SINGULARITIES OF SOLUTIONS OF LINEAR PARTIAL •✓

DIFFERENTIAL EQUATIONS

by

Karl Gustav Andersson

fil . lic., Vrml.

By due permission of the Faculty of Mathematics and

Natural Sciences of the University of Lund to be

publicly discussed at the Institute of Mathematics

TVv

on April 3, 1971 at 10 a . m. , for the degree of doctor

of philosophy .

TvV

g 1~~10Tl'-iiEK M/\THEM -~Tl~•H C~till>II.\JM ,, .,. - '

•

Karl Gustav Andersson

SINGULARITIES OF SOLUTIONS OF LINEAR PARTIAL

DIFFERENTIAL EQUATIONS

Lund 1970

Printed in Sweden

Studentlitteratur

Lund 1970

Introduction

The purpose of this paper is to sum up the main points of

[1] and [ 2]. This will be done in the form of a short review

of old and new results concerning related problems . The general

problem of describing the structure of the singularities of

solutions of linear differential equations has a long history

and the exposition will be split into four sections corre­

sponding to different aspects of the problem . Inside each

section the results appear in chronological order . Section 0

is of an introductory nature and contains basic concepts and a

precise formulation of the problem of singularities . My own

work [1] and [ 2] will be treated in sections 2 and 3 . It has

been largely influenced. by my teachers Lars Garding and Lars

Hormander and I wish to take this opportunity to thank them for

their support and guidance .

O. Bicharacteristics and the problem of singularities

Bicharacteristics . Let n be an open subset of in and let

P(x , D) = a (x )D 0

a

be a linear differential operator in n with C00

- coefficients .

The principal symbol

Pm(x,I;) = l: I a I =m

which is a function on the cotangent bundle Tx(n) of n, is

assumed to be real . If

gradf;;Pm(x , E;;)-/-. 0

t ➔ (x(t),E;;(t))

( 1 )

( 2)

( 3)

when

in

P is of principal type, i.e.

E;; c JF.n = in \ O , then the solutions

of

are called bicharacteristic strips of P and their projections

t ➔ x(t ) bicharacteristic curves of P . The strips and curves

are invariantly defined in Tx(n ) and n respectively (see

[16] p . 31 ). When Pm has constant coefficients, the bi ­

characteristic curves are straight lines. For operators P of

principal type with real principal part, the bicharacteristic

4

curves (strips) are the adequate invariants of Pm required to

describe the geometrical structure of the singularities of solu­

tions u of P(x,D)u = o. When P is no longer of principal

type the situation is more involved. For operators P(D) with

cons tant coefficients, Hormander [17] has extended the concept

of a bicharacteristic line in the following way.

Denote by L(P) the set of differential operators Q(D)

which may be obtained as a limit

( 4) Q(s) = lim c(n)P(s+n)

where c ( n) E: 0:: and n -+- 00 in lE.n. The minimal linear sub­

space A'(Q) of lE.n along which Q(D) operates is a 'bi ­

characteristic space ' through the origin. Note that, when P

is of principal type and p m is real, then

L(P) = {a < grad Pm(s) ,D> + b; Pm(s) = O, a,b E: O::}

and the bicharacteristic spaces are the usual bicharacteristic

lines.

Fundamental solutions. A distribution E(x,y) in n x n is

said to be a (right) fundamental solution of P if

P(x ,Dx) E(x,y) = o(x - y) .

When P has constant coefficients , we shall only consider

fundamental solutions E(x , y) of the form F(x-y) where F

is defined in all of lE.n and P(D)F(x) = o(x). Such an F

will also be called a fundamental solution of P(D).

The problem of singularities. If u is a distribution in n,

we denote by a.s.u and s.s.u the smallest closed subset of n

outside which u is analytic or infinitely ditterentiable

respectively . We shall consider the following two prob lems

(corresponding to a = a and a= s respectively):

( 5 ) a

Given two closed subsets x1 ,x 2 of n, find the

smallest set X such that

a . s . u c X 1 , a . s . Pu C X 2 = > a. s . u c X , u E :;I) ' ( n) .

Note that if P is elliptic, i .e . Pm(x,I;) f- 0 when E; E Rn,

then X = x1 n x2 in (Ss) . The same result is valid for (Sa)

provided that the coefficients of P are analytic. This is

the classical result on the regularity of solutions of elliptic

equations . Taking x 2 to be a point one is lead to study

fundamental solutions of P and have the natural problem of

constructing fundamental solutions for which the set of singu -

larities is as small as possible . Such fundamental solutions

make it possible to obtain good majorants of x1

for a general _choice of x 2 (see e.g. Lemma 5 . 2 . of [2]) .

6

1. Fundamental solutions of hyperbolic operators

Operators with constant coefficients . Fundamental solutions of

the wave operator in various dimensions were constructed by

Kirchoff, Volterra and Tedone [ 20 , 31 , 29]. For operators of

higher order, Zeilon [32, 33] obtained fundamental solutions of

arbitrary, real homogeneous operators P(D) of principal type

in three variables. Following earlier work by Fredholm [7] in

the elliptic case, Zei l on expressed his fundamental solution of

P(D) as a sum of abelian integrals on the curve P(s) = O

relative to its intersection with <x ,s > = O in complex pro-

jective space. Zeilon's main contribution to this branch of

mathematics is [34]. Here he extends his previous results to

the case of four variables and arrives at a complete analysis

of the support and the regularity of the fundamental solution

of the equations of bi-axial crystal optics. These equations

correspond to a non-strictly hyperbolic operator P(D) o f

degree four. For this operator the local cones K(P ,N) n

(see

below) are three dimensional if n is parallell to any of the

two optic axes of the corresponding crystal. This f act

describes the phenomena of conical. refraction .

The equations of crystal optics were studied independently

by Herglotz [15], which also derived formulas, similar to those

of Zeilon , for fundamental solutions of a large class of

nurnugeHeuu::; , ::; Ll' .Lt.: L .ty ny veL' uu.t.tc u 1-'e L·o. LuL·:::,.

[26] generalized Herglotz ' formulas to an arbitrary homogeneous ,

strictly hyperbolic operator P(D) with constant coefficients .

If P ( D) is strictly hyperbolic with respect to N and F is

the unique fundamental solution of P(D) with support in

H = {x ; <x , N> ~ O} , then Petrovsky proved that F is analytic

outside B n H. Here B = B(P) denotes the conoid generated

by the bicharacteristic lines of P ( D) through the origin .

Starting from this resu l t , Petrovsky then gave a very precise

description of the suppor t of F . The work of Petrovsky has

r ecently been extended to the gener al , non- strictly hyperbolic

case by At iyah , Bott and Garding [3] . As we have already re ­

marked in section 1 , the bicharacteristic lines are only rele ­

vant for operat or s of principal type . To examine the singu ­

larities of fundamental solutions of non- strictly hyperbolic

operators the localizations (4) of P have to be utilized .

Atiyah , Bott , Garding only consider ' homogeneous localizations ':

lim c(t)P(E; +tn) t -->- co

If Lh(P ) denotes the set of such localizations it is proved

tha t if P(D ) is a homogeneous operator, hyperbolic with

res pect to N, t h en every element in Lh(P) has the same

prope r ties . Thus one may de f ine the local cone r(P , N), i.e . n

the compon e nt of { x E ]Rn;

K (P ,N) = { x E: JR n ; <x , E;> > n

P (0:;t:O} wh ich conta i ns N, and its dual n

0 when [, E f( P , N) } . Obvious l y n K(P ,N)

n i s c ontained i n the ' bicharacteristic s pace ' A' (P ).

n The union u

n c in K(P , N )

n is d e no t ed by W( P , N). If P( D)

hyperbo lic with r es pec t t o N, it is proved in [ 3] t hat the

8

is

unique fundamental solution F with support in {x; <x,N> > O}

is analytic outside W(Pm,N). Moreover, when P is homo ­

geneous , formulas similar to those of Zeilon, Herglotz and

Petrovsky hold. These formulas form the basis of a detailed

examination of F close to W(Pm,N).

Operators with variable coefficients. Starting from previous

work by, primarily, Riemann , Kirchoff and Volterra (see

[27, 20, 31]) Hadamard in [10, 11] constructed a fundamental

solution of any second order, hyperbolic differential operator

P(x,D) with analytic coefficients. Let B denote the conoid y

generated by the bicharacteristic curves of P through y and

put H = {x; <x- y,N>.:. O} . y If p is hyperbolic with respect

to N, then, for a suitable neighbourhood U of the origin,

Hadamard defines a fundamental solution E in U x U which is

analytic at (x,y) if x J. B n H . ,- y y In [1 2] he extends these

results to equations with non-analytic coefficients and in [13]

the globalization of the previous results is discussed. The

general reference to Hadamard's work is [14]. Hadamard's

results were extended in 1956 by Courant and Lax [s] (see also

Lax [21] ). For any first order system A(x,D), strictly hyper­

bolic with respect to N, Courant and Lax constructed a local

parametrix, i . e. a distribution E(x,y) such that

(A(x,D)E(x,y) - I·o(x-y)) E C00

• Here I denotes the unit

matrix. Furthermore they proved that, close to the origin in

E.n X E.n' E lS infinitely differentiable at (x,y) if

X ~ B n H As before B denotes the conoid generated by the y y y

bicharacteristic curves through y. Ludwig [24] generalized

this result and also discussed its globalization.

2. Construction of solutions with singularities on

a bicharacteristic

As mentioned in section 1, the results in the previous

section on fundamental solutions of hyperbolic operators may be

used to obtain majorants of X in the problem (50

). The con­

verse task of finding minorants of X lead to the problem of

finding solutions u of Pu= D with a small non-empty set of

singularities . In 1960 Zerner [ 35] proved that , if P(D) is a

second order operator of principal type with real principal part

and 1 is a bicharacteristic line of P(D), then there is a

distribution u satisfying P(D)u = D with s . s . u = 1 . This

result was subsequently generalized by Hormander ( [1 6] p . 22 1)

to arbitrary operators P(D) of principal type and by Zerner

[36] to a local resu lt for operators of principal type with

analytic coefficients and real principal part . ( [ 36] also con ­

tains certain results for the case of complex coefficients .)

In [17] extensions of these results to arbitrary operators with

constant coefficients are given. For example it is proved that

for every Q E L(P) there is a so lution u of P(D)u = D with

s . s .u = A'( Q) . In [17] Hormander also generalizes Zerner ' s

results in [36] to operators with C00

- coeffi cients . The corre ­

sponding global results will appear in a forthcoming paper by

Hormander and Duistermaat .

10

The theorems above only apply to the problem (Ss ). In the

problem (Sa) one should construct a solution u of Pu= 0

which is singular on a bicharacteristic and analytic elsewhere, .

For operators P(D) of principal type with real principal part ,

the following result is proved in [2] :

For any closed interval I contained in a bicharacteristic

line 1 of P(D) , there is a distribution u such that

a.s . u = s . s . u = I and P(D)u is analytic except at the

(finite ) endpoints of I .

Generalizations of Zerner ' s results to the analytical case

are also valid for the general class of locally hyperbolic

operators studied in [2]. Finally we should mention that Sato

[28] has announced that for operators with real principal part

and analytic coefficients re sults , concerning analyticity ,

corresponding to those of Zerner have been obtained by Kawai

within the framework of hyperfunctions.

3 . The problem of singularities for operators with

constant coefficients

We shall now discuss the general problem (50

). In the case

is compact and x2 convex , John and Malgrange

[19, 25] proved that X = x2 in (Ss) for any operator P(D)

with constant coefficients . The same result for (Sa) is due to

Boman [4] . On the other hand, when p ( D) m

is real and of

principal type but x1 and x2 arbitrary, the results from the

preceeding section give the following necessary condition on

X:

(6) X ~ x1 n x 2 and, for every bicharacteristic line 1,

the set X contains any component I of

1nrn , x1 n x2 ) suchthat I C X1 .

That this condition on X is sufficient in (Ss) was proved by

Grusin [9] in 1963. Let J be the closure of a relatively open 0 0

subcone J of the bicharacteristic one B such that J con-

tains one point on every bicharacteristic line of P(D) through

the origin. Grusin proved that for any such J there is a

fundamental solution F of P(D) with s.s.F CJ. The results

obtained by John, Malgrange and Grusin are all generalized by

Hormander in [17], where the . problem (Ss) is investigated in

detail for arbitrary operators with constant coefficients . In

12

that paper the'bicharacteristic spaces' A'(Q), which we have

described in section 1, are defined . If we denote the union

U A' (Q) by B(P), then Hormander proved that there is a QE":L(P)

fundamental solution F of P(D) with s.s.F c B(P). When

L(P) contains hyperbolic operators this result may be improved.

In particular the results of Grusin [9] are included. Gabrielov

[8] has shown that B(P) is a semialgebraic set of codimension

> o.

For non - hyperbolic operators, the first general investiga­

tion of the analyticity of fundamental solutions is due to

Treves and Zerner [30]. They prove a general but rather inex­

plicit criterion ensuring the existenc e of a fundamental solu­

tion analytic outside a certain algebraic conoid (see [30]

p. 178). For homogeneous operators P(D) such that

grad Re P(s) and grad Im PCs) are linearly independent at

some point s E: in satisfying P ( s) = 0 this criterion does not

apply. This is an immediate consequence of Lemma 3 . 1 in [2] .

When grad Re Pm CU and grad Im Pm(U are linearly indepe n-

dent at every point s E": lRn with Pm(U = 0 it is proved in

[1] that P(D) has a fundamental solution analytic outside

B(P). In this case the A'(Q):s ~ {O} are the planes generated

by grad Re Pm(s) and grad Im Pm(s) for some sf: in satis­

fying Pm(s) = O. The proof of the existence of such a funda­

mental solution is the main result of [1], where al so a

simplified account of the explicit results of [30] is given. In

particular it is proved that, if p ( D) m

is real and of princi-

pal type, then there is a fundamental solution F of P(D)

analytic outside B(P) . The question now arose whether it

would be possible, in the analytical case, to sharpen this to a

r esult analogous to that ot Gru§in in the c - case . Inis tasK

is c arried out in [2]. A main tool here is the partition of

unity treated in paragraph 2 of [ 2]. The construction we give

there was suggested by L . Hormander. The techniques may be

extended to a quite general c lass of operators which we have

called locally hyperbolic. In this extension, methods from [ 3]

play a decisive role. In particular all homogeneous operators

P(D) satisfying the criterion of Treves - Zerner are locally

hyperbolic. Outside the class of locally hyperbolic operators

the situation is still unclear, except in the special case

mentioned above . New problems enter which do not occur in the

CCX)-case of [1 7].

14

4 . The problem of singularities for operators of

principal type with variable coefficients

The first general results seem to be due to Delassus and

Le Roux [6, 22, 23] . They study equations P(x,D)u = f in two

variables x = (x1 ,x 2 ), where f and the coefficients of

p(x , D) are analytic functions. If Pm(x , D) has real coeffi -

cients and is of principal type Delassus proves, using the

method of majorant series, the following result:

Suppose that ~(x) = O is a smooth curve dividing the

connected open set n into two connected parts n+ and n

corresponding to ~(x) > O and ~(x) < O. Suppose also that

u is an analytic solution of P(x , D)u = f in n+, which can­

not be continued analytically from n+ into n- . Then ~(x) = 0

has to be a bicharacteristic curve.

The constructions, due to Hadamard, Courant, Lax and Ludwig,

of fundamental solutions of hyperbolic operators with variable

coefficients immediately give information about the pr oblem of

singularities for such operators. For general operators P(x,D)

of principal type with real principal part, Hormander proved the

following theorem in his book [16]:

Let t ➔ x(t) be a bicharacteristic curve of P tangent

to the surface ~(x) = ~(x 0 ) at x 0 = x(t 0 ) . If

L LI d ~(x(t))/dt t=t 0

> 0

for every such bicharacteristic curve, then there is a neigh-

bourhood of such that, if

then every u belonging to J}• (n) n C00

(n+) such that

Pu E C00

(n) belongs to C00

(n) .

[16] also contains extensions to operators with complex co­

eff icients. However , this theorem is not as precise as that of

Grusin for operators wi th constant coefficients . In [17]

Hormander announced the extension of Grusin ' s results to the

case of variable coefficients . A very elegant proof , using a

more detailed definition of the singular support of a distri ­

bution u E J)• (n) as a subset of Tx(n), was given by

Hormander in his Congress lecture [18] . For operators with

analytic coefficients , Sato [28] reported that similar results

concerning the analyticity of hyperfunctions had been obtained

by Kawai , Kashiwabara and himself.

16

References

[ 1J Andersson , K. G., Analyticity of fundament al solutions ,

Ark . Mat . 8 , 7 3 - 8 1 ( 1 9 7 0 ) .

[ 2] Propagation of analytic i ty of solu-

tions of partial differential equations with

constant coefficients , to appear in Ark. Mat.

8.

[3J Atiyah, M. F ., Bott , R. , Garding, L., Lacunas for hyper-

bolic differential operators with constant

coeff icients I , II. Part I has appeared in

Acta Math . 124 , 109 - 189 (1970) .

[4] Boman, J . , On the propagation of analyticity of solu -

tions of differential equations with constant

coefficients , Ark . Mat . 5 , 271 - 279 (1964 ),

[5] Courant , R., Lax, P. D. , The propagation of discon -

tinuit'ies in wave motion , Proc. Nat . Ac. Sci .

U . S . A. 42, 872 - 876 ( 1 956) .

[ 6] Delassus , Et. , Surles equations lineaires aux derivees

partielles a caracteristiques reelle s , Ann. Ee .

Norm . Sup . (3 ) 12 , Supplement, 53 - 12 3 (1895).

[ 7] Fredholm , I . , Surles equations de l ' equilibre d'un

corps solide elastique , Acta Math . 23, 1-42

(1900 ).

[ 8] Gabrielov , A. M., On a theorem by Hormander (Ru ssian ),

Funkt. Ana l . i ego pril . 4 , 18 - 22 (1 97 0 ).

L9J Grusin, V. V., The extension ot smoothness ot solutions

of differential equations of principal t ype ,

Soviet Math. 4, 248 - 251 (1963) .

[10] Hadamard, J., Recherches sur les solutions fondamen ­

tales et l ' integration des equations lineaires

aux derivees partielles, Ann . Ee . Norm . Sup.

(3) 21, 535-556 (1904) ; (2e Memoire) 22 ,

101-141 (1905).

[ 11]

[12]

[13]

[14]

Theorie des equations aux derivees

partielles lineaires hyperboliques et du

probleme de Cauchy, Acta Math. 31, 333-380

(1908).

La solution elementaire des equations

aux derivees partielles lineaires hyperboliques

non- analytiques, C. R . Acad . Sc . Paris 170,

436-445.

Principe de Huygens et prolongement

analytique, Bull . Soc. Math . Fr . 52, 241 - 278

(1924).

Le probleme de Cauchy et les equations

aux derivees partielles lineaires hyperboliques ,

Hermann 1932 .

[15] Herglotz, G., Ober die Integration linearer partieller

Differentialgleichungen mit konstanten Koeffi ­

zienten I -III, Berichte Sachs . Akad. d . Wiss .

78, 93 -1 26, 287 - 318 (1926) 1 80 , 69-114 (1928 ).

[16] Hormander, L., Linear partial differential operators,

Springer 1963.

On the singularities of solutions of

partial differential equations, Comm . Pure

Appl. Math . 23, 329- 358 (1970) .

Linear differential operators, Proc.

Int. Congr . Math. Nice 1970.

18

[19] John , F ., Continuous dependence on data for solutions

of partial differential equations with a

prescribed bound , Comm . Pure Appl . Math . 13 ,

551 - 585 (1960 ).

[ 20J Kirchhoff , G., Zur Theorie der Lichtstrahlen ,

Sitzungsber . der K. Akad . d . Wiss . zu Berlin

1882 .

[ 21 ] Lax , P . D., Asymptotic solutions of oscillatory initia l

value problems , Duke Math. J . 24 , 627 - 646

(1957 ).

[ 22] Le Roux , J ., Surles integrales des equations lineaires

aux derivees partielles du second ordre a deux

variables independantes , Ann . Ee . Norm . Sup .

(3 ) 1 2 , 227 - 316 (1895) .

[ 2 3] Surles equations lineaires aux derivees

partielles , J . Math . Pure et appl . (5 ) , 4 ,

359 - 408 (1898) .

[ 24] Ludwig , D. , Exact and asymptotic solutions of the Cauchy

problem , Comm . Pure Appl . Math. 13, 473 - 508

(1960 ).

[ 25] Malgrange , B., Sur la propagation de la regularite des

solutions des equations a coefficients con­

stants , Bull . Math . Soc . Sci . Math . Phys . R .

P. Roumaine 3 , 433 - 440 (1959 ).

[26] Petrovsky , I . G., On the diffusion of wave s and the

lacunas for hyperbolic equations, Mat. Sb . 17,

289 - 370 ( 1945) .

[ 27] Riemann , B., Ueber die Fortplantzung ebener Luftwellen

von endlicher Schwingungsweite , Gott . Abhandl .

8 , (1860 ).

[28] Sato , M., Hyperf unctions and partial differential

equations , Proc. Int . Congr . Math . Nice 1970 .

2 2 m 2 2 a Cf/at - Z: • a cp/ax . = 0 , Annali di Ma t €mat ica 1

l l

(3) 1, 1-23 (1898) .

[30] Treves, F., Zerner , M., Zones d ' analyticite des

solutions elementaires , Bull . Soc. Math . Fr.

95 , 155 - 191 (1967) .

[ 31] Volterra , V., Sur l es vibrations des corps elastiques

isotropes, Acta Math. 18, 161-232 (1894).

[ 32] Zeilon, N., Das Fundamentalintegral der allgemeinen

partiellen linearen Differentialgleichung mit

konstanten Koeffizienten , Ark . Mat . Astr . Fys .

6 , No 38 , 1- 32 (1911).

[3 3]

[3 4]

Surles integrales fondamentales des

equations a characteristique reelle de la

physiqu e mathematique, Ark . Mat . Astr . Fys . 9 ,

No 18 , 1-7 0 (1913) .

Surles equations aux derivees partielles a quatre dimensions et le probleme optique des

milieux birefrigents, Acta Soc . Scient .

Uppsaliensis (4) 5 , No 3- 4 (1919 -1 921).

[35 ] Zerner, M., Solutions de l ' equation des ondes presentant

des singularites sur une droit, C. R. Acad . Sc .

Paris 250 , 2980 - 2982 (1960).

[ 3 6] Solutions singulieres d ' equations aux

derivees partielles, Bull. Soc. Math . Fr . 91 ,

203 - 226 ( 196 3 ).

r '( 3'-{/ 'Li 20