . T he p rob lem in i t s p re s en t fo rm was inves t iga ted by D . C .

SPENCER and J . J . K O H N by means o f in teg ra l equa t ions . The p re s en t

author [I3] solved this problem for the special cases of 0-forms and

z — 1-forms ( i.e. forms of th e typ es (0 ,0) an d (0,1) in our cur ren t no tati on )

on ce r ta in ' ' t ubu la r" man i fo lds and us ed thos e re s u l t s to p rove tha t any

compac t rea l -ana ly t i c man i fo ld can be ana ly t i ca l ly embedded in a

Eucl idean space of suff ic ien t ly h igh d imens ion . Unfor tuna te ly there is

an erro r in t h a t p ap er w hich is corr ected in § 8 .2 by us ing the re sult s of

K O H N presen ted in th is chapte r . These resu l ts apply to forms of a rb i t ra ry

t y p e [p, q) an d th e solutio n form s a re show n to be of c lass C° on th e

c los ed man i fo ld p rov ided the me t r i c , bounda ry , and non -homogeneous

t e r m ^ C^ t h e r e . R e c e n t l y H O R M A N D E R [2] has ex tended these resu l tsus ing Z2-n ie thods and cer ta in weight func t ions . He was ab le to demon

s tra te exis tence ( in the sense treated in §8.4 below) of forms of type

{p, q) in cases where the Levi form ((1.2) below) ei ther has a t leas t q + \

nega t ive e igenva lues or a t leas t n — q posit ive e igenvalues . This is a

much less res t r ic t ive condi t ion on h M than our condi t ion of pseudo-

convex i ty . F ina l ly , K O H N and L . N I R E N B E R G have obta ined a s t i l l

g rea te r s impl i f ica t ion of th is theory . I am presen t ing here essen t ia l ly

the i r s impl i f ica t ion bu t conf ined to the 5-NEUMANN prob lem, whe reasthey have appl ied the i r method to a more genera l c lass o f p roblems .

In h is recent papers , K O H N sketched applications of his results (a)

to th e s tu dy of th e 3-cohomology th eor y , (b ) to th e s tu dy of deform at ions

o f complex s t ruc tu re s , and (c ) to obtain a new proof of the result of

N I R E N B E R G a n d NEWLANDER which s howed tha t a complex ana ly t i c

s t ru c tu re cou ld be in t rodu ced on an in teg rab le a lmos t -com plex man i fo ld .

However , par t o f the in te res t in th is p roblem to those work ing in par t ia l

d i f fe ren t ia l equa t ions l ies in the fac t tha t the problem is no t a regula r

bounda ry va lue p rob lem in the s ens e o f A G MON -D OU G LIS-N IREN BERG

([1] and [2]), B R O W D E R ([1], [2]) LOPATINSKY, etc . ) . We shall give an

example be low a f te r we have in t roduced the no ta t ions and s ke tched

the resu l ts ; we sha l l a lso show the connec t ion wi th the 5-cohomology .

W e a s s u m e t h a t M = M \J h M i s a com pac t com plex -ana ly t i c

m a n i f o l d h a v i n g b o u n d a r y h M oi c lass C^. We assume tha t we a re

g iven a he rm i t i an m e t r i c

( 8 . 1 . 1 ) ds^ =g^pdz''dz^ 1 {g^oc=gocp, ^ , ^ 5 = 1 , . . . , ^ )

which is of class C^ on M. W e s uppos e tha t the func t ion r ^ C^ {M) a n d

equa ls the nega t ive of the geodes ic d is tance to b M for po in ts wi th in a

d i s tance —SQ of b M, SQ <, 0. I t is c lear that there exis ts a s l ightly larger

such manifo ld M' s u c h t h a t M C M' a n d t h a t t h e m e t r i c r can be ex-

1 R epea ted Greek ind ices are sum me d f rom 1 to v.

3 1 8 T h e ^-NEUMANN problem on s t rongly pseudo-convex mani fo lds

t end ed to ^ C^ {M') s o t h a t r is the geodesic dis tance from b M on

M' — M. The s t rong ps eudo-convex i ty o f the bounda ry h M impl ies

th a t the re i s a co ns t an t co > 0 such tha t a t an y poin t Po on h M (where

r = 0) we ha ve

(8.1.2) r^rzv T^ Tr > c^g^y T^ Tv

for all complex vectors (T^, . . . , T^) s u c h t h a t

(8.1.3) rz^T^=.o.

I f / i s any o ther rea l func t ion of c lass C^ n e a r b M s uc h t h a t V / ¥" 0 and

f = Oonb M, a n d / < 0 o n M n e a r b M, then the posit iveness of the form

/ / ^ y T^ fy for T such that/^/S T^ = 0 fo l lows . In the above and through

ou t th i s chap te r we a s s ume tha t the ope ra to r s dldz°' a n d dldz^" a re

def ined by

z< ^ =z x^" + i y'^, z"" = x"" — i y"".

W e let 91 de no te th e set of all ex ter io r differen tial form s of class

^^[M) (i.e. C on M) an d de no te by 9l2^'« th e set of all tho se w hi ch

are of type [p,q), i .e . which can be expressed in any local analytic co

ord ina te sys tem in the form

(8.1.5) ^^-^'"^^^

W e a b b r e v i a t e t h e n o t a t i o n t o

(8.1.6) (p = Z ^ijdz^ AdzJ;

w hen we use th is no ta t ion / an d / wi ll a lw ays s ta nd for increas ing

sequences as in (8 .1 .5). However, we shall often wish to have the (pi,ji,. ,jq

defined for all sequences of indices ^i . . . jq ; in th is case , we assume tha t

t h e (p's are def ined so as to be an t isy m m etr ic in the y- ind ices . W e sha l l

a t t imes wish to do the same with the / ind ices and sha l l somet imes wri te(pi,0LR w he re i? = (ri, . . . , r^_i) w it h /'i < • • • < rq_\ a n d oc runs f rom 1

t o V i n d e p e n d e n t l y oi R.

We shall wish to consider M (or M) as a rea l manifo ld wi th metr ic

given in [x, y ) •= {x^, . . ., A.'', y i, . . . , 3; ) co or di na te s b y (1.1) w hi ch

becomes , on se t t ing dz^" — dx°^ + i dy"- a n d dz^ = dx^ — i dy^,

giocpidx'^dx^ + dy^dy^) + Ig^^^dx^dy^,(8.1.7 , .

T h e n t h e d u a l *(p would be defined by firs t express ing cp in terms of real

differentia ls dx°' a n d dy"^ and then t ak ing the o rd ina ry rea l dua l o f the

rea l and imaginary par ts . This p rocedure in t roduces a fac tor 2^+^ in to

t h e c u s t o m a r y i n n e r p r o d u c t

(8.1.8) (99,^) = f(pA*ip

8.1. In t ro d u c t i o n 3 1 9

of two forms of the same type. Along with most workers in this f ie ld we

omit these fac tors . The space S^ '^ i s the comple t ion of the space %V^

using the inner product (8 .1 .8) and S is jus t the H I L B E R T space sum of a l l

the S^^ . For two forms of the same type , i t i s convenien t to def ine the

poin t func t ion (cp, ip} b y

((p,y)ydM = q)A'^yf, <(p,^y = \(p\^

w h e r e dM is the e lement of vo lume on M, The formulas for ((p, ip), dM,

a n d (^(p, ip} a re

(8.1.9) {(p,yj)=l{(p,tpydMM

where in any ana ly t i c coo rd ina te s ys tem

dM = r(x, y ) dx dy(8.1.10) ^ ^^ _

if (p is given by (8.1.6) and y ) i s correspo ndingly def ined . H ere F i s the

V X V d e t e r m i n a n t o f t h e gx^, g^^ is the p X p de te rminan t o f the

^^v^^ a n d g'^^ i s the q X q de te rminan t o f the g^a^^. I f we use the an t i

s y m m e t r y o f t h e (p's a n d y)'s in a l l the i r ind ices , we may wri te

<^ ' ^> ^ ^ ^ * ' ' * ' • • • ^^^**^^^"i^i • • • g^^^"^ 9h • • • ivh -'-kX

XWkl-' 'kph' • . Iq -

For forms in 51 we def ine the opera tor d as follows: If 99 is of type (p, q)

w it h ^ = r, we define ^(p = 0',iiq <Cv a n d cp is given by (8.1.6), we define

(8.1.11) d(p == 2J(pij-,o^dz^ A dzi A dz^.

For forms in 51 we define b^? as follows: If 99^ 51^'^ an d ^ = 0, we d efine

b(p = 0; other wise w e define b99 as th a t form of ty p e {p, q — 1) such tha t

(8.1.12) . (b(p, f ) = {(p,dyj)

for every ip in 9][2>.«-i w it h co m p ac t s u p p o rt in M. As is seen by inte

gr at in g by p ar ts (see § 3), th is leads to a form ula of th e form

(8.1.13) {b<p)iR = ( - 1 ) ^ + i r ^ ( ( ^ / , a i ? . , ^ + ^ f ^ ^ ( P S , . T ) ,

for suitable functions A^^^, and to the genera l fo rmula

(8.1.14) ((pjtp) = (b(p,f) + f^co,y)ydSbM

wh ere ^ 5 is the inva r ia n t surface e lem ent on 6 M an d

(8.1.15) a}=v(p, (joiR= [—^Y g°'^(pi,o^Rrz^.

From (8 .1 .15) , we may a lso der ive the formula

(8.1.16) {d(p,^p) = (<p,by)) + j(cp.vxpy dS.bM

W e let 5lo de no te th e sub set of 99 in 51 for w hic h v q) = 0 on b M.

3 2 0 T h e ^-NEUMANN problem on strongly pseudo-convex manifolds

I t fo l lows immedia te ly f rom (8 .1 .11) and the an t isymmetry of the

e x t e r i o r p r o d u c t t h a t

(8.1.17) dd(p = 0, (p^%.

F r o m (8.1.14) an d (8.1.16) if follows t h a t if ^ a n d t^^ 51, th en

{b(p, dy )) = {(f, ddip) — J (v(p, dip} dS

(8.1.18) ^^= (b hep, w) + j i'^'^^y ipydS.

bM

By f i rs t le t t ing ^p be a rb i t ra ry wi th compac t s uppor t in M a n d t h e n

le t t ing i t be a rb i t ra ry we f ind tha t

(8.1.19) bbq) = 0 on M and vb(p = 0 on 6 M if cp^'Sio.

T h e 5^-NEUMANN problem is to show the ex is tence and regula r i ty of

the so lu t ions of the complex Poisson equa t ion

(8.1.20) nq bdf + db(p -= € 0

s ub jec t to the bounda ry cond i t ions

(8.1.21) vq) = vd(p = 0 on bM.

This boundary va lue problem is seen to a r ise formal ly f rom the var ia

t iona l p roblem of min imiz ing the in tegra l

(8.1.22) d{(p,(p) — 2Re(co,(p), d[(p,\p) = {d(p,dip) + [bip^btp)

among all 99^ 5Io ( 9? = 0 on & M). If co and 99^ C'^{M), we see that 99

satisfies

(599, dip) + i^cp, bip) — {cD, ip) = 0

= {\I](p — co,ip) + f {(v dcp, ip) — (Sep, Vyj)} dSbM

for all tp^^o', the second line follows from (8.1.14) and (8.1.16). Since

yip = 0^ we see from (8.1.23) (and kno w n formu las-see en d of § 3) th a tthe cond i t ion vd(p = OonbMisdi n a t u r a l b o u n d a r y c o n d i t i o n .

8 .2 . Resu l t s . Examp les . Th e an alyt i c emb ed d in g th eorem

In order to get a complete pic ture of the results , we define

(8.2.1) D{(p,ip)=d((p,y^)+{cp,yj) {(p^yj^^o)

w h e r e d{(p,ip) was defined in (8.1.22). We define ^ as the closure of 5Io

with respect to the norm corresponding to the inner product D, The following pre l im inar y the ore m is p rov ed in § 8 .4:

Theorem. If (p ^'^, then cp ^ H\{Ms) for each s < 0 and if ip ^% and

we form d(p, b(p, dtp and by ) as in (8.1.11) and (8 .1.1 3) using the strong

derivatives then the inner product D((p, y j) is still given by (8.2.1). Moreover

^ ^ € HIQ(M). Finally, if cp^ H\[M ) andv(p^H\^{M), then 9 ? ^ ^ .

8 .2 . R es u l ts . E x am p l es . T h e an a l y t i c emb ed d i n g t h eo rem 3 2 1

Then we def ine the opera tor L as follows: (p^^(L) <^q)^% a n d

there ex is ts an A: in S s uch tha t

d((p , f) = [a,\p), ip^'i^;

if 99^ ® (L), we define Lcp = oc. We define § as the set of all (p in % s uch

t h a t d(p = b(p = 0. We define ^PQ = ^ PI S^^^, ^^^ = § H S^^ . In

§§8 .4 and 8 .6 , we prove the fo l lowing pr inc ipa l theorem:

Theorem, ( i) L is self adjoint, ^[L) = S © § , and § is closed

(ii) If co^ S 0 $ , d ^ unique (p^'^[L) Pl ( S 0 | ) ) ^Lq) = co.

(iii) If we define N w = Ofor co^ § and N w as the solution in (ii) if

<^€ S 0 § , i^^'^ ^ ^^ completely continuous.

(iv) If q > 1, ^'P^ is finite-dimensiona l.

(v) IfM^%, thenNw^%.(vi) Ifq'>i, § ^ ^ C ^ r -

P a rt s (i)— (iv) are pro ved , for forms of ty p e (p,q) w i t h ^ > 1, i n

§ 8 .4 . O nly p ar ts of the resu l ts a re prov ed th ere for forms of typ e {p , 0 ) ;

the s e a re t r ea ted comple te ly in ^S.6. The smoothness resu l ts in (v) and

(vi) are proved in ^S.6. The pr inc ipa l too ls used a re (a ) the importan t

form ula (8.3 .15) for in teg ra t io n by pa r ts , an d (b) th e "s - no rm s" in t rod uce d

in §§8.5 and 8.6.

Be fore p roceed ing , we in t roduce s ome add i t iona l no ta t io ns : Th emanifo ld Mg = M(s) for s < 0 con sists of all po in ts P on M for which

r (P) < s. An ana ly t i c coo rd ina te pa tch wi th doma in G an d ran ge 9^ is

sa id to be tangential a t s ome po in t PQ of b M ^ SL p a r t g of b G con ta in s

th e or ig in and correspon ds , und er th e m ap pin g f rom G to 91, to 9^ H & M ,

the or ig in corresponding to PQ, and at the origin gocp{0) = doc^ a n d t h e

ex te r io r no rma l to M a t PQ corresponds to the pos i t ive y ^ axis (i.e.

ryv{0,0) = 1). I n cas e r = (r^, . . ., r^) or ( n , . . ., tr ) is a set of indices,

Ty de no tes th e set (r^, . . ., T^~^, T^+^, . . ., T^). If oc = {oci, . . ., av) is a

sequence of non-nega t ive in tegers , then D^" m e a n s D^] . . . D^l. If 99 isa vec tor func t ion V^^ deno tes i t s g rad ien t .

Nex t , we g ive an example to i l lu s t ra te the fac t tha t the 5 -Neumann

problem is no t regula r , except in the case where q = v when i t reduces to

the DiR icHLET problem s ince dcp ^ 0 a n d v q) = 0 on b M '^ (p ^=^ 0 on

b M in th a t case . In th e case ^ = 0 , the proble m is obviou s ly no t re gula r

s ince the nu ll spac e is ju s t t he sp ace of holom orp hic form s §2^0. To show

th a t th e prob lem is no t regula r for 1 < ^ < r — 1, we t ak e , as an

e x a m p l e , v = 2, M the un i t ba l l in i^4 , the metr ic Euc l idean , and se t

cp = (pi dz^-\-(p2 dz'^, (pi=—z^0y (p2 = z^0,

0 = ^ ~ / ' ^ ' A (z), r^ = z^z^ + z^z^,

6

w h e r e A (z) H\[M) and is ho lomorphic on M(0) bu t i s no t in Hl{M).Morrey, Multiple Integ rals 21

3 2 2 T h e ^-NEUMANN problem on strongly pseudo-convex manifolds

T h e n

r • vcp = z^cpi + L^(p2 ^ 0, d(p = CO dz^ A dz'^

(O = 992-1 - 991-2 = Z^ 0-1 + Z^ ^-2 + 20 = (\ - T^) A (z) H\^{M)

A(pi=: —Z Â0 — 0z^, A(p2 = Z Â0 + 0zl

A0=--^(zÂzi + z^ Az2 + 2A)^L2(M).

I t fol lows eas ily that ^^^{L) b u t 99 d o e s n o t ^ H^{M) as i t would if the

p rob lem were regu la r .

W e now p rove a theo rem ind ica t ing the connec t ion wi th the d-

cohomology theo ry :

T h e o r e m . If (p ^ ^^^ with q "> i and if dcp = 0, there is a harmonic

field 990 (€ ^ ^ ^ ) such that (p — cpo =~d 0 for some 0^ S^v,q-i,

Proof. Fi rs t of a l l, sup pos e / ^ C ^ on Ri, and we define

01 = f{r) {v 99), / (O ) - 0 , /MO ) = 4 , f{r) = 0

for y < So < 0 for so m e su ch SQ. A compu ta t ion s ome th ing l ike tha t in

(8.4.11) and (8.4.12) below shows that

v{(p — d0i) = 0 on & M .

Hence we may assume tha t 99^3lo^ . Then , le t 0 = N b(p. Then i tfollows from Theorem 8.6.3 that 0 , d0, a n d 1 0 all ^ 5Io, so that db0

i s o r thogona l to b{(p — d0) (see (8 . I .14)— (8 . I . I6 )) a nd a lso equa l to i t

an d h enc e zer o. T h u s (since ^99 = 0)

b{(p — d0) = d{(p — d0) = O or (p — d0^ ^^^.

The following analog of the K O D A I R A decom pos i t ion th eor em is of

s ome in te re s t :

Th eor em . S = § © '^ ' © K where ^ has its usual significance, ® ' is

the totality of forms of the form d(pfor some 99^ ® , and K is that of forms of

the form b\p for ip in%. If the given form in S $ 31, then its projections on

§ , ® , and g also ^ 31.

Proof. I t is c lear that if A ^ § an d 99 a n d W ^^, then the fo rms h,

d<p, an d b! F (see (8.1.14) — (8.1 .16) and app rox im at ion ) a re m ut ua l ly

or thogona l . S ince ^'P^ jus t consis ts of the holomorphic forms of degree p

an d th e §2^^ w ith §' > 0 are f ini te-dim ensio nal , i t follows th a t § is

closed. If c o ^ S © § , le t 0 = No), (p = b0, a n d y ) = d0. F r o m

Th eo rem 8.6.3 , i t follows t h at 99 an d y)^^ 0 ( S © § ) an d t h a t

(8.2.2) d(p + bip = 0) a n d bq) = dip = 0.

I t i s c lea r f rom the f i rs t s ta tement above tha t the decompos i t ion is

un i que . T h a t © an d 5) ' a re c losed fo llows f rom our pr inc ipa l resu l ts an d

the part icular choice of 99 and ^ in (8.2.2).



8.2. Results. Examples. The analytic embedding theorem 3 2 3

In case co ^ ^ , we f i rs t sub trac t off the form coi defined (different

n o t a t i o n u n f o r t u n a t e l y ) in the proof of Theorem 8.4-4, so t h a t coo

= CO — coi^^Q. If we then def ine 0Q, (po, and ipo as a b o v e in t e r m s of

coo, we see that (8 .2 .2) holds for COQ and t h a t 9 0 and ipo^ 5to-

W e can now p r e s e n t a s implif ication and correc t ion to the proof given

in the p a p e r , M O R R E Y [I3], m e n t i o n e d a b o v e of the poss ib i l i ty of

embedd ing ana ly t i ca l ly a rea l -ana ly t i c abs t rac t man i fo ld in E u c l i d e a n

space . The error in t h a t p a p e r was in the proof of T h e o r e m C of t h a t

pape r wh ich was g iven in §§ 8 — 1 1 . The e m b e d d i n g t h e o r e m was p r o v e d

by BocHNER for compac t man i fo lds in 1937 ([!]) a s s u m i n g the exis tence

of an ana ly t i c me t r i c ; th i s r e s u l t was e x t e n d e d by MALGRANGE in 1957

to the case of non-compac t man i fo lds . The re s u l t of M O R R E Y [13] was

genera l ized to mani fo lds wi th a coun tab le topo logy by G R A U E R T us ingm e t h o d s of the t h e o r y of func t ions of severa l complex var iab les and

some resu l ts of R E M M E R T w h i c h had not been pub l i s hed at t h a t t i m e .

W e now ou t l ine our m e t h o d of proof. F i r s t of all, on a c c o u n t of

B O C H N E R ' S resu l t , it is sufficient to s how the following:

T h e o r e m A. With each point PQ of the given real analy tic compact

manifold MQ there are associated v functions Wy which are analytic over the

whole of Mo and have linearly independent gradients at PQ.

F o r the grad ien t s wi l l r ema in l inea r ly independen t in s ome ne ighbor h o o d of PQ and t h u s MQ can be covered by ne ighborhoods 9 ^ , q = 1, . . . ,

Q, w h e r e the func t ions Wqy, y — 1, . . ., r are ana ly t i c ove r MQ and h a v e

l inea r ly independen t g rad ien t s ove r 9^^, q = \, . . . ^ Q. The m a p p i n g

Wqy = Wqy(P) m a p s Mo ana ly t i ca l ly in to Euc l idean s pace of Q^ d i m e n

s ions ; the m a p p i n g may no t be 1 — 1 in t he la rge but is locally 1 — 1

and non-s ingu la r and the Euc l idean me t r i c induces an a n a l y t i c m e t r i c

on MQ.

T o p r o v e T h e o r e m A, we f i rs t embed MQ in an open complex ex tens ion

M (see M O R R E Y ([13], § 2, S H U T R I C K , or W H I T N E Y - B R U H A T where this

e m b e d d i n g is discussed for mani fo lds wi th coun tab le topo logy) . Let PQ

b e any p o i n t on MQ, let TQ be a complex -ana ly t i c coo rd ina te pa tch wi th

d o m a i n Go con ta in ing the origin and r a n g e ^0 c o n t a i n i n g PQ, in which

PQ and the orig in correspond and the p a r t of GQ in R^ (i.e. for which

y = 0) co r re s ponds to Sto H MQ, and choose a Hermit ian metr ic (8 .1 .1)

w h i c h is of class C^ (M), w h i c h is rea l on M o (i.e. goc^ is rea l on M o ) , and

for which we h a v e

(8.2.3) go:^ (x, y) = docp (x, y) on Go,

with re s pec t to the coord ina te s ys tem To.

F o r p o i n t s P on M n e a r Mo, we define r'{P) to be the geodesic

d is tance f rom P to Mo. It is eas i ly shown (see M O R R E Y [I3], § 3) t h a t

the func t ion K(P) = [y '(P)]2/2 is of class C^ in a n e i g h b o r h o o d of Mo

21*

3 2 4 The d-NEUMAN]sf problem on strongly pseudo-convex manifolds

inc lud ing all po in t s whe re r ' (IP) < RQ. We define MR as the c o m p l e x

ana ly t i c man i fo ld r' (P) < R. It is easy to compute that (^ajS (x, 0) is

real)

(8.2.4) 4X,a^^(:^,0) T-T^ = K^-yP\x,0) T-T^ = g,^[x,Q) T-T^for any complex -ana ly t i c pa tch wh ich ca r r i e s the p o i n t s (x, 0) i n t o MQ.

T h u s if 0 < i? < 2^1, b MR is regu la r and of class C^ and MR is s t rong ly

p s e u d o - c o n v e x ; the func t ion r (P ) used in th is paper reduces to r' (P) —

— R on MR and the pseudo-convexity follows from (8.2.4). In fact, s ince

r{P) =r' (P) — R and K{P) = \r'{P)'\'^j2 on MR, we see by following

t h r o u g h the proof of T h e o r e m 5.8 (of the c i ted paper) as far as e q u a t i o n

(5.18), t h a t we o b t a i n

dR[(p) ^ IR[(P) + j Z^""' ^^"^ g""^ g''r,^',yCpio.SCpK6TdS[P)hMiR)

> -Ci((p^rp)R+C2R-^l\(p\^dS, (P^^(MR)

bM{R)

(cf. 8.3.15 be low) . Thus we o b t a i n

l\cp\^dS^CR[dR{(p) + ((p,(p)R], 0<R<Ri<Ro

where C is independent of R.

W e now s ke tch the proof, given in § 7 of the c i t ed pape r ( M O R R E Y

[13]), of the i m p o r t a n t i n e q u a l i t y

(8.2.6) [(p,cp)R<.CR'^dR{cp), 0<R^R2^Ri, (P^^^^HMR), q>\.

Inc iden ta l ly , th i s s hows th a t ^^^[MR) consists only of the zero element if

q > 1 and R is smal l enough. We conc lude f i rs t tha t there is an 7 3,

0 < Rz < Pi, s u c h t h a t M (P3) can be covered by a f in i t e number of

ne ighborhoods 3 f each of which C 9^^, the r a n g e of a complex ana ly t i c

c o o r d i n a t e p a t c h n of the t y p e of To (i.e. rea l on Mo) w i t h d o m a i n Gt,

a n d is the r a n g e of <3: '-quasi-geodesic" no n- an aly tic (but C^) coord ina te

s y s t e m rf w i t h d o m a i n of the form Got X P( 0, P3) , where G^t is a

d o m a i n of class C^ in rea l r -space R^ and Gof C C^ Pi R^. If P $ 9 f, its

quas i -geodes ic coord ina tes (f, rj) with re s pec t to rf are d e t e r m i n e d as

follows: There is a un ique geodes ic th rough P which is o r t h o g o n a l to

M o at s ome po in t of MQ. Let its e q u a t i o n s be

^ « ^ ^« (^) ^ ya ^ ya (;,) ^ 0 < r < r' ( P ) ,

in the {x , y) coord ina te s of T^ We definet{P) = x^{0), C°'(P) =- y (0) 0 - dyjdr).

Since the m e t r i c is rea l a long Mo, we see t h a t

ds'^ = gio:§{x,0) {dx^dx^ + dy'^dy^),

= gi «^ (x , 0) (di°^ d^^ + dC"" dC^), a long Mo.

8.2. Results. Examples. T h e analytic embedding theorem 3 2 5

By us ing the Lagrange method of reduc ing a quadra t ic form to a sum of

s qua re s , we in t roduce the rf b y

where the d"^^ C°° and the matr ix i s non-s ingula r so tha t

(8.2.7) ds^ = ^ia^(f, 0) ^ | « ^ 1 ^ + (5«^ drj^" drj^ a long MQ.

From the cons t ruc t ion , i t fo l lows tha t

(8.2.8) K(P) = \f,(P)\^l2, r'{P} = \f,{P)\.

I t is shown in M O R R E Y [ I3] 3 , th a t th e coo rd ina te s a re C .

W i t h e a c h t and each R,0 < R ^ R^, we def ine an ana ly t ic manifo ld

MtR as f_ollows^ Let %R = r * [Got X B(0,R)l and le t GtR = r^^ {%R) ;

c learly Got = GtR D R^- Choose pos i t ive numbers at a n d At such tha t i fwe define

Ft=Ff\ Ff:\x''\<At, dc=\,..,,vt h e n

GtR,CFlo^xB{0,atl2)^ GotCFl'K

E x t e n d t h e m e t r i c gtup to be of class C^ for all (:^, y ), to be periodic of

per iod 2 At in each x^" to be real ii y =^ 0, and s o tha t gt<xp(x, y) = d<xp

for all X on and nea r dFt and all y with | y | > 'iatjA- Then, if % is small

enough, the quasi-geodesic ( | , rj) coord ina tes can be ex tended to a l l(f, Yi) w i t h \rj\ ^ at to be periodic of period At in eac h f . W e th en let

MtR be the set of all {x, y ) corresponding to the ( f , rj) w i t h \rj\-^R, a n y

two po in t s (xi, y) a n d {x2, y ) where each x^ — x^ = 2At n'', n"' an in teger ,

be ing iden t i f ied .

Now, s uppos e (f^'^'P^ {MR), 0 < i^ < R^. L et { C J, s = 1, . . . , S

be a pa r t i t ion o f un i ty s uch tha t e ach f s € G^^H^R^ and has s uppor t in

MR^ n 91* for som e t (Cs ne ed n ot va ni sh on h MR^) and le t (ps = Cs (f -

Th en , b y ap pr ox im at in g 99 by sm oo th forms , as we m ay on ac cou nt of

our pr inc ipa l resu l ts , we see tha t each (PS^^^^{MR). But now, we may

assoc ia te each (ps with a form (pst on MtR by def in ing the components o f

(fst on GtR to agree with those of (ps there and to vanish e lsewhere on

MtR. Then, c lear ly ,

dR{(ps) = dtR[(pst), {(fs, (PS)R = ((Pst> (Pst)tR,

dR[(ps) < C[dR{cp) + (g?,(p)R]

since the Cs do not depend on R. Accordingly^ it is sufficient to prove

(8.2.6) for iovms^^PQ(MtR), s ince, if th is is done, we would have

(?>, <p)r < 1 (^.. <PsYd' <C-S-R-[dR (^) + (9, g,U]Ws = l

fro m w hic h th e res ult follows easily if 0 < i^ < i^2 < ^ 3 -

So we consider some MtR an d dr op th e i . W e first p ro ve (8.2 .6) for

fo rms ^0 € ^20 (MR) if 0 < i^ < 2^3 < ^ i -

We sha l l hereaf te r denote the c o m p o n e n t s (pu s imp ly by cpK The i i

(g o, (fo) = j J ((po> (pQ)dS dr

(8.2.9) ^ '"^/^^Cfr^-^drfdSf2\9U^>lO)\^d2;(0).

0 F 2 ^'

w h e r e ^ = bB{0,\) and 6 deno te s coo rd ina te s on bB(0,'[), {^,rj) be ing

the quas i -geodes ic coord ina tes , and {r, 6) be ing po la r coo rd ina te s in the

?y-space. Since 990 (R, f, 0) = 0, we h a v eR

(8.2.10) ' B

I <pi(r. 1,6) |2 (i? - r) / 1 ,pl{s, 19) |2 dsr

Su bs t i tu t in g (8 .2 .10) in t o (8 .2 .9) and us ing th e condi t ion s s > r, we ob ta in

(^0, (foJR < CR^ • ((990, qô))R = CR^{{(po, (PO))R2 < CR^ dEs{(po)

= CR^dE{(po)

(for the def in i t ion of the s tr o n g n o r m ((99, 99))!'2 jn HI, see § 8.4) if we

define 990 = 0 for jR < y < i? 3 and use Theorem 8 .4 .1 for M(Rs). T h i s

is (8.2.6) for 990.

F r o m t h i s , it fo l lows tha t if 99$'5), H a unique 990^ HIQ{MR) w h i c hmin imizes the in teg ra l dR(q)o — 99) among all such 990 if 0 < i^ < Rs. I t

follow s t h a t 990 — 99 is a ha rmon ic fo rm H and t h a t

(8.2.11) dR((p) = dR((po) + dR{H)

To prove (8.2 .6) for a h a r m o n i c H, we sha l l p rove for any h a r m o n i c

H, whether in %^Q(MR) or not, that

(8.2.12) (iJ, H)R<C'RfiH,HydS (for R smal l enough)bM(R)

from which (8.2.6) follows, using (8.2.5). We may a s s u m e H^ ^^{MR).

To prove (8.2 .12), we see as in § 5, t h a t the rea l and i m a g i n a r y p a r t s W

of the componen ts s a t i s fy a s ys tem of dif fe ren t ia l equa t ions of the form

in t e rms of the quas i -geodes ic coord ina tes , where

(8.2.14) a'^^dO) =gf(i,0), b^^{i,0) = 0, c° '^ ( | ,0 ) =:d^^.

L e t us now take s phe r ica l coo rd ina te s in the ?^-space as above . Then the

equations (8 .2 .13) are seen to be e q u i v a l e n t to

Htr+ {v - i)r-^m + r-ÂêH^ + 2cyH^y + r-i Cy'Hiy,6 +

(8.2.15) + 2rB''Hlôc + ZB'^yH^^^y +A<'^H^^^6 + 2D{^H^â +

- 1 - EiH^; + Ir-ÊiyH^y + Fim = 0

8 .2 . R esu l t s . Exa m ples . The analy t ic embedding theorem 327

where A^e denotes the Beltrami operator on the unit sphere and all the

coefficients ^ C°° in {r, f, 6). We define the positive form Q by

Q = ^ [(^tf + ^-2 I V , H^ |2 + 2 C Hr H,y + r-^ Cr^ H{y H^s +

(^-^•^^) '"' + 2rB^H\ocHl + 2B-yH\o:Hiy+A'^^H\ocHy-\

(8.2.17) F{^)=\i: [H^s, f, e)fdZ{0).

Then we see easily that

F'{s)==2l ^ W (s, I 0) Hi(s, f, 6) d^d 2: {0)Fxi: ?

(8.2.18) r'{s) = 2 / ZH^HU+[H^:)^]dHZ

FX2^ ?"

F'(0) = 2 f ZHHi . 0) Hi{0,1 6) dSdZ = 0.Fxi: ?•

Using (8.2.15) to eliminate the Hl^, integrating by parts with respect to

the f* and Qy , and using the fact that

juA^Qud'^ = —f \ ^e'^f^'EE E

we find that

F"{s) = 2nQ + 2:[-{v-\)r-^WHi+2ClyWHi +

(8.2.19) + r-1 C l W Hiv + 2rB%o. W Hi + 2 B'^y W H\u +

+ ^pf H^ H\f\ - 2D{^ W H\. - etc.l d^dZ{B),

Using the positivity of Q and the simple device \2ah\ <, e a^ + e-'^ h'^,

we conclude from (8.2.19) that

F" (s) > - (r - 1) s-i r (s) -X^F (s)

where X depends only on bounds for the coefficients and their derivatives(i.e. on the metric). Thus if R is small enough ( < A~i i)

(8.2.20) F{s)<2F(R), 0 < s < i ^ .

thusR

(H, H)R < C lf^-^F(r) dr < CR^F{R) < CRJ{H, H)dS0 hM(R)

as desired.

The inequality (8.2.6) states that the constant C3 in Theorem 8.4-5can be replaced by C R^ for the manifolds MR. Thus, from Theorem8.6.4 we conclude the following theorem:

Theorem B'. If w ^"^ [LP^^) on MR and R < R2. then

(8.2.21) \\NR LRW\\<. C5 -R2 \\LRWI

where 11 11 denotes the norm in S^^'^^.

http://lrwi/

http://lrwi/

3 2 8 T h e d-NEUMANN problem on strongly pseudo-convex manifolds

W e now s how how to p rove Theorem A us ing Theorem B\ We firs t

cons t ruc t , fo r each R, func t ions W2Ry , y = \, . . . , w h i c h $ ^ ( L ^ ' ^ ) ,

are of class ^ ^ [M R ) , which a re ana ly t ic a t leas t in the ba l l B[PQ, R)

w i t h'^2Ryz^{0) = dy^ (with respect to TQ)

(8,2.22) I LR W2Ry I < Z i JR^ (on MR)

h = [ r /2] , 0 < i^ < 7^4 < ^ 2 , Z i independen t o f R .

T o do th is we firs t define wiy = zy in B {PQ, R2) with respect to To,

e x t e n d wiy to be of class C°^ on MQ U B(PQ, R2). W e t h e n e x t e n d wiy

in to s ome MR2 us ing W hi tney ' s ex tens ion theo rem, a s s ign ing the va r ious

de riv ativ es of order < ^ + 1 w ith resp ect to th e y °' in each complex-

ana ly t ic pa tch , rea l on MQ, i n s u c h a w a y t h a t t h e C a u c h y - R i e m a n nequa t ions and a l l the i r der iva t ive of o rder < h hold a long MQ. T h u s t h e

second condit ion in (8 .2 .22) is sa t is f ied. The functions W2Ry are con

s t ruc ted to ^ "S) (L^R^) by a method l ike tha t in the proof of Theorem

8.4.4 which re ta ins the second condit ion in (8 .2 .22). Then if we set

wsRy = NR LRW2Ry, wc scc f rom Theorem B' t h a t

II t 3i?y II < Ci^2 11 2 2i?y II <Z2i^^+2+W2.

But a lso w^Ry = W2Ry — w^Ry i s ana ly t ic on MR, S O wsi^y is analytic on

B{PQ, R). F ro m t he ineq uali t ies of § 2 .2, i t follows th a t

IV wsRy (0) l^ZsRK k= \ + (r/2) + [r /2] - r = % r 1.

Hence if R i s smal l enough, the grad ien ts o f the wsRy are so smal l tha t

those of the W4Ry (0) are l inearly independent a t 0 .

8 . 3 . S o m e i m p o r t a n t f o r m u l a s

In case q)^^^^, d(p w as defined t o be 0 if ^ = r an d wa s defined in

(8.1.11) otherwise . Start ing from that definit ion, we obtain

= ( ^ ^ ^ 2 ' ^ g 9 ^ / , ; i . . . V ^ 4 ^ ^ ^ A dz- A dzh A . . . A dP'a +

+ Z (— ^)^ dzi A dzh A . . . A dzh A dz"" A dzh+i A . . . A dz^Xd=i J

= {-\)^ Z ^S^-'^y-^ 9i,M'-z^ydzi ^dzM{\M\^ q + \,q<v).I,M y = l

(8.3.1)The form (8 .3 .1) has the advantage tha t the coeff ic ien ts a re an t isymme

tric in a l l the indices mi, . . ., mq_^i .

W e s ha l l be conce rned wi th bounda ry in teg ra l s ove r the man i fo lds

b Ms. W e no te tha t the func t ion r defined in § 8.7 is a re al- va lu ed func

tion of class C^ {M ) s u c h t h a t | V r | (as measured on ikf) = i nea r

8.3. Some imp ortant formulas 3 29

bM,r=:0 on bM, and ^ < 0 on ilf nea r b M. If we let d M (P) denotethe volume element and dS{P) denote the surface element along someb M s at P, then

dM (P) = r\x, y ) dx dy , dM{P) =dS (P) dr (P)'^ ' dS(P) = \Vr(x,y )\-^r(x,y )dS{x,y ),

(x, y ) corresponding to P in some coordinate patch; here | V r {x, y ) \denotes the gradient of ;' with respect to the coordinates (x, y ) anddS (x, y ) denotes the surface element of the surface of integration in the(x , y)-space. Let G be the domain of an analytic coordinate patch havingrange 3J such that "St f) b M is not empty and let g be the part of b Gwhich corresponds to ^ C) b M . Th en, if 99 or ^ v anishes on and nea r

b G — g, we note that

/ (p xpzP r{Xy y )dxdy = i (pxp'fzP' \V r[x, y ) |- i r{x, y) dS {x, y) —G g

( 8 . 3 . 3 ) -fy^[r-i^^rcpyrdxdyG

and the integral over g can be expressed in the form

(8.3.4) j (piprzPdS[P).

Next, suppose that (p^ %v<i, y j^ ^l^^-^-i, q >i, (p is given by (8.1.6)and y) is given by a similar formula. Then, using (8.3.1), (8.1.10), and theantisymmetry in the indices, we obtain

(7), dyj}=-\2J 2^g^^ghî . . . gJa'â X^' I,K U),{m)

(8.3.5) X I i-"^)^-^''-^ (piJi.,.ky'K,m'y Z ^v

I,K B,L a,j8

If we cover M with coordinate patches and use a partition of unity {Cs},each with support in one patch, let 99 = f§ 99, integrate by parts using(8.3.3), (8.3.4), and (8.3.5), and add up, we obtain

(8.3.6) {(pjy ^)=l Z Ig'''g''''ojiRy jKLdS{P) + (b(p,f)

where co = 1 99 is given b y (8.1.15) a nd 699 is given by (8.1.13) w here th eîRjS ^re determined so that

{b(p,yj) ^f^gîgRL{b(p)iRfKLdM{P)

( 8 3 7 ) ^

M

for a l l t / ;^5(2>.^-i .

N o w , le t (p and ^ ^ 9l2>Q'; we wis h to deve lop an i m p o r t a n t f o r m u l a for

(8.3.8) d[(p,xp) ={h(p.hxp)+{^d(p,dxp).

Clear ly , we may wri te cp and \p each as s u m s of fo rms each hav ing compac t

s u p p o r t and s u c h t h a t if the s u p p o r t s of (p^'^) and ' ( ) in te rsec t , the i r

un ion l ies in one coord ina te pa tch . So we as s ume 99 and ip h a v e s u p p o r t s in

one pa tch . Suppo s e cp and tp are given by (8.1.6), q <C v, and Q = d(p,

a — dip; t h e n Q and a are given by formulas like (8.3.1), so

{dq.Jip)=j ~ I I i 5;(-l)r+^^^^ X( 0 . 3 . 9 ) r i . . . r « + i = l

X gmm . . . gma+ir^+i cp^^^^-my ipK /sz'd dM = h + h

w h e r e 7i is the p a r t of the sum w h e r e d = y und I2 is the r e m a i n d e r . We

ob ta in^

(8.3.10) Ii = f2;g'''g^^g°'^(pij-,ocWKLz^dM.

Us ing the a n t i s y m m e t r y of the ind ices we see t h a t

w h e r e m^^ d e n o t e s the m s equence wi th my and ms b o t h o m i t t e d , etc.

T h u s , we o b t a i n

M

Using (8.3 .7), we thus ob ta in ( in te rchang ing (oc, y) and (/ , d) in (8.3.11))

d{fp,w)-h+flg'''g^''g''^g''\.-Î,.S'zyfK,6Tz0 +

(8.3.12) + ((piôcSzP + ZAYl^ (pu,ocv) {y^K,6T~zy +

+ ZÎTV Ww,dx)'] rdxdy = / i + /g .

N o w we define the fo rms '% and 'co ^ ^2?, ^-1 by

'X = I'Xis^^^ A dzs = (~i)Pv^,

(8.3.13) 'co = {—i)Pvy)=-2'cOs:Tdz^ A dz^

'%IS = g^'^^zP (pieces, 'CO^T = g^^rzyipKdT'

N e x t , we n o t e t h a t

'^(q>i,<xSzyWK,dTzy — 9i,o:s'iyWK,dTz^) =j^[(pi,^sWK,dTzy —

— 9l,o^S~zyWK,6T) + -^r:^[fpi,<xSz^WK,6T — (pi,ocS fK,dTzP)

(8.3.14) ^^

g^^r^p (pj,s-y = 'xiszy - {g^^^^z^y + gt^zP) cpi,.s

gy^r-yipjK^dTz^ = 'coKTzS — {g^^r^P-.Y + g^pr-v)ipK,6T'1 Strictly speaking, the integrand in Ii is not invariant under changes of coordi

n a t e s ; however, the final result in (8.3.15) is invariant.

8.3- Some important formulas 3 3 1

W e t h e n i n t e g r a t e Is by pa r t s un t i l the re a re no t e rms Uke (pu,<xvz^ a n d

Ww^dx'zV ^^ the rema in ing in teg ra l ove r M. The result is of the form

M

+ 'o)KT[e[h <p)is + BY^^°'(pu,ccv] — g^^ 'XIS'Z^WK,6T —

( 8 . 3 . 1 5 ) ( ^ = ( - 1 ) ^ + 1 ) .

In the case q = v, dq) = dy j — 0; a s pec ia l compu ta t ion l eads to theresult (8.3.15) in this case. In the case q ==:: 0, b(p = btp = 0 = 'x = '^

an d (8 .3 .15) ho lds wi th ou t a bou nd ar y in tegra l . In th e case ^ = r i t

fo l lows tha t vq) = 0 on bM^ the componen ts o f 99 = 0 on 6 M .

From the result (8 .3 .6) i t fol lows that (p^'$in%< ^vq) = 0 on bM.

But now i f (p an d ^ ^ 91 0 ®, the b ou nd ar y in teg ra l in (8 .3.15) , i s seen to

reduce to tha t o f the las t te rm s ince 'x a n d 'co v a n i s h on b M and hence

(8.3.16) 'Xiszy = îsr-^Y a n d 'COKTZ^ =p'KTrzP on bM

for suitable functions A/^ and JUKT- From our hypo thes i s o f p s eudo-

convex i ty o f b M, i t fol lows that

^/gyT^r^" > Cg^yT^ry, C > 0 , wh eneve r rgpr^ = 0 .

Cons equen t ly , ii ip = (p an d 99^ 91 Pi ®, th en

fgKIgSTr^p-^gcc^gyS^j^^^^j^.^dS

(8.3.17) ^^^ C f g^^ g^"^ g-' (pi,.s^KST dS = Cf\(p\^ds.

bM bM

Now, for forms cp ha vin g sup po rt in th e ran ge 5)1 of some co or din ate

p at ch , it is som eti m es des irab le to in tr od uc e a n ew bas is f , . . ., C*',

C^, . . , c,^ for the 1-forms, g iven by

^^^ctd^, dzy = dyc°'(8.3.18) -^ ' C'^^c^dzy, dzy^diC^"

and to in t roduce the corresponding d i f fe ren t ia l opera tors

U y = d%Uz0^y UzOi — C'Û^y,

(8.S.I9)Uy = d'^ûioi, uôc = clu-

where of course the matr ices {cfj a n d [d^) are inverses of one another as are

the matrices (c*) and (5j;) . The exterior mult iplication al lows us to express

any form in terms of the C^ and f^:

w = — L- 2 ' ^ n . . . i p h . . . k ^ ^ * i A . . . A dzh A dzh A . • • dzU^ p\q\ ii...ip

(8.3.20) . '""''

= ^Vq^Î^n...i,h.^.kdl[- • • diid^ . . . i f i A ... A C^^

? i A . . . A ? .In case the bases a re in t roduced so tha t

(8.3.21) g-nyc$ = Ay ^ a n d g,pd^^dl = Ay 3 , y ,d=h.. . ,v

we see tha t

((p, tp) = 2 "puWiJ^ if

I,J I,J

We call a bas is in which the cl satisfy (8.2.21) an orthogonal basis. Such

bases were used by K O H N ([1], [2]).

In terms of such a bas is , we see that

y = l

QIR = ( - 1 )^ " ^ ^ 2 ' 9^'/,«i2,a + 2 ; ^ F / > C ^ , « Fe x .

where the A's a n d B's are su i tab le C^ func t ions .

Such bases a re more usefu l in bo un da ry n e ighb orhoo ds 9^ ( in w hich

"^ n b M is no t e m pt y) . In case cr is a tangential analytic coordinate patch

w i t h d o m a i n G\J g an d ran ge 9^, we m ay , by ta ki ng a smaller 3^ if

necessary , choose an or thogona l bas is C such tha t

(8.3.24) 2ig^Prz^ = d^.

It is also possible, choosing ^ smal le r i f need be , to in t roduce non-

ana ly t i c bounda ry coo rd ina te s {t, r), t = (t^,.. ., ^2»'-i) of cl as s C^ whichrange over some GR U OR and wh ich a re s uch tha t the me t r i c t akes the

form

(8.3.25) ds^ = i:\At,r) dt'dtf + dr^, a, (0,0) = d,,.

Now, s ince the basis is complex-orthogonal, each ^,y == (^Dû — i" D'û)/!

a n d u^- = ['Dy u + i" Dy ^^)/2, w h e r e 'D y a n d "D y are rea l opera tors

wh ich a re , in fac t , d i rec t iona l der iva t iv es a long rea l un i t vec tors e^ a n d

e^ in which a l l 2v a re mu tua l ly o r thogona l and e^ |= iV r- jThus , in te rms

of the {t, r) coord ina te s

u,y = 2êy Ut^, y <v (Cy c o m p l e x ) ;A

(8.3.26) u,v = êlutx — ( 72) Ur [el real);

8.4. The HiLBERT space results 3 3 3

Fina l ly , by us ing the re la t ions impl ied by (8 .3-20) be tween the com

ponen ts wi th re s pec t to the {dz^, dz^") basis and the orthogonal (f*, f«)

bas i s , we find eas ily that

(8.3.27) VCp = 2^C07ijC^ AC^, COIR = (~nH-^l2) (Pl,vR

pr ov id ed t h e (C^, f' ) basi s satisfies (8.3.24) on th e b o u n d ar y ne igh bo r

hood 9^.

To see that (8.1.23) for all ip^^o imp l ie s tha t v d(p = 0 on b M in

case (p ^ 5Io, we note that if ip has s uppor t in a bounda ry coo rd ina te

p a t c h , € 0 — vdcp, an d (C, f) is an o rth og on al basi s satisfy ing (8.3.24), an d

we wr i t e

CO = 2^COIJC^ AC-^, ip = ZfuC^ AC-^*

t h e nJ <oj, ip)ds = j ^ coijipij ds.

bM b3I ^J

If (8.1.23) holds, it follows that all the cou = 0 unless y^ = v. But , f rom

(8.3.27) , we conc lude tha t

coij = (-\)^{-il2){^(p)i,,j

and so cou aut om at ic a l ly = 0 if y^ = v.

8.4 . The Hilbert space resu lts

In this section we prove the f irs t four principal results s ta ted in

§ 8.2 for for m s of ty p e {p, q) w ith ^ > 1. O nly a few results are pro ve d

for {p , 0) forms here , they are treated fully in § 8.6. We begin with prelimi

na ry re s u l t s inc lud ing thos e s ta ted the re .

L e t M be covered by the ranges ^s of ana ly t ic coord ina te pa tches T^

w i t h d o m a i n s Gg, where each Xs can be ex tended to a doma in FsZ^Gg.

F o r eac h s , suppo se f § = 1 on Gg, Cs € C"^ (Fs), Cs has s uppor t on Fg, a n d

satisfies 0 < ^^ < 1 on Fg. For forms of a definite type (p, q), we define

((cp, ip)) = {cp, f) + Z fCs I {9ijx-¥ij^- + (pijy-fiJv-) ^^ dy >

(8.4.1) ' ^ ''''

s jV Ija

w h e r e (p^/j a n d ip^/} are the components o f 99 a n d ip with respec t to tg . F o r

compos i te forms , we def ine these inner produc ts by summing over

(p,q). E v i d e n t l y t h e space/f| is the c losure of 5t under the s trong norm

defined by the inner product ((99, y))).Theorem 8.4.1 . Hl^ C^.Ifcp^ Hl^, then

(8 .4 .2) (((^ ,99))<C((99,(^))- .

7 / 9 9 ^ ® , then v<p^ H\Q and

(8.4.3) [{(p,<p))i<CD(<p.cp), ({v(p,v(p))<C(v<p,v(p)i<CD{q>,(p).

3 3 4 The d-NEUMANN prob lem on s t ron gly pseud o-con vex manifolds

Proof. The f i r s t s t a tmen t isobv ious . If (8 .4 .2) were not true , there

would ex is t as equence {q)n} ^(pn^ ^7i ) w i t h

.g . .. {((Pn, (Pn)) = 1 and (((pn, (pn)h - > ^

(pn-7(p in HIQ.

T h e n 99^ - > 99 in Sso 99 = 0. Also, for each s

(8.4.5) ICsI:0.

B u t , ifwe wr i te (p=(p i + i(p2 and assume 99^ C^ir), t h e n

f ^{(Pz^'l^d^dy = f C[((pixo^ — (p2y^Y + (9^2 a;* + Cpiycc)'^] dx dyr r

(8.4.6) = ji^{\(px^\^ + \^yct\^)dxdy +f [Cx<^{(pi (P2y^ — (p2 (piy^^) +r r

+ Cy<x{<Pixo^(p2 — (p2x°'(pi)]dxdy.

Applying the result (8 .4 .6) to (8 .4-5) and us ing the s trong convergence of

<pn t o 0in S, w e find t h a t ((99^, 99^)) - > 0 w hic h c on tr ad ic ts (8.4.4).

If 99^ 9lg^ w i th ^ > 1, V (p ^ HIQ s ince it 51 an d van ishe s on b M.

The result (8.4-3) for such 99 follows from (8.3.15) with y) =cp a n d co ' = %'

= 0on b Mand from (8.3.17), from which we obtain

d{(p,(p) >I((p,(p) + ciJ\(p\^dS, I{cp,(p)>{{(p,(p))-^—C[(p,(p), c i > 0 .bM

I (99, 99) being the integral over Min (8.3.15)- Moreover ifco == i 99, t h e n

((co, ft))) < C((co, a>))- < C((99, 9?))- <CD((p,(p),

T he re su lts for 99 in S) follow b y an eas y lim it pr oce ss. If = 0, then

D{(p, (p) a n d ((9?, 99))- yiel d eq ui va le nt no rm s.

Be fo re p rov ing the nex t theo rem, we conclude from (8.1.13) and

(8 .3 .1) and the definit ion of exterior mult iplication that

(8 4 7) ^^^ (p) =r]dq) + drj A(p, b (rj q)) =rjb(p — (o

coiR = {-'i)^g''^VzP(Pi,.Rs rj^W^.

From (8.4.7) and (8.1 .15) we see that if rj =f{r), t h e n

(8.4.8) (D==f(r)v(p.

Theorem 8.4.2. If cp ^%, then 99 ^H\{Ms) for each s < 0. / / each

<pn^ 5(o < < (pn -^(p, d(pn - ^ ^ , (^nd b(pn -^X> ^^^^ ip =d(p and ;^ = 699

where d(p and b(p are computed from (8.3.1) and (8.1.13), respectively, the

derivatives being the strong derivatives. Finally

{{(py (p))s < Ci s-^D((p, cp), So < s < 0.

Here (( ))§ denotes the strong inner product on M(s).

Proof. For each s, choose rj ^ C^ (M) w i t h ? = 1 on Ms a n d rj =f(r)

for s < r < 0. Then if 99, is a sequence as above , rj(pn^ HIQ for each n

8.4. The HiLBERT space results 3 3 5

and , f rom Theorem 8.4.1 , we see tha t

{(rj (fn, n (p n)) < C{{ri (fny fj (pn)y^ < CD (r] (fn, f] (fn)

Thus we s ee tha t rjcpn-^rjip in H\Q S O that the results follow s ince wem a y a s s u m e t h a t | / ' W | < 2 | s |~ i .

Theorem 8.4.3. (a) L is self-adjoint.

(b) § is closed.

(c) (p^&Q^(L) < ^ ^ ^ ^ ( L ) andL{(p) = 0<^(p^^.

Proof, (a) Since D(q), cp) > (99, cp), i t follows t h a t L + / is 1 — 1

an d (L + I)~^ is defined ev ery w her e an d has bo un d < 1. Sup pos e co

a n d 7}^i an d le t 99 = (L + / ) - i co an d ^ = (L + / ) ~ i rj. T h e n cp a n d

\p^^ and we see from the definit ions of L a n d D t h a t((L + / ) - i a>, .y) == [cp, (L + / ) r ) = ^ ( ^ , V) = {{L + I) % y>)

= {co,{L + I)-^rj).

T h u s (L + / ) ~ i i s se l frad jo in t so th a t L + I and L are a lso.

(b) Suppose cpn^^ for each n an d 99^ -^99 in £ . Since 399^ ^ 0 an d

599^ - ^ 0, it follow s t h a t 99^ S) a n d ^99 = 599 = 0.

(c) The firs t s ta tement follows from H I L B E R T space theory . The second

s ta tement fo l lows s ince 99 ^ ^ (L) with L 99 = 0 <^

^(99, \p) = 0 for each \p^%.

Theorem 8.4.4. Suppose 99 $ ^ ^ ^ and ^ > 1. Then

(8.4.9) j\cp\^dS <.C2D{(p,(p), s o < s < 0 , s o < 0 .hM{s)

Proof. I t is suffic ient to prove this for 99 ^ % . L et co = ^99. F r o m

Theorem 8 .4 . I , we conc lude tha t oy^H\Q a n d

(8.4.10) ((a),co)) <.CD{(p,(p).

Next, le t us define(8.4.11)

T h e n

(8.4.12)

y j = (p-4Q, Qjj^^^

Q - l

r = i

OJIS — 4-^"^^^2)8

== 0 (n ea r b M ]

--\)y ^r-iyO)i,/^.

^^a CO IS +

( - l ) ^ r ^ ^ 2 ^ < ^ / , a s ; , =^^^'^^2/5^20 99/ , ,^ ,^ = 0

on accoun t o f the an t i s ymmet ry o f 99 in the indices s a n d oc. T h u s i; = 0

n e a r b M . Hence for \s \ sufficiently small, (8.3.I5) and (8.3.17) with

(p = ip yie lds

/o / .^ ^s{f,w) =Is['(p>w) + / 2g^^g^^r,nyg^"^g''^Wi,ocsWK,dTdS(8.4.13) bMis)

> (^2((w,w)hs— C(y ),y ))s + c j\\p\^ds, c > 0 , C2 > 0 .6M(s)

From (8.4.10) and (8.4.11), one sees eas i ly that Q^ HIQ and

(8.4.14) ({Q^Q))^CD{cp,(p).

Since g = 0 on b M, we see t h a t

I \ Q{S, P)\^ dS (P) < 21 \Q{S, P)\^ d 2 {P )(8.4.15) bMis) bMis)

< C\s\' J\Qr{s,P)\^dM < C\s\'D((p,(p), Si < s < 0 , si < 0.M-M (s)

The result follows from (8.4.II), (8.4.13), (8.4.14), and (8.4.15).

Theorem 8.4.5. Suppose each q}^^%'PQ with q>\ and suppose

(fn —7 q> , ^(pn -7 W, ^'^(^ ^<Pn—7 % ^^ S- Then (fn -^(p in Q, (p^'^, y ) = dcp,

and ^ = hq). Also ^2?a has finite dimensionality . If (p^ S ^ ^ ^ Q ^ ^ * , then{(p,(p) < C^d[(p,(p).

Proof. Clearly 99 ^ ^ w i t h d(p = y ) and hep = Xy s ince the manifo ld

of triples (99, Sep, b (p) is closed in S ® S © S. To p r o v e the s t rong con

ve rgence , let {;'} be any s ubs equence of {n}. F r o m T h e o r e m s 8.4.1 ,

8.4.2, and 3-4.4, it fo l lows tha t there is a fur ther su bsequ ence {s} such

that 99s -> s ome 99', w h i c h m u s t be 99, on each Mt with ^ < 0. Now,

choose € > 0. From Theorem 8 .4-4 it fo l lows tha t there is a 2 < 0 and

s o s ma l l tha t

(8.4.16) / \(ps\^ dM < C2\t2\ 'D((ps, (ps) < (£ /3 )2 .M-Mit2)

F r o m the weak convergence , it follows that (8.4.16) holds also for 99.

Since 99 ->99 on M{t2), we conc lude tha t

(8.4.17) f\(p-qs\^dM<(el})\ s> s^.Mih)

I t follows t h a t 99s - > 99 on M. I t follows t h a t 99^ - > 99 on M.

F r o m t h i s , it follows eas ily that if W is any closed l inear manifold in

^TP^y t h e r e is a form cp^^'P^ which min imizes d[(p, 99) among all (p^%'P^

n W for which (9?, 99) =:: 1. We may le t W\ = S>P9 and 991 a minimizing

form in SKi, 9K2 be those forms in S^^ o r thogona l to 991 a n d 992 a m i n i m i z

ing form in ^ 2 , etc. Clearly 0 < d((pi, 991) < d{(p2, 992) < * * *. Now,

s uppos e all the ^(99^;, 99 ;) = 0. Since each (99 ;, 99 ;) = 1, we ma y e x t r a c t

a subsequence {99^} s u c h t h a t (pn —7 (p,^(p -^W {= 0), and 599^ ->;^ ( = 0).

B u t t h e n (pn -xp- But th i s is impossible s ince the 99^ form a n o r m a l

o r thogona l s e t . Hence §^ ^ has f in i te d imens iona l i ty and

d{(p,(p) >C((p,(p), C > 0 , if ^ ^ ^ 3 ? f f n ( S 2 ^ « © § ^ « )

from which the las t result follows.

W e can now comple te the proofs of our princ ip a l resu l ts (i )— (iv) for

fo rms in S ^ Q with $' > 1:

8.5. The local analysis 3 3 7

Theorem 8.4.6. (a) 7 / ^ > 1, 'SiiL'PQ) =^VQQ^VQ,

(b) If co^&^^Q ^^^ and $ ' > 1 , there exists a unique solution

(c) / / we define N^^co as this solution if o)^ S>^^Q ^^^ and N^^co

= 0 if w^^'P^, then N^^ is completely continuous [q>\).

Proof. By v ir tue of Theorem 8 .4-5 , i t fo l lows tha t there i s a un ique

9? in ®2?« n ( S ^ ^ © §2>«) wh ich m ini m ize s

d[cp, (p) — 2Re{co, cp)

among a l l s uch (p.* I f co^ S^^ © §^^ , then 99 satisfies

(8.4.18) d{(p,ip) = {oy ,tp), y )^%

s o t h a t (p^%{LPQ) a n d L(p = co . From (8 .4 . I8) and Theorem 8 .4 .5 , we

c o n c l u d e t h a t

il 9'!^ < Cz d(f, <p) = Csico. <p) < C3 1 a>l| • i| ^^H,s o t h a t

| | | | , d(<p.<p)^CzM\^

SO N i s bou nde d . F ina l ly , s uppos e con-yco i n S ^ ^ © ^^^^ Then, i f q)^

= Ncofi a n d q) = Nco, (pn—7 ^, d(pn—7 ^(p, a n d hcpn—^hcp, s o t h a t

99^ - > 99 by Theorem 8 .4-5 . The comple te cont inu i ty of N^^ follows.

8 . 5 . Th e loca l an a lys i s

In this section, we consider an equation of the form

(8.5.1) =IJ^ii>(t,r) + Zf,4i'r) +Mt,r)+g{t.r)

( = ^ 1 , . . . , t^^-i, (p^^ = dipldt^", etc.)

in which the coefficients are real and of class C^ for all (t, r) w i t h

— jR < r < 0 an d a re per iod ic of per iod 2 i^ in each t°^ and we a s s umetha t th is i s t rue of each f'^^, /* , / , an d g. Actua l ly , we a s s ume tha t the

a°'^ depend on R (as do the other functions , and satis fy

\S/P e'^^(t,r)\ <K pR^~^, ^ = 0, 1, 2, . . .

^^•^'^^ £«^(0 ,0) -= 0, 8°^^(t, r) = 3°^^ - a^^[t, r) .

Notations. In the rema inde r o f th i s chap te r GR deno te s the hype rcube

l^ 'l <R, oc = \, . . .,2v — 1, an d Gij den ote s the celU ^o-ij , —R <r < 0 .

W e s ha l l be in te re s ted in func t ions /^^ , /* , / , g, a n d u wh ich ^ C°° {GR)

and van i s h nea r r = —R an d /« = ± i^ an d we wish to ob tai n bo un dsfor ce r ta in norms which we now def ine : We def ine

(111^111, )2 = ( 2 i ? ) 2 - l / 2 ' [ ( l + | m | 2 ) ^ | ^ ^ M | 2 +

(8.5.3)

+ (1 + \m\^Y-^R^\u^(r)\^] drMorrey, Multiple Integrals 22

8.5. The local analysis 339

Proof. E x p a n d i n g a and u in Fourie r se r ies , we o b t a i n

(1 + \m\^Y'^[au)m[y) = 2an{r)vn,m(r),

( 8 . 5 7 )

Vn,m(r) = (1 + \m\^rÛm-n{r)a n d t h i n k of the an{r) as sca la rs and the Vn,m (and others below) as

c o m p o n e n t s of a vec tor func t ion %. We now w r i t e

Vn,m(r) =Vn,m(r) +W n,m(r), F^,w = (l + | ^ - ^ |^)'^^ ^m-n(r)

Wn,m(r) = [(1 + \m\^Y^^ - (1 + \m-n^Y^^]um-n{r).

(8.5.8)

Us ing the inequaUt ie s

1 5 — b^\ < |5| '{a^-l + 6^-1) \a — b\, a>0, 6 > 0,

2-1(1 + | w | 2 ) - i ( i J^\m — n\^)î+\m\^<2{\ + \n\^)(\ + \m—n\^),

we f ind th a t

(8.5.9) \Wn,m(r)\ < C 2 ( ; ^ , s ) - [ 1 + | m - n | 2 ] ( ^ - i ) / 2 | ^ ^ _ ^ ( ; . ) | ^

I C2 (n , s)\ < Ci (n , s).

The result follows easily from (8.5.7), (8.5.8), (8.5.9), the definit ion of the

p r i m e n o r m , and the fact that ' | | |F , j | | | s i j = '||| u J\SR for each n.

Theorem 8 .5 .3 . Suppose that u, v,f, and g ^ C^ for —R-<r<,0,all are periodic with period 2R in the f^, u and v vanish for r = —R and

0, and

Au =f, Av—gr.

Then there are absolute constants C3 and C4, such that

III 111 si? < CsR^'\\\f\\\s-2,R, \\\v\lsR < C4i^-'|||^|||si?.

Proof. The u^, satis fy the cond i t ions

(8.5.10) < ( r ) - 7r2 \m\^R-^Ûm (r) =fm{r), Um (-R) = Um(0) = 0

Mul t ip ly ing bo th s ide s of (8.5.10) by —R^(2R)^''-^(i + \m\^Y-^ Uni(r)

a n d i n t e g r a t i n g by pa r t s y ie ld s0

(2i^)2»'-i 1 2 ^ (1 + I m |2)«-i [ 7 1^ I w |2 I um \^ + R^ \ < j ^ ] dr

(8.5.11)= - i ^ 2 ( 2 i ^ ) 2 v - i / ' 2 ^ ( ^ +\m^^)s-if,riir)un,(r)dr,

-R m,

Since Um{—R) = Um(0) = 0, one eas i ly conc ludes tha t0

(2i^)2f-i 12^(1 + 1^12)5-11^^(^)12^;--R "

0

<ZiR2{2R)^- if 2:(i + | m | 2 ) s - i K ( r ) | 2 ^ y ,

(8.5.12) -^

22*

3 4 0 Th e d-NEUMANN problem , on s t ron gly pseu do-c onv ex man ifolds

Combining (8.5.11) and (8.5-12) yields

(8.5.13) 111 ^ | | | f ^ < Z i 7^2(22^)2 / 2 ^ ( 1 + \m\^Y~^fm{r)urr,{r)dr

from which the first result follows using the Schwarz inequality. In thecase of V, we obtain the result (8.5.13) with/^^ replaced by ^^. The second result then follows by integrating by parts.

Theorem 8.5.4. Suppose that u and the functions Sjf, f°'^, /*, / , andg^ C^ for —R^r< ,0, are periodic of period 2R in each fy, and vanishalong r = — R. Suppose also that the e%^ satisfy (8.5.2) and that u satisfies(8.5.1). Then there is a number RQ, depending only on v and the Kp anda constant C, depending only on v such that

- 1 / 2 , R +M U E<c{s'\\\f^^\\U E + R2'\\\f\

0<R<Ro.

Proof. Let H be the unique harmonic function of Theorem 8.5.I for

which H(t, 0) = u(t, 0) and let U = u — H. Then U satisfies all the

conditions with different f'^P, etc., and also vanishes along r = 0. We

may write the equation (8,5.1) for C7 in the form

AU = 8%^ U^^p + e\^H,o.^ +f% +f% +fr + g

= (/«^ + e%^U + 8l^H),o.p + ( / - 2e%%U- lel^^^H),, + fr +

+ g + et.p U + 8%%^ H = F% + F^,+Fr + G.

Clearly

(8514) U^V% + V:^ + V + W.^y oc^^pocp^ ^y a^pu AV = Fr, AW = G

and V°'P, F", V, and W all vanish along r = —R and r = 0.Now, from Theorem 8.5.2 we conclude that

(8.5.15) l l le f t^ i | | i /2< ^S ^ ' l! | f / | t | i /2+ C-^K s) ' | | |C7 | | |_ i / ,

where A^^ and C°'^{a, s) are the constants of Theorem 8.5.2 for 8%^ = a""^

and where we have left out the subscript R on the norms. But, from(8.5.2), (8.5.4), and differentiation, we see that

2\m\^^\8%l{r)\^<ZlKlR^

m(8.5.16) ,p = 0,i,2,...

I\m\^p\'e'A{r)\^<ZlKl.m

Thus, from (8.5.15) and (8.5.16), we see that

(8.5.17) 'l<fiV\li^^n<A''l>-R -'\lVly ^_ j,, V = UovH.

8.6. The smo othness resul ts 3 4 1

where ^«^ depends on ly on the K^. From (8.5.14), (8 .5 .17) and Theorem

8 .5 .3 and Lemma 8.5.I, i t fol lows that

111 U 111,/,,;,, < C^ \Z ['||i/«^ III,/, + i?^-^C|| | U III,/, + 'I ^ III,/,)] +

(8.5.18) +«2'['ill/11l-i/2 + ^''{'!llf^lil-i/2 +'111 ^1-1/2)] +a

+ R '\y\Ui^ + R^ ['III g 111-3/2 + R-^ A ('ill [/ i||_3/, + '\\\H \U„)]]

where C i = C i ( r ) and the ^ « ^ , A"^, an d ^ dep end only on th e Kp. T h e

resu lt follows from (8.5.18), T he or em 8.5.1 , and the fac t tha t u = U + H.

A s imila r ana lys is can be carr ied th rough for func t ions u^ C'îCR)

w h e r e CR is the full cube

CRilt^^l <R, a =- 1, . . ., 2v.

We are in te res ted in the so lu t ions of the equa t ions

( 8 . 5 . 1 ) ' z a ' ^ n t ) «,a? = zf% + if,^ + / .where we have rep laced rhyt^^ and the coeffic ients and functions ^ C^

and are periodic of period 2i? in each P" a n d t h e a^"^ satisfy (8.5.2). We

def ine the s -norms by

u\ m

u(t) = Ûm e^'''^'^'^, (m = mi,.. ., .m2v).m

Theorem 8 .5 .5 . Lemma 8.5.I holds if (x = [CKI, . . ., a^v). Theorem 8.5.2

carries over without change. In Theorem 8.5.3 ^^ ^^^<^ retain only the case

A u =f. In Theorem 8.5.4 we need consider only the cases where u satisfies

Au = e^û,ufi+f%+f% + g,

w h e r e oc, ^ == i, . . ., 2v, in which case the result is

llkHI|i/2<c[2'1!l/°'^|||i/2 + i?2^1ll/«||U2 + i^2^lk o<R<Ro.[a,/5 a j

8 .6 . Th e sm oo th n ess resu l ts

Until further notice , we shall be concerned with forms of a given

t y p e (p, q) in w hich ^ > 1. W e beg in b y defining the s-norm s for forms

(p^WP^. E ac h inter ior p oin t P of M is in a neig hb orh ood 3J:p w hich is

the image of a cube CR with re s pec t to a ho lomorph ic coo rd ina te pa tch

T. Each boundary po in t P of M is in a ne ighborhood "^p on M VJ h Mwhich is the image of GR D OR u n d e r a n o n - a n a l y t i c [t, r) b o u n d a r y

coo rd in a te sy s tem of the t yp e descr ibed a t t he en d of § 8 .3 . Le t us

choose a covering It of M U b M by means of such ne ighborhoods

9^1, • • ., ^T and choose a par t i t ion of un i ty {rjt}, t = 1, . . ., P, of class

C^, each having support in the corresponding 9^^ In each 5ft^ , le t us

3 4 2 The d-NEUMANN pro blem on s t ro ng ly pseu do-c onv ex man ifolds

choose an or th og on al b asis (C, f) as in § 8.3 w hic h satisfies (8.3.24) if ^t

i s a bounda ry ne ighborhood . W e then de f ine

(8-6.1) lll<?'!llf = 2 ' 2 ' l h * ^ i * i l l l f

t IJw he re th e 99J*} ar e th e co m po n en ts of 99 w ith re spec t to th e bas is (C, f) in

9^^. Of course the norm depends on the 9^^, the coordinate patches rt,

and the ba s e s {Ct, Ct) but any two such norms , for a f ixed s, are topologi-

cally equivalent . We choose one f ixed set of Sl^ f]t, a n d {Cty Ct)- T h e n w e

have the following fact:

Lemma 8.6 .1 . \\\(p\\\s ^'s non-decreasing in s and if rj ^W^'^, \\\'yj(p\\\$

W e beg in by p rov ing the fundamen ta l inequa l i ty :

Theorem 8.6.1. If (p ^ 9lg^ {and q > 1), then

(8.6.2) \M\m^Cî^>9^)>

where C depends on M and the metric and the choices of 91^, rjt, and {Ct, Ct)-

Proof. I t is c lear, from Theorem 8.4.1 . t h a t , for a gi ve n 99 in 5Ig^

there is a un ique (po^ HIQ which min imizes D{q) — cp,q) — (p) among a l l

q) in H\Q. Since this jus t lea ds to th e Dir ichlet p rob lem , 990 an d hen ce

H = (p — (po^ ^0- Clearly, also, 990 satisfies

(8.6.3) D((po-(pyip)= -D{H,y))=0, y j^Hl^

s o th a t we h ave

(8.6.4) [L + I)HÔ, D(<p ,(p)=D ((po,(po)-\-D{H,H).

F r o m T h e o r e m 8.4.1, the def in i t ions , and Lemma 8 .6 .1 we conc lude tha t

(8.6.5 ) lll o |!!?/2 < il ô li f < Z i • ((^0, ^0)) < Z2D((po, (po) < Z2D{(p, <p) .

Thus we need only prove (8.6 .2) for H^SHio and satis fying (8.6 .4).

From the form of the in tegra l in (8 .3 . I5) i t fo l lows tha t the compo

n e n t s Hij of H in any ho lomorph ic coo rd ina te s ys tem s a t i s fy equa t ions

of the form

[%.6.6) g^P Hjj-x^p + (lower ord er te rm s in a l l th e HST) = 0.

T h e c o m p o n e n t s Hjj with respec t to an or thogona l bas is , be ing l inear

combina t ions of the Hjj (wi th C^ variable coeffic ients) , sa t is fy s imilar

equ a t ion s . I f w e m ul t ip ly by 4 , wr i te g°'^ = g^^ + i g^^, and exp re s s

the der iva t ives in te rms of those wi th respec t to x^" a n d y°' , t h e e q u a t i o n s

(5.6.6) t ake the fo rm

{S.6.7) gf UU ,p + gf Uioc i3 + ^ f ^ ; a ,/? + gf Ui,ccyfi+ •••=0

where we have deno ted the componen ts by uK

N o w w i t h e a c h b o u n d a r y p o i n t P we choose a {t,r) b o u n d a r y c o

ord ina te p a t ch Tp f rom GA U CA ô M [J b M so tha t the or ig in corre

sponds to P and the metr ic has the form

(8.6.8) ds^ = ao,^{t,r) dif^ dt^ + (dr)^, a«^(0,0) = ^«^ .

8 .6 . The smoo thness resu l t s 3 4 3

We also choose a (C, C ) orthogonal bas is sa t is fying (8.3 .24). The equa

t ions (S.6 .7) then t ake the fo rm

It i s c lea r tha t the mappings may be chosen so tha t A a n d t h e b o u n d s

for the coeffic ients and al l their derivatives are independent of P

Hav ing chos en xp , we define

< (/, r)^b-^-\- hn [ t , r) [a-^ { t , r) - d-^], { t , r)^Gn,

(8 .6 .1 0 ) 2 ^ -1

h R { t , r)=h[2R-^ r] [J h[IR-^ tr],

w h e r e h^ C ^ ( i ^ i ) , h{t) = 1 for |^| < 5/4, h[t) = 0 for \t \ > 7/4,0 < ^(/ ) < 1 for all t\ then we ex tend a^ to be periodic of period 22^ in

each P, We define J^^jj, e tc . , s imi la r ly . Then the a"^ satisfy (8.5.2) with

K^ in de pe nd en t of P . T he n we can choose an i^i , 0 < jR i < c i?o (Theo rem

8.5.4) an d a f ini te cove ring of th e s tr ip ~ P i < y < 0 b y th e pa rt s of the

^P co r re s pond ing to Gpi^. In l ike manne r , we may cove r M_p^ b y t h e

pa r t s correspo nding to C7j/2 of ho lo m orph ic coo rd ina te pa tch es w i th

0 < P < P o so th e af/ satisfy (8.5.2).

Now, s uppos e H is harmonic and in 9lo and le t i] have s uppor t in one

of these smal l ne ighborhoods , say a boundary one . Then i f W are the

componen ts o f K, w ith re spec t to (f , f) and W ^ = r j W, t h e u^ satisfy (since

the ^«^ - al^ in GR,^)

/ ? = (V icR + 2(5;jt al^ rjj) HK fi = {v h'lcR + 2djjc V r ) H^

gj = bl CjjcR + djjc(rjrr + a'lfrj^^^) — in ^IkR + 2^3Jc ^lfv»p) ,« —

— (V^JfcR + 2dj]cr}r)r\H^'

From th is and Theorem 8 .4-4 wi th c p = H, we conc lude tha t

i : ' \ m n - ^ i : ' m u + ' \ \ \ s n \ \ \ i + 2 f \u ^ ^ d t < c D { H , H ) ,where C depends on rj (poss ib ly P) , and the K > p a n d t h e n o r m s m a y b e

t a k e n o n 50 1 or on GR as des ired. Thus , from Theorem 8.5.4 , i t fol lows that

lr,H\\l,,^<.CD{H,H).

A s imila r resu l t ho lds for the inner ne ighborhoods . I f the rit form a

par t i t ion of un i ty , the resu l t fo l lows .

W e have a l ready s een tha t [L + / ) - i is a bou nded ope ra to r de fined

eve rywhere on ^'P^. We sha l l p rove the fo l lowing theorem concern ing

the s o lu t ions c p of

(8.6.11) {L + I)(p = co, (p^'^m, or (p = (L + I)-^ co.

W e let §« de no te th e c losure of 51 w ith resp ect to th e s-norm .

3 4 4 T h e d-NEUMANN problem o n strongly pseudo-convex manifolds

Theorem 8.6.2. / / CO^SP^, then c p £ ^^^^ and

(8 .6 .12 ) | | | 9^ | | | i /2<C | | co | | .

I f o ) ^ '^^-^l^for an integer s > 1, then cp^ §«+i/2 ^^^

(^•6.13) |||9^|||5+l/2 < C(s)'|| |o>|||5_l/2.

Ifoy^^thencp^S^Q.

Proof. W e shall show firs t th a t if c o^ 5t, th en (p^%Q and a l l the in

equali t ies (8 .6 .12) and (8.6 .13) hold. The firs t s ta tements then follow by

ap pro xim at io ns . To do th is , we assume co^ 51^^ an d for each e > 0 , we

le t 99g be th e func t ion min im iz ing

(8.6.14) D,{cp,cp)-2Re{m,cp), cp^S^o, D,((p,ip) = e(((p,y ^)) + D((p,y j),

Th is m in im um exis ts and is un iqu e an d ^ 5lo as i s seen f rom t he the oryof § 6.4 (as lo ng a s £ > 0). It satisfi es

(8.6.15) DM>w)-ico,w)> V ^ € 3 lo .

We sha l l show by induc t ion tha t i f ( p ^ satisfies (8.6.15), then

(8.6.16) e III 9P, lllf+i + III < p , If+y^ ^ C(s) ' \ \ \ c o \\\l_„„ s = 0 , 1 , . . .

We note f irs t that (8 .6 .16) holds for s = 0, s ince

*• III % illf + III 9 s lllf/2 ^CD^(9?„ 9?,) = C ( m , < p , )

^^•^•'^^ ^ C ' W C O | | | _ x / 2 • 111 9 - . I I I 1 / 2 < \ III C P e l l l f / 2 + C 'III « , r-V.

(us ing Theorem 8.6.1) from which the result follows. From the defini

t ions in (8 .6 .1) and from Lemma 8.5-1, we f ind t h a t

e III /r . Ill 2 _J_ III /y, III 2 V V fj:" lll-y) m(»*M I|2 _1 _ ||| oo Ar,(w )'||l2 "] I III ^ IIU+i i ^ III 9 llls+1/2 — Z. Z i l^ III Vn (Pij llls+1 ^ III Vn (Pij |||s+i/2J ^

n IJ

(8.6.18) + 2 2 SC.[s\\ \Dtf,n<F'^J\\ \l + l\Dtf]n<p?J\il^(Rlnr^''^n IJ \(x\<^s

w h e r e n indexes the coo rd ina te pa tches and the no rms on the r igh t in(8.6.18) are those in GR as defined in § 8 .5- O ur m et ho d of show ing thi s

will be to use the f irs t inequali ty in (8 .6 .17) for the tangentia l derivatives

D°'i^n(P£) and then use results l ike (8 .6 .15) for these derivatives .

We begin by cons ider ing what happens in one of the coord ina te

pa tches wh ich we take a s a bounda ry pa tch , the t r ea tmen t fo r in te r io r

pa tch es be ing s imila r an d s impler . W e f i rs t rep lace th e subscr ip ts / / ,

e t c . , by a s ingle superscript . Then, if 99 or ^ has support in the range of

a coo rd ina te s ys tem wi th doma in G {= GR),

(8.6.19) ^

whe re we have pu t r = t^^ a n d D^ i s he rmi t i an s ymmet r ic , i . e .

(8.6.20) A^,t = Aff, Cu = Cii.

8 .6 . The smoothness resu l t s 3 4 5

Next we prove by induc t ion tha t ( i f cp^ an d ^ = 0 ou ts ide our pa tc h)

(8.6.21) D,[D' f,, xp) = (-\r\DA< Ps, D'f) + 2 " ^ ^ ( 0 " <p,. <p),

where a l l the D^^ a re g iven by he rmi t i an s ymmet r ic in teg ra l s l ike D^,

w h e r e D^ a n d Df^ a re t angen t ia l de r iva t ive s , and whe re

(8.6.22) fji <X m ea ns 0 <^ â < / la , oc = \, . . . ,2v — i a n d | / / | < | / i | .

(8 .6 .21) is ev ide nt for |A| = 1. To com ple te the indu ctio n, we ass um e

(8.6.21) and use th a t for |^ | — 1 to ob ta i n

(8.6.23) D,{D^^' 99, ^0 = - D,{D' 99, D^ xp) + DU[D' cp,yj) cp = cp,.

Applying (8.6 .21) to the f irs t term on the r ight in (8 .6 .23), we obtain

D,{D^+^(p,y)) == (-1)1^1+1 A ( ^ > ^ ' ' ' ' ^ ) -(8.6.24) _ ^D',^{D^cp,D^xp) + DU{D'cp, xp).

The resu l t fo r A + ^ follows b y appl yin g (8.6 .21) for 1^ | = 1 ba ck w ar ds

to the te rms in the middle sum on the r igh t in (8 .6 .24) .

Now, i t is easy to see that we may choose a real function Y\ of class

C^ and vanish ing outs ide our pa tch (see (8 .6 .10)) such tha t

( 8 . 6 . 2 5 ) 1 V ^ ( ^ ) | 2 < C 7 ^ ( ^ ) , t^G.

W e n o w a s s u m e t h a t 99 (E^ (p^ satisfies (8.6.15) and then set(8.6.26) xp ^Yi^D'^cp, A g iven

in (8 .6 .21 ). B y us ing the he rm i t i an s y m m et r y , the reade r m ay ve r ify th a t

ReD^{D ^(p,ri^D^(p) = D^{rjD^ cp^rj D^ cp) - j Alli^,o^Yi,p0^0Ut,G

(8 .6 .27 ) ( -1 ) ' ^ ' D,{(p, D^ xp) = ( - 1)l^l(co, D^ xp) = (D^ (o , >y2 D^ cp),

0i = D^(p\ (p = (p^^

usin g (8.6 .15). In o rde r to ha nd le th e var iou s inte gra ls JD^^, we le t /denote any hermit ian symmetr ic in tegra l wi th coeff ic ien ts a*f , b'^p Cfj.

We not ice tha t , fo r any vec tors Q a n d a,

I(u,r]v) = I (rj UyV) + ^ [{dj u, v^) — (u^, dj v)]

(8.6.28) ''

dju = a'tfrj^û]^ + îYj,oûK

W e a l s o no te tha t

(8.6.29) djT] u = T] djU + a'lf rj,ocr]j ^^*.

No w, suppose ^ i s a m ul t i - i nde x w i th |^ | = 1 . Th en , by apply ing(8.6.28) twice and (8.6.29) once, we see that

(8.6.30) I(xp,rj^xp,Q) =I{rj^xp,xpe) + 2 ^ [(dj xp , rj x/P^^) - {y)^\ 1 ^ dj xp ^ g)].j

I n t e g r a t i n g / {y f xp, xp^ g) by pa r t s and t rans pos ing one in teg ra l , we ob ta in

(8.6.31) 2ReI(xp,rj^xp,g) = Re{-2l{rj rj,gxp,xp) — leiv^V,^) + *}

3 4 6 The d-NEUMANN prob lem on s t ron gly pseud o-con vex manifolds

w her e * den ote s t he sum in (8 .6 .30) an d IQ deno te s the in teg ra l w i th

coefficients a^^l^, etc . Applying (8,6 .28), (8 .6 .29), e tc . , we obtain

2ReI [ i p , r \ ^ ^ , g) = — / ^ ( j ] y},r]y)) + L Q ( i p , ip ) +

(8.6.32) + 2Re(-I(rj^,y j,r]7 p) + Z [{^3V,Qy ^>W') - iV^Q^^^Jf)] +

?•

w h e r e LQ is an integral like that in (8.6.27) with coefficients a^l^.

We now complete our proof of (8 .6 .16) by induction on s: W e s uppos e

it is tr u e for all s < ^ an d co nsid er t h e case s = ^ + 1 • Suppos e | X \

= k+\. Fr om (8.6.17 ), (8 .6 .21) w i t h ^ = ^ 2 / ) A ^ / a n d fr om ( 8.6 .2 7) ,

we conc lude tha t

G

Using (8.6 .25), (8 .6 .32) and the facts that

(8.6.33) |(/ ,^) |< Ill/Ilk 'llkllk ., ' | | |v ^ |l | ,< c j^ -i ll l^ | |U ,for all T,

w h e r e Vu i s the vec tor func t ion (u^a), o c — \, . . ., 2v, t^^ = r, we find

tha t ( tak ing a l l norms on Wl an d a l lowing th e C ^ to depen d on R)

1 jAl^ri,o.Yi,^D^(piD'pdt\ < C | | | Z ) ^ ^ ^ | | | _ i / 2 - | | | ^ D ^ 9 ? l | | i / 2

(8 .6.34 ) '<^ '| ( 7 y Z ) ^ c o , 7 y Z ) » | < C | | | i ) ^ a > | | | „ i / 2 | h Z ) ^ ^ | | | i / 2 .

R e g a r d i n g t h e in t e g ra l s D^^ w ith | / / | < |A | — 2 as sum s of a pp rop r ia te

inne r p roduc t s one can ea s i ly ob ta in the bound

i i ? . z )^ - ^ (Z ) ''( ? > „ , j 2 D ^ - V . III1/2 +(«-6-35) + | i |Z) / '9 , J | | i ,2 - | l | ,yZ)^9 , , | | |_ i /2 ] , \X-^\>2.

O ne m ay p roceed in m uch th e s ame w ay fo r thos e in teg ra l s wi th | A — //1

= 1 ; how ever , one s ta r ts wi th (8 .6.32) . W e obt a in , for e xam ple

\I(r]eDf'(p,7]Df'(p)\ < C'\\\\7 fje D f " (p\\\_i,2' '\\\V r j D^ (p\\\ii2 + '"

< C WlrjgDf' (f\\\ii2' \\\rjD^ (p\\\ii2 H (q) = %)

\(D^(p^ ,rjdjD'(p)\^C\\\D^(p \\\il2'\\\rjD'q>\\\il2 + "-

where the do ts denote te rms of lower order . The o ther te rms can be

hand led s imi la r ly . By us ing the common dev ice tha t

\ 2 a b \ <ia^ +i-^b^, ^>0, a,h real , | small

and by s umming ove r a l l X w ith |A| = ^ + 1 we find t h a t

Z Q[eIIIriD'<pll+ 11 nD'<p\\U < : c 2 "Q ||hz)^o)r_i/, +U|=fc+1 |A|=fc+l

+ CZ\\D''vrm-\fi\^k

8.6. The smoothness results 3 4 7

Then by s umming ove r the pa tches and add ing in the lower o rde r t e rms

acc ord ing to (8.6.1) and L em m a 8.5.1 we ob ta in (8.6.16) for s = ^ + 1.

Th is comple te s the induc t ion .

By repeated use of the differentia l equations (8 .6 .9) we find in each

p a t c h t h a t all th e der iv a t ives of th e q){ sa t is fy s imi la r inequa l i t ies and

(8.6.12) and (8.6.13) hold for (p^, independen t ly o f e. T h u s a s £ ^ 0 i t

follows eas ily that q)e->(p toge the r wi th a l l i t s de r iva t ive s . Thus (p^ô

if CO ^ 51 a n d (8.6.12) an d (8.6.13) ho ld .

W e can now p rove ou r de s i red s moo thnes s theo rem:

Theorem 8.6.3. If (p ^ "^{LPQ) and 5 > 1, then cp, d(p, and b«+^) and

(8.6.36) LV'Q+^d(p = dLPQ(p, LP^Q-^hq) = bL^^cp.

^ / ^ € d ^^ and ^ > 1, thencp^ % .

Proof. We f i rs t p rove the smoothness resu l ts . Le t L^^ cp = L (p

= co^ ^ . Th en ^ ^ S2^« an d

{L + I)(p = CO + ( p .

H enc e 9? an d therefo re co + (p^ ^^^^ by Th eore m 8 .6 .2 . U s ing ind uc t io n

and Theorem 8 .6 .2 repea ted ly we see tha t 99^ § +1 2 for every s . Using

th e differentia l eq ua tio ns , we find t h a t a l l th e de riv ativ es of 99^ §^+1^^so (p^%o (Sobolev lemma, Chapte r 3 ) . Tha t dcp^ô follows from the

na tu ra l bounda ry cond i t ion (8 .1 .23 ) . Tha t bq)^ô follows from (8.1.19).

Since Lip = dbip + bîp for ip a n d 5 ^ ^ %Q, (8.6.36) follows easily from

the fac t s tha t dîp = 0 and b b ^ — 0 for ^ ^ 51.

N o w , s u pp o se c o $ S ^ ^ G § ^ « a n d L(p ==^w. L e t a)n€ 51^^ 0 ( S ^ ^ ©

§ ^ ^ ) , (pn^^{L)y I^(pn = (JOn, and s uppos e con^co in S^^ . For each n,

d{dq)n) = 0 , b(b(pn) = 0, bdcpn + db(pn = (On,

{bd(pn,df) = {db(pn,hx) = 0, yj^Sav^Q-^ x^'ô''^^

T h u s dq)n a n d bq)n sa t is fy the equa t ions

{dd(pn,dip) + {bd(pn-Wn,bip)=^0, '^^ îg'^+l

(a b9 9^ -c o^ ,5 ; ^ ) + (bb99^ ,b ;^ ) = 0 , x^'^l'"'^ •

I t fo l lows tha t

d{d(pn, d(pn) + d(b<pn, bcpn) =- {(^n, 0)n)

f rom which the remain ing s ta tements fo l low (by tak ing cp ^ = N con so

that 99^->99).

We can now comple te the proofs of our pr inc ipa l resu l ts in the cases

w her e ^ = 0.

Theorem 8.6.4. (a) m (Lv o ) = g p o Q | , p o .

(b) / / co^ S ^ O 0 i ^ ^ there is a unique (p^% {L^^) Pl [S^^©S>^^] such

that L^o (p=.o}.

(c) / / we define N^^ c o as this solution when C J O ^ S > ^ ^ Q $ ^ ^ andN^^ co

— 0 when co^ ^2?o^ ^/^^^ jsivQ is completely continuous. Moreover

| | i V ^ « l l < C 3 ,

C3 being the constant in Theorem 8.4-5-

(d) / / CO ^ %v^, then m^ w ^ ^v^.

Proof. Suppos e co^^^^. Then Sco ^ ^P^ an d if ^ 6 § ^ ^

(5 0),y)) = ^ J <ft), V y ))dS = 0bM

so ac o£ £2 ^1 0 ©^^. So le t 0 = A^^co. T he n ( 9 0 $ ® (L^^s) Q ( S ^ 2 0 |^ 2 > 2 )

(as above) and L^^ d't-' = ddco == 0 so d0 = 0 and hence

(8.6.37) d(b0-co) = O, 5 0 - 0 ) ^ §2^0.

O n th e o the r han d if co$ S ^^ © §2?o and ip^ ^P^, t h e n

(8.6.38) ( 5 0 - CO, i p ) s = ( 0 , dip)s - (co, y j ) s - f <y 0,ip} dS,bM s

By averag ing th is over a smal l in te rva l (t , 0) and us ing the fac t tha t thefunc t ion v0^C^(M) and vanishes on bM, we f ind tha t the r igh t s ide

of (8.6.38) -> 0 as ^ -^ 0~ . Thus we mus t have

(8.6.39) h0 = o), 3 0 = 0.

Since 0 {= N dco) ^ S^^iQ § P I , we m ay def ine

(8.6.40) Q = N0, (p = bQ.

Since 5 0 = 0 , we aga in ha ve dQ = 0, so tha t

As in (8 .6 .38), we see a lso t h a t ^ ^ S ^ ^ O Q ^ J J O .

I f there were another so lu t ion in % [ L ' P ^ ) , the difference (p \ w o u l d

be in ^[L^^) and we wou ld have

(pi^'^'P^, d{cpi,ip) = {dcpi, dip) = 0, f^ ^2^0

so that 991^ ^P^, s ince we could take ip = c p i . F r o m (8.6.39) an d (8.6.4O )

we see tha t N'P^ is bo u nd ed . Fi na lly , if co^ - ^ co, it w ou ld follow t h a t

h0n—yM, d0n = 0, a n d 0n -^ 0, at leas t in a subsequence, s ince, byTheorem 8 .4-5 , D{0n, 0%) is un i fo rmly boun ded . Thu s , by th a t theo rem

0y^ - > 0 a n d so ( p n -> ( f y since

[ ( p , ( p ) = (bQ, bQ) = d{Q, Q) < C3 \\LQf = Cs(0, 0) < C | d{0,0)

= Cl\\oyf.

http://lqf/

http://lqf/

http://lqf/

9.1 . In t ro d u c t i o n . Pa ramet r i c i n t eg ra l s 349

Chapter 9

Introduction to parametric Integrals;

two dimensional problems

9 . 1 . In trod u ct ion . P arametr ic in tegra l s

An integral (1 .1 .1) issaid tobe inparametric form, and we say f is

the integrand of aparametric problem, if and on ly if the v a l u e of the

in teg ra l is u n c h a n g e d by an a rb i t ra ry d i f feomorph is m with positive

Jacohian from G o n t o a n o t h e r d o m a i n G'; if we a s s u m e t h a t / {%, z, p)

i s def ined everywhere , we see t h a t we m u s t h a v e

j f[x\ z'{x'), \/z'(x')] dx'= j f[x, z{x), Vz{x)] dx,

z(x) = z'[x'(x)], z'p(x) = ^ ^ ^ . - ^ - ^

for any G', and v e c t o r z'^C^(G'), and any pos i t ive d i f feomorphism

x' =x'(x) from G o n t o G\ By t a k i n g XQ, ZQ, 'p,and XQ as a r b i t r a r y c o n

s t a n t s and defining

'^^ = 4 + ' ^ i ( V - ' ^ S ) , 'x-^'xl + a1(xf^^xi),

^^•^•^^ d e t l | a | l > 0

a n d t a k i n g G =B (XQ, Q) and then l e t t ing ^ -> 0, we o b t a i n(9.1.3) det lla^ll 'f[x^, zo,p[,.. .,pl] - / [ ^ o , ô, aîPa,.. ., ;]

for all se ts ofc o n s t a n t s as i n d i c a t e d . T h u s , / m u s t be i n d e p e n d e n t of x

a n d we m u s t h a v e iV > r.

In case N = v + 1, we now s h o w t h a t / m u s t h a v e the form

f(z,p)==F(z,Di,.,.,Drî),

(9.1.4) A =

' ^l'"Pi-^PX''-"Pi

w h e r e F is pos i t ive ly homogeneous ofthe first degree in the Di. T h i s is

easily seen as follows: (9.1.3) implies that if the pm a t r i x is mul t ip l i ed

on the left byany r X r m a t r i x w i t h p o s i t i v e d e t e r m i n a n t , t h e n / is

mul t ip l i ed by t h a t d e t e r m i n a n t . T h u s , ifZ)r+i > 0, we may m u l t i p l y the

p m a t r i x on the left by the inverse ofthe m a t r i x ofDvî. The resu l t is

t h a t the new ^ - m a t r i x has the i d e n t i t y for its first v c o l u m n s and the

e l e m e n t s in the las t co lumn are j u s t ^^DijDvî. T h u s , for s ome F,

(9.1.5) f{z,p) D. ,-F(z, , ^ \ ) , Z), i>0.

If D+i < 0, we begin by changing the sign of 1 row and obtain

(9.1.5) /( . ) = - ^ ^ . i ^ ( ^ . - ^ . - - - . - ^ . - l ) ' ^^.KO

3 5 0 Pa ra m et r i c i n t eg ra l s ; t wo d i men s i o n a l p ro b l ems

where the two F's jo in up cont inuous ly a long Dv+i = 0. I t is no t nec

essar i ly (and usua l ly no t) t rue tha t

(9.1.6) F{z, -Di, . . ., - D . + i ) = -F(z, Di, . . . , Z),+i).

I t i s c lea r f rom th is d iscuss ion tha t an in tegrand of a parametr ic p roblemis ordinari ly not of c lass C^ on the set of p fo r wh ich th e ^ - m a t r i x ha s

ra nk < i^ and , m oreov e r , a t po in t s w h e r e / i s of c la s s C^, th e q ua d ra t i c

form in (1.10.7) a n d (1.10.8) necessar i ly has vO e igenva lues and the

b iquadra t ic form in (1 .5 .5) degenera tes .

W e h av e a l ready seen th a t th e a rea in teg ra l (1 .1 .4) i s in p ara m etr ic

fo rm. Us ing ou r cu r ren t no ta t ion

(9.1.7) f(z,p) = yni + Dl + D| .

U nt i l H . A. Schw arz show ed how to inscr ibe a po ly hed ron of a rb i t ra r i lylarge area in a port ion of a cylindrical surface, i t was believed that the

area of a surface could be defined as the sup. of the areas of inscribed

poly hed ra , b y a na log y w i th t he def in it ion of th e leng th of a curv e . T his

upse t t ing d iscovery was the mot iva t ion for a lengthy s tudy of surfaces

and the i r a reas . The pr inc ipa l surv iv ing def in i t ions a re those of the

Hausdorff two-d imens iona l measure of a po in t se t in R^^ N > -_ } , a n d

of the Le besg ue a rea of a Frec he t surface. The H ausdo rff m easu re /12

is gen era t ed in th e usua l sense of m easu re theo ry by the ou te r m easu re/12 * defined by

(9.1.8) yl2* (5) = lim Al * ( 5 ) , A^ * (S) = ini 2J 7t rfQ-*-o n^Q

for a l l coverings of 5 by a countable number of balls B[xi, ri). To define

the Lebesgue a rea , we mus t def ine a Freche t surface .

The idea is to invent a notion of surface which is independent of i ts

parametr ic represen ta t ion . In case we have a po in t se t 5 which is the

homeomorphic image of a c losed c i rcu la r d isc , we would na tura l ly ca l l S

a surface a nd a ny ho m eom orp hism from a c losed Jo rd an reg ion onto 5

a parametr ic represen ta t ion . But there a re t imes when i t i s des i rab le to

consider more general surfaces . We therefore proceed formally as follows:

Definition 9.1.1. Let ^1 and Z 2 be con t inuous vec to r func t ions wi th

d if feomorphic i com pac t dom ains Gi and G2 which mus t be f in i te un ions

of d is jo in t manifo lds , wi th or wi thout boundary , o f c lass C^ a t leas t .

We def ine the distance

(9.1.9) D(zi,Z2) = inf ( m a x U i (:^i) - Z2[r{xi)]\\

1 His torical ly , homeomorphic domains and homeomorphisms were al lowedinstead of diffeomorphic domains and diffeomorphisms. In the one and two d imens ional cases , the res t r ict ion which we are making resul ts only in the res t r ict ion todomains of class C^; in higher dimensions , Milnor has shown that two manifolds ofc lass Cl ma y be hom eom orphic w i thout be ing d i ffeomorphic. O ur res t r i c t ion resu l t sin a great s impl i f icat ion of the theory of integrals over Frechet variet ies .

9.1. Introduction. Parametric integrals 3 5 1

for a l l d iffeomorp hism s T from Gi o n t o G2. If D (zi, Z2) = 0, we say tha t

zi is equivalent t o z^ and wr i t e zi ^ ^2-

Theorem 9.1 .1 . (a) The distance D so defined is non-negative, sy mm e

tric and satisfies the triangle inequality .

(b) The equivalence relation defined above is sy mm etric and transitive.

Definition 9.1.2. We define a Frechet variety S as a class of con

t inuous vec tor func t ions which cons is ts o f a l l such which a re equiva len t

to some given one. Any vector function in S is called a parametric

representation of 5 . Th e ran ge 9 i (5) is th e com m on ran ge of the pa ra

m e t r ic rep re s en ta t io ns of 5 . The topological type ofS is th e c lass of d om ain s

of the vectors in 5 and the dimension of S i s the common d imens ion of

t h e s e d o m a i n s .

Definit ion 9.1 .3 . If 5i and S2 a re F reche t va r ie t i e s o f the s ame topo logical type, we define the distance

D {81 ,82) =D(zi,Z 2)

w h e r e Z k i s a pa rame t r ic rep re s en ta t ion o f SA:, ^ = 1, 2.

Theorem 9.1.2. (a) With this definition of distance, the Frechet varieties

of a given topological ty pe form a metric space.

(b ) If 8 and 8n are all of the same topological type, z is a representation

of 8 with domain G, and D {8, 8n) -> 0 , then H ^ representation Z n of 8n,with domain G, such that Z n-^z uniformly on G.

(c) If z and Z n are representations of 8 and 8n, respectively , all having

domain G, and if Zn —> z uniformly on G, then D {8, 8n) -> 0.

Proof, (a) and (c) are evident. To prove (b), le t Z n be a rep re s en ta

tion of 8n h a v i n g d o m a i n Gn. From the definit ion of dis tance i t fol lows

tha t there ex is ts , fo r each n, a diffeomorphism Z n from G o n t o Gn such

t h a t

\Zn(^) —Z(x)\ <.D (8,8n) + \ln, Zn{x) =Z n[rn{x)], X^G.

The result follows.

N o w LEBESGUE [1] was familiar with the following fact: Suppose

Z n a n d z are each p iecewise l inear homeomorphisms on a po lyhedra l

d o m a i n G and s uppos e Z n converges un iformly to z (no th ing be ing sa id

about der iva t ives ) on G. T h e n

(9.1.10) ^ (^) < lim inf ^ (^^ ).

A (z) a n d A (%) being given by the integral (1 .1 .4) and being the e lementa ry a reas of the corresponding polyhedra . In the next sec t ion , we sha l l

p rove a genera l lower-semicont inu i ty theorem from which we can con

clude that (9.1.10) holds if G is of class C^, z i s L ipsch i tz , a nd each

Z n^ C^{^)'> in fact z m a y m e r e l y ^ HI (G) in the two d imens iona l case .

Assuming these resu l ts , the theory of Lebesgue a rea proceeds as fo l lows :

3 5 2 Parametric integrals; tw o dimensional problems

Definition 9.1.4. Given G of class C^ a n d z ^ C^ (G), we define

(9.1.11) L(z,G) = inf lim inf A (zn, G)

for a l l sequences {z^^ in which each z^ ^ C^ (G) a n d Z n -^z un i fo rmly

on G.

The fo l lowing theo rem fo l lows immed ia te ly :

Theorem 9.1.3. We suppose that v = 2.

(a) L(z,G) is given by ( I . I .4 ) ifz^ H\[G) H C^[G).

(b) L {z, G) < lim inf L {zn, G) if Z n-^z uniformly on G.

(c) Given z^C^ [G), there exists a sequence {0^} in which each Zn ^ C^ (G),

Z n-^z uniformly on G, and L(zn, G) ->L{z, G).

(d) L(zi) = L(Z2) if D(zi, Z2) = 0.

Proof, (b) and (c) are evident, (a) follows since the integral is lower-

semicont inuous accord ing to Theorem 9-2 .1 so tha t L(z, G) < the in te

g r a l A (z, G). O n t h e o t h e r h a n d if Zn->z uniformly on G and s t rong ly

in H\{G), t h e n A {zn, G) ^ A (z, G). To prove (d), le t Zn -^Z2 u n i f o r m l y

on G2 s o t h a t A{Zn, G2) ->L{z2, G2) a n d Z ^ ^ C^(G2). There ex is ts a

s equence {xn} of diffeomorphisms from Gi o n t o G2 s u c h t h a t Z2 [tn (xi)]

-^zi{xi) uniformly on Gi. Thus , i f Cniî) = ^nltniî)], t h e n Cn->î

un i fo rmly on Gi and each Cn € ^^ (^1) so that

L (zi, Gi) < lim inf A (Cn» Gi) = l im A {Zn, G2) = L ( 2, ^^2) •

T h e a r g u m e n t i s s y m m e t r i c .

Remark. The foregoing def in i t ion and theorem can be carr ied over to

th e case wh ere G is a man ifo ld by us ing coord ina te pa t che s and pa r t i t io ns

of un i ty .

Definit ion 9.1 .5 . We define the Lebesgue area L(S) of the surface 5

as the common va lue of L {z, G) for i t s represen ta t ions z.Theorem 9.1.4. (a) Parts {b) and (c) of Theorem 9.1-3 ôld with {z, G)

and (zn, G). replaced respectively by S and Sn-

(b) / / 5 possesses the representation z^H\{G) C\ C^[G), then L(S)

= A (z, G).

Now, there are some anomalies . If we define z = z (x'^) for {x^, x^)

on G = 5 ( 0 ,1 ) , the n i t i s obv ious th a t L{z, G) = 0 even if the range of

z f il ls a cu be . Bu t one mi gh t hop e th a t if 2: is a ho m eom orp hism w ith

range 5 , then L{z, G) would be equa l to the Hausdorff measure A^{S).However , th is i s no t the case . In fac t , BESICOVITCH has g iven an example

of such a z and S for which L {z, G) is f ini te and the three dimensional

measure of 5 > O! O n the o the r ha nd f rom an o ld theo re m o f the au t ho r

[3], i t fo l lows tha t th is S possess a ' 'genera l ized conformal" represen ta

t ion z = 2:*(A;*) on 5 (0 ,1 ) = G in w hich 2:* is co ntin uo us an d in H\{G)



9.1. In t ro d u c t i o n . Pa ramet r i c i n t eg ra l s 353

a n d , m o r e o v e r ,

£ * = G*, F* = 0 a.e. L (z, G) = L (^*, G) = f^* "^ ^* dx*

( 9 . 1 . 1 2 ) G

£ * = | ^ * l * | 2 , F^ - = 4 l * - 4 2 * , G* = 1^*2* |2.

W e n o t i c e t h a t the a r e a i n t e g r a n d

(9.1.13) ]/£* G* - (F*)2 ^E*+G* ^^^^^^^^ ^ * _ ^*^ F * - 0

s ince

(9.1.14) ^ C - ^ 2 + | ^ j l ^ \ ^ 4 _ 5 2 _ / ^ 1 ± _ £ \ ^ ( ^ _ £ * ^ B=F*

C =G^).

H o w e v e r , the r e a d e r can eas i ly p rove tha t if 2: is a dif feomorphism of Go n t o S, t h e n L {z, G) = A^ (S),

In deahng wi th more gene ra l in teg ra l s

(9.1.15) If{z,G)= jf{z,p)dxG

in which / satisfies (9.1.3). it is des i rab le to i n t r o d u c e the no t ion of

oriented Frechet variety. T h i s is done by defining the d is tance D (zi, Z 2)

in Definit ion 9.1.1 allowing only positive (i.e. with pos i t ive J acob ian )

d if feomorphisms r and requ i r ing the d o m a i n s Gi and G2 to be or ien ted .Then Theorem 9 .1 .1 ho lds and we may define an or ien ted F reche t va r ie ty

a s in Defin i t ion 9.1.2 us ing the ' ' o r i en ted d i s tance" above . We may t h e n

define Z)(Si, S2) for orien ted var ie t ies as in Defin i t ion 9.1-3 and t h e n

n o t e t h a t T h e o r e m 9-1.2 holds .

F o r the special class of in tegrand func t ions / (2 ' , ;/>) of (9.1.15) which

is descr ibed in the next sec t ion , we can e x t e n d the no t ion of Lebes gue

a r e a to t h a t of an in tegra l over a F r e c h e t v a r i e t y .

Definit ion 9.1 .4 ' . Su ppo se / satisfies (9.1.3) and the cond i t ions

decs r ibed in § 9.2 below. Given G of class C^ and z^ C^{G) we define

S / (^ , G) = inf {lim inf If{zn, G)}

for all sequences {zn} as in Definit ion 9.1-4.

The following theorem follows as before us ing the t h e o r e m of the

nex t s ec t ion :

Theorem 9 .1 .3 ' . If f satisfies the conditions above (with v ^ 2) then

Theorem 9.1.3 holds with L replaced hy ^f and A [z, G) by If{z, G), e x c e p t

that we must require z^ C^ [G) if v '> 2 in part (a).M a n y p e o p l e h a v e m a d e i m p o r t a n t c o n t r i b u t i o n s to the t h e o r y o u t

l ined above . We m e n t i o n o n l y E. J. MCSHANE ([1], [2], [3], [4]), T. R A D O

([3], [4]), L. C E S A R I ([1], [2],' [3]) and the aut hor ([1], [2]). L . C .Y O U N G

([2], [3]. [4], [5]) i n t r o d u c e d the idea of a ' ' genera l ized surface" ana logous

t o MCSHANE'S idea of a genera l ized curve ; th is idea was deve loped at


3 5 4 Parametric integrals; two dimensional problems

some length by Y O U N G , W . H. F L E M I N G and others ([1], [2], [3]). The

idea is sugges ted by the fo l lowing: Suppose z^ Hl(G). T h e n the i n t e g r a l

(9.1.16) IF[z(x),Di{x),. . .,Drî{x)]dx,

G

w h e r e F is h o m o g e n e o u s of degree 1 in t h e Di which are defined in (9.1.4),

is a linear functional of F. Any l inear func t iona l of F, w h i c h is > 0

w h e n e v e r F is , is called a generalized surface. U n f o r t u n a t e l y , an a r b i t r a r y

generalized surface does not arise from a vec tor func t ion z by m e a n s of

a formula (9.1.16) even if fa i r ly genera l vec tors z and ' ' genera l ized

J a c o b i a n s " are a l lowed . The ' ' i n teg ra l cu r ren t s ' ' i n t roduced by F E D E R E R

a n d FLEMING seem l ike a m ore prom is ing too l in t h e s t u d y of the ex is tence

and d i f fe ren t iab i l i ty proper t ies of s o lu t ions of v a r i a t i o n a l p r o b l e m s in

volv ing such in tegra ls .

9 . 2 . A l o w e r s e m i - c o n t i n u i t y t h e o r e m

In § 4.4 , we s aw th a t i f / s a t i s f i e s (9.1 .3 ) and (9.1-4) with N = v + 1,

t h e n / i s w e a k l y q u a s i - c o n v e x i n p if and on ly if F [z, X) is c o n v e x in the

X]c (we sha l l use Xi i n s t e a d of Di from now on) which , in tu rn , imp l ie s

t h a t / is q u a s i - c o n v e x i n j ^ . In tha t c a s e , if F^C^ for all [z, X) w i t h

X ^ 0, t h e n / s ati sf ie s the cond i t ions of Theorem 4 .4-5 and hence the

in tegral (9 .1 .15) is lower - s emicon t inuous if z^-^z uniformly on G and

weakly in Hl{G). This resu l t , however , is no t sufficient for the t h e o r e m s

of the preced ing s ec t ion . So we s ha l l have to p r o v e a special lower-

s e m i c o n t i n u i t y t h e o r e m for s uch in teg ra l s . We sha l l use the m e t h o d of

MCSHANE [3]. T h i s method can be e x t e n d e d to certain integrals (9-1.15)

in which iV > r - j- 1 as well as to all s uch in teg ra l s if TV = r + 1.

Lemma 9.2 .1 . Suppose f{z, p) ^ C^ for all (z , p) for which the p-matrix

has rank v and suppose f satisfies (9.1.3)- Then if (ZQ, po) is such that the

rank of po is v, there are unique constants ai(I = ii, . . ., ip with ii < 2< • • • < 4) such that

f (20. zo) =Z<nH: XI = detI

1 i)h . . h^v 1

A-Af{zo.P)-:ZîX'] = 0 for all ( -.a)

âidzi = AdI:^ ... f" {dI zi = dzh /\ .

(9.2.1)

for some linearly independent linear functions C'îz), . . ., C^(z).

Proof. W e f i r s t no te tha t if w e m a k e a r o t a t i o n of axes in the ^-space

a n d m a k e the co r re s pond ing change in t h e ^ - m a t r i x :

zi = c] 'zK PI = cYpi, f(z, p) = f {z\ p')

9.2. A lower semi-cont inu i ty theorem 3 5 5

t h e n the X^ t r a n s f o r m in the co r re s pond ing way,

(9.2.2) XI = Z^jX'^, Oj = d e t c j ^

J

a n d the condit ions (9 .2 .1) are pre s e rved in the new va r iab le s . We maythe re fo re a s s ume tha t

(9-2.3) po = \

\pi,...pi,o...olF o r p n e a r po, we use (9-1-3) to c o n c l u d e t h a t

/i...o^r'---^f\f(zo.p)=Xy.--'f[z^,p], p^{ •_• ••••_

(9.2.4) \o. . .opi+K..p^/

X^ = Xi---''Xi, : / > l = ( Z i . . - ) - i d e t ' • • ' ^ ^

_ \pi---pr'Pipr'-th e XI being formed f rom the p m a t r i x .

From (9 .2 .4) , we conc lude tha t

f{zo,Po)-Xl"-^f{zo,Po)

- P I J

(9.2.5) fpi(zo,po) =

f{ô,po) dX^'-^

fp i {ô> Po) dpi

\ <:i <v

+ 1 < ^ < iV

where po is, of course , the matr ix ob ta ined f rom p by s e t t i n g all the Pi,

w i t h ^ > r, = 0. If we define

(9.2.6) fi(zo.p)==^ZaiXi=:^X^--'-2Jî^'>I I

we ob ta in , u s ing the formulas (9 .2 .5) for / i , tha t

f l (ZO,po) = ^ l . . . r ^ o " • ^ fliÔ^Po) = ai, . .v

/dX^'"^'

fipi{ô,po)

(9.2.7)

^ 1 . dpi

2 ; ( - i ) ^ - ^ ^ i . . . < 5 _ i , 5 + i , . . . , Hdx^-

dpÎ > V.

So, from (9.2.5), (9.2.7) and the first two l ines of (9.2.1), we conc lude

t h a t we m u s t h a v e

( 9 . 2 . 8 ) ^ i . . . v = / ( ^ o , ^ o ) , ^ i . . . . 5 - i , 6 + i , . . . , H = ( - 1 ) " ~ V 2 ? j U , ^ o ) .

N o w , let us i n t r o d u c e l i n e a rl y in d e p e n d e n t C* by

(9.2.9)7 = V+ 1

23*

Then, if we define cj = (5j- for 1 < / < v, th e c -m atr ix has th e sam e form

as the p m at r ix . The th i r d hne of (9-2,1) , th en requires th a t

(9.2.10) âjdzi =A2J d e t

^ ^ l • • • ^ r1 'v

Equating coefficients in (9-2.10) for I = (\, . . .,v) or (1, . . . , ^ — 1,

(5 + \, . . .,vi) yie lds

(9.2.11) ai,,,r=Â =f{zo,Po ), c\ = [f{zo,por^fvi{zo> Po)

s o tha t the rema in ing aj a re un ique ly de te rmined . The C* are , of course ,

de te rmined on ly up to a l inea r t r ans fo rma t ion among thems e lves wh ich

affec ts on ly th e fac tor ^ .

In order to ca ry over MCSHANE'S proof, we impose the following

r e q u i r e m e n t s o n / :

General assumptions on / . We assume that

(i) / is continuous for all {z, p) and ^ C^ for all [z, p) for which the

rank of the p-matrix is v;

(ii) f satisfies {9A.}) for all {z,p), etc.,

(iii) f(z, p)>m \X\, m>0,\X\ = Z ( ^ ^ ) ^

1(iv) if the rank of pQ is v and the ai are the constants of Lemm a 9-2.1for [ZQ, po), then

/ ( ô , p)^2:îX'^ h (^0, P) = f{zo, po) +Iaj {XI - XI).I I

Remark 1. li N = v -j- \, we have seen tha t these condi t ions (except

for ( i i i )) are necessary and suffic ient for the parametric integrand to be

quas i -convex in p. F or iV > r + 1, i t is no t kno w n th a t th e con dit io ns

above a re necessary for quas i -convexi ty .

Remark 2. W e no te th a t i f / m e r e l y sa tis fies (9-1-3) the n(9.2.12) f[z{x),p{x)] dx = (plz(x)Mv(x)] ' dA [z)

for an y C^ m app ing z = z{x), Uv be ing the o r ien ted t angen t r -p lane .

This fact makes i t eas ier to define I{z, G) w he re G is a C^ ^-manifo ld.

W e mus t now in t roduce the no t ion o f the order of a point in v-space

with respect to an oriented [v — \)-manifold. We sha l l be concerned pr in

c ipa l ly w i th such manifo lds of the topologica l ty pe oidG for some domain

G in Rv of c lass C^. H ow eve r, th e gener alizat ion to o the r orie nt able

manifo lds wi l l be ev ident . We a l low dG to consis t of several dis jointmanifo lds , each or ien ted as a par t o f ^ G in the u s ua l way . liv = 2 a n d

dG is jus t a Jordan curve , then th is o rder i s jus t the winding number of

the po in t wi th respec t to the curve .

Definition 9.2 .1 . Suppos e G is of cla ss C^ an d ^ ^ C i(G ), 99 = (99I,

. . ., qf) and le t r '. z"" = (p°'(x), oc = 1, . . . , r. Su pp os e z fr{dG). T h e n

9.2. A lower semi-continuity theorem 357

we define the orde r 0 [z , r {d G)] by

(9.2.13) 0[z,r(dG)]= f ^ Ko,o.[<p(x) - z]'{- \r^^dx^^

6G

w h e r e KQ is the e lemen ta ry func t ion for LAPLACE'S e q u a t i o n , as defined

in (2.4.2), and

(0 2 id) ^ y g ^ a(y i , . . . , y«- i / r+ i , . . . ,<y , r )

Lemma 9.2 .2: Suppose r : z°^ = (p'^(x) is of class C^ on the open set Q in

Rv. Let W he the set of x for which dcpjdx = 0. Then

\r(W)\ =0.

Proof. S u p p o s e XQ $ W. T h e n we m a y w r i t eT : " - ^g = [ c | + e^(x, xo)] (x^ - x^),

^0 = (p'îô)> c^ = (p°!^(ô), \e(x,xo)\<e ii \x — xo\<,d.

T h u s

(9 2 15) ^''-4 = <Û^^ - 4 ) + ^ ^ ( ^ , ^ o ) , \r{x,xo)\ ê'.\x-xo\,

ii \x — xo \ ^d.

T h e i m a g e of B (XQ, r) unde r the Hnea r pa r t of t h e m a p p i n g is an elHpsoid

s in s ome p lane of lower d imens ion . From (9 .2 .15) , it fo l lows tha t thet o t a l i m a g e of JB (:ô, ^) C (s, er) ii r < (5. Since the de r iva t ive of the 99*

a re un i fo rma ly bounded on any compac t s ubs e t of Q, the result follows

easily .

L e m m a 9.2.3. (a) 0[z,r{dG)] depends only on the values of the qf on

d G and the values of their directional derivatives along d G. Moreover, it is

independent of the parametric representation of dG. It is as slimed in this

lemma that q)^ C^ (G) and that G is of class C .

(b) 0 [z, T{dG)] is continuous in z if z i T ( 5 G ) .

(c) If(p(x) ^ z for any x^ G(i.e. G), then

(9 .2 .16) 0[^ ,T(aG)] = 0 .

(d) / / the closed regions Gi, . . ., GR < ^ disjoint and C G, all deing ofK K

class C^, and if F = G — U G^ and z ir(dG) U r(dGjc), then

(9.2.17) 0[z,r{dG)] =0[z,r{dr)]+ 2;.0[z,r(dGjc)].

(e) If z ir{dG), then 0 [z, T(dG)] is an integer and

(9.2.18) N [z, r(G)] > 10 [ , r(dG)] \ (a lmos t everywhere) ,

iV [z, T (G)] being the number of x in G such that (p (x) = z.

(f) / / 99 {x, t) ^ C^ (G) in X and is continuous on G x [0, 1] and if

{z i xt (dG) for any t on [0,1], then 0 [z, Tt(dG)] is constant with respect to t,

• xt : z = (p{x, t).




(g) The functions N[z, r(G)] and 0 [z, r{dG)'] ^ Li{Rv) and

(9.2.19) J0[z,x{dG)]dx= f^dx, fN[z,r(G)]dz=f\^\dx.

Rv G Rv G

Proof, (a), (b), and (d) are e v i d e n t . To prove (c) , we n o t e t h a t

G a

(9.2.20)

since the f irs t term on the r igh t van i s hes by Lemma 4 .4 -6 .

T o p r o v e (e), we begin by l e t t i n g W he th e set of A; in G for w h i c h

dcpjdx = 0. Since the (p'^ can be e x t e n d e d , it follows from Lemma 9.2.2

tha t | T ( P F ) | ^ | T ( ^ G ) | = 0 and t(W) and r{dG) are compact . Now if

z^r{G) - [T(W) U r(dG)] t h e n N[z,r(R)] is finite {R = G(0) - W)

a n d d(pldx 9^ 0 at each x for which (p {x) = z. In fact , if ^ is a s ma l l doma in

of class Ci, its counte r image cons is ts oiK = N[z,r{R)'\, s ma l l dom a in Q

Gi,, . .,GK and the res t r ic t ion of 99 to Gjc is a dif feomorphism of Gjc o n t o

Q for each k. If ^ is in te r io r to Q, all of i ts coun te r images are in the Gjc,

on e in each , so t h a t if F is the d o m a i n in (d) , then

( 9 . 2 . 2 1 ) 0 [ ^ , T ( a r ) = 0, 0[z,r(dG)] =2;o[z,r{dGjc)].k= l

T o e v a l u a t e one of th e t e r m s on the r igh t in (9.2.21) we beg in by

rep lac ing KQ in (9.2.20) (G = Gjc) by Koa w h e r e

(9.2.22) Koa {y ) = fKo (y - rj) tp^ [rj) drj, ^* (rj) = g-' ip [q-^ 7 ]),

Bi6,a)

w h e r e ^ is a moUifier . Thus , for o" > 0 but smal l ,

JKOO,. [cp (x)-z](-\ r ^ 1 ^ dx'p = f rpt [(p [X ) ~z\^£dx

QG H GJC

= ^g^ ( S ) / ^ * ( - ) ^c = s^^ff •Q

The re s u l t (e) follows. Moreover, we e v i d e n t l y h a v e

K

fo[z.r{dG)]dz^^ f^dx

K

JN[z,r{G)]dz^ j\^^\dx.

(9 .2 .2?) ^

fc = l Tic

the l eng ths l[Tn{dR)] a n d l[r[dR)] a re un i fo rm ly bo und ed so th a t

|T^(c? i ? ) | = |T(a7? ) | = 0 . Th en , d^g^in 0\z,rn{dR)] ^0[z,r(dR)] for

a lmos t a l l z a n d

(9.2.26') j I 0 [z, xn (dR)] i dz < (j-^Y" • I [xn [dR)].e

Thus the resu l t fo l lows .

R em ark . Th e on ly th in g th a t p re ven t s u s f rom a l lowing q? ^Hl {G) for

a n y v is th a t w e d o no t kno w th a t |T(^i^) | = 0 even if 99^ Hl{dR) a n d

is th e s t ron g l imi t in HI {dR) of cpn in C^. Th is is on acco un t of th e ex am ple

of B E S I C O V I T C H .

W e can now p rove MCSHANE'S p r i n c i p a l l e m m a :

L e m m a 9.2.5. Suppose (p^C\ (G), R is a cell for which (9.2.24) holds,<Pn € C^ (D) fo^ some D with R(Z D C C. G, and (pn converges uniformly to

(p on R. Then there are measurable subsets VnC R such that

hm f^dx=f^dx.

liv = 2, the same resu l t ho lds licp^ C^{G) H Hl{G).

Proof. L e t £ n = m a x \q)n(^) — (fi^) \ for x g dR. For each n, le t Wn

be the subset of R w h e r e dcpnidx = 0. Cover almost all of the set of z

w h e r e 0[z,r(dR)] ^ 0, z irn(Wn), an d 2: is at a dist an ce > En from

T (dR) by smal l d is jo in t domains Qni, for each of which, r^^ (Qni) = U Gntkk

a n d Tn \ G nijc is Si diffeomorphism from GniJc o n t o Qni- T h e n

(9 .2 .28) 2 ' ( ~^dx = ^ (0[z,r[dR)]dzi,k ^ -' ^ ^ i -

GriiJc Qn i

s ince 0[z,rn{dR)] = 0[z,r(dR)] for 2:^ U Q!^^. Since th e r ig ht s ide of

(9.2.28) tends to r 0 [z, r(dR)] dz, w e m a y t a k e Vn = U Gnik to ob ta inthe resu l t .

Theorem 9.2 .1 . Suppose f satisfies the general cond itions, G is of class

C i, z = {z'^, . . ., z^)^ ^ i (^ )> ^n^ ^-^{Q* ^ ^ ^ Z n-^z uniformly on G.

Then

(9.2.29) // (z , G) < lim inf // [zn , G) .•J^-••c>o

If V = 2, the result holds if z merely ^ C^{G) 0 H\[G).

Proof. We sha l l p rove the f i rs t s ta tement ; the proof of the second iss im ilar . If th e r ight s ide of (9 .2 .29) is + 00, th e resu lt follows. O therw ise ,

by t ak ing a s ubs equence if necess a ry , we m ay a s s ume th a t If{zn, G)

< w M s o th a t

(9.2.30) A(zn,G) = j\Xn[x)\dx<.M , # = 1 , 2 , . . .

9-2 . A lower sem i-cont inu i ty theorem 3 6 1

Fr om the rep re s en ta t ion (9 .2 .12 ), the d i f fe ren t iab ih ty cond i t ion o n / , an d

the compactness of the to taHty of n .o . se ts in RN, i t fol lows that there

ex i s t s a number L and a func t ion co(d), d e p e n d i n g o n l y o n / a n d A, s uch

t h a t

\a\ = \2:ajW^<L , \f(zi,p)-f(z2,p)\^oj(d)\X\,

(9.2.31) _ lL_^^k^B{0,A)> p i — ^ 2 | < ( 3 , lim(o(d) = 0

w h e n e v e r aj i s the s e t of con s tan t s of Le m m a 9 -2 .1 f o r / a t s ome po in t

(-2 0,i o) w h e re z^s^Bi^, A) and the rank of ^o is v, A be ing a common

bound for a l l the Zn {pc).

Next we not ice tha t there a re se ts Za of measure 0 such tha t i f no

face of the cell R l ies a long a plane x'^ == c°'^Zx, then (9.2 .24) holdsfor every set Z ^, / = ^'i < • • • < iv . For a lmos t a l l XQ

Urn \R\-^jf[z[x),p{x)] dx =f[z(xo),p{xo)]

(9.2.32) .lim\R\-'^l\X(x) -X{xo)\dx = 0,

R^XQ ^

Le t £ > 0 . Th en we m ay cov er a lm os t a l l of G by a coun tab le n um be r of

cells Rj, avo id ing the Za , hav ing m ax im um edge < twice min im um

edge , s o s ma l l tha t

(9.2.33) \z(x) - z(xj) I <co(d) < £ / 4 M , x, xj^Rj

a n d s o t h a t

I f \f[^{^)>P{^)] -f[zM>P(xj)]\dx<el4

(9.2.34)

2 ' f \X{x)-X{xj)\dx<el4L.

We may choose a f in i te number of these Rj s o t h a t

(9.2.34) Z ff[^{^j)>PM '] ^^ > If{^> Q - 2£ /4 .?• R^

Now, if for some j , the rank of p (Xj) < v, t h e n

lim inf f f[Zn {x), pn (^)] dx >()

(9.2.35) ''^•

= j f[z[xi),p{xj)]dx {X{xj)=0).R J

O therwise , if rjn = m a x | Zn (x) — z(x)\,

ff[Zn(x),pn{x)]dx^ff[z{x),pn{x)] dx — Co{f]n) A {Zn, Rj)Rj Rj

(9.2.36) ^lfi,^xi),p„{x)] dx - [co{ri„) + e / 4 M ] A (z„. Rj).

3 6 2 P a r a m e t r i c i n t e g r a ls ; two dimens ional p rob lems

N o w , for each fixed j forwhich X{xj) ^ 0, letai bethe c o n s t a n t s of

L e m m a 9.2.1 and let fi, . . ., C** hethe l inear func t ions ofz d e t e r m i n e d

a s in tha t l emma . Then , f rom our gene ra l a s s umpt ions and L e m m a 9 . 2 . 5 ,

i t fol lows that there are se ts Vn CRj s u c h t h a t

I f[^(^j)>Pn(x)] dx> If[z(xj),pn{x)'\ dx

(9.2.37) >/ 2 ^^^^W ^^ = / ^ ^ ^ ^ - ^ / ^ ^ ^ ^ ^Vn I Vn Sj

= / ^ ai X^ {x) dx> f[z (xj), p (xj)] dxR j I Rj

-L(\X[X) - X{xj)\dx.

From (9.2.30) and (9 .3-33)— (9.2.37), we o b t a i n

lim inf / / { z n , G) > lim inf ^ f f[zn (^), pn (^)] dx

^2 ffl^M^Pi^j)] ^^ - 2£/4 > // (^ , G) - e.^ Rj

The theorem fo l lows .

9 .3 . Two d i m e n s i o n a l p r o b l e m s ; i n t r o d u c t i o n ; the c o n f o r m a l

m a p p i n g of su rfaces

The s imp le s t pos s ib le mu l t id imens iona l pa rame t r ic p rob lem is the

p r o b l e m of P l a t e a u , i.e. the p r o b l e m of p r o v i n g the exis tence (and

differentiabil i ty) of a surface of leas t a rea having a g i v e n b o u n d a r y .

T h i s was so lved in the one contour case by J. DOUGLAS and T. R A D O

[3] in 1 93 0— 3 1. The nex t decade saw the so lu t ion (see M O R R E Y [8] formore literature of this period) by D O U G L A S , R . C O U R A N T , M C S H A N E ,

and o the rs of th i s p rob lem among s u r faces of higher topologica l type

b o u n d e d byone ormore con tou rs . The so lu t ion was grea t ly fac i l i ta ted

b y the use of the t h e o r y of confo rma l mapp ing wh ich enab led one to

rep lace the a rea in teg ra l by the D IRICH LET i n t e g r a l ; as we have seen in

(9.1.13) and (9.1.14) we see t h a t

(9.3.I) A{z,G)<'-D(z,G),

the equa l i ty ho ld ing if and only ifz is a confo rma l rep re s en ta t ion of the

given surface . Thus themin imiz ing vec to rs areha rmon ic . Th is fac t

enabled M. M O R S E , ([4]) C. B. T O M P K IN S ( MO R S E and T O M P K I N S [1] —[4]),

C O U R A N T ([1]), M. S H I F F M A N , andothers (see M O R R E Y [8]) to apply the

M O R S E cr i t ica l po in t theory toobta in in te res t ing resu l ts concern ing the

9.3- Two d imens ional p rob lems; conformal mapping 363

ex i s tence o f ' ' uns tab le ' ' min ima l s u r faces . F ina l ly the au tho r ( M O R R E Y

[8]) generalized COURANT'S solution in the case of k contours , the surface

be ing of the type of a domain bounded by k circles, to the case of surfaces

ly ing in aRIEMANNIAN mani fo ld . The con fo rma l mapp ing theo rem s t i l l

holds for such surfaces but the work of B O C H N E R ( [2]) on ' 'ha rmonic ' '

vec tors in RIEMANNIAN spaces was no t suff ic ien t to enable the au thor

to ca r ry ove r COURANT'S work . I t was necessary to ca rry over the theory

of functions of class H\ to vec tors in a RIEMANNIAN manifo ld in order

to ob ta in the " in the l a rge" s o lu t ion o f the min imum p rob lem. By a s s um

ing tha t the man i fo ld 501 (of c lass C^ at leas t) is "ho m og ene ou sly reg ula r" ,

i . e . tha t there ex is t positive n u m b e r s m a n d M, independen t o f PQ, such

t h a t e a c h p o i n t PQ of^ is in the range of a coord ina te pa tch wi th

d o m a i n ^ (0, 1) for whichw | f | 2 <^g^^{x)ii^3' < M | | | 2 , x^B(0,i), f a r b i t r a r y ,

the au tho r cou ld s how tha t the min imiz ing s u r face i s con t inuous . If Wl

is of class C^igij^ C^), the d i f fe ren t iab i l i ty resu l ts fo r v — 2, N a r b i t r a r y ,

as s ta ted in § 1.10 im ply th a t th e so lu t ion harm on ic vec to rs ^ Q for

any / / , 0 < /^ < 1; if 501 is of class CJJ, 0 < < 1, for some n> 3, then

t h e h a r m o n i c v e c t o r s ^ C^; ii Wl is C°° or ana ly t ic , so a re the harmonic

vec t ors . T his w ork is p re sen ted in § 9-4 be low.

In § 9 .2 , we ex ten de d MCSHANE'S proof ([3]) of lower semicontinuityfo r more gene ra l pa rame t r ic in teg ra l s . Sho r t ly a f t e r the wa r , CESARI

(W > [2], [3]) extended MCSHANE'S theorem to h is more genera l in tegra ls

(we do not d iscuss these) and a lso showed tha t the convexi ty condi t ion

on F (z, X) was- anecessary, as well as a sufficient condition for lower-

s e m i c o n t i n u i t y . B u t itwas not un t i l the concurren t resu l ts o f CESARI

[4], DANSKIN a n d SIGALOV [2] th a t an ex is tence the ore m for th e in tegra l

/ / of § 9 .2 , w ith r = 2 an d iV == 3> wa s pro ve d. T he diff iculty w as th a t

the lower s emi-con t inu i ty theo rem requ i re s un i fo rm conve rgence (a tl e a s t on in te r io r doma ins ) and even though con fo rma l mapp ing kep t the

D I R I C H L E T in teg ra l bound ed , th i s is no t enough t o ens u re eq u icon t inu i ty .

Bu t the imp l ied cond i t ion

(9.3-2) m\X\<f{z,p)~F{z,X) <MX |

impl ies an a prior i H O L D E R condi t ion , on in te r ior domains , fo r the so lu

t ion vector. The author ([17]) was able to give a short proof of this

t h e o r e m b y m i n i m i z i n g a d o m i n a t i n g i n t e g r a l / for f among a l l z for

wh ich ano the r in teg ra l / {z, G) < K, ob ta in ing bounds independen t o fK, and t he n le t t ing X -> o o . W e presen t th is in § 9 .5-

The au tho r i s no t aware ofan y genera l d i f fe ren t iab i l i ty the ore m s

for the so lu t ions which have been shown to ex is t . However , i f the ra t io

Mjm above is sufficiently small and if

(9.3.3) \X\Fx7x6{z.X)^y '>mi[\t?-\XmX'm

364 Parametric integrals; two dimensional problems

w h e r e mi is suffic iently large, then the domina t ing func t ion /o de f ined by

(9.3.4) [/o (z , p)f = [/(^. p)]^ + ( ^ ) ' + F^

{E , F, G defined in (9.1.12)) is convex in all the pi{(x == 1,2). To be s u re ,/o does not ^ C^ for any p for which X ^ 0 bu t the second der iva t ives

a r e b o u n d e d and the ana lys is of § 1.11 and §§ 5.2 — 5-5 progresses far

enough to s h o w t h a t the so lu t ion vec tor z^Cj^ for any ju , 0 < ^ < 1.

Since the so lu t ion vec tor is obvious ly conformal , the higher differenti

abil i ty follows in the u s u a l way nea r po in t s x w h e r e not all the de r iva t ive s

z\ {x) are z e r o ; at s uch po in t s /o ^ C^ and so it is not l ikely, in genera l ,

t h a t d e r i v a t i v e s of orde r > 1 will be c o n t i n u o u s at s uch po in t s . For the

case of minimal surfaces , we shall see t h a t t h e y may h a v e b r a n c h p o i n t

s ingula r i t ies .

F o r the gene ra l pa rame t r ic in teg ra l w i th v = 2, the a u t h o r has con

s t r u c t e d a d o m i n a ti n g f u n c t i o n / * * , c oin cid in g w i t h / w h e n E = G arid

i^ =: 0, w h i c h is regula r on ly if the general sense (see § 9-5 be low) .

H o w e v e r , by m a k i n g c e r t a i n a pr io r i a s s umpt ions abou t the so lu t ion

(i.e. that it has the form z =f(x,y)^ C^, etc.) J E N K I N S and S E R R I N

have ob ta ined in te re s t ing re s u l t s conce rn ing the s e s o lu t ions .

Moreove r , in 1957, a s t u d e n t T. C. K I P P S , in his thes is , der ived some

a p r io r i bounds for the H O L D E R c o n t i n u i t y of the f i rs t der iva t ives of aconfo rma l rep re s en ta t ion of a so lu t ion surface , assuming tha t these

de r iva t ive s we re con t inuous on an in te r io r doma in and a s s u m i n g t h a t the

p r o b l e m was regu la r . We p r e s e n t his re s u l t s in §9.5- St rang ly enough ,

h e was not ab le to use these resu l ts to d e m o n s t r a t e the c o n t i n u i t y of the

de r iva t ive s , in s p i t e of th e fac t tha t his b o u n d s do no t d e p e n d on the

m o d u l i of c o n t i n u i t y of the d e r i v a t i v e s .

W e s ha l l now s ke tch p re l imina ry re s u l t s , inc lud ing LTCHTENSTEIN'S

c o n f o r m a l m a p p i n g t h e o r e m for surfaces , leading to the so lu t ion of the

p r o b l e m of Pla teau wh ich wi l l be p r e s e n t e d in § 9.4- We shall confine

ourse lves here to surfaces of type k, i.e. thos e hav ing the topologica l

t y p e of a p l a n e d o m a i n b o u n d e d by k dis jo in t J O R D A N curves ; surfaces

of h igher topologica l types are t r e a t e d in the b o o k [3] by COURANT. We

shall not a s s u m e any t h e o r e m s on confo rma l mapp ing ; the s e wi l l be

p r o v e d ; we sha l l , how ever , use the MOBIU S ( l inear, fractional) tra ns fo rm a

t i ons . We shall assume, a lso, if G is a d o m a i n of t y p e k and class CJ, t h e r e

is a dif feomorphism of class CJ of G o n t o a circular domain B, i.e. one

b o u n d e d by k dis jount c irc les , in s uch a way t h a t the o u t e r b o u n d a r i e sco r re s pond .

Lemma 9.3 .1 . Suppose {zn} is a sequence of vectors such that each

Zn^ C]^(Bn), where Bn is a circular domain of type k. We suppose that

Cni = dB{0,\) is the outer boundary of B^ for each n and € ^2, • • •, Cnk

are its other bounda ries, that Fi, . . .,r^ are oriented F R E C H E T curves

9.3- Two dimensional problems; conformal mapping 3 6 5

whose ranges are disjoint J O R D A N curves, and that the F R E C H E T curves

Fni, each of which is defined by Zn restricted to Cni, converge respectively to

Fi as ^ -> oo. We suppose also that there is an L (<i ^ oa) such that

(9.3.5) D(zn,Bn)<L, n=\,2,...Finally, we suppose that the Cni converge to Ci, where each Ci is a circle or

a point, and that C\, . . ., Ci, I <,k, are circles and the Ci with I <. i ^ k

{if any ) are points.

Then the Ci, . . ., Ci are disjoint.

Proof. If this were not so, two of the Hmiting circles Ci a n d Q ( 1 0) b e

the min imum d is tance f rom a po in t on any Fp to one on a different Fq

and choose i ? > 0 b u t sm al l enoug h so th a t e very c i rc le dB{xo, r) w i t h0 < r < i^ int ers ect s Cf a n d Cj. Now, choose d, 0 <C d <C R. T h e n , t h e r e

is an N such that if ^ > AT then every c irc le dB{xo, r) w it h (5 < r < i^

con ta in s an a rc y n r which is in Bn and has i t s endpo in t s on two d i s t inc t

Cnp a n d Cnq ( ' ' u s ua l ly" C^i a n d Cnj, but there may be c i rc les Cns w i t h

/ < s < ^ close to XQ). Thus for such r

f\ Zn^[r, ^) |2 d^ > (2JT)-1 \ j \ Zn& \ d&

SO t h a t

2 d ^

fr-^f\zn4r, &) |2 d&dr > ^ l o g ( J ) , n>N,

S ynr

Since d i s a rb i t ra r i ly smal l , we a rr ive a t a contrad ic t ion .

As a result of this lemma, we eas ily conclude the following corollary:

Corollary 1. Suppose that the sequences {zn}, {Bn], {C ni}, {J^ni}, C i,

and Fi satisfy the conditions of Lemm a 9.3-1- Then it may also be assumed

that Cni = Cifor each i <,l and, moreover, this assumption may be made

without altering the values of Urn inf D{zn, Bn).Now suppose tha t a l l these condi t ions a re sa t is f ied . S ince the Fi a re

J o rdan cu rves and the F reche t cu rves Fni ->-A for each i, it is easy to see

tha t the re ex i s t con t inuous vec to r func t ions Cni((p) which a re per iod ic

of per iod 2 j r , a re no t cons tan t on any segment , and which converge

un ifo rm ly t o f (99) w he re ^ = f* (99), 0 < 99 < 27r is a top olo gica l rep res en

ta t ion of Fi, e x c e p t t h a t CHO) = C'(27r) (Theorem 9.1.2(b)). It follows

eas i ly tha t the re a re non -dec rea s ing con t inuous func t ions (pni s u c h t h a t

(i) Cm[(fniiO)] = ini{0) = Zn [xi + ri COS0, y i -f- ri sinO],^^^^, Ci = dB{xi,yi,ri), i=\,.,,J;

(9.3.0)

(ii) (pni{^) — 6 is periodic of period 2jr ;

(iii) —7t <(pni{0) < J r .

W e may immed ia te ly conc lude the fo l lowing co ro l l a ry :

Corollary 2. Suppose the sequences [zî^, e tc . , of Corollary 1 satisfy

the conditions of that corollary and that the ^ni> d, Sni, < < ^ni ^<^^^ ^^^^

chosen as above. Then asubsequence, still called {n}, m ay be chosen so

that the functions (pni{Q) converge to non-decre asing limit functions cpi{d).R em ark . Thes e l imi t func t ions need no t be con t inuo us bu t we p rove

the fo l lowing lemma:

Lemma 9.3.2. Suppose the sequences [zî], {q)ni], e tc . , satisfy the con

ditions above. Then, for a given j , 1 <.j^l, and do, either cpj is continuous

at do or else

(9 .37 ) (pj(do + ) - ^ i( ô - ) =271:.

Proof. Suppose for some i an d ô , we h av e

T h e n dlfpiiOo)] a n d Ci[(pi(Oo)] a re d i s t inc t po in t s ; l e t

\Ci[<Pi{e^)]-Ci[cpi{eo)] = d>o,

L e t po be the point of Ci co r re s pond ing to do- From the un i fo rm con

vergence of Cni{(p) t o Ci((p) and the po in twise convergence of (pni(0) to

(fi (9), a l l be ing monotone , i t fo l lows tha t H a ^ > 0 and an N s u c h t h a t

Ci div ides the d isc B {po, S), B {po, d) i n t e rs e c t s n o o t h e r Q w i t h ; < I, a n d

ify{r)

is the arc ofdB{po, r)

which conta ins po in ts o fBn

t h e n(9.3.8) Zn [7 {r)] is of length > i / 2 , 0<r <d.

I t is c lear that (9 .3-8) certa inly holds even if that arc contains points of

Cni with ^' > Z i f we in te rpre t y{r) = yn{^) as the par t o f tha t a rc in Bn-

T h u s , for each e > 0

(9.3.9) L>jr- j\zn{f)fdy>> \o i -y n>N.

e Yn (r )

This i s imposs ib le .We wish now to prove the fo l lowing theorem of LICHTENSTEIN [2] :

Theorem 9.3.1. Any locally regular^ surfaces of ty pe kand of class Cj^

possesses a representation z which is conformal and of class Cj^ (B) where B

is a domain of type kbounded by circles. Three given points on one of the

boundary curves of S can be made to correspond, respectively, to three given

points on the outer boundary of B w hich may be taken as dB{0,\). Incase

the surface is ofclass C^ for n>1, C^, or analy tic near a point, the

representation z has the corresponding class near the corresponding point.If the point is on the boundary of S and d S has the same class near that

point, then z has the same class near the corresponding point on dB.

Prepa ra to ry remarks . As was po in ted ou t by COURANT in [3], this

theorem is no t necessary if one wishes only to prove the exis tence of a

1 See (9.3.10) et seq. below.

9.3. T wo d i men s i o n a l p ro b l em s ; co n fo rma l ma p p i n g 3 6 7

minimal surface ( in the sense of d i f fe ren t iaLgeometry) wi th g iven bound

ary . But the theorem is in te res t ing in i t se l f and is he lpfu l in proving the

exis tence of a surface of least area having a given boundary which surface

is simultaneously a minimal surface. We sha l l p rove th is for the genera l k;

LiCHTENSTEiN [2] p ro v ed it for ^ = 1. Su pp os e t h a t

(9.3.10) S:z = z{u,v), (u,v)^G, Z uXz^^O,

w h e r e G is of type k and of class CJ and z^ Cl{G). Using invers ions and

s imp le t r ans fo rma t ions , we may a s s ume tha t the ou te r bounda ry o f G

corresponds to tha t bounding curve of 5 conta in ing the th ree g iven

poin ts . We f i rs t p rove severa l lemmas .

Lemma 9.3.3. There exists a mapping

(9.3-11) r:u = u{x,y )y v — v{Xyy ), [x,y )^B

from a circular domain B of ty pe k onto G in w hich u and v ^ H\ (B)

n C^{B), and which minimizes

D{Z,B)= n[\Z^Y^ + \Zy\^)dxdy,( 9 . 3 . 1 2 ) B

among all such mappings which carry the three given {distinct) points on

the outer boundary Ci = dB{0,\) of B into the three given points on theouter boundary / \ of G, where dG = FiU . . - v Tjc.

Proof. Fi rs t of a l l , i t is c lear th a t th er e is a diffeomorph ism from

s ome c i rcu la r doma in B' with ou te r bounda ry C i on to G. A map of the

des i red type i s then ob ta ined by p reced ing th i s wi th a Mob ius t r ans

format ion of B (0,1) in to i tse lf to es tab l ish th e th ree po in t corre spon den ce ;

s uch a t r ans fo rm a t ion ca r r ie s c i rcu la r dom a ins in to c i rcu la r dom a ins .

W e n o t e t h a t

D(Z , B) = J J[E(u, v) [ul + ^il) + 2F[ux v^ + % Vy ) +B

(9.3.13) +G [vl + vl)-\dxdy

E {u, v) = Z u ' Z u, F ^ Z u ' Z't,, G = z^ ' Z'i^

w h e r e E, F, a n d G^ G^^(G) a n d t h e r e a r e n u m b e r s m a n d M s u c h t h a t

w(A2 + / / 2 ) < £ A 2 + iFXfjii + Gfji\^M{X'^+ fjil), 0<m<M.

(9.3.14)

To show the ex is tence of the min imum, le t {tn} be a min imiz ingsequ ence . F ro m (9 .3 . I I)— (9-3 .14) , i t fol lows th a t { r^} sa tis fies th e con

d i t ions of Lemma 9 .3-1 and so , by choos ing a subsequence , we may

assume tha t our min imiz ing sequence sa t is f ies a l l the condi t ions of Coro l

la ry 2 . From the th ree po in t condi t ion and Lemma 9-3-2 , i t fo l lows tha t

th e res t r ic t io n of r^ to Ci converges un ifor mly t o a rep rese n ta t ion of / \

3 6 8 Pa r am et r i c i n t eg ra l s ; t wo d i men s i o n a l p ro b l ems

which sa t is f ies the th ree po in t condi t ion . There a re now three types of

degene racy wh ich can occu r wi th re s pec t to the bounda ry cond i t ions :

case i: ^ > 1, 1 < / < ^, one or m or e Q w it h i ^ I

are po in ts in te r ior to the Hmit ing domain B bounded by Ci, . . . , C^;case ii: some such Ci is a point on some Q with j <,l;

case i i i : some cpi w it h ' < / satisf ies (9.3-7).

W e firs t consider case i . Le t i^ > 0 be chosen sm all eno ugh so th a t

B{po,R) C B and con ta in s no o the r Cp than thos e wh ich reduce to

d = pQ. L et 0 < ^ < i^. T he n th er e is an N so la rge tha t a l l the Cwp

which a re approaching ô l ie ins ide 5(^0 , <5). Thus , fo r each r,d <,r <,R,

Tn\_dB[pQ,r)'] enc loses a l l the corresponding F^ an d so has leng th >

some ^ > 0 so th a tR

(9.3.15) D{Z n, Bn) >m jr-'^{d^l2n) dr = {m d'^jln) log{Rjd)

6which is impossible s ince d i s a rb i t ra ry . The reasoning to e l imina te case

i i i s the same. So we may assume tha t I = k and a l l the Bn = B.

We now suppose tha t case i i i ho lds . Le t po be the po in t o f Q having

co ord ina te ô- Choose jR > 0 so th a t B {po, r) H B is bounded by an a rc

y ' of Q and an arc y " of dB(po, r) for each r d. {y U y '" = Q )

But , f rom Lemma 9 .3-2 and the convergence of (pnt to cpi, i t fol lows that

d ia m T ^ j( y "') < ^ / 2 so l[rn(y")]'> djl, d<r< R, n>N.

As in (9 .3 .9) we arrive a t a contradict ion. Thus for each i, t h e (p i a re

cont inuous so the convergence of Zn is uniform on each Ci to a rep re s en ta

tion of Fi.

Fina l ly , we no te th a t th ere is an i? > 0 , ind epe nd ent of p, s u c h t h a t

ii p^B, then e i the r B (p, r) G B or B ( P , r) O B is bounded by an a rc 7 '

of some one Ci and an a rc y " of dB(P, r) w h e n e v e r 0 <ir <. R] in the

former case , we set y " = dB(P, r). N ow , let £ > 0 an d ch oose ^1 > 0

s o s ma l l tha t d iam Tn(7 ') < e/2 whenever r <C di; th is is poss ible on

account o f the equicont inu i ty . Now, suppose for some d, 0 <C d <. di,

tha t o s c Tn on B{p,d) f) B > s. T he n, for ^ < r < di, the length of

Tniy') > £/2 so t h a t (see (9-3-15))

(9.3.1.6) D(Zn, Bn) > mJr-îe^lSn) dr = {m e^jSn) log{dild).

dThe equ icon t inu i ty o f the Tn now follows from (9.3-16) and a subsequence

converges un iformly to a min imiz ing map r .

9.3« Two d imens ional p rob lem s; conformal m app ing 3 6 9

O ur a im now is to d iscuss th e m in imiz ing m ap r . S ince a s im i la r i ty

t rans fo rma t ion o r , more gene ra l ly , a MoBius t r ans fo rma t ion in the (x, y)

plane carr ies one c i rcu la r domain in to another and leaves the Dir ich le t

in teg ra l inva r ian t , we s ee tha t ou r r min imizes D (Z , B) among a l a rge r

c lass of mappings f rom c i rcu la r domains in which the ou te r boundary

need no t be dB{0,'[) and three po in ts do no t need to be f ixed . So ,

s u p p o s e t h a t

A '^ ' = l ( ^ , y ; ^ ) , y' = r ] ( x , y ; X ) , m < A o ,

(9.3.17) i{x,y;0)=x, r](x,y;0)=y,

fA(^%y;0) =v{x,y), rj2,{x,y;0) =a){x,y)

is a family of diffeomorphisms, of class C^ in (x, y, A), of 5 in to a circ ula r

d o m a i n B;^ for each A. Then v a n d co must sa t is fy the fo l lowing boundary

condi t ions on each Cj, j = \, . . ., k:

{V, co) = {n, COi) + (V2, CO2) + {V3 , CO3) ,

vi cosO + coi sind = 0, (v2, 0)2) = (oj, dj) = cons t .(9.3.18)

vs + iws = (cj + ifj) (cos0 + i sinO), {c j + ifj) = cons t .

Cj : X = Xj + Tj COS0, y = yj - \- TJ s i n 0 .

Moreover, i f we set

cpW= D{Z ',B,), Z '[x\y ')=Z [^'{x\y '-X), rj'[x\ y '• K)],

al^ :x = ^'{x\y '',X), y = rj'{x\y'; X),

t h e n cp' (0) = 0 since 99 (A) tak es on i t s mi n im um for A = 0 , s ince r (Tj"

is an admiss ible map of B;^ o n t o G. By express ing D (Z\ B^ as an in tegra l

ove r B, we ob ta in

(9 20) ^''^ ^ ^~^ ^^"^ ^^ ~ ^^ ^""^' ^^ ' ^ ^ " M - ^ ^ ^ 2/ + ^y f J >J = ^x'r]y — ^y rjx-

^(A ) = / / [ g ( l | + t , ) - 2g(f^|^ + ^^7^^) + (5^(f| + rjl)]J-idxdy .

(9.3.21) ^ e = | Z r . | ^ g - Z ^ - Z ^ , , (l = |Z2/|2

®, g , and ^ be ing ind ep end ent of A and in Li{B), Diffe ren t ia t ing wi th

respec t to A and s e t t ing A (a nd 1 / ^.nd ?^ ) = 0, w e o b ta in

— (p'{0) = f f [U(vx — coy) — V{vy + Wx)] dxdy = 0 ,(9.3.22) -^ i

t / = g - ^ , F = - 2 g .L e m m a 9.3.4. Suppose v and co ^ C'^(B) and satisfy the boundary

conditions (9.3-18). Then there exists a family a^ of maps satisfy ing the

conditions in and following (9.3-17).

Proof. B y us ing a pa r t i t io n of un i t y of c lass C ^, we m ay w ri te an y

s uch (v , co) as a sum of one which ^ C^ (B) and three for each Cj w i t h

Morrey, Multiple Integ rals 2 4

y > 1, each of which van i s hes excep t nea r Q . Corresponding famihes ai

in (9.3•'17) can be found s imp ly by s e t t ing

^(%^ y] X) = X + Xv{x, y), rj(x, y; 2) = y + Xco(x, y)

for the {v , co) ^ C^ (B ) or of the t y p e s {v2, C02) and (vs, C03) for each Q .The rema in ing ones are h a n d l e d by po la r coo rd ina te s and the s e l a t t e r

each ca r ry B i n t o itself.

F r o m t h i s l e m m a and the duscuss ion preceding it, we o b t a i n the

following corollary:

Corollary. / / ( £ , ^ and @, as defined w (9.3-21), correspond via (9.3-13)

to our minimizing map r, then (p' (0) = Ofor all {v , co) ^ C^ (B ) and satisfy

ing (9.} AS).

L e m m a 9.3.5. If the conclusion of the corollary holds, then(9.3.23) u=v=^o.

Remark. The essen t ia l idea of the l a s t pa r t of th is proof is due to

H . L E W Y ; the proof given here follows in a gene ra l way the a r g u m e n t

g iven in COURANT'S book [3] , pp. 1 6 9— 1 /8 .

Proof. Suppose , f i rs t , tha t v and o) £ ^^{B), le t -y; be a moUifier, and

le t VQ and OJQ be the ^-mollif ied functions . Then, defining JJ = V =^ v

=z CO = 0 ou ts ide B and keep ing Q smal l , we o b t a i n

/ / [U(vex — ojQy) — V(vey + ajex)] dxdyB

^ If II ^ ? (^"~ ^ ) ^ (^ ' ^ ) (^^"" ^ "~ ^ ( ' ^ ' ^ ) ( ^ + ^^^^ ^ ^ y ^^ ^

= / / [UQ{VX - o)y) — Ve{vy + ojx)] dxdy

B

= fJ[oj{UQy+ VQO^) -v(Ue^ - Vgy)]dxdy,B

(9.3.24)

T h u s if th i s = 0 for all s uch (v , co), we s ee tha t UQ + i VQ is a n a l y t i c for

each ^ > 0 and hence we h a v e

{9.3.25) 0{z) =: U + iV a n a l y t i c on B.

Now, s ince U and V^ Li(B), we see t h a t

(9.3.26) lim f[U(vdy + codx) — V{cody~vdx)] = 0

for any (v , co) ^ C^(B) and satisfying (9-3.18) (BQ is the set of (x, y) in B

s u c h t h a t B{x,y;Q) C B). S e t t i n g(9.3.27) V + ia) + P{r) [ci + i di + (ei + ifi) e^^], 1 - A < r < 1

w h e r e P{r) = i n e a r r = i and P{r) = 0 n e a r r = i — h, v + ico

be ing 0 elsewhere, (9.3.26) for all a, di, ^ i , / i , impHes tha t

(9.3.28) j0(z)dz = Iz0{z)dz = O, 0 < Q< Qo^

9.3. T wo d i men s i on a l p ro b l em s ; co n fo rma l map p i n g 3 7 1

By choosing the corresponding variations for each Q, we obtain

^^'^^^^ ^^^ ^ - 0 < ^ < ^ o ,

CjQ being the circle in B concentric with and at a distance Q from Q.From (9.3-29), we conclude that there is an analytic function ^(z) onB such that

W(Z) =0(Z) .

Since dW(z)ldx = 0(z) and dW(z)ldy = i0(z), it follows thatW^H\{B) and so has strong boundary values in Li(dB). Thus W iscontinuous on B.

Now, let us choose

(9.3.30) V = -P(r)A(e) sine, co = P(r) A(6) cos0

where P(r) was defined above (in (9.3.27)). Then (9.3.26) yields

(9.3.31) - l i m fA(e)Im{W[(\ - Q) e^^]e^^^}dO = 0.

Inte gra ting by pa rts once, we m ay th en let ^ - > 0 in (9.3-31) to o btain

(9.3.32) ImjW '{e^^)[ie^Â'{e) - e^Â(e)]de = 0.

If we break this into two integrals, change the variable to 99 in the second< p

integral, write A((p) = A{0)-\- f A'(6) dO, and then interchange the0

order of integrations, (9.3.32) becomesf 271 r 2n 1 271

I ml (A [6) i 6'^ W {e^^) - fw'(e^^) • e^^d(p dO + iA(0) f-^ W(e^^) dcplo L e J o

(9.3-33)

in which the coefficient of A (0) vanishes. Since A' is an arbitrary realfunction ^ C^ except that

(9.3.34) fA'{e)de = o,0

it follows from the Lemma of Du Bois R A Y M O N D (Lemma 2.3.1), thatthe imaginary part of the quantity in the bracket in (9-3-33) is a constant. Carr^dng out the integration, we see that

(9.3.35) Im[iz W\z) — i Wiz)] = const, on Ci,so that z W{z) — W(z) = ^{z) where ^ is analytic across Ci. Thus wesee tha t W^ C°° along C\ (since ^îs continuous). Differentiating (9.3.35)with respect to d we obtain

(9.3.36) Imliz^JizW -iW]\ = ~-Imz^0{z) = 0 .

24*

3 7 2 Pa ra m et r i c i n t eg ra l s ; t wo d i men s i o n a l p ro b l ems

B y repea t ing th e a rg um en t fo r each Q , we ob ta in

(9.3.37) Im[z — Z j)'^0[z)=::O, z^Cj, j=2,...,k.

Accord ingly we see tha t 0 can be ex tended ana ly t ica l ly accross each

boundary c i rc le of B.Now, on Ci, le t f(6) = z^0(z). Then f (6) is real and it follows from

(9.3-28) that

27 1 2JI 2TC

(9.3.38) lf{e)de = lz0(z)'zdO= -ifz0(z)dz = O,

0 0 0Since / is an aly tic an d periodic , i t fol lows th a t / , an d hence 0 h a s a t

leas t two zeros on Ci. By using (9.3 .29), we conclude also that 0 has a t

leas t tw o zeros on each Q . N ow, e i ther0{z) ^ 0

onB

or else it has af in i te and non-nega t ive number of ze ros in B and no poles there . The

n u m b e r n of zeros of 0 in te r io r to B is given by

Hz)( 9 . 3 . 3 9 ) "'^^B'g

-l^lim fZ^dz.

B'^ = B-U[BnB(Cs,Q)].

t h e Cs bein g th e zeros of 0ondB a n d Q be ing s u i t ab ly s ma l l . B u t th e va lue

of the integral in (9.3.39) is easily seen to yieldn <, — k

s ince the re a re a t leas t tw o Cs- on each Cj,j=\,...,k. T h u s w e m u s t h a v e

0(z) ^ U + i V -^^O and the l emma i s p roved .

Lemma 9 .3 .6 . Suppose z = z(w), w = {u, v) ^ G, is a diffeomorphism

of class C^, G being of class C ^ (z = (z^, . . ., z^)). Suppose S is the [unorien-

ted) Frechet surface having the representation z and suppose z = Z{X),

X = {%, y ) ^ B, is a representation of S in which Z ^C^ (B) n HI {B)

where B^ C^. Then ^ a unique m apping r : w = w{X), X^ B, such that

(9.3-40) Z{X)=z[w{X)-\, X^B,

T ^ C^{B) n Hl(B), and 3 ^ sequence r^ : w = Wn(X) of diffeomorphisms

from B onto G which converge uniformly to r.

Proof. L e t \S \ denote the range of S, i . e . the ranges of z a n d Z . Since

^ is 1 — 1, z~ ^ i s a hom oem orp hism from 151 ont o G. Since Z is a represen

ta t i on of 5 , H a sequen ce {xn} as above s uch tha t z [xn {X)] ^ Zn {X)

converges un iformly to Z(X). If we define r{X) = z~'^[Z (X)], we see

that (9 .3 .40) holds and also that

rn(X)=Z-^[Zn{X)]

s o t h a t Tn conve rges un i fo rmly to r on B. Clear ly r is un i que ly de te rm ined

by (9 .3 .40) . To show tha t r^H\[B), le t XQ^B, le t wo = r{Xo), a n d

ZQ = z{wo) = Z(Xo). I t i s c lea r tha t z(u^, u^) can be ex tended to a



9.3 . Two dimensional problems; conformal mapping 3 7 3

dif feomorphism C('^^> • • •> ^•^) from a neighborhood in Rjs^ of (u], u^,

0, . . . , 0) where WQ == (ul, u^) on to a neig hbo rho od of 2-0 in i? ^ . Th en

T ( Z) = c - i [ Z ( Z ) ]

for X in -B near XQ. T h a t r ^ H\{B) follows from Theorem 3-1.9.

L e m m a 9.3.7. Suppose r \w = w{X) satisfies the conclusions of

Lemma 9.3-6. Then the Jacohian Ux Vy — Uy Vx has the same sign almost

everywhere in B.

Proof. From Lemma 9-2 .4 , i t fo l lows tha t there a re se ts Ji a n d J2, of

1-dimensional measure 0 such that if none of the s ides of the cell R lie

along l ines x"- = C^ w h e r e c'^^Ja, oc = 1, 2, th en (9.2.24) ho lds . F ro m

L em m a 9.2.5, it follows t h a t if i^ is a cell for w hic h (9.2.24) ho lds, th er e

ex is ts a sequence of measurab le se ts VnG R s u c h t h a t

Vn R

I t is c lear, s ince we may confine ourselves to a subsequence that we

m ay a s s ume th a t a l l the J aco b ian s Unx'^ny — '^ny '^nx have the s ame

s i g n t h r o u g h o u t B. The result then follows using (9.3 .4I) .

W e can now p rove the con fo rma l mapp ing Theorem 9.3-1.

Proof of Theorem 9.3 .1 . W e le t r be the mapp ing o f Lemma 9 .3 .3 -

From Lemma 9 .3-5 , i t fo l lows tha t

® — ^ + 2 ^ 5 = 0 = = £ : {ux + i % ) ^ + 2F {ux + i Uy) x

X {vx + i Vy) + G{vx + i VyP;

he re (£, g , an d @ ar e defined in (9-3-21) an d E, F, G in (9-3-13)- It follows

from (9-3-42) that

Vx=- — G - 1 (F Ux± H Uy ) y Vy = G - 1 ( i t HUx — F Uy),

H =^]/EG-F^.

From Lemma 9-3 .6 and 9-3 .7 , i t fo l lows tha t the ^^ s ign may be rep laced

b y k, w h e r e k = ^ i and is constant on B. T h u s

(9.3-43) Vx =—G-^(Fux + kHuy), Vy = G-^(kHUx — FUy ), or

rr. . . .^ % "= —k(bux + cvx), Vy = k(aux + bvx),(9.3.44)

Since T is co nti nu ou s, th e coeffic ients in (9 .3.43) are con tinu ou s an d

(9-3-43) impl ies (since r ^ ^ K ^ ) a lso) th a t

flG~^[Cx{kHux-Fuy)+Cy (Fux + kHuy )]dxdy = 0, ^^C\{B),B

(9-3.45)

F ro m th e theo rem of § 5 .5 , i t fo llows f irs t th a t T ^ Hl(E>) fo r each p > 2

and each D C CB. T h i s i m p l ie s t h a t T $ C|J, (Z)), that the coeffic ients

^ Cl(D), a n d h e n c e t h a t u a n d v^ CJ(D ) for D C C B. The h ighe r

d if fe ren t!ab ih ty resu l ts on the in te r ior fo l low as usua l .

L e t {xo,yo)^dB a n d (uo,vo)=T{xo,yo). A ne ighborhood '^ of

(^0, ^o) may be mapped by a diffeomorphism of c lass C^:

U = p{u,v), V = q(u,v), u = P{U,V), v==Q {U,V)

onto a d isc B(0,0,R) so (^0,^0) corresponds to (0 ,0) and ^ddG

co r re s ponds to U = 0. B o u n d a r y c o o r d i n a t e s {t, 6), w h e r e t = logr a n d

(r, 6) are polar coordinates with pole a t the center of the c irc le oi dB

c o n t a i n i n g {XQ, JQ), may be chosen and the equations (9-3-43), (9-3-44),

an d (9.3.45) go over in to equ ati on s of th e sam e ty pe s . Since f/ = 0 wh en

t = con st . , we conclu de from t he th eo ry of § 5-5, f irs t th a t U and hence

V^ e v e r y H\[D), U an d F ^ eve ry C^„ th e coeffic ients $ C^, an d henceU srndV^ C],[D) for D a ne ighborhood of {x^, yo) on B. T h us T $ C]^{B),

Again the higher differentiabil i ty results follow as usual .

9 .4 . Th e p rob lem of P la teau

We now resume our d iscuss ion of the problem of PLATEAU. L e t F

deno te the s ys tem {Fi, . - ., Fjc), each Fi being an o r ien ted c losed F R E C H E T

cu rve wi th range 17 . |.

Definition 9 .4 .1 . We def ine

/ ( r ) = i n f { l i m i n f L ( ^ ^ , G ^ ) }

d{F)= inf {hm inf D (%, Gn)}

for all sequences {zn} where Zn^ C^(Gn), each Gn is of type k and of

class C^ w ith b o un da ry curv es C^i, . . ., CnJc, each or ien ted as a par t o f

dGn, a n d Z n re s t r i c ted to Cnt i s a represen ta t ion of Fni w h e r e Fni - > / \ - ,

i =^ \, . . ., k; in the case of d(F) we requ i re the Z n also ^ H\{Gn), W e

also define/ * ( r ) - ^ * ( r ) = + 00 if ^ = 1

^^'^'^^ l*(F) - m i n 2 ' ^ ( r ( * ) ) , d^{F) = mm ^d{Fi))

for a l l poss ible sys te m s / ' (^) , . . ., 7 (2?) w he re each F^'^^ consis ts of the curves

Fj f o r w h i c h y $ Ti w h e r e [J Ti = {\, 2, . . ., k}.i=\

Lemma 9.4 .1 . (a) In the definitions of l[F), d{F), l'^(F), and d*{F)

we may assume that each G^ is of class C\ and that each Zn ^ CJ [Gn) and is

locally regular.

[h) d[F) = 2l[F), d*(F) = 2l*(F).

(c) d(F) <d*{F).

Proof. All the s ta tements in (a ) a re ev ident f rom the def in i t ions

except poss ib ly the one concern ing loca l regula r i ty . To see th is , le t

3 7 6 Pa ram et r i c i n t eg ra l s ; t wo d i men s io n a l p ro b l ems

oriented bounda ry circle Ci of B is a representation of Fi, Moreover z is a

representation of a surface of least area', i.e. L(z, B ) = l{r).

Proof. L e t {zn} be a sequence ^ L{zn, G^) -> l{r). W e may f i r s t

a s s ume each Gn^ CJ and each Zn^ Gl{Gn) an d loca l ly regula r . O n

account o f Theorem 9 .3-1> we may assume tha t each Zn^ C]^[Bn), each

Bn being a c ircular region, in which we have

(9.4.?) D{Zn,Bn)^d{r).

F i n a l l y w e m a y a s s u m e t h a t t h e Z n a n d t h e Bn satis fy a l l th e con dit io ns

in Lemma 9.3-1 and i ts two corollaries as well as (9 .4 .3)- Then there are

th re e poss ible ty pe s of deg ene rac y as in th e proof of Le m m a 9-3-3- W e

sha l l show tha t each of the cases cons idered there impl ies tha t d{r)

= d*(r) and is there fore imposs ib le . Le t M = s u p D{zn, Bn).Suppose case i holds and le t XQ = \Ci\. Choose 6 > 0 b u t so sma l l th a t

B (XQ, (5 /2) (2 B and conta ins no | Cj \ other than those for which | Cj \

= XQ. F o r n'> N, all the \Cnj\ w i t h | Q | = XQ are ins ide B(xo, d). F o r

s uch n, s ince

M > / lr-^\zne{r,0)\^drde,6 0

i t fol lows that there is an fn, d ^fn ^ d^l^ s u c h t h a t

f\zne(fn,6)\^d6<2MI\logd\

0a n d Zn(rn, 0) is A.C. in 6. L e t Bni = BnU B{xo, fn), B'^^ be the doma in

ou ts ide the Cnj in te r ior to B[x{i,rn), a n d Bn2 b e (yn[B'^<^, w h e r e On is

the invers ion in dCnt followed by the reflection in the x'^ axis . Le t Hni

be the ha rmon ic func t ion in B{XQ, rn) coinc id ing on dB(xo, rn) w i t h Z n

and le t Hn2 be the t rans form of Hni unde r the inve rs ion in dB [XQ, rn).

Define

{x ) , X^Bn — B{XQ,fn)

i[x), x^B(xo,rn)

x^B'^^nBi

^)> x^B'^^ — B(xo,fn)

Z n2 {x) = < 2 [O'n ^ (^)] > ^^Bn2-

Then we see f rom Lemma 9-4 .2 tha t

D {Z ni, Bnl)+D {Z n2, Bn^) < D [Zn, Bn) + 4 M / | l o g ^ | .Since d can be a rb i t ra r i ly smal l , i t fo l lows tha t d*{r) <,d(r).

Suppose case i i holds and le t XQ = Ci, XQ^ CJ, and le t ZQ be the image

of XQ unde r the l imi t ing con t inuous map o f Q on to Fj def ined by the

func t ion 0j(d) of Lemma 9.3-2 and equation (9-3.6) (recall that \ < j

^ l < i). We may choose a sequence {dn} in which each dn> 0 a n d

, , , Mx)

9.4. The problem of PLAUTEAU 377

^n ->0, and a s equence fn w i t h dn ^ n ^ ^n^ s u c h t h a t dB(xo, d]l^)

in te rsec ts on ly Q and B{xQ,d]l'^) con ta in s on ly thos e Cqn for which

Cq =XQ, all of these be ing ins ide B{XQ, dn) and s u c h t h a t

(9.4.4) / I Z ne {fn, e) |2 de< 2M/I log^^ |.

Bnf]SB {xo,rn)

Since the Z n converge un iformly a long Qwe see t h a t

mRx\zn{x) — ZQ\ ioT x yn = [C j n B(xo, r^)] U[dB{xo, fn) Pi 5] < e i,

(9 .4 .5) l im£^ = 0.

L e t Rn- 0 w i t h enIRn -> 0 and define the m a p p i n g

(9.4.6) (J0n(z) = Z Q + 0n{\z - Zo\) ' \z - ^ o | - ^ {z - ZQ)

w h e r e 0n C^{Ri)> 0nis non-dec rea s ing , and

0^ (s) = 0 for s < £^ and 0{s) = s fors Rn,

s - i 0n{s) and 0;(5) < RnliRn - 2sn) for alls.

If we then def ine

(9.4.8) z'^(x) =con[zn(x)]

w e see that (9.4-3) still holds and z'^[x) = ZQ in anopen set D y n- Con

s e q u e n t l y , for each n, the re ex i s t s a s imple c losed curve Cjc+i, n of class

CJ wh ich is in th is open set, is inB, and encloses the Cpn for w h i c hCp =XQ. Let usdefine B i = BnU UCpnU U Bpn w h e r e Bpnisthe

in te r io r ofCpn, and let Bn2 he the ex te r io r of the Cpn and define

\z^[x), x Bni and ou ts ide Cjc+i^nZni(x)

\zo , X on and inside C]c,i n-(Q . 4 . Q )

{ z^{x), X ins ide Q+i nand ou ts ide and on the Cpn,

0 , oc outside and on CA:+I,%.

Then , aga in(9.4.10) D{Znl, Bnl) + D{Zn2, Bn2) = D {z Bn)

a n d we see t h a t ^ * [F] = d(r).

F i n a l l y , we suppose case iii h o l d s ; we as s ume tha t 1 <,j <.l and t h a t

the func t ion 0j of L e m m a 9.3.2 satisfies (9.3.7). Let XQ be the

p o i n t on Q co r re s pond ing todo and letZQ = 0j (6) for 0o < 6 < ^o +

+ 2jr. T h e n , asa b o v e , we may choose dn and fn sothat (9 .4-4) holds ,

dB(xo,dj/^) in te r s ec t s Q only , and the Cnp forw h i c h \Cp\ =XQ are

all ins ide B{xo,dn). This t ime (9.4 .5) holds for allx on yn = [C; —— B(xo, fn)] U[dB(xo, Yn) Pi B]. We again choose Rnand define o)n{z)

by (9.4.6) and (9.4.7) and ^ by (9.4-8) and see that (9 .4 .3) s t i l l holds

a n d z^ [x) = ZQ on an open set D y n- Cons equen t ly for each n, t h e r e

ex is ts a J O R D A N c u r v e Cjcj^i^n of class CJ in th is open setand in Bn

which enc loses Qand the Cpn for w h i c h Cp =XQ (if a n y ) . T h e n we may

define Bni — B^U Q U U CpnU (their interiors) and 3^2 as the

exterior of Q and Cpn and can define Zni and Zn2 as in (9.4-9) and find

again that ^* (r ) = ^( r ) .

Thus none of the degeneracies occur, En = B for every n, and the Znconverge uniformly on dB. The result follows by replacing each Zn by

the harmonic function with the same boundary values and using the

lower-semicontinuity. The conformality follows from Lemma 9-4.1 .

As was mentioned above, the author solved the PLATEAU problem,

in the ^-contour case, for surfaces situated in a "homogeneously

regular" Riemannian manifold ( M O R R E Y [8]); these were defined in the

first paragraph of § 9-3. We now present a simplified proof of this result.

We shall assume that this manifold W is of class CJ,, 0 < /^ < 1, at

least and that W is connected. It follows that M is separable. We shall

consider vector functions z from various sets having as values points

of m.

Notation. If zi and Z2 are two points in ^R, we let 12:1 — 21 be the

geodesic distance between them, i.e. the inf of the lengths of all paths in

W joining them.

Definition 9.4.2. A vector function z defined on a measure space

{E, (j) is said to be measurable-JJL iff for every open set D C 391, the

subset of X in E where z[x)^ D is measurable-//.

Lemma 9.4.3. If zi and Z2 are ju-measurable on E, then \zi(x) — Z2{x) \

is measurable on E.

Proof. If z is measurable-// on E and 5 is a B O R E L subset of 9K, the

subset of X where z(x)^ S is measurable. For each n, divide 9)1 into at

most a countable number of disjoint B O R E L sets Snj, each of diameter

< n~ ; it is easy to see how to do this using coordinate patches. In each

Snj, choose a point Znj. Let Enjk be the subset of x in E for which Zi (x) ^

Snjand

Z2{x)^Snk-For each

n,the

Enjjcare disjoint and measurable

and E = U Enjki' fixed). Define 0n(x) = \znj — Znk\ ior x^ Enkj- Then

each 0n is measurable-// on E and 0n (^) -> I ^1 ( ) — ^2 ( ) | for each

x^E,

Definition 9.4.3. A function z (into 9)1) defined on a measure space

(£,// ) is said to be of class Lp[E,[ji) <^-^ is measurable-// on E and

\z{x) — ZQ\^ Lp{E, fj) (p > 1) for each ZQ in 3K.

The following lemma is elementary and we omit its proof.

Lemma 9.4.4. / / zi and Z2 ^ Lp(E, //) then \zi{x) — Z2(x) \ ^ Lp{E, ju).

If we identify equivalent functions and define the distance

Ip (^1, Z2) = lf\ zi (x) — Z2 (x) \P d/A

the resulting space is a complete metric space ( > 1)

liv

0.4 . The problem of P L A T E A U 3 7 9

L e m m a 9.4.5. Suppose z is of class Lp{E) for some measurable set E

in Rv, r > 1. Then there exists a function z defined almost every where on E

and coinciding with z almost every where on E such that

l im \m (^)]~^ / k (^) — ^ (ô) \dx = Q

for every XQ where z {XQ) is defined and every regular family of sets e G E

about XQ.

Proof. L e t {zn} be a dense subset of M. Since \z{x) — Z n\ ^ Li(E)

for each ^, 3 a set Z w i t h m{Z) = Oîi XQÊ — Z , t h e n

(9 .4 .11) " ^ ^' '"'

lim{2h)-^ f \z(x) — Z n\dx = \z{xo) — Z n\

l in i [m(e n E)lm(e)] = 1

for every n and every regula r family of se ts about XQ. It follows easily

t h a t i f Xo^ E — Z, then (9.4 .11) holds with Z n rep laced by any ZQ.

Definit ion 9.4 .4 . A ve cto r z ^ Cl{G) (G C Rv, 0 < ^ < / / ) <^ for

each XQ in G, t h e c o m p o n e n t s z^ with re s pec t to a coo rd ina te pa tch hav ing

range D z (XQ) are of c lass CJ in some neighborhood of XQ on G, A v e c t o r

z is absolutely continuous (A.C.) on [a, b] <=» for each £ > 0 , J ad ^ 0

s u c h t h a t

(9.4.12) 2\^K') - ^ K ) I ^ ^ w heneve r 2^ {^7 - ^d ^ ^>

and the in te rva l s (xp x^) a re d i s jo in t . W e s ay tha t z is of class Hl(G) ^ z

is continuous on G, ^ is A.C. in each x"" for a lmost a l l values of the other

va r iab le s , and the pa r t i a l de r iva t ive s \zôc\ ( i .e . the derivatives of the

arc - length a long x'^ = cons t , wi th respec t to x'^) ^ Lp(G).

Lemma 9.4 .6 . z is A. C. on [a, b] {of class Hl(G)) ^ for each XQ on

[a, b] (XQ on G), the components z^ with respect to a coordinate patch with

range D z(xo) are A.C. {of class Hi) in some neighborhood of XQ on [a, b]

{G). If z is of class Hl{G ) the quantities

(9.4.13) Go.p(x)=gij[z{x)]z%(x)zi^(x), \z,4x)\ = l/Ga«(^)

^ Lpi2 {G) and are independen t of the local (z)-coordina tes used. If z is of

class Hl{G) and x = x(y ) is a \ —- \ J^'-LIPSCHITZ map of F onto G, then

w(y ) = z [x (y)] is of class HI [F) and the derivatives transform in the

ususal way.

Nota t ion . For vec tors def ined on domains in R2, we us e the no ta t ionsE, F, a n d G ins tead of Gu, G12, a n d G22, o r p e r h a p s ,

(9.4.14) \z^\^ = E, Z cc'Z y = F, \zy\^ = G.

Definit ion 9.4 .5 . We carry over the definit ions of D{zi, Z2), or ien ted

a n d n o n - o r i e n t e d F R E C H E T va r ie ty , pa rame t r ic rep re s en ta t ion , e tc . a s

3 8 0 Pa ra m et r i c i n t eg ra l s ; t wo d i men s io n a l p ro b lems

given in Def in i t ions 9.1 .1 , 9.1.2, and 9.1 .3- For a vector z^ C'i(G), G of

class C^, we define

(9.4.15) A(z,G) ^liyW G'^^^J^dxdy, D{z, G) = fj (E + G) dxdy ,

G G

E, F, a n d G being defined locally by (9.4-13) and (9.4.14). Then, for

z^ CO(G), we define L (z, G) as in (9.1.11) and L (S) as in Definit ion 9.1 .5 .

Lemma 9.4.7. The theorems of §9 .1 generalize in the obvious way to

vectors z and surfaces S with ranges in W . We interpret the space H^{G)

n C^{G) to he our new space H\ [G). Lemmas 9.3 .1— 9-3.7 cmd their corolla

ries and Theorem 9.3-1 <^^^ hold for vectors z with ranges in 3Jl.

Definit ion 9.4 .6 . We define /(F), d{r), Z*(r ) , and i* ( r ) a s in De f in i

t ion 9 . 4 . 1 ; in the definitions of d[r) a n d d*[r)y we requ i re tha t theZn^H\[Gn).

Lemma 9.4.8 . (a) Lemma 9.4.1 holds.

(b) Instead of Lemm a 9.4.2, the following is true: Suppose ^ is A.C . on

dB(xo, R) with27 1

(9-4.16) f\C'{d)\^de<mln,0

Then H a vector H of class Hi [B {XQ, R) J which coincides with f on

dB [xo, R) for which2 . 71

(9.4.17) D[H,B[xQ,R)]<.K']\C{Q)\^dQ, K = Mjm.

0

(c) Instead of Theorem 9.4-1, the following is true: Suppose that F

satisfies the hy potheses of Theorem 9-4.1. Then H a circular region B of

type k and a sequence {2' } of vectors of class H\ (B) such that D {zn, B)

->d(F) and the restrictions of Zn to the properly oriented Ci converge uni

formly along Ci to continuous representatives of Fi, i = i, . . ., k.

Proof, (a) is evident except for the local regulari ty . But here again

w e m a y e m b e d W in the manifo ld Tl X R2, the metr ic in any loca l

coo rd ina te s ys tem be ing g iven by

ds^ = ZgiA^^' . - -, ^ ) dzi dz^ + m [(^^^+1)2 + (^^^+ 2)2],i, ? = 1

m = ( w + M ) / 2 ,

the re la t ion be tween any two coo rd ina te s ys tems be ing

'zi=f{z^,.. .,z^), i=\,...,N, 'zi==z\ i = N + \,N + 2.

the /* being defined as for W. Clearly W i s hom ogeneo us ly regula r w i th

t h e s a m e n u m b e r s m a n d M.

(b) Extend f to be per iod ic in 6 and choose ^o- There is a di w i t h

\6 i - 0 o| < 7 r suc h t h a t |C(l9) - C{6o)\ < \C(6i) - C(0o)\ for all (9.

9.4. The problem of PLATEAU 381

F ro m (9.4.16), i t follows t h a t

( 9 . 4 . 1 8 ) | C ( 0 i ) - C ( 6 > o ) | 2 < j r / | r ( < 9 ) P ^ ( 9 < w .

0

Now, s ince W i s homogeneous ly regula r , we may choose a spec ia l coord ina te pa tch in wh ich f (do) is the origin and

(9.4.19) ni\i\^<.gij(z)^i& ^M\^\^, z^B(0,\).

From (9.4.19), i t fol lows that for any path z^ = z^{s) (s a rc l eng th ) ,

I I

/ ]/gij [^(s)] 4 4 ^^ ^ y ^ / \zs\ds>:l]fm0 0

so tha t the range of th is pa tch conta ins a l l po in ts wi th in a d is tance ] /wof C(^o) and hence all the C{6). So, if we define H as tha t vec tor on

5 (XQ, R) w h o s e c o m p o n e n t s H^ with re s pec t to th i s pa tch a re the ha rmo

nic func t ions co inc id ing w i th C^(6) on dB(xo, R), we see , us ing Lemma

9.4 .2 , tha t

D[H ,B{xo,R)]=f jgij[H{x,y )]{HiHi + HiHi)dxdy

2 7 1 2 n

^ ^ 2 j l\VHi\^dxdy<Mj^\Ci\^dO^Kf\Ce\^de .* B(x,B) 0 * 0

(c) The proof of Theorem 9-4.1 carries over r ight down to the las t

p a r a g r a p h . T h e m a p p i n g s co^ are defined as follows: Since Wl is of class

C^, there ex is ts a coord ina te pa tch (Z ) in which ZQ corresponds to the

or ig in in the Z-space and gij (0) = dij. Then, for n suffic iently large we

define con as the iden t i ty ou ts ide the range of tha t pa tch and def ine i t a s

in (9.4.6) a n d (9. 4.7) in te rm s of th e Z coord ina te s in s ide the pa tch .

In order to complete the proof of Theorem 9.4.1 for surfaces in W,

the au thor found i t expedien t to in t roduce a theory of func t ions ofclass HI (G) with va lues in Wl. Since ^ is not a l inear space, some of the

theo rem s of cha pte r 3 do no t ma ke sense an d new proofs hav e to be found

for mos t o f the remain ing theorems . A usefu l too l i s the wel l -known

device , used ex tens ive ly by E. SILVERMAN, of mapp ing W i sometr ica l ly

in to the s pace m of a l l bounded sequences . Al though th is space is l inear ,

i t is no t separab le an d so th is dev ice can no t be used to prove a l l theorem s .

Remark. Many yea rs ago , H . W H I T N E Y [1] p ro ved th a t any manifo ld

W of class C^ could be mapped by a d i f feomorphism 0 of class C^ onto

an ana ly t ic manifo ld W in som e Rp (ac tua l ly P m a y b e t a k e n — 2N + 1).

I f 5K is a R iem ann ian m anifo ld a nd th is ma pp ing could be done in such

a w a y t h a t t h e r e w e r e n u m b e r s mi a n d Mi w ith 0 < Wi < Mi s u c h t h a t

t h e geodesic distances a long W a n d W satisfied

mi\p-q\ <\0{p) -0{q)\ <Mi\p-q\,

i t would then fo l low tha t a v e c t o r z(x) would be A.C. on [a, b] <=» X(x)

= 0[z(x)] is A.C. on [a, b] and we w o u l d h a v e

mi\z'(x)\ <\X'{x)\ <.Mi'\z'(x)\.

In th is case , it w o u l d be sufficient to define z to be of class HI (G) <=» 0 (z )

^ HI (G ); then the whole theo ry of these func t ions would be ava i lab le .

A c t u a l l y , J. NASH ([1], [2]) has proved that 0 can be t ak en to be isometr ic

(i.e. wi = Ml = 1) if P is suffic iently large. But all these fac ts require

proof and the l a s t - m e n t i o n e d t h e o r e m is somewhat d i f f icu l t . There

fore, in orde r to keep the expos i t ion re la t ive ly se l f -conta ined , we p r e s e n t

a n a l t e r n a t e d e v e l o p m e n t of the t h e o r y of HI spaces us ing SILVERMAN'S

idea and t e r m i n a t i n g w i t h the D IRICH LET growth Lemma 9 .4 .18 .

Definition 9.4.7. A vec tor func t ion X(x) (= {X^(x), X^(x), . . .})i n t o m is A.C. <^ (9-4.12) holds with 2: rep laced by X ; he re \X(^) — X((x)\

= sup \X^^) ~ X^oc)\. X^Lp(G), GC Rv, each X^ is m e a s u r a b l e

on G and \X{x)\= sup \XHx) \ ^ Lp(G).i

Lemma 9.4.9. Suppose X is A. C. on m and 0{x) is the length of the

arc X = X[t), a •<t <, X. Then 0 is A.C. on [a, b] and

(9.4.20) 0' {x ) = s u p I Xi {x) I a.e.i

If the X^ are A.C. on [a, h] and s\xp\Xl{x) \ ^ L\{\a, 6]), then X is A.C.

on [a, b].

Proof, (a) T h a t 0 is A.C. is i m m e d i a t e . T h e n if a <, x h-^\Xi(x + h) -Xi{x)\, ^ ' - 1 , 2 , . . .

f rom which it follows that 0' (x ) > s u p | X j . (x ) |. O n t h e o t h e r h a n d if thei

X^ are A.C. on [a, b] and sup |XJ , (A;) | ^ Li([a, b]), theni

\X^^) -X^oc)\ < f \Xi(x)\ dx < f sup \Xi{x)\ dxa a *

f rom which the absolu te cont inu i ty fo l lows eas i ly .

Definition 9.4.8. A vector function z from G d R into Wl is of class

Hl(G) ^ (i) z^ Lp(G), (ii) z is e q u i v a l e n t to a func t ion ZQ which is A.C.

in each var iab le x"^ for a lmos t all va lues of the o the rs , and ( i i i ) the part ia l

de r iva t ive s \zo,oc(x)\^ Lp(G). A vec to r X from G i n t o m is of class

Hl{G) <^ each X^^HKG) and if \X(x)\ - s u p | X ^ ( : ^ ) | and \X,4x)\i

= s u p I XJa (^) I € ^2? (^ ) • Let {zn} be a countab le dense subse t oiW; wei

define the m a p p i n g r from W i n t o m by r{z) = {\z — zi\, \z — Z2\,. . .},

Lemma 9.4 .10. (a) If z ^ HI (G), then z^Hl (D ) for each D C G.

(b) / / z^Hl{D) for each DCCG and z and \zo,oc\^ Lp(G), t h e n

z^HliG),

9-4. The problem of P L A T E A U 3 8 3

(c) The mapping r is isometric,

(d) A vector z into Tl is A.C. ^ the vector X = r z into m is A.C. A

vector z into m^H l{G) ^ X = r z^ HI(G).

(e) If z^Hl(G), X = x(y ) is a hi-Lipschitz map from F onto G, and

w{y ) = z [x (y)], then w^H l (F).

(f) / / z^H\[G), then z is equivalent to a function z which has the

absolute continuity properties of the ZQ in definition 9-4.8 and, in addition,

if X =^ X [y] is a hi-Lipschitz map ofF onto G and we define w[y ) = z [x(y )'],

then w also possesses those absolute continuity properties. In fact z may be

chosen as the function of Lemm a 9.4.5-

Proof, (c) Clearly \\z' - Z n\ -\z" - Z n\ = \Xn{z') - X'^{z")\

^\z' — z"\ for every n. B u t , b y t a k i n g Zj n e a r z' a n d zi n e a r z" , we see

t h a t \X{z') - X{z")\ = sup|X*'(;2') - X^(z")\. (a) and (b) are evident.i

(d) follows from (c) and Theorems 3.1.2 (g) and Lemma 3-1-1.

To p rove (e), i t is suffic ient to use (d) and then Theorem 3.1.7 for

e a c h c o m p o n e n t X^. If we let Y^{y) = X* [:v(y)], w e h a v e! y , « ( y ) I = s up I .Y > \x[y)-\ • xl{y) \ < M2'{sup | X\^ [. (y)] | } ,

M = sup I x ^ ^ (y) I.

Clearly each I Y a | € ^2)(-/").

Part (f) follows from (e) and Theorem 3-1-8 for each X^. The la s ts t a temen t in ( f ) mus t be p roved by re tu rn ing to the vec to r z in W a n d

repea t ing par t o f the proof of Theorem 3 .1 .8 .

I t i s exceedingly importan t to ob ta in the usua l formulas for the

der iva t ives in the change of var iab le theorem. We prove what i s neces

s a ry in the nex t l emma .

L e m m a 9 .4 .11 . Let z(x,y ) ^Hl(G), let z(x,y) be the equivalent func

tion of Lemma 9.4.10(f), let E,F, and G be defined by (9.4.13) and{9AAA)

in terms of the partial derivatives of the z^, let a : x = x{s,t), y = y[s, t)be a bi-Lipschitz map of F onto G, and letw{s, t) — z [x{s, t), y (s, t)]. Then

\ws{s, t) |2 = E{x, y ) xl + 2F Xs, y s + G yf

\wt{s, t)\^ = E(x, y ) xf + iFxtyt + G y ^ (a.e.),

\ws\'^ and \wt\'^ being defined similarly in terms of the partial derivatives

of the w^.

Proof. Since the express ions for E, F, G, \ws\^, a n d \wt\^ are in

dependen t o f the coo rd ina te s ys tems (z ) on W, we confine ourselves to

the range 9^ of an admiss ib le coord ina te pa tch . Le t 5 and 5 ' be therespective sets of {x, y ) and (s, t) w h e r e z(x, y) a n d w{s,t)^'^; c lear ly

S = a(S'). If z(xo, y) a n d z{x, yo) a r e A . C , a n d (XQ, y o) $ S , then H an

oc (XQ, y o) 5 {xo, y ) a n d {x, y o) $ 5 for all x an d y 5 | ^ — : o | < ^ an d

\y ~ yo\ <i(x^' Le t e > 0. Th ere exis ts a co m pa ct sub set ^ of 5 such t h a t

a l l the func t ions z^, 4 , a n d z^ a re def ined an d con t inuou s on ^ , w^, w],

3 8 4 Pa ram et r i c i n t eg ra l s ; t wo d i men s io n a l p ro b l ems

'^t> s, y s, ^t, a n d y t a r e a ll c o n t in u o u s o n ^ ' = (y~^(2^), ni(S — 2^) <Ce,

^ ^ ^ z^ (x + h,y) — z^ (x, y) = p j {x, y ) + s{ (x, y, h)] h

z^ {x, 3/ + A) — * [x, y ) = [z^ {x, y ) + 4 ( > y ^ ^)] ^

wh eneve r the po in t s invo lved a re in ^ ; we a ls o have

\4{^'y*^)\> \4{^>y'^^)\ < e W > Hmsih) = o, {x,y)^2!'

Now, fur ther , i f (XQ, y o) is no t in a sub set Z of ^ of m eas ur e 0 , th e l in ear

me t r ic dens i ty a t (XQ, yo) of th e p a rt of ^ on th e lines x = XQ a n d y = y o

i s 1 an d the l inear metr ic dens i ty a t (SQ, ^0) = or-i (XQ, y o) of the part of

^' on the l ines s = SQ a n d t = to is 1 . Consequently the projection of

a [^' (to)] on at least one of the lines x — xo oi y = yo has me t r i c dens i ty 1

a t {XQ, y o); he re ^ ' ( o) den ote s the p ar t of ^' on the l ine t = to. I f tha t ony = y o has th a t p ro pe r ty , the in te r sec t ion of ^ ( y o ) and the p ro jec t ion

^2/0 ['^{2' (^0)}] still has metric density unity at {xo, y o) and so the subse t

A of ^ ' (^ 0 ) s uch th a t Py^laiA)] l ies in the intersection above s t i l l has

m e t r ic dens i ty 1 a t (SQ, ^0). T he u sua l re la t ion s

^ l (5o , ^0) = zi(xo, yo) ^s(so, to) + 4(^0 , yo ) ys{so, to)

follow by le t t ing s ->so on zJ . The formula for Wt and hence thos e

in the theorem follow s imilarly . The lemma follows s ince e i s a rb i t ra ry .

Defin ition 9.4.9. A do m ai n G C i *' is said to be of class D' <^ it isthe union of a f ini te number of diffeomorphic images of r-s implices which

are joined together as are the s implices in a s implic ia l manifold.

L e m m a 9.4.12. Suppose z^H\[G), G being of class D\ Then the

function z of Lemm as 9A-^ oind 9-4 .10( / ) is defined almost everywhere on

dG and ^ Lp [dG ). If G is the cell [a, h], then

(9.4.21) lim (\z{x'',x^) —z{a'^,x^)\r>dx^ = 0, oc=\,...,v.

If ^ooci ^01, ni(Z o:) = 0, then the line x^ = x^^ intersects dG in a finite

number of points, at each of which d G has a tangent plane which does not

contain the line, and z{x'^, XQ^) is A.C. in x°^ on each closed segment of the

line in G.

Proof. In or der to pr ov e th e f irst tw o s ta t em en ts , i t is suffic ient t o

prove them for a cell [a, b], on account of Lemmas 9.4.10(f) and (9.4 .11).

In this case , choose oc a n d e x t e n d z across the face x"^ — a°^ by reflection

o b t a i n i n g ZQ w ith th e usu a l A.C. p rop er t ies . Th en ^0 is def ined a lm os t

eve rywhere on x^" = a"^ a n d fo(<^°', ^a ) = lim ^(: \;°', x^) a.e. Then (9.4-21)

follows from equation (3.4 .10) which reduces in our case to

(9.4.22) j\z{x^,x^) - ^ ( a ^ O • 2 ? ^ A ; ; < | A ; « - a « | 2 > - i | j \z,o.[x)\v dx.ai , ace cf'

9.4. The problem of P L A T E A U 3 8 5

The f i rs t par t o f the las t s ta tement (about d G) i s wel l known (see Lemma

9.2.2). The rest follows, since if x^^ is not in Za, a neighborhood % of a

p o i n t XQ = (XQ, %oa) on ^G can be mapped by a diffeomorphism of the

form

yi = < . y ''= -- /« ( ^ ^ 0. < € K , ^ a ]

onto a ce l l in such a way tha t dG O'Sl corresponds to a p lane y^"

— c on st .

The fo l lowing subs t i tu t ion lemma is an immedia te consequence of

L e m m a 9 . 4 . 1 2 :

Lemma 9.4.13. Suppose G and A are of class D' with A C G, suppose

z^Hl{G), w^H l{A), w{x) =z(x) a.e. on dA, and Z [x) ==w {x) on A

and Z{x) = z{x) on G —A . Then Z ^ Hl(G) and Z {x) — z{x) a.e. on

dA U dG.

Lemma 9.4.14. Suppose {z\ is a family of functions in Hl(G), G

Lipschitz, such that Dp (z, G) is uniformly bounded. Then

(i) if f \z{x) — ZQ\^ dx is uniformly hounded, so is ( \z{x) -— ZQ\^ dS)0 d o

(ii) if J \z{x) — zo\^ dx or f \z(x) — zo\^ dS is uniformly boundedX a

for some cell r dG or some open set G on dG, then (\z[x) — zo\^ dx is

uniformly bounded. ^Proof. Since G i s the un ion of a f in i te number of b i -Lipschi tz images

of cells , it is sufficient to prove this when G a n d a are cells . Then (i)

follows immediately from (9.4-22). Moreover, i f the integral over a (pa r t

of a face of dG) i s un iform ly bou nd ed , (9 .4-22) shows th a t the in tegra l

over a cell r ad jacen t to a is also . If T = [y, d], we see by apply ing

(9.4.22) with a = 1 th a t the in teg ra l ove r ri : a' <, x' < . b', x[^ [y [, d[]

i s un iform ly bo un ded . The resu l t (ii) fol lows by app ly ing th i s p roced ure

for a = 2, } , . . . ,v in t u r n .

Definition 9.4 .9 . We say tha t Z n tends weakly to z{zn -^ z) in Hl(G)

^ each Z n a n d z^ tI\{G), Dp{zn, G) i s un i fo rmly bo und ed an d Zn-^z

in Lp [G); he re we assume G of class C\. In case p =^ 1, we assume also

th a t th e se t func t ions Di {z^, e) are un iformly abso lu te ly cont i

n u o u s .

L e m m a 9 .4 .15 . / / G is of class C\ and Zn —7 z in Hl{G), then Z n-^ z

in Lp(dG). Moreover D p (z, G) < lim inf Dp{zn, G).

Proof. I t is c learly suffic ient to prove this for G a cell . To prove the

convergence in Lp of Znia^", x'„) to z{a°', x^, we le t \r) be a subsequenceof {yi) an d le t £ > 0 . W e ma y choose a subseq uence {z^ s u c h t h a t

Zs{^'^y â) -> ^( ^* , â) i ll ^v\.^'oi^ â] ^^"^ a lmos t a l l x"^. From (9.4-22) and

the weak convergence (and the un iform abso lu te cont inu i ty in the case

^ = 1), i t follows th a t w e m ay choose an x°^ s u c h t h a t Zs(x°', x'^)

->z(x^, x^ in Lp and the integrals on the left in (9-4-22) (with z ^ z ox

Z s and x'^ = %"•) are all < (e /3)^. I t follows eas ily that

{ \\zs{a'',x'^)-z{a'',x'^)\vdx\ <e, s>S(e).

U IT h u s Zsia"^, x^)->z(a°', x^) and, since {r} was any s ubs equence , it

fo l lows tha t Znia"", < ) - > 5 ( a « , < )

T o p r o v e the lower semicont inu i ty , choose a s ubs equence {zr} so

^p(>2^w, Q - ^ l i m i n f Z)j,(^^, G), and so th a t ^ r (^ ^ ^oa)-^^(^ '^> ^oJ inn

Lp [a"^, b^"] for almost all J Q^ , ^ = 1, . . ., r. In case

J\zr,oc(x'',XQ„)\^dx°'

ao c

i s un i fo rmly bounded for s ome s ubs equence , the conve rgence of Z r to z

a long the l ine is un i fo rm, so t h a t

(9.4.23) / I ^ , a ( ^ ^ A;;J l^ ^:^°^ < l i m i n f / I f r , « ( ^ ^ ^ ^ J |^ ^^°^;

of course thi s holds if the r igh t s ide = + 00. The result follows by integra

t ion wi th respec t to XQ^ and s u m m i n g on oc.

Lemma 9.4.16. Suppose Zn € Hl(G) for each n,p > \, G is of classCJ, and Dp(zn, G) and f f \^n{Xy y) — ^ o|^ dx dy are uniformly hounded.

G

Then a subsequence converges weakly on G to some z in H\{G).

Proof. It is sufficient to prove th i s for the cell G: a'<x<,b,c<,y

< d. Let Unic be the set of y s u c h t h a t Zn[x, y) is A.C. andh

( 9 . 4 . 2 4 ) / [\Zn[^. y) - Zo\P + \-Znx[x> y) | ^ ] dx < k.

a

T h e n m{Unk) '> {d — c) — Mjk, M be ing the obv ious bound . Let

N= l n=N fc=l

T h e n Ujc consis ts of all y such that (9 .4 .24) holds for in f in i t e ly many n

a n d U consis ts of all y for which (9.4.24) holds for in f in i t e ly many n and

s ome k. Clearly m(U) = d — c and if y ^ U, t h e n the Zn(x, y) are equ i -

con t inuous wi th |Zn{ x , y) — zo\ uniformly bounded ( for some subsequen

ce). T h u s we may find a s ubs equence zis converg ing uniformly on [a, b]

to Ci(^) for y = yi, yi be ing in the midd le th i rd of [c, d]. By r e p e a t i n gt h e w h o l e a r g u m e n t on the s ubs equence ^ i^ , we can find a second sub

sequence Z2 s conve rg ing un i fo rmly to C2 {x) ony = y 2 in the midd le th i rd

of [c, 3 1]. By con t inu ing th i s and t h e n t a k i n g the diagona l sequence ,

we ob ta in a s equence Zg s u c h t h a t Zs (x, yq) converges un iformly to s ome

tq[x) for each yq, yq being dense in [c , d].

9.4. The problem of PLATEAU 387

W e n o w s ho w t h a t {zs} is a C auc hy sequen ce. Le t e > 0. Fr o m (9.4.22),

i t follows t h a t th er e is a f ini te subse t T {s) of the y q suc h th a t if c < y 5 ( £ ) , q^T(e).a

Combining (9.4.25) and (9.4.26) for s and t, we f ind th a t

b

J \zs(x, y ) — Z t{x, y )\P dx < eP, c <y ^da

f rom which the resu l t fo l lows .

Lemma 9 .4 .17 . Suppose z ^ H\[B{pQ, a)], po = {xo,yo), with

rD[z,B{po,r)] ^k^(r), j Q - ^ k{Q) dq = K(r) < o o , 0 < y < a .

0Then Z(XQ, y o) is defined and if w(r, 6) = z(xo + r cosO, yo + ^ s in0) ,

then w is A.C. in r on [0, a] for almost all 6 and

1

\z[x, y) — f (A;O, yo) | < ^ / / [ ( 1 — t) X(i + t x, (\ — t) y ^ + t y ] dt,

0/(x, y ) = [E [x, y ) + G [x, y)] i /2 , r^ = [x - xo)^ + {y - yo)^

for almost all [x, y) .

Proof. Fr om th e change of var ia b le the or em , we conc lude th a t i f we

let L (r, 6) = l{xo + r cos0 , yo -\- r s in6) , then

L(r,d) > 0 , L^{r,e) == \wr{r,e)\^ + r-^ \we{r,d)\^.

I^efine ^ 2 .

h(r)=ffQy^L(Q,e)dgdO,0 0

T h e nh(r) <(2 j r ) i /2y i /2y^ ( r )

I f L{Q,e) dgdO = f Q-^l^ h'{g) dg0 0 0

r

^ r-il2h(r) + ~ f Q-^l^h{o) dg < [2n)^l^ [k{r) + K[r)] - > 00

as r ->0 , Thus , s ince L{r, 6) > lwr(? ' , 0) | , r~^ [weir, d)\, i t fol lows that

w is A.C. in y on [0, a] for a lmost a l l 0 w i t h w(0,6) in Li and there ex is ts

a seq uen ce r^ - ^ 0 5z^6i(^w, ^) - ^ 0 in Li s o tha t w(0,d) i s a cons tan t .

The result follows eas ily .

25*

Lemma 9.4.18 (DIRICHLET g r o w t h l e m m a ) . Theorem 3.5.2 holds in

the case _ = r = 2 and u replaced hy z.

Sketch of proof. For if {xi,yi) a n d {x2,y'^ are any poin ts in B{xo,y{i\

R), z{xi,yi) a n d z{x2,y2) are bo th def ined and the resu l t o f Lemma

9.4.17 may be used for (XQ, yo) = {xi, yi) a n d (X2, ^2) in tu rn and a lmos t

all (x,y). Then the proof of Theorem 3.5.2 carries over with \S7 u\

rep laced by L.

We can now prove our vers ion of the D I R I C H L E T p r o b l e m :

Theorem 9 .4 .2 . Suppose -2:* ^ Hl{G) and G is of class CJ. Then there

exists a z^H\{G) such that ^ = f* a.e. on dG and minimizes D(Z , G)

among all such vectors. If 9K is of class C^, then z ^ Q , {D) for every

D CLdG and every /^', 0 < // ' < 1. / / 9J is of class C ^ for some ^ > 3,

then z^ C^{D) for such D. IfW is of class C°° or analy tic, so is z on eachsuch D. If G is a circular region and 2:* is continuous along dG, then z is

continuous on G.

Proof. The ex is tence of a min imiz ing z i s im m ed ia te , u s ing a min imiz

ing sequence and Lemmas 9 .4 .14 , 9 .4 .16 , and 9-4 .15 . Now, suppose

B{xo, yo; a) C B and le t w{r, 6) = z{xo + r cos0 , yo + r s i n 0 ) . F o r

a lmos t a l l r, w is A.C. in 0 with \WQ\ in L2. Let

0 (r) = Z) \_z, B (xo, y o; ^ ) ] , ^ = m a x ( M / w , n d{z*, B)jm).

Th en , f rom Le m m as 9 .4 .8(b) a nd 9 .4 .13 . i t fol lows t h a t

0[r) <. Lr0'{r), 0(a) ^d(z*,B) =D {z,B)

for a lmost a l l r. H e n c e z satis f ies the hypotheses of Lemma 9-4.18 and

so z^ C^ (D) for any D G G B, The differentiabil i ty results then follow

from Theorem 1.10.4 s ince the cont inu i ty a l lows us to conc lude tha t z is

an ex t rema l fo r the in teg ra l I {z, D) w h e n e v e r D is small enough for

z (x, y ) to ^ the ran ge of a coo rd in a te p a t ch of Tl fo r {x, y )^D, where

f{x, y , z, p, q) = gij(z^, . .. ,zN) (pi pi + qi qJ).

Now , le t -B be a c i rcu la r dom ain an d supp ose (xo, yo) ^dB. Suppos e

t h a t R i s smal l enough so tha t B(xo,yo',R) in te r s ec t s no pa r t oi dB

except the c i rc le C conta in ing (xo, y o) and define

D [z,B(xo, y o', R)] == e^(R)l27 i, w(r, 6) ^z(xo-rr cosO, y o + r sin(9).

I t fol lows that there is an f s u c h t h a t

l\we(f,e)\^dd<D [z,B(Po,R)], Rle<f<R, po = (xo, y o) ^

BrtdB { P Q , r)

a n d z is A.C. a long B Pi dB{po, f) (so w is A.C. in d up to dB). F r o m t h e

ScHWARZ inequa l i ty and the cont inu i ty of z* a long dB, it follows that if

C* = f on 5 n dB(pOy f) an d = ^* on B(po,f) n dB, t h e n

(9.4.27) osc C* on d[Bn B(Po, r)] <e(R) + oj(R)

9.4. The problem of PLATEAU 3 8 9

w h e r e CD (R ) is the oscillation of z* on B(po, f) H dB. N o w , B pi B(po, f)

can be ma pp ed b y a conformal m ap on the c losed un i t d isc B(0, 1) so

th a t a g iven po in t pi of B O B (po, r) is carried into the origin. By

us ing maps of the form w — WQ = (z — zo)°' to get r id of the cornersand the n us ing The orem 9 .3 -^ we s ee tha t the re s u l t ing m ap is ana

ly t ic on the c losed domain except a t the corners . Le t f be the t rans

form of C* and Z be tha t o f z. T h e n Z i s c o n t i n u o u s o n ^ ( 0 , 1 ) ,

Z ^HI[B{0,\)], Z = C a.e. on dB(0,\), a n d Z min imizes D[Z\B(0,\)]

among a l l such func t ions . I f R i s smal l enough, D[z,B(po,R)] ^mjn so

t h a t

0{r) <Lr0'(r), 0 < y < 1 , 0 ( 1 ) <8^(R)l27i,

0(r) =D[Z,B(0,r)].

By the method of p roof of Lemma 9 .4 .17 , we f ind tha t

0(r) < (27r)-i e^(R) • (rlR)^f^ - k^(r), Iju = 1/L,

( 9 .4 .2 8 ) 2 jr _ 1 271

f\C{0) -Z (0,0)\de <I jL(r,d)drde <e(R)(\ +/^-^),

0 0 0

Since oscC = oscf* and (9.4.27) and (9.4-28) hold we see that

m a x IZ (0 , 0) - C(^) I < £ (^) (1 + /^-^)l2n + e(R) +a) (R).e

Since Z(0, 0) = ^(pi) a n d pi w a s a r b i t r a r y , t h e two dimensional con

t inu i ty o f z a t {xQy yo) follows.

W e can now p rove ou r ma in theo rem.

Theorem 9 .4 .3 . Under the hypotheses of Theorem 9.4 .1 , H a vector z of

class H\{B), B being a circular region of type k such that

D [z, B) = d[r) a n d L (z, B) = l(r);

z is (generalized) conformal and satisfies the boundary conditions as inTheorem 9.4-1. The differentiability results for z are those of Theorem 9.4.2.

Proof. L e t {zri] be the min imiz ing sequence of Lemma 9-4-8 (c ) . Then

a subsequence converges weakly in H\ (B) to a vec tor Z which satis f ies

the bounda ry cond i t ions and i s con t inuous a long dB. From the lower -

semicont inu i ty , i t fo l lows tha t D(Z,B) <d(r). From Theorem 9 .4 .2 ,

i t fol lows that H a z^H\{B), z = Z a.e. on dB, a n d z min imizes D{z', B)

among a l l such z' . Moreove r z is continuous on B and so ^ H\[B). Con

s e q u e n t l y

d{r) ^D(z,B) ^D (Z , B) < d(r)

s o t h a t z is our des ired solution. Since d{r) <2l(r) a n d D{z',B)

> 2 L{z\ B) for every z'^ B[\{B), i t fo l lows tha t z i s conformal .

Remark. Z is a lso a min imiz ing func t ion and so has the proper t ies

of z.

9.5. The g e n e r a l t w o - d i m e n s i o n a l p a r a m e t r i c p r o b l e m

We f i rs t p resen t the author ' s s impHfica t ion ( M O R R E Y [17]) of the

existence proofs of C E S A R I , D A N S K I N , and S I G A L O V referred to above.

We sha l l assume tha t the in tegrand func t ion sa t is f ies the genera l assumpt i o n s o n / g i v e n in §9.2 (just before equation (9.2.12)) and , in add i t ion

the cond i t ion

(9.5.1) M '\X\ >f{z,p) {>m\X\ by (iii), 0 < m < A f ) .

W e reca l l tha t if iV = } (and v = 2) t h e n f{z, p) = 0(z, X) and the

cond i t ions ho ld if 0 ^ C^ for X :^ 0, 0 convex in X, a n d 0 satisfies

(9.5.2) m\X\ < 0 ( ^ , X ) < M | X | , X^ = [p^ q^ - p^ q^),

Z 2 = (^3 ql _ pi qS) ^ x^ = (^1 q^ ~ p^ q^).

We shall confine ourselves to the cons ide ra t ion of surfaces of t y p e k

w i t h k = 1, i.e. of the t y p e of the disc .

Definit ion 9.5 .1 . If F is an orien ted c losed F R E C H E T c u r v e , we define

(9.5.?) d(r) = inf j l im inf ^f(zn, Bi)}\ Bi = B(0, 1)

for all s equences {z^} which conve rge un i fo rmly on ^ i to a r e p r e s e n t a

t ion of r.

The fo l lowing lemma is i m m e d i a t e :Lemma 9.5 .1 . (a) If Fn-^ F in the sense of Frechet, then

d(F) < l i m i n f ^ ( A ) .

(b) If d[F) <ioo^ ^ a sequence {J!^} 5 ^cich Fn is regular and of class

C°^, Fn -^F, and d(Fn) -^d(F). If F is a Jordan curve, the Fn ntay be

chosen to be regular Jordan curves of class C°°.

(c) / / F is a regular Jordan curve of class C^,

d[F) = mi^f{z,B{)for all locally regular z^ C ' ^ ( 5 i ) for which the restriction of z to dBi

furnishes a representation of F.

R e m a r k . In orde r to p r o v e the loca l regula r i ty in (c) a b o v e w i t h o u t

increas ing the n u m b e r of d imens ions N, which wou ld requ i re the e x t e n

s ion of/, one can begin by a p p r o x i m a t i n g by p o l y h e d r a Fin w i t h b o u n d a

ries Fny s p a n n i n g Fn and /^ by a piecewise regu lar C°^ ba nd , and t h e n

r o u n d i n g off the ve r t i ce s and edges .

I t is c o n v e n i e n t to h a v e at our disposa l the ' ' d o m i n a t i n g f u n c t i o n ' '

0 defined in the fo l lowing lemma. I t is not neces s a ry for the exis tence

t h e o r e m ; the rough func t ion W defined by

See Equation (9.1.15) and Definition 9.1.4'.

9.S. T h e g en e ra l t wo -d imen s i o n a l p a ram et r i c p ro b l em 3 9 1

i s suff ic ien t fo r tha t purpose . However , i t y ie lds a s imple proof tha t the

solu tion ve cto r is (generalized) co nfor m al a nd is a lso helpful in prov ing

the h igher d i f fe ren t iab i l i ty resu l ts once the so lu t ion vec tor i s shown to

^Cl(D)ioTDCCG.

Le t u s s uppos e tha t f{p, q) is of class C^ when the {p, q) m a t r i x h a s

rank 2 and suppose we perform a ro ta t ion of axes in R^ and define

'pi = Ciip. Y = Ciiqi, 'f{p',q')=f{p,q).

Then we obs e rve tha t

(9.5.4) = 2 ' y^' Jiii^ + ^fîJiiy + Ji^P^) + /^ ' Jm '^' ' 1 ^

T hu s if / i s w eak ly qu as i -co nvex in {pyq), J is also in i^p, 'q). If {pQ, qo)

i s any poin t where the {p, q) m atr ix ha s ran k 2, we m ay pe r fo rm a ro ta

t ion of axes so tha t

(9.5.5) %-'qi = 0 for . • > 3 , 'Pl'ql-'Pl'ql> 0.

Let us consider the set of a l l i^p, 'q) satisfying (9.5-5) and let us drop

the p r imes . The re la t ion (9 .L3) becomes

(9.5.6) f{^p+pq, y P + Sq)={ocd-^y )f(p,q), ocd-^y >0.

Differentia t ing (9-5.6) with respect to p^ a n d q^ and so lv ing , we obta in

MP>Q) = dfpi - y U, MP , Q)^- ^fpi + ocU

fpipi(P,Q) = A-^[d^fpipj — y S(fpiqi +fqipo) + y'^fqiqi]

(9 -5-7) fpiqj(P>Q ) = ^~^ [— pSfpipJ + Ocdfpiqj + ^yfqipj — OCy fqiqj]

fqiqi{P,Q) = ^~^W ^fpip^ — 0C^(fpiq7 + fqipi) + OC^fqiqf\

P = ocp+Pq, Q = y p + dq, A=ocd-^yy U==fAP>q)> e tc .

T h u s , if w e le t î = (1 , 0, . . ., 0) , ^2 = (0, 1, . . ., 0) a n d de fin e

/^ r- ON fp^pj(^ly ^2) = îj, fpiqj{ei, ^2) = hjy fqiq^(^l> ^2) = Cfj,

fpi{ei, 62) = di, fqi{ei, 62) = e\, f(ex, ^2) = ^0

we find that, on the space (9-5-5)

f{p.q) = h^{PH^-pH ^), fpi=dtq^--e[q\ f^,= -d^p^ + e,p\

Qif) = K-îiaiAq^^-P^H)^ - (hj + hi) {q^^-P^fA X(9.5.9) X ( ^ u - p^ fi) + cij (^1 A-p^/^)2] ii iJ

K - ]lE G-F^ =.piq^ ~ ^ 2 ^ 1 .

Q{f) bein g th e form in (9-5.4). T hu s th e form Q necessar i ly degenera tes

an d has ra nk < iV — 2 in f fo r a ll (A, / / ) with X^ -^^ fjfi = 1.

9.S- The general two-dimensional parametric problem 3 9 3

A s t r a i g h t f o r w a r d c o m p u t a t i o n le a d s t o t h e r e s u lt t h a t

Q(0) = (1 + CO - TO ),) (A2 + /<2) If |2 + T-2(Tft), - h CO^) X

X D-i(GX^-2F?./A + Efi^)\d\^ + (o„X^ +

- f 2 T - l K f t - T-1 COft) X y + T-2 Wftft r 2 + 7-2 o)^ D - 1 X

(9.5.14) ifX 2" [«jj e^ - (^« + *j«) e <^ + c«; (T2] I< f

Q = Xq^ — jup^, a = Xq^ — [xp^.

X and y are seen to be b o u n d e d and defined ii D ^ 0, x i=^ 0, and the

l a s t t e rm in the e x p a n s i o n for Q (0) is b e t w e e n

(9.5.15) miT-^conW and Mir-^wuW, W = D~^(GX^ - 2FXju +

Clear ly we may a s s u m e

(9.5.16) 0 < m i < 1 .

Accord ing ly , we w a n t t h e r e to exis t an W2 > 0 s u c h t h a t

(9.5-17) l + c o — T c o ^ > m 2 > 0 , r co ^ — hcofi'> ~ niicoji.

W e mus t a l s o have

(9.5.18) 0 = D{\ + co) ^f=rhD, co(\,h) = h - \,

the equa l i ty ho ld ing on ly if T = 1 (corresponding to E = G, F = 0),

W e no t ice a l s o tha t

(9.5.19) (l - ] / l - T 2 ) | | | 2 < T F < ( I + 1 / I - T 2 ) | f | 2 .

We def ine

CO(T, h) = [\ + {h -2)T](p{u), u = {h- 1) T/[1 +(h — 2) r ] ,

(9.5.20) 0 < T < 1 , 2 < A , T = fi/[(A — 1) — (A — 2) w],

T h e n it fo l lows tha t u increases from 0 to 1 as T does , for each h, and

co^=[[h — \) — {h— 2) u] cp'(u) + (h — 2) f [u)

r co^ — o) = u (p ' (u) •— (p (u), coh = r[(p (u) -\ - (i — u) cp' (w)]

(9 21) "^<^r — (^ — ^ i ) oiji = T{[(2 — wi) — (1 — mi)] 99'[u) —

— (2 — w i) 99 [u]]

0)^^ (Dnh ~ {corh — T-i co^)2 = 0

0)^^ = [{h — 1) — (A - 2) w] (p"(u) (duldr) > 0 <^ " (^) > 0.F o r T = 1,(1 +co) — T h = 0 and

^ [(1 + co) — T A] =- co^ — /j < 0 ^/ ^ " {u) > 0

A ^ ^ = [( / ,_ 1) _ ( / , _ 2) ^] (^"(^) > 0

3 9 4 Pa ra me t r i c i n t eg ra l s ; t wo d i men s io n a l p ro b l ems

SO 0)^. takes i t s max. for r= \ where i t s va lue is

co,{\,h) = q)'{i) + h - 2 .

But for T = 1, we must have ((9.5-17))

(jo^{\, h) — co(1, A) < 1 — m2 (p'('i) +h — 2<h — m2

^^•^•^^^ (p'{ ) <2-m2.

T h u s a> ^ <,hand so (9.5-18) holds. The first inequality in (9-5-17) holds

if the last one in (9-5-22) does. If 99"(^) -= 0for ^ < (1 — w i )/ (2 — w i )

a f te r which (p" (u) > 0, a l l the above hold and rco ^ — {h — mi) cohÔ

so we see tha t

(9.5.23) (p'W > 2 - m i .

So we take 99 = 0for 0 < w < (1 — wi ) / (2 — mi), choose cp^C^ [0, 1]

and ana ly t i c wi th q)" [u) > 0 for (1 — wi ) / (2 — m-i) < w < 1 and so

that 99(1) = 1, 2— w i < 99'(1) < 2 — ^ 2 w h e re 0 < m2 <mi.

Definition 9.5 .3 . W e define the in teg ra l

G

J(Z, Bi) =fj(\zaâ^\^ + 2 1 2,12 + \zyy\^) dx dy,

By.

Remark. W e n o t e t h a t

I(z, G) =^f{z, G) ^ z Hl(G) a n d z i s conformal .

L e m m a 9.5.3. Suppose F is aregular Jordan curve of class C°^ and

z =C(p), p^dBi is aregular representation of F of class C . Then, for

K sufficiently large, theclass g(K ) of vectors z H\ {Bi) O HI {Bi) for

which

(9.5.24) J{z. Bi) < K, z{p) =C (P ) for p= (1 ,0), ( - i , ± ^ ,

is not empty . For each such K, there is a vector ZK if i 3{K) which minimizes

I (z, Bi) among all z in (K). Moreover

(9.5-25) d{F) =\im I(zK^Bi).

Proof. The f i r s t s t a temen t is evident s ince there is a loca l ly regula r

Z $ ^ ^ ( ^ i ) su ch t h a t z =Z(p), p^dBi, gives a r e p r e s e n t a t i o n of F;

the th ree po in t condi t ion may be secured by performing aMobius t r ans

format ion , the t rans form of z being in C^ (Bi). Now le t {zp} be a min imizing sequence for / in Q (K) . Then the no rms o f the Z p in Hl(Bi) are un i

fo rmly bounded s o tha t the Z p a re equ icon t inuous on î by Sobo lev ' s

le m m as (see § 3-5)- T hu s, asubsequence converges weakly in Hl(Bi) a n d

uniformly on 5i to some func t ion Z K in HI {Bi). Since th is convergence

impl ies weak convergence of the s econd de r iva t ive s and s t rong con -

9.5- The general two-dimensional parametric problem 3 9 5

vergence of the f i rs t der iva t ives in L2(Bi), we see t h a t Z K is admiss ib le .

Since / and / are bo th lower - s emicon t inuous (/ involves only f irs t

de r iva t ive s ) , ZKis a min imiz ing func t ion . The las t s ta tement fo l lows

from the fac t tha t we may select a sequence {zp} of regular surfaces

of class C^, each bounded by F, such that ^f{zp,Bi) -^d(r); each of

the s e may be rep re s en ted con fo rma l ly by some admiss ib le vec tor z*.

Lemma 9.5.4. For each K, Z K satisfies the condition

(9.5.26) D [zK,B(Po. R)] <.D[ZK,B{P,, a)] - g ) \ X = ^ ,

0 <R <a,

for every circle B (Po, a) C Bi. Fhus the ZR satisfy (if K is large enough)

a uniform Holder condition on EA, which depends only on A, m, M, andd{F), for each A <. \ and are equicontinuous along dBi. There exists a

vector z^ Hl(Bi) such that z = z(p), p^ dBi, gives a representation of F

which satisfies the three point condition in (9.5-24) and for which ^/(z, Bi)

= d(F); the vector z satisfies the Holder condition satisfied by the Z K on

each BA' Moreover z is [generalized) conformal.

Proof. The e q u i c o n t i n u i t y on each BA follows from the f irs t result

a n d the wri te r ' s ' 'D i r i ch le t g rowth theo rem" (Theorem 3 -5 .2 ) . The equ i

con t inu i ty a long dB\ follows from a wel l known lemma of C o u r a n t(Lemma 9.3.2) s ince we h a v e D(ZK, BI) un i fo rmly bounded and h a v e a

th ree po in t cond i t ion .

To prove (9.5-26), we use (9.5-12) and the m i n i m i z i n g p r o p e r t y of Z R

t o s h o w t h a t

'-mD [ZK, B (Po, R)] < / (ZK. B ( P Q , R)] <I[Z,B ( P Q , R)]

(9-5.27) .

<^^-MD[Z,B(Po,R)], (m = 2)

w h e r e Z is the b iha rmon ic func t ion s uch tha t Z — z^^ HIQ[B(PO, R)],

We sha l l omit the rou t ine jus t i f ica t ion of the following formal calcula

t i o n s .

L e t (r, 6) be polar coord ina tes wi th po le at Po and let

ZK (r, 0) = — ^ + 2J [^n (r) cosnO + bn(r) sinnO], 0 < r < ^,2 n= l

Z(r,Q) =^~- + 2^[An(r) cosne+ Bn(r) sinnQ], 0<r<R.

Since Z is b iha rmon ic and has the s ame Di r ich le t da ta as 2: on 5 (Po, R),

we ob ta in

/ r \n ,y\n+2 /yxn /y\n + 2

w h e r e Cn, dn, %, and fn are cons tan ts def ined by

2Cn = (n + 2) ocn— ^n, 2dn = ^n — nan,

and s imila r formulas ho ld for en a n d fn for ^ > 0. T h u s if we set

W{R)=D[z,B{Po,R)],

equa t ion (9 .5 .27) and a computa t ion of D [Z , B(Po, R)] s h o w s t h a t

W{R) <LD[Z,B{Po,R)]R oo

0 n-l

<2LRW{R) , 1 = — .

The result (9.5-26) follows.

Thus f rom the Z K, we may ex t rac t a s ubs equence {zp} which con

verges weakly in Hl(Bi) and un i fo rmly a long dBi and on each EA w i t h

^ < 1 to some z^H\ (EA) for each A <i \. Since b o th / an d 3 / ^ire

lower semicont inuous wi th respec t to th is type of convergence , (Theorem

4.4-5) we conclude that

^f{z,Bi)^I{z,Bi)<d{r).

But suppose we def ine ZR = z on ER a n d Z R to be the harmonic func

t ion on Bi -— ER which co inc ides wi th z on dBiU SBR. Since each Z R

i s con t inuous and ^/(ZR, BI) -> Sf(z, Bi), e tc . , we f ind tha t

d{r) < ^f(z, Bi) < I(z, Bi) < d{r).

From th i s we conc lude tha t z i s conformal (on the in te r ior) . From the

lower-semicont inu i ty of / on each domain D of type k, we ob ta in

I {z, G) = l im / (zp, G).

B u t now , f rom th e lower semic ont in u i ty of Z), we h ave

1 1

-mD(z,G) < - m lim inf D (zp, G) < lim / (zp , G)(9.5.28) ^ ^ ^'"^'^ ^"''^

= / ( ^ , G) <I(H,G) <^MD{H,G),

H being the usua l harmonic func t ion . The cont inu i ty of z at points of

dBi now follows as in the proof of Theorem 9.4-2. The result (9.5-28)

g ives the Holder cont inu i ty of z on in te r io r doma ins .

W e can now p rove the ex i s tence theo rem:

Theorem 9.5 .1 . Suppose F is a Jordan curve for which diT) < 00 .

Then H a continuous vector z^H\ (Ei) such that

(9.5-29) ^f{z^Bi) = d(r).

Moreover z is conformal.

9.5- The general two-dimensional parametric problem 397

Proof. Choose {Fn} so that each Fn is regular and ofclass C^ and so

that Fn ^ rand d(Fn) ->d{F) (Lemma 9.5-1). Choose regular representa

tions z =^n(p), P^dBi, ofclass C of the Fnsothat the Cn(p) con

verge uniformly on dBi toC{p) where ^ = f (^) is atopological represen

tation of F. For each n, let Znbe the minimizing vector ofLemma 9.5-4.

From Lemmas 9-4-2 and 9-5-4, weconclude that a subsequence, still

called {zn}, converges uniformly along dBi and oneach BAas in the

proof ofLemma 9.5-4. Byrepeating the argument in the last part of

that proof, we find that z{= limzn)^ Hl(Bi), that (9.5-29) holds, and

the z is conformal.

We conclude this section with the results ofK I P P S mentioned above.

In fact, we generalize his results somewhat.

Lemma 9.5.5. Suppose pand q H\ [^(Po, < )].

(a) Then, foralmost all r, 0<Cr <i a, pand q[Lemma 9-4-5) < ^ A.C.

on dB(PQ, r) with pe and qe^L^y < ^

2 / / ipxqy — pyqx)dxdy = J {pdq— qdp).B(Po,r) SBiPo.r)

(b) If, -also, jrand x^Hl[B[PQ, r)] for such anr with jt— p and

K-q^H\^[B{Po,r)-\,then

J J {jtx^y — JtyXx) dxdy = J J (px y— pyqx)dxdy.B{Po,r) B{Po,r)

(c) If p=z Zx and q=Z y for some z^ Hl[B(Po, a)], then qx = Py a.e.

on B{Po, a).

Proof, (a) and (c) areproved byapproximating by the mollified

functions. To prove (b) write n = p-\- TIQ, K=q+XQ', then TCQ and XQ

may be approximated strongly in HI [B (PQ, r)] by functions

^ C [B (PQ, r)] and p and qmay beapproximated similarly by func

tions ^ C°° [B(P,r)].

Lemma 9.5.6. (a) The conformal minimizing vector zof Theorem 9.5-1

satisfies

(9.5.30) fJiCifv +Clfai +C'fzi)dxdy=0, C^Hl,(B,).

" Bx

(b) / / also, pand q^H\\B{PQ, a)] for some B{PQ, a) C CBi, then

fpi, fqi, and fzi ^HI [B [PQ, a)] and z satisfies

(9.5.31) Lr f fpr + jar-fzr=0 a.e. on 5 (PQ, ).

(c) Foralmost all [x, y) where p =q=0, we have fp==fq=fz =

and the derivatives of p, q, fp, and fq vanish a.e. onthat set. Moreover p

and qsatisfy the relations

(9532) P 'P^ — ^ ' ^ =q 'Px +P'q^ =P ' py — q'qy

= (l'Py+p-<ly = a.e.

3 9 8 P a r a m e t r i c i n t e g ra l s ; two dimens ional p rob lems

(d) If N = } and we define X = p x q and f{z, p, q) — 0[z, X), then

^fpr + -^-U+fzr = kr[{Ap^ + B Py + B q^ + C qy)' k - HD],

(9.5.33) k = \X\-^X, A=0xQXop^p'. B = 0xQXop^q^,

C = 0XQXO q^q\ H = 0XQ ZQ

for almost all (x, y) where X 0, i.e. {p, q) ^ (0, 0).

Proof, (a) follows since z min imizes the n o n - p a r a m e t r i c i n t e g r a l /

a n d so satisfies (9.5-30) with / rep laced by 0. Since 0 > / e v e r y w h e r e

w i t h the equa l i ty ho ld ing a long our so lu t ion ,

(9.5.34) <>pi= fpi, 0qi = fqi, 0zi - fzi a.e.

a long our solution. Since 0 is homogeneous of degree 2 and the d e r i v a

t ive s 0ppy 0pq, and 0qq are c o n t i n u o u s and b o u n d e d if {p, q) ^ (0, 0)

a n d the de r iva t ive s 0pz and 0qz are con t inuous eve rywhere , we easily

s ee tha t fp = 0p and fq = 0q HI [B(Po, a)]. We have seen e lsewhere

(§. }A) t h a t the de r iva t ive s of ^, q,fp,fq, and/^ = 0 a lmos t eve rywhere

on the set w h e r e ^ = q = 0, The relations (9-5.32) follow by dif fe ren t ia t

ing the conformal i ty re la t ions .

Th e re la tions in (9.5-33) follow by s imp ly ca r ry ing out the differentia

t ions in (9-5-31) and us ing the form of / : Since qx = py, we see, for

a l m o s t all {x, y) where X i 0, t h a t

Lr = %rsp% + "^rsiPl + q%) + ^rsty + "^rsp' + ^rsq' - 0ZT

^ r s = (pXQX<^ Xlr X^s , S&rs = (fXQX*^ X r X^s , Krs = (pXQX^ X^r XJs ,

5)r5 = q^XQzs Xlr , ©rs = (fXQzs X^r, {f(z, p,q) ^ (p (z, X)) .

Us ing the confo rma l i ty re la t ions , one easily sees that

k^X^r^ -p^, krXlr= - qK

Since

prXlr = qrXlT=X^, pr Xlr = ^^ Zgr = 0,

the re s u l t s in (9.5-33) now follow easily from the h o m o g e n e i t y of 99 as a

func t ion of X,

O u r m e t h o d is to set up a n o n - p a r a m e t r i c i n t e g r a l of the type d i s

cussed in § 5.4 with unknown 3-vec tor func t ions n and x which is m i n i m i

zed by t a k i n g [n, K) = (p, q). The Holder cont inu i ty then fo l lows f rom

the resu l ts of tha t s ec t ion .

Definition 9.5.4. Given a confo rma l vec to r z ^H\ (Bi) w i t h p and q

also $ HI [J5(Po, ^)], B{Po,a) C ^ 1 . We define

(9.5.35) I* [^, ^, B(Po, a)] = j J W(x, y, n, x, Jix, ^x, ^y> >Cy) dx dy

JB(Po.«)

9.5- The general two-dimensional parametric problem 399

where , for X i^O, i.e. [p, q) i=- 0,

W=\ny- K:c\^ + 1 X | - 1 {[p-n^-q- K^)^ + [q ' n^: + p ' x^)^ +

+ {P'Tly- q- HyY + {q'ny+p' Hy)'^] + J [(A ' Tlx + B ' Jty +

+ B ' Kx + C ' Ky ) ' k — H(p ' n -\- q- >c)j2\'^ — 2l{7ix ' Ky — Tiy - Kx)

+ \n-p\^ + \K-q\^, k = \X\-^X, / = (2 - ]/2)/(1 + W ^ i )

(9.5.36)

w h e r e A, B, C, and H are defined in (9.5-33) and mi and Mi refer to

(9.5-9). If ^ -= == 0, we define

W = m2{\7Zx\^ + \7iy\^ + \xx\^ + \xy\'-) + ITT -P\^ + \X- q\^,

(9.5.37) m = (W2 + M 2 ) / 2 , m2 = I mijMi,

M2 = la rger of 2 + ]/2 — / an d / Mijmi.

Theorem 9 .5 .2 . HWQ denotes the sum of all the terms in W which are

qtiadratic in jix, 7 i y , Xx, d'yid Xy, then

(9-5-38) msd ^P + \7iy\^ + \xx\' + \xy\^) < ^ o < M2{\7ix\^ +

+ \ny\^ + \^x\^ + \>cy\^) > m2 + M2^2.

Suppose that z is the minimizing vector of Theorem 9-5A and that p and

q^Hl[B(PQ, a)], [B(PQ, a)] C Bi. Moreover the pair of vectors (7c,x)

==(p,q) minimizes I'^[7i,x, B(Po, a)] among all such pairs (7 z,x)^H\[B{PQ, a)]^n — p and x — q^H\Q[B{PQ, a)]. Thus, p and q

^ C^ [B (PQ, R)] for each R < a. Moreover the Holder condition depends

only on the bounds m, M, mi, and Mi and not on any bounds or moduli of

continuity for z, p, and q other than those holding by virtue of the minimiz

ing property of z. If 0 is of class Q ( C ' ^ , or analytic) for \X\ ^ 0; then

so is z away from the locally compact subset where p ==z q = 0,

Proof. At po in t s whe re {p, q) ^ 0, let us define the n.o. set {i,j, k) by

i=\P\-lf, j=\q\-U. k=\X\-^Xa n d let us set

Ttx = ocii + ^ij + yik

ny = a2i + ^2 j + y^k

T h e n , for s uch po in t s .

Wo = {oc2 - ocs)^ + ih - iffs)

(9.5.39) + ( 3 + i5l)2 + (^2 - i^4)2 -

+ By2 + Bys+ C y^)^ + I [(<X2 ocs — ai 0C4)

+ (y2 73 — 7 1 7 4 ) ] .

A s u rp r i s ing ly s imp le compu ta t ion s hows tha t the cha rac te r i s t i c roo t s

are (some are mult ip le roo ts )

1 ,1 + 2, —l + 2±y2, 2 — 1, lAjC, and I CIA,

Xx

Xy

{h

= ocsi

= a i

- 73 )2

(^4 + hY

-Oil

+ PsJ

+ ^d

' + ( 1

^ AC

^4)+(i^2/?3-

+ 73^

+ 74^.

~N' +

( ^ 7 1 +

-PiN +

4 0 0 T h e higher dimensional plateau problems

Since the la t te r two a re be tween ImijMi a n d I Mijmi, (9.5-38) follows.

In case TZ = p a n d x — q, we conclude from Lemmas 9.5-5 and 9-5.6

t h a t

(9.5.40) 'P=-2l{Pcc'qy-py'qx) (A ) (0,0).

F o r o t h e r {n, x) we see tha t

(9 54 ^ ) ^ > -2l[nx'Ky -Tly'Kx),

/ * [TZ, X, B(Po, a)\> — 21 j J (px ' qy — py' qx) dxdyB(Po,a)

s ince (9 .5-40) and the f irs t inequali ty in (9 .5-41) hold a lmost everywhere

w h e r e [p, q) = (0, 0) or, respectively, (jr, x) ^ (0, 0 ) . H ence t he min imiz

ing s ta te m en t fo llows .F ina l ly , if z is the min imiz ing vec to r , we know tha t {p, q) ^L2 (Bi)

and also satis fy aDirich le t g rowth condi t ion ofthe form

/ / {\P\^ + \q\ )dxdy D[z,Bi)'[rjh) -i \ 5= 1 - |OPi|.B{Fi,r)

T h u s t h e i n t e g r a n d W satisf ies th e co ndit ions in § 5 .4 wh eth er pa n d q a re

cont inuous or no t . The remain ing resu l ts fo l low.

Chapte r 10

The higher dimensional plateau problems

10 .1 . In trod u ct ion

Unti l recen t ly , no genera l resu l ts had been obta ined concern ing the

exis tence and/or d i f fe ren t iab i l i ty of the so lu t ions of p a r a m e t r i c p r o

b l e m s inmore than two var iab les . The grea tes t s ing le s tumbl ing b lock

was the non -ex i s tence of a useful generalization of a c o n f o r m a l m a p to

higher d imens ions . Now, by imi ta t ing the proof ofthe au tho r ' s o ld con -

fo rma l mapp ing theo rem ( M O R R E Y [3]) , one can prove that a " n o n -

d e g e n e r a t e ' ' F r e c h e t v a r i e t y of the topologica l type of the I'-ball (i.e. a

Freche t var ie ty which possesses a r e p r e s e n t a t i o n on^ ( 0 , 1 ) in w h i c h

n o c o n t i n u u m iscarr ied in to a point) which possesses a r e p r e s e n t a t i o n

of class Hl[B{0,i)] possesses such a rep re s en ta t ion wh ich min imizes

f \v z\^ dx a m o n g all s uch . Howeve r , one can notconc lude tha t5(0,1)

the va lue of th i s in teg ra l <, C • L[z,B{0,\)'] or e v e n t h a t L [z,B[0,\)']

is given by the a rea in teg ra l for such a rep re s en ta t ion . So the me

thods which had been success fu l inthe two d imens iona l p rob lems d id

not lead to resu l ts in the h igher d imens iona l cases .

10.1. In t ro d u c t i o n 40 1

Almos t s imul taneous ly , resu l ts on the r -d imens iona l PLATEAU p r o b l e m

with V > 2 were obtained by D E G I O R G I [2], R E I F E N B E R G [1], F E D E R E R

and F L E M I N G and F L E M I N G [2]. D E G I O R G I proved that a portion of

m i n i m u m a r e a of a p a r t of the ^^-dimensional boundary of an open set in

(v + 1)-space is a regula r ana ly t ic manifo ld . R E I F E N B E R G p r o v e d t h a t if

A is any compact point set in R;^, t h e r e is a compact point set X in R^

w h i c h is b o u n d e d by ^ in a certa in sense (see Definit ion 10.2.6) which

min imizes A*{X — A) a m o n g all such X and which has the a d d i t i o n a l

p r o p e r t y t h a t e a c h p o i n t ^ of ^ — ^ , not in a re la t ive ly compac t s ubs e t

Z oi X — A w i t h /[*' (Z) = 0 , is in a ne ighborhood on X — A w h i c h is a

topo log ica l r -d i s c. R ecen t ly ( R E I F E N B E R G [2] and [3]) he has p r o v e d

that these topological i^-discs are in fac t ana ly t ic . Very recent ly , s ince

de l iver ing the Colloquium Lec tures before the A m e r i c a n M a t h e m a t i c a lSoc ie ty in Augus t , 1964 , the a u t h o r ( M O R R E Y [21]) has found tha t the s e

re s u l t s may be carr ied over to se ts X and ^ in a R i e m a n n i a n m a n i fo l d

of cons iderab le genera l i ty (see be low) . A lmos t concu r ren t ly , F E D E R E R

and F L E M I N G ( F E D E R E R [2], F L E M I N G [2], F E D E R E R and F L E M I N G ) have

approached th i s p rob lem us ing the i r in teg ra l cu r ren t s and have ob ta ined

re s u l t s more or less comparable wi th those of R E I F E N B E R G . Since F E D E

R ER is w r i t i n g up the i r resu l ts in book form, we sha l l p resen t the a u t h o r ' s

ex tens ion to R i e m a n n i a n m a n i fo l ds of his s implif ication of the w o r k of

R E I F E N B E R G .

S u p p o s e t h a t 9Jl is a R i e m a n n i a n m a n i fo l d w i t h o u t b o u n d a r y of

class C**, w > 2. Let P o € ^ - It is wel l known tha t the re is a (non -un i

que ) coo rd ina te pa tch r of class C^h a v i n g a d o m a i n c o n t a in i n g the origin

s u c h t h a t

(10.1.1) hi^{0) = dij, hiJy;Jc(0) = 0,

t h e hij be ing the c o m p o n e n t s of the metr ic t ens o r wi th re s pec t to T . T O

ob ta in s uch a m a p p i n g , let co be any c o o r d i n a t e p a t c h of class C^ w i t hrange con ta in ing PQ ; we may c lea r ly a s s ume tha t its d o m a i n D in the

y-s pace con ta in s the origin. By l e t t i n g T be a proper ly chosen l inear

t rans fo rma t ion f rom the y-space to the <2:-space, we may a r r a n g e t h a t

gij(0) = dij, the gij be ing the c o m p o n e n t s of the metr ic t ens o r in the z

c o o r d i n a t e s y s t e m . We o b t a i n r by l e t t i n g

(10.1.2) wi = zi + a)T^z^z^

w h e r e the a)^ are c o n s t a n t s to be de te rmined . D i f fe ren t ia t ing the re la t ion

wi th re s pec t to z^, s e t t ing 2 = 0, and using (10.1.1) and (10.1.2), we see

t h a t the a | j . m us t sa t is fy

(10.1.3) «5ft + «b = ^«M(0).Morrey, Multiple Integrals 26

4 0 2 T h e h i g h e r d i men s i on a l PLATEAU p ro b l ems

The reade r may ea s i ly ve r i fy tha t we may take

Definit ion 10.1 .1 . A coord ina te s ys tem co of class C^ (at leas t) is a

normal coordinate system centered at a p o i n t qonWl^ t h e d o m a i n G of

0} conta ins the or ig in , a> (0) = q, gij (0) = dij, t h e gij b e i n g t h e c o m p o n e n t s

of the metr ic tensor wi th respec t to co, th e a rcs on 50 wh ich corr espo nd t o

s e g m e n t s in G t h r o u g h 0 are arcs of geodes ies , and the d is tance a long

s uch s egmen ts equa l s the corresponding d is tance a long the geodesic

a rcs .

L e m m a 10.1 .1 . Suppose W. is of class C^. Then

(a) Each point q of W is the range of a normal coordinate system of

class C^ centered at q,(b) If 0)1 and co2 are two such sy stems both centered at the same point q

and both having the domain B{0,R), then OJ^^ co i is the restriction to

B (0, R) of an orthogonal transformation.

(c) If oy is such a system and gij(z) are the components of the metric

tensor with respect to oj, then

gij(tX) ^n^ = \ if 2* W ^ = ^ ^^^ 0<t<R;(1 0 .1 .4 ) ^=1

gijzJc(0) = 0 for all i,j, k.(d) Given a point PQ and a mapping r of class C^ satisfying (10.1.1),

there exists a number Ro'> 0 and a family cop of normal coordinate

sy stems, one o)p being centered at each point p in B (PQ, iô/3) [on W) and

having the domain B(0, R Q), such that the vectors in the tangent space to

at p which correspond under ojp to the unit vectors ei, . . ., e^ in RN CLY^

obtained from those corresponding to î, . . ., ^jv under x by the Gram-

Schmidt process. If p and q^B(Po, i?o/3), the mapping

(10.1.5) ^=U{y;p,q)ôj-ncop{y)]is of class C2 for {y, p, q) ^G x 3 1 X "SI where G = i5 (0, Roj}) C RN ^ ^ ^

W=B{Po,Rol3).

Proof. Clearly part (a) follows from part (d) . Part (b) is evident s ince

t h e r e is a un ique geodes ic pa s s ing th rough a g iven po in t and hav ing a

given d i rec t ion . The f i rs t equa l i ty in (10.1.4) follows since the l ines

z^ =z X^t correspond to geodes ies and t is the arc length a long the geodesic

a n d

(10.1.6) (dsldt)^=gij'{tX)Xn^^ 1.By d if fe ren t ia t ing th is wi th respec t to t a n d 2fi we ob ta in

gijzic{o)xnn^ = 0 (aiu)( ^- ' ^ gijzk (0) A* A* + 2 l im ^ - i [gjcj (t X) - djcj] X^ = 0

for all A. T h e seco nd r esu lt in (10.1.4) follows from (10.1.7).

10.1 . I n t r o d u c t i o n 4 0 3

To p rove (d) , we f ir st n o te th a t theE uL ER equa t io ns fo r min im iz ing

the l eng th in teg ra l become , if t is a param eter propor tional to arc length

(10.1.8) " ^ "^U == :^^^^[h}wTc + hikwj — hjkwi]'

So, le t PQ and r be g iven as s ta ted . From the ex is tence theorem for

sys tems of o rd inary d i f fe ren t ia l equa t ions as appl ied to the sys tem

(10.1.8), we see tha t the so lu t ion func t ion IX (^; y, z) of the sys tem

nU(t;y ,z)+HUm t;y,z)]nin^ = 0

U**(0; y , z) = z\ Ui(0; y , z) == a}(z) yj

where the vec to rs a}{z) ei are the vec tors in RN co r re s pond ing unde r rto the vec tors in the tangent space a t ^ = T (Z ) which a re ob ta ined f rom

thos e co r re s pond ing unde r r to ei, . . ., e-t^hy the Gram-Schmid t p roces s .

Tha t i s , the a}{z) a re un ique ly de te rmined by the cond i t ions tha t

(10.1.9) hij{z) ai(z) al(z) = djcu 4 ( ^ ) = 0 for i > k.

I t i s easy to see tha t IX ^ C^{Q) for some open set Q containing (0; 0 , 0)

a n d t h a t

(10.1.10) Vi{t',ry,z) == Vi[tr]y,z) so Vi{t]y,z) = X X ( l ;/ y , ^ ) .

If we define

(10.1.11) U{y, z) = 1X(1; y, ^ ) , a>^(y) = r{Uly, r-^p)]}

we see tha t 17 ^ C^ in a dom ain of th e typ e des i red and t h a t t he cop

sa t is fy the condi t ions s ta ted .

Remark. Of course IX is of class C^ in t and, s ince the 7^^ ^ C^ (and, in

ge ne ral , no th in g m or e ca n be sai d if 3)1 is on ly of class C^), we see tha t

th e fi rst a n d sec on d p a rt ia l d er iv at iv es of XX, XX , a n d XI ^ w it h r es pe ct to

y a n d z are of class C^ in (t, y, z). Since the hfj are of class C^, so are the

a}{z).

W e now s ta te ou r gene ra l a s s umpt ions on Tl:

Genera l assumptions on 501. We assume that W is a separable Rieman-

nian manifold without boundary of class C^ and that there are positive

numbers RQ, CQ, f]o, KQ , and K i, which are independent of PQ, such that

each point PQ of Wt is in the range of a coordinate system r of class C^ which

satisfies (10.1.1), has domain B(0, 4Ro), is such that T (0 ) = PQ, and for

which

2 (\V^ hij(w)\ + \V^ hij(w)\) < Ko.

We suppose also that RQ, C O , and KQ are so related that

(1 + Cor^)-^<hij(w)2,n^ <(i + Co/'2)2,

(10.1.12) Nr=\w\, 2 ' ^ ^ = ' ' . (1 + l 6 C o iR g )2 < 5 /4 .

26*

4 0 4 The h igher d imens ional PLATEAU p ro b l ems

With each PQ, we associate a definite r as above, and w e associate a family

{cojj} of normal coordinate sy stems related to T(PO) as in Lemm a 10.1.1;

we assume that the number RQ mentioned there is the same as that mentioned

above. Instead 0/(10.1.12) we assume that, for each fixed p, cop satisfies

(1 +r]or)-^^gij{y;p)?i'X3'^(\ +r)or)^, 2 " W = ^

^^^'^'^^^ | r ; f e ( y ; ^ ) A V ^ > ^ l < ^ o | A | - | / ^ | ^ 0^\y\Lr^Ro,

the gij {y ; p) being the ojp components of the metric tensor and the Fjj^ {y , p)

being obtained from the gij by the corresponding formulas (10.1.8).

Finally, we assume that the second gradients of the function U (y , p, q) in

(10.1.5 ) are uniformly bounded by Ki for each PQ and all {y y p,q) in

Remarks 1. It i s p robable tha t many of the resu l ts can be carr ied

ove r to man ifolds of c lass C^ w ith n <i 4 ( 0 < / / < l ) . How ev e r , a

re laxa t ion of the d i f fe ren t iab i l i ty requirements appears to necess i ta te a

d i f fe ren t method of proof. I t is shown in § 10.7 that the topological ^-discs

(on t h e m in im iz in g set) a re of class C^ for a n y //, 0 < // < 1, if 3}l is of

cl as s C4, a re of cla ss CJJ if W is of class Q w it h n '> 4, and are of class

C^ or ana ly t ic i f M is .

Remarks 2. I t i s seen tha t any compact manifo ld of c lass C^ satisfies

all the cond i t ions .Lemma 10.1 .2 . Let co be a normal coordinate sy stem with domain

B(0, RQ) and suppose that the gij {z) are the components of the metric tensor.

Then, writing z = [z^, % ) , we obtain

(a) gNm{z^, 0)=dNm, ^ ^ | < i^O i

(b) the angle on W at p between an arc y through p and the geodesic

from q = (0(0) to p is the same as the angle at y = CJO~^ (p) between co~^ (y)

and the line 0 y.

Proof. S e t t i n g z^ = tX^ in (10.1.6) and us ing the fac t tha t l inesthrough the origin are solutions of (10.1.8), we obtain

(10 .1 .14 ) gij(z) zizJ= \z\^, r;j^{z)z3 z^~0, i= \,. . ., iV.

S e t t i n g 2:* = 0 for * < iV an d usin g th e form ulas for F^^, we ob ta in

( 1 0 . 1 . 1 5 ) gNN[z^> 0) = 1, 2gNizN[z^, 0) - gNNzi[z^, 0) == 0.

Differentia t ing (10.1.14) with respect to z'^, s e t t ing z^ =^ 0 for i < N,

and then us ing (10 .1 .15) , we a rr ive a t the equa t ion

^^^[^^? i^m(^^ ,0 ) ] = 0

which leads to (a) , (b) a long the z^ axis follows from (a) and along any

line through the origin follows by a rota t ion of axes .

Lemma 10.1 .3 . For each ?y > 0, there is an RQ with 0 <, RQ <. Rol3,

which depends only on W (i.e. RQ, C O , TJO, KQ , and Ki) and rj and which has

1 0 . 1 . I n t r o d u c t i o n 4 0 5

the following property: Suppose 0 < R <, RQ, B{p,r) C B(q, R), W is

a normal coordinate system centered at q , andy ' is any point on a>~i [dB{p, r)],

then

BiCco-^[B(p,r)]GB2where Bi and B^ are the halls in R^ of respective radii r{\ + ^ ) ~ - ^ ( ^ ^ ^ ^

r[\ + rj) whose boundaries are tangent to co~i [dB{p, r)] at y ' .

Proof. 'Letp' = c o {y'). Since our resu l ts a re una l te red by or thogona l

t rans fo rma t ions in R^, we may s uppos e tha t c o = c o q , a n d t h a t ojq, Wp,

an d a>^, are all re la te d to a single r a s in Le m m a 10.1.1 and o ur g ene ral

a s s u m p t i o n s . L e t a be the non -homogeneous l inea r t r ans fo rma t ion on

R N which carr ies y ' in to the or ig in and which oscu la tes a)~} M Q a t y '; w e

w r i t e a in the formz = (T{y), a{y') = 0.

The angle a t the or ig in in the ^-space be tween two a rcs which in te rsec t

there is the same as the angle a t p' between the a rcs which correspond

u n d e r t h e t r a n s f o r m a t i o n c o q a~^. Also l ines in the y-space correspond to

those in the 2:-space.

L e t y be the d i rec ted a rc in B (0 , r) which s ta r t s a t the po in t W Q

— oj~^(p') ^dB(0 , r) and is such tha t G a)~'^a)p(y) i s a segment s ta r t ing

a t 0 and mak ing an ang le 6 < 7r /2 wi th the inner normal to a a)~^cop[dB{0,r)]. From Lemma 10 .1 .2 , i t fo l lows tha t y makes the angle 6

with the rad ius vec tor f rom W Q to 0. Since U(y ; q, q) = y and the second

gradien ts o f U ( inc lud ing mixed ones ) a re bounded by Ki, we see tha t

( 1 0 . 1 . 1 6 ) ( i + i ^ ^ ^ ) - 2 < | i M < ^ ( 1 + i f , 7 ? ) 2 ,

w be ing the coo rd ina te s in the cop s y s t e m .

Now, le t us in t roduce the Euc l idean metr ic in the ^-space and le t

JV/ N X 7 (>2^ ^z^

Then the a rc y is a so lu t ion of the equa t ions

(10.1.17) - _ + r 'o , ,( t. ) — ^ = 0

w h e r e t h e F^jj^ are defined as usual in terms of the gQjcj a n d t is the dis

tance in the 2:-space. From our assumptions i t fol lows that

( 1 0 . 1 . 1 8 ) \ r i , , ( w ) x i , x i i ^ ' ^ \ < z ^ { m ) \ x \ - \ i , \ \

Now def ine

v{t) = [d{Pt,pW = Z (wi)^.

4 0 6 T h e h i g h e r d i men s io n a l PLATEAU p ro b l ems

Let t ing do ts denote d i f fe ren t ia t ion wi th respec t to t and using (10.1.17),

we ob ta inN

v{0) = r^, v{0) = 2 2Jw^{0) w^O ) = —2r \w(0)\ cos ^

(10.1.19) '^^v(t) = 2 2! (ze'«)2 — 2^! rijj, (w) w^ w3 w^,

Since t is distance in the ,2:-space, (10.1.16), (10.1.18), and (10.1.19) yield

r^-2{\ +Ki R)^tr cosd + {\ + Ki R)-^ (1 - Z i R) t^ <v(t)

(10.1.20) < ; ' 2 _ 2(1 +KiR)-^trcosd + (1 + J ^ i 7^)4(1 + ZiR)P,

1 - ZiR>Q.

B y le t t in g ^* be th e f irs t posit iv e valu e of t for which v{t) — r^, we findfrom (10.1.20) that

2 (1 + K i J R ) - 6 ( 1 + Z iR)-^rcose <t*

(10 .1 .21) ^ ^ ^ ^ +KiR) {\ -ZiR)-^rcosO.

This p roves the theo rem wi th a)~^ [B(p, r)] rep laced by a co~i [B(p, r)].

Call the balls in the ^-space B[ a n d B^^ Their respec t ive rad i i a re

(10.1.22) (\ + Ki R )-^ (\ + Zi R)-^ ' r a n d (\ + KiR)^ (\ - Z iR)-^-r,

a n d w e h a v e

a-HB',)Cco-nB{p.r)]Ca-HB',)

where a~^ (B[) a n d a~^ (^g) ^^^ el l ipsoids with min. and max. radii

a-^(B[): (1 + Ki R)-» (1 + Z i R)-^ r an d (1 + Kx R)-^ (1 + Z i J R ) - i r ,

cy- i (^2) : (^ + Ki R)^ (1 - Z i i? )- i y and (1 + Ki RY (1 - Z i i ? ) - i r ,

respec t ive ly ; these e l l ipso ids a re tangent to

co~^[dB{p, r)] a t co~'^{p') = y ', If R is sufficien tly s m all, th e ellipsoids

B\ and ^g ^^^ nearly balls and balls B\ a n d B^ can be chosen wi th

B\ C ^ i a n d B^ D ^g^ which ha ve resp ectiv e r adi i (1 •\- r\)~^Y a n d

(1 -^ Y()Y and wh ich a re t angen t to o\^\dB(p, r)] a t y '. T h e l a s t s t a t e

m en t is p ro ved s imi la rly .

Defin ition 10.1.2. Given a set 5 , we define (5 , ^) as th e set of all

po in ts wi th in a d is tance ^ of S ; i . e .

( 5 ,e ) = U S ( P , e ) .

If S\ and 52 are compact sets , we define their point set distance D (Si , S2)as the s ma l le s t number Q s u c h t h a t Si C (5*2, Q) an d 52 C (5 i , ^ ) .

Definit ion 10.1.3 . ^ geodesic k-plane centered at a point P 0 / 9K is a

locus of the form colliD B(0 , R Q ) ] where 77 is a ^ -p lane in R^ t h r o u g h

0 and f t) i s a n orm al coord ina te sy s tem w ith do m ain B (0 , RQ) for which

co(0)=P.

10.2. V surfaces , their b o u n d a r i e s , and their H A U S D O R F F measures 407

Lemina 10 .1 .4 . Suppose co is a normal coordinate sy stem with domain

B (0 , i^o) ^^^ range B (PQ, RQ) for which co (0) = P Q . Let P be a point in

B(0, R), let Si and S2 he compact subsets of B{0, R), and let a){P) = Q

and w (Sjc) = Tjc, k = 1,2, 0 < J R < RQ. Then

(1 + i?o R)-^ d{P, Si) < d(Q, Ti) < (1 + Tyo R) d{P, Si)

( IO . I.23) (1 + rjo R)-^D(Si, S2) < Z ) ( r i , T2) < (1 + >;o R) D{Si, S2)

(1 + rjo R)-^Q{Si) <Q(Ti) < (1 + 70 R) Q{SI)

where Q(S) denotes the radius of the smallest sphere containing Si [compact).

Proof. If Si cons is ts of a s ing le po in t , the first line of (10.1.23) follows

from our assumption (10 .1 .13) . There is a p o i n t Pi in 5i s u c h t h a t

d{P, Pi) = d{P, Si). If we let Qi = co(Pi) , then

d{Q, Ti) < d(Q, Qi) <.{\+f)o R) d(P, Pi) = {\ + rjo R) d{P, Si).

The o the r inequa l i t i e s are proved s imi la r ly .

We conc lude th is sec t ion wi th the fo l lowing lemma:

Lemma 10 .1 .5 . There are constants RQ = RQ (W) <, Roj} and K2

= K2 (W) with the following property : Suppose that ^ is a geodesic k-plane

centered at a point p^ B{q, R) where 0 < P < RQ. Suppose co is a normal

coordinate system with domain B (0 , PQ) for which co (0) = q and suppose

^0 is the k-plane in P^v which is tangent at 0)~^{p) to co~^(^). Then

co-i [ZnB (q , P ) ] C (Zo, K2 P2)

co-^[i:r\B{p,r)](Z[Zo,K2r^), r ^ RQ .

Proof. If the coord ina te s in the s y s t e m co are w and the F^j^ are

defined as us ua l , the arcs in B (0 , PQ) co r re s pond ing to the geodesies are

so lu t ions of the system (10.1.17) with F^^j^ rep laced by F^j^. Since the

geodes ies making up ^ H P (q , R) co r re s pond to a r c s t a n g e n t to ^ 0 at

co-^ip) and s ince ^ fl P ( ^ , P) C ^ Pi P ( ^ , 2P) so t h a t no such arc is

of length > 2 P , the result follows from the assumptions (10 .1 .13) . The

las t s ta tement fo l lows s ince the second der iva t ives of t h e t r a n s f o r m a t i o n sco^^ cooq, e tc . , are u n i f o r m l y b o u n d e d .

1 0 . 2 . V su rfaces , th e ir b ou n d ar ies , and th e ir Hau sd orf f m eas u res

In th is sec t ion , we define the r -d imens iona l Haus dor f f meas u re s of

se ts in a metr ic s pace and prove ce r ta in theo rems abou t the s e meas u re s .

Mos t of the l e s s s t anda rd theo rems p re s en ted he re are due to R E I F E N -

BERG [1]. W e also in tro du ce our no tio ns of r -surfaces and the i r ' ' a lgebra ic

b o u n d a r i e s ' ' . M o s t of the topologica l resu l ts needed for the r e m a i n d e rof th is chapte r are p r e s e n t e d in the following section. The p r e s e n t a t i o n

the re g iven makes ex tens ive u s e of the gene ra l theo ry as pre s en ted in the

book ''Foundations of Algebraic Topology'' by E I L E N B E R G and S T E E N -

R O D , espec ia l ly Chapte r 1, §§1 — 14, Chap te r 9 , §7, and C h a p t e r 10,

§§2 and 5.

Definit ion 10.2 .1 . Suppose 5 is a se t in a metr ic space and tha t v > 1

an d 6 > 0. If S is em p ty , we define *^ ^ (S) == 0; oth erw ise, we define

*Al (S) as the inf. of ^ yv rl for a l l coverings of 5 by finite or countablei

families of balls {B (Pi, r^)} in w hic h ea ch n < d. We then def ine theHausdorff outer measure *yl*' by

(1 0.2 .1 ) * yl''(S ) = l i m * / l S ( 5 ) .f5-»-0+

A'-measurahle sets are th e n defined as us ua l. If r = 0, we define */lg (S)

as the inf. of the number of balls of radii < d required to cover S a n d t h e n

define *ylo(5) by (10.2.1).

The fo l lowing lemma is wel l -known and we omit the proof:

L e m m a 10.2.1. (a) Borel sets are measurable A^ for each v.(b) Any set is measurable A^ and A^{S) is the number of points in S.

(c) / / 5 is a set in a metric space ^ and r : S -> ^' is a mapping in

which W is a metric space and d[x{p), r[q)'] <.2 .d{p,q) for p and q on

S, then

*A'[r{S)] <A» '* /L^(S) .

Definit ion 10.2.2 . Su ppo se G is a dom ain on a m anifo ld X of class C^

and s uppos e r ^ C 'i(G) whe re r i s a m app ing in to the R iem ann ian

manifo ld Tl of class C^. We define

L,(T) =jF{p;r)dS(p)G

where if cr : F ->G a n d co : Q -^Tl a re coo rd ina te pa tches wi th doma ins

FG Rp a n d Q C RN and ranges con ta in ing p a n d T {p ), respec t ive ly , we

h a v e

F{p\T)dS{p) = [y [x)']-^i'^dx, y {x) =det(y ap{x))

dS(p) = fe(^)]i/2 dx, g{x) = de t(g.^{x))

w h e r e Gij(z) a n d gocp{x) are the components o f the metr ic tensors on W

a n d X with re s pec t to the coo rd ina te pa tches w an d cr, resp ectiv ely, an d

z(x) = o)~^T a(x).

Lemma 10.2 .2 . Suppose that G, X, 90^, and r have their significance as

in Definition 10.2.2 and suppose that r is a diffeomorphism. Then

Lp(r)^A-[r{G)].

This i s p roved by f i rs t p roving the formula

lf[T(P)] •F[j>;r] dSip) = ff{q) dA'iq)G r(G)

for / ^ C'^[r{G)]. This i s p roved f i rs t fo r func t ions wi th suppor t in a

smal l ne ighborhood of a g iven poin t . The proof in tha t case is e lementar5 \

10.2. V surfaces, their boundaries, a n d their H A U S D O R F F measures 4 0 9

Definit ion 10.2.3 . A fa m ily g of sets T is said to cover a set S in the

sense of Vitali iff e ach p oi nt P of 5 is in a set T of g of ar bi tr ar il y sm all

d i a m e t e r .

Definition 10.2.4. Let g be a family of sets T. By (7(g) , we mean the

un io n of all th e se ts T for T ^ g . In ca se f is co un ta bl e an d g = {B{Pi, ri)}

we s t i l l use the no ta t ion

am = UB{P i,ri).

L e m m a 10.2.3 . (cf. M O R S E , A . P . [2]) Suppose that W satisfies the

conditions o/§ 10 .1 . Suppose that S is a bounded set and that % is a family

of closed halls in 9}l which covers S in the sense of Vitali. Then there is a

countable disjoint sub-fam ily {B(Pi, ri), i = 1,2 , . . .} such that

S -U B(Pi,ri) CU B(Pi, Sn), k = \,2,...

Proof. W e s uppos e tha t 5 C B{PQ, ro) and le t ^' be the family of a l l

ba l ls B(P, r)^^ s u c h t h a t B{P, r) C B(Po, ro) a n d B(P, r) 0 S is

not empty. Then g ' s t i l l covers 5 in the sense of Vita l i .

We now def ine the rad i i Ri, the ba l ls B {Pi, ri), and the families î b y

ind uc t ion as fo llows : iî — sup r fo r B (P, r) ^ ^\ B (Pi, ri) is a ba ll of g '

in which ri > iî/2 , an d g i consis ts of tho se balls of g ' w hich do no t inte r s ec t B(Pi, ri). Having de f ined Ri, . . ., Rjc, B[Pi, Vi) for * = 1, . . ., k,

and gi , . . . , %jc, we define Rjcî = snpr for B(P, r) ^ ^jc, B(Pjc+i, rjc+i)

as a ball in ^jc for which rfc+i > Rjc+il^, a n d ^k+i as those balls in Â;

which do no t in te r s ec t B{Pjc+i, rjcî). W e s ee by induc t ion t ha t an y ba l l

in g ' — g i mu s t lie in B(Pi, Sri) and , in genera l , any ba l l in ^jc — ^jc+i

must l ie in B(Pjc+i, Srjc+ij^ By induc t ion , we conc lude tha t the B{Pi, r^)

a re d i s jo in t and tha t

a(^jc)CUB(Pi,Sri)GUB(Pi,Sri) if j < k + \ ,i=Jc+l i= j

Now, s uppos e P^S — \J B(Pi,ri). Then, s ince the f ini te union isi = l

c los ed P ^ s ome ba l l B{P',r) which does no t in te rsec t any B[Pi,ri)

w i t h i < k. T h u s B{P', r) ^ % k a n d he n ce P ^ a{%ic). The result follows

since

5 - u5 ( P ^,n) = n S-\jB{Pi,ri)

Definit ion 10.2.5 . Suppose 5 is a se t , P is a point , and /7 is a ^-plane,

a l l being in Rjsf. We def ine the cone C ( P , S) to consis t of a l l the segments

P Q ioT Q in S. We define C(77, S) to consis t of a l l segments P Q w h e r e

P ^S a n d Q is its projection, on 77. If M satis f ies the condit ions of § 10.1,

CO i s a normal coord ina te sys tem with or ig in a t P and domain conta in ing

4 1 0 T h e higher dimensional PLATEAU problems

C(P,S), we define the geodesic cone C(P', S') == (o[C (P, S)], where

P ' = co(P) ^nd S' = a)(S).

Theorem 10.2.1. Suppose S C RN, S is a Borel set, and A^~^(S) < oo .

Then,for any P,

^^ [C (P , 5 )] < r - i r / l " - ! (5) if S C B{P,r).

The proof i s s imi la r to bu t s impler than tha t o f the fo l lowing theorem

an d is left to the rea de r.

Theorem 10.2.2. Suppose S C RN> A^~'^{S) < oo, and IT is a p-plane

with p <. N. Then

AnC(n,S)}^Ci{v)'r'A^-HS) if S C {n,r) ^ U B{P,r).

pen

Proof. The theo rem i s ev iden t ii v = \. So we supp ose r > 1 . W ecover 5 with a countable family of balls {B (Pi, r^)} such th a t

(10.2.2) 2y v-i^r^<^'"'^{S) + £ ' n<d, i=\,2,... ,i

e a n d 6 being a rb i t r a ry pos i t ive num be rs . Le t P^y , 0 < y < ( r — ri)lri

be the po in t on the s egmen t I (Pi), jo in ing Pi to i t s p ro jec t ion Qi o n / 7 ,

which is a t a d is tance j Ti from P^. li P^B (Pi, fi), we see tha t

/ ( P ) C UB(Pij,2ri).1

Accord ingly , s ince j < rJTi, we ob ta in

Al, [C (77 ,5)] < 2 ^ y^i^^i)' ^ ' ' (2" Y'lY'-i) • ^ y - i rV'

<c[(v)-r-[Ar'{S)+e].

The result follows from the arbitrariness of d a n d e.

Theorem 10.2.3 . Suppose 9)1 satisfies the conditions o/ § 10.1 , S is

A^-measurable with yl*'(5) < oo , 5 C G C 501 where G is open, and

U^C ^(G), S/U ^ 0 on G and U satisfies a Lipschitz condition with

Lipschitz constant M. Then

oo/ y l ^ - M ^ n Ch) dh<MA''(S),

—ooCji being the subset of G where U (P) = h.

Proof. I t i s suff ic ien t to prove th is for 5 compact ; then \VU(P)\

> c > 0 for P on 5 . L et e a n d 6 be smal l pos i t ive numbers and cover S

with a family {B (Pi, ri)} of ba l ls such tha t2r^n<A',{S) + s, B{Pi,n)(lG. {ri<d).

i

Now, for each i,

Ar' [Sh n B (Pi, n)] < y._i [Qi (A)]-i, Su==sncn,

w h e r e Qi(h) is the radius of the smalles t c losed ball containing Ch

OB (Pi, ri). This in te rsec t ion is non empty on ly for

U(Pi) -[ri'\V U(Pi) I + 0(d)] < A < U(Pi) + [n I V U(Pi) I + 0 ( ^ ) ] .

10.2. V surfaces, their boundaries, an d their H A U S D O R F F measures 4 1 1

F o r s u c h h,

Qi{h) = ir^,-h'\ A' = [| V U{Pi) i + 0 (a ) ] - i • [h - U[Pi)].

T h u s

JAr^Su) dh<Z jl^^ U(Pi) I y ,_ i [Qtih)]"--^ dh' + 0(d)

^MZyvrl + 0(a) < MAl(S) + 0{d) + e.

i

The result follows.

Definition 10.2.6. A 7^-surface is merely a compact set (in W) X. I n

case ^ is a co m pa ct sub set of X a n d v ^ \, we define the algebraic

boundary b{X, ^ ) of X with respect to A (more p rope r ly b[X, A,G), G

bei ng a gr ou p of coeffic ients; b u t we shall sup pre ss G) as the kernel of the

h o m o m o r p h i s m i^ : Hv-i (A) -> Hv_i {X) (i.e. Hv-i {A,G) ->Hv^i (X, G));

he re i i s the inc lus ion mapping f rom A i n t o X a n d i^ is the corres

pond ing homomorph is m. The Cech homology theo ry i s u s ed .

R em ark s . In case X is a com pac t o r ien tab le r -man ifo ld of class C^

w i t h o r d i n a r y b o u n d a r y t h e c o n n e c t e d (v — 1)-manifold A of class C^,

t h e n b{X, A) = Hv-i(A) (for any G). Of course our definit ion a l lows X

a n d A to be a rb i t ra ry compac t s e t s w i th A G X.li A = X, t h e n b {X, A)

= 0, an un in te res t ing case . But i f L is a non-zero subgroup of Hv-i{A),the c lass {£{A, L) of sets X for which b((X, A) Z> L i s an in te res t ing

class and, as we shall see , large enough to contain an X s u c h t h a t X — A

h a s m i n i m u m H A U S D O R F F / l* ' -measure . The topologica l theorems which

ensure th is a re the fo l lowing:

(i) Each class ^{{A,L) is closed under point set convergence (see

D e f in i ti o n 1 0.1 .2 ) of X ^ t o Z (i.e. if e a c h Z ^ $ e : ( ( ^ , L ) , t h e n X $ e : ( ( ^ , L ) ) .

(ii) Suppose X^{i[A, L),_G is open (in Tl),GnAis empty, XDdG

= B, U is compact, U CG , Ur]dG = B, and b(U, B) Z ) B(Xi, B)

where X\ = {X r\G). Then the surface X\ obtained from X by replacing

XnGby UnG, also ^^(A, L),

( i) is jus t Theorem 10.3-16 and (i i) is Theorem 10.2.5 below. The theorems

in § 10.3 and m an y o th ers were pro ved b y J . F . ADAMS i n t h e A p p e n d i x

to the paper [1] of R E I F E N B E R G m e n t i o n e d a b o v e . T h a t A p p e n d i x c o n

ta in s m an y fu r the r exam ple s i l lu s t ra t ing the no t ion of a lgeb ra ic bo un da r y .

Lemma 10 .2 .4 . Suppose A is a compact set in R]>^, P is a point in R^,

and X = C{P, A). Then b(X , A ) = H^_ i{A) if v > 1.

This follows from Theorem 10.3-2 s ince X i s contrac t ib le .The fo l lowing isoper imetr ic inequa l i ty of R E I F E N B E R G i s i m p o r t a n t :

Theorem 10 .2 .4 . Suppose W and RQ satisfy the conditions o/ § 10.1.

Then there are constants C2[v,W) and C^{v,W), 2 < r < iV, with the

following property: If A is compact, Ad some ball B(Po,Ro), and

y l ' ' - i ( ^ ) = / " - i < + o o, there exists a surface X such that b(X,A)

41 2 The h igher d imens ional PLATEAU problems

D Hv-.i(A), X is in the geodesically convex hull of A and within a distance

< C2I of A, and

(10.2.3) A^X) ^Cs'l'-

Proof. B y vi rt ue of th e resu lts of § 10.1 and L em m a 10.2.1 (c), i t is

suf fici ent t o pr o v e th is for 90 = i jsr in wh ic h cas e RQ m a y b e a r b i t r a r y .

Accord ing ly we a s s ume tha t A C RN anxi ca rry ou t the cons t ruc t ion in

RN-

W e prove th i s by induc t ion on v. We define

0 0

F r o m T h e o r e m 10.2.3, i t fol lows that

i . i z

J(p(a)da <A'^{A) = I.0

Accordingly there is a value of a for which (p(a) < 1 an d hen ce 0. B y

repea t ing th is cons t ruc t ion a long each ax is , we see tha t we may d iv ide

R^ into cubes of side 1.1 I, no one of which contains a point of A

on i t s boundary . Of course on ly a f in i te number conta in po in ts o f A ; call

them iî, . . . , Rg. In each Rs, select a Ps in te r ior to the convex hul l o f

A n Rs and define

X = UC(Ps,As), As = AnRs^

T h a t b{X, A) = Hv_i{A) follows from Theorem 10.3.3. The o the r p ro

pe r t i e s a re ev iden t .

Suppos e , now tha t v> 2, N '>v, and tha t the theo rem has been

proved for a l l (v', N') in which 2 < 1 ' < v and N' > v\ L e t A be given

and def ine

T he n the re is a va lue ^ i , 0 < ^^ < / , such th a t 99I (a^) < l^-^. L e t

n]i = m. 1 and D\x = An n\i. T h e n

From our induc t ion hypothes is i t fo l lows tha t we can f ind {v — 1)-

surfaces B\i C n\i s u c h t h a t h{B\i, D\i) = J^,_2(Dii) , ^î is in the convex cover of D\i an d wi th in a d is tance C /î < C Z of D\i and satis f ies

(10.2.3) for r — 1. If we let ^^î be th e p a r t of A for which a^ + / (s^ — 1)

< :x;i < a^ + / s i a n d de fin e

C i =- ^ U U B\i , C\i = B\i_^ U A\i U B\i , (/ii)-2 _ ^ . - 2 p i ^ ) ,

si

10.2. V surfaces , their boundaries , and their H A U S D O R F F meas u res 4 1 3

th en th e hyp oth eses of Th eor em 10 .3 .11 a re sa t is f ied (wi th r = s^ +

+ som e integ er) an d also

c^IA^-HDi^y--l)/(»'-2) < C / ^ - 1 .

For each s^, we replace A b y C\i and repea t the cons t ruc t ion above

a long the x^ ax i s ob ta in in g the su r faces D ^ 2 s panned a s above by B^^ ^

satis fying (IO .2.3). W e le t A^^l^^ be the par t o f C\\ for which ^ 1 / +

+ Z(s2 - 1) < ::t;2 < ^12 + / s^'and define

Again, the hypotheses of Theorem 10.3.11 are sat is f ied (with r = s^ +

+ som e integer) and also

T he n, for each (s^, 5^), we p erfor m th e sam e co nst ruc tio n along th e x^

ax i s wi th A rep laced by O-^ ^. This process i s cont inued unt i l a l l the

axes a re exhaus ted .

Then each CJi - T ^ is in the c losed cube

^1 + Z(sl — 1) < :: t: l < a i + Zs^, a^f + / ( s 2 — 1) < ^ 2 < ^ 12 ^ / ^ g , . . .,

of side I, For each Qi--^j^, select a p oi nt Pgi'V. ' in te rio r t o th e c on ve x

co ve r of C,V::5^ a n d defin e Z ,V ;;5 ^ = C( P,V ;:5^', CJ,-;;. ^)

S ^ S i

Th en , b y us ing Le m m a 10 .2.4 f i rs t and th en Theo rem s IO .3 .9 , IO .3 .IO ,

and 10 .3 .11 repea ted ly , wi th A = Qi*;;. s& a n d Agk+i = Clxii%t}u ^ s + i= Xli;;^gt}i, we see in tu rn tha t

b{X,A)=H,_i{^)

and tha t a l l the o ther condi t ions a re sa t is f ied .

Theorem 10 .2 .5 . Suppose L is a subgroup of Hv-i{A), b{X, A) D L ,

and G is an open set such that G C\ A is empty . Define Xi=^ [X r\G),

Ai = Xi — G, and y = Y1 U ^ 2 , where Y2 =^ X — G and Y\ is anysurface such that 6(Yi , ^1 ) D 6 (Xi , ^1 ) and YiCW^G A^. Then

b{Y,A):DL,

Proof. Define Z 2 - y 2 , A2 = A{jAi = B, U = b{Xi,Ai), U

==b(X 2,A2), L[ = b(YiMi), and L^ = L2 = ^ ( ^ 2 , ^2 ) . Th en the

hy po the ses of bo th Th eore m s IO .3 .9 an d 10 .3 .10 a re sa t is f ied b y {X, Xi,



414 The higher dimensional P L A T E A U problems

X2) and (Y, Yi, Y2). T h u s

b(X, A) = i{B, A)^ni{B, Ai)^ Li + i{B, A2U U]

h{Y, A) = i{B, A)-^ [i{B, Ai)^ L[ + i(B, A2U L',].

Since L[ D Li and V^ = L2, it follows that b{Y, A) DbiX, A).

10.3. The top o log ica l resu l t s of A d a m s (see R E I F E N B E R G [1])

W e sha l l a s s ume th a t all sets are s ubs e t s of a H A U S D O R F F space . {X, A)

is a pa i r (of compac t s e t s in which A G X). bA (X) or b{X; A) — Ker i^:

: Hv-.i(A) -^Hv-i{X) w h e r e i : A -^X is the inclus ion map. We sha l l

d e n o t e the inclus ion map of A i n t o X by i{X, A) : i{X, A) (a) = a if

a^ A. Let PQ d e n o t e a ' ' b a s e p o i n t " and also the set {Po}- Given X, wele t fx'.X ->Po be defined by fx(x) = PQ for x^X. We let ex (or

s(X)) : Ho{X) ->G be the map y / x* w h e r e y is the i somorphism from

HQ(PQ) o n t o G (Actua l ly , in the b o o k by E I L E N B E R G - S T E E N R O D , G is

defined as Ho(Po), in which case y is j u s t the i d e n t i t y ) . € x is called the

' 'a u g m e n t a t i o n h o m o m o r p h i s m " by A DA MS (R EIFEN BERG [1]) . Theorems

1—8 below are j u s t the r e sp e ct iv e L e m m a s 1 A — 8 A of the p a p e r

R E I F E N B E R G [1].

Theorem 10.3.1. If X = A, then b{X, A) = 0.Proof. For i{X,A)^: Hv_ i{A)-> Hv_ i(X) is the i d e n t i t y so no

e lemen ts a ^ 0 in A are carr ied in to 0.

Definition 10.3.1. A set X is sa id to be contractible (oti itself to a

po in t ) iff H a c o n t i n u o u s map h \ X x I -^X s u c h t h a t h {x, 0) = A; and

h{x, \) = X{i for all x on X, XQ be ing a f ixed po in t .

Theorem 10.3.2. IfX is contractible, then b {X, A) = HP_I(A) if r > 1,

and b(X, A) = Ker SAifv= 1.

Proof. F r o m E I L E N B E R G - S T E E N R O D , C h a p t e r 1, T h e o r e m 11.5,

i t fol lows that Hq(X) = 0 for e v e r y q ^ 0 and HQ{X) = 0 (see C h a p . 1,

§ 7) and Ho {X) is i s omorph ic wi th G. By definit ion

b{X, A) = Ker i(X, A)^ = Hv.i{A) ii v>i

s ince Hp_i {X) = 0 if r > 1. In case v = \

b (X, A) = Ker i^ (X,A) = Ker SA

s ince SA = y/x* H {^> ^) ^^^ y / y * is an i s omorph is m.

nTheorem 10.3.3. Suppose that X = U Xr where the Xr are disjointr=l

and contractible. Define Ar = A 0 Xr, Sr = e(Ar), and

Ko = SH^> ^ r ) * KersrCHo(A),r

Then b(X, A) = Hp_i(A) if v > 1 and b(X, A) =Koitv= 1.



1 0 .3 - T h e t o p o l o g i c a l r e s u l t s o f A D A M S 4 1 5

Proof. F r o m E—S^, Chap. 1 , Theorem I3 .2 , i t fo l lows , by rep lac ing

the genera l pa i r {X , ^4) by (^, 0) and (X , 0) , tha t

r = l rand , in fac t , each a in -ô(^) i s un ique ly represen tab le in the form

(10.3.1) a = 2^ i(A,Ar)âr, ar^Ho(Ar)

and the correspoinding resu l t ho lds for X a n d Xr. If r > 1, it follows as

in Th eor em 10 .3 .2 th a t each Hvî(Xr) = 0 so th a t Hvî(X) = 0 a n d t h e

theorem follows in that case as before .

N ow , let (5 an d 6 b e t h e i s o m o r p h i s m s a b o v e f r o m ^ o ( ^ ) o n t o

^ Ho(Ar) and f rom Ho(X) o n t o ^ Ho(Xr) and le t / be the homomor-r r

phism from ^ Ho{Ar) t o 2! Ho(Xr) in wh ich Xr ~ i[Xr, Ar)^(ar).

Then, i t foUows that i{X, A)J=d-Îd. T h u s a ^ h { X , A ) ^ i ( X , A)^ a

= 0 4=> i{Xry Ar)^ Ur = 0 for r = i, . . .,n. But s ince € r = y fxri^ i{Xry

Ar)^ a n d y fxr^ is an isomorphism, i t fol lows that a^ hiXÂ) ^ Sr(ar)

= 0, r = i, . , .,n. But th i s ho lds

^ a^2 i(A, ^ r ) * Ker Sr = KQ.r

Theorem 10.3.4. Suppose that X is an v-disc with boundary A. Let Ldenote Hp_i(A) ifv'>\ and Ker SAifv= 1. Then b{X, A) Z ) L.

Proof. This follows from Theorem 10.3.2 s ince X i s con t rac t ib le .

Theorem 10.3.5. If A is a Jordan arc, then Hi{A) == 0.

Proof. This follows s ince A is contractible .

Theorem 10.3.6. Suppose that f: (X, A) -> (Y , B) and LA is a sub

group of Hv_ i{A). Let LB = {f\A)^LA and suppose b{X, A) D LA. Then

b{Y,B)DLB.

Proof. Fr om Th eorem 4 -1 , Cha p te r I of E— S and i t s proof, it followstha t the fo l lowing d iag ram i s commuta t ive :

Hv.i{A) ^—Hr.ii^)

/ l * • , / 2 *i ^* i

where we have u s ed the no ta t ion o f tha t theo rem. In pa r t i cu la r

W e a re g iven tha t i^(LA) = 0 and th a t LB = / 2 * ( ^ ^ ) . I t fo llo ws t h a t

i^ (LB) = fu (0) = 0 wh ich is to say t h a t LB C Ker i(Y, B)^ = b{Y, B).

Theorem 10.3.7. Suppose L is a subgroup of Hv_i{A), thatL C b(X, A),

and that Y D Z . Then LCb(Y,A).

^ i . e . E I L E N B E R G - S T E E N R O D .

41 6 The higher dimensional PLATEAU problems

Proof. Th is fol lows f rom The ore m 10 .3 .6 by t a k i n g / as the inc lus ion

m a p .

Theorem 10 .3 .8 . Suppose N = v and A is the unit {v — \)-sphere in

Rr{A =dB{0, 1)). Then

(a) IfXD^^), b{X,A) = H,_ i{A) if v > \ or b{X,A):^Ker

SA if V = \.

(b) If X does not contain B{OA ),h{X, A) = 0,

Proof. P a r t (a) follows from T he or em s 10.3.4 an d IO .3.7 . In p a rt (b) ,

there i s a po in t in ^ (0 ,1) — X, s ince X i s com pac t . L e t XQ be such a

po in t and l e t Y consist of all points (1 — t) x + t^, 0 < / < 1 , x^X,

where f is the intersection of the ray [x^s x) w i t h A. T h e n {A, A) is a

s t rong de fo rma t ion re t rac t o f {Y,A). B y T heo rem 11 .8 , Ch apte r 1 of

E—S, the homology sequences for (Y, A) a n d (A, A) a re i s omorph ic .

Accordingly, in considering the ^ 'j , . -parts of these sequences which are,

re s pec t ive ly ,

HmMY) < - — ^ m - i ( ^ ) a n d Hm-i{A) ^^^H^_i(A),

we conc lude tha t Hm -i{A) :^ Hm -i(Y) so b{Y, A) = 0. H e n c e b(X,A)

= 0 b y Th eore m 10 .3 .7 .

Theorem 10.3.9. {R \^, L e m m a 11 A ) . Suppose that X = U^ Xr.

Suppose ArdXr and A (Z X and we define B = A\J U Ar. Supposer

L and Lr are subgroups ofH v-i (A) and Hv-i [Ar), respectively, and suppose

b(Xr, Ar) Z ) Lr. Suppose also that

(10.3.2) i(B, A)^LCZ ^ {B. Ar)^ Lr.

Thenb(X,A)DL.

Proof. Suppos e h^L. T h e n

i(X, A)^ h = i(X, B)^ i{B, A)^ h^ i(X, B)^ Z HB, ^ r ) * Lrr

= 2 " i(X. Ar)^ Lr = Z i{X. Xr)^ i{Xr, Ar),. L ,r r

= Z i(X,XrUO = 0r

s ince b {Xr, Ar) D Lr. T h u s h^ Ker i(X, A )^ = b(X, A).

L e m m a 10.3.1. Suppose that X = U^Xr, B = UÂr, Ar C Xr for

each r and Xr C[ Xg =^ Arf\ Ag whenever r ^ s. Then, for each q, each

u ^ Hq {X, B) can be represented uniquely in the form

(10.3.3) U^îrÛr, Ur^H q(Xr,Ar), ir : (Xr, Ar) C (X, B ).

rProof. We sha l l p rove th is ior n = 2; the theorem can be proved for

a n y n b y i n d u c t i o n .

To do this we consider the tr iad (Yi U Y2; Yi, Y2) and the inclus ion

m a p s i^2:(Yi, Y i 0 Y2) C (Yi U Y2, Y2) an d h: (Y2, Yi 0 Y2) C (Y i

1 i. e. R EIFENB ER G [1].

41 8 The higher dimensional PLATEAU problems

Th e m ark ed is omorph is m fo llows f rom L em m a IO .3 .I and 2J ^ d e n o t e s

th e tra ns fo rm ati on defined b y *"

12 ^] (î> • • •> '^n) = {dui, . . ., dun), e tc .

If ^ ^ Hv_i(B) a n d i(X, B)^ ^ =- 0 , then k^Ker i{X, B)^ = dHy{X, B).

This fo l lows f rom the exac tness of the Cech homology theory over the

ca tego ry ^ c of com pac t pa i r s an d map s of s uch {E—S, Chap te r 9 ,

Th eore m 7 .6) . T hu s H a # in Hv{X, B) ^du = k. F r o m t h e m a r k e d

isomorphism, i t fo l lows tha t u i s un ique ly represen tab le in the form

(10.3.3) with q rep laced by v. T h e n

% , Ur^ Hv[Xr, Ar), ir '• (Xr, Ar) -> (X, B).r

T h e n du = ^ diri^ Ur = 2 i{B, Ar)^ hr, hr = durr r

as des ired and s ince each of the rows

— ^ H, (Xr, Ar) — - > H,_ i (Ar) -^^ H,_ i (Xr) >

is exact , i t fol lows that i(Xr, Ar)^ hr = 0 SLS requ i red .

Theorem 10.3 .11. Suppose thatn

A = U Ar, Ar n Ar+i = Dr for r = \, . . .,n ~ \,

ArH As = 0 if \r — s\> \.

Let Kr denote Hv-2 (Dr) if v > 2 and Kr = Ker s : HQ (Dr) ->Gifv = 2.

Suppose that

b(Br, Dr) 0)Kr, r = 1, . . ., n; BQ = Bn+i = 0

C := AUU Br, Cr = Brî U ArU Br, r = \, . . . ,n+ i,

AnBr = Dr, r = 0, . . ,,n+ \.Then

2:i(C, CrU H,_ i{Cr) D i{C, AU H,_i{A).

Proof. W e prove th is for w = 2 ; the proof for the gene ra l n is by in

duc t ion . In th is case we change our no ta t ion as fo l lows :

AÂiUA2, AinA2 = D, Ci = AiUB, C^Â^^JB,

C = Ci\J C2 = A\JB, AnB:=D, b(B,D)DK

w h e r e K = H^_ 2(D) ii v > 2 smd K = Ker s : Ho(D) - > G if r = 2. T he

resu l t i s p roved by d iagram chas ing in the fo l lowing d iagram:

Hv-i (Ai) -r-> H,_i (C i) — > H,_i (Ci, B) ^ — F . _ i ( ^ 1 , D)

^ ' i i -I' 1Hrî(A) — - > i / . _ i (C ) —>Hv-i(C,B) ^—H,_i(A,D)-^-^H,_2(D)

^ * A j / * A ''^ A A

Hv-1 (A2) — > H,_ i (C2) - ^ H,_ i (C2, B) < - — H,_i (A2, D) ~ 1^2* J2* T2



10.3. The topological results of ADAMS 419

Since C — A = B — D =^ Ci — Ai = C2 — A2 and is open on C,

it follows from the s t rong exc is ion theorem (E—S, T h e o r e m 5-4, p. 266)

t h a t the m a p s T , T I , and T2 are i somorphisms (on to) . Clear ly the i m a g e s

in Hvî(A,D)

of the Hv_i(Ar) are inc luded in thos e of the Hv_i{Cr), i.e.Or r~^ jr^ ir^ [Hv-1 (^r)]

= T-iy* Qr ir^ [i^v_l(^r)] C T' j^{Qr[Hv-l{Cr)]] •

N e x t , the sequences

^ . - 1 {Ar) ^-^^^-t Hrî {Ar, D) -^^ H,_2 {D), r=\,2,

H,_i (A) :^:lhl% H,_i (A, D) — ^ H,_2 (D),

a re exac t , so the i m a g e of Hv-.i(Ar) in Hv_i{Ar, D) consis ts of e x a c t l ythos e e lemen ts Ur of the l a t t e r g r o u p for which dur = 0, r = \, 2; and

the co r re s pond ing s ta temen t ho lds for the image of Hv-i {A ) in Hv_i(A, D).

Fina l ly , f rom Lemma 10.3.1, it fo l lows tha t each e lement u of Hv-i(A, D)

i s un ique ly rep re s en tab le in the form

u = aiui -{- 02^2, Ur^ Hvî (Ar, D).

Tak ing inve rs e images unde r j'^^ x, it fo l lows tha t

i(C, A)^ H,_i{A) C i(C^ Ci)* H,_i{G) + i(C, C2)* H,_i(C2)

w h i c h is the des i red resu l t .

Theorem 10.3.12. Suppose that X = 1 xY and ^ = (0 X Y) U

(1 X y) . Let lo and I\ he the natural embeddings of Y in A as [0 X Y)

and (1 X Y), respectively. Then h{X,A) D the totality of elements in

Hvî (A) of the form If h-I^hforh^ H^-\{Y).

Proof. Let / o = i{Xy A) IQ and / i = i(X, A) Ii. T h e n /o and Ji are

h o m o t o p i c a l l y e q u i v a l e n t in X. T h u s / i * = / o * [E—S, C h a p t e r 1,

A x i o m 5), so t h a t

i(X, A)^ (h^ h - /o* h) - / u A - Jo* = 0 for all h^ H,_i(Y),

T h i s is the resu l t .

Theorem 10.3.13. Suppose that X' = I xY, AQ=0 xY, A[

= 1 X Y, A' := AQKJ A\ and suppose that f: X' -^X is continuous and

that fiA^) = Ao, f{A[) = Ai. Suppose that fo=f\ A^, that fi=f\ A\,

and that fQ is a homeomorphism. Suppose K = h[X, A) and LQ is a sub

group of Hvî(Ao). Then there is a siibgroup of Li of Hv_ i(Ai) such that

(10.3.5) K + i(A, Ao)^ Lo = K + i{A, Ai)^ Li.

Proof. Let io: Y -> AQ and iii Y -^ A[ be the h o m e o m o r p h i s m s

defined by io{y) = (0, y) and ii{y) = (1 , y), re s pec t ive ly . F rom E—S,

C h a p t e r 1, § 5, it follows that/o^jt, ^'0*, and 'i, are i s omorph is ms . Thus we

define Li as the t o t a l i t y of e lemen ts of the form

(10.3.6) hi =fi^ ii^ h w h e r e h = V*Vo* 3^0 and ho^ LQ.

27*

4 2 0 The h igher d imens ional PLATEAU problems

Now, s uppos e k^K a n d h^^ LQ. There ex is ts a un ique h in Hvî{Y)

defined b y (IO .3.6); le t h be defined by (10.3.6). Then

i{A, ^ 0 ) * ho =i{A, ô)*/o* ^ '0* h = Fo^h,

i{A, Ai)^ hi = i{A, ^ i ) * / i * ^ 1 * h = Fi^ h,Fo:Y-Â, Fo = i(A,Ao)foio.

Fi'.Y-Â, Fi = i(A,Ai)fiii.

B u t n o w t h e m a p s Go = i{X, A) Fo a n d Gi = i{X, A) F\ are seen to be

homotop ica l ly equ iva len t in X s ince we may define

Gt'.Y^X b y Gt{y)=f{t.y), 0 < ^ < 1 .

Thus f rom E—S, C ha pte r 1 , A xio m 5, i t fol lows th a t

i{X,A)^[i[A,Ao)^ho~i[A,Ai)^h{\ = 0, or(10.3.7) i[A,Ao)^ho=-k' + i{A,Ai)^hi, k'^K = b{X,A), or

k + i(A, A oU ho = k + k' + i(A, Ai)^ hi^K + i{A, A^) U.

In l ike manne r ii k^K a n d hi^Li, there is a t leas t one h and hence

one ho satis fying (10.3.6) and hence (10.3.7). Thus

k + i{A, Ai)^ hi = k-k' + i[A, Ao)^ ho^K + i[A, Ao)^ U.

Theorem 10.3.14. Suppose that IJ is a v-plane [in R^) through P, that

A =nndB(P,r), and that D = U r\ B{P,r). Then h (D, A)~ Hvî(A) and if X is any surface I ) A for w hich b(X, A) = L where L

is a non-zero subgroup of Hv-i(A), then

A'{X) >yl»'(Z)) =yrrr

Proof. T h a t b{D,A) = Hv_ i(A) follows from The ore m IO .3.8. Le t

f = Pn ^nd Y = / ( Z ) . T h e n / : ( X , A)-^{Y, A) an d /2 - / U is t h e

i d e n t i t y . F r o m E—S, Ch apte r 1, § 4 , the d ia gra m

H,_i(A)~^*^H,_i(X)I/2* l / i*

Hr_i(A)^H,_i{Y)

is co m m ut at iv e an d /25J- is ju s t th e ide nt i ty (Axiom 1). No w, sup pose

y^ b(X, A). T h e n y^ Hp_i{A) a n d i^(X, A) y = 0. So

n(y) =fuHf2iy = ^'T h u s b(Y, A) D b(X, A) ^ 0. The theorem then fo l lows f rom Theorem

1 0.3 .8 a n d t h e f ac t t h a t y l ' ' ( Z ) > / l » ' ( Y ) .Theorem 10.3.15 {R \, L e m m a 21 A ) . Suppose Xr D Xr+i and

b(Xr, A) D Lfor r = \,2, . . ., L being a subgroup of Hv-i{A). Let

X = C\Xr. Then b {X, A) D L.

Proof. Suppos e h^L, T h e n i{Xr, A)^ h = 0. By the con t inu i ty o f

the Cech homology {E~S, pp . 260— 261) th e in jec t ions i{Xr,X) yield

10.4. The min imiz ing sequen ce; the min im iz ing se t 4 2 1

an isomorphism of Hvî{X) onto the inverse Hmit g roup l .imHv_i(Xr).

This i s omorph is m maps the e lemen t i(X,A)'^h in to the sequence

{i {Xr, ^ ) * ^} w hich is a sequen ce of zeros an d is th e zero e lem ent of

the inve rs e l imi t g roup . Thus i(X, A):^ h = 0. Since h i s a rb i t ra ry

b(B,A) Z)L.

Theorem 10 .3 .16 . Suppose b{Xr, A) Z ) L for r = \,2, . . ., L being a

subgroup of Hv_ i{A), and suppose Xr -^X in the point set sense. Then

b{X,A) D L .

Proof. F ro m Th eo rem 10 .3 .7, i t fol lows th a t b[Yr, A) D L for each ;',

w h e r e Yf = \J Xg. Clearly Yr D Yr+i a n d pi Yr = X so the result

fo llows f rom T heo rem IO .3.15 .

1 0 . 4 . T h e m i n i m i z i n g s e q u e n c e ; t h e m i n i m i z i n g s e t

In th is sec t ion we cons t ruc t a min imiz ing sequence and a min imiz ing

se t and prove a f i rs t smoothness peoper ty of the min imiz ing se t . We con

s ider compact se ts X a n d ^ on 50 in w hi ch

(10.4.1) ACX, b{X,A)DL, L a s u b g r o u p o f ^ r _ i ( ^ ) .

We def ine , fo r a g iven compact A a n d s u b g r o u p L,

(10.4.2) d(A, L) = iniA'îS) for all X w i t h b{X,A)DL, S ^X - A.W e s ha l l a s s ume tha t 9D satisfies th e con dit ion s of § 10.1 a n d

( 1 0 . 4 . 3 ) d(A,L) < + 00.

We now def ine R(P) = m i n [ ^ ( P , A), RQ]. li b(X, A) ^ L, we define

r

(p{r, P]X) = fA'-^ [X n dB{P , t)] dt

(10.4.4) ,R{P)>0, 0<r<R(P).

y){r,P]X) =Â''[Xr\B{P,r)-\L e m m a 10.4 .1 . Suppose b{X, A) D L. Then

cp{r^, P; X) - (^ (n , P ; X) < ^ ( r 2 , P ; X) - y>{r,, P ; X ) ,(IO .4.5)

o < n < ^ 2 < P ( P ) ,

w{r\P']X)<w(r' + \PP'\,P]X), r' <R(P'),10 .4.6 ) ^ ^ / r v i I h , / , V y.

y + \PP'\<R{P),

and (p and ip are non-decreasing on r for each P i A,

Proof. (10.4.5) follows from Theorem IO .2.3. w i t h U(Q) = d(P, Q)

t a k i n g S = Xr\[B{P,r2) -BiP.n)'}. (IO .4.6) follows since B{P\r')

GB(P,r') + \PP'\).

Lemma 10.4 .2 . Suppose b(X, A) Z ) L and

(10.4.7) A^(S) =d(A,L) + s.

4 2 2 The higher dimensional PLATEAU problems

For 0 <r <R(P), define

^{r, P; X) = m a x [ 0 , (p{r, P; X) - e],

wir, P; X) = msix[0, w(r, P; X) — s],( 1 0 . 4 . 8 ) '^v . , ; I > w \ ' > J J ^ ^

Q {P , X) = s u p r for those r for which ^ (r , P; X) = 0

Q*(P, X) = supr for those r for which y){r, P; X) = 0.

Then, for almost allr, 0 <,r < R(P),

Cs[(pr(r,P;X)rli^^) + 8.(10.4.9) y^(r,P;X)< , ,^ ^ . ^ / p VN i

\y-ir{i + hr) (pr(r, P; X) + e,

where C3 denotes the constant of Theorem 10.2.4 and h is chosen so that

(1 + rjo r)^"-^ < (1 + / ^) for 0 < r < J^o. Also

g*(P)<Q(P), ^r{r,P;X)^^r{r,P;X) {riZ{P)))

(10.4.10) ^{r2,P;X) -^{ri,P;X) <y ^(r2,P;X) -y ^{ri,P;X),

0<.ri<r2<R(P);

(10.4.11) ^-"(1 + hrY^(r,P;X) and ^-"(1 + hryipir, P; X)

are non-decreasing;

(10.4.12) ^{r,P;X)^H[r -Q(P,X)]\ Q(P, X) <r < R{P) ,

X = ! ; - » ' Q - " > 0 ;

(10.4.13) w{r,P]X) <e + v-^\ -k)-^[i +Q(P,X)]S,

0 <r ^kQ{P,X), 0<k<\.

Proof. For a l m o s t all r < R{P)

(pr{r, P; X) = A'-^ [X 0 dB[P, r)].

From Theorem 10 .2 .5 wi th G = B{P, r), Ai = X H dB(P, r), and Yi

any s u r face wi th b(Yi,Ai) D Hv_i(Ai). I t fo l lows tha t [X — B ( P , r)]

U Yi = Y is a surface wi th b{Y, A) D L. By Theorem 10 .2 .4 we mayt a k e for Yi a surface wi th A^'iYi) < C3[(pr(r, P] X)fl(''-^). T h u s if the

f i rs t inequa l i ty in (10.4.9) did not ho ld , we w o u l d h a v e

A^(S) > A'(Yi) + A^'IS -B(P, r)] + £ > d(A, L) + s {S = X-A).

The o ther inequa l i ty fo l lows s imila r ly s ince we may a l s o t ake Yi

= CO [C (0, ^10)] w he re to is a n o r m a l c o o r d i n a t e s y s t e m w i t h d o m a i n

^ ( 0 , RQ) such that a>(0) = P , ^10 = c o ~ i ( ^ i ) . T h e n we would a r r ive at

the s ame con t rad ic t ion , s ince in th is case

A'{Yi) < ( 1 + 7 y o ^ ) M » ' [ C ( 0 , ^ 1 0 ) ] = ( 1 +r^^ry'v-'^rA'-'^{A^o)

< ( 1 +r}or)^''-^v-'^r(pr{r,P;X),

as is seen from our assumptions (10 .1 .13) , Theorem 10.2.1, and L e m m a

10.2.1(c).

The re s u l t s in (10.4-10) follow from (10.4-5) and (IO .4.8).

10.4 . The minimizing sequence; t h e minimizing s e t 4 2 3

F ro m (10.4.5) and (10.4.9), i t fol lows t h a t

(10.4.14) ^ ( r , P ; Z ) < | {riZ{P))

fro m w hich (10.4.12) and th e firs t in eq ua li ty in (IO .4.11) follow. To pro ve

(10.4.13), we assume 0 < ^ < 1 , 0<r<ks, S<Q{P,X) and us e

(10.4.9) to obtain

^ ( r , P ; X) <ip(k s, P ; X) < - - ^ - / % - ! (1 +ht) cpr{t, P ; X) dt-^ e

< [1 +hQ{P, X)] r - M l - k)-^(p[s. P)X) + e

< e + r - i ( 1 - k)-"^ [1 + Q{P. X)] S

from w hich (IO .4.I3) follows.

To prove the second inequali ty in (10.4.11), we define

X(r. P ; X)=Jipr(t^ P ; X) dt, co(r, P ; X) =ip{r, P ; X) - f (r, P ; J ^ ).0

From (10.4.10) and the definit ions i t fol lows that

co(r2) -win) >j[q>r{r, P, X) -y >r(r, P, X)] dr. 0<n< n <R(P)-r i

But s ince co is singular, i .e. co ' (r) = 0 a.e. , it follows that co i s non-decreas

ing. From (10.4.9) and (10.4.10) and the definit ions , we obtain

y^[h(r)]-^ip(r,P;X) > 0 , h{r) =yvr^\ +hr)''', or

A [h{r)]-ix{r. P; X) > A [h(r)]-^co(r, P ; X)

= - {[h (r)]-^y co{r,P; X) (r ? Z 2 (P))

[h{r2)]-^X(r2) - [h{ri)r^x{ri) > -fco(r, P ; X) d[h(r)]~i

h~^ ( 2) CO ( 2) — h-^ (ri) CO (ri) — f h'^ (r) dco (r)

{co{r) =€ o(r,P;X))

from w hic h th e desired ine qu ali ty follows, since co is non-d ecre as ing .

Now, s uppos e tha t ^ C SO^ a n d t h a t {Xn} i s a min imiz ing sequence ;

i.e. b{Xn, A) Z^ L a n d A^[Xn — A) — d{A, L) + en for each n, w h e r e

£7i - > 0. L et {^^} be a co un ta bl e dens e su bs et of 9fi. T he re is a su bs eq ue nc e,

still called {X^, s uch tha t the func t ions -y; (r , Qf.Xrij converge for each i

42 4 The higher dimensional P L A T E A U problems

and each r, 0 < r •<R(Qi) to func t ions ^ (r , Qi), N e x t , we define (for

t h a t s u b s e q u e n c e {Xn})

y )+ {r, Q) = Hm s u p ^ (r, Qi), ip~ {r, Q) = hm inf ^ (;', Qi)

9^" i^y Q) = 1™ sup99 (f, Q]Xn), (p- {r, Q) = hm inf99 {r, Q;Xn).

T h e n we h a v e the following results :

T h e o r e m 10.4 .1 . (a) (p^ and ip^ are all non-decreasing in r and y)'^{r, P)

is upper-semicontinuous in (r, P).

(b ) y)-{r, P) < lim miy)(r, P\Xn) <. lim s u p ^ ( r , P\Xn) <. y)+{r, P).

(c) f- (^2, P) > yj+ {ri, P) whenever 0 ^ ri <i r^ <i R (P). Thus

yj+ (r, P) — ip- (r, P) whenever either y)+ or ip~ is continuous in r.

(d) (p-(r, P) < ^+(r, P) < \p+[r, P), 0 < r < R{P).

(e) ^+(V', P') <y)-(r, P) if r' <R{P') and r' + d[P, P') < r

<R{P).

(f) For each P in ffl - ^ , [h(r)]-^ip± (r,) and [h{r)]-^ (p±{r, P) are

a l l non-dec res ing in r and

y)(P) = lim [h(r)]-i ^+ (r , P) (h{r) = y^ r"" {\ + h r)-")r->0+

is Upper-semicontinuous.

(g) There is a positive number ^ such that tp (P ) > ^ whenever ip (P ) > 0,so this latter set is closed in'^ — A,

(h) (p-{r, P) >ip{P) ' h[r), 0<r< R(P).

(i) If y }(P)=0, then y )+(r,P) = 0 for 0 <r <Q{P) for s ome

p o s i t i v e n u m b e r Q ( P ) .

(j) / / we define S as the set of P in W — A for which ip(P) > 0, then

S is bounded and XQ = S U A is compact.

(k) An altered sequence, still called [Xri], exists, which has the same

functions \p^, for which the new functions 'cp^ are still non-decreasing in r

and satisfy the conditions in (a), (f), and (h), and for which D {Xn, XQ) ->0.

(1) IfP^S,0<,r<: R(P) and {B(Pi, ri)} is a finite or countable

disjoint family in which each Pi^S and each B{Pi, ri) C B{P, r), then

i

Proof, (a) If / i and f^ are c o n t i n u o u s and non-dec rea s ing on an in

t e r v a l I, t h e n the func t ions msix[fi(x), f2(x)] and min[/i(:\;), /2(:\:)] are

easily seen to be non-decreas ing . This resu l t is eas i ly ex tended to i m p l y

t h a t the sup and inf of a coun tab le fami ly of non-decreas ing func

t ions (cont inuous or not) are non-dec rea s ing . Thus (a) follows easily.

The proof of (b) uses the s ame ideas as t h a t of (c). So we p r o v e (c).

Choose ^ > 0 so 4^ < ^2 — ^ 1 - We h a v e

ip-(r2, P) > in fv^( / , Qi), y)+(ri, P) < supip{/\ Qj).r'>r2—Q r"<ri+e

d(P,Qi)<Q d{P,Qj)<Q

10.4. The min imiz ing sequenc e; the min imiz ing se t 4 2 5

But now, i f r' >r2 — g, d{P, ft) < Q, r" <ri + Q, d{P, Qj) < Q, t h e n

^ ( f t . Qj) < 2 e a n d

y^{r\ Qi) >ip[r' - d{Qi, Qj), ft] îp(r2 - }Q, QJ) >ip[r", ft-),

using (10.4.6) and a limit process. Parts (d) and (e) follow from (10.4-5)and(10.4.6) and l imit processes . Tlia tA~^99± and/^~i^± are non-decreas ing

follows from (10.4.11) and the argument in (a) .

To p rov e (h), we fi rst no te th a t

(pr{r,P\Xn) '>vr-^{\ +hr)-^[f(r,P;Xn) - Sn].

By in teg ra t ing wi th re s pec t to r b e t w e e n TQ an d r, 0 < ro < ^, an d t he n

le t t ing ^ -> 00 f i rst a nd t he n TQ - ^ 0 , w e o b t a i n

r

(p-(r,P)>:fvt-^(\+h t)-^ yr [t, P) dt0

from which the result follows, us ing parts (f) , (b) , and (c) .

To prove (g) and (i) , we consider two cases :

Case 1 l i m ^ ( P , Xn) = 0. Then it follows from (10.4.12) that

(p-{r,P)^^y vr'', 0<r< R(P), ^ = Ky -\

Case 2 l im Q ( P , Xn) = Q(P) > 0 as n runs th rough a s ub -

W->>oosequence. In that case we see from (10.4.13) by le t t ing n-ôa t h a t

y )~ (r , P ) = 0 for 0 < f < Q(P) S O t h a t y){P) -= 0.

To prove ( j ) , le t us suppose tha t S is no t bounded . Then there ex is ts

a dis jo int c ou nta ble fam ily of balls B{Pi, RQ) w i t h Pt^S a n d B(Pi, RQ)

n A empty fo r each i. B u t t h e nd{A,L)= limA^{Sn) > lim inf Z ^' l^n 0 B{Pi, R Q ) ]

> lim inf 2 " y> {Ro, Pi •,X„)>Z <p- (Ro. Pi) > ^ N h(Ro)

for every N. This i s imposs ib le .

To prove (k), choose for each p a f in i te number of regula r domains

of class C2 whose union Gp con ta in s (XQ, \j2p) and is conta ined in

(Xo, \lp). Let G^ be the set (G^, Qp) w h e r e Qp is so small that G^ and

rp = G'^ — Gp are of class C\ and define Up on Fp by def in ing Up [x)

= t for x^dGpt th e paralle l surface *7 of the w a y " from dGp t o dG'^.

T h e n Up^ Cî^p) a n d \J Up^ 0 there . Clearly i t fol lows from part

( i ) and the compactness of Fp that / l* ' (Pp Pi Xn) -> 0. Thus , for each

py we may choose an ftp an d a ^^, 0 < ^^ < 1, so th a ty l" " ! [dGpt^ C\ Xn^< \lp^-'^. F ro m T heo rem 10 .2 .4 i t fol lows th a t th e pa r t of Xn^ ou ts ide

Gptj, can be replaced by a set Yp s u c h t h a t b(Yp,Ap) D Hv_ i(Ap),

Ap = dGpt^ n Xn^, Yp C {Ap, C i ^ - i ) , a n d A^Yp) < C^p-r Clearly

also , if P ^ 5 , th ere are poin ts of th e new (and of th e previou s) Xn^ con

ver gin g to P as ^ ^ cx D . T he re sult follows.

To prove ( / ) , we note tha t

ip+ {r, P) > l im s u p ^ (r, P;Xn)> lim inf/l»' [Xn D B ( P , r)]

> Urn inf 2 " A' [Xn O B {Pi, n)] > Z {lim iniA" [X^ D B (Pi, n)]]


Lemma 10.4.3 (cf. R E I F E N B E R G [1] , Lemma 5*) . Suppose Xn, XQ, S,

and all the various other functions have their significance above. Suppose

p '> V >}. For each ri'> 0, there are positive numbers e (rj, ^, p, v, W),

r\ (rj, ^, p, V, W), and A(r], /5, p, v, W) with the following property . If P' ^ 5 ,

0<r'< R[P'), and K[P ', r') = S r\ B(P\ r') C {U', ri'r') for some

geodesic p-plane U' centered at P' and if

(10.4.15) P < [h(r)]-^(p-(r, P) < [h{r)]-'îp-^{r, P) < ^ + e'

[h[r) =y^r'[\ + hr)-'')

for every B(P, r) C B{P\ r') for which P^S and 0 <r <R{P), then

there exists a P * in S such that K(P*, 1 r') C [11, rj X r') for some p — \

dimensional geodesic plane U centered at P * and B(P*, X r') C B{P', r').

In fact we may take e, rj', and X all of the form C{^, p,v,W) rj^,

h = h{p,p,v,m).Proof. We f i rs t show tha t there i s an R' = c{p,v,Wl)r\ c > 0, such

t h a t t h e r e is SL Q in LP s u c h t h a t B [Q, R') C B [P', r'jl) a n d K(Q, R') isempty . I f th is were no t so , every sphere B[Q,R') w i t h QÛ' would

con ta in a po in t Q' of S and so every sphere B{Q, 2R') would con ta in a

s phe re B{Q\R'). But the re a re a t mos t Z i{p,m) - (r'jRy dis jo in t

s phe re s B(Qi, 2R') C B(P',r'l2) (QtÎI'), so tha t ( reca l l Theorem

10.4.1(1) an d th e fact th a t i? (P ) < RQ for every P^m~ A).

ip + 8') h{r'l2) ^ ^+ g , P') > 2^W-{2R\ Qi) > Z i. [^Y ' p • h{2R').i

T h u s , i f we take e < /5, we conc lude tha t(10.4.16) (r'IRy-'' < 21-2*'• Z^ ;! • (1 + 2hRy.

Now, le t B(Q, R) be the la rges t sphere in B(P\ r'l2) w i t h Q on 1 1 '

s u c h t h a t K{Q,R) i s em p ty . Le t P^S ndB{Q,R) and le t PQ b e a

poin t on Q P p r o d u c e d b e y o n d P a n d l e t ^ = | P P o | - L e t In ^ ln(P, r)

^ Xn D dB{P,r), le t Ce be the geodesic half-cone consis t ing of the

geodes ic rays th ro ug h P w hich m ak e an angle < 0 a t P w i th the geo

desic QPPQ, and le t Ine^ lne{P,r) ^ ln(P,r) f) Ce. For a lmos t a l l

r a n d 6

(10.4.17) AnC(PoJn)]Â-[C(PoJne)]+AnC(PoJn-lne)]^

Le t T be a no rm a l coo rd ina te pa tch cen te red a t P . Th en

/ . ^ . . o x lneC:B(Po,g), r-Hln6)CB{Po,Qo), ln-lneCB(Po,a)(10.4.18)

ln-lneCm -B(Q,R), Ql = r^+ x^-2rxcosd.

10.4. The minimizing sequence; the minimizing set 4 2 7

From Lemma 10.1.3, it follows that B[T-^(Q'), Rjl] CT-^[B(Q, R)]C B[T~^(Q"), 2R] where Q' and Q'' are on the ra y P ^ at respective distances Rl2 and 2R from P (this assumes that RQ is small enough).Accordingly, if we let Q and a be the smallest radii satisfying (10.4.18),we must have

^2 < (1 + Zi r)2 (y2 j ^ x^ ^2rx cosO),

(10.4.19) (72 < (1 + zi rf (r2 -\- %^ -\- 2r^ xjR)

0-2 > (1 + zi r)-^(r^ + x^), Zi < 2rjo ii x <, r{i2 — l)

on account of Lemma 10.1.3. From the minimizing property of the Xn,

we conclude, as in the proof of Lemma 10.4.2, that

A^{Kn) < / t ^ [ C ( P o , In)] + Sn. Kn ^ Kn[P. f) ^ Xn f^ B{P, r ) ,

A^{Kn) < v-ife(1 + hQ) A^'-^lne) + a(i + ha) A'-^{ln - ke)] + en.Solving for A^-'^[lne), we obtain

(1 0.4 .2 0) y l» '- l( 4e ) < ( ( T - ^ ) - l [ o r ( 1 +ha)A'-^{ln) - V A^ {Kn) + V Sn].

Now, let us choose ro, ri, x, and cos^ as follows:

MO A 21^ 0 < r o = (ro/i^)i/i2 ; £' = m in [^ ,( ro /ie)] .

We shall consider only values of r with

^1 < ^ < ^0 .

Then, using (10.4.19) and (10.4.21), we obtain (using ]/l ± y < 1 db y /2,

etc.)(T2 - ^2 > _ [(I _!_ ^^ ;.)2 _ (1 4_ 7 ^ ;.)-2] (^2 _^ ; 2) _|_

+ 2r x{\ + Z i r ) 2 c o s e > ( 1 + Z i r ) [—AZ ir{r^ + x^) +

+ 2r :v COS0] > (1 + Z i r) • 2r ro{{rolR)^^f^^ -

- 2Zi ro [1 + (ro/i? )]} (Zi < 27 0)

(T + ^ < ( 1 + Z i . ) . [ 2 + | - ^ c o s 0 + 5 ] ( J < ^ c o s 0 )

< ( 1 + Z i r ) . 2 r [ l + i ( V i ^ ) i i / 6 ] .

. • . (T - ^ > i ro (ro/i^)ii/i2 if 2fjQR^ h and /-o/i^ < Ao.

Now, since yl" (X^) - s^ < T -I r (1 + / ) A"-^ (In) and yl*' (iC«)> P h[r) by Lemma 10.4.2, we may use the inequalities above to conclude that (if TQIR < ^0, etc.)

0^a{\ +ha) / t ' ' - i ( y -vA'(Kn) ++ vsn(a<.r(\+Z ir)(\ + 2rolR) < 2ro)

, .^ . . . . < ^ ( 1 + Ziro) (1 + 2ro/i^) (1 + 2hro)A--Hln) -(10.4.22) _ ^ ^ ^ ^ ^ . ( ^ j^kr)-' + ven

< ^0 [(1 + 3 0/i ) ^ ' '"MM - / ^ 'W] + ^^n,{h{r) =y^f^{\ + Ay)-").

Using (10.4.22), in tegrating (10.4.20), and l e t t ing n -^^^, we o b t a i n

lim s u p / y l ' ' - i ( 4 e ) dr<.{a- Q)-^ roh{ro) [(1 + }rolR) ( + a) -

~P + ^h(n)lh{ro)] < Z2(ro/i^)i/i2 h(ro)

s ince r > 3 so h(ri)lh{ro) < Z3 • {roIR)- Us ing a s imila r p rocedure we

o b t a i n

lim sup A^'iKn ( P , ro) H Ce] < lim supA^ [Kn (P, ^0)] -

- lim inf A" [Kn (P, ro) n (2)1 - Cg)]

To(10.4.23) < (i + e') h (ro) - lim inf / A""^ [In) dr +

0ro ri

+ lim sup JA''-^ (In d) dr + lim sup J yl»'-i (4) dr

n 0

< Z 4 ( r o / i e ) i / i 2 A ( r o ) .

L e t ITi be the geodesic (AT — 1 )- pl an e t h r o u g h P±QP. Let A = ro /2y ' .

T h e n

(10.4.24) r o / / - 2A < ro/i^ < (ro/f') (r'lR') < Z5(i^, r, 501) • A.

W e now s h o w t h a t if A, or ro/i^, is chosen smal l enough, then K (P, roll)= K(P, Xr') ~SnB(P, Xr') C (77i, T A / / 2 ) . For s uppos e P'^^KIP, roll)

a n d is at a d is tance > rjrolA = rj Xr'll from 77i. Let P^ = T ~ I ( P * ) ,

(Jo = '^~^(Q')> nio =r-^(IIi), and Ceo = r-^(Ce). Since r is cen te red

a t P, T - i ( P ) = 0, T-i [5(P, f)] =B(0, r) if r < iô, /T io is an a c t u a l

(A^ — 1)-p lane th rough 0, and Ceo is an ac tua l ha l f -cone whose gene

r a t o r s m a k e an angle 6 w i t h the n o r m a l O P Q O to i7i. Poo = 't~^(Po)'

F r o m the definit ion of Q' given jus t below equation (10.4.18) it

fo l lows tha t B(QQ,RI2) is t a n g e n t to Uio at the origin. We h a v er e m a r k e d a b o v e and can a lso conc lude f rom Lemma 10 .1 .3 tha t

B(QQ,RI1)CT-^[B(Q,R)]. F i n a l l y P * is at a d is tance Q from 77io

where (Lemma 10 .1 .4)

(1 + r]o roll)-^ fj rolA < ^ < (1 + ô 2 ) r] fo/4.

Since P* is not mB(Q,R), P^ is not in B(QQ, Rll) and so, since it is in

B (0, roll), it m u s t be on the same s ide of Uio as is Ceo if ô/^ is s m a l l

e n o u g h (<rjl(i +7joRI6)). By pas s ing a 2-p lane th rough 0, Poo, and

P ^ , we see t h a t the dis tance f rom P^ to Ceo is

d = Q sec(p sin(0 — (p) > (f = angle Poo 0 P * .

I t is easy to see t h a t d t a k e s on its m i n i m u m w i t h Q g iven when P ^

$ 55(0 , roll). For t h a t v a l u e of (p .

d = Q s in0 — COS0 ]/(rg/4) — ^^.

10.4. The minimizing sequenze; the minimizing set 429

Using Lemma 10 .1 .4 aga in , we see t h a t B[P*,{r] TQ — }rocosd) 16]

GB(P, TO) n Ce which implies (see (10.4.23)) that

Ph[{r) ro - }ro cos6>)/6] < Z^h{ro) • {roIR)^/^^

w h i c h is false if ro/R is smal l enough.

Now, s uppos e K(P', r') C {U', rj ' r'), let P be the (geodesic) projec

t ion of P on IT', and le t 77* be the geodes ic ^ -p lane th rough P s u c h t h a t

T-i(77*) is para l le l to 77^, the tangent p lane at PQ = T~^{P) to r-^iU').

Since B[P,Xr')(ZB{P',r') it follows that K[P,}.r') C.{n',rj'r') so

that d[P,P) <,rj' r\ From Lemmas 10 .1 .4 and 10 .1 .5 we conc lude tha t

T - 1 [ K ( P , A O ] C ( T - I ( / I X ( 1 +r]o^r')rj'r')

C ( T I ; , (1 +rjo^ r') rj' r' + K21^ r'^)C (/7o*, 2{\ +r]o?i r') rj' r' + K^ l^ r'^).

Cons equen t ly K(P, Xr') C {IT' , }rj'r') if X is smal l enou gh. S ince

d ( P , P)ld{Q P) < ri ' r'lR' <.ZQr]' < 1/2, it follows that 77* and 77i are

a lmos t o r thogona l so t h a t K ( P , A r') C (77, rj X r') if

^ ' < ^ A / 4 and ?^'< 1/2Z6, and 1 1 = 11'^ Dili .

The fac t , tha t e' ty', and A may be t a k e n in the form C(p,p,v, W) r]^

follows from an inspec t ion of the proof.Theorem 10 .4 .2 . (cf. R E I F E N B E R G [1], L e m m a 6 * ) . For each | > 0,

there exists an so(S, ^,v,M) > 0 and a v{$, p,v,Wl) > 0 such that if

P'^S and r' <:R [P') and

(10.4.25) p < [h{r)]-'^cp-{r, P) < [A(r) ] - iy ;+(r , P) < / 5 + £0

for every (r, P) with P^S and B(P,r) G B{P\r'), then to each such

B ( P , r) will correspond a P*^S and a geodesic v-plane IT through P *

such that

(10.4.26) B(P*,vr)CB{P,r) and K(P^, v r) C {IT, ^ v r).

Moreover so and v may he taken to have the form C {^, v, W) • f ^ ,

h = h(^,v,m).Proof. K(P,r) lies at a d is tance 0 from the whole of W. T h e n a p p l y

L e m m a 10.4.3-

Lemma 10.4 .4 . Suppose G is open and does not intersect A and suppose

S C] G is not empty and y = the inf 'y;(P) for P^ S O G. Then if SQ > 0,

there exists a sphere B ( P i , ri) such that

W'-(r>P)^(y + eo)h(r), P^S, r<R{P), B{P,r) C B(Pi,ri).

(10.4.27)

Moreover, for all B{P, r) ^B{P,r)CG, P^ S, and r <R{P), it follows

that

q)-{r, P) ^y ' h{r).

Proof. Choose P i in S fl G s o tha t 'if (Pi) <y +eo/4, choose ^2 > 0

s o t h a t B(Pi, 72} CG a n d y)+(r2, Pi) <(y + eojl) h(r2), and choose n,

0 < /'I < r2y s o tha t

(y + £o) h{r2 - ri) >{y + £o/2) h{r2).T h e n , if for s ome B(P,r) in B{Pi,ri), w i t h P^S a n d r<R{P),

(10.4.27) does no t hol d i t follows from T he or em IO .4.I th a t (s ince r < r i —

- l ^ i ^ l ){y + eo/2) h{r2) >ip+{r2, Pi) '>ip~{r +r2 — n, P) > ip+{P, ^2 — n)

> ( r + eo) h{r2 - r i) >{y + eo/2) ^(^2)

which is imposs ib le . The second s ta tement fo l lows f rom Theoremi0 .4 .1(h) .

T h e o r e m 10.4.3 . v^(P) > 1 for all P S.

Proof. L e t ^ =iniy)(P) for P S. Choose f > 0 and define s a n d vas in Theorem 10 .4 .2 . There is a s phe re B{Pi,ri) w ith P i ^ S a nd

n < it! (P i) such th a t ( IO .4 .25) ho lds . W e m ay ap ply Theo rem 10 .4 .2 to

conc lude tha t e ach B{P, r) G B(Pi, ri) w i t h P^S c o n t a i n s a p o i n t

P'^^S a n d t h a t H a (geodesic) i^-plane 77 th ro u g h P * so t h a t (10.4.26)

h o l d s . We sha l l assume tha t the min imiz ing sequence , ca l led {Xn},

sa tis fies the a ddi t i ona l condi t ion in Th eor em IO .4.I (k ) . Th en , for la rge n

and fixed P a n d r

Kn ( P * , vr)C.(n,2^vr) , A' [Kn ( P * , v y jl)] > 'h(v r/2)(10.4.28) (p(vr, P * , Xn) <(p + 2£o) ' h(vr),

(A- [Kn(P^ r)] =y)(r, P * ; Xn), e t c . ) .

Hence, for each n, there i s a Qn, vrjl <,qn vr s u c h t h a t

( 1 0 . 4 . 2 9 ) A'--^ [Xn n a ^ ( P * , Qn)] < 2 ( ^ + 2£o) (?^ ^ ) - l h(vr).

Let us choose a normal coord ina te sys tem r cente red a t P* and def ine

yô - T-i [x^ n ÎP*7^)] , Ano = T-1 [x^ n a^(P*, ^^)],

//•Q =r-Hn), rnO=fn[C(no,Ano)], A^^^fnTzAnO-rnoO no,

^nO ^=ÂnOÂ^Q, Yn='TYnO, An=^TAnQ, In^=TInO,

A^ =rA^Q, A^ = r(A'^^

w h e r e ndenotes the pro jec t ion ( in RN) in to 77o , and /^ deno te s the rad ia l

p ro jec t ion on to dB(0,Qn) ou ts ide B{o,Qn'^\ — 25^2) and the expan

s i o n f r o m P ( o , Qn^\ — 2512) on toP(0 , ^^) o therwise . S ince Kn(P'^,vr)

C(n,2ivr),it follows that

Yno C(77o, 2(1 +riovr)Svr)C (/7o, S^Qn).

From Lemma 10 .2 .2 and Theorem 10 .2 .2 , it follows that

(10 4 30) A''(rn)<Z,{[i+2eo)-h(vr)-i,

i f we assume tha t 1 +rjoQn < 5/4.

10.4. The minimizing sequence; the minimizing set 43 1

N o w , s u p p o s e t h a t 7i(Yno) does n ' t con ta in the po in t QnÛo

nB{o,Qnp - 2 5 f 2 ) , l e t Qn=fn{Qn)^nonB{0,Qn), and le t gn{Q)

for Q ^ Q'^ be the rad ia l p ro jec t ion f rom Q'^ o n t o dB(0,Qn)' Let 92^

= gnfnTt and le t Zno = (fniyno)- W e no te a l s o tha t A^Q = (pni^no)-In the next paragraph , we sha l l apply Theorems 10 .3 .9 and 10 .3 .10 wi th

/ .^ . . .^ ^l = Z nO, ^ 2 = ^ w 0 , X = Z noUrnO, Ai = A ^ Q ,(10.4.31)

A2 = AnoU A*Q, AÂnO

t o s h o w t h a t b{Z noU Fno, Ano) D b(Yno, ^no)- From this , i t fol lows

from Theorem 10 .3 .6 tha t

(10.4.32) b{ZnU rn,An)Db(Yn,An), Z n = T{Z no)-

From (10.4.32) i t fol lows that we may replace Yn b y Z n U Fn a n d o b t a i nanother surface in (^(A, L). B ut , s ince Zô C H^ D dB(0, Qn), we see

t h a t A^Z no) = A''{Zn) = 0 and so

A-[Zn U Fn) = AîFn) <. Z î^ + 2£o) ' h[v r) ' ^

A^[{Xn - Yn) UZ nU Fn] < d{A, L) + En - 2 - " - ! •^'h{vr) +

+ Z2'(p+ 2fo) h{v r)'^< d(A, L)

if I an d £0 ar e sufficiently sm all an d n is suffic iently large. Thus , for such

I, £0, and n, n{YnQ ) D FTQ D B{O, Qn ] /l — 2 5 |2 ) , so

/ l ' ' ( Y « o ) > r . ^ : ( l - 2 5 | 2 ) - W ^

A-{Yn) > r . ^ ; ( 1 - 25 |2)-W2(i _|_ ô ? ; r ) - ^

Th e the ore m th en fo llows from th e a rb i t ra r i ness of | a nd SQ by le t t ing

n -^<=>o th rough a s ubs equence so Qn -^Q and then us ing the a rb i t ra r ine s s

of r.

We not ice f i rs t , us ing Theorem 10 .3 .12 tha t

b(I X Xno,Dno) D{h'\ K (h)

Dno = (0 X A„o) U (1 X A„o)

where 7»o and I„ i are def ined in tha t theorem. Then , us ing Theorem

10.3.6 with F„:(I X Ano, Dno) ^ ( A o , A„o U A*„), w h e r e

Fn(i,Q)=fn[{^-t)Q + tn{Q)], QÂ„ o,

we s ee tha t

b(rno.A„oD A*o)D {k\k=Jni^(h)-Jno*{h). heHr_ i(Ano)} = L2,

Jnl(Q)={<Pn\Ano){Q), < ? € 4 « 0 , Jnl{Q) € AnoU A^.

Jno{Q) = Q , Q€ A„o. J«o(Q)ÂnoUA*^.

Also , us ing the s ame theo rem wi th F„ rep laced by (p„, we ob ta in

b{Xi. Ai) = b(Zno, A*^) D (fn I Ano)* [b(Yn, Ano)] = Lx.

In o rde r to prove (10.4-32), we conclude from Theorems 10.3-9 and

10.3.10 that i t is suffic ient to show that

[b(YnO,Ano)]

(10.4-33) C i{Ano U ^*o> ^*o)*(9^^ I ^no) [b(Yno, Ano)] + L2,

o r / w O * [ ^ ( ^ 7 i O , ^ w o ) ] CJnl^[h[YnQ,AnQ)] + L .

But i f h^h{Yno, An^) (C Hv^i[Ano)), we see tha t

so th at (10.4.33), an d henc e (IO .4.32) holds .

Theorem 10.4.4. The set X minimizes A^{X' — A) among all X'

^^{Ay L). Moreover ip (P ) — 1 almost everywhere on S and

xp-{r,P) - : y l ' ' [ 5 n 5 ( P , r ) ]

for all r<C R (P) not in a countable set.

Proof. We have seen, us ing Theorems 10.4-1 (k) and Theorem 10.3-16,

t h a t Xo^^(A, L). For each ^ > 0, we can find a dis joint family

{B (Pi, ri)}, w h e r e 5 ^ < ^, Pi € S> a n d B {Pi, ri) f] Ai s empty for each i,

s u c h t h a t

(10.4.34) S(Z\JB{Pi,ri)\J \jB[Pi,3ri), k = Q,\,2,...

( reca l l Lemma 10 .2 .3) . From Theorem 10.4-3, we ob ta in

l r . ^ ^ < ( 1 +hQ y 2:h[ri)^{\ + hQY Z w-{n,Pi)i= l i=\ i l

0 0

<{^ +hQyZ lim inf ^(r^-, P^; Xn)

(10.4-35) ' ^ ' ** 00

<{\ +hQy Hm inf 2 ' Win. Pi', Xn)n i=l

<(i +h QY \imA''(Xn) = (i + h QY d(A, L).n

From (10.4-35), we see that the series y^rj + . .. conve rges . F rom th i s

and (10.4-34) we see that(10.4.36) Al[S) <Zr'rl<(\+hQYd{A,L).

T he first resu lt follows b y le t t in g ^ - ^ 0 in (10.4-36).

By rep lac ing 5 by 5 H P (P , r) and requ i r ing the B {Pi, ri) C B ( P , r) ,

we conc lude b y the a rgum en t above th a t

(10.4.37) A'[Sr\B (P , r ) ] < ^ - {r, P) < lim inf A' [Xn H B ( P , r ) ] .n

If, now, P^S, r<R{P), a n d A'[S 0 dB{P,r)] =0 a n d A^[Sn

ndB{P, r)] =0 for every n, we f ind by repea t ing the a rgumen t fo r the

se t S - B{P,r) t h a t

(10.4.38) A''[S - B ( P , r)] < lim inf A' [Sn - B ( P , r)] .

10.4. The min im iz ing sequenc e; the min imiz ing se t 4 3 3

Since A^'iXn — A) -Â^'iXo — A) = / l* ' (5 ) , the equa l i ty mus t ho ld in

(10.4.37) and (10.4.38). Consequently

ip(P) =lim{y ,r^)-Â'[SnB(p,r)], P^S.r->0

S u p p o s e AîZjc) > 0 and choose s w ith 0 < e < A''{Zjc)l2k. Cover Z jc

by an open se t G s o t h a t A^ [G H (S — Zjc)] < a. Again we may f ind a

d is jo in t family {B(Pi, rf)} in wh ich P^$ Zjc, ri < R(Pi), 5n <Q , a n d

B(Pi, n) C G such that (10.4-34) holds with S replaced by Zjc . Clear ly

t h e B{Pi, Yi) cover all of Z jc except poss ibly a set ^]c w i t h /1''(CA;) = 0-

T h e n

A^[Zi,) + 8 > i A - [ S n 5 ( P ^ , n ) ] > ( 1 +hQ)-^{\ + ^-1) Z yv A

> ( l + ^ - l ) ( l + A ^ ) - 4 ( Z ; , ) .

Sin ce th is ho lds for eac h e > 0 an d ^ > 0, it follows t h a t / l " (Z^)

> (1 + k-^) A^'iZjc), s o t h a t A^'iZjc) = 0. Since this is t rue for each ^, the

resu l t fo l lows f rom Theorem IO .4.3.

O ur a im now is to show tha t S = XQ — A sa tis fies R e i fenbe rg 's

{e, Ri) cond i t ion a t an y po in t PQ w h e r e ip{Po) < 1 + £0 wh er e eo

depends on ly on v, N, W, a n d s (of course e, £0 > 0 ).

Definit ion 10.4.1. A locally co m pa ct set 5 is sa id to sat is fy R eifen

b e r g ' s (e, Ri) cond i t ion a t PQ <^ ther e is a nu m be r P i > 0 5 to eachp o i n t P $ P ( P o , 2 P i ) a n d ea c h P R ) n P ( P , P ) ] < £ p ,

D[sn P(Po, 2Pi),2; n P(Po, 2Pi)] <sRi.Lemma 10 .4 . 5 . With each £ > 0, there is associated a positive number

RQ, with PQ < Po , which has the following property: If 0 < / < P Q ,

0 <Q^d<.do,andB {Q, Q) C B ( P , r) and P i B {Q, g), then

ArHS)<{\+e)ArHS') , S = B{Q,Q)ndB{PJ), S ' = S n n

n being any geodesic v-plane centered at P and containing Q. Also

(10.4.39) l^'-^\nr\B{Q, Q) n a P ( P , O l dt

= A^[nnB(Q ,Q)nB(P,r2)] -AnnnB{Q,Q)nB {P,ri)].

Remark. lfTl = RN, Ar^{S) = A^^S') withou t re s t r i c t ion on r.Proof. The la s t s t a temen t i s ev iden t s ince IT O B(Q,Q) f) dB(P , t)

m ov es pe rp en dic ula r t o itse lf on 30 as ^ var ies . T o pro ve th e f irst s t a t e

m e n t , l e t CO be a no rma l coo rd ina te s ys tem cen te red a t P with -2:-^ axis

t h r o u g h QQ = co-^{Q). Let us wri te coord ina tes of po in ts on the z^

axis in the form {a , 0) . Le t 77 > 0 an d choose PQ to sa t is fy the condi t ion

4 3 4 T h e higher dimensional P L AT E AU problems

of L em m a 10.1.3 for ^ ins te ad oi sirjisto be chosen la te r . Le t So = co^^ (S),

SQ = co- i (S ' ) , an d i7o = co~^(n). T h e n IT o i s a r -p lane th rough the z^ '

axis and 5Q = 5o Pi UQ. Sinc e ^ < (5, it is cle ar t h a t

(10.4.40) Ar' (S) = y r.i [Q ( S ) ] - i , Ar' {S') = r . _ i fc ( S ' ) ? " ^where ^ (S ) was de fined in Le m m a 10 .1 .4 . F r om th a t l em m a , we ob ta in

(10.4.41) Q(S) < (1 + rjor) ^ ( S o ) , Q{S',) < (1 + rjor) Q(S')

If we write Qo — (c, 0) , c > ^ , we see from L em m a 10.1.3 th a t

S^ C 5o C S+, S-^- = UB[c + kQ, 0; (1 - TJ) Q] n dB(0, t)

S+ = dB{0,t)r\B[c + riQ,0;{\ +ri) Q\C\ B[c-riQ,0;{\ + rf) Q].

(10.4.42)

From (10 .4 .41 ) , i t is poss ible to conclude, for ri s ma l l enough , tha t

Q[S^] < Q ( 5 o ) <.Q{St), Q{Si)<.{\+Zri)Q (So-)

e(So- n/7o) = e(5o-) <e(s') <.Q{st n T I O ) = e(5o+)

Q{S^)^[\+Zri)Q[S'^).

The desired result follows from this , (10.4-40), (10.4-41) and proper

choices of 7] a n d RQ .

Lemma 10 .4 .6 . Suppose A*{X) < 00 and £ > 0, suppose RQ satisfies

the conditions of Lemm a 10.4.5> <^^^ suppose that, r <,RQ, 0 > 0, le > 0,and 0 < i{\ -i-rjodo) < 1 — c os 0 . Suppose that for arbitrarily small

^ < ^0 that there exists a disjoint family of halls {B (Pf, ri)} such that

B{Pi,ri)CB(P,r), n<d (for each ^•)

h{ri) ^A'[K[Pi,ri)-] <. Mh[ri), ^r^K > ^0i

P^B{Pi, ri) for some i, Pi a n d P^X for each i

K[Pi, n) C (i7^, f ri) (K{Q, Q)^XnB {Q, Q))

where ITi is a geodesic v-plane centered at Pi and m aking an angle > 0

there with the geodesic P Pi. Then

cp[r,P)X) < ( 1 + e)[\ +r]or)^tp{r,P;X) -

-2-^(\ +r]or)-^[\ -cosO - i ( \ + > 7 o ^ o ) ? ' ^ 0 .

Proof. Let us cover K(P,r) — UK{P i,ri) by a countab le family

{^ {Q h Qj)} of balls C B ( P , r) a n d s u c h t h a t

(10.4.43) gj<d, 2:y^Q'i^^nK{P,r)-UK{Pi,ri)]+d.

0 iNow, us ing Lemma 10 .4 .5 wi th 1 1 j cen te red a t P and pas s ing th rough Qj,

we ob ta in

/ Ar^ [B (Qi, Qi) n 1 {P. t)] dt^(\+ s) A' [77; n B (Qi, Q))](10.4.44) 0

< (1 + e) (1 + riorYy .Q'i (1 {P. t) = X n dB(P, t)).

10.4. The min imiz ing sequen ce; the min imiz ing se t 4 3 5

If P iB{Pi, n), we note first that B{Pi, n) 0 l[P, t) is empty fort < Si, where

(10.4.45) Si = d(PPi) -ncosO ~rii(\ -\-rjod).

This expression for Si is arrived at by choosing a geodesic coordinatesystem centered at Pi and using the result of Lemma 10.1.4 in this

system. Using the same idea, we conclude that B [P i, tj) C B {Pi, Vi)

r\B[P ,t) for a suitable point P^ on PPi, where

/ . . . .^^ 2 z =^Si- {\PPi\ - n) =-ri[\- cose - f (1 + riod)] +(10.4.46)

-{-IriO, t>Si.

Thus , if 77* is centered at P and passes through Pi, we conclude th at

r

jAr"- [B{Pi, n) n 1{P. t)]dt<(i + e)x0

X / / I ' - i [ / 7 f n B {Pi n) n / ( P , <)] <<

Si

(10.4.47) = (1 + e) yl- [ilf n B (-Pi. n) ] - (1 + e) • '([ ilf n B (P,, n )

n 5 ( P , s«)] < (1 + e) (1 + rjQr)'Y,r\ -

- ( 1 + e)(1 +J7o>')-' '(T''r.rfusing (10.4.46) and Lemma 10.1.4.

Since the B{Pi, ti) are disjoint, at most one contains P and

(10.4.48) / A'f^ [B {Pio, n^) n I (P , t)]df< A' [K {Pi,, ri,)] <Mh (no).0

Adding th e results in (10.4.44), (10.4-47), and (10.4-48) and using (10.4-43),we obtain (since y,» rj < (1 -{- hSy h (vi))

jAr' [l{P,t)]dt<(i+e)(i+rjo ry A^ \K [P, r) - U K {Pi, n)] +

+ (1 + £) {^+rjory{\ +hdyA^\UK{Pi,ri)] -

- (1 + e) (1 + Vo y y o^ ee

from which the lemma follows by letting ^ -^ 0.Definition 10.4.2. Suppose ZT is a geodesic _/>-plane centered a t P and

suppose f/ is a set C B{P, R Q ) . We define the projection pn{U) of U onn as follows: Let r b e a norm al coordinate system cen tered at P. Then wedefine

pn{S) =T^pon^r-^{S)

where po denotes the ordinary orthogonal projection into r~'^{n) (whichis an actual ^-plane in Rjsf). From Lemma 10.1.1(b), it follows thatpn{S) is independent of the particular r chosen.

28*

http://-/-rjod

http://-/-rjod

http://-/-rjod

T h e m e t h o d of proof of Theorem 10 .4 .3 leads to the fo l lowing lemma:

Lemma 10.4.7. Suppose that X ^ S ( ^ , L),A''{X — A) < d{A, L) + rj,

Q •<RQ, B(P, Q) D A is empty, JJ is a geodesic v-plane centered at P ,

K(P, Q) =XnB{P, Q ) , 1{P, Q)=Xr\ dB[P, Q ) , and

K(P,Q)C{n,iQ), A-[_K[P,Q)]>:^h{Q), Z | < l / 2 , Z = = l + 7 y o ^ ,

71 + C i | ^ ( 1 +rioQ)^'{\ -ZHY^I^A^-^[l(P,Q)]<ph{Q)

C i being the constant of Theorem 10.2.2. Then

Pn[K[P, Q ) ] Dnr)B{P, Q l/l - Z 2 f 2 ) .

Proof. For, o the rwis e Y = K{P,Q) U 1{P, Q) could be rep laced by

FU Z, c o n s t r u c t e d as in the proof of T h e o r e m 10.4-3, in which yl" (Z) = 0

a n d/ i " ( r u z ) + / t ^ ( r ) < C i I e (1 + 7 0 ^)2»' (1 - Z21^2)-W 2 x

XA^-^[1(P,Q)]<A^{Y) -7J.

Theorem 10.4.5. There are positive constants ci, C2, h'> 1, and RQ

with the following property: Suppose XQ is a minimizing set as in Theorem

10.4.4 and suppose PQ^S ^ XQ — A. Then, if

0<r2<.Ro, r2<R (Po ) , 2 < oi £*, £2 < 2 (e*)^, and

(10.4.49) h{r) < y)+{r, P) <h(r){\ + 2^2) for all

B{P,r)c:B(Po.r2),

S satisfies an (e*, Ri) condition at Pofor any Ri < ^2/32.

/ / jp (Po) < 1 + £2, there is an ^2 such that 2 < RQ and r2 <. R (Po)

such that the second line holds in (10.4.49).

Proof. The proof of the l a s t s t a t e m e n t is l ike tha t of Lemma 10 .4 .4 .

Suppos e £2 > 0, I > 0, and RQ is chosen so th a t L em m a 10.4 .5 ho lds

w i t h e = 1/4, say. We s u p p o s e t h a t f2 0 s uch tha t e ach B[P, r)

C B (Po, r2) c o n t a i n s a p o i n t P* in 5 th rough wh ich pas s e s a geodesic

I'-p la ne ^ ( P * ) , c en t er e d at P*, s u c h t h a t

B[P^,vr) C P(P, r), K(P^, v r) ~ S D B{P^,vr) C (2{P'')> ^ ^ ^ ) -

Suppos e , now, t h a t

P'^S, P ( P ' , / ) C P ( P o , ^ 2 ) , Qj^K{P\ri4), j=\,...,v + 2.

W e s u p p o s e t h a t 0 < ^ < (5o, w h e r e

A ^ ( 1 + 7 y o ^ o ) < (16/15)2 .

F r o m L e m m a 10.2.3, it fo l lows tha t we may find a dis joint family of

bal ls {B {Pi, ri)} s u c h t h a t

K{r, r'l4) C U K(Pi, Qi) U U K{Pi, SQi), ^ - 1, 2, . . .

P i€ S, 5ei<d. B(Pi.5ei)CB{P',r'j4).

10.4. The minimizing sequence; the minimizing set 437

Since A^[K(Pi,Qi)] '>h{Qi), it follows that the series of / [• ' -measures

conve rges . Hence we may choos e a f in i te subfamily such tha t

Â'K{Pi,Qi)>\A'K{P',r'lA)

(10.4.50) Zh{Qi) > 2-1(1 + 2e2)-Â'K{P, r'jA)

' > 2 - i ( l + 2 £ 2 ) - i A ( / / 4 ) .

Fo r each / < v + 2 and each i,

B {Pi, Qi) C B {P\ r'lA) C B (f t- , / /2) C B {P\ / ) .

T h u s we h a v e a set of bal ls {B[Pfy v Q I ) } s u c h t h a t Pi"^ S and t h e r e

ex is ts a geodes ic r -p lane ^ (Pf) cen te red at Pf s u c h t h a t

K[Pf, Qi) C iZ(Pf), SvQi), B {Pf, VQt)GB {Pi, Qi).

Now s uppos e 0 > 0 and s uppos e

(10 .4 .51 ) (1 -cos 6>) : -3Af, X=\+rjodo.

L e t Oj be t h e set of i 5 the geodesic Pf Qj m a k e s an angle > 0 at P f w i t h

^ ( P f ) . Suppos e for s ome j and s ome // > 0 t h a t , for a rb i t ra r i ly s ma l l d,

we wou ld have

(10.4.52) 2JyvV-Qlêe=jLcv^(r'l4Y, JLOO.ieOj

Then , f rom Lemma 10 .4 .6 and (10.4.51), it would fo l low tha t

(10.4.53) (p{r'l2,Qj) < v ^ ( / / 2 , ft) -eeX'^'{\ + £i) (1 + ^ o / / 2 ) - ^

But, from (10.4-49), we h a v e

^p[r'l2, ft-) - 99( / /2 , ft-) < 2^2(1 +hy'l2)-'y,[y'l2Y

which contradicts (10.4-53) if

(10.4.54) £2 < 2-i-» ' / i (A?; | )» ' (1 + £i) (1 + c o / 2 / 4 ) - v ( i + hr'liy.

T h u s the reve rs e inequa l i ty mus t ho ld in (10.4-52) for all sufficiently

s m a l l d.

Since there are on ly (r + 2) ft-, we may choose a ^ so s m a l l t h a tt h e r e is an io w h i c h is no t in an y Cj, p r o v i d e d m e r e l y t h a t [ji is chosen

s m a l l e n o u g h so t h a t

(10.4.55) [v + 2)fi< 2-1(1 + 2£2)-Ml +^r'lA)-''yv\

th is follows by comparing (10.4-52) (with the inequa l i ty reve rs ed ) and

(10.4.50). Thus for s ome i^ the geodesic Pf^ ft- m a k e s an angle < 0 at

P^* w i t h ^{Pf^ for eachy. Thus, us ing this fact , (10.4-51), and the proof

of Lemma 10.1.5 (the c u r v a t u r e of arcs corresponding to geodesies) , we

h a v e s h o w n t h a t if ft, . - -, ft+2 are any {v + 2) p o i n t s in B[P\ r'jA),t h e r e is a geodes ic r -p lane ^ cen te red at a p o i n t of K{P', r'jA) s u c h t h a t

each ft- is w i t h i n a d is tance

(10.4.56) ^ '^[svae + \K, g ) ] ^ [ | [U^l^ + \K, (-)] ^ ar'jl

otZ(=I{Pt).

Let T be a geodesic coordinate system with dom ain B (0, RQ) for

which T(0) = P'. Choose (?*, . . ., (?*+! in K{P',r'l4) so that yl''(zl*) is

a m axim um , J * being the r-simplex in R^^ having the r~ [Qf) as vertices.

Let Q be any other point of K{P', r'jA), le t ^ be the geodesic 7^-planeof the preceding paragraph determined by Q and the Qf, suppose ^ is

centered at P{^K[P', r'jA))y and suppose ^ o is the tangen t r-plane in

Rjsf to T~ ( ^ ) at T~i (P ). Then it is easy to see, using L em m as 10.1.4 and

10.1.5 th a t the distances from ^ o of T ~ ^ ( 0 and the r~'^{Qf) are all

< ( 1 + ^0 74) c r / /2 +7^2/2 /1 6 = a ' / / 4 ,(10.4.57)

o'' = 2(1 + ^o / /4 ) (T + i ^ 2 / /4 .

Let AQ be the [v + 1)-simplex having Zl* as one face and T~'^[Q) asopposite vertex. From our construction it follows that the A* measureof each face of Zlo is < JD = yl*'(Zl*). Consequently the A^ measure of theprojection of Zio on ^ o is < (r + 2) D and hence

*'+M o) < 2 Z ) ( T ' / / 4 .

Since Zl* is one face of Zlo, it follows that the distance of r~ {Q) from ther-p lan e of zl* is < (r + 2) a' r'l2. Since Q is any point of K[P', r'l4), itfollows that T"i {P') = 0 is at a distan ce < (r + 2) a' r'j2 from that

plane. Thus T ~ ^ ( 0 is at a distance < (i + 2) o* '/ from the planethrough 0 parallel to that of Zl*. The image of this plane under r is ageodesic v-plane ^ through P' and we have shown that

( 1 0 .4 .5 8 ) i ^ ( P ' , / / 4 ) C ( 2 ^ ,£ ' / / 4 )

provided that

(10.4.59) 4(1 + ^0^74) [v + 2) or' < £ '.

Now, from (10.4.49) and (10.4.58), it follows that there exists an x

between ;'78 and ^74 such that^' ZL"-! [/ (P' , x)]<{\ +2e2)h [r'jA) or Zl^-i [I {P\ x)] < Z i x-^ h {x),

A'[K(P', x)] >h{x), K[P\ x) G (2; , 2e x).

Since Xo is minimizing, it follows from Lemma 10.4.7 that the projection of K[P', x) on 2^ contains 2^r\B (P ' , x j / l — 4^2 e'2)^ 7 being thenumber (near 1) mentioned there, provided merely that f is small.

We now show that 5 satisfies an (e*, R{) condition at PQ if

(10.4.60) P i < f2 /3 2 .

Let P^K(Po, 2 Pi ) and R < Pi . Then B (P, P) C P (Po, 2/8). If we takeP' = P and / = 8P, then B(P\ r') C B(PQ, r2) and we may take2 ' ( P , P) = 2 '- U sing (10.4.58) we find t ha t

K{P, P) = K(P\ / /8 ) C (Z (P> ^ ) . 2 e' P ) .

10.5- The local topological disc property 43 9

F r o m the p r e c e d i n g p a r a g r a p h we conc lude tha t the pro jec t ion of

K{P, R) on^ = ^{P,R)z:>2^r\ B{P, R ]/i - 4Z 2 e'^), so t h a t

Z{P, R) n B{P, R) C[K{P, R), 2e' R + R{i - ] / i - 4^2 s'^)]

(10.4.61) C [K(P,R),'}s'R]

( l - ]/l - 4^2 e '2) < £'.

The cases where P = PQ are t aken ca re of in a s imila r way by s e t t ing

P ' = P o , / = \6Ri.

N o w , by following back through equations (10.4-61), (10.4-59),

(10.4.58.), (10.4.57), (10.4.56), and by choos ing do and RQ S O t h a t

A = 1 + 7 7 0 ( 9 o < (1 6/1 5)2 , 7 y o i ^ ; ' < 1 , Ro<Ro>

w e see t h a t it is sufficient to h a v e

(v + 2) (100fi/2 + 105i^2 ^ 8 ) < £*,

and f rom Theorem 10 .4-2 , we may take £2 of the form Z • ^^.

1 0 . 5 . The local topological d isc property

In th i s sec t ion , we pro ve wh at is proba ly the mos t in te re s t ing theo rem

in R E I F E N B E R G ' S first paper:

Theorem 10.5 .1 . There are positive numbers EQ — SO{N,W) and RQ

= RQ (N, W) < RQ such that if 0 <, s <, SQ and if the locally compact set

S satisfies the {e , Ri) condition at PQ for some positive number Ri < RQ,

then a neighborhood of PQ on S is a topological disc.

Definition 10.5 .1 . If ^ and ' are r -p lanes in R^ we define

w h e r e ^ and ' are the re s pec t ive ly pa ra l l e l r -p lanes th rough 0. If c

a n d c' are or thogona l ma t r i ce s , we define

\c-c'\=\Z\ci-'cim^ {c = {ci),etc.).

W e no t ice the following facts :

L e m m a 10.5 .1 . (a)If c is any orthogonal matrix, |c | = ^N.

(b ) / / Si and S2 are compact sets, P is a point [in W), R, rj, r\ > 0,

and SiHB [P, R) C (52, r)') f) B {P , R)> then

(5 i , ri)nB [P, R) C (S2, r) + r|')f^B (P , R).

(* ) U2 <^ ^ .Z " ^'^^ v-planes through P in RN and if D [^ H B ( P , R),

2;'nB ( P , R)] =rjR, then do {Z > I') = V-

(d ) If c and c' are orthogonal matrices and c ^ ^ ^ 2c' ^^^ v-planes in

RN s panned re s pec t ive ly by ci, . . ., c and 'ci, . . ., 'cv, then

do{2e,Zc')<\o-c'\ {c = ( 4 ) , c^ = {cl,...,c^).

Proof. Parts (a), (b), and (c) are easily verified by the reader. To prove(d), we first note that if ^ c and ^g , pass through 0, then^ o(^c> 2^) i^just the max imum distance fr o m ^ c of any point of 2Jc' ^ dB{0, 1). Let

us now write(10.5.1) c^ = cj + 2eôjc.

k=l

A point P will be on ^Jc^ C) dB{0, A) <^

(10.5.2) o^p^;^^xic] ^tVo + il^î o^ '4, I'% I = 1.9 = 1 3 ^1 L . P = l J

If we let d = d ('xp) denote the distance of the corresponding point Pfrom ^, we see that

(10.5.3) d^ = ;^{xlY, ^p = 2 ' 4 H ' ^ > ^ + l .

From (10.5.1) and (IO .5.3), we obtain (by taki ng t he max.)

j = l fc=v+l 1,k=l

from which the result follows.Lemma 10.5.2 (Extension lemma), (a) There is a constant C^{^) > 1

with the following property : Suppose {j)i\ is a finite set in 3JI, r >• 0, > >• 0,

the fi are vectors in some RQ and we haved{Pi,p,,)>r if i'î

\fi—fi,\<.rj if i'î and \pi — pi.\ <. 6r.

Then there is a vector function f (into RQ) ^ C^ on U B{pi, 5^/2) such thati

fiPi) = fi /<^^ ^^^^ ^'\^f{p)\^C^Vlr, | V2 / ( ^ ) | <C 4 - 7 7 / r 2 , pÛB(Pi,Srl2),

If \fi I < M for all i, we may take f so that \f{p) \ < M.(b) There exist constants C5 (M) > 1 and di (M) > 0 such that theextension theorem of part {a) holds for orthogonal matrix functions providedthat T) -< di (M) and we replace C4 by C5.

Proof, (a) Let ^(5) ^ CîRi) with (p'(s) < 0, 9?(s) = 1 for s < 0, and^ (5) — 0 for s > 1, and define

(10.5.4) h{p>) = ^[ j ^ J,

Mp)=Up)n[\-%[p)'\, k{p)=k{p)llMp)'i+i i

I t follows tha t

}^{p) = 1 if d(p, Pi) < rl4. k{p) = 0 if d{p, PJ) < r/4, j i- i,(10.5.5) k(p) = 0 if dip.Pi) > i3r/4, \v^MP)\ <î(5K) •»•-*.

Sk{p) = 'i, 2 ' V * A j ( ^ ) = 0 , pÛB(Pi,5rl2), k = i,2.

10.5 . The local topological disc property 441

Th e bo un d for |VA ^(^) | does n o t dep end on th e nu m be r of po in ts pi

s ince the number of non-zero te rms in the express ions for S7Xi(p) a n d

VXi[p) for agiven pis <t h e m a x i m u m n u m b e r of pj in the ba l l

B{p, 13W4).Then we def ine

(10.5.6) mIfi'kiP), pS=UB(P,Srl2).i

Now, s uppos e pSand ca l lp o the nea re s t pi a n d / o = f(po). Then s ince

2! V 2.i{p) ^0, we have

V m - K f i - /o) V kiP) = 2:{fi- /o) V k(p).i d(2>.2>i) '>13r/4

Since po i s th e neares t pi to p, w e h a v e d(p, po) <5 ^ /2 s o th a t th e n um be r

of te rms in the la t te r sum is aga in <m a x i m u m n u m b e r o f pj in B(p,

13^/4) and this depends only on 30^. Clearly, ifd(p,pi) <13; ' /4 , then

^{po,pi) <6^ s o tha t e ach \f i ~ / o | < ' ^ . T h e f ir st r e s u lt f ollo w s a n d

the las t s ta tement follows from (10.5.6).

To pro ve (b) , we mere ly e x te nd each of the vec tors Cj as in (a). HT]

is suffic iently small , the ci, . . ., cjsf as ex tended a re l inea r ly independen t

on 5 and may be rep laced by an o r thonorma l s equence by app ly ing the

G r a m - S c h m i d t p r o c e s s .

Definit ion 10.5.2 . Given an n.o . se t c =(ci,..., C;^) a nd a r - p l a n e în jRiv- Let 1 1 be an {N ~r)-plane J_ ^ an d sup pose th a t cf, . . ., c*

are the respec t ive pro jec t ions of ci, . . ., Cv on ^ a n d c^+i, . . ., c^ a re

those of Cvî, . . . , Cjsf on U. For those (c , ) for which c*, . . . , c* span

^ a n d c*+i, . . ., Cj sp a n i7 , we def ine

where c^, . . ., c^ an d c^+i, . . ., c^ are ob tai ne d from th e r espe ctive sets

cf , . . . , c* and c*+i, . . . , c^ by the Gram-Schmidt process .

Definit ion 10.5.3 . A finite set (^^}is onr-setfor asetS ^d(pi, p^,) >rif ^'^ 9^ ^' and 5 C UB{pi, r).

L e m m a 10.5.3 . There are numbers d^lW) >0 and CQ{^) >1 SWC/J

that

\A{c.Z) -A{c', 2 " ) i <C 6 [ | c - c'l +< ^ o ( 2 ' . 2 " ) ] .

provided that

Ic — c'I < 2, ô ( X m ^2, ô ( X -Z'c) < ^2,

ô(2" ,2 ' ; ' )<^2Z£ A ;' ^ c OLnd ^ ^ , <^re if/ ^ v-planes through 0spanned hy (c i , . .., Cy) a^ t^

( q , . . . , c^). In fact, Ais analytic in (c, ^ ) z£Â ;' ' is defined.

Proof. Th is l a s t s t a tem en t mea ns tha t if we s uppos e tha t c" = A [c, ^ ) ,

w h e r e ^ i s s panned by the vec to rs c^ \ . . ., cJ', then the vec to rs c/ a re

an aly tic fun ction s of the co m po ne nts of th e vect ors (c i, . . ., cjsf, c{', . . .,

4") . Since the d o m a i n of definit ion of these func t ions is open and con

t a i n s the c o m p a c t s u b s e t of vec tor sequence (ci, . . ., c^, c^\ • • •, c'J') in

which ci, - . ., Civ is an n.o. set and c^' z= Cj, j = 1, . . ., v, the l e m m a

follows.

L e m m a 10.5.4. There exists a positive number RQ < RQ with the

following property. Suppose S is a locally compact set on ^ , YQ^W,

S C B(YO,RQ), and r is a normal coordinate system centered at YQ with

domain B(0, RQ) {andr(0) = YQ). Then

(i) if S satisfies an {s , Ri) condition at a point P and B ( P , 3 Ri) C

B{YQ, RQ), then r~^(S) satisfies an [e', iî/2) condition at r~^{P), where

e' = 2e + 4KR;

(ii) if So ^ T~^ (S) satisfies an (e, Ri) condition at a point PQ andB (PQ, 3iî) C B (0, RQ), then S satisfies an (s', iî/2) condition at P = T ( P O ) .

Proof. We choose RQ < RQ so t h a t

:^d{p, q) < d[T{p), r{q)] < ]/2 d(p, q)

w h e n e v e r p and q are in B (p , RQ) and T is a n o r m a l c o o r d i n a t e s y s t e m .

S u p p o s e t h a t So satisfies an (e , Ri) cond i t ion at PQ and t h a t B {PQ, 3 i ? i )

C -6(0 , RQ). T h e n r[B(Po, 3i? i ) ] C B(Yo, RQ). Choose R, 0 < R < RxjZ,

anAQ^B (P, Ri) w h e r e P = T (PQ) • T h e n

SnB{Q,R)cSn r[B{Qo.Ry2)] n B(Q, R) (Qo = T - I ( 0 )

= r[SonB{Qo.R]/2)]nB(Q.R)

CrlZoiQo, R]f^),sRy2\nB{Q.R)

(z{r[Zo{Qo.R]f2)].2sR}nB{Q,R)

CiSiQ.R). 2sR + 4KR^]nB(Q,R)

= [S(Q.R).e'R]

2^(Q,R) be ing the geodes ic p lane t angen t at Q to r\2o{Qo, Ry^)] (see

Lemma 10 .1 .5) . L ikewise ,

2 ' ( a R) nB{Q, R) c r{T-HS(Q, R)] n BiQo.Rfi)] n B(Q, R)

C r{i;o{Qo.Rf2) nB{QoRf2). 2KR^} 0 B(Q, R)

C r{So n B{Qo,R]f2), 2KR^ + 8R']f2} C\B{Q, R)

C{Sns'R)nB(Q,R), 0<R<Ri!2.

T h e r e m a i n d e r of the proof of (ii) and t h a t of (i) is s imilar .

We now reca l l the d e v e l o p m e n t s of § 10.1. We s u p p o s e t h a t P o € 501,

t h a t T is the assoc ia ted coord ina te sys tem of class C^ and satis fying

(10.1.1) , and t h a t coop is a, norma l coo rd ina te s ys tem cen te red at p, the

coop a n d r being re la ted as in Le m m a 10 .1 .1 (d) an d sa t is fy ing th e b ou nd s

10.5- The local topological disc property 443

in our genera l condi t ions on 3 R. The va r ious Z i will depend only on v, N,

a n d Wl, unless o therwise spec if ied , and may denote d i f fe ren t cons tan ts

in different proofs .

Lemma 10 . 5 . 5 . There are positive numbers rjQ{Wl), RQ(TI\), Kz(Wj,and K4(W ), with RQ < iô ^fid rjQ<, 1/2, with the following property:

Suppose p and q^B(0, Roj}), d{p, q) == ri <RQ, 0<r2<.Ro, ^ ^

^ {py 2Q) = ^3 ^ ^ 0 ' ^2* Suppose olso that ô is a v-plane Rj:^ and ^p

= coop(2o)' ^ ^ ^ 2Q =o)oq (2Jo)- Then

(10 .57 ) D [Zv n B[p, r^), Z ^nBip, rg)] < Ks{ri + r2) r^ + K^ r^.

Proof. L e t ^ o p b e t h e r -p l an e i n Rjsi t a n g e n t a t ^ o = ^ôqiP) t o ^ J ^

= o)Qq{Zp)' Clearly V 0)^^ ° coo^(O) differs from the identi ty matrix by

a ma t r ix o f no rm ^ Zid(p, q). T h u s(10.5.8) d{2;op.Io)<=Z2ri, d(Po. ô) <. (^ +Von) rs.

Let us le t F= COQ^ [B(p, ^2)] and let ^Q be the i^-plane | | ^ o t h r o u g h po.

Th en, we conc lude f rom (10 .5 .8) an d Le m m a 10 .1 .5, e tc . , t h a t

D[2:op nr,Zor\r]<Zznr^, D[Z' nr,2*0n r j <z^r^D[Zopnr,z$^nr]<K2ri

I t fo l lows tha t

D[ISvr)r,Zonr]<Zsnr2 + 2 1 + 74^3.The result follows s ince ojoq{2^^^ 0 F) =21 P H B{p, ^2), 0)0^(2^^0 0F)

= lQnB(P,r2).

Definit ion 10.5 .4. The geodes ic r -p lanes ^ â n d ^ âre said to be

parallel with respect to r and the family {coop}.

Lemma 10.5 .6 (The c - lemma). There are positive numbers QO(M) < 1/2

and ds {W) such that to each pair of numbers {rji, Q)for which 0 < ^ < qî^)

and 0 <,rji •< dz (W) Q, there correspond positive numbers ei [rji, Q, Tl)

and RQ' {fji, Q, W) such that if the locally compact set S satisfies an {e, Ri)-

condition at apoint PQ, with 0 <i e <, si and 0 <C iî < RQ', and the

coop have their significance above, then there exists asequence {cj[p)] of

orthogonal matrix functions with Cj^ C^(S'^, 3^;/2) such that

(10.5.9) (i) \ cj{p)\ ^7]irr ;

(10.5.10) (ii) \cj(p) - Cj_i{p)\ < Ce(2e +^1/4^) + (1 + Q)r]ilS;

( 1 0 . 5 . 1 1 )(iii)

ifp^S^\y > 1, and

IHP) =a)op[2;i{p)]>

ZKP)

being the v-plane inRN through 0 spanned by Cji{p), . . ., Cjv(p), then

D[Sn B(p, 6rj), Z HP) n B{p, 6rj)] < r j i rj.

Here, we define

(10.5.12) ro =RilS, rj =roQ^, S* = S n B{Po, 2Ri - 6ro).

Proof. We def ine CQ{P) as a con s ta n t m at r i x su ch t h a t co i , . . ., cor

s p a n s O)QP^{^). Since ^l[p) = ^^PO(2J)> ^^ follows from the (e, Ri) con

d i t ion an d L em m a 10.5 .5 (wi th ri = d(p, PQ) <, \0ro, ^2 = 6^0, ^3

= ^(P>2) < 8 £ ? ' o ) t h a tD[Sr\ B(p, 6ro), 2nB{p, 6ro)] +

+ D[2;n B{p, 6ro), Z îP) n B[p, 6ro)] < X 3 • l6 ro • 6^0 + K^ - Sero

w hic h implies (iii) for y = 0, pro vid ed th a t

(10.5.13) 96K2ro + SK4 e < 7 y i .

O bvio us ly (i) ho lds .

N o w s u p p o s e t h a t Cj_i(p), / > 1, ha s be en defined to satisfy (i) an d

(iii) above. We define Cj(p) as follows: Let {pjt} be an TJ set (Def. 10.5-3)for

S' = Sr]B{Po,2Ri),

F o r e a c h i, select a geodes ic r -p lane ^ | = coop.^ (Zlo) t h r o u g h pji s uch

t h a t

(10.5.14) D[Sn B(Pji, Srj), Z i n B{pî, 8r;)] < S s r , - 1 .

I t fo l lows tha t

Z l n B[Pn, Srj) C (5, 86 rj) D B{P ji, Srj)

(10.5.15) C (S, Ssrj) n B{pji, 6rj_ i) D B[pî, 8rj)

C il^-HPji), Serj + rjirjî) H B{Pji, 8rj).

A p p l y i n g COQ^.^ an d Lem m a IO .5 .I to (IO .5.15) , we f ind t h a t

(10.5.16) doiZio^ ItHPji)) < (1 + 8 7 0 ,-) {s + ml^Q) <2e + ml'^Q-

iZio = (ôpu{Z i)> e tc . ) p rov ided tha t

(10.5.17 ) 87^0^0 < 1,

s o tha t we mus t have

(10.5.18) 28 + r]il4g<d2.

If (10.5.18) holds , we may then define

(10.5.19) Cj-{Pji)=A[c^_i(Pji),Zio\'

From Lemma 10 .5 .3 and the fac t tha t

(10.5.20) oj_ i(Pj'i) = A[cj_ i(pji), Z trîPji)],

we conclude, us ing (10.5.16) a lso, that

(10.5.21) \c3 {pH)-oj_ i(Pji)\ <C e(2s + rjil4Q).

1 W e shall omit t h e bars over t h e various se t s .

10.5- The loca l topo log ica l d isc p ro per ty 4 4 5

U sin g (10.5.14) an d L em m a 10.5.5 w it h î < 6^^, r^ = 2rj, a n d

rs = SeTj, we find that if d(pji, pji') < 6^; , then

2^1 n B[Pji, 2rj) C (5 , Serj) H 5 ( ^ n , 2^,-)

C (S, 8erj) O ^ ( ^ ^ ^ 8^,-) 0 5 ( ^ ^ , 2;^;)

C ( 2 ' l ' . ' l 6 ^ r ; ) n ^ ( ^ ^ , 2 r , . )

C (2 ' l s ^6 £^ ; + X3 • I6r? + X4 • 8£r, -)

^ f be ing the geodes ic r -p lane cen te red a t pji a n d p a ra ll el to ^ j ' ( w it h

respec t to {coop}). Proceeding as in (10.5.16), we obtain

(10.5.23) (do 2'lo> Ih ) < (^6 + SKi) s + I6 i^3ry

pro vid ed th a t ( IO .5 .17) ho lds . Fr om (IO .5 .9) fo ry — 1, we f ind th a t

(10.5.24) \cj_i(pji) — cj_ _ i{pji')\ < ^ ^ ^ i if d{pji,pji') < 6rj.

Therefore we conc lude f rom Lemma 10 .5 .3 and equa t ions (10 .5 .19) ,

(10.5.23), and (10.5.24) that

(10.5.25) IcîPu) -Cj[Pn')\ < C 6 [ ^ ^ ^ i + \6s + %K ê+\6Kzr,^

p r o v i d e d t h a t

(10.5.26) ^ ^ ^ 1 + ^ ^^ + 8 i ^ 4 e + 1 6X 3^ 0 < ^ 2 .

If (10.5.26) hold s we conc lude from Le m m a 10.5.2(b) th a t th er e exis ts a n

or thogonal matr ix func t ion of c lass C'^[S', '}rjl2) which is an ex tens ion

of Cj as so far defined with

IV c^{p)\ < C5 Cî^^Qrii + 16£ + 8X 4 £ + l6 i^ 3ro ) rr^

(10.5.27) ^ 1 _i

if we choose

(10.5.28) C5 C^{6Qrii + 3 2 0 £ + 1 6 0 ^ 4 ^ + 320X 3^0) < >7 i

(C5, C 6 > 1 ) .

We mus t now prove ( i i i ) . Le t p^S^, Th en th ere is a ^ ;^ such th a t

^(^» A'^) ^ ^ j - Using Lemma 10 .5 .5 wi th ri < r^, ^2 = 6^;, a n d r 3 = Se r^,

w e f in d t h a t ( ^ | b e in g {coo p}-parallel to ^ \ t h r o u g h p)

(10.5.29) D[2\ n ^ ( ^ 6r,-), Z \ n ^ ( ^ 6r,-)] < K4 • 88 ;^ ; + 42 i^3^ , ? .

Us ing Lem m a 10 .5 .1(d) an d (i ), we conc lude th a t

(10.5.30) D[2\^B{p, 6rj), Z np) n B(p, 6r; ) ] ^±rj^rj(\+6rjorj)^

U sin g (10.5.29), (10.5.30), (10.5.17) a n d (IO .5.28), we see th a t

(10.5.31) D[Zi n B{p, 6rj)^Z^(p) n B(p, 6rj)] < { ^ ^ i ^ / .

T h e nSr\B{j>, 6rj) C S n B{pî, 7Yi) n B[p, 6rj)

(10.5.32) ^ ^^^^ ^^^^ ^ ^^^ ^^^^ ^ (inp),nirj) n B(p, 6rj),

In l ike manne r , we s ee tha t

ZHP) n B{p, 6rj) C ( 2 ^ 1 , { ^ ^ 1 rj) H B{p, 6r^) 0 ^ t o ^ , Jr^)(10.5.33) ^^^

P a r t (iii) follows from (IO .5.32) an d (10.5.33).

Since d{p,pji) <.-rj, we obtain, from (i) and (10.5-21),

(10.5.34) \cj{p) - Cj_i{p)\ < Ce{28 + rjil4Q) + (1 + ^) ml^

from which (ii) follows.All th e condit ions in eq ua tio ns (IO .5.I2), (10.5.17), (IO .5.I8), (IO .5.26),

and (10.5.28) can be satis f ied by requiring

Tyo iî < 1, I8 C 5 Ce ^ < 1, (960 + 48O K4) C5 CQ e^rji

(^^•5.35) rj^^d2Q. \20Kz C5 CQRi<.rji

s ince Q < 1/2.

Proof of Theorem 10.5 .1 . W e s uppos e tha t the loca l ly compac t s e t S

satisfies an (a , i ? i ) -condi t ion a t PQ a n d t h a t Q, rj2, a n d 61 are pos i t ive

numbers s a t i s fy ing(10.5.36) 0<Q<Qo{m)<^, 0 2ô being the i^-plane in

Rx^ s p a n n e d b y ei, . . ., ev. W e s uppos e t h a t 0 < 171 < d3(W) Q, 0 <. e

< £i(>yi, Q, 9Jl), 0 < iî < i? o(î, ^, ^ ) , an d t h a t {cy(i^)} is a seq ue nc e

of or thogona l matr ix func t ions sa t is fy ing the condi t ions of Lemma 10 .5 .6 ,

t h e Tj being def ined in (10 .5 .12) . We re ta in the no ta t ions of tha t lemma.

For each j , we define

(10.5.37) A; = r]2roZ Q^> Sj = S"^ 0 B(PQ, 2RI - 6ro - Aj), ; > 0 .

For eachy , we choose the rj-set{pji} for 5* of Lemma 10.5.6 and define

t h e m a p p i n g s aj^ C^(S'^, 3 r; /2) b y

ajip) = coj^lrjjip)], rjjip) = Z hiiJ>) S [^^;(^^)? ^^>

(10.5.38) i ^="+1

cojpiy) =a}op [cj{p) y ] , ; > 1 ,

where , fo r eachy , the ^ji{p) are defined by (10.5.4) in terms of the^^^. We

then de f ine the mapp ings TJ^ C^{TQ) and the Tj b y

ro = o)oPo\Tl rjâjTjî, Tg = I ' D H ^ ( 0 , 2 i? i - 6 ro - Ao)

(10.5.39) T,- = rjm) = aj{T^_ i),

We div ide the remainder of the proof in to severa l par ts .

http://20kz/

http://20kz/

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10.5. The local topological disc property 447

Remarks. To orient the reader, we notice that if 9!K == RN cind SG 2

•= ^Q, then we may take

COOP, = I, cooz{y) =y + z, coQ^(y)=y — z, Cj{z)=I.

Then, since the pji == Zji^^^ all the ^^^ = 0 if > v, so that

i k=v+l k=v+l

since 2 ^Jii^) =" '^' ThusN V

aj(z) = z— 2J zê}c =2! zêjck=v+l fc=l

and is the projection of 2: on 5.

Part 1. Ifrji and hence e and Ri are sma ll enough, we verify the following facts hy induction on j

(i) d[aj{p), p] < ri2 r^ fo r p T^_i, ; > 1;

(ii) TjC(Sj,r]srjî), rjs =rj^H <^196, y >0

(10.5.40) (iii) the angle at pbetween ^ (p) and Tj is < 0i if p Tj

(iv) Uip, q) <d[aj{p), Ojiq)] < 2d(p, q) if

Proof ofpart 1. For j =0, TQ =2J C) B (PQ, 2Ri — 6ro — Ao) and

So = S n B{Po, 2R1 — 6ro — Ao) so (ii) follows provided merely that

(10.5.41) SsQ-^<rj3,

To prove (iii), we use the normal coordinate system a>oo = cooPo- Clearly

(10.5.42) cOo-oMô) = T«, coôniHP)] = coîmp{T'o)-

Thus O)OQ[Z^(P)] is part of the locus of the equation

(10.5.43) w=U(y;p,Po), y IlFr om § 10.1, we conclude th at

<:Zid[p,Po), \X\ = \ \ =1

I l=v+l m=l I

SO that (iii) follows provided merely that

(10.5.45) Z2d{p,Po) ei.

Suppose now that (10.5.40) is true for y — 1. Suppose _^^ Tj_i . From(ii) for y — 1, we conclude that there is a point p' ^ Sj C S* within a

distance < rjs rj oip and hence one of the^^-^-, call it q, is within a distance(1 + ^3) ^j ofP' From the (s , iî)-condition and theconstruction in

Lemma 10.5.6, we see that

(10.5.46) 5* n B{q, Srj) C(2'^'(^), Ssrj) H B{q, Sr^)

4 4 8 The higher dimensional PLATEAU p ro b l ems

SO t h a t all th e pî in B {q , STJ) ^ (^^' {q),Se Vj) and (since p^{S*, r}^ T J ) )

(10.5.47) PîU'iq)^ êrj + ri^fj).

So we use the norma l coo rd ina te s ys tem cDjq and define

îai^P) = ^> ^no[z) = hi{P)^ ^Jo{z) = Cj(p)

(10.5.48) _ N Vtj{z) = co^'qâj[cojq{z)], tjo = z — 2 zêjc = Z ^^ ^Jc-

k=v+l fc=l

We sha l l show in p a r t 2 b e l o w t h a t

(10.5.49) i9i^)=i3o{z)+tf{z), tf {0) = 0, \^tf{z)\

From (10.5.47), it follows that the d is tance

(10.5.50) d{z, Zo) < (1 + ^srj) (Ssrj + rjsrj) < 2r]sr^

if the l a s t inequa l i ty ho lds . Thus \tjo(z) — z\ = d{z, ^ g ) , so t h a t

\tj{z) -z\<{r]4 + 2rj3)rj(10.5.51)^ ^ d[p, (Tjip)] < (1 + Zsrj) {r)4 + 27^3) r^ <rj2rj

if the las t inequa l i ty ho lds . S ince d[aj(p), q] < (1 + ? 2 + ^3 + Se) r-;

< Svj, we see from (10.5-46) and the c o n s t r u c t i o n t h a t

d[(yj{p)> S*] <D[S^n B{q, Srj),2;nq) r\B[q, Sr^) +

(10.5.52) + d[a^{p),2nq)] < 8srj + (1 + Zsrj)d[tr{z), 2'g]

< 8£ry + (1 + ^ 3 rj) rj^ Yj < ^3 ^;+i

(since tj{)[z) ^ ^ g) p r o v i d e d t h a t

(10.5.53) r H ^ £ + ( i + ^3^0)^4] <^3.

T h u s (i) and (ii) are verified.

To verify (iii), we note f i rs t tha t the angle at ^ b e t w e e n 2j^~'^{P) ^^^

Tj_i is < 01 so the angle at z b e t w e e n the t a n g e n t p l a n e s to coj'^ [ ^~^ {p)]

a n d o)Y^(Tj_i) at 2: is < 20i . Now, ev ident ly , coj'q [2^^~^ {p)] is th e locusof the equations (see (10.5.38) and § 10.1)

(10.5.54) w = a),-/co;_i,2,(y) = [oj{q)]-Û[c^_i{p) y',p, q], y^Zo-

F r o m § 10.1 , we see f i rs t tha t

U\m[cj_i(p) y;p, q]=di+ ei[cj_i(p) y;p, q], \8\ < Z^r^.

L e t t i n g S/wand e d e n o t e the m a t r i c e s dw^jdy^ and e^, respec t ive ly , we

o b t a i n

Vw = fe-(^)]-i[/ + e\ cj_i(p) = [cj(q)]-Uj.i{P) + £1(1^11 <^5^;-)

= ^+ [(0j{q)]-Hcj{p) - cj(q)] + fe-(^)]-i [c,'_i{p) - Cj(p)] + si.

Since, on our locus coj'^ [2'^~-^{P)^' J ranges on ly over ^ g , i t fo l lows tha t

the ang le be tween tha t locus and g is

(10.5.55) < ^6 \r^ + ( W 2 0 ) + C6(28 + 1 1/4 ) + (1 + ^) 8 ] < 6)1

10.5- The local topo log ical d i sc p rop er ty 4 4 9

i f the las t inequa l i ty ho lds . Thus the angle be tween 0)^^ (Tj_i) a nd ^ g

is < 3 ^ 1 .

Now le t dy be a vec to r t angen t a t z t o co^^(Tj_i) and le t dw he t h e

co r re s pond ing vec to r t angen t to o)J^{Tj) a t tj[z). Let us wri te dy= dyi + dy 2 w h e r e dy i is the projection of ^y on ^Q. T h e n

dw = dyi + V tf (z) dy = dy i + dw"^, \dw*\ < ? ^ 4 \dy\

(^0.5.56) 2-1/2 | ^ y | <^cos}di'\dy\ <\dyi\ <\dy\

s ince 3 0 i < jr /4 and th e ang le be tw een ^ g and t he t an ge n t p lane to

cjoJg^(Tj_i) a t ^ i s < 3 ^ i - A n easy a rg um en t , us ing (10 .5 .56) shows t h a t

2 -1 \dy\ < |dfz£ | < 2 \dy\,

s o th a t ( iv ) ho lds , an d the ang le be tw een ^ g an d the t an gen t p lane to

coY^(Tj) a t tj{z) is < arcsin rj4^ ]/2. Now, co,^^[^^*(^')] is seen, as in

(10.5.54), to be the locus of the equations

(10.5.57) t^ = [cj{q)rÛ[cj{p')y ;p\q]^ y ^Z l P'-ÂP)-

T h u s t h e a ng le b e t w e e n ^ g a n d t h e t a n g e n t p l a n e a t tj[z) t o ooy^ [.21 KP')\

is < Z7 rj i so tha t the angle a t p' b e t w e e n T ; a n d 2HP') i^

(10 .5 .58) < ( 1 + Z zTj) ' Z ^rji + arcsmrj^]/! <ei

i f the las t inequali ty holds . This proves ( i i i ) .

Part 2 . P r o o / 0 / ( 1 0 . 5 . 4 9 ) . F o r ^ a n d q fixed, we defineu(y, z) = oyT^ o)jp(y) = cr^Ô) U [cjo{z) y\ cojq(z), cojq(0)]

y := V(w, z) <^ w == u(y, z),

v[w, z) = coJ-^[a)jq (w)] = cj^{z) U [cjQj{p) w] cojq(0), a)jq{z)]

(see (10.5.38), (10.5.54), an d § 10.1). T h en we see t h a tN

rjjo(z) = ^ 2.jio(z) 2J '^^(^ji> ^) ek ( = m{P))i k=v+l

(10.5.59) ^(y ô)=y , u(0,z)=z, v{z,z)=0, v(w,0)^wtj(z) = (oJ-^^Mjplrjjoiz)] = u[r]jo{z), z], Z ji = oyJ^{pn)-

From (10 .5 .59) and the un iform bounds for the second der iva t ives of

U(Y;p,q) with re s pec t to i t s a rgumen ts , and f rom the bounds in

(10 .5 .46) an d Le m m a 10 .5 .6 an d the ex tens io n lem m a (Lem m a IO .5.2) ,

we f ind th a tN

r]jo(0) = 1 • 2 " ^^(0 , 0) A; = 0 (zji = 0 for Pji == q)fc=v + l

VJo(z) = Z ^•*o(^) 2 [v^[zji, z) - v^{0, z)] eic + Z ^ ^ 0 , ^) eici fc=v+l fc=v+l

^-iz'êu^ vfo W , Vfo (^) = i { « 5 | ( z ) + S k 3 i o (^) [^ i +fc=v+l k=v+l i

Morrey, Multiple Integra ls 2 9

dl(z) = — [v^{z, z) — v^(z, 0) — v^{0, z) + v^{0, 0 ) ] ,

d\^,{z) = [v^ZJ^, z) - v^zji, 0) - v^0, z) + vÔ, 0)]

i^^.l ^Zserj; |V tyfo(^) I < ^9 ^ + ^ lo ^^

V e tj(z) =1 + VzTijo {z) + e2{z), ^y(O) = ^ ( 0 , 0) = 0 ,

1 21 <Ziirj

tj(z)=tjo(z)+tf(z), \Vtf{z)\ < Z i 2 y y + Z i 3 e

which p roves the re s u l t .

Pa r t 3 . The maps tj, defined in (10.5.39), converge uniformly to a

homeomorphism T from T^ onto a topological v-disc S (Z S*. From (10.5.39)

and (10.5.40), i t fol lows that

^ ^ ; (y ) > T^j-i (y)] = ^ [(^3 b^j-i (y) ] . T^J-I (y)] < v^ ô ^*

s o tha t the TJ converge un iformly on 7^ to a map T and

(IO .5.6O ) d[Tj(y),r{y)]<Pij(=rj2roZQ'')'

Suppos e , now, tha t T{X) — T(y). Then, f rom (10 .5 .60) , we obta in

d[rj{x), riiy)] < 2Ay = 2r]2roQ^+^l{\ - Q).

But then it follows from (10,5-40) (iv) that

d[Mx),ro(y)] ^7 j2ro(2Qy +W -9)

for e very y. T hu s x = y , sin ce ^ < 1/2.

P a r t 4 . If e and R\ are small enough, S D K{PQ, 4 ô)- F ro m Le m m a

10.5.4, we c on clu de t h a t caô (5) satisfies an (e ' , R'^ condi t ion a t 0 . By

a s s u m i n g Ri s ma l l enough , we may a s s ume tha t CO^Q (5) satisfies an

[e', i ? i ) -condi t ion a t 0 , where we assume

3 6 e ' < 1 , ^ < ^ o .

We sha l l d rop the pr ime on e and assume tha t a l l the se ts a re in RN ; w e

a s s u m e ^ — ^ g a n d TQ — T^ and us e the s imp le r no ta t ion .

In o rde r to pro ve th e result in thi s pa rt , w e select a y ^ S Pi -S (Po, 4ô)

and def ine a cont inuous map cp with domain To as fo l lows :

I't{x) if \r{x) — y | < 2^0

X if \r{x) — y | > 3^0

^^^ { [ k W — y | — 2^0] ^ + [3^0 — \t[x) — y l] T(:V)},- otherwise.

Since 99{x) is on the segment from x tor[x), we see th a t

\(p{x) — r[x)\<^rQ, \(p{x) — x[^ro,

on acc ou nt of (IO .5.6O ). A ccord ingly if x^dTo =^ Z ^ dB{Po, 2Ri —

— 6ro — Ao), then

\x(x) — y \= \\x(x) ~ x'\ -\- {x — y )\'>\x — y \ — \'t[x) — x\

> iR x — 7^0 — 4^0 — ^0 = 4^0

10.5- Th e local topological disc pro pe rty 4 5 1

SO th at (p{x) = X on ^TQ. Since \(p{x) —r(x)\ < r o , it follows that if

\(p(x) — y \ < 0 then \r{x) — y \ < 2ro so (p(x) =r(x). Thus

(p{To)nB(y,ro)CS.

Now, suppose y i S . Then, for some n,

(10.5.61) (p(To) n B(y, 2-^ro) is empty.

If this holds for n = 0, then B (y , TQ) 0 (p (TQ) is empty. But since (p (x)is between x and r (x) and T (: ) ^ 5* C ( ^ , 8 £ ro), it follows that (p (To) C( X Seô) C (^/'^ l 6e ro ), 2 " being n i ; through y . Since I6 e < 1/2, itfollows that if ji denotes the projection on ^, then 990 = ^ 9 is a continuo usmap from TQ into 2J which is the identity on d TQ and whose range contains no points in B (yo, ro/2) Pi ^ C th e disc TQ (yo = ny) . But this isimpossible. If (10.5.62) holds for some w > 0, then

(p(To) n B{y, 2t) = (p{To f) [B(y , 2t) -B(y ,t)], t = 2-^ro.

But , if 2 ( y , 2t) denotes a r-plane th rou gh y whose existence is guara ntee dby the (e, JR i)-condition, the n

(p(To) n B(y , 2t)CSn B(y , 2t) C (Z{y> 2^), 2t s) a[B^t - Bt)

and this set m ay be m appe d continuously onto a subset of ( ^ ( y , 2^), 2t E)

ndB2t by a mapcon

which is the identity outside and on dB2t' Thus(setting Br = B(y, r), etc.)

((On (f) (To) nB4t=^ {(p(To) n (^ 4 ^ - ^2 ^ )} U Vn-l

Vn-i = con{(p(To) n (^2^ - ^ C (Z (y ^ 2t), 2t e) 0 dB^t.

But now, using the (e, iî)-condition, we find that

(Z( y> 2^), 2t e) n dB2t C (5, At s) 0 dB^t = (S, At s) H B^t O dB^t

C(Z(y>4t),St8)ndB2t.

(p(To) n B^t C (2(y, 4t), At e) n B^tfor an appropriate ^(y, At). Since s is small, we see that that the set

(con cp) (To) n B4t can be mapped continuously into a subset of (^(y, At),

At e)ndB4t by a map co^_i which is the identity outside and on dB^f

This procedure may be repeated and we conclude finally that

(coi. . . con cp) (To) n B(y , ro) C (2 /(^ -^0) , ^0 £) Pi dB(y, ro)

for an appropriate 2^(y, ro). Now

(2'(y> ^0), ro s) n î5(3;, ro) C (S, 2ro e) O B(Po, 8ro) 0 ^^(y, ro)

C ( X Ô ro e) n B(Po, Sro) H aJ5(y, ro) C (2", ^Sero) H aJ5(y, ro),

2J' being | | 2 throu gh y. Since coi . . . co j is the ide nt ity outside and on

dB(y, ro), it follows that ncoi. . . (J0n(p has the. same pro perty as th e

former mapping 990 = JT 99. Since this is impossible, 5 D 5 fl B(Po, 4ro).

29*

In order to prove fur ther smoothness proper t ies o f these d iscs , i t i s

necessary to cons ider the i r r -d imens iona l Lebesgue a reas .

Lemma 10.5.7. Let us suppose that S, S, the Tj, the TJ and T, the Cj,

the pji, the 2^{p), and the positive num bers £, Q, di, RQ, rji, and rj2 have their

significance above and that e, Q, 6i, RQ, TJI and rj2 are sufficiently small. Then

for each rj6'> 0, there is a d ^ 0 such that

(10.5.62) hm miLv(rj | r ) < (1 + ê) ^* ' [T (r) ] if A^[r(dr)] = 0

provided that all the numbers above are <id, F being any open set for which

r C, To, Lv denotes the Lebesgue area.

Proof. For a f ixed y and each i, we define Dji as the set of all p ^ (S*,

^rjl2) s u c h t h a t d[p,pji) < d{p,pjr) for all i' and le t 2 ' ; ^ = SHPji)-

E a c h Dji i s bou nde d b y po r t ion s of ^ ( 5 * , }rjl2) and loci of equations of

the form d(p, pji) = d(p, pj^). Thes e l a t t e r loc i a re app rox ima te ly

p lanes wh ich a re app rox ima te ly no rma l (depend ing on rji a n d rj) to the

geodes ic r -p lane ^ji', we call these loci the projecting faces.

Since {pji} is an rj-set for S*, B(pji, rjj2) G Dji. O n t h e o t h e r h a n d

s ince any p ^ Dji O SÛ B(pjfc, rj) w i t h d{p, pji) < eve ry d(p, pjjc), w e

o b t a i n

(10.5.63) B[Pji, rjjl) C Dji, Dji Pi 5 C B[Pji, rj).

F ro m (10.5.63) an d (IO .5.40) (iii), etc . , it follows t h a t Tj 0 Dji i s a lmos t

para l le l (depending on di, rj2, a n d rj) t o 2Jji C\Dji. Since we have seen tha t

the pro jec t ing faces a re a lmos t normal to 21 H> we conc lude tha t

(10.5.64) L,{Tj n Dji) = A'(Tj n Dji) < (1 + r]e)îÂ'(Dji 0 Zn)

if Oi, Yji, rj2, an d jR i, an d hence e (if Q is chos en < ^0 (9K)) are sm all en ou gh .

Moreover if B(pji, rj) H 5 i s no t empty and B(pji, rj) f) dS i s emp ty , i t

fo llows f rom L em m a 10.4-7 an d Le m m a 10 .5 .6 th a t the pro jec t ion of5 n B(pji, rj) = the projection of 5 pl B{pji, rj) (as defined in Defini

tion 10.4.2)

D Z H n B{pji, rj 1 / 1 - Z 2 | ^ f / 3 6 )

Z being the cons tan t in Lemma 10 .4 .7- Thus , s ince 2!H H B(pji, rj) D

2Jji n Dji an d the pro jec t ing faces a re nea r ly no rm al to 21 ji, we conc lude

t h a t

(10.5.65) A^(S n Dji) > (1 + rje)-^'Â-(Dji n ^Ji)-F ro m (10 .5 .64) and ( IO .5 .65) we dedu ce th a t

(10.5.66) Lr{Tj n Dji) < (1 + ^6) A'{5 n Dji).

L e t rt = rJ^{TjC\Dji), Fji = T-^(S D Dji). Since r and TJ a re

homeomorphisms , i t fo l lows tha t a l l the Fji, for those i for which /^^ H F

10.5 ' The local topological disc property 4 5 3

i s no t empty , l ie in (F, Sj), w h e r e sj - ^ 0 . T h u s , s u m m i n g o ve r s u ch i a n d

using (10.5.66), we obtain

i i

<{i+7je)A'{r(r,sj)}

f rom which th e lemm a fo llows by le t t ing y -> o o .

We wish to conc lude th is sec t ion wi th the fo l lowing theorem:

Theorem 10.5 .2 . Suppose that e, Q, 61 , RQ, rji, and rj2 dye- fixed suffi

ciently sm all num bers, that S,S,r, and the tj have their previous significance,

and that (10.5.62) holds for some fixed (finite) TJQ > 0. Then

Lv{r I r) < y l ' ' [ T ( r ) ] , if T i s o pe n a n d T c T g.

We sha l l p rove th is theorem af te r s ta t ing severa l def in i t ions andproving severa l lemmas . The methods of p roof a re sugges ted by those

of some deep results of F E D E R E R ([1], [2]). W e pr es en t simplified proo fs

of some ad di t i on a l resu l ts , which we do n ' t require he re , in § 10 .8 .

Definition 10.5.4. Given a ^ Rjsi an d £ > 0. W e le t C^ (a , e) d e n o t e t h e

to ta l i ty of iV-cubes of th e form \z^ — a^ — e d^) \ < e, wh ere ^ is a seque nce

of even in tegers . We le t C^_i(^ , E) be the to ta l i ty of the [N — 1)-cells

on the boundaries of the iV-cells of C^(^, e), and deno te by C^-^ia, e)

the to ta l i ty o f [N — 2)-cel ls on th e bou nda r ies of t he [N — 1)-cells of

^'N-I(^> £), e tc . Similarly we le t C'^{a, e) denote the totality of A/'-cells

of the form \z^ — a^ — e6^\ •< e where 0 is a sequence of odd integers ,

and let C^_i(<3!, e) deno te the to ta l i ty o f {N — 1)-ce lls on th e bou nda r ies

of the cells of C^(a, e) , e tc . We somet imes a l low C'j^(ci, s) or Cj^' (a, e) t o

denote the un ion of the po in t se ts corresponding to the co l lec t ions . I f

a == 0 an d e = 1, we de no te C^(0, 1) b y C^, et c.

Lemma 10.5 .8 . (a) yjcq is a cell of Q <^ it is of the form

I-s:* — (5 I < 1 , i^sjcq and z^ = 6^ if i ? Sjcq

sjcQ being som e set of k distinct integers < N; here the d^ are even and the 6^

odd. The corresponding statement holds for the cells of each C^.

(b ) For each k,v-\- 1 < ^ < iV, and each q, the center ofy^g^ C^_^_i .

If k = V + 1, yjcqC) C"^-!,-! is the center of y jcq. If k <^v + \, then

ykqC\ C^_v-i ^'s empty.

(c) / / we define rjc : C^ — C^_y_i -> C^_i by defining it on each

y kq — C^_y-i as the radial projection from the center ofy jcq onto its bound

ary, then rjc(C^ — C^-^.J C C ;_ i — Q % - i , k>v + i.

Proof, (a ) i s eas i ly proved by induc t ion downward . To prove (b) and

(c ) we may a s s ume tha t y j^q = y^, w h e r e

fc : |> *| < 1 for i = \,. . .,k\ z^ = \ for k + i <,i <,N.

Now if y^-v-i € ^N-v-v YN-V-I li^s a lo ng a pl an e 2:* = d^, d^ even, for

V -\- i values of i. I f y ^ - ^ - i O y^. i s no t to be empty , these i must a l l be

< ^ . Fo r each s uch s e t of i, the re a re 2^"*'-i cells of C^_y_i which in te r s ec t

yjc, in which the z^ must sa t is fy

1 ' — 0 ^ ' | < 1 , e^=^±\, 1 < ; < ^ , y ^^ a n y ^ ' .

ThusyJJ. n C^_y_i consists of all the (^ — r — 1)-cells for wh ich 2: +1 = • • •=: z^ = 1, -2:* = 0 fori^ + 1 va lu es of i < k, a n d 12:* | < 1 for th e rem ain

i n g j . Parts (b) and (c) now follow easily .

Definition 10.5.5. W e define

Qv : RN — C';^-v-i -^RN b y £ v = ^f+i 0 . . .or^

w h e r e t h e rjc are defined in L em m a IO .5.8 (c) . W e define B as the un i t

cube in Rj^, i.e. B == {y : \y^\ < \, i = \, . . ., N, and define

I4v(y ) = (\yh — Sh\, . . ., \ yU+i •— (3v+i|)

w h e r e d i s th a t sequence of even in tegers such th a t y — d^B and (^'i, . . .,^V) is a pe rm u ta ti o n of (1 , . . . , iV) for w hic h

I y h _ ^h I < I /yi2 _ ^^2 I < * * * < | y% — S^N \ .

G i v e n a a n d e, we def ine the maps co , ta , and r^ by

0}{y ) = ay , ra(y) =y + a, ra = raOa)OQpO o)-^ 'r_ a.

L e m m a 10.5.9. (a) If rj ^Ri — C ^ _y _i , then Qvirj) ^ C^ a^ ^ *s 0^ the

boundary of the N-cell of the form rj — d^B in which rj lies.

(b ) J \u,(v - r])\-^ dA^ (rj) = C7 (r, N) A^ {B).(c) ^Tj^R^- C^_,^i ^ w.(i ) 7^ 0 .

(d) Ify = a-\-£d-{-srj, where rj^B and 6 is (as usual) an N-

sequence of even integers, then ra [y) is the point on C [a, e) given by

ra{y ) = a + ed + eQv{r]), Qv(r))^dB,and

\Dra[y)\<.C^{v)'\u,{ri)\-^

where D denotes the maximum directional derivative.

Proof, (a) This follows from Lemma 10.5.8 and i ts proof. To p rove (b ) ,w e n o t e t h a t u is periodic of period 2 in each Y}'^. T h u s w e h a v e

j\nv[v-Yi) Y dA^(7 ]) = f Iuvirj) \-' dA^(r]).B B

The resu l t fo l lows by d i rec t eva lua t ion which is made easy by the sym

metry, (c) follows from the definit ions . To prove (d), w e n o t e t h a t

ra{y ) = a + £ [d + Qv(r])'], r]^B.

From the homogene i ty , we s ee tha t

\Dyra(y)\ = \D^Q47])\.

To calculate a bound for the la t ter , i t is suffic ient , because of the sym

m e t r y , t o a s s u m e t h a t 0 < ? ^ < T ^ < • • • < iy-^ < 1. T h e n

10.5- Th e loca l topo log ica l d i sc p ro per ty 4 5 5

we mu st have rf^^ '> 0 ii rj i C^_y_i. In our case

and the bound Cg \ Uv (rj) \~^ is easily obtained for \D o^(rj)\.

Lemma 10.5.10. Suppose G is a hounded open set in Rp, r is a homeo-morphism from G into W, A''\T:[G)] < O O , A'\T{dG)'\ = 0, r[G) CB (Po, R), R •< RQ, and r = co o z where co is a normal coordinate systemwith domain B(0, R) and co (0) == P Q . Suppose also that there exists asequence {r^}, each r^ = co o Zn^ C^{G), such that Xn ->t uniformly on Gand such that, for some rj '> 0

UmZv (r^ I T ) < (1 + rj) A' [r (P) ] , if T (ZG and A' [r (^P)] = 0.

Let 99 : 9K ^ 501 he a piecewise smooth mapping with domain D T {G) suchthat \D (p(p)\ <.%(p))y where % is continuous near r(G). Then

Lr{(por)^(\+rj)f[x{p)rdA{e^),r{G)

Proof. Let & be those cells r in Pjv such that A^lziG) H dr] = 0.For each or > 0, we m ay cover z {G) with a finite number of closed cellsTi of @ of diameter < cr, the open cells r f being disjoint. For each i forwhich rt H z(dG) is empty, let Ft = z-^ {r^f^). Since z and co are homeo-

morphisms, the diameters of the P^ and of the T(P^) - > 0 as a->0.Moreover, for each such i, A^'lridTi)] = A^'lcoidrt D z{G)} = 0. Forsuch i and all n

U ((pOTnl ri) =A'[(pO Xn (P*)] < IX {pi)r A- \Xn (P^)]

(10.5.67) - \x[pi)'rU{rn I Ti) < (1 + 7y) [;^(^)]M•'[T(P,)] + ^n.

where x(Pi) denotes the maximum value oi x(P) for ^ ^ T ( A ) . The result

follows by adding the results in (10.5.67) for those i for which rf n z(dG)is emp ty, tak ing th e lim inf a nd then letting cr -> 0.

Lemma 10.5.11. Suppose G is a hounded open set in Rv, r is continuouson G, Lv (T) < 00, r{G) C B (Po, R), R < Po, ^'^d r = co 0 z where co isa normal coordinate sy stem with domain B (0, R) and co (0) = P Q . SupposeZ is the union of a finite numher of{v — \)-planes in R^, and Q = z~^ [Z Oz(G)]. Then

Lr{r\G-Q)=L,(r).

Proof. It is sufficient to prove this for Z a single (v — 1)-plane. Letcooo be the norm al coordinate system related to co by a rotation of axessuch that Z is part of the {N -]- \ — v) plane z^ = z^, i = v, . . ., N. Byapproximations from the interior, we may also assume that G is the unionof a finite number of domains of class C^ which have disjoint closures.Moreover, for each e an d each o*, 0 < cr < 1, we can find an open set

r (ZG — Q s u c h t h a t Fconsists of a f ini te number of domains of c lass C^

with d is jo in t c losures and

I v ( T | r ) > Z V ( T | G - ( ? ) - e /2 , r[T{x)']<al2 for x^G-F,

r(p) being the geodes ic d is tance f rom^ toW = (Ooo{Z), W e m a y a s s u m e

o) =coooi for each q^ B{Po, iô/3)> we let COQQ be the no rma l coo rd ina te

s ys tem cen te red a t q and rela ted to cooo as in § 10.1. To points pn e a r W

we may ass ign coord ina tes was follows: Let q =nw{p), the (geodesic)

p ro jec t ion oi p on W\ t h e n oy^Q (q ) — {w'^, . . ., w^-^, z^, . . ., z^). T h e n p

m u s t lie on the geodesic (iV + 1 — r ) -p l an e W t h r o u g h q so t h a t

p =a)Qq{0, . . .,0,w^, . . ., w^). We define oy ^ hy p— co* {w ); clearly to*

is of class C2 ne ar Z .

N o w , wedefine a seq ue nc e {C;}, ea ch C;$ C^(^) ^ ^ ^ € î(^) ( i-^-L ips ch i tz ) s uch tha t Cy - > T un i fo rmly on G and

(10.5.68) Lr(Cj\F)- L,(r,F).

T o doth is , we f i rs t approximate un iformly toz on F by {C*}, each

C*€ Ci(^)> so th a t (10.5.68) ho ld s w it h ( y = co C* an d th en ap pr ox im ate

un i fo rmly to onG by any s equence {rjj}, each rjj ^ ^ ( G ) . N o w , for

each y, f * — ^ ; I ^ can be ex tended to sa t is fy aLipschi tz condi t ion on G,

L e t f be such an ex tens ion and def ine

if \if{x)\ < 2 c o , -

\-îf(x), if | ff(^) |>2a>,-=

co^ == m ax 1 I f (A;)

xeF

Qj ( ) = %• ( ) + f ; ( ) > C; (: ) = CO 0 ^; (:\;).

Clearly QJ [X) = Cf (A;) on Z' andeach QJ is LIPSCHITZ on G. Since

ly; | r - > ^ | r , it follows that ^f\F->0 socoy - > 0 an d I;/ - > 0 , so t h a tQj - > 2: (a l l convergence uniform) so [l,^] satis f ies the condit ions .

Finally, let us define 99a : 9}l -> 9)1 b y

if r{p) < (T,

if c 7 < ; ' ( ^ ) < 2 ^ ,

p ' be ing the po in t A of the way from nw (P) along the geodesic to pw h e r e

h == { — or)~^ [^(P) — o"]- W i t h t h e a id oft h e co"^ coord ina te s ys tem, it

i s easy to see tha t (fa i s p iecewise smooth and tha t

\Dq)a(p)\ <.ioy %a = const., lim;^cT = 1a->0

^[ô(p) , p]<.o, bo th for ^ ^ 9K.

1 Incase Cf— iii\r =0 on JT, set | ; (;ir) = |*(;t^) = 0on G.

10.5- Th e local topo log ica l d i sc p ro per ty 4 5 7

We now def ine T; = (fa O Cj and we see that if j is large enough for

I^j{P) ~ '^(P)\ ^ < /2> t h e n tj i s L i p s c h i t z a n d r j ( x ) ^ W i o r a l l x ^ G — F.

Therefore , fo r such j ,

L,{Tj)^L4ri\r)<xlLr{Ci\r)

The lemma follows from (10.5.69) and (10.5.68) and the arbitrariness of

a a n d e.

Proof of the theorem. Since 5 minimizes A^ (X) a m o n g a l l c o m p a c t

X^{£{dS,L), w h e r e L = Hv_i(dS) (Th eorem s IO .3.4 an d 10.4.4), i t

fo l lows tha t ip{P) = 1 a lmos t eve rywhere on 5 . F ro m Theore m IO .4 .I ,

i t fol lows that ip i s uppe r s emicon t inuous . Le t rj > 0 and choose pos i t ive

n u m b e r s Q, di, RQ , rji, r]2, a n d e s o s ma l l tha t the rjQ of Lemma 10.5-7 is

< 7 ]. F ro m T heo rem 10.4-5 i t fo llows th a t we m ay ta ke eo > 0 b u t so

sm all t h a t if y^(P) < 1 + £0, th en 5 satisfies a n (e , JR 2)-condition a t P

w i t h R2<, RQ, if P i s in te r ior to 5 . Le t W be the subset of S — dS

wh ere ^ ( P ) > 1 + £o /2 ; th en W i s c lo s ed in S — a 5 , A^iW) = 0, a n d

X = r-^{W) is closed in B(0, R) ~ (T^Y^\ R = 2Ri ~ 6ro - AQ. W e

a s s u m e t h a t Po^S and tha t 5 C B(Po, 2 P i ) a n d t h a t coo 0 i s a no rma l

coo rd in a te s y s tem w i th dom a in B(0,Ro) and cooo(O) = Po- W e le t

Z = 0)^0 (W) — z(X). L e t G be any open subset of T^Y^K W e cove reach XQ oi G — X by Si ne ighborhood "^ w i t h "^ d G — X s u c h t h a t t h e r e

ex is ts a sequence {tn}, each Zn ^ C^ {^) w i t h Tn = coooo Zn, which con

ver ges un ifor m ly to T | 9^, an d for w hich

l i m Z . ( T ^ i r ) < ( 1 +fj)A^[r(r)], if(10.5-70) ^-^^

P C ^ , Anr(dr)] =0.

li xo^X, we take 9^ = G.

L e t F be any compact subse t o f G. W e m a y c o v e r z(F) by a finite

number of c losed cells rf with d is jo in t in te r iors such tha t A^ [z{G) H dri]

== 0 and Vi f] z(dG) i s empty for each i, each also being of diameter < a.

Since r a n d z a r e h o m e o m o r p h i s m s , w e m a y t a k e a so small that each of

the s e t s Ff = z'^ (ri) is in some one neighborhood 9^, provided F^ f) F

i s no t empty . So we le t G' be the union of these Ff; c learly G' D P. I t i s

suffic ient to show that Zr(T, G') = A^[r(G')] for such G\

L e t Y be the union of these dfi. T h e Fi have the proper ty (10 .5-70)

possessed by the 9^. Let d be an a rb i t ra ry pos i t ive number and choos e ain Rjsf w i t h | a* | < h, wh ere ^ > 0 b u t so smal l t h a t

f [Xo(z)rdA^(e,)<d, Vo={Y ,SNh), Xo = Xa,h

Xo{z) = {C9 \u,[h-^z - a ) ] | - i - \}q2(z) + 1, C9 = Cg + 3 -

q2[z) = 1 on ( y , ANh) and qîz) = 0 on RN - (Y, SNh)

0 < ^ 2 ( 2 ) < 1 on (Y,SNh) -{Y,4Nh),

and q2 has Lipschitz constant (Nh)~^. To see that this choice is possible,

we note that (using the notation of Lemma 10.5.9)

/ dA^(a) I [xo(z)rdAê^)

= h^ f dAN (b) I Cl\u, (/z-i z~h) \-^ dA' [e^)

^ VonziGn

= A^ • Q • C7(v, N) A^^B) A'[Vo 0 z(G')]

= A^ [CO (B)] 'C'Cl'AnVoDz (G')]

using Lemma 10.5.9. Since /l*'+i [ (C)] = 0, it follows also that yl^+i[P*^ 0 z(G')] = 0 for each (i^ + 1)-sequence / . Thus the A''-measureof the totality of (iV — r — 1)-planes of the form z^ == a^ which intersectthe set z [G') is zero. Hence we may also choose a so that z [G') C RN —— C^_y_i [Uy h). Moreover by choosing a small enough, we m ay ensure th ateach Fi which intersects X is such that r^ C a neighborhood U ofz[X 0 G'] such that

(10.5.71) f^\xo{^)\'dAê^)<d,UnriGn

Having chosen e and a as above, we define qi (z) as we defined q2 (z)except with 4N h and SN h replaced, respectively, by }N h and 4N h.

Then we define

n i y ) = y + ^ i ( y ) [^ « (y ) — y ] > C = ( p o o z , r' - = a > o o o c .

We note that

(foiy) =y for y^RN — (Y, 4iVh)l9ô(y) — y | < A ' A , y^RN

^^^•^•^^^ \U^)~-z{x)\<Nh

\D(p^{y)\ < 1 +{Nh)-^'Nh+\ -^\Dra{y)\ < 1^0(^)1, y^RN^

Now, let Q be the collection of components P of G' — ^-'^ [C^.^ {a, h)\

Then we conclude from Lemma 10.5.11 that

(10.5.73) U[r'\G')=ZLv{T'\r).

If r does not intersect z-^(Y) nor X then it is part of one of the Ff C

some ^ with "^G G — X, and we conclude from Lemma 10.5.10 that

(10.5.74) L,(r'\r)<L(\ + rj) f \xo(z)\^ dA^(e^),z{r)

10.6. The R E I F E N B E R G cone inequality 459

s ince

T' =(por, (p =COOO(POWQ^, \I^(p{P)\ < (1 + ^Vo i) \I^ (po[cooi(p)]\,

\D(p{p)\ = i, piV, V =cooo{Vo), L==:(\ +2rjoRi)^^.

If r does in te rsec t z-^{Y), le t xo m z-^{Y). T h e n z(xo)^ Y, s o tha t

(by (10.5.72)) C(xo)^(y,Nh) and hence ^ ac o m p o n e n t r cube (of side 2h)

E of Cl(a, h) — C^_i (a, h). L e t xi be any point of Fand jo in XQ t o xi b y

a p a t h xu 0 <,t <,\. Ks long as C (^^) ^ (Y, 3iV A), t (^^) € E, and as long

a s ^{xt) ^ E, it $ ( Y , '}Nh). Cons equen t ly ^[E] C EC.{Y,'i N h) and

z {F) (Z [Y, AN h) GFQ. For s uch aF, we can conc lude on ly tha t

(10.5.75) L,{x' \F)^L(\+ r]Q)f \xo(z) \' dA'(e,).

If F in te r s ec t s X b u t n o t z-^(Y), then ^(J*) CC/ an d we can conclu de

tha t (10 .5 .75) ho lds . Subs t i tu t ing the resu l ts (10 .574) and (10 .575) in

(10 .573 ) , we ob ta in

LAr' I G') <(^+v) ^ ^ ^ ( G ' ) ] + {2 +7 j+rje)'L -fjxoiz) l^^dA^ie^) +

VonziGn

(10 .576 ) + (1 +V )J\xo(z)'rdA'(e,)

Unz(Gn

< (1 + rje) A^riG')] + (3 + i? + 2rje) • ^ • L .

m a x \r'{x) — r(x)\ <,Nh.

Since , for a g iven G, rj, a n d d, hm a y bearb i t ra r i ly smal l , we obta in

(10.5.76) w ith T ' repl aced b y T. T he n th e result follows for G' f rom the

a rb i t ra r ine s s o f 7] a n d d.

10.6. Th e Rei fen b erg con e in eq u al i ty

Suppose 2: is a polynomial A^-vector which defines a mapp ing f romB (0, R) o n t o S, s u p p o s e t h a t ZQ = Po RN> a n d R(PQ) is t h e m i n i m u m

v a l u e of \z(s) — zo\ fo r son dB(0, R). L e t

U(s) = \z(s)-zo\^;

t h e n t/ is apo lynomia l . F rom recen t re s u l t s by W H I T N E Y [3], it follows

t h a t the locus of U^oc(s) = 0, oc = \, . . .,v, hasonly f in i te ly many

c o m p o n e n t s . F r o m a theorem of A. P . M O R S E [1], itfollows that U is con

s tan t on each of the s e componen ts . Thus , excep t fo r a f in i te number of

v a l u e s of r, 0 < / <R{Po), the locus l(r, PQ, Z) = SH dB(Po, r) cons i s t s of a f in i te number of regula r o r ien tab le ana ly t ic manifo lds wi thout

b o u n d a r y . T h e r e m a i n d e r of th is section isd e v o t e d tothe proof of t h e

fo l lowing theo rem:

Theorem 10.6.1 (The R E I F E N B E R G cone inequa l i ty ) . The reexistpositive

numbers QQ, eo, and k, k<, \, with the following property: If z, r, and

I (y, P o , z) have their significance above, r is not exceptional, and there is

a v-plane ^ through PQ such that

/ l " - ! [/ (r , Po, ^)] < 1 (1 + € o) yv r'-^, a n d

D[l{r,Po,z),ZndB(Po,r)]<Qor

then there is a set Y * with b(Y'^, A) D Hv^i(A), A == l(r, PQ, Z ), such that

(10.6.2) A^{Y^) < (1 - ^) v-^rA^-^ [l(r, PQ, Z ) ] + ky^/^

Suppose we le t Yr be the cone wi th ver tex Po and base l{r, PQ, Z ).

Then the inequaHty (10 .6 .2) becomes

(10 .6 .2 ') / [ " (Y*) < ( 1 -k)A^(Yr) +kyrrr

O n accoun t of the homogen e i ty , we m ay a s s ume th a t r = \.W e ma y le t SR i be the co l lec tion of regula r an a ly t i c manifo lds , w i th ou t

b o u n d a r y , i n B (0 , R) which is the counte r image of / ( I , Po , z). We choose

a n (x, y ) coord ina te sys tem with or ig in a t Po and x"^, . . . , x^ axes i n ^ .

W e m ay then rep re s en t Y i pa ram e t r ica l ly on / x Tli, I — [0, 1], by

(10.6.3) xr=rl{p), y = rf]{p), 0 < r < l , \i\2 + \r]\^ = \, p^W i.

I t i s convenien t to le t Y denote the in f in i te cone ob ta ined by ex tending

the rays of Yi to in f in i ty and to represen t Y by

x = ri(p), y=:rrj{p), r^O,

W e m a y a p p r o x i m a t e t o f a n d rj (or ^ and rj) un i fo rmly together with

their first derivatives b y p iece wise an a ly t ic func t ions ^^ a n d rjn, each pa i r

of which is such tha t the cone Yn is composed of a f ini te number of

"s implic ia l cones ' ' , each of which is the union of a l l rays joining Po,

her eaft er called 0, to t he p oi nts of a (v — 1)-s implex in a (r — 1)-plane

not pass ing th rough 0 . We ca l l such a locus Yn a polyhedral cone. O u r

method of proof of the theorem will consis t in constructing a set Y* as

des i red in the theorem for each Yn and then s howing tha t the Y* tend

to Y* in the point-set sense, in such a smooth way that yl* '(Y*) ->/l* '(Y*).

We may assume tha t each s impl ic ia l cone in each Yn has a 1 — 1 p ro jec

tion on the ;^;-plane ^ .

O n a po ly hed ra l cone Y, we sha l l den ote by x{P) a n d y{P) th e x

a n d y coord ina tes of the po in t P on Y. We have seen tha t each of the

manifo lds compos ing Y is o r ien tab le ; we suppose tha t each such manifold is oriented. This induces an orientat ion on each s implic ia l cone. If

P is inte rio r to such a s implic ia l cone , we define y (P) = ± 1 accord ing

as th e s imp lic ia l cone and i ts pro jecti on on ^ are s im ilarly or opp osite ly

or ien ted . O n. an y s impl ic ia l cone of Y, we m ay cons ider the y ^ as l inear

func tions of th e x^ and we define \Jy{P) ^ as the sum of the squares of

10.6. T h e R E I F E N B E R G co n e i n eq u a l i t y 4 6 1

the Jacobians of all orders < min(r, N — v) of the y ^ with respect tothe xK The area element on Y at P is then

(10.6.5) ]/^ + \Jy (P)\^d2:{P), d2:(P) = \dx(P)\

where | ^ A ; ( P ) | is the v-area element on ^ . We also define

N-

^ 2J \bx0j Dy[P)\^= W ^ ^ ' ^ ^

(10.6.6) ? = i fc=i

= / Vl + \Jv{P) f dZ(P) - y. = A'{Y^)Yi

where Yi is the part of Y in the cylinder |^(P)| < 1^. Comparing this

notation with that in (10.6.4), we see that

(10.6.7) x{P)=ri(p), y(P)=r7](P).

We shall prove the theorem for all Q and e which are sufficiently small,

Q being a bound for | y (P) \ on Yi; we may also assume that

(10.6.8) ^ < ^ 6 ^

We first notice the following fact:

Lemma 10.6.1. Suppose Y is a poly hedral cone of the type describedand suppose we define

(10.6.9) y {x) = Z r{P)y{P)x{P)=x

for points x not on the image of the boundary of any simplicial cone of Y.Then y coincides with a single-valued function which is continuous on Rpand linear on each of a finite number of simplicial cones. Moreover

(10.6.10) Zy{P)=>^y a.e. on Y,K=±\.

x{P) = XiProof. The projections on the ^v-space of the simplicial boundaries of

all the simplicial cones of Y divide that space into infinite conicalregions which can be further subdivided into a finite number of simplicialcones.

Let us consider two of these simplicial cones Zli and A^ which havean {v — 1)-dimensional simplicial cone J in common. The poin ts P of Ywhose projections lie in A\ form a finite nu m ber ^ly, / = 1, . . ., / i , of

simplicial cones, each of which is a part of one of the original ones of Y.The same is true of ZI2; let d^j, j = 1, . • ., J2, be those correspoinding toZl2- From the nature of Y, it follows that each dij abuts on a unique ^2;or another dij and the same is true of each ^2;. Thu s, we m ay assume (byreordering if necessary) that each dij ab uts on ^2;' for y = 1, . . . , / 3 and

^ T h e e here has no connect ion with the e ' s used in other sect ions .

http://bx0j/

http://bx0j/

http://bx0j/

2:y(P})y{Pi)±i[y{Pjz^2,)-? = 1 Q= l

x{Pj)=x, 2A=Js-Ji,

-y(Pj2+2p-i)]>

- y{Pjz2q-l)],

2B = Js-j2.

xÂ,

XÂ2,


then that ^1,^3+22>-i abuts on ^1,^3+22) for p = \, . . ., {Ji — Js)l2, and

(d2,Jz+2q-i abuts on d2,j^+2q, ^ = 1, . . ., (73 — 72)/2. Then

y(x)

(10.6.11)

Clearly y(Pj) has the same value on î^ and 62 j but has opposite sign onPJB+2P and Pj^^2p-i' Consequently as A;-> a point xo on A, the secondsums -> 0 and the first sums approach the same value. Since y (x) is

obviously linear interior to Zli and A2, the continuity of y follows.Since y{Pj) has the same value on a dij and ^2^ and has the oppositesign on Pj2+2p and Pj^+2p-i and on Pjs+2q and Pj^+2q-i, we see that

(10.6.12)

Iy (Pj) = Iy {Pj)^ ^ € ^ 2 .

It follows that the sum in (10.6.10) has the same value K on Ai and A2and hence for almost all x in R^ From the hypothesis (10.6.1) (whichholds for Yn on account of the uniform convergence of the derivatives)it follows from Lemma 10.4.7, etc., that ;><;=: - j ^ 1.

Remark. Hereafter we assume that Y (and the Yn) cire oriented so that

(10.6.10') 2y (P) = +'^ a.e.

For Y a polyhedral cone as above we define u(0) = 0 and

(10.6.13) u{x) =f(pî,i(t - x)y {t)dtB{X,Q\X\)

where 99 is a non-negative molHfier ^ C^ [B(0, 1)] C C^(i?v) and

(10.6.14) ^tiy) = Q~"îQ-^y )

(see Theorem 3.1.3). If we represent Y by

(10.6.15) x = X(s), y =Y {s), s $ ^ = i ? + x M i ,

we see from (10.6.9) and the definition of y tha t

(10.6.16) u(x) =f< p^^^^[X(s)-x] Y(s)dX(s).

In the form (10.6.16), we see that as Yn ^Y, Un-û uniformly withall its derivatives on any bounded closed set in Rv which does not contain the origin. Clearly u is homogeneous of the first degree in x and$ C ^ ( 7 ? , - { 0 } ) .

(10.6.17) v^-.r

fl


F o r e a c h Yn and for Y, we cons t ruc t our des i red

Y^ = Vnl U Vn2 U Un3 U F ^ 4 , Y* = F i U F2 U F3 U F4

w h e r e Vi i s the par t o f Yi ou ts ide the cyHnder \x \ = 1/2 a n d F2 ca n be

rep res en ted on [1/4, 1/2T x ^ 1 b y

r l ( ^ ) , 1 / 4 < ^ < 1 / 2 , p^Wi

Z {r,p) ^{Ar - i)rri(p) + {2 - 4r) u[ri{p)].

I t i s show n in L em m a 10 .6 .12 be low th a t if | V ^ | i s suffic ien t ly sma l l

( th is i s shown in Lemma 10 .6 .8 be low) then there is a nearby Car tes ian

coord ina te s ys tem ('x, 'y), which depends s moo th ly on V u (eva lua ted on

\x\ = 1, say, since u is homogeneous of the f irs t degree) such that if U

denotes the cone(10.6.18) U:y = u(x) or 'y ='u('x)

t h e n

^p^dx' = 0, ^ 3 = 5 ( 0 , 1 /8 ), i<.i<v, 1 < ^ < A / '

(10.6.19)

Th en F3 is th e par t of U for which |^ ' | > 1/8 and \x \ < 1/4 a n d F4 is

def ined by

^ . n ^ . «^ ^ 4 : 'y =^v('r,'e) ^'u + ( 8 y ) 2 [ ' ^ ( 1 / 8 , '6) - 'u],(10.6.20)^ ^ 0 < V = |':;t;| < 1 / 8 , 'd=^yx\-^'x.

Corresponding formulas ho ld for Vni, e t c . T h e p a r a m e t r i c r e p r e s e n t a

t ions of the Vnj conve rge un i fo rmly on the i r r e s pec t ive doma ins toge the r

with the i r f i rs t der iva t ives to those of the Vj s o t h a t / 1 * ' ( Y * ) ^ / l * ' ( y * ) .

I t i s shown in Lemma 10 .6 .11 be low tha t

A^{Un n TF) < Ai(^, £) • £ + A^{Yn n W), \imhi{Q, e) - 0 ,

fo r any bounded meas u rab le W. Using th is , i t i s shown in Lemma

10.6.10 below that /1*'(F^2) < ^^[Q, S) • 6 + /t» ' (co rres po ndi ng p a rt of

Yn). Fina l ly , in Lemma 10 .6 .14 be low, i t i s shown tha t

(10.6.21) yl»'(F^ 4) < J^^ ^ A'' (corresponding par t o f Un).

T h u s , whereas rep lac ing the par t o f Yni ou ts ide \'x\ = 1/8 by Vni

U Fw2 U F^3 m ay increase th e a re a b y h{Q,s) • s w h e r e h(g, s) ->0 as

Q,e -^0, rep lac ing the par t o f Yn ins ide \'x\ = 1/8 reduces the area by

a n a m o u n t A (v, N) - s, A > 0 . Th is leads to the theo rem .H o w e v e r , Wi ^ F3 U F4 is ev id ent ly a r -cel l w i th bo un da ry

(10.6.22) Ci:y = u(x), |A; | = I / 4 ,

a n d Wo = F i U F2 m a y be rep res en ted on [0, 1] x Wi by equa t ions l ike

(10.6.23) Wo:x= X(a,p), y =Y(a,p), a ^ [0, 1], p^ Mi

in s uch a way tha t a = 0 y ie ld s a homeomorph is m o f Wi on to / {r, PQ, Z)

^ Co an d a = 1 y ie lds a m ap pin g on to Ci which , however , need not be

a homeomorph is m. W e wis h to s how tha t

(10.6.24) b(Y^, Co) D H,_ i{Co), Y^ = WoUW i.To do this , we firs t apply Theorems 10.3-9 and 10.3.10 with

A = Co, B=Âi.

Those theorems y ie ld the resu l t (s ince b(Wi, Ci) = Hv_i{Ci) b y T h e o r e m

10.3.4), , , ^'(Co U C i, Co)* b(Y^, Co) = b(Wo, Co U Ci) +

(10.6.25)^ ^ + ^ ' ( C o U C i , C i ) * i / . _ i ( C i ) .

I f we now ap ply Theo rem IO .3 .13 w i th

X' = IxW, A'^ôxmu A[=:ixWi, A'==A'ûA[,

Ao = Co, Ai = Ci, X=W o,

t h e m a p p i n g / : X' ->Wo being given by (10.6.23), we conclude that if

LQ === Hv_i(Co) the re i s a s ubg roup Li of Hv_i(Ci) s u c h t h a tK + i(Co U Ci, Co)* Lo = K + i{Co U Ci , C i )* L i ,

K==b(Wo,CoUCi).

Tha t i s , a h^ Hv_i(Co) an d /^ i$ i^ , the re ex is ts SL k2 in K a n d a n I in

^» '_ i (C i ) s uch tha t

^(Co U Ci, Co)* h = (k2- ki) + t(Co U Ci, Ci)* /

which proves (10.6.24) by showing (cf. (10.6.25)) that

i{Co U Ci, Co)* ^v_ i(C o) C b{Wo, Co U Ci ) + ^*(Co U C i, C i)* H,_i(Ci)

the r ight s ide being that of (10.6.25). Of course this is a lso true for each n.

We now prove the inequa l i t ies which we have ment ioned for a po ly

he dra l cone Y i . W e use th e no t a t io n of (10 .6 .5)— (10.6 .7) .

L e m m a 10.6.2. (a) ]l\ +af + h < - 1 ^ 2 ' [ l / 1 + ^f - 1j

(b) |/ i + [Aa + (1 ~ A) b]^ < A | / l + ^2 + (1 - A) 1/1 + ^^2,

0 <;c < 1,

(c) 1/1 + A2 a2 - 1 ^ A [ ] / I + a2 - 1], 0 < A < 1 .. n

(d) yi + («i + . . . + a„)2<2:1/1 + «?.Proof, (a) F o r ^ > 0 an d J5 > 0, w e define

f(A, B) =(\ + ^ ) l / 2 + (1 + 5)1/2 _ (1 _|_ ^ _ j_ J5)l/2 _ 1

an d show b y d i f fe ren tia t ion th a t / i > 0 so / > 0 . Th e resu l t for m ore

terms follows by induction, (b) jus t follows from the fact that ] / l + x^

10.6. The R E I F E N B E R G cone inequality 4 6 5

is a convex func t ion , (c) follows by s e t t ing b = 0 in (h). To p r o v e (d) set

f{ai, .. .,an)=Z y^ + af - j / l + (ai + • • • + a^)2.

i

I t fol lows eas ily that each fai < 0 and t h a t f(ai, . . ., Un) -> 0 as allUi -> -\- CO.

L e m m a 10.6.3 . Suppose f is homogeneous of degree zero. Then

l { I 1/Bi {B{X,Q\X\)IO

r ' l ^ r l / L l ^ -:^)^ + ?j]\dA dt\dx < C i o W r \f{x) I dx,

J IProof. Let g[x, r, ?.) = \f{x + XQ\x\r)\. T h e n the in teg ra l above is

dr}dx

Bi U i LO

= / { / [ jimldrjUAdr. G{r,^)CB(0.i+kQ),Bx lo [ (T.A) J J

w h e r e , for r and X f ixed, G (r, A) is the i m a g e of Bi u n d e r th e 1 — 1

t r a n s f o r m a t i o n7 } ^ X -\- XQT \X\.

T h a t t h i s t r a n s f o r m a t i o n is 1 — 1 follows since

1^2 — ^11 -{1 — ^ ^ 1T|) < |?y2 — : ^ I | ^\X2 — xi\ • (1 + A ^ | T | ) .

The result follows eas ily .

Lemma 10 .6 .4 . Let us define [almost everywhere)

V{X)=Z\Y{P)\<

X{P)=X

define Xi as the subset of Bi where v[x) =^ \ and X^ as the subset of Bi

where v{x) > 1. Then

Jv(x) dx < 2e, f [v{x) — \]dx = ( [r(:v) — \'\dx <, e.

^2 Bi Xz

Proof. Clearly

y r + 8= A^'iYi) > jv{x) dx = yr + f [v{x) — 1] dx.

Bi Xz

The result follows from the fac t tha t

v{x) < 2[v(x) — 1] for v(x) > 2.

Lemma 10 .6 .5 . / / \x\ < 1, then \u{x)\ < C i i ( r , N) Q [if q < 1/2).

Proof. F r o m the h o m o g e n e i t y , it fo l lows tha t is is sufficient to p r o v e

t h i s for |;t:| = 1 — ^, if ^ < 1/2. From (10.6.9) and the definit ion oiv[x)

w e see t h a t

(10.6.26) \y[t)\^qv[t).

From (10 .6 .13) , it then fo l lows tha t

\u{x)\ <Q + ZiQ'-*'f Q'[v{t) - i]dt<:Q[\ + ZiQ-^'e]

Xz

466 The h igher d imens ional PLATEAU p ro b l ems

from which the result follows, since e < Q^^. Z\ depends also on themoUifier ^.

Lemma 10.6.6. Suppose fx is a measure over a set E, iLt{E) is finite, andf is non-negative and integrable (/j) over E. Then

min I CM (E)]^ [ ffdnf] < 3 [/* (£)] • / (Vl +P -i)dfi.

Proof. From the Schwarz inequality with

/ = {fiyi\ + / 2 + 7 ) • Vl'l + / 2 + 1, we obtain

ffdf,f < IJiYT+P - l)^/^]. [/(l/l + /2 - 1 + 2)df,

<A^ + 2AB, A=: f{-]/\+p-\)dft, B=fx(E).E

li A^ + 2AB^'}AB, the lemma is t rue. liA^ + 2AB> }AB, then^ > ^ a n d 5 2 < }AB.

Lemma 10.6.7. f\u[x{P)] - y[P)\^ dj;[P) < Ci2{v,N) - Q^ e.

Proof. Let us denote the integral by / . Then

/ < / l + / (ClO + 1)2 Q^V{X) dx<^h + Z2 {V, N) Q^ 8,

(10.6.27) , 'h = J \u{x) — y{x)\^ dx,

using Lemmas 10.6.4 and 10.6.5 and (10.6.26). Now ii x^Xi

B(X,Q\X\)\'\U(X) — y(x)\

<Z3'Q\X\'J \f\Dy[{i -X)x + ^t]dA\dt

IZs{(p,v,N)' Q\X\'\B{X, Q\X\)\,

on account of Lemma 10.6.5. Using Lemma 10.6.6 with E the set {t, X)with t^B{x, Q\X\) and 0 < A < 1 and d/bt = dt dX, we obtain

/ . n ^ . . ^ B[x,Q\x\)\'\u[x)-y{x)\^(10.6.28)

< 3 Z f ^ 2 | ^ | 2 | U\^J\^-\Dy[[\ -K)x + Xt]\^- \\d?\dt.B{X,Q\X\) U J

Using Lemma 10.6.3, we therefore conclude that(10.6.29) f\u{x) — y {x)\^dx^Z4.Q ^f{]/i + \Dy{x)'f - l ) ^ ^

X i

But, from (10.6.9) it follows that

(10.6.30) Dy{x)\<2:\Dy(P)\.

10.6. The R E I F E N B E R G cone inequality 4 6 7

T h u s , f rom L em m a 10 .6 .2 (d), we conc lude th a t

Bi Yi

(10.6.31)Th e resu l t fol lows f rom (10 .6 .2 /)— (IO .6 .3I) .

Lemma 10.6 .8 . (i) \Du(x)\ < Cisiv, N) e^l^, if Q < 1/2;

(ii) \u(x)\ <\x\ '\Du{x)\.

Proof, ( i i) fol lows from (i) and the homogeneity . From (10.6.13), i t

fo l lows tha t

u{x) = f(p(X) y[x + XQ\X\) dl (A = ( \A)~^{^ - A)Bi

s o t h a t

(10.6.32)

Bx

^y I x ^ - ^ B ^ y \ I 1 ^ dX

from which i t fol lows, by sett ing t =^ x -\- Q\X\X, t h a t

(10.6.33) \Du{x)\^{\+ZiQ)j 99*,,,(^ - x) \Dy[t) \ dt.B{X,Q\X\)

Since D u a n d D y are homogeneous of degree 0, i t is suffic ient to prove

(i) for \x\ — \ ~ Q. Then, s ince B[X,Q\X\) <Z Bi, we conclude from

the Schwarz inequa l i ty as in the proof of Lemma 10 .6 .6 tha t

j^%\if-^)\i)y{i)\dtfBI,X,Q\X\)

(10.6.34)X9^*i . , (^ - x)[^J\ + |Z)3^(^)|2 - 1) ^^ ]

B{X,Q\X\) J

/ vtU^ - *)(yi + l-DyW^ - 1 + 2) dt]iB{x,Q\x\) J

<.ZQ-^e{2 + ZQ-^e).T he res ult follows from (IO .6.34), (10.6.8), an d (IO .6.3I) .

Lemma 10.6 .9 . /* I/1 + \Du{x)\^v{x) dx < f ^ l/l + \Dy{P)\^ x^ ^ x{F)=x

Xdx + C Qe, B = Ba — Bi,, 0 <b <a<^\.

Proof. Since D u, v, a n d Dy are homogeneous of degree 0, i t is suffi

c ien t to p rove th i s wi th B rep laced by Bi.

From (10.6.33) and the definit ion of y , we obtain

\Du{X) I < (1 + Zi ) l \Dy(P) I df,(P;X)Y{X,Q\X\)

(10.6.35) - (^ + Z iQ) f < Ptui{i - X) I\Dy{P)\dtB{X,Q\X\) x{P)=t

{d/,(P;X) = cptui[^{P) - X]dZ {P))

where Y{X,Q\X\) consis ts of a l l P on Y such th a t X[P)^B{X,Q\X\).

30 *

Us ing the dev ice of (10 .6 .34) wi th 5 ( Z , ^ | Z | ) r e p l a c e d hyY{X,Q\X\)

a n d IZ) y ( ) I rep laced by | JD y (P) | , we find t h a t

[ ( ] / l + \Du^~ ^)dx<^f\Du(X)\^dX

Bi Bi

< i (1 + Z i ^)2 fw{X) [w{X) + 2co{X)] dX,

Bx

(10.6.36) w{X) = f <ptui(t- X) J V [ l / l + l - D y ( P ) | 2 _ i]]dt

B(x .e ix | )

B{X,Q\X\)

Clea r ly w a n d co a re homogeneous of degree 0 s o we may e s t ima te them

a s s u m i n g t h a t \X\ = \ ~ Q. Then, s ince \D y{P)\ < \Jy{P)\, we see

f irs t , us in g (IO .6.36). L em m a IO .6.4, an d (10.6.8), th a t

w {X) < Z2 o-*' e, o) (X) < 1 + Z2 p-" • 2s( 1 0 . 6 . 3 7 ) V / — 1 t^ w{X) + 2co{X) < 2 ( 1 + Z 3 ^ )s o t h a t

(10.6.38) / ^ | ( l / l + \Du\^ ~ \)dx< (1 +Z4Q) fw{X)dX.

BX BX

If, in (10.6.36), we set t = X + Q\X\r, in teg ra te , and change the

o r d e r of in teg ra t ion , we ob ta in

lw(X)dX = fcp{r)\f/ Z \^\ + \Dy{P)\^-\KdX\dr.Bx Bx \BX \^^P)=X+Q\X\r / J

(10.6.39)

If we set, for each r in i5i, rj ^= X -\- Q\X\r,we s ee th a t the t r ans fo rm a

t ion and i t s inverse a re Lipschi tz and the Jacobian \dXjdYi | < 1 + Z5 ^;

m o r e o v e r , for each r the i m a g e of Bi C ^ i+e - Acco rd ing ly we o b t a i n(since D y is h o m o g e n e o u s of degree 0)

(10.6.40) / < ( 1 + Z 6 ^ ) r Z [l/l + | Z ) y ( P ) | 2 - l ] ^ ^ .

Now , u s ing Le m m a 10 .6 .8 , we m ay wr i t e

J y i + \Du[x)\^v[x)dx<, j ^ \ + \Du{x)\^dx +

\[\+Z,Q)j[v{x)-\]dx^y, +j\ Z l/l + | D y ( P ) | 2 | x

Bx Bx U(P)-n JX drj — f v(x)dx + J [v(x) — \]dx -{ -

Bx Bx

+ ^ 6 ^ / 1 2 ' [^J^ + \Dy [F)f-\]\d7 l + Z ,Q([v{x)-\\dxBx V<^)=»? J Bx

from which the result follows {\Dy[P)\ < | / y ( P ) | , L e m m a I O . 6 . 4 ) .

http://dxjdyi/

http://dxjdyi/

http://dxjdyi/

Lemma 10.6.10. Suppose Z = F2 and is defined by (10.6.17). Then

A^'iZ) <yl^(Y2) +(p(Q,s)' £, lim(p{Q, s) = 0

where Y% is the part of Y for which 1/4 < \x \ < 1/2.

Proof. O n each simplicial cone of Y, we may consider 3/ and z as function of X. From (10.6.17) and the notation of (10.6.5), etc., we obtain

z[x) = (4^ — 1) y{x) + (2 — 4^) i^[x)

g = (4^ - 1) f j + (2 - 4r) U + d\„ d\, = A{y ^~ u^) [x^jr)

b(x\ XO) ~ ^^' 'l b (Xi, XO) ^ ^^^^'

d(Ar*i, xH, xh) ^^ ' d (;»;*!, ;ir*2, x^^) ~^ 3 1 ' • • •

where each d^p is a sum of Jacobians, of order <^p, oi the y ^ with respect

to the x^ multiphed by products of components of D u and y — u .

R egarding th e Jaco bians, etc., as components of a vector, we obtain

\Jz(P)\^{4r-\) \Jy{P)\+{2-4r)\Du[x(P)]\ + \d{P)\.

Then, for each ?y > 0, we have

\Jz{P)\^ <{i+ri) {{4r - 1) \Jy{P)\ + (2 - 4r) \Du[x(P)]\}^ +

-\-{i+r)-i)\d{P)\K

Hence (refer to Lemma 10.6.2)/ [ I / 1 + \jz{P)|2 - i]d2:</[l/i + (1 + v-^)\d(P)|2 -i]d2 +

Yz Fa

+ / [ y i + (1 + rj) {{Ar - 1) \Jy[P) I + (2 - 4y )\Du[x[P)-\ \f - \]dZ

< (1 + »j-i)i/2 / [yi + i^( P) |2 - 1 ] ^ 2 " +

+ {^+V?'^l[]/^+{{4r-i)\Jy {P)\ + (2-4r)\Du[x(P)]\}^-i]dSY2

< (1 + »?-l)l'2 / []/l + |i ( P ) |2 - 1] (^2" +2

+ [(1 + »?)i'2 - 1 ]/ (4 r - 1) / [1/1 + 1/y (^) P - 1] ^ 2 " +I r^

+ (2 - 4;') / l/[l + IZ) « [ ^ (P)] |2 - 1] <f2l +y^ J

+ ( 4 r - l ) / [ f l + \Jy {P)\^ - \]dZ +Y2

+ (2 - 4r) f [|/l + \Du[x(P)]\2 - 1 ] ^ 2 -

. • . / ] / l + | 7 0 ( P ) | 2 c ^ 2 ^ ( 4 r - 1 ) / ] / l + | 7 y ( P ) | 2 ^ 2 ++ (2 - 4^) / 1 / 1 + ID M [^(P)] |2 c? 2 + £• 9 ' i(e. £)

Fa

from which the result follows, using Lemma 10.6.9.

Lemma 10.6.11. / / U is the cone given by (10.6.18), then

AÛ n W) </ l» ' (Y nW) + h{Q, 8) • e, lim^(^, e) = 0

for any bounded Borel set W.

Proof. It is sufficient to prove this for W the cylinder \x \ < 1. Since\J u\^ = \D u\^ plus the squares of all the Jacobians of the u^ withrespect to the x^ of order two or higher, we see from Lemma 10.6.8 that

A^{Ui) = / l / l + \Ju{x)\2dx<f]/\ + \Du(x)\^dx + Z s'^l^

< f 2^ Y^ + \Dy {P)\^dx + Z £ ^l^ + CQs<A ''{Yi)+Z e^^^ + CQe

using Lemma 10.6.9.

Let us now consider the cone U given in th e {x, y) system by (10.6.18)where u^ C' {Rp — {o} and u is homogeneous of the first degree. Wewish to prove the following lemma:

Lemma 10.6.12. / / |Z)^(%)| 5 sufficiently small, there is a nearby{'x, 'y) Cartesian coordinate sy stem which depends smoothly on \7 u{x),evaluated on \x\ = \, such that

(10.6.41) fy^^dx'Ô, i = i,...p = N -V, k=i,.. .,v,Bi

Remark. I t is clear th a t (IO .6.41) will then hold with Bi replaced byBa = B (0, a ) for any a> 0 .

Proof. Suppose we let

(10.6.42) 'x^ = c^x^ + d^yr, V = elx^ + / j y ' *

be a positive orthogonal transformation. Then U is given param etricallyin the ('x, 'y)-system by

(10.6.43) 'x^ = c^x^ + dûr{x), 'ui = elx^+fiur(x).

Then(10644) -^^^^— V ( ^.,^rd'ui^dr.\,../.^-i/.^î,,../x^)[d.-.-i

Thus, the e quation s (IO .6.4I) become

/(10.6.45) _ r y i . ^ , , ^ - .. V ^ r ^ i , . . . / ^ ^ - i/ ^ ^ ^ i , . . . /^ v )

~JjiLj^ ^ d 'xi d{x^ ,...,xi-^,xi+ ^,...,x^) ^ ^ - ^B*

where the derivatives d' u^dx^, d' x^jdx^, etc ., ma y be found from (10.6.43)and 5 * is the region in the ^-space correspoinding to the un it sph ere B inthe ';v-space by the first equations in (10.6.43). To determine the matricesc, d, e, and f, we have the iV(iV + l)/2 equations which require the m atr ix

(10.6.46) (^ ^ ) (iV = i . + ^ )




to be orthogonal. Next we have the vp equ ation s (10.6.45), which a re ofthe form

(10.6.47) '^l{c,d,ej)u) =Q , i = \,.. .,p, k=\,.. .,v

where g | is a polynom ial in the c, d, e, f with coefficients which are integrals over 5* of various jacobians of the u'^ with respect to the xK Thisleaves

{v + p)^ —V — p , v^ — V p^ — p— V p - \ —

equa tions which we can use to prescribe the cf with / > ^ and the / jwith j '> i.

By replacing uhy u -\- Xv.itis easy to see that the Frechet differen

tials%i{c,d,e,f]u,v)

(10.6.48) _ ^.^^_^ ^^^^^^ ^^ ^^^. u^Xv)-%i[c,.. „ / ; u)]

are continuous for || u || sufficiently small, where

(10.6.49) 1 11 = mdiK\Du[x)\,\x\=\

We have already noted the dependence of ^^ on {c, d, e,f). The integrand

in (10.6.45) is

/I + cj + d]iii^ cl + dfui,.. .el + {6} +f})ui^.. . c\ + d]u\\

(ietf 2 + ^Û ^ + ^1 + d]u\^...e' + (^j + / j )^ !2 -- . 2 + ^>;2 j

c\-\-d}u\, cl + dû\,... el+[d]+ f;)ui,..A+cl + d)u[J)

(10.6.50)

we have replaced 4 by 1 + c\ and f) by d) + f] .If we now regard all the c, d, e, f as small and neglect products of two

or more of them and two or more of the ^-derivatives, the determinant in(10.6.50) reduces to that of the matrix

(10.6.51) (4 + ^ : f c + / > y -

If we also replace 5* by B and define

(10.6.52) Ui = \B\-^Ju%(x) dxB

the equations for the c, d, e, f become

4 + ui+f}Ui = o, c |= / | = o

(10.6.53) 4 = - cf if ki^l, fi = - / j if ; V i

4^-î' i,j=i,....p, k,l=\,...,v.

4 7 2 T h e h i g h e r d i men s i o n a l PLATEAU p ro b l ems

It is clear that we may prescribe the f) wi t h / > i and the cf w ith / > ^and the remaining c\ and /} are determined. Then the e\, are determinedfrom the U^ and f] and then the d\ are determined. Since the equations(10.6.53) 9.re obtained by linearizing the actual equations, and sincethe derivatives and F R E C H E T differentials are continuous, it follows thatthe equations (10.6.47) together with the orthogonality relations yielda unique solution in which the matrix (10.6.46) is sufficiently near theidentity provided that the prescribed ei and f) are sufficiently near 0 asis II ^||.

We now prove:Lemma 10.6.13. Suppose that u ^ C^(Rv — {O}) and is homogeneous

of degree 1. Suppose also that

(10.6.54) fu%{x)dx = 0, i=i,.,.,p, k=\,.,.,v.B

Let V he defined by

v(r,d) =u(0) +r^[u (\,d) — u(0)],(10.6.55) r

u{0) = \Z \-^fu(\,d)d2;{e),

Then

(10.6.56) f\Dv\^dx< ^^l.^ f\Du\^dx.B B

Proof. Vr = 2r [u{\, 6) — u(p)]

ur.^r.^ r-^Vev(r,e)=rVeu(\,e)(IO .0.57)

\Dv\^== 4r^ 1^(1, 6) -u (0) |2 + ^21 Veu{\, d)\^.

From homogeneity,

u{r,d) =ru (i,e), Ur{r,e) = ^ ( 1 , 6 ) , r-^V6u{r,e) = \7eu{\,d).

Hence

j\Dv \^dx = - L _ | ( 4 1^(1, 6) - f (o) |2 + I V6u(\, 6) |2}^2'(^)B ±

/ ' |Z)^ |2^ :^ = 1/'{|^(1,0)|2 + iV0^(l,6l)|2}^2'{6>).

B 2:

If we now introduce the spherical harmonics qnk (see after Lemma

10.6.14 below) with^01 = 12^1-1/2, qijc=C,jc'\x\-^xK k = \,...,v;

our hypotheses and definitions imply that

n,]cn ^ 2

V + 2 -


It follows (see below) that

B \ n,k I

I w 2 Jj\Dv\^dx = ^{^\anjc\n4 + n(n + v-2)]\

B \ n,k IU ^ 2 J

U^2 J

< 7 - f - r i + - - ^ 1 . / ' | D W | 2 ^ A ; = - ^ flDul^dx.— (v + 2) L 2v 4- 1J j I I 2v + 1 j 1 I

Lem ma 10.6.14. TA^r^ 's <3 num ber r]{v,N) '> 0 such^that if u ^C^(R — {O}), \D u\ ^rj,u satisfies (10.6.54), and v is defined by (10.6.55)then

(10.6.58) f{\/\ + \Jv\^ - i ) ^ ^ < - ± ^ | ( i / i + \Ju\^ ~\)dx,

B B

Proof. For if \Du\ <r], it follows from (10.6.57) that \Dv\ < 2r} onB. Thus

Ay^ + 1 7 ^ 1 ^ - ^)dx<^^f\Jv\^dx<^(\ +Zirj^) f\Dv\^dx

B B B

— 2 V + 1 j ' ' ( 2 X ' + 1 ) ( 1 - Z 2 i y 2 ) J Vy r i I J

B B

since

l / l + f 2 _ 1 > i f 2 _ ^ | 4

for all f a nd |Z) ^ | < | / ^ | . The result (10.6.58) follows easily.The spherical harmonics. We begin by noticing that iin ^ p and Hnand Hj, are homogeneous harmonic polynomials of respective degreesn and py then

0 = J HnAHpdx = —• f Hn,ocHp,ocdx + J HnHprdS

(1 0 .6 .5 9 ) 5 B 6 ^

= / {H nH p r — H jp H n r) d S .

d B

From the homogeneity it follows that

Hnr = rHnr = nHn, Hpr == r Hpr = r Hpr = p Hp if r=\.

Thus we see that

(10.6.60) JiP — n) Hn HpdS = 0 or J HnHpdS = 0.dB dB

4 7 4 T h e h ig h e r d im en s io n a l PLATEAU p r o b lem s

So we define the qnk on dB = 2^ ^^^ each n, to be a complete n .o . se t ,

each of wh ich is th e res tr ic t ion to ^ of a (homog eneous) ha rm on ic po ly

nomial of degree n. Then all fhe qnk for all n form a complete n .o . se t for

^2(2')-Le t u s s uppos e tha t u is representable as a f ini te sum of the form

(10.6.61) u{r, 6) = Z^nk(r) qnJc{6), anjc{r) = r^bnic(r)

where each hnTc[y) i s an even polynomia l in r. Since r'^ qnk ^^ a h a r m o n i c

polynomia l , ^ i s a po lynomia l . Then

J \Du\^dx = J {u^ + r~2 l"^ Q u\^) dx = J u Ur d^ — f uAu dx.B B dB B

(10.6.62)

If is giv en b y (10.6.61), it follows t h a t

, _ ^ ^ ^ ^ l^r = 2 <lc{^) ^nk(6)(10.6.63) n.fc

A U = Urr + (^ — 1) ^~-^ ^r + ^~^ Zl2(9 ^

whereZI20 i s the Be l t ram i ope ra to r o n ^ . Ap p ly ing (10 .6 .63) to u = Hnk

= ^^ qnky we ob ta in

, , ^ ^ , , 0=^n[n + v~2)r^-^qnk + r'^-Â2eqnk.(10.6.64)

Aêqnk = —n{n + v — 2) qnk-T h u s

(10.6.65) ^ w - = 2 ' K f c + ( ^ - i)r-âlj,-n{n + v - 2)r-ânk]qnk-n,k

S ub st i tu t in g (10.6.61), (IO .6.63), an d (IO .6.65) in (10.6.62), we ob tai n1

f\Du\^dx = fr^-^f (u^ + r-^\Veu\^)d2^drB O S

1

= fr'-^ S K l + n{n+v-2)r-^ 4,^] dr.

0 n,kFrom the a rb i t ra r ine s s o f n, i t fol lows that

f\Veu{\,e)\^d2 = In(n + v-2)alj, (1).

i w.fc

10 .7 . T he loca l d i f ferent iabi l ity

In th is sec t ion we prove the fo l lowing main theorem:

Theorem 10.7.1. Suppose XQ = S \J ^ ^s a minimizing set as in

Theorem 10.4.4- There exists an £0 > 0 with the following property: If

PQ^S =^ XQ — A andip{Po) < 1 + £0, there exists for each/bc, 0 < // < 1,a neighborhood of PQ on S which is a regular v-cell of class C^. Thus the

subset of points P^ S w here ip[P) = 1 is open on S and ip(PQ) = \. If

W is of class C^for some n > 4 and some [x, 0 <^ [x <C \,the v-disc may be

taken to be of class CJJ. If W is of class C°° or analy tic, the v-disc may be

taken to be of class C"^ or analy tic, respectively.

10.7- The local di fferent iabi l i ty 475

From Theorem 10 .4 .4 , it follows that y){P) (see § 10.4) = 1 a lmos t

everywhere on S. From Theorem 10.4-5, i t fol lows that if-y; (PQ) < 1 + £i

(for £i = ei (£ * ,r , SJ ) suffic iently sm all) , t he n 5 satisfies an (6*,Ri)

cond i t ion at PQ. If £* and Ri are suffic iently small , it fo l lows tha t a

neighborhood of PQ on S is a topologica l r -d isc 5 . From Lemma 10 .4 .4 ,

i t fol lows that iir]o'> 0a n d si a n d RQ a re s ma l l enough , we may take 5

s o s ma l l tha t y)(P) < 1 + TO for P on 5 . Le t

(10.7.1) r:B(0,R) o n t o 5 , r = cooooz,

w h e r e COQ o is anorma l coo rd ina te s ys tem cen te red a t PQ. Then we con

c lude f rom Theorem 10 .5-2 tha t

(10.7.2) Lv(r)^A {S), U{r\r) <A''[r{r)], ifTis

o p e n a n d r c 5 ( 0 , i e ) .

L e m m a 10.7.1. The equality holds in (10.7.2).

Proof. For suppose for some F tha t the inequa l i ty ho lds . Then H a

s equence {r^}, w h e r e Tn 6 C'^ [P ) a n d rn->r uniformly on P s u c h t h a t

l i m L , (rn\r)=d< A'' [r{P)];

w e m a y a s s u m e t h a t e a c h Z n is loca l ly regula r . Then it fo l lows tha t

Lvlxn I P] =A^[rn(r)]- W e may now s e t up the func t ions y){r, P , T%),

q){r, P , Tn) and the i r l imi ts jus t as in§ IO .4 . Th e deve lop m en ts of t h a t

sec t ion may be repea ted and lead to a se t XQ for which A^ (S') = d, w h e r e

S ' =^ XQ— r{dP). W e now mod i fy th e tn as follows: We select a sequence

o f ana ly t i c doma ins {Gn} ^GnG Gn+i foreach n andU Gn — P. By

going a long the inner normals to ^G^ a shor t d is tance Qn, we fo rm ano the r

a n a l y t i c d o m a i n G'^dGn. We define r'^[x) =rn(x) for x^G'^,r^(x)

= r{x) ioT x P ~ Gn, a n d

T * [ ( 1 -t)x' + tx] = C O o o [( 1 -t)Zn(x') +tz{x)], 0 < ^ < 1 ,

'Vn =COQoZn,

w h e r e x' a n d x are ex tremit ies o f an o r m a l s e g m e n t t o dGn and cooo is a

norma l coo rd ina te s ys tem cen te red a t P Q . Th is a l te ra t i on does no t a f fec t

the func t ions ip (r , P , Zn) a n d q? (r, P , Zn) ifwe mere ly res t r ic t r <, R (P) —

— an w h e r e On - > 0 . B u t t h e s e t s T * ( P ) a n d r(P) now a l l have the same

b o u n d a r y . T h u s XQ h a s thes a m e b o u n d a r y asr(P). But , s ince d

<A^[r(P)], th i s con t rad ic t s the min imiz ing cha rac te r o f XQ and hence

r(P). The proof for the whole surface is the same.

Proof of the theorem. S u p p o s e t h a t P Q 6S> y^(^o) < 1 + £2, and 5

is ar -d isc conta in ing P Q for which (10.7-2) holds. Since ip i s upper semi-

cont inuous , we may f ind an ^3 > 0(see Lemma 10.4-4) such that

(10 7 3) ^ ^ ( ' ' ' -P) = ^'' [-5 ^ -S (P , r)] < (1 + 2s2) h{r).

From Theorem 10.4.5> i t fol lows that there is an. r^ , 0 < ^4 < ^3 su ch

t h a t w i t h e a c h P^B(PQ, ^4) a n d ea ch ?', 0 < r < ^4, th e re is a ge ode sic

^'-plane 2J (P> ^ ) th rough P s u c h t h a t

(10.7.4) D[SnB(P,r), 2 ' ( ^ ^ ^ ) n 5 ( P , r ) ] < ^ o r / 3 ,

^0 be ing the co ns ta n t in th e cone ine qua l i ty .

Le t u s s uppos e t ha t 5 i s r ep re s en ted a s in ( lO . / . l ) . Th en we m ay

a p p r o x i m a t e u n i f o r m l y t o z on B(0,R) by po lynomia l s Z n s o t h a t

limLv{Tn) = Lv(T) = A^(5), rn = COQQO Z n.

Choose P in te r ior to 5 , le t OJP be a no rma l coo rd ina te s ys tem cen te red

at P, and le t us define

Sn = rn[B[0 , R)], l(r, P , Sn) =5nn dB{P, r),

lo{r.P,Sn)=cojni{r.P;Sn].

From the cone inequa l i ty i t fo l lows tha t there a re pos i t ive numbers ^0

(above) an d £3 w i th th e fol lowing pro pe r ty : Fo r each r for which there

is a I'-plane ^ in RN t h r o u g h 0 s u c h t h a t

D[lo [r, P , Sn). 2 ^ 0 dB(0 , r)] <Qor, a n d

A'-^ [lo ( / , P , Sn)] < (1 + £3) r 7 . r^-^,

the re ex is ts a se t Y^ w i t h b{Y'^, An) D Hv_i{An), w h e r e

An = l(r,P,Sn), Y * = c o p ( y * ) ,s u c h t h a t

A-{Y^) < (1 - ^) r - i r / t - i [ / o ( r , P , Sn)] + ky rr\

From the uniform convergence and (10.7.4), i t fol lows that if we choose

ri a n d ^2, w it h 0 < ^i < ^2 < H, then (if 1 + yo ^ < 3/2)

D[l{r, P , 5 ^ ) , 2 ^ ( P , r) n a 5 ( P , ^)] < 2^ 0^ /3 n < r < r 2

Z )[Z o(r,P ,5 ^), 2 ' n a 5 ( o , r ) ] < ^ o r ^ > A r .

If we let dn be the inf. of A" {X) for all sets X for which b {X, An) D

Hv_i{An), i t then fo l lows , as in the proof of the lemma above tha t

dn ->A''{S). Then, we conc lude f rom the cone inequa l i ty , the inequa l i t ies

in § 10 .1 , an d the a rgu m en t in th e proof of L em m a 10 .4 .2 t h a t

jrjn + t-^r(\ + hr) cpr{r. P , 5^) + kyvf''[\ + >yo^)^ if (a)

\r]n + v~'^r[\+hr) (pr(r, P , Sn), in all cases

r^i < r < ^2, ^ = (1 — k)-^ V, n>N, P^B (Po, n), lim? 7^ = 0

(a) (fr {r, P , 5 ^ ) < (1 + ss) V y r r""! {\ + rjo r)^~%

[\ +hr->(\ +r]or)^''-^.

F o r n and P fixed, define kn {r) = max [0 , y) (r , P , Sn) — rjn]-

^(r,P,S^)<f

(IO.7.5)

10.7- The local di fferent iabi l i ty 477

Since cpr and % (below) are > 0 an d (pr < ipr ( L e m m a 10.4.1, 2, etc.) and

99 is A . C , we not ice f i rs t tha t

(10.7.6) kn[r) = qn{r) + Sn{r), s^{r) = 0 a.e.

w h e r e qn is A.C. and % is n o n - n e g a t i v e and non-decreas ing . Then , f rom(10.7.5)> we conclude that (a .e . )

(10.7.7) ^ ^ ^ - 1^-1 r{i +hr)k:^ {r), in aU cases

^ (r) = y, 7- (1 + Tyo r)\ (D {r) = {i + £3) v y y r^ - i (1 + ^0 ^ ) ^ - ^

Since (1 + TJQ r )2 ' ' - i < 1 -f / r, it follows that

(10.7.8) t-'^r[\ -f hr)co{r) + kx<v~^r(i + hr)a){r).

W e now define J^7^(/') as the un ique s o lu t ion in C^ of the e q u a t i o n

(ob ta ined by requ i r ing the e q u a l i t y in (10.7-7))

itr-^(\ +hr)-^lpn — kx), if pn<t-^r(\ + hr)co +kx

Pn= lvr-^(i + hr)-^pn, if pn >v-^r(\ + hr)a)

(10.7.9.) \cjo(r), if t-^r{\ + hr)co + kx <pn <v-^r{\ + hr) o)

Pn{r2) = kn(r2).

By cons ide r ing all possible cases , we conc lude tha t

(10.7.10) k'^{r) ^p'^{r) whenever kn{r) >pn{r), ri^r<r2, n>N.

So , if we define u^ {r) = kn (r) — pn ( )> we h a v e

(10.7.11) Unir^i) = 0 and u^{r) > 0 w h e n e v e r Un{r) ^0, n < r < ^ 2,

(n > iV). I t follows that Un(r) <. 0, ri <^ r <, r2, n N; for if, for s ome

n and r w i t h 'i < r < ^2, we w o u l d h a v e Un (r) > 0, t h e n Un {r) would

b e > 0 for r in an i n t e r v a l [f, r + 6] (on a c c o u n t of (10.7.6)) in w h i c h

i^n{r) > 0 ^'^' From (10.7.6), it would fo l low tha t r + ^ = 2 w h i c h

w o u l d be impossible s ince Un ( 2) = 0.

If the first line of (10.7.9) holds for ri < r < ^2, t h e n

(10.7.12) Pn-X-t-^r(i+hr) {f, - Xn) + [\ - k) y. ^^^^ z{r),

z{r) = (1 -f Tyo rY'"^ (^ + ^0 + 2hrjo r).

In teg ra t ing (10 .7 .12 ) , we o b t a i n

Pn ir) -x{r)< ( 7 ^ ) ' • ( ^ y [^» (''2) - X (^2)] +

(10.7.13) + Z i "' r>+i-«

if i < v + 1.

Now, s in ce 1 -\- hr >{i + rjor)^'~^,

(10.7.14) :f-i>-(1 +hr)co -{\ -k)x^(i -k)ezx{r)-

478 The higher dimensional PLATEAU problems

Now, the right side of (10.7.13) will be < that of (10.7.14) iff

f{r) =: [ht (/-)]-! X{r), ht(r)=rt{i+h r)-K

Now

(10.7.16) m=^^-'{' + i f b ) - ^ - Mi + ^ ^ ) -

= _ tr-i{i + A,)-i [1 _ (1 _ k) (1 + hr) ( ^ ^ ^ ^ ) ] <Oiir<Z .

So f(r) takes on its minimum for r = r2. Then (IO.7.15) holds for all

^ < 2 if

(10.7.17) Pn{r2) < [1 + ( 1 -k)sz]x(r2) -Zirl+HA +hr2)-K

Since (Lemma 10.4.2), [h(r)]~^ kn(r) is non-decreasing for each n andkn{r)-^y){r,P) and [h(r)]-^ ip{r, P) N\ Accordingly kn{r)

<=pn[y), wherepn is bounded as in (10.7.13). Passing to the limit, we see

that (since ri was arbitrary)

(10.7.18) yj+{r, P) < x{r) + {i - k £3) [ht{r2)]-^[ht{r)], 0<r<r2.

From (10.7.18) and (10.7.5), we conclude that there are numbers Zi and

^, depending only on v and 9)1 such that

h{r) < ip+[r, P) < (1 + Zi fP) h[r) for all

(10.7.19) ^^p^ ^ ^ ^^p^^ ^2;ô), 32/0 < min (rg, 4)

and ro is sufficiently small to satisfy the further conditions in (10.7.21)

and (10.7.22) below. From Theorem 10.4.5, it follows that S satisfies an

(^0' ^0) condition at P Q with

e"^ = larger of 32 cf ^ ro and Z2 r^,

(10.7.20) Z2 = (2-i -c î -Z i -3 20i / ^, oc = Plh>0

provided that TQ satisfies the following conditions ( 2 = 32?'o)

(10.7.21) 32ro<Ro\ 32 fo <Cl € ^ £2 = 2- i -Zi -32^ -rg< C2-( 4)^ .

We see that ej = Z2 r^ provided that

(10.7.22) yJ-'^<32-iciZ2.

Now, suppose 0 < A < 1. For each k, we conclude from (10.7.19) t hat

(10 7 23) ^(''^ - ^ ^ ( ' ' ' P) < (1 + Z i . 32Â^^/g) ^(/) for allB(P,r)cB[Pjc, 32A^ro) where Pjc^B(PQ, 32ro - 32A*ro).

Thus, from Theorem 10.4.5 and equations (10.7.21) and (10.7.22) it

follows that S satisfies an (e*, A TQ) condition at any Pjc^B(Po, }2ro —

— 32A^ro), where

(10.7.24) s^ = Z2^^''r^.

10.7. The local differentiability 479

Us ing the definit ion, we see t h a t for each k, the following is t r u e : For

each P in any B {Pjc, 2X^ ro) and each R '<X^ TQ. There ex i s t s a geodesic

r - p l a n e ^ (P, R) cen te red at P, s u c h t h a t

(10.7.25) D[SnB(P,R), 2:{P>R)nB(P,R)]<8^'R.F o r a given k, U B{Pjc, 2^^ ro) = B{Po, 32ro — 30A^ro) and the i n t e r

sec t ion of the s e for all ^ > 0 is j u s t B(Po, 2ro). If P^B(Po, 2ro), the

conc lus ions above hold for all k. So, if we a s s u m e t h a t

^k+i ro<R<^^ro

w e see t h a tE^<ZzR\ Zz = l-^Z^.

Since this is t r u e for each h, we conc lude tha t

( 1 0 7 2 6 ) ^ [ 5 n 5 ( P , i ^ ) , 2 ' ( ^ > ^ ) n 5 ( p , i e ) ] < Z 3 i ^ i + « ,

P ^ J B ( P O , 2ro), 0 < i ^ < r o .

F r o m the proof of Lemma 10 .5 .4 , it follows that (10.7.26) holds with S

a n d ^{P, r) rep laced by CO^Q (5) and COQQ [ ^ ( P , r]), respec t ive ly . Accor

d ing ly , for the r e m a i n d e r of the proof, we s ha l l a s s ume tha t 5 s t a n d s for

o)^^ (S ) and t h a t ^ ( P , r), etc., are ac tua l r -p lanes in Rj^.

N e x t , we observe, us ing (10.7.26), that

2;{P, rl2) n B(P, rj2) C [5 n ^ ( / / 2 ) , Z3(r/2)i+«] n B{P, r/2)(10.7.27) C [5 r\B{P, r), Zs(rl2)^+-] H 5 ( P , r/2)

C IZ ( P . r)nB ( P , r), Zsr- + Z3(r /2)- ] 0 B ( P , r / 2 ) .

Acco rd ing ly

(10.7.28) d[2(P,r),2{P,rl2)]<^Z4r-, Z^ = 2Zs{i + 2~^-).

R e p la c in g r by 2-'^ r for n = \, 2, . . ., in (10.7.21), we conc lude tha t

t h e r e is a )^-plane ^ ( P , 0) t h r o u g h P s u c h t h a t

( 1 0 . 7 . 2 9 ) ^ [ 2 ; ( P , ^ ) , 2 ' ( ^ > 0 ) ] < ^ 5 ^ ^ Z 5 = = Z 4 ( 1 + 2-« + 2 - 2 - + . . . ) ,

s o t h a t

z)[5n ^(P,^), ^(P, 0) n 5(P,r)] <Z 6r l+^ Ze ^ Zs + Zs0<r^r2, P^B{Po,r2)

prov ided tha t ^2 is s ma l l enough .

N e x t , if P and P ' $ 5 ( P o , 2) and | P P ' | = r / 2 , we see, by proceed

in g as in (10.7.20) and usi ng (IO .7.23)

ZiP', 0) nB{P\ rl2) C [5 n B{P\ r /2) , Z6(r/2)i+«] fl ^ ( P ' , ^/2)

c Lr(P. 0) n 5(P, r), Ze ri+- + Z6(r/2)i+-] n P(P', ;'/2)C [2 ; ' (P> 0) n -B(P , r)l 2Z6 fi+- -f 2Z6(r /2) i+-) fl P ( P ' , r/2)

s ince P ' ^ [^/(P, 0) H 5 (P , r), Ze ri+'^ + Ze (^ /2) i+^; here 2 " (^. 0) is

t h e r - p l a n e t h r o u g h P ' \\2(P, 0) . Thus

(10.7.31) ^ [I(P, 0 ), 2 ' ( ^ ' , 0)] < Z^(rl2Y.

From (10.7.23), it follows that if ^2 i s smal l enough and lip is the (N — v)-

p l a ne t h r o u g h P J _ ^ ( P , 0) a n d 7 7 ' is a n o t h e r {N — r ) - p l a n e t h r o u g h P

with ô(7J ' , Up) < 1/2, say, then P i s the on ly in te rsec t ion in B{P, r^)

of1 1 '

a n dS.

B y t a k i n g1 1 '

as the p lane th rough P | | i7po> we see tha tS n B {PQ, r2l2) has a 1 — 1 p ro jec t ion on ^ ( P o , 0 ) . I t fol lows ea si ly

t h a t 2J (P> 0) is th e ac tu al ta ng en t r-p lan e to S a t P . If we choose a (f, rj)

coord ina te s ys tem wi th o r ig in a t P Q and f axes in ^ (Po> 0 ) , we may

represen t the par t o f 5 near PQ i n n o n p a r a m e t r i c f o r m rj =r](i). F r o m

(10.7.24) it follows that rj^ CI. The higher differentiabil i ty follows from

th e r esu lts of §§ 6.4 a n d 6.6 s ince the E U L E R equa t ions form an e l l ip t ic

s y s t e m .

10.8. A d d i t i o n a l r e s u l t s o f F e d e r e r c o n c e r n i n g L e b e s g u e i ^ - a r e a

In this section we present an account of those results of F E D E R E R

which the wr i t e r o r ig ina l ly though t we re re levan t fo r ou r p rob lem bu t

which he la te r found not to be essen t ia l . S ince the resu l ts a re importan t

and not eas i ly access ib le , we inc lude them there toge ther wi th the neces

s a r y b a c k g r o u n d m a t e r i a l .

O ur pre sen ta t io n leans heav i ly on th e no t i on of th e order of a po in t

w i th respe c t to a m ap pin g wh ich was in t ro du ced in § 9-2 . Th e use of

a lgeb ra ic topo logy has been reduced to a m in im um . W e beg in b y

extending the def in i t ion of the order func t ion and in t roduc ing F E D E R E R ' S

essen t ia l m ul t ip l ic i ty func t ion .

Definit ion 10.8.1. W e s u p p o s e t h a t W is an oriented i^-dimensional

manifold of c lass C^ which is d i f feomorphic to an ana ly t ic manifo ld , G

is a do m ain on 3K th e c losure of which , G, C 3J l , and we suppose tha t

J : z ^^ z(x) i s a cont inuous mapping f rom G in to P^ , and .2:0 ^ Pv- If

ô^TîdG), we define 0 [ZQ, T{dG)] == 0; otherwise , we define

(10.8.1) 0[zo,r(dG)] = limO[zo,r{dGn)]

w h e r e {Gn} is any sequ ence of d om ain s of c lass C^ suc h th a t Gn C Gn+i

for each n a n d U Gn = G; itis c lear tha t the l imi t i s independent o f the

sequence . We def ine

M (zo. r) = su p 2 ; 10 [^0, T {d Gi)] \i

for a l l f ini te or countable sequences {Gi} of dis joint subdomains of G.

L e m m a 10.8 .1 . Suppose G, T, and each T « satisfy the conditions ojDefinition 10.8.1 and Xn converges uniformly to r on G. Then

(a ) 0 [z, r(dG)] is constant o n each component of Ry — r{dG).

(b) 0[z,r(dG)] =limO[z,rn(dG)], zir(dG)

(c) M{z, r) < li m inf ikf (2:, Xn)

10.8. Additional results of F E D E R E R concerning LEBESGUE I'-area 481

(d) L,(z,G) >JM(z,r)dz

, is a homotopy, z ^ Up, and{%\ xt (x) = z}

=-{x\ro (x) = z}, then M[z, n) = M{z, TQ).

Proof, (a) and (b) follow from Lemmas 9.2.3 and 9.2.3'. To prove (c),

we let Z)i, . . ., Dm, be any finite sequence of disjoint sub-domains of G.

Then, since 0 [z, T{dD)] = 0 ii z^r(dD),m

Z\0[z,r(aA)] I < lim inf 2" 10 [2, T „ ( 5 A ) ]

<liminfM(^, T^).

The result follows. To prove (d), we choose {z of class C on a domain

D G (on W) so that Zn^^z uniformly and Lv[zny G) ->Lv{z, G). For each

n, the inequality holds (use (9.2.19)) and it holds in the limit by lower-

semicontinuity.

To prove (e), let Di, . . ,, D^ be disjoint subdomains of G such that

z i To (dDi) for any i. Then r^'^iz) OU dDfis empty and

2\0[z,rt{dDi)]\i

is continuous in t. Thus M (z, TI) < M{z, To). The argument is symmetric.

Lemma 10.8.2. Let aiy = C{^) ^^ ^ continuous map from dB{0,i) into

dB{0, 1), B{0, 1) C Rjc, k 2, let yo and yi^dB{0, 1) with yi =^ yo, lety = a~ (yo), and let F he a domain on dB(0, 1) such that y C. F hut

yiia{r). Suppose also that r:z — rj(y) is a diffeomorphism from

dB{0, 1) — {yi} onto Rjc^i. Finally, suppose that 0 [0, a{dB{0, 1)}] = 0.

Then

(10.8.2) 0[ryo,ra{dF)] - 0 .

Proof. Suppose Cn converges uniformly to f on dB{0,\), each Cw

being a simpUcial map (see below). If n is large enough the hypotheses

hold for {an, Cn) • It is clear, then, that we may as well assume that C is

already simplicial. This means that the domain and range spheres are

each subdivided into small curvilinear simplices Ai and Gj, each of

which is the radial projection on ^5(0, 1) of the actual simplex having

the given vertices. The simplices are supposed to fit together in the usual

way as do the simplices of a polyhedron; the subdivision is then com

pletely determined by its vertices. Finally the map C maps each simplex

Ai of its domain berycentrically onto some simplex Gj on the range sphere

(several different At may be mapped onto the same Gj); that is the points

of the flat simplices corresponding by radial projection correspond inthat way.

If we introduce local coordinates 6}, . . ., 6^-^ on each domain simplex,

the formula for the order becomes (since |C| = 1)

0[o,<y{dB{o, m=rj;^Sf^°'i-^r-Hdadet)ddt.^ At

M o r r e y , M u l t i p l e I n t e g r a l s 31

4 8 2 The higher d imens ional PLATEAU p ro b l ems

The quanti ty (—' )°'~^(dCuld6i) is proportional to the normal to the

(k — 1)-surface ^ = C(6^). Since this lies on the (k — 1)-sphere dB(0, 1),

the normal either points towards or away from the origin; the direction

does not change within any simplex Ai. Thus0 [0, a{dB (0, 1)}] =rj^^Z f ^i {dlldS) dS, £ = ± 1

where dS and d^ are the [k — 1)-area elements on the domain and rangespheres, respectively. Clearly, we may write

0 [0, a{dB(0. 1)}] = r,-i 2" *-^ {G}) • <«i

for each y interior to Gj, coj is the number of x such that C( ) = y,

each point being counted with its multiplicity. We shall now show thateach 0)^ = 0 [0, a{dB (0, 1)}].

To do this, let G and & be two adjacent range simplices having the

(k — 2)-simplex y in common. Let Zl^, . . . be those At carried into G and

Zl'i, . . . be those carried into G\ From the nature of the mapping, we

see th at we may order these simplices so tha t Zl ? and Zl'^ are adjacent for

_> = 1, . . ., Pi, say; the remaining Z|2> can be arranged in adjacent pairs,

the same being true of the remaining A'^. It is clear that e^ = e'^ for

each p '<P\ and tha t the e'^ and e^ for each of the remaining adjacentpair have opposite signs. Thus

Px Pi

Hence all the coj are equal and hence equal to 0 [0, a(S)], S =^ dB(0, \),

In our case, the order vanishes.

Now,if we

considerthe

piecewise analyticmap rcr, we see

againthat if yo i T(dGj) for any y, the number of t imes that r a(x) = yo withpositive multiplicity equals the number of times this happens withnegative multiphcity. Approximating to F by smooth regions and usingLemma 9.2.3 and its proof, we obtain the result (10.8.2). If 3^0€ somer(dGj)y the same result holds by continuity.

W e now prove the following special case of H O P F 'S extension theorem(ALEXANDROFF-HOPF, pp. 498—508):

Theorem 10.8.1. Suppose that G and r satisfy theconditions of Defini

tion 10.8.1 with r > 2, and suppose that

yoiridG) andO{yo,T(dG)] = 0,

Then 3 ^ map r"^ :G ~> Ry , such that ^* (x) = z(x), x^dG, and y o i T* (5).

Proof. We shall prove this and the following statement A by induction on r > 2:

10.8. Addit ional resul ts of F E D E R E R concerning LEBESGUE v -a rea 4 8 3

A-Suppose a: y = C(x) is a map from dG into S = S*'-i = dB(0, i)

(B(0, \) CRv,v > 2) such that 0 [0, a{dG)] = 0, G being diffeomorphic

to B(0, 1). Then H a homotopy atiy = h{x,t), where

h{x,0) =C(x), h(xJ)^S, h(x,\)=y , x^dG, 0 < ^ < 1 ,

y being a fixed {independent of x) point of S.

We firs t show that , for each given v, s t a t e m e n t A imp l ie s the theo rem.

Clearly , w e m a y assu m e t h a t the ma nifold 9)1 of De finit ion 10.8.1 is

ana ly t i c and in s ome R^. Let us choose ana ly t ic domains Gt w h e r e

GQG G a n d d G t i s ob ta ined by proceeding a d is tance Q t a long the no rma ls

t o dGo, Q bei ng sm all , 0 < ^ < 1, GiG GQ. W e may choos e z^ ana ly t i c

a n d n e a r z on ^ i and then def ine

zi(x) = z(x) on G — Go, Z i(x) = z* {x) on Gi,

zi{x) =(\ — t) z*(x) + tz(x) on Gt, 0 <t<i\

G — Go is supposed to be so c lose to dG a n d zf so close to z t h a t y o t

r(G — Go) U r(dGi). If dzijdx^ 0 on Gi, then ri(Gi) is of measure 0

and we may f ind a y i arb i t ra r i ly c lose to y o which is not in n (Ci) and

T* m a y be form ed b y following TI by a rad ial projec tion of B{yi, QI) —•

— {y i} o n t o dB[yi,Qi) w h e r e yo^B{yi,Qi) and ^ ( y i , ^ i) f i T ( ^ G ) is

em pt y . O therwise we can find a y i a rb i t ra r i ly c lose to yo such th a t

Tf ^ (yi) is a finite set of points C G\. W e may then choos e a doma in F

of clas s C i 5T5:^(yi) C - T c ^ C G i, a n d P i s d i f feomorphic wi th 5(0 , 1 )

(begin by pass ing an arc of c lass C^ through the points , going s l ightly

bey ond th e f ir st and l a s t po in t s , the n beg in wi th a th in tu be ab ou t th a t

arc). O n Z ', we in t roduce po la r coo rd ina te s r, 0 < r < 1, a n d p on

dB{0, \) b y m ap pi ng on J5(0, 1). C hoosing ^i as abo ve, we define

2-* [x) = zi (x) ou ts ide P a n d

z* (r, p) = yi -\- Qi h[p, \ — r).

W e n o w p r o v e A for r = 2 . Clearly we m a y assum e th a t G = B(0, 1).

Then , i f we in t roduce angula r coord ina tes 6 a n d <p on the doma in and

range c i rc les , respec t ive ly , we may wri te

a:(p = (p{d), O < 0 < 2 j i ; .

li (p^ Ci, then the proof of Lemma 10 .8 .2 shows tha t

(10.8.3) ^ [ ^ ' ^ l ' ^ ) ] = ^ / (p'(0)de=[q){27 t) - (p{0)]l27 z = 0

{S = dB{0,\)),

If (p(6) i s mere ly cont inuous , we see tha t (10 .8 .3) ho lds by approximat

ing . We may then def ine

(rt: (p = (^ - t)<f[Q)+tcp, ^ = ^ I cp{d)dd = <p(d).27 1

0

31'»

Now, l e t u s a s s ume tha t s t a temen t A and our theorem are t rue for

2 < r < y an d le t us consider s t a t em en t ^ for r = ^ + 1. Ag ain, we m a y

a s s u m e t h a t dG = S — dB{0, i). We f i rs t show tha t there i s a map

ci* y = Ci W from 5 to 5 which is hom oto pic to r and w hich is such th a t

s ome po in t yo € >5 is not in R(ri). Then i t i s c lea r tha t c i i s homotopic to

t h e m a p 0*2: y = y where y is on 5 and diametrically opposite to yo- If

a a l ready omi t s s ome po in t y o, we m a y ta k e cxi = a. O the rwis e , l e t

<T* (• y = C* (^)) be an a na lyt ic m a p from S to 5 such th a t |f* {x) — C{x)\

< 1/3 for A;^ 5 . T h e h o m o t o p y

shows that f* is homotopic to C. If dC^^jdx^ 0, the range of or* has

m easu re 0 an d we m ay tak e f i = C*- O therw ise , a'^(Z) has meas u re 0 , Z

being the se t where d^^ldx = 0, and we may choose y o a n d y i in 5 wi th

y o 9^ yi so t h a t (T*~i(yo) is a finite set . T he n we m a y choo se a sm oo th

d o m a i n F Z) <y*~-^(yo) s u c h t h a t a* {F) does not contain yi . If we now

m a p vS — {yi} diffeom orphically on to Rjc b y a m a p CJO: z = rj(y), i t

follows from Le m m a 10.8.2 th a t 0[co yo, co a{dF)] = 0 s o th a t the re

e x i st s a m a p n : F -^Rjc s uch th a t n and co a coincide on dF a n d

co(yo) i Ti(/^). If we define

ai(x) = o)-^ri{x), x^F, ai{x) = a'^ix), x^S— F,

we see tha t yo iai (5). If we define

a{xj) =a*{x), x^S ~F

a(xj) - = c o - i [ ( 1 --t)coa'^(x) +tTi{x)], x^Fj

we see tha t ai is homotopic to a* and hence to or.

Lemma 10.8.3 . Suppose G satisfies the conditions of Definition 10.8.1,

U CRv, r > 2, U = {y\ | y"'| < 1, ^ = 1, . . ., r} and: G -^U is a

mapping 5 0 ^r(dG).(a) Suppose 0 [0, r{dG)] = 0. Then H ^ w<3 i> ^ i : Q —> dU such that

ri[x) = r(x) for x^dG n r-^(dU).

(b) Suppose O[0,r{dG)] = xm, K = ± \, 0 < w < + 0 0. Then

H disjoint compact sets Qi, . . ,, Qm ^^ G ^^<^ ^ mapping TI : G - > £7 $ TI (A;)

= T(J%:) O # ( ^ G n r~^{dU), (TI | ft) ' ^ diffeomorphism from Qi onto U

with Jacohian having the sign of x, i = i, . . ., m, and

ri[G-UQf^]<ZdU.

i

Proof, (a) Th e m app ing n m ay be ob ta in ed f rom th a t of Theo rem

10.8.1 by following i t with a radial projection from 0 onto U; clearly if

r(x)^d U, n (x) = r(x).

(b) W e choose the an a ly t ic d om ains Gt as in the f irs t part of the

proof of Theorem 10.8.1 and approximate c losely on Gi to r by an analy-

i o < / < i ,

10.8. Additional results of F E D E R E R concerning LEBESGUE v-area 4 8 5

t ic T ' ; a s before we c ons t ruc t r* = r on G — Gi , r * = T ' on GQ a n d

r*{x) = {\—t)r(x) + tr'{x) o n dGt» B y t r an s la t in g r ' s l igh t ly , if

neces s a ry , we may ens u re tha t 0 ^ T * (Z), Z being the subset of Go where

dr'^ldx — 0 ; s in ce m ^ 0, dr'^/dx =^ 0 . Then , T * ~ ^ ( 0 ) = {x[, . . ., x'^

a n d dr'^jdx ^ 0 s± each x^ . There mus t be a t l e a s t m of these , call them

^1 , ' • -/^m* Q-t ea ch of w hi ch ^ T * / ^ A ; has the sign of x. If we choose Q,

0 < Q < 1, small enough and set UQ = {y\ {y^^l <:Q, oc = i, . . . , r} ,

then T*~^{i[7e) DU Qi where T*( f t ) = UQ a n d A;^$ (2 •, ' = 1, . . ., m.

If we le t r = G — U ft, w e see th a t 0 [0, r * ( 5 r ) ] = 0 so we co nclude

from T heo rem 10.8.1 th a t the re i s a m ap r^'- T -^R^ — {ff) which co in

cides with T* on dF) by fo l lowing th is wi th a map which is a rad ia l p ro

jec t ion f rom 0 on to dUqiiy^ UQ and is the iden t i ty ou ts ide UQ, w e m a ye n s u r e t h a t r2'- T -^Rv — UQ. L et T3 = T2 on /^ a n d T3 = r * on each Qi.

W e th en ob tai n n b y following T3 b y o" wh ere a i s the rad ia l expans ion

from UQ on to U' for J^UQ and (7(3/) is the radial projection of y on ^ C7

i iy ^U - UQ.

Definit ion 10.8.2 . For each sequence I = {ii, . , ., iv) in wh ich th e 4

are inte ge rs an d 1 < n < 2 < * * * < ^ < A , we define th e pro ject ion

P^ from Rx^ (N > r) on to Rv b y

(10.8.4) ze °'(2:) = z^x, oc ="[,.. .,v, w h e r e w = P^[z),

We now reca l l the no ta t ions in t roduced in Def in i t ion 10 .5 .4 and

Lemma 10 .5 .8 . In add i t ion to the s t a temen ts p roved in Lemma 10 .5 .8 ,

we prove those in the fo l lowing lemma:

L e m m a 10.8.4. Suppose G satisfies the usual conditions

(a ) If w: G -^ C^. — C^_^_i andy for some XQ and I, P^ 0 a){xQ) = 0,

P^ 0 rjc [cjo {xo), t] = 0 for 0 < ^ < 1, where

(10.8.5) rk(coJ)= ^(\ -t)co + trjc{a)),

fjc being defined in Lemm a 10.5.8.

{h) If co: G -^ C^ — C ^_ y_ i and, for some XQ and I, P^ 0 co(xo) ^ 0,

then Pi 0 r]c[(jo[xQ), /] 9^ 0, 0 < ^ < 1.

Proof, (a ) W e a s s ume tha t co{xo) ^ some yjcq, w h e r e

(10 8 6) r^ f f ' l ^ ^ ' -^ ^ ' l < 1> j=ji,---Jk, o}3 = ei, j:^^nyjp,

d^ even , 6^ odd , 1 < 71 < y2 < • * • < jk < N,

Si nc e co*"«( o) = 0, t h e 4 ar e a m o n g th e jp . We may, wi thout loss ofgene ra l i ty , a s s ume tha t the d^' = 0 a n d t h e 6^ = 1 . By renumber ing the

a x e s , w e m a y a s s u m e t h a t y jcq — yk , as defined in the proof of Lemma

10.5 .8(a ) , and assume tha t ioc = oc. But , s ince a){xo) i G^_^_;^, we must

h a v e co^{xo) ^ 0, i = v + \, . . .,k. Clearly the f irs t v coord ina tes of

^k [co [ X Q ) ] will s t i l l be 0 and the others wil l not .

To prove (b) , weaga in a s s ume tha t a){xo)^ s o m e yjcq asdefined in

(10.8.6). If theioc are not inc luded in t h e jp, t h e n co^oc(^xo) =6^ and

rk[o)(^o), Q h a s ^ - c o o r d i n a t e =6^ ^ 0for 0 < ^ < 1. If t h e 4 are in

c luded int h e jp but a l l the dhare not 0 , the result s t i l l holds . But, if

all the dj = 0, we m a y a g a i n a s s u m e t h a t 4 = oc for 1 < < r a n d

jp =p^ \ <,p '<k. Since P^ o (^{XQ) i 0, not all the co*a(:vo) are zero

and th is s t i l l ho lds a f te r apply ing r]c{(Dy t).

Lemma 10.8 .5 . Suppose G satisfies the conditions of Definition 10.8.1.

Suppose a^Rjsf, e > 0, z: G->Rx^ — C^_y^^{a, e) is a mapping

^z(dG) CRN — C'jl^_^,(a, s), and A denotes the totality of v-termed

sequences dof even integers.

Then ^ aLipschitz map u: G ->RN such that

(10.8.7) l^(^) — ^(^)| < 3ey^^, x G,

(10.8.8) Ly (u , Q < 2^ 2^ 2" £" M [Pi 0 z, Pi[a) + 86].I 6eA

Proof. If the r igh t s ide is + oo, there i s no th ing to prove . In v iew of

an obv ious reduc t ion by h o m o t h e t i c t r a n s f o r m a t i o n s , we may also

a s s u m e t h a t a = 0and £ = 1.

B y v i r t u e of Lemma 10 .8 .4 and ou r hypo thes i s , it follows that the

m a p s my. G -> C'j^_^ — C^_v_i given by

co j = r -u 0 ' ' ' oTjsf 0 z, j = i, . . .^ N — V

are a l l def ined and cont inuous . If we define w— co^_^, it follows from

L em m a 10 .8 .4 an d from L em m a 10 .8 .1 (e) th a t

(10.8.9) M(Pi ow,d) = M{Pi 0 z; 6), w{G)(Z C;

for all / a n d 6.

L e t % be the family of a l l componen ts F' ofw-^{C'^ — C^_i) . Since

the in te r s ec t ion ofC^-^ wi th one of the open i^-cells i of C^ — C^_i is

jus t the center of that cell and s ince w i s cont inuous on acompac t s e t , it

fo l lows f rom the Heine-Bore l theorem tha t the sub-family ^ i o f those F'

which con ta in po in t s of the c o m p a c t set w''^ [C'^-v H (C^ — C'^-i)] is

f in i t e . Each componen t F' of %i is open on& b u t it c a n h a p p e n t h a t

F' r\dG IS n o t e m p t y , inwhich case w[F' C\dG) C s ome K\ o the rwis e

w[dF') G dK. Let us define F = F' — dG] t h e n F is also a d o m a i n

(= J " i f JT' n ^G is empty). Since 2:(^G) HC^_v is empty , itfollows from

Le m m a 10 .8 .4 th a t th e cen te r CR of Kis not on w(dG).

So let us choose aF, letKbe theopen r-cell of Q — C^_i con

t a i n i n g w{F), let / be theun ique r - s equence ^P^(K) is a r-cell Ud(= {co I Ico* — S'^l < 1, 6°" even , oc= \,. . ., v}), d =P^ (CR) and let

cor be defined by cor (x) =P^ 0 co(x) — P^ CR ; clearly wp'-T ->tJ,

U = Uo. Then we note f i rs t ofall (see 10.8.9) that

(10.8.10) Z\0[0,cor(dF)]\ ^M{Pioz,d)

10.8. Addit ional resul ts of F E D E R E R concern ing LEBESGUE v-area 487

where 5( / , d) is the set of all r^w(r)CK for some K^Pi K= Us.From Lemma 10.8.3, it follows th a t we may choose a m ap vr : T -^ K ^ (a)if 0 [0, cor(dr)] = 0, then Vr(r) C dK, and (b) if 10 [0, coridF)] \ = m,

0 < m < + oo, the re are dom ains <2i, . . ., Qm with the Qi disjoint andin r such that Vr{r) cR, Vr\ Qi is a diffeomorphism of Qi onto Kând Vr{r— \J Qi)CdK, In both cases, Vr{x) = w{x) f o r A ; $ ^ / ' nz£;-i(îL). Let us define v'{x) = Vr(x) for x on each /^ of g i and i;'(A;)— w{x) elsewhere on G. Then, since if F^ g — gî, it is clear th a t w[r)does not contain the center CR of any cell K in C^ — Ci_i, it follows t h atv'[G) contains no such CR- SO , let us define v{x) =rv[v'[x)]y n beingdefined on each J? — cx as the radial projection from CR onto dK. Then

(10.8.11) \v{x) -z[x)\<.2]lN, x^G,since if z{x) ^ some K^^ C'j^, v{x) is on ^Xjv"- Moreover there are a

finite number i^^, * = 1, . . ., P , of domains such tha t th e Ri are disjoint

and C G, and v\Ri is a, diffeomorphism from Ri onto some cell i^ of C^.

Finally

v{x)^C,_^ f o r : ^ ; $ G - U i ^ ^

(10.8.12) *

Unfortunately v is not Lipschitz (probably). To remedy this, webegin by extending ?:; to a slightly larger domain D Z) G. For each sufficiently small Q, we form the ''mollified" functions VQ by defining a function KQ (X, I) on M for x and | near D so that

iê(:t:,f) > 0 , jKg(xJ)di = \, Kg{x,i) = 0 if | ; ^ - | | > ^ ,

where KQ^ C^ in (A;, | ) and C^ in %, and |^ — l| denotes the geodesicdistance along W. We then define

vi{x)=fK,(xJ)vî)di, x^G,m

Th en , for ^ > 0, each v^ ^ C^, VQ converges uniformly to v on G, and V VQ

converges uniformly to V t" on each compact subset of each Rf. Moreover,for each ;> > 0, we can choose a sufficiently small ^ > 0 such t h a t Vg

furnishes a diffeomorphism from some RI C Ri onto the correspondingKi^, the cell of side 2 — 2;<; concentric with Ki{= v{Ri)), and carries all

points outside the R^ into the set (Cl^^,x). We then define u = rorpî0 . . . 0 r^ 0 VQ where r is defined on each K^C'^ as the expansion fromK^ onto K for y on R^ and define r (y) as the rad ial projection of y fromCK into dK, a y ^K ^ R^. Then u has all the preceding properties of v

and is also Lipschitz. That u is the desired vector follows from its form,(10.8.12), and Theorems 9.1.3, 9.1.4, and 9.2.1.

Lemma 10.8.6. Suppose G satisfies the conditions of Definition 10.8.1,z : G -^ R x^, A has its preceding significance, £ > 0, and R is the set ofa^Rx^^

(10.8.13) Z Z ^''^"M\.Pôz,Pia + ed]<.Z (M[Pi oz,y )dy.

ThenA^(R) > 0.Proof. Letting 5 and T be the cubes in i?jv and Rv, respectively, with

center at 0 and side 2 s, we obtain

f2j2^£ ''M[Pôz,Pi{a) + ed]dA^(a)

- 2^2»'£»' f ^M\Pioz,Pi{a) + sd]dA^{a)

I s ^

= 2:2^6^ f 2^M[Pioz,y + sd]dA^'(y )I y a

= Hz f M[Pi oz,y ]dy]dA^'{a).

The result follows.Theorem 10.8.2. Suppose G satisfies the usual conditions, z:G-^ R^^,

N '>v, and

(10.8.14) /l»'+i[^(G)] = 0 and A^lzidG)] = 0,

Then

(10.8.15) L,{z,G) <2Lr{Pôz,G).I

If N = V, then

(10.8.16) L4z,G) = f M{z,y)dy .

Proof. Let e>0. From (10.8.14) we conclude that A^'+ÎPJ0 z{G)]= 0 for each (v + 1)-sequence / , so th a t th e AT-measure of th e set of alljV — r — 1 dimensional planes of the form x*^ = a^ which intersectz (G) is zero. Consequen tly th e se t of all a R^^ ^^(Q O C^-^-i {a, e) isnot empty has measure zero. In like manner, the set of ^ in R]sf^z[dG)n C'^-v {(^, e) is not em pty has N measure 0. From Lemma 10.8.6, it then

follows that there is an a in RN such that (10.8.13) holds and z{G) DC^_y_i [a, e) and z[dG) H C'^-v (^, fi) are bo th em pty. From Lem ma10.8.5 and (10.8.13), it follows that H a Lipschitz map u : G -^RN suchthat (10.8.7) holds and

U{u,G) <Zf^\Pôz,y]dy.

10.8. Additional results of F E D E R E R concerning L E BE S G U E v-area 489

Since th is uexis ts for each e > 0, it follows that

(10.8.17) U{z, G)<2 lM{Pioz, y) dy.

In case N =v, (10.8.16) follows from (10.8.17) (/ m u s t be (1 , 2, . . ., v))

an d L e m m a 10.8.1 (d). T he n (10.8.15) follows from (10.8.16) and (10.8.17).

Definition 10.8.3. By an admissible measure 0 we m e a n one w h i c h is

the difference of two non-nega t ive meas u re s , for each of which Bore l se ts

a r e m e a s u r a b l e ; a set 5 is m e a s u r a b l e ^ for each e > 0, 3 a c o m p a c t

set F and an open set O^FGSCO and 0± (0) - 0±{F) < e, 0+

a n d 0~being def ined as follows:

\m{e)= sup 2 mei)\

ifor all m e a s u r a b l e p a r t i t i o n s e = \J ef. We then def ine

0+ (e) = [||0 il (e ) + 0 (e)]l2, 0- {e) = \\\ 0 \\ (e) - 0 (e)]/2 .

Definition 10.8.4. A c o n t i n u o u s m a p p i n g / : X -> Y is> said to be

m o n o t o n e 4 ^ / - i ( y ) is ac o n t i n u u m or ap o i n t for each y^f{X). A con

t i n u o u s m a p p i n g f: X - Y issaid to be light f- {y ) is c o m p l e t e l y

d i s connec ted foreach y^f(X). Suppos e z:X->Z is con t inuous , X

be ing a connected metric s pace . Let Y be the collection of c o n t i n u a of

c o n s t a n c y ofz and, if yi and y^ are two such, define dz (71, 72) =infd i a m [z (7)] for e v e r y c o n t i n u u m y which in te r s ec t s bo th yi and 72-

Lemma 10.8.7. With this definition, Y is a metric space, the mapping

m : X onto Y, defined by the condition that m(x) be the continuum of con

stancy containing x ismonotone, and the mapping ^ :Y ->Z defined by

^(rj) == z [m~^ (rj)] is light. If ^ : Y ~ Z is a light mapping, Y and Z

being compact and metric, then Ha function 99 (^) - ^ 0 as Q - 0 if y is

a continuum in Zof diameter < g, then any component of f "^ (y) has dia

meter <=(P{Q).

Proof. The f i r s t s t a temen ts are e v i d e n t . To p r o v e the las t , let {yn}

b e as equence of c o n t i n u a in Zwhos e d iame te r s -> 0 and, for each n, let

rjn be ac o m p o n e n t of C~'^ (yn) and suppose diam ly^j > Co > 0. By choos

in g a subsequence, s t i l l cal led {n}, we may a s s u m e t h a t yn ->y and

'i]n -^rj' Clearly rj isR c o n t i n u u m of d i a m e t e r > 0*0 and, by c o n t i n u i t y ,

^{rj) =y. B ut s ince diam yn^O we see t h a t y is apo in t . But th is con

t r a d i c t s the l igh tness of ^.

Definition 10.8.5. We s u p p o s e t h a t G satisfies the cond i t ions of Defin i

tion 10.8.1 and t ha t Z:G->RN, LV{Z,G)<C^, A''+^[Z[G)'] = 0,

A^ [z[dG)'\ — 0, and z— C 0m w h e r e m:G o n t o X ism o n o t o n e , and

^ : X -^RN is l igh t . Let (3 be the t o t a l i t y of cells r in Rjs^ s u c h t h a t

A^[z(G) C) dr] =0; ev iden t ly all cells r, n o n e ofwhose faces lie a long

a c e r t a i n c o u n t a b l e n u m b e r ofhype rp lanes be long to (S. Let us define

Qo as the t o t a l i t y of c o m p o n e n t s off- (r) and Uo as the t o t a l i t y of com-

Morrey, Multiple Integrals 3 ^ ^

ponen ts o f z-'^(r) — dG for al l r ^ ®. W e define Q{ as the to ta l i ty of

com pone n ts of (P ô f ) ~ i ( r ) and Ui a s the to ta l i ty of com ponen ts of

{Pi oz)-^{r) f o r a l l r C i ^ v .

Lemma 10.8.8. For each / , there is a unique measure 0^, defined on X,

such that, for each open set co d X,

(10.8.18) 0J(a>) = jO\y ,Pioz{dU)'\dy , U = m-^{a)) -dG.

We assum e that G, X, z, m, and f satisfy all the conditions above.

Proof. We begin by def in ing 0l{o)) by (10.8.18) for co ^ i ^ o s o t h a t

U^ Ho- Now if CO, coi, . . . ,con a l l ^ i ^ o , U = m-^((jo) ~ dG, Ui = m-^((jOi)

— dG, t h e co i a n d Uf are dis joint , andco = L)coi, U = U ' Ui, t h e n w ei

see by choos ing a smal le r smooth domain in each Ui t h a t , ii y i P^ oz(dU) yjpioz{dUi),

n

0[y,Pioz{dU)-\ ^ZO[y,P'oz{dUi)],

(10.8.19) *; '

From this , i t fol lows that , i f we define

i<îM = sup2'l*SWIi

for genera l ized par t i t ions of co l ike those above, | |ôl | sa t is f ies (10.8.19)

for the same types of par t i t ions . Next we may def ine

0lHw) = \\0i\\ (CO) + 0l{w)]l2, 0i-{co) = [im («>) - 0l{w)]

a n d t h e n define (co^ dis joi nt a n d ^ i3o)

n

0i± (oj) = su p ^ 0 J ± (co^), U6)iCo), CO open ,

n0l± (a) =. inf 2J 00^ (oJi), a C Uwi, a closed.

Since C is l ight and there are arbitrari ly small cells r^ ^, there a re CO^QQ

of a rb i t ra r i ly smal l d iameter . Thus i f a is c losed (and hence compact) and

CO is open, it follows easily that

0l±{co ~ a) ^ 0l±(co) - 0i±{a).

We then def ine the ex te r ior measure 0^^ (5) as the inf of 01 ^ (co) for

open CO D S and the in te r io r meas u re s 0^ f (S ) as the sup of 0^^ [a) forclosed a C S and the measures can be deve loped c lass ica l ly .

Lemma 10.8.9. Suppose G, z, m, C, ^'^d X satisfy all the conditions

above. For each I, let Z ^ be the set of ^^X'^^^ a continuum y of positive

diameter such that P^ o ^ (y ) is a point. Then Z ^ is of Borel ty pe Fa and

(10.8.20) A'[{Plot) [ZI)-] = 0 , \\<l>i\\[{C'rH^''ZI)}] = 0 .

10.8. Addit ional resul ts of F E D E R E R concern ing LEBESGUE v-area 491

Proof. If we let Z ^ be the set of | ^ J^ 5 f € 3- c o n t i n u u m y of d iame te r

> l/yfe su ch t h a t P^ 0 ^(y) is a point , then Z]c is seen to be closed. For a

give n / an d a given y^[P^ o C) (Z^), we see that C(^^) 0 {P^)-^(y) D

a con t inu um of pos i t ive d iam eter , so th a t A'^[I^{X) pi [P^)~'^{y)] > 0.

H e n c e A^ [[P^ o f) (Z^)] = 0, for othe rw ise it wou ld follow from T he or em

10.2.3 th a t yl»'+i[f (Z )] = /L^+i[^(G)] > 0. L et Z i be defined as above,

s uppos e CO is open on X a n d co D (f^)~-^ {C" (^i)} ^-nd sup po se U — m~^ (co) —

— dG. Then f rom the def in i t ions , we f ind tha t

II n II m-^ \J:' {Zi)]} <:\\ll{oy)<JM[y,{Pioz)\U]dyR

<jM[y,Pioz]dy.

{Pioz){U)

From Theorem 10 .8 .2 , i t fo l lows tha t M{y , P^ oz)^ Li{Rv) so the las t

result follows by expressing (C^)~^{C^(Z|)} as the intersection of a

decreas ing sequence {co } of open sets.

Since our goal in this section is really Theorem 10.8.3 below, we con

ten t ourse lves in what fo l lows immedia te ly wi th s i tua t ions in which

G = G is a compact manifold of class C^ without boundary .

Lemma 10.8 .10. Suppose G = G is a compact manifold of class C^

without boundary , z : G ->Rv is continuous with Lv(z, G) < oo, and otherwise suppose, z, X, f, and m satisfy their usual conditions. Suppose Zn -> z

uniformly on G with Z n^ C^ (G) and Lv [zn, G) < Kfor all n and suppose (p

is continuous on X. Then

(10.8.21) lim [ (p[m\x)-\^ dx = f (p(^) d0o{ei),

G X

00 being the measure 0^, I = (\, 2, . . ., v), of Lemma 10.8.8.

Proof. Since the l imi t depends on ly on the l imi t vec tor z, it is suffic ien t to prove th is for some subsequence . We may there fore assume tha t

t h e m e a s u r e s Wn defined by

"Pnie) = 1bzn

d xdx

converge weakly to some measure S^ . Le t % denote the ce l ls o f cont inu i ty

oiW. li r^(^ an d £ > 0, we m ay choose r' a n d r" w i t h f' Crdf Cr"

s u c h t h a t

(10.8.22) f M{y ,z)dy <s, Wn{r' '— r') < s for all #.

L e t CO be an open subset of C^^ir) which consis ts of components of

C-^{r), and le t U = m-^{w) = z-^{r). T h e n z{dU)Cdr ^nd z{U)Cr.

31a*

Also, s ince Zn(O) C r" a n d Z n(d U) C r" — r' for n la rge

f^dx=f0[y,z„{dU)]dy+l0[y,z„{dU)]dy, n>N.

0o(ct>) = jO[y. z[dU)] dy + fOly, z{dU)] dy.

Since 0 [y, ;^^(^ 17)] = 0 [y, ^(^C7)] for y^r\ it follows from (10.8.22) that

. ^X — 0o(co)OX ^ '

< 2e for n > N.

Since C is l igh t i t fol lows from L em m a 10.8.7 th a t we m a y write X = U co^,

th is be in g a finite un io n of co w he re th e cot have a rb i t ra r i ly s ma l l d ia m e t e r a n d t h e cof are d is jo in t and of the type d iscussed above . The lemma

follows.

Theorem 10 .8 .3 ( F E D E R E R ' S convergence proper ty [2]) . Suppose

G = G is a compact mam fold of class C^ without boundary , suppose

z : G -^ Rjsf is continuous with Lp (z, G) < oo and, suppose z, X, m, and C

satisfy their usual conditions. Suppose that Z n->z uniformly on G with

Lv {zn, G) ^K for all n, each Zn ^ C^ [G). Suppose that the (pi are continuous

on X, Then

(10.8.23) lim fy<pi[m{x)]^^dx=-Jycpj(^)d0l{e,),

where the 0^ are the measures of Lemm a 10.8.8.

Proof. Let us fac tor P^ o ^ = r]^ o h j and le t hi{X) = Xj. T h e n hjom

i s monotone f rom G t o Xj a n d t]^ is light from Xj to Rv a n d P^ o z

= r}-^ 0 {¥ 0 m). L e t 0^^ be the measure of Lemma 10 .8 .10 on Xj. T h e n ,

s ince Lv (P^ o z, G) < K for each / , i t fol lows from that lemma that , i f

t h e (p f are cont inuous on Xj,

(10.8.24) lim f ^ cpf [(hi 0 m ) [x)] ^ dx = ^ f <pf ( f ) ^ 0 * ' ( ^ l ) •

'^~*'^G I I Xi

Moreover , for a subsequence still called Zny we have some m e a s u r e s 0^ on

X such tha t , i f the q)i are cont inuous on X,

(10.8.25) lim f y cpi[m{x)]^-^dx = f y cpj{i)d0i{e,).

N ex t w e no te th a t if co is a com pon ent in Xj of rjj^ (r) w h e r e r^î

a n d t o Pi (^ (so to speak), then hj'^ (co) consis ts of a f ini te number of

dis joint open sets coi in X, and a comparison of the def in i t ions shows

t h a t 0^^w) =Zô{î)' I = follow s t h a t

(10.8.26) ^0*^ W = ^ 0 lÎ^ W ] > ^ a B O R E L se t CXj.



10.8. Additional results of F E D E R E R concerning L E BE S G U E v-area 493

In add i t ion , by choos ing the 99/(f) =(pi\hi{^)] in (10.8.25), we con

clude th a t (10.8.26) holds w ith 0J rep laced by 0 ^ . But no w , since | |0J | | (Z^)

~ 0 and hi is topologica l off of Z^, it fo l lows ra ther eas i ly tha t 0 = 0 .

Since the l imi t in (10.8.23) is i n d e p e n d e n t of the s ubs equence {zn}y the

theorem fo l lows .

I n the p a p e r [2], F E D E R E R proves the following two t h e o r e m s w h i c h

we sha l l mere ly s ta te , s ince we h a v e n ' t f o u n d a way to s implify their

proofs and do not n e e d t h e m in the i r fu l l genera l i ty .

Theorem 10.8.4. Suppose G = G, z, X, M, and ^ satisfy their usual

conditions and that the 0Q are the measures of Lemma 10.8.8. Then H a

simple v-vector v^ (f) defined for \\ 0 \\-almost all i on X for which \v{S)\ is

an integer and

01 {A)^fvi(i)dS {es). S{e)= A' [C (e)],A

where A and e are Borel subsets of X.

Theorem 10.8 . 5 . Suppose G = G, z, X, m, C, and 0Q satisfy the con

ditions above. Then, if A is any open subset of X,

\\0[A)\\ = Lr[z\m-^[A)],

That is, Lebesgue area is additive.


http://-almost/

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— [2] t Jber zweid imens ionale regu lare Va r ia t ionsprob lem e. Math . Ann . 101, 620—632 (1929).

— [3] O n the p rob lem of l eas t a rea an d th e p rob lem of Pla teau , M ath . Z . 32 ,7 6 3 -7 9 6 (1 9 3 0 ) .

— [4] Le ngth a nd area , Am er . M ath . Soc . Colloq . Pu bl . Vol . 30, Prov idence , R . I . ,1948.

R A D O , T . , an d P . V . R E I C H E L D E R F E R : Cont inuous t rans format ions in analys i s .Berl in • G ott ing en • He idelbe rg: Springe r 1955.

R E I F E N B E R G , E . R . : [1 ] So lu t ion of the Pla teau p rob lem for w-d imens ional sur facesof vary ing topolog ica l type . Acta Math . 104, 1— 92 (I960).

— [2] An ep iper imet r ic inequa l i ty re la ted to the ana ly t i c i ty o f m in ima l sur faces .Ann. of Math. 80, 1 — 14 (1964).— [3] O n th e anal yt ic i ty of m inim al surfaces , An n. of M ath . 80, 15— 21 (1964).RE L L ICH, R . : E i n Sa t z i ib e r mi t t l e r e Ko n v erg en z , Nach r . Ak ad . Wi s s . Go t t i n g en ,

M ath . - Ph ys . Kl . , 30— 35 (1930) .R O S E N B L O O M , L . C. : Linear par t i a l d i f feren t ia l equat ions . Surveys in Appl ied Math .

5, New York , 1958 .



5 0 2 Bibl iography

SAKS, S.: Theory of the integral , 2nd revised ed. , Eng. t ranslat ion by L. C. Young.

Hafner Pub . Co . , New York .ScHAUDER, J . : U ber l ineare ell ipt ische Different ialgleichungen zwei ter O rdnu ng.

M ath . Z. 38, 257— 282 (1934).

ScHECHTER, M. I O n LP es t ima tes and regu lar i ty , A m .r . J . M ath . 85 , 1 — 13 (1963).S E E L E Y, R . T . : [1 ] S ingular in tegra l s on comp act m ani fo lds . Amer . J . M ath . 81 ,

6 5 8 — 6 9 0 (1 9 5 9) .— [2] R egular i sa t ion of s ingu lar in tegra l opera tors on com pact mani fo lds . A mer .

J . M ath . 83 , 265— 275 (196l ).— [3] Un p u b l i s h ed .S E R R I N , J . : [1] O n a funda m enta l the orem of the calculus of varia t ions , etc . . A cta

M a t h . 102, 1 — 32 (1959).— [2] O n the def in i tion an d proper t i es of cer ta in var ia t ional in tegra l s . Tran s .

Amer . Math . Soc . 101, 139— 167 (1961).SHIFFMAN, M.:

Different iabi l i ty and anlyt ici ty of solut ions of double integralvar ia t iona l p rob lem s . Ann . of Math . 48 , 274— 284 (1947).SHUTRICK, H . B . : Complex ex tens ions : Qu ar t . J . M ath . O xford , ser. (2) 9, 189— 201

(1958).SiGALO V, A. G .: [1] R eg ula r do uble in tegr als of the ca lculus of va ria tion s in no n-

pa ram et r ic fo rm. Dok lady Akad . Nau k SSSR 73 , 891— 894 (1950) (R uss ian).— [2] Two d imens ional p rob lems of the ca lcu lus of var ia t ions . Uspehi M at . N auk

(N. S.) 6 , 16— 101 (1951). Am er. M ath. S oc. Tra nsla t ion No . 83.— [3] Two-d im ens ional p rob lems of the ca lcu lus of var ia t ions in non -param et r ic

form, t rans form ed into pa ram etr ic form. Ma t . Sb. N . S. 38 (80), 183— 202(1956). Amer. Math. Soc. Translat ions , Ser . 2 , 10, 319— 340 (1958).

SILVERMAN, E. : Defini t ions of Lebe sgue area for surfaces in me tr ic spaces , R ivis taM at . Univ . Pa rm a 2 , 47— 76 (1951) .

SMALE, S. : Morse theory and a non-l inear general izat ion of the Dirichlet problem,mimeographed no tes . Columbia Univ . , New York .

SoBOLEV, S. L.: [1] O n a the ore m of function al ana lysis. Ma t. Sb . (N. S.) 4, 471— 497(1938).

— [2] Appl ica t ions of funct ional analys i s in m athe m at ica l phys ics . Trans l . M ath .Mon. Vol 7, Am er . Math . S oc , Procv idene , R . I ., 1963 .

STAMPACCHIA, G. : [1] I problemi al contorno per le equazioni differenziali di t ipoel l i t tico, A t t i V I Congr. Un . M at . I tal . , Naple s , 21— 44 (1959); Ed i t ion C remo-

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Bibl iography 5 0 3

ToNELLi, L .: [8] L 'e s t re m o assoluto degl i in te gra l! dop pi . Ann. Scuola Norm . S up.

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Index

Admiss ib le boundary coord inates y s t em, 30O

Admiss ib le coord inate sys tem, 300Admiss ib le measure , 489

Al g ebra ic b o u n d ar y b{X, A), 411

Area in tegra l , 1

Basis (for co mp l ex 1 forms), 331

Basis , or thogonal , 332

B o u n d a r y b M oi M, 30O

Bo u n d ary o p e ra t o r , 319

Bo u n d ary v a l u es , 76

Bounded s lope condi t ion , 98

CALDERON'S ex tens ion theorem, 74

CALDERON-ZYGMUND inequalities, 56, 58

CAUCHY inequali ty, 3 5

Class 6:(^, L), 411

Closed forms (p{d(p — O), 299

Coerciveness inequal i ty , 253

d cohomology , 322

Complement ing condi t ion , 212

Complex-analy t i c mani fo ld , 317Com plex DiR iCHLET integ ral d{q),xp),

32 0Complex improper in tegra l , 174

Co n t rac t i b l e , 414

Convex funct ion, 21

Convex set, 21

Cover (in the sense of Vital i ) , 409

d{r),d*{r), 374, 380

DiNi condi t ion, 54Di rec t me t h o d s , 16

DiRiCHLET data, 252

DiRiCHLET growth condition, 32

DiRiCHLET growth lemma, 79

DiRiCHLET integral , 1

DiR iCHLET int eg ral for forms, 291, 320

D I R I C H L E T ' S principle, 5

DiRiCHLET problem, 44

Di s t an ce b e t ween F R E C H E T var ie t i es ,351b e t ween map p i n g s [D {zi, ^2)], 350

do{i:,U'), 439

geodesic, 378

p o i n t set, 406

Dis t r ibu t ion der iva t ives , 20

D o m a i n BfiR. 174

BohR> 181

Gfir, 181

Lipsch i tz (of class Cj), 25, 77of class C , CJJ, C°°, an a l y t i c , 4

of class D\ 384

s t rongly Lipsch i tz , 72wi t h r eg u l a r b o u n d ary 72,

E l em en t a ry fu n ct io n , 43

El l ip t i c sys tems (of differentialequat ions ) , 210

E U L E R ' S equations, 2, 7, 8

Exterior co-differential , 290

E x t e r i o r d e r i v a t i v e , 29O

Exterior di fferent ial r -forms, 288

E x t r e m a l , 32

F E D E R E R ' S convergence proper ty , 492

F E D E R E R ' S mul t ip l i c i ty funct ion , 480

Field of ex t r ema l s , 14

Firs t d i fferent ial , 31

Fi rs t var ia t ion , 7.8

Form, closed, 299

co mp l ex , of t y p e (p, q), 318even , 288

exterior di fferent ial , 288

in H^, 2p, 288

22-flat, 299

n o r m a l p a r t of a, 301 302

odd , 288

multiple integrals in the calculus of variations oct 2008

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