Recent Titles in This Series 83 Carlo s E. Kenig, Harmoni c analysi s techniques fo r secon d orde r elliptic boundary valu e

problems, 199 4 82 Susa n Montgomery, Hop f algebras and their actions on rings , 199 3 81 Steve n G. Krantz, Geometri c analysis and function spaces , 199 3 80 Vaugha n F. R. Jones, Subfactor s an d knots , 199 1 79 Michae l Frazier, Bjom Jawerth, and Guido Weiss, Littlewood-Pale y theor y an d th e stud y

of function spaces , 199 1 78 Edwar d Formanek, Th e polynomial identitie s and variant s of n x n matrices , 199 1 77 Michae l Christ, Lecture s on singular integral operators, 199 0 76 Klau s Schmidt, Algebrai c ideas in ergodic theory, 199 0 75 F . Thomas Farrell and L. Edwin Jones, Classica l aspherical manifolds, 199 0 74 Lawrenc e C. Evans, Wea k convergence methods for nonlinear partial differential equations ,

1990 73 Walte r A. Strauss, Nonlinea r wave equations, 198 9 72 Pete r Orlik, Introductio n t o arrangements, 198 9 71 Harr y Dym, J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d

interpolation, 198 9 70 Richar d F. Gundy, Som e topics in probability an d analysis , 198 9 69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

superalgebras, 198 7 68 J . Willia m Helton , Josep h A . Ball , Charle s R . Johnson , an d John N . Palmer ,

Operator theory, analytic functions, matrices , and electrical engineering, 198 7 67 Haral d Upmeier, Jorda n algebra s in analysis , operator theory , and quantu m mechanics ,

1987 66 G . Andrews, ^-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,

combinatorics, physics and computer algebra , 198 6 65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o

differential equations , 198 6 64 Donal d S. Passman, Grou p rings, crossed products and Galois theory, 198 6 63 Walte r Rudin, Ne w constructions of functions holomorphi c i n the unit bal l of C , 198 6 62 B6I a Bollobas, Extrema l graph theory with emphasis on probabilistic methods , 198 6 61 Mogen s Flensted-Jensen, Analysi s on non-Riemannian symmetri c spaces, 198 6 60 Gille s Pisier, Factorizatio n o f linear operators and geometry of Banach spaces , 198 6 59 Roge r Howe and Allen Moy, Harish-Chandr a homomorphism s fo r p-adi c groups, 198 5 58 H . Blaine Lawson, Jr., Th e theory of gauge fields i n four dimensions , 198 5 57 Jerr y L. Kazdan, Prescribin g the curvature of a Riemannian manifold , 198 5 56 Har i Bercovici, Ciprian Foia§, and Carl Pearcy, Dua l algebras with applications to invarian t

subspaces and dilation theory , 198 5 55 Willia m Arveson, Te n lectures on operator algebras , 198 4 54 Willia m Fulton, Introductio n t o intersection theor y i n algebraic geometry, 198 4 53 Wilhel m Klingenberg, Close d geodesies on Riemannia n manifolds , 198 3 52 Tsit-Yue n Lam, Orderings , valuations and quadrati c forms, 198 3 51 Masamich i Takesaki, Structur e o f factors an d automorphism groups , 198 3 50 Jame s Eells and Luc Lemaire, Selecte d topics in harmonic maps , 198 3 49 Joh n M. Franks, Homolog y and dynamica l systems , 198 2 48 W . Stephen Wilson, Brown-Peterso n homology : a n introductio n an d sampler , 198 2 47 Jac k K. Hale, Topic s in dynamic bifurcation theory , 198 1

(Continued in the back of this publication)

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Harmonic Analysis Techniques for

Second Order Elliptic Boundary Value Problems

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Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Serie s in Mathematic s

Number 8 3

Harmonic Analysis Techniques for

Second Order Elliptic Boundary Value Problems

Carlos E . Keni g

Published fo r th e Conference Boar d of the Mathematica l Science s

by th e American Mathematica l Societ y

Providence, Rhod e Islan d with suppor t fro m th e

National Scienc e Foundatio n

http://dx.doi.org/10.1090/cbms/083

Exposi tory Lecture s

from th e N S F - C B M S Regiona l Conferenc e

held a t th e Universit y o f Missouri , St . Loui s

J u n e 3-7 , 199 1

Research part ia l l y suppor te d b y

Nat ional Scienc e Foundat io n Gran t D M S 920090 8

1991 Mathematics Subject Classification. Primar y 35-02 ; Secondar y 42B20 .

Library o f Congres s Cataloging-in-Publicat io n D a t a

Kenig, Carlo s E. , 1953 -Harmonic analysi s technique s fo r secon d orde r ellipti c boundar y valu e problems : dedicate d t o

the memor y o f Professor Anton i Zygmund/Carlo s E . Kenig . p. cm . — (Regiona l conferenc e serie s i n mathematics , no . 83 )

Includes bibliographica l references . ISBN 0-8218-0309- 3 1. Harmoni c analysis—Congresses . 2 . Boundar y valu e problems—Numerica l solutions —

Congresses. 3 . Differentia l equations , Elliptic—Numerica l solutions—Congresses . I . Zygmund , Antoni, 1900 - . II . Title . III . Series . QA1.R33 no . 8 3 [QA403] 510 s—dc20 94-1496 0 [515/353] CI P

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the America n Mathematica l Society' s T j X macr o system .

10 9 8 7 6 5 4 3 2 0 1 0 0 9 9 9 8 9 7

Dedicated to the memory of Professor Antoni Zygmund

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Contents

Introduction x i

CHAPTER 1 . Divergence for m ellipti c equation s 1 §1. Preliminarie s 1 §2. Th e classica l Dirichle t proble m 5 §3. Estimate s fo r harmoni c measur e 8 §4. Fato u typ e theorem s 1 3 §5. Are a integrals , BM O (duj) an d Hard y space s 1 9 §6. The Neuman n problem : variationa l an d wea k formulation s 2 6 §7. Th e Dirichlet , Neuman n an d Regularit y problem s wit h LP data .

Formulations o f the problem s 2 8 §8. Som e genera l consequence s o f (R) go an d {N) qo 3 1 §9. Counterexample s base d o n th e Beurling-Ahlfor s theore m fo r

quasi-conformal mapping s 3 8 §10. Som e approximatio n result s 4 0 §11. Note s 4 2

CHAPTER 2 . Some classe s o f examples an d thei r perturbatio n theor y 4 5 §1. Th e Dirichlet , Neuman n an d Regularit y problem s fo r th e

Laplacian o n Lipschit z an d C 1 domain s 4 5 §2. Th e metho d o f laye r potential s fo r Laplace' s equatio n o n Lipschit z

and C 1 domain s 5 0 §3. Hard y space s o f harmoni c function s o n Lipschit z domain s 5 7 §4. The Dirichlet , Neuman n an d Regularit y problem s fo r operator s

with radiall y independen t coefficient s 6 3 §5. Th e multilinea r singula r integra l approac h t o th e radiall y

independent cas e an d it s perturbatio n theory . Extension s t o th e complex coefficien t cas e an d it s connection s wit h th e boundednes s of th e Cauch y integra l an d Kato' s squar e roo t proble m 6 7

§6. Som e analogie s betwee n th e perturbatio n theor y fo r th e Dirichle t problem an d classica l differentiatio n theor y 7 5

§7. Perturbatio n theor y fo r th e Dirichle t proble m 8 3 §8. Perturbatio n theor y fo r th e regularit y an d Neuman n problem s 9 7 §9. Som e example s relate d t o th e relationshi p betwee n (i?) p, {N)p an d

(D)p> 10 8 §10. Note s 11 1

x CONTENT S

CHAPTER 3 . Epilogue : Some furthe r result s an d ope n problem s 11 4 §1. Genera l divergence for m equations 11 4 §2. Othe r result s and open problem s fo r constant coefficien t operator s

on domain s 11 5 §3. Othe r result s and open problem s for variabl e coefficien t equation s 12 7

References 134

Introduction. In recent year s there ha s been a great dea l of activity in the study o f boundar y

value problems, wit h minima l smoothness assumption s on th e coefficients , o r o n the boundar y o f th e domai n i n question . Thes e problem s ar e o f interes t bot h because of their theoretica l importance , and in view of their applie d implications , and the y hav e turne d ou t t o hav e profoun d an d fascinatin g connection s wit h many area s o f analysis . Technique s fro m harmoni c analysi s hav e prove d t o b e extremely useful i n these studies, both a s concrete tool s in establishing theorems , and a s model s whic h sugges t wha t kin d o f result s ma y b e true .

The mai n purpos e o f thi s monograp h i s t o describ e thes e development s an d connections fo r th e cas e o f secon d orde r ellipti c equation s i n divergenc e form , and t o sho w that , i n spit e o f th e extraordinar y successe s encountere d s o far , many interestin g problem s remai n open .

In Chapte r 1 , w e star t ou t wit h a discussio n o f th e (b y no w classical ) loca l and boundar y propertie s o f solution s t o suc h equations , wit h real , symmetri c coefficients, an d formulat e an d stud y th e classica l Dirichle t proble m fo r them . We the n procee d t o a detaile d stud y o f estimate s fo r th e associate d harmoni c measures, an d giv e application s t o Fato u typ e theorems , th e stud y o f squar e functions (are a integrals ) an d Hard y spaces . W e then se t u p th e variationa l an d weak formulation s o f th e Neuman n proble m i n thi s context . Th e res t o f th e monograph is , roughl y speaking , devote d t o th e stud y o f L p estimate s fo r th e Dirichlet an d Neuman n problems , an d o f L p regularit y i n th e Dirichle t problem . To finish th e chapter , w e introduce a clas s o f examples , whic h arise s fro m con -sidering quas i conforma l mappings , which shows that , i n general , suc h estimate s fail.

Chapter 2 i s devote d t o th e stud y o f classe s o f example s fo r whic h th e L p

estimates d o hold . W e star t ou t b y considerin g th e Laplacia n i n bounde d Lip -schitz domains , fro m severa l point s o f view , includin g th e classica l metho d o f layer potentials . W e the n tur n t o operator s i n th e uni t bal l whos e coefficient s are independen t o f th e radia l direction , an d thei r perturbations . W e first stud y the perturbation s usin g th e metho d o f multilinea r singula r integrals . Thi s al -lows for comple x coefficients , bu t s o far, applie s only t o the 'smal l perturbation ' case. Thes e result s hav e interestin g connection s wit h wel l know n problem s i n harmonic analysi s (th e boundednes s o f th e Cauch y integral ) an d operato r the -ory (Kato' s squar e roo t conjecture) . Next , w e develop a n analog y wit h classica l differentiation theory , whic h i n th e cas e o f rea l coefficient s lead s t o ver y shar p results i n th e perturbatio n theor y o f L p estimate s fo r th e Dirichlet , Neuman n and regularit y problems . Finally , w e presen t som e example s whic h clarif y th e relationship betwee n thes e problems .

Chapter 3 is devoted t o the descriptio n o f some further result s connecte d wit h the main themes of the monograph , and t o the discussion and formulation o f open problems. W e have chose n problem s whic h w e find particularl y challenging , an d which w e feel wil l lea d t o furthe r importan t development s i n th e subject .

As fa r a s expositio n goes , som e o f th e result s ar e prove d i n full , whil e fo r others th e proof s ar e merel y sketched , o r omitte d completely . I n th e las t tw o cases, appropriat e reference s ar e given . Sometimes , i n order no t t o disrup t th e

xi

xii ACKNOWLEDGMENT S

flow of the exposition , biographica l reference s ar e relegated t o two sections enti -tled 'Notes' , a t th e ends of Chapter s 1 and 2 .

Acknowledgments. This monograp h i s a n outgrowt h o f th e note s fo r a series o f CBMS lecture s

that I presented a t the University of Missouri, St. Louis, in June, 1991 . I am very grateful t o Gran t Wellan d fo r organizin g th e conferenc e an d t o th e staf f o f th e Department o f Mathematic s a t th e Universit y o f Missouri , St . Louis , especially to Ms. Delori s Licklider, for al l their efforts i n running the conference. I am also very grateful t o al l th e participant s fo r thei r interes t an d encouragement .

Throughout th e year s I hav e ha d th e goo d fortun e t o benefi t fro m man y conversations an d collaboration s wit h a numbe r o f outstandin g colleagues . I have no t onl y profite d fro m thei r generou s sharin g o f thei r ideas , insight s an d enthusiasm, bu t als o fro m thei r friendship , suppor t an d encouragement , whic h helped m e get throug h some very difficul t times . I feel especiall y indebte d t o B . Dahlberg, E . Fabes, R. Fefferman, D . Jerison, J. Piphe r and G. Verchota. Than k you!

Carlos E . Keni g Chicago, October 199 2

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Recent Titles in This Series (Continued from the front of this publication)

46 Edwar d G. Effros, Dimension s and C*-algebras , 198 1 45 Ronal d L. Graham, Rudiment s of Ramsey theory, 198 1 44 Philli p A. Griffiths, A n introduction t o the theory of specia l divisors on algebraic curves,

1980 43 Willia m Jaco, Lecture s on three-manifold topology , 198 0 42 Jea n Dieudonne, Specia l functions an d linea r representations of Lie groups, 198 0 41 D . J. Newman, Approximatio n wit h rational functions , 197 9 40 Jea n Mawhin, Topologica l degree methods in nonlinear boundary value problems, 197 9 39 Georg e Lusztig, Representation s of finite Chevalley groups, 197 8 38 Charle s Conley, Isolate d invariant set s and the Morse index, 197 8 37 Masayosh i Nagata, Polynomia l rings and affine spaces , 197 8 36 Car l M. Pearcy, Som e recent development s in operator theory , 197 8 35 R . Bowen, O n Axiom A diffeomorphisms, 197 8 34 L . Auslander, Lectur e notes on nil-theta functions , 197 7 33 G . Glauberman, Factorization s in local subgroups of finite groups, 197 7 32 W . M. Schmidt, Smal l fractional part s of polynomials , 197 7 31 R . R. Coifman and G. Weiss, Transferenc e method s in analysis, 197 7 30 A . Pekzyiiski, Banac h space s of analyti c function s an d absolutel y summin g operators ,

1977 29 A . Weinstein, Lecture s on symplectic manifolds , 197 7 28 T . A. Chapman, Lecture s on Hilber t cube manifolds, 197 6 27 H . Blaine Lawson, Jr., Th e quantitative theor y of foliations , 197 7 26 I . Reiner, Clas s groups and Picard groups of group rings and orders , 197 6 25 K . W. Gruenberg, Relatio n module s of finite groups, 197 6 24 M . Hochster, Topic s in the homological theor y of modules over commutative rings, 197 5 23 M . E. Rudin, Lecture s on set theoretic topology, 197 5 22 O . T. O'Meara, Lecture s on linear groups, 197 4 21 W . Stoll, Holomorphi c functions o f finite order i n several complex variables, 197 4 20 H . Bass, Introductio n t o some methods of algebraic AT-theory, 1 974 19 B . Sz.-Nagy, Unitar y dilations of Hilber t spac e operators and related topics , 197 4 18 A . Friedman, Differentia l games , 197 4 17 L . Nirenberg, Lecture s on linear partia l differentia l equations , 197 3 16 J . L. Taylor, Measur e algebras, 197 3 15 R . G. Douglas, Banac h algebra techniques i n the theory of Toeplitz operators, 197 3 14 S . Helgason, Analysi s on Lie groups and homogeneous spaces, 197 2 13 M . Rabin, Automat a on infinit e object s and Church' s problem, 197 2 12 B . Osofsky, Homologica l dimension s of modules , 197 3 11 I . Glicksberg, Recen t result s on function algebras , 197 2 10 B . Grunbaum, Arrangement s and spreads , 197 2 9 I . N. Herstein, Note s from a ring theory conference , 197 1 8 P . Hilton, Lecture s in homologica l algebra , 197 1 7 Y . Matsushima, Holomorphi c vecto r fields on compact Kahle r manifolds , 197 1 6 W . Rudin, Lecture s on the edge-of-the-wedge theorem , 197 1 5 G . W. Whitehead, Recen t advances in homotopy theory , 197 0 4 H . S. M. Coxeter, Twiste d honeycombs , 197 0

(See the AMS catalog for earlier titles )