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 ).
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