Introduction to the Theory of Linear Partial Differential Equations
STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 14
Editors: J. L. LIONS, Paris G. PAPANICOLAOU, New York R. T. ROCKAFELLAR, Seattle H. FUJITA, Tokyo
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
INTRODUCTION TO THE THEORY OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS
JACQUES CHAZARAIN and ALAIN PIRIOU Professors at the Universite de Nice
English version edited, prepared andproduced by TRANS-INTER-SCIENTIA P. 0.Box 16, Tonbridge, T N l 1 8 0 Y , England
1982
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
North- Holland Publishing Company, I982 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any f o r m or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN 0 444 864.520
Translation of: Introduction a la Theorie des Equations aux Derivees Partielles Lineaires Bordas (Dunod), Paris, 198 I
Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors f o r the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 5 2 VANDERBILT, NEW YORK, N.Y. 10017
L i b r a r y 0 1 Congress Cataloging in Puhlication I)ata
C h az a r a in, J a c q u e s . I n t r o d u c t i o n t o t h e t h e o r y of l i n e a r p a r t i a l d i f f e r e n t i a l equations. ( S t u d i e s i n mathematics and i t s a p p l i c a t i o n s ;
14 T r a n s l a t i o n 9 f : I n t r o d u c t i o n a l a f h i o r i e des Gquations aux d e r i v e 6 s p a r t i e l l e s l i n e a i r e s Bibliography: p. Includes index. 1. D i f f e r e n t i a l e q u a t i o n s P a r t i a l . I. P i r i o u , A l ai n . 11. T i t l e . 111. S e r i e s . QA374.C4513 1982 515.3'53 82-8226 I S B N 0-444- 86452-0 (Else v i e r )
.
PRINTED IN T H E NETHERLANDS
TABLE
FOREWORD
OF
CONTENTS
................................................
xiii
............. .........................
Chapter 1 : D i s t r i b u t i o n s a n d o p e r a t o r s
. 2.
Spaces of distributions
1
Convolution and Fourier transformation of distributions
................................... 3 . Singular spectrm of a distribution ............. 4 . Operators and kernels ........................... 5 . Operators and support properties ................
.
1 1 13 21
23
30
6
Differential operators with constant coefficients
33
. 8.
Operators and distributions on a manifold
.......
44
7
9
.
Operators and kernel distributions on a manifold
59
Regular open subsets of lRn and manifolds with boundary
62
........................................
.
10
.................. Paley-Wiener-Schwartz theorem ............ Homogeneous distributions ............... Fourier transform of eiXx / 2 ............ Fundamental solution of the Cauchy-Riemann operator ................................. Fundamental solution of the Laplace operator .................................
Additional notes and exercises 10.1
10.2
10.3 10.4
10.5
10.6
Fundamental solution of the heat operator -v-
71 71 72 73
74 74 76
vi
CONTENTS
10.7
Fundamental solution of the Schrgdinger operator
10.8
Fundamental solution of the wave operator
10.9
............................... ...............................
Inverse image of a distribution
........
77 78 80
....... 1. Dirichlet's principle ......................... 2. The spaces Hs(mn) and HS (X) ................ loc s S 3. The spaces H (X) and Hloc(X) ..................
104
4.
112
S o b o l e v s p a c e s and a p p l i c a t i o n s
Chapter 2 :
.................. 5 . Application to the Dirichlet problem .......... 6. Sobolev spaces and regularisation ............. 7 . Additional notes and exercises ................ 7.1 Poincar6's inequality ................... Trace theorems. spaces HS (X) 0
7.2 7.3 7.4 7.5
7.6
Invariance of H
S
under diffeomorphism
...
....................... ..................... inequalities ................
Strict inclusions
Lax-Milgram theorem Hbnander's
.
1
. 3. 4.
2
83 85
122
126 131
131 132
133 133 134
Local solvability for differential operators of principal type with real principal symbol
135
Symbols. o s c i l l a t o r y i n t e g r a l s and st a t i onary-phase theorems
139
........................
Chapter 3 :
83
............. Introduction .................................. Symbols ....................................... Elliptic symbols .............................. Asymptotic expansions of symbols ..............
139 140
149 152
CONTENTS
.
vii
................. 6. Various generalisations ....................... 7. Oscillatory integrals ......................... 8. Integral operators associated with a phase and an amplitude .................................. 9 . Stationary-phase theorem ...................... 10. Additional notes and exercises ................ 5
Topology on the symbol spaces
10.1 10.2
10.3
10.4
Symbols with uniform upper bounds with respect to x in Bn
............ Oscillatory integrals and decomposition into plane waves ....................... Lacunas of the fundamental solution of the wave operator ......................
. 2. 3. 4. 5
.
6.
182 182 183
184
Stationary-phase theorem for a surface integral
188
...............................
196
...
216
Pseudo d i f f e r e n t i a l o p e r a t o r s
Symbolic calculus of p.d.0.'~ on a manifold Elliptic p.d.o.'s
........ 10. Friedrich's lemma and generalisations ......... 9.
167
187
............................. 8. P.d.0.'~ and Sobolev spaces .................... 7.
159
......... Definition .................................... A characterisation of p.d.0.'~ ................ Symbol of a p.d.0. ............................ Algebra and symbolic calculus of p.d.o.'s ..... P.d.0.'~on manifolds .........................
Chapter 4 : 1
'157
Fundamental solution of the wave operator for space dimension equal to 1. 2 or 3
...........................
10.5
155
Elliptic complexes and Hodge's theorem
191
205 207 211
221 227 243 250
viii
CONTENTS
.
11
................ P.d.o.'s with uniform symbols .......... P.d.o.'s with uniform symbols and dependent on one parameter ............. Second proof of the estimate (7.6.1) of Chapter 2 ........................... Convolution with certain homogeneous distributions ..........................
Additional notes and exercises
256
11.1
256
11.2 11.3
11.4 11.5
Endomorphisms of complexes and traces
11.6
Cotlar-Knapp-Stein lemma
11.7
Chapter 5 :
............... Calder6n-Vaillancourt theorem ..........
Elliptic boundary-value problems
.....
..................................
.
Introduction
2
.
Regularity of the potential at the boundary
3
.
The Calder6n projector
1
4.
..
...
........................
...................................... 5 . Examples ...................................... 6. Additional notes and exercises ................ 6.2
6.3 6.4
6.5 6.6 6.7 6.8
Example of a p.d.0. which does not satisfy the regularity property of Theorem 2.5
............................ Cauchy's formula and the Hilbert operator ............................... Weakening of assumption (2.1) .......... Local regularity in the neighbourhood of a point of an
264 264 264 266 267
271 271
Application to elliptic boundary-value problems
6.1
261
....................... Global regularity ...................... Converse of the regularity theorem ..... Case of systems which are elliptic in the sense of Douglis-Nirenberg ......... The oblique derivative problem .........
275 290 295 302
307
307
308 310
310
310 311 312 316
CONTENTS
6.9
Chapter 6 :
.
1 2
.
. 4. 3
ix
Elliptic boundary-value problems with interfaces
........................
320
....................
325
Evolution equations
The Cauchy-Kovaleski and Holmgren theorems
....
325
Necessary condition for the Cauchy problem to be well posed
337
Hyperbolic operators with constant coefficients
344
Hyperbolic Cauchy problems with variable coefficients
359
.................................
.................................. 5 . Parabolic Cauchy problems ..................... 6. Semigroups of operators and applications ...... 7. Additional notes and exercises ................ 7.1
Counter-example to the property of finite propagation speed in the non-differential case
....................
7.2 7.3 7.4 7.5 7.6
Generator of a strongly continuous group
....... . Wave equation with Neumann boundary conditions ............................... Equations of the Schradinger type ....... Further discussion of Theorem 6.9
Energy inequality for the wave equation
............. 1. Introduction .................................. 2. Operators and spaces used ..................... 3. The uniform Lopatinski condition .............. 4. Energy inequalities ........................... 5 . Construction of the symmetriser ...............
Chapter 7 :
M i x e d h y p e r b o l i c problems
378 384 402 402 402 403 404 405 407
409 409 411 421 440 448
CONTENTS
X
6
.
S o l u t i o n of t h e problem without i n i t i a l conditions
.................................... 7 . S o l u t i o n of t h e mixed problem ................. 8 . F i n i t e propagation speed ...................... 9 . A d d i t i o n a l n o t e s and e x e r c i s e s ................ 9.1 Spaces eyt H ' ......................... Y 9.2 Energy i n e q u a l i t y i n t h e s c a l a r case .... 9.3 Algebraic p r e l i m i n a r y t o E x e r c i s e 9.4 ... 9.4 P o s i t i v e symmetric systems .............. 9.5
Example of a boundary-value problem f o r a f i r s t - o r d e r d i f f e r e n t i a l system s a t i s f y i n g t h e uniform L o p a t i n s k i condition
...............................
Chapter 8 :
. 2. 3. 4.
1
5
.
Microlocalisation
......................
General p r o p e r t i e s of t h e s i n g u l a r spectrum (WF)
...................... .........................
460
478 483
491 491 493
494 495
499
501 501
The fundamental theorems
508
The c a s e of manifolds
523
..... ................
The c a s e of p s e u d o - d i f f e r e n t i a l o p e r a t o r s
534
A d d i t i o n a l n o t e s and e x e r c i s e s
539
5.1 5.2 5.3
5.4 5.5 5.6
Conclusion of t h e c a l c u l a t i o n of w F ( G ( x ' ) 0 Y ( x n ) )
...................
539
Conclusion o f t h e c a l c u l a t i o n of WF'(v(x-y)) for v E a'(iRn)
540
Construction of a d i s t r i b u t i o n whose s i n g u l a r spectrum i s a d i r e c t i o n i n T*(X)\O
540
.........
.............................. I n t r i n s i c c h a r a c t e r i s a t i o n of t h e s i n g u l a r spectrum ....................... Trace c a l c u l a t i o n s ...................... The Lefchetz-Atiyah-Bott formula ........
541 543
545
CONTENTS
Bibliography
Index
....................................
...................................................
xi
549
555
This Page Intentionally Left Blank
FOREWORD
Since t h e f i f t i e s , t h e t h e o r y of p a r t i a l d i f f e r e n t i a l equations has undergone c o n s i d e r a b l e development , y e t it i s c e r t a i n L y one of t h o s e f i e l d s where w e f i n d b o t h t h e l a r g e s t number o f r e s e a r c h a r t i c l e s and t h e s m a l l e s t number o f works of s y n t h e s i s . The s p e c i a l i s t can come t o terms w i t h such a s i t u a t i o n , b u t t h e student who d e s i r e s an i n t r o d u c t i o n t o t h i s t h e o r y f i n d s it r a t h e r a handicap.
Thus, i n t h e p r e s e n t book, which i s not a
t r e a t i s e b u t an i n t r o d u c t i o n , we aim t o p r e s e n t a r e a s o n a b l y l a r g e range of r e c e n t methods:
pseudo-differential operators,
o s c i l l a t o r y i n t e g r a l s , s t a t i o n a r y phase expansion, m i c r o l o c a l i s ation
....
We show how t h e s e methods permit t h e s o l u t i o n o f
numerous c l a s s i c a l problems i n an e l e g a n t and g e n e r a l manner: e l l i p t i c e q u a t i o n s , boundary problems, e v o l u t i o n e q u a t i o n s , mixed h y p e r b o l i c p r o b l e m s , . . .
, We have a l l o t t e d c o n s i d e r a b l e
room t o t h e e q u a t i o n s of Physics because, h i s t o r i c a l l y , t h i s has been t h e o r i g i n of important problems i n p a r t i a l d i f f e r e n t i a l equations and remains t o t h e p r e s e n t day t h e p r i n c i p a l s o u r c e of interesting questions.
A s r e g a r d s t h e form of t h e book, we have endeavoured t o g i v e very d e t a i l e d p r o o f s i n such a way a s t o r e n d e r t h e work accessi b l e t o a r e a d e r a t post-graduate l e v e l .
With t h i s i n mind, we
have sometimes p r e f e r r e d t o r e s t r i c t t o some degree t h e g e n e r a l i t y of c e r t a i n results i n o r d e r t o avoid o b s c u r i n g t h e ideas- under-
-xiii-
xiv
FOREWORD
l y i n g t h e t e c h n i q u e s b e i n g used.
Each c h a p t e r b e g i n s , i n
g e n e r a l , w i t h a b r i e f d e s c r i p t i o n of t h e s u b j e c t m a t t e r and t h e methods which w i l l b e developed t h e r e i n , and ends w i t h supplements i n t h e form of e x e r c i s e s t o g e t h e r w i t h h i n t s f o r s o l u t i o n ( a n a s t e r i s k i n d i c a t e s t h o s e which w i l l b e u t i l i s e d i n t h e t e x t which follows).
For t h e convenience of t h e r e a d e r , t h e main e x t e n s i o n s
a r e l i s t e d w i t h i n t h e t a b l e of c o n t e n t s .
i s deliberately restricted:
The b i b l i o g r a p h y given
anyone w r i t i n g on such a s u b j e c t
r e f e r s t o such a l a r g e number of s o u r c e s t h a t an e x h a u s t i v e l i s t of t h e s e would be unwieldy.
It seems t o us t o be p o s s i b l e t o use t h i s book a t two l e v e l s ; on t h e one hand t h e f i r s t f o u r Chapters ( D i s t r i b u t i o n s and Operat-
ors; Sobolev Spaces; Pseudo-differential
Symbols and O s c i l l a t o r y I n t e g r a l s ;
O p e r a t o r s ) can s e r v e as t h e b a s i s f o r an
introductory post-graduate course.
On t h e o t h e r hand, t h e l a s t
f o u r Chapters ( E l l i p t i c boundary problems; Mixed h y p e r b o l i c problems;
Evolution equations;
M i c r o l o c a l i s a t i o n ) a r e aimed more
p a r t i c u l a r l y a t Ph.D. s t u d e n t s and a t r e s e a r c h workers who w i l l f i n d c o n t a i n e d t h e r e i n numerous r e s u l t s f o r m e r l y d i s p e r s e d amongst s p e c i a l i s e d j o u r n a l s .
CHAPTER I
D I S T R I B U T I O N S AND OPERATORS
1.
SPACES OF DISTRIBUTIONS The c o n s i d e r a b l e expansion o f t h e t h e o r y of p a r t i a l d i f f e r -
e n t i a l e q u a t i o n s d u r i n g t h e l a s t twenty y e a r s h a s , i n l a r g e measure, i t s o r i g i n i n t h e development by L . Schwartz of t h e theory of d i s t r i b u t i o n s .
T h i s t h e o r y c l a r i f i e s and u n i f i e s
s e v e r a l concepts which were p r e v i o u s l y r a t h e r u n c l e a r , l i k e f o r example t h e concepts o f :
weak d e r i v a t i v e , g e n e r a l i s e d F o u r i e r The t h e o r y of d i s t r i b u t i o n s i s
t r a n s f o r m , fundamental s o l u t i o n . nowadays u n i v e r s a l l y known;
however, f o r t h e convenience of t h e
r e a d e r , we s h a l l h e r e summarise t h e elements of t h i s which will The most s t a n d a r d of t h e p r o o f s w i l l b e
prove u s e f u l l a t e r .
o m i t t e d , t h e s e b e i n g a v a i l a b l e i n SCHWARTZ [11, HORMANDER [11,
TREVES [ll. Letting
c1
.. , a n )
= (a1,,
E
W n be a m u l t i - i n d e x , we p u t
with
= - l-
D
j
I.\
=
a,
+
...
+ a"
9
1
a! = (a,!)x
a
i ax
...
a
x (wn!);
2
DISTRIBUTIONS AND OPERATORS
i f x = (xl,
...,
n x n ) E IR we p u t 1x1 = (x:
We l e t X be an open s u b s e t o f IRn
, and
+
(CHAP. 1)
... + x2n )1 / 2 .
we suppose m
E
IN u { +
3.
m
m
We denote by C ( X ) t h e space of complex-valued f u n c t i o n s whose d e r i v a t i v e s of o r d e r l e s s t h a n or e q u a l t o
m
a r e continuous.
The f a m i l y of semi-norms
where K i s a compact s u b s e t o f X and
j
i s an i n t e g e r such t h a t A subset
0 5 j S m y e q u i p s C m ( X ) w i t h a Frgchet-space t o p o l o g y .
B of C m ( X ) i s s a i d t o b e bounded i f , f o r any semi-norm p have
Sup cp E B
p (cp) K, j
c
+
OD
K , j y we
.
The s u p p o r t of a f u n c t i o n cp
E
C o ( X ) i s by d e f i n i t i o n t h e
s m a l l e s t c l o s e d s u b s e t o f X o u t s i d e o f which t h e r e s t r i c t i o n o f
i s z e r o (or a l t e r n a t i v e l y , t h e c l o s u r e of t h e s e t of p o i n t s of X where Q i s n o n - z e r o ) , and we denote it by supp
C i orBK)t h e
.
If K i s a
o r b K ( X ) (or more con-
compact s u b s e t of X, we denote by C:(X) cisely
cp
c l o s e d t o p o l o g i c a l subspace of C m ( X ) formed
by t h e f u n c t i o n s w i t h s u p p o r t i n K.
The v e c t o r subspace o f
m
C ( X ) formed by t h e f u n c t i o n s w i t h compact s u p p o r t i s denoted by
C t ( X ) or b m ( x ) (or more c o n c i s e l y & ( X )
when m =
+
m);
e q u i p it w i t h t h e i n d u c t i v e l i m i t t o p o l o g y of t h e C"(X)
K
v a r i e s over t h e f a m i l y of compact s u b s e t s of X.
a l i n e a r mapping from
c:(x)
we when K
In p a r t i c u l a r ,
i n t o a t o p o l o g i c a l v e c t o r space F i s
Q
3
SPACES OF DISTRIBUTIONS
(SEC. 1)
m
continuous i f and only i f i t s r e s t r i c t i o n t o each C,(x)
i s contin-
ous, s o it i s s u f f i c i e n t t o b a s e o u r r e a s o n i n g on sequences. bounded s u b s e t o f $ ( X )
i s a bounded s u b s e t of C:(X)
A
for a certain
compact s u b s e t K .
A distribution
DEFINITION 1.1 :
continuous linear form on
c ~ ( x =)
u
on the open s e t X i s a i . e . a linear form u
&(XI,
on B(X) such t h a t f o r any compact subset K C 2 0, j
IN with \ u ( y ) I
E
C
5
The i n j e c t i o n of & ( X )
p
K,j
there e x i s t s
(Q) f o r a l l Q
Bm(X)
into
c X,
E
i s continuous, and a l s o
o f dense image ( c o n s i d e r a r e g u l a r i s i n g sequence, i . e . of f u n c t i o n s p
where p
E
&(IR")
I p ( x ) dx = 1;
i s such t h a t
r
%
+
=
Ju(y)
u in
&'Im(X)
of
a sequence
K o f t h e form
we c o n s i d e r t h e r e g u l a r i s a t i o n s u
u,(x)
&,(XI.
pk[x-y)
Bm(X)i f
k
Bm(X)i s
+
dy
+
k = u
*
pk o f
suppose u u
E
Bm(X);
d e f i n e d by
and we can e a s i l y show t h a t
a).
Thus t h e d u a l t o p o l o g i c a l space
i d e n t i f i e d w i t h a v e c t o r subspace of 8(X)
( w e i d e n t i f y a continuous l i n e a r form on
% . m( X )
with i t s r e s t r i c t -
ion t o & ( X ) ) .
a m ( X ) i s termed t h e space o f d i s t r i b u t i o n s on X
of o r d e r 5 m.
I n p a r t i c u l a r t h e d i s t r i b u t i o n s of o r d e r 5 0 are
measures.
For example, f o r a
E
X , we d e f i n e t h e Dirac measure
4 8 Q
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS E$'(X)
a E
a
a t the point I Let Lloc(X)
B0(X).
< 6 a , Cp > =cp(a) f o r
by
be t h e space of c l a s s e s o f f u n c t i o n s
which a r e i n t e g r a b l e over any compact s u b s e t of X for t h e Lebesgue measure dx = dxl f dx
E
,&''(X)
.. .
dx
n
i n X.
For f
E
1 Lloc(X),
t h e measure
i s i d e n t i f i e d w i t h t h e d i s t r i b u t i o n ( a l s o denoted
f) d e f i n e d by
by
I n p a r t i c u l a r we s h a l l c o n s i d e r t h e c a s e where
i s t h e Heavi-
f
s i d e f u n c t i o n Y i n IR d e f i n e d by Y(x) = 1 i f x 2 0 , and by Y(X)
=
o
if x < 0.
We now i n d i c a t e w i t h o u t proof s e v e r a l t o p o l o g i c a l p r o p e r t i e s
of a ( X ) ,
a'(X )
which w i l l be u s e f u l l a t e r on.
n a t u r a l t o p o l o g i e s on 8 . ' ( X ) :
t h e topology
ence, and t h e s t r o n g d u a l t o p o l o g y ,
"e,, of
There a r e two
rso f
simple converg-
&(XI, i . e . t h e t o p o l -
ogy of uniform convergence on t h e bounded s u b s e t s o f &(X)
(since
a bounded s u b s e t o f b ( X ) i s r e l a t i v e l y compact, t h e t o p o l o g y o f uniform convergence on t h e compact s u b s e t s of 8 ( X ) c o i n c i d e s with
rb). When we
equip a ( X ) with topology
rb,t h e
strong
d u a l of ,#( X ) i s i d e n t i f i e d a l g e b r a i c a l l y and t o p o l o g i c a l l y w i t h
(X).
Furthermore,
a(X)i s
t h a t t h e bounded s e t s of
rb,and t h a t
&'(X)
a b a r r e l e d s p a c e , which means a r e t h e same f o r
Ts
and
t h e y c o i n c i d e w i t h t h e e q u i c o n t i n u o u s s u b s e t s of
"es
We t h e r e b y deduce t h a t
,@'(X).
5
SPACES OF DISTRIBUTIONS
(SEC. 1)
yb c o i n c i d e
and
on t h e
I n p a r t i c u l a r , i n t h e c a s e of a
bounded s u b s e t s of #(X).
sequence of d i s t r i b u t i o n s , convergence i n t h e s e n s e o f convergence i n t h e s e n s e of
b
c o i n c i d e , and we t h e n speak of a
convergent sequence of d i s t r i b u t i o n s .
,
from a m e t r i z a b l e space i n t o #(X) depend upon t h e t o p o l o g y (
Thus, i f A i s a mapping
t h e c o n t i n u i t y of A does n o t
r sor "e,) w i t h
which we e q u i p #(X).
m e same i s a l s o t r u e i f A i s a l i n e a r o p e r a t o r from
a'( X) , where
zs and
B ( Y )i n t o
s i n c e t h e c o n t i n u i t y of
Y i s a n open s u b s e t of IRp
A i s e q u i v a l e n t t o t h a t of i t s r e s t r i c t i o n t o each
BK(Y).
This
b e i n g t h e c a s e , we s h a l l simply r e f e r t o a continuous l i n e a r i n t o #(X).
o p e r a t o r from &(Y)
,&(X) i n t o &(X)
I n p a r t i c u l a r , t h e i n j e c t i o n of
i s c o n t i n u o u s , and we s h a l l s e e t h a t it i s o f
s e q u e n t i a l l y dense image ( s e e Remark 2 . 3 ) .
We can g e n e r a l i s e t o d i s t r i b u t i o n s c e r t a i n o p e r a t i o n s on functions.
I f Y i s an open s u b s e t of X and i f u
ine the restriction u C ulYg cp
> = < u,
distribution u ion a u
E
a'
P(x,
uI
f o r a l l cp
>
#(X)
8 ' ( Y ) of
u E
Ip
> = (-
D) = ,
F
111~1
u
E
U,
C"(X) i s t h e d i s t r i b u t acp
> for all q
E
&(XI.
i s t h e d i s t r i b u t i o n on X d e f i n e d by
c u,
a ( x ) Da
Ilcrcm a c o e f f i c i e n t s aa E c"(x),
The product o f a
&(Y).
and a f u n c t i o n a
aa u of
fl(X), we def-
t o Y by
a ' ( X ) d e f i n e d by e au, cp > = c
The d e r i v a t i v e
<
E
Q
E
1Y
E
a"
cp > f o r a l l q e &(XI.
If
i s a d i f f e r e n t i a l operator with
we p u t
~ ( x D , )U =
6
( C W . 1)
DISTRIBUTIONS AND OPERATORS
L e i b n i z ' formula g i v e s D.(~u) J
D. a , u
=
J
+
D.
a .
U
J
and r e a s o n i n g by r e c u r r e n c e over t h e e x t e n t o f t h e m u l t i - i n d e x
By
we o b t a i n
( 1.1 2.)
( 1)
where
ci
j
$(a
-<
j
(!)
=
u)
.
DPa-.
Q l B . . and where
=
....,
f o r j = 1,
ci 5
Dcr u
6 means t h a t
n.
We term t h e complete symboZ of t h e d i f f e r e n t i a l o p e r a t o r P ( x , D) t h e f u n c t i o n (polynomial i n C,) P(x, where
5a
=
a1
5,
..I
all
Sn
.
=
F
\asm
aJxl
P Y
We term t h e principal symbol o f P t h e
f u n c t i o n (homogeneous o f degree aa(X]
5)
.
'5
m
i n 5)
We denote by F ' ( a ) ( x , D) t h e d i f f -
e r e n t i a l o p e r a t o r w i t h complete symbol
a ci5
P(x,
c),
and we have
t h e Leibniz-Harmander formula, which can b e deduced from (1.1.2) by l i n e a r i t y :
(1.1.3.)
P(x, D)(au)
=
E$
(PcQ)(x, D)a)
, DQI u
tY
For example, i f a ( x ) = e x p ( i x . 6 ) where x.5 = x,sl we have
+
... +
xn%,
(SEC. 1)
-> c
u
We d e f i n e t h e c o n j u g a t e
c u,
7
SPACES OF DISTRIBUTIONS
cp > =
u,
E
#(X)
of a d i s t r i b u t i o n u E
9
for a l l
t h i s means t h a t given
an a r b i t r a r y c o v e r i n g o f X by open s u b s e t s V
&'(V.)
E
such t h a t u
j
J
unique u
E
\
in V
such t h a t u = u
&'(X)
in fact that Q
=
j
j
j
jy
and d i s t r i b u t i o n s
f o r a l l j.
If we suppose
i s a p a r t i t i o n of u n i t y i n X subordinate t o t h e
giiren c o v e r i n g , i . e . a f a m i l y of f u n c t i o n s supp 'pj c V
j
n Vk, t h e n t h e r e e x i s t s a
in V
j
by
&(X).
E
The d i s t r i b u t i o n s form a s h e a f on X ;
u j
B'(X)
q~ i s
t h e f a m i l y supp
j
Q
j
E
C"(X) such t h a t :
locally f i n i t e ( i . e .
for
any compact s u b s e t K of X , t h e s e t o f i n d i c e s j such t h a t supp
qj n K
*
E
a'(X) such t h a t u
= I
, and
cp. J
2
O. We can
cp. u defines a distribution J j = u f o r a l l j. The uniqueness of u
t h e n show t h a t t h e formula u
u
ff cpj
@ is finite),
I
J
Pj
j
is o b t a i n e d i n t h e same manner by decomposing each t e s t f u n c t i o n
T into
cp =
'pj
cp
.
j T h i s s h e a f p r o p e r t y shows i n p a r t i c u l a r t h a t i f a d i s t r i b m
u t i o n i s z e r o ( r e s p : C ) i n open s u b s e t s V m
( r e s p : C ) i n t h e i r union; DEFINITION 1 . 2 :
We term the support (resp:
o r t ) of a d i s t r i b u t i o n u outside of which
E
.&'(X)
If
singular supp-
the smallest closed subset of X
u i s zero (resp:
supp u (resp: s i n g supp u ) .
denote by &;(XI
of X , t h e n it i s z e r o
j t h i s a l l o w s us t o s t a t e :
F
C").
We denote t h i s by
i s a closed subset of X , we
the subspace of P ( X ) formed by the d i s t r i b u t i o n s
a
DISTRIBUTIONS AND OPERATORS
(CHAP. 1)
v i t h support i n F.
m
I f we l e t P = P(x, D ) be a d i f f e r e n t i a l o p e r a t o r w i t h C c o e f f i c i e n t s on X, it i s c l e a r t h a t
(1.2.1.)
SUPP P U
c supp
u
The i n j e c t i o n of Cz(X) = &(X)
,
c s i n g supp u
s i n g supp PU
m
i n t o C (X) i s c o n t i n u o u s , and a l s o c o n s i d e r an exh-
This i s proved by t r u n c a t i o n :
of dense image.
a u s t i v e sequence o f compact s u b s e t s of X, i . e . a sequence of comp0
act subsets K
j
u
of X such t h a t K j c Kj+l,
j
K . = X. J
K i s an a r b i t r a r y compact s u b s e t of X, t h e n K c K iently large.
Let t h e r e b e f u n c t i o n s
(12.21
If
QE
XJ
xj
= 1
E
C:(X)
x.Q J
for j suffic-
such t h a t
Kj
in
Cm(X) it immediately f o l l o w s t h a t
j
Note t h a t i f
m
-f
J
QI i n
c (X).
Thus t h e d u a l d(X) o f Cm(X) i s i d e n t i f i e d w i t h a v e c t o r subspace o f fl(X).
We can show t h a t t h i s subspace i s t h e subspace
o f d i s t r i b u t i o n s w i t h compact s u p p o r t i n X, and t h a t t h e d i s t r i b ution
u
w i t h compact s u p p o r t i n X i s i d e n t i f i e d w i t h t h e c o n t i n m
uous l i n e a r form on C (X) d e f i n e d by p
.+
< u,
8g >,
where 8 i s
m
an a r b i t r a r y f u n c t i o n i n C ( X ) w i t h v a l u e one i n t h e neighbour0
hood o f supp
Cp
E
We s h a l l a l s o denote by
t h e v a l u e a t
u.
Cm(X) of u
E
e'(X). More g e n e r a l l y , i f u
E
8'(X) and
Q E
are such t h a t
C"(X1
K =
(1.2.3.)
we put
9
SPACES OF DISTRIBUTIONS
( S E C . 1)
< u,
n
supp u
supp cp
i s compact,
m
> = < u, eg >
where 9
E
C (X) h a s t h e v a l u e 1 i n 0
t h e neighbourhood o f K .
For t h e p a i r C m ( X ) , &'(X) we have t h e same t o p o l o g i c a l prope r t i e s as t h o s e which have been d e s c r i b e d f o r t h e p a i r a ( X ) ,
a'(XI. x.
Let
J
n. be open s u b s e t s o f IR
t h e t e n s o r product u u
j
E
#(X.) J
1
0
F i r s t l y , we have
The space of f i n i t e linear combinations of
functions of the form Q ,
ially dense i n &(X,
x
@
X2),
cube K o f s i d e T such tha.t a l l y , with period T, i n t o
(where
Q2
Qj
Q E
Supp
&(X,
x
0
ej E c~[RnJ] ( j =
with
g2 = 1 i n supp c p .
F o u r i e r s e r i e s expansion of
.&(x.)) i s sequentJ
Q c
We extend. cp
K.
8 can
Q
S(X,).
IK
periodic-
Suppose we have
1, 2 ) s u c h t h a t supp(gl
We have
@
Consider a c l o s e d
X2).
@ E C"( B n l + n 2 ) .
functions
E
and we denote it by 0 6 ( X , )
I n f a c t , suppose w e have
@
We can d e f i n e
u2 of two d i s t r i b u t i o n s
(j = 1, 2 ) .
LEMMA 1 . 3 :
(j = 1, 2).
= 9i
be w r i t t e n
0
0
02@
9,
) c K,
, and
the
DISTRIBUTIONS AND OPERATORS
10
-ix a1 2n/T -i?.Cu;? 1’ e e
T
(CHAP. 1)
2n/f dxld%
Q(xl,
By i n t e g r a t i o n by p a r t s , we s e e t h a t for all N t h e r e e x i s t s
a constant C
N
such t h a t
n
for a l l
cyj
E Nj
(j =
I,
2)
.
where
and (1.3.1) shows t h a t t h i s s e r i e s converges i n .&(X, hence t h e lemma.
We c o n s i d e r t h e b i l i n e a r mapping
x
X2),
11
SPACES OF DISTRIBUTIONS
(SEC. 1)
This extehds i n unique manner t o a continuous b i l i n e a r mapping from 8(Xl) x
B'(X2)
into 8'(Xl
x
X2), which we a l s o denote by
Uniqueness i n f a c t f o l l o w s from Lemma 1 . 3 .
In order t o
prove e x i s t e n c e , it may be shown t h a t we can p u t , f o r
q,
E
ax,
x
x2): rp>
=
=<%1*.2>
for
SUPP lu, 69
1.4
"2)
= supp u1
E
x
axj) ,
j=
"2
.
supp
I,
2
m
JUMP FORMULA:
and we denote by u
0
j that
We c o n s i d e r a f u n c t i o n u
t h e f u n c t i o n which is e q u a l t o
x 2 0 and t o 0 for xn < 0. n integer
rp. J
where
E
C ( B n1,
u for
We can show by r e c u r r e n c e on t h e
.
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
12
More g e n e r a l l y
suppose :
m P(x, D )
Pj(x,
=
(with
0')D;
D' = (D,,
j=o
i s a d i f f e r e n t i a l operator with C
m
..., DWl 1)
c o e f f i c i e n t s i n 7Rn.
We have
t h e "jump formula"
1.5
DISTRIBUTIONS REGULAR WITH RESPECT TO ONE VARIABLE: U E
Suppose we have
Co(R, &'(fi"l))j
we can show t h a t w i t h it we
can a s s o c i a t e i n j e c t i v e l y t h e d i s t r i b u t i o n u
< "nl,L&., Such a d i s t r i b u t i o n
u
",I
&'(IRn ) by p u t t i n g :
E
> dxn
for
cp E
i s s a i d t o be continuous w i t h r e s p e c t t o
xn w i t h v a l u e s which a r e d i s t r i b u t i o n s w i t h r e s p e c t t o x' and we d e f i n e i t s " s e c t i o n a l t r a c e on {x = O}" n
yo u = U(0).
We n o t e t h a t
aa' U.
with t h e function u
yo
IJ
E J'(R"I)
by
a ax ,t u i s t h e d i s t r i b u t i o n a s s o c i a t e d If U
E Ck(R, ,p1(RW1))
we s a y t h a t
i s of class Ck w i t h r e s p e c t t o xn w i t h v a l u e s which a r e d i s t -
r i b u t i o n s w i t h r e s p e c t t o x'.
05 j 5 k
, a,j
u
We n o t e t h a t , for
i s t h e d i s t r i b u t i o n associated with t h e
CONVOLUTION AND F.T.'s
(SEC. 2 )
function <('),
yj
IJ
13
and we d e f i n e t h e s e c t i o n a l t r a c e
E 8'(RW1)
by yj u =
y,(D:
u)
.
In p r a c t i c e we s h a l l
make no d i s t i n c t i o n between t h e n o t a t i o n s U and u.
2.
CCSV0LU'TO:I AXD FOUEIIE3 TFW:SFO3??T;CS OF >ISTRIBUTIO!:S
The a b e l i a n group s t r u c t u r e on Bn allows t h e d e f i n i t i o n of a d d i t i o n a l o p e r a t i o n s on a(IRn =& )' ( w e
s h a l l omit t h e i n d i c a t -
ion of t h e open s u b s e t X i n t h e spaces of d i s t r i b u t i o n s when
x
= Bn).
2.1
CONVOLUTION:
If we have u
E
L
1
,v E
W
Co
, we
define
t h e convolution product
and we show t h a t
(*-1 2.)
flu * vll
2
5
I14L,
l!VllL2
;
consequently
L ( 2 . 1 . 1 ) remains meaningful (almost everywhere) f o r v ,-
have u ++vE L'
.
I f furthermore
Q E
& , we
E
L2 and we
can show t h a t
In view o f ( 1 . 2 . 3 ) t h i s formula l e a d s us t o s t a t e
14
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
Two d i s t r i b u t i o n s u, v
DEFINITION 2.1:
convoZvabZe i f the mapping (x, y ) b-% supp u
x
n
supp v i n t o IR
convoZution product u
x+y
i s proper. v of
f a r e said t o be
from
lhder these conditions the
and
u
E
v i s the d i s t r i b u t i o n on
IRA' d e f i n e d by < U @ V , Q O s >
< U * V , Q >
for a l l
Q
E&
The p r i n c i p a l p r o p e r t i e s o f c o n v o l u t i o n a r e summarised i n t h e following :
utions.
u,v are two convoZvabZe d i s t r i b -
Suppose
PROPOSITION 2.2: Then =
u * v aa(u
*
s ~ p p(u
v + u
=
V]
*
aa c
v)
u
*
SUPP u
u
=
v
+
* aa
supp v
v
, where
the right-hand
side i s cZosed.
(2.2.1.)
u
(TX
*
;1(Y1
v
E C" mad
=
REMARK 2.3:
If u
E
a', we
towards
u
in
(u
Y
*
v)(x)
=
<
u, T~ v
>
, where
4X-Y)
Let p
j
be a r e g u l a r i s i n g sequence ( s e e (1.1.1)). m
can show t h a t t h e C
8';
suppose
xj
functions u n p
j
converge
i s a sequence of t r u n c a t i o n
CONVOLUTION AND F . T . ' s
(SEC. 2 )
f u n c t i o n s as i n ( 1 . 2 . 2 ) ;
Pj
*
E&
( X . u)
J
we can show t h a t t h e f u n c t i o n s
converge towards
sequential density of & i n
2.4
15
u
i n &'
.
This proves t h e
B'.
FOURIER TRAnTSFORMATION:
If
f E L1 = L1(Rn)
,
its
A
F o u r i e r t r a n s f o r m i s denoted by 9 f where f is d e f i n e d by
Let us now d e f i n e a subspace o f 'L
which i s i n v a r i a n t under
Fourier transformation:
We denote by 8 the space of functions
DEFINITION 2.4:
cp
E
'h(V1
C"( IRn ) such t h a t , f o r any i n t e g e r h,
supRn (1
=
1.1
+
1x1')~
1
aa cp(x)l
the quantity i s finite.
o
is
Ih
called the space o f functions which are rapidly decreasing, along with a l l t h e i r d e r i v a t i v e s .
We e q u i p 8 w i t h t h e topology d e f i n e d by t h e f a m i l y o f seminorms q h ( h
E
IN ) .
the following :
8 i s a Fr&chet space.
We r e c a l l without proof
16
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
THEOREM 2.5:
The Fourier transformation 'jl e s t a b l i s h e s a
topological vector isomorphism of 8 onto i t s e l f , with inverse ~ r 's a t i s f y i n g
(T'g l ( x 1
where $ 5 = ( 2 ~ ) -d ~5
.
=
I
eiy*c g(c1
for g E 8
cp, Q E 8 , we have
If
(2.5.1 .)
If we have cp E
B , D e f i n i t i o n ( 2 . 4 . 1 ) f o r + shows t h a t it e x t e n d s
i n t o an e n t i r e f u n c t i o n on C
n
d e f i n e d by
We s h a l l now show t h a t t h e behaviour of
@
i n Cn d e t e r m i n e s t h e
convex envelope o f t h e s u p p o r t o f c p .
To do t h i s , we d e f i n e t h e i n d i c a t o r f u n c t i o n IK of a compact s u b s e t K of R n by
For example, i f K i s t h e b a l l B w i t h c e n t r e 0 and r a d i u s R , we
The i n d i c a t o r of K d e t e r m i n e s , for
ach hyperpl
t h e s m a l l e s t c l o s e d h a l f - s p a c e which c o n t a i n s K .
direction,
Consequently,
t h e Hahn-Banach theorem shows t h a t IK d e t e r m i n e s t h e convex n
envelope K of K and IK =
%.
17
CONVOLUTION AND F . T . ' s
(SEC. 2 )
We can s t a t e t h e f o l l o w i n g :
THEOREM 2 . 6 :
(Paley-Wiener).
Let K be a convex compact
subset of B n . An e n t i r e f u n c t i o n U on Cn i s the Fourier transform of a function belonging t o &K i f and only i f , f o r a l l N,
there e x i s t s CN such t h a t
6=5+i7l E C Proof of n e c e s s i t y .
n
.
For any m u l t i - i n d e x a , t h e e q u a l i t y ( 2 . 5 . 1 )
implies t h a t
which g i v e s t h e upper bound.
and ( 2 . 6 . 1 ) f o l l o w s from t h i s immediately.
Proof of s u f f i c i e n c y .
The upper bounds ( 2 . 6 . 1 ) imply t h a t U(c)
i s of r a p i d d e c r e a s e ;
it i s t h u s t h e F o u r i e r t r a n s f o r m of a C
m
f u n c t i o n Q , g i v e n by
(2.6.2
.) I n o r d e r t o show t h a t supp Q c K , i t i s s u f f i c i e n t t o v e r i f y
t h a t i f # { K we have cp(xo) = 0 .
By Hahn-Banach,
there exists
18
(CrnP. 1)
DISTRIBUTIONS AND OPEFATORS
a hyperplane which s e p a r a t e s x
I n o t h e r words, t h e r e e x i s t s
q
0
0
and t h e compact convex s e t K . E
lRn \ 0 and c
E
lR
such t h a t
A f t e r a l i n e a r change of c o o r d i n a t e s , w e can assume rlo
= (0,..., 0, 1); we put 5 =
( E l ,
6,).
For t > 0 , Cauchy's theorem allows t h e i n t e g r a l ( 2 . 6 . 2 ) t o be r e p l a c e d by
Taking N = n + l , w e deduce from ( 2 . 6 . 1 ) and ( 2 . 6 . 4 ) t h e upper bound
0
Since ( 2 . 6 . 3 ) i m p l i e s xn t e n d s t o 0 when t -+
+
m;
> IK(qo), t h e hence
cp(xo]
right-hand s i d e o f ( 2 . 6 . 5 ) =
0
.
I t can e a s i l y be seen t h a t t h e i n j e c t i o n s
B C>
8 &>
caD
a r e continuous and of dense image ( t o prove
t h a t & i s dense i n 8 , we proceed by t r u n c a t i o n u s i n g f u n c t i o n s
CONVOLUTION AND F.T.'s
(SEC. 2)
x
xj(x) = x(x/j), where
Ix I
B(IRn)i s such t h a t x(x) = 1 f o r
E
S is
Thus t h e d u a l s'of
1).
S
vector subspace o f
19
8' c o n t a i n i n g
n a t u r a l l y i d e n t i f i e d with a
d.
8' i s termed t h e space o f
tempered d i s t r i b u t i o n s .
For u
E
O'and
, and
u atq,
E 8 , we a g a i n denote by t h e v a l u e of
Q
for the pair 8 ,8'
analogous t o t h o s e of t h e p a i r b
we have t o p o l o g i c a l p r o p e r t i e s
, s'
.
Formula ( 2 . 5 . 1 ) p e r m i t s
t h e F o u r i e r t r a n s f o r m a t i o n t o be extended t o 8 ' :
For u
DEFINITION 2.7:
Fu
=
G
E
u by< SJ,$I
E 8 , of
Fourier transformation
81, we define the Fourier transform
>=
c u, Sg > f o r a l l J,
E
The
8 .
3 e s t a b l i s h e s a topo ZogicaZ vector isomor-
phism of 8 , (equipped with one of the two topologies
y so r
'%
onto i t s e l f . .
I n t h e c a s e o f d i s t r i b u t i o n s w i t h compact s u p p o r t , we have:
2.8:
THEOREM
If u
E
&, then Zu is the Cm f u n c t i o n
defined b y
[
~
t
=j
<
U,
e-ixc >
for
g E R"
3 u extends the e n t i r e f u n c t i o n on Cn defined by
[su)(G)
=
<
u, e- i ~ * 5>
for
5 E R"
.
3 u i s a sZowly increasing f u n c t i o n i n B n J i . e . there e x i s t C
and u such t h a t
)
DISTRIBUTIONS AND OPERATORS
20
d
More generaZZy, i f B i s a bounded subset o f
u such t h a t (2.8.1) is vaZid f o r aZZ u
E
( C H A P . 1)
we can f i n d C and
B.
We point out merely that the upper bound (2.8.1) derives from the upper bound of the linear form u on C
m
*
by a semi-norm p
K,!J,
furthermore, we can employ an identical semi-norm for any u because a bounded subset is an equicontinuous part of
E
B
6.
We have the analogue of Theorem 2.6 as follows:
THEOREM 2 . 9 :
(Paley-Wiener-Schwartz). Let K be a convex
compact subset of IRn.
An e n t i r e f u n c t i o n U i s the Fourier trans-
form of a d i s t r i b u t i o n of
i f and onZy i f there e x i s t C and
N such t h a t
We shall not have to make use of this theorem and we postpone a proof until Exercise (10.1).
The principal properties of Fourier transformation in 8'are summarised in the following theorem:
(SEC. 3 )
21
SINGULAR SPECTRUM
Suppose we have u
THEOREM 2.10:
(2.10.1 .)
S(U
*
3(Da
v)
=
U)
=
E
$'and v
E
&'
.
Then
FU.KJ
Fu
2 3 e s t a b l i s h e s a linear homeomorphism of L and we have Parseval's
formula:
Formula ( 2.10.2 ) shows t h a t
, by
Fourier transformat ion , t h e
d i f f e r e n t i a l o p e r a t o r P(D) i s transformed i n t o t h e o p e r a t o r of m u l t i p l i c a t i o n by P([).
This p r o p e r t y makes F o u r i e r transform-
a t i o n a b a s i c t o o l for t h e s t u d y o f d i f f e r e n t i a l o p e r a t o r s .
REMARK 2.11:
Sometimes convolution products can be d e f i n e d
without a s u p p o r t c o n d i t i o n . just as in (2.2.1), (u
u n v
E
C"(IRn),
*
nus, for u
v ) ( x ) = C U,
E
"cx
and we can show t h a t u n v
S' and v
>. E
E
S
, we
put,
We have
8' and t h a t
formula ( 2 . 1 0 . 1 ) remains v a l i d .
3.
SINGULAR SPECTRUM OF A DISTRIBUTION From t h e d e f i n i t i o n o f t h e s i n g u l a r support of u
E
8 ' ( X ) , we
22
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
have t h e equivalence
x
0
4
1
s i n g supp u
t h e r e e x i s t s an open neighbourhood V of
xG such t h a t f o r a l l a
we have au
E
E
1
a(V)
&(V)
and by F o u r i e r t r a n s f o r m a t i o n , t h e c o n d i t i o n ciu E & ( V ) i s equiva l e n t t o t h e e x i s t e n c e f o r a l l N of a c o n s t a n t C such t h a t
Iasll
(3.1 .I.)
5
c
(1 +
ICll4
5
for
F?
E
I n t h e l a t t e r form, we can r n i c r o l o c a l i s e t h e concept of s i n g u l a r support b y d i s t i n g u i s h i n g t h e d i rec ti on s have an upper b o w d o f t h e t y p e ( 3 . 1 . 1 . )
A point ( x o , g o )
DEFINITION 3.1.
5
i n which we
. E
X x ( IRn
-
no t within the singuZar spectrum o f the d i s t r i b u t i o n u
0
1) is
E
&'(X)
i f there e x i s t s a neighbowhood V o f xo and a neighbourhood W of
5' such t h a t (3.12.)
for a21
c1 E
IG('T?,)I
B(v) and a22 5
c
7
N there e x i s t s C such t h a t
for
T
2
1 and 5
E
w.
The singuZar spectrum o f u i s a closed subset o f X x ( En - { 0 I ) denoted by m u . This concept w a s i n t r o d u c e d and i n v e s t i g a t e d by HORMANDER [ 5 l (under t h e name "wave-front s e t " ) f o l l o w i n g t h e i n t r o d u c t i o n by SATO of t h e concept of t h e " s i n g u l a r spectrum" of a hyperfunction ( s e e for example SATO 111).
See a l s o
DUISTEWT [l] and GUILLEMIN-STERNBERG [11 f o r t h e concept of a s i n g u l a r spectrum. Knowledge o f t h e s i n g u l a r spectrum determines t h e s i n g u l a r s u p p o r t because
4)
(SEC.
rx (WFu) = s i n g supp u where
(3.1.3.)
'* x x m n \ 0 ) 3 x , & ) c - - , x
77 X
In f a c t , i f x (xo, x
0
4
0
4
E
x.
s i n g supp u , it i s c l e a r t h a t
g o ) 4 W F f~o r a l l g
E
zn - i o I .
r,,(WFu), we a s s o c i a t e w i t h a l l g
/El =
23
OPERATORS AND K2RNELS
Conversely, i f
Bn
-
{ 0
and
0
i n which we have an
E
A
1 neighbourhoods Vt and Wi o f x
i n e q u a l i t y of t h e t y p e (3.1.2).
0
1
such t h a t
If we e x t r a c t a f i n i t e c o v e r i n g
t h a t w i t h t h i s choice o f V we have t h e c o n d i t i o n ( 3 . 1 . 1 ) and consequently x
0
4
s i n g supp u.
EXAMPLE 3 . 2 :
I<
-ix 5
&af cy e
show t h a t
4.
The immediate e q u a l i t i e s
>I
=
WF(6,)
< Y,
lcy(a)l,
=
{a] x
emix5 >
cy
=
and WF(Y] = {O] x
(R" \ 0)
(R \
0)
OPEWTORS AND KERNELS. n Suppose X , Y a r e open s u b s e t s of lR , Bp; we i n v e s t i g a t e con-
tinuous l i n e a r operators f r o m B ( Y ) i n t o
EXAMPLE
4.1
:
a' ( x ) .
Let K be a f u n c t i o n i n C"(X
x Y).
Then
t h e i n t e g r a l o p e r a t o r A which a s s o c i a t e s w i t h each $ E ~ ( Y t) h e f u n c t i o n A$ d e f i n e d by (ASHXI
i s continuous from.f?(Y)
=
\
K(x9 Y ) $ l Y I dY
i n t o C"(X).
Moreover, A extends i n t o
. L
a l i n e a r o p e r a t o r A f r o m c ' ( Y ) i n t o C"(X)
when w e p u t
U
( A v ) ( x ) = < v , K(x, . ) > f o r v m
'H,j
E
&(Y).
i s a semi-norm on C (X), we have
More p r e c i s e l y , i f
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
24
and s i n c e t h e f u n c t i o n s 3
c4
X
K(x, . ) d e s c r i b e a bounded s u b s e t of OI
m
C ( Y ) when x d e s c r i b e s H , we s e e t h a t A i s a continuous l i n e a r
o p e r a t o r from
& (Y)
r b ) i n t o C"(X).
(equipped w i t h
More g e n e r a l l y , l e t us c o n s i d e r a d i s t r i b u t i o n K
E
With it we a s s o c i a t e an o p e r a t o r A = Op(K) which
r ( X x Y).
i s continuous and l i n e a r from .&Y)
by p u t t i n g
< A$, ep > = c K, ep e3 6 > f o r a l l J, E
(4.1.1.)
&(X
x X)
.
1
x
E
Bcxl
X 1
We denote by 6D t h e d i s t r i b u t i o n on
be t h e d i a g o n a l of X x X.
< 6,,, i >
X x X d e f i n e d by
. a ( y > , rp E
Let D = d i a g ( X x X ) = { (x, x)
EXAMPLE 4 . 2 :
0 E
i n t o &'(X),
@ [ x , x ) dx
L
for a l l
Then it i s c l e a r t h a t Op( 6 ) = I d e n t i t y .
D
EXAMPLE 4 . 3 :
Suppose we have T
E
&'(En).
With it we
n n a s s o c i a t e t h e d i s t r i b u t i o n on IR xJR denoted, by an abuse o f n o t a t i o n , by T(x
< T(x
- y),
-
y ) and d e f i n e d by
i > =
( t h u s 6D = 6(x
-
< T(x),
Then, Op ( T ( x
y)).
convolution w i t h T: $.
@ ( x+ yly)
+
T
*
-
> dy
i E J(F? x R")
9
y ) ) i s t h e operator of
$.
We l e a v e t o t h e r e a d e r t h e t a s k o f v e r i f y i n g t h a t t h e sing-
ular s u p p o r t of a d i s t r i b u t i o n o f t h e form T(x
-
y ) i s equal t o
t h e set
(4.3.1 .]
C
(x, y) E
R" x R"
I
(x
- y>
E s i n g supp T I .
4)
(SEC.
OPERATORS AND KERNELS
25
We now r e t u r n t o t h e l i n e a r mapping K + A = Op(K) from &(X
x Y ) into&@(Y),
B'(X))
d e f i n e d by ( 4 . 1 . 1 ) .
This mapping
We s h a l l now s e e t h a t it i s
i s i n j e c t i v e i n view of Lemma 1 . 3 . surjective: THEOREM
4 . 4 (Sehwartz's kernels theorem)
tinuous linear operator A f r o r n & ( Y ) unique d i s t r i b u t i o n K
E
&'(X
:
For any eon-
i n t o &'( X) , there e x i s t s a
x Y) such t h a t A = Op(K);
K is
ealZed the kernel of the operator A. PROOF :
The b i l i n e a r form
B[x)
X
B(y) 3
(Cp,
$1 ->
< A$,,cp > E c
induces a l i n e a r form K on t h e a l g e b r a i c t e n s o r product
ax)
@
afY).
(X x Y ) we have
E x p l i c i t l y , for 0 N
(4.4.1 .] where N
(4.4.2.) i s a decomposition o f @,
and
(4.4.1) i s
independent o f t h e choice
of t h e decomposition of 0 .
To conclude, it f o l l o w s from Lemma 1 . 3 t h a t it s u f f i c e s t o show t h a t K i s a c o n t i n u o u s l i n e a r form on topology induced b y a ( X x Y ) .
ax) f23 aY) f o r
the
L e t t i n g H , L be compact s u b s e t s
o f X , Y , we show t h a t t h e r e e x i s t s a semi-norm p such H x L,m that
(4.4.3.)
I<
K,
Let H1, L
B 1
>I
5
C p,,
L,m ( 9 )
,
f o r CP E f l y ) w i t h supp 0 c H x L .
be compact neighbourhoods o f H , L ; by t r u n c a t i o n ,
DISTRIBUTIONS AND OPERATORS
26
we can always w r i t e ( 4 . 4 . 2 ) w i t h (pj €
BH,(X)
(CHAP. 1)
$ j EBL [y) 1
if
supp
I
,
H x L
C
m
Suppose we have f E Co(X) e q u a l t o 1 on H1 and we p u t H2 = S ~ P Pf ; a3
l i k e w i s e suppose we have g
E
Co(Y) e q u a l t o 1 on L1 and we p u t
c o n s i d e r i n g t h i s i n t e g r a l as an i n t e g r a l o f a f u n c t i o n w i t h values i n b
A I)j
=
( Y ) , t h e c o n t i n u i t y of A p e r m i t s us t o w r i t e
.i;d l .,
fis
eiY.7)
f(x1 eix**
.
i j ( s > A! ,
Likewise, by w r i t i n g
we o b t a i n
where
The c o n t i n u i t y o f
< A$,
[cp, I)) &%(X)
x &,(Y)
(p
A i m p l i e s t h a t t h e b i l i n e a r mapping
>
i s s e p a r a t e l y c o n t i n u o u s from
i n t o C;
a n a l y s i s , it i s c o n t i n u o u s .
t h u s , u s i n g a r e s u l t from f u n c t i o n a l
A l s o , t h e r e e x i s t s C , h , II such
that
I n p a r t i c u l a r , it f o l l o w s from t h e d e f i n i t i o n of F t h a t t h e r e
(SEC. 4 )
27
OPERATORS AND KERNELS
e x i s t s C such t h a t
c
IF(%, S)l
(4.4.5.)
for
(5, 7 ) E R”
By l i n e a r i t y , we deduce from
< K,
[4.4.6. ]
@
>
+ ] 5 l I a (1 +
x RP ,
( 4 . 4 . 1 ) and ( 4 . 4 . 4 ) t h a t
F(5,
=
m
We choose an i n t e g e r
(1
1) k,a) 8 5 8 s
such t h a t
h+&m
c
-
I
;
fwp)
4
i n t e g r a t i n g by p a r t s i n t h e i n t e g r a l d e f i n i n g 9, we o b t a i n :
which a l l o w s us t o m a j o r i s e
( 4 . 4 . 6 ) and o b t a i n ( 4 . 4 . 3 ) by v i r t u e
of ( 4 . 4 . 5 ) .
4.5
TRANSPOSE AND ADJOINT OF A LINEAR OPERATOR: The t r a n -
spose ( r e s p . t h e a d j o i n t ) of a c o n t i n u o u s l i n e a r o p e r a t o r A : ay)+ a ( X )
3 (X) i n t o &‘(Y) (4.5.1.)
C tA
i s t h e l i n e a r o p e r a t o r tA ( r e s p . A*) from
d e f i n e d by
9, d( >
=
C cp, A $
>
( c A* 9, for
(P
T >=
c
Cp,
5 >)
E 4 x 1 , $ E ay’)
We immediately have:
PROPOSITION 4 . 5 :
*
The operators tA and A are continuous
.
Linear operators from B(x)i n t o 8 ( ~The kernel of tA
DISTRIBUTIONS AND OPERATORS
28
(CHAP. 1)
(resp. of A*) is deduced from the kernel K of A (resp. of
i?) by
We have
interchanging the r o l e s of the variables x , y. * I
t(tA)
and (A )
= A
= A.
The operator A is continuous from &(Y) i n t o C"(X) if and only if t A extends i n t o a continuous operator from
(X) i n t o 8(Y) (both
equipped with the topology 'C,,).
For example, i f A i s a d i f f e r e n t i a l o p e r a t o r
,aE
c o e f f i c i e n t s i n X , t h e n tA i s t h e
w i t h C"
rn ).(,a
f ( a a ( x ) . ) and A* i s t h e
d i f f e r e n t i a l operator
IaE
rn D
d i f f e r e n t i a l operator
I& 4.6
*
~
~
.
~
.
rn
REGULARISING OPERATORS:
DEFINITION 4.6: A continuous linear operator A : R(Y)
->
&(XI
i s said t o be regularising if it extends i n t o
a continuous operator from &'(XI i n t o C " ( X ) , equipped w i t h
where &'(Y) is
Zb).
We have t h e f o l l o w i n g c h a r a c t e r i s a t i o n :
THEOREM 4.7:
An operator A i s regularising if and only i f
i t s kernel KA is i n C"(X PROOF:
x
Y).
We saw i n connection w i t h Example
4 . 1 t h a t t h e condition
4)
(SEC.
OPERATORS AND KERNELS
29
To show t h a t it i s n e c e s s a r y , we c o n s i d e r t h e
is sufficient. mapping
x
x Y
3
(x, y )
->
Y)
, . [ a
By h y p o t h e s i s , t h e f u n c t i o n a i s C
=
[A 6 y ) ( x )
E
C
.
m
s e p a ra te l y with r es pe ct t o
x and y , and t h i s b e i n g s o t h e theorem a r i s e s out of
LEMMP,
PROOF:
4.8:
The function a
We begin
by showing t h a t
is in C"(X
x
Y ) and KA = a.
i s continuous.
a
5
7'.
Furthermore, t h e c o n t i n u i t y o f A i m p l i e s t h e e x i s t e n c e o f Q'' > 0 such t h a t
(4.8.2.)
sup Ix-xo)I
I(A 6 y ) ( x )
T'
- [A
6
)(x)l
I
I Y - YoI The c o n t i n u i t y o f
5
when
YO
a
5
7"
.
f o l l o w s from (4.8.1) and ( 4 . 8 . 2 ) .
I n o r d e r t o prove t h a t
a
is C
m
,
it i s s u f f i c i e n t t o reason
30
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
i n l i k e manner for t h e f u n c t i o n s
. I f we denote by A
regularising; multi-index
1
by c o n s t r u c t i o n , we have for any y
E
Y and any
6
which proves t h a t A = A i o n s of t h e t y p e 6 " ) Y
REMARK
a , it i s t h u s
t h e operator with kernel
4.9:
1
s i n c e l i n e a r combinations o f d i s t r i b u t -
6
a r e dense i n
Theorems
Y).
4.4 and 4 7 a r e i n f a c t a p p l i c a t i o n s
of t h e t h e o r y of n u c l e a r s p a c e s ;
for t h i s approach, s e e f o r
example TREVES [11.
5.
OPERATORS AND SUPPORT PROPERTIES
PROPOSITION 5 . 1 :
K~ E P ( X X Y ]
, we
If we l e t A be an operator w ith kernel have f o r v
E
C;(Y)
there e x i s t s such t h a t
PROOF:
(x,y)
y
E
E
supp K~
supp v
S i n c e supp v i s compact, t h e s e t supp KA
0
supp v i s
(SEC. 5)
OPERATORS AND SUPPORT
c l o s e d in X. those u
supp u
E
31
< Av, u > = 0
We now have t o show t h a t
for
C”(X) which a r e such t h a t 0
n
means t h a t
(supp KA
supp
0
[supp u
4
=
@
now, t h i s c o n d i t i o n
;
supp v) fl supp KA
x
=
, hence
g
the
result.
We denote by p [ r e s p . p
X
Y
) t h e p r o j e c t i o n of X
x
Y on X
(resp. Y ) .
We say t h a t the operator A with kerneZ
DEFINITION 5.2:
KA E R’(X x Y ) i s proper i f the projections an d
a r e ;sroper.
p y l s ~ p pKA
p x l ~ ~ pKAp
We have
Suppose A i s a proper operator;
PROPOSITION 5.3:
i ) For any compact subset K
Y, the s e t
c
supp KA
i s a compact subset of X and consequentZy A sends 0IK,
0
K
then =
K’
a K (Y) t o
(x).
i i ) For a n y compact subset K 1
c
X, there e x i s t s a compact subset
K of Y such t h a t : [SUPP
v
n
K
=
6)
[SUPP
AV
n K*
=
ei).
Consequently, the operator A transforms a ZocalZy f i n i t e suni of functions i n &(Y)
i n t o a ZocaZZy f i n i t e swn o f d i s t r i b u t i o n s i n
32
( C H A P . 1)
DISTRIBUTIONS AND OPERATORS
&' (X). Furthermore, i t extends uniquely i n t o a continuous l i n e a r mapping from C " ( Y ) i n t o &'(X)
PROOF OF i ) :
which again s a t i s f i e s (5.1.1).
We have by d e f i n i t i o n
=
K'
n
pX tP;'c~l
supp K ~ I
so t h i s i s a compact s u b s e t of X because
i s proper. pylsupp KA
PROOF OF i i ) :
K =
I f we p u t
py [ p i ' ( K ' )
'n
, this
supp K,,)
is
a compact s u b s e t o f Y and, by c o n s t r u c t i o n , it i s c l e a r t h a t i f supp v does n o t i n t e r s e c t K t h e n supp Av does not i n t e r s e c t K'.
Let Cp,
(9,) be a
p a r t i t i o n of u n i t y on Y by means of f u n c t i o n s
Ecp1
For v
m
E
C ( Y ) , we have a l o c a l l y f i n i t e sum
; t h u s t h e sum E 'pav U l o c a l l y f i n i t e and we denote it by Av. v
A['?,
=
v)
is also
cy
We have t h u s d e f i n e d an
e x t e n s i o n of A i n t o a l i n e a r mapping, a g a i n denoted by A , from into a ' ( X ) .
C"(Y)
The c o n t i n u i t y o f t h i s e x t e n s i o n f o l l o w s immed-
i a t e l y from t h e d e f i n i t i o n o f t h e s t r o n g t o p o l o g i e s .
The unique-
n e s s comes from t h e d e n s i t y o f C"(Y) i n C " ( Y ) . 0
F i n a l l y , by u s i n g (5.1.11,we have for suppAv C U SUPP A ( ( P ~ tljCU [SUPP KAosupp a cy
ve
v
E
C"[Y)
:
V ) = S U P ~KA
0
s ~ p pv
For t h e s i n g u l a r s u p p o r t w e have
PROPOSITION 5 . 4 : KA
E
.P(X x Y j
.
Let A be an operator with kernel We aesume t h a t A extends i n t o a continuous
(SEC. 6 )
33
D.O.'s WITH CONSTANT COEFFICIENTS
D'(x) such t h a t
operator from &(Y) i n t o A(C:(Y))
c"[x>
C
(5.4.1.)
We then have t h e incZusion
,
s i n g supp A V
s i n g supp K A
C
o
.
v f &"Y) PROOF:
Suppose we have x
hand s i d e o f
(5.4.1);
([x,}
x
0
,
s i n g supp v
which does not belong t o t h e r i g h t -
i n o t h e r words,
s i n g supp v)
n
s i n g SUPP KA
#
=
Since s i n g supp v is compact, t h e r e e x i s t open neighbowhoods U , V of xo and s i n g supp v such t h a t
n
(U x V) Suppose w e have ec s i n g supp v. s i n g supp.
Since
E
C"(V) 0
We w r i t e
AV
K AJU x
v
=
s i n g supp KA
,
=
.
@
i d e n t i c a l t o 1 i n t h e neighbourhood o f
v =
QZ/
+
(I-crJv
, and
s i n g supp A ( W ) s i n c e
we have
E c~(Y)
(I<)v
i s Cm, we t h e r e b y deduce t h a t A ( K v )
IU
.
is in
C m ( U ) which proves t h a t xo does n o t belong t o t h e l e f t - h a n d s i d e
of
6.
(5.4.1).
DIFFERENTIAL
6.1
OPERATORS WITH CONSTANT COEFFICIENTS
FUNDAMENTAL SOLUTION:
acyD"
Let P(D) =
be
finite
a d i f f e r e n t i a l o p e r a t o r w i t h c o n s t a n t c o e f f i c i e n t s ao(
E
C.
We
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
34
say t h a t a d i s t r i b u t i o n E
E
.&'(En) i s a fundamental s o l u t i o n of
P(D) i f
.>
P[D).E
(6.1 .I
=
.
go
Fundamental s o l u t i o n s have l o n g been known f o r t h e more c l a s s i c a l o p e r a t o r s , and we t a b u l a t e t h e s e below: Operator
(6.1
A fundamental s o l u t i o n E
P
a -
2.1
axn
(6.1.3.)
(6.1.4.
j
A =
a,2 +
... + an
2
*-
~ x / * - ~, n
2
on-1 = a r e a o f t h e u n i t sphere in I R ~
(6.1.5.)
at
-
A
(heat)
(6.1.6.) I
(6.1.7,.)
We postpone t h e p r o o f s of t h e s e r e s u l t s u n t i l t h e e x e r c i s e s .
3
(SEC. 6 )
D.O.'s WITH CONSTANT COEFFICIENTS
35
I n t h e g e n e r a l c a s e , w e have
THEOREM
6.1:
Any non-zero d i f f e r e n t i a 2 operator P(D) a h i t s
a fundamenta 2 s o l u t i o n . This g e n e r a l theorem h a s been proved by EHRENPREISS and by MALGRANGE, and we g i v e h e r e a c o n s t r u c t i v e proof which i s due t o HORMANDER [11.
o
If P ( C ) #
for
5
E
it i s n a t u r a l t o t r y t o d e f i n e E as
B~
the inverse Fourier transform of
In t h e g e n e r a l c a s e , t h e i d e a i s t o i n t e g r a t e i n t h e complex
6 E cn
domain i n o r d e r t o a v o i d t h e zeros o f P(c),
.
We u t i l i s e
the following:
If we have P(D) # 0 , there e x i s t s a sequence
LEMMA 6 . 2 :
Bk
E
Pk E
lekl
Bn with
,
Cm(Rn)
(6.2.1.1
1, a p a r t i t i o n of u n i t y ( p k ) with
k Ek
and a constant c
I P ( ~z+ ek)I 5 E
for PROOF:
5
Letting
P
supp
2
c
Pk
,z E
C
and
> 0
such t h a t
1.
= 1
.
be t h e f i n i t e - d i m e n s i o n a l v e c t o r space on
C
of t h e polynomials Q i n n v a r i a b l e s o f degree 5 m y we equip it with t h e norm
N(Q)
=
(E\Q(')(D] cy
12 11/2
.
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
36
We can show without t o o much d i f f i c u l t y t h a t t h e f u n c t i o n
r(aj
=
e i s a l s o a norm on
;
(
sup
lek
I
Q(ze))
inf It1 = I
E R"
z E C
consequently, t h e r e e x i s t s a c o n s t a n t
C1 > 0 such t h a t
For Q f i x e d , t h e d e f i n i t i o n of r ( Q )shows t h a t t h e r e e x i s t s some
e
E
IR", l e i s 1, such t h a t Q(ze)
(62.2.)
2
c,
,
N(Q)
z
E c
1.
,
= I
.
We t h u s deduce t h a t t h e r e e x i s t s some neighbourhood o f Q i n which I
(6.2.2) remains t r u e w i t h t h e c o n s t a n t C2 = C 2 1' For a l l 5
E
lRn
P6(6) = P ( 5 e x i s t s a 2'
5
, we
d e f i n e t h e t r a n s l a t e d polynomial
+ 5 ) , and
and a neighbourhood V
IP(T)+ z e$l
(6.2.3.)
t h e preceding d i s c u s s i o n shows t h a t t h e r e
5
2
subcovering V,
k '
1
o f 5 such t h a t we have
"pT))
From t h e c o v e r i n g of IRn by t h e V compactness o f t h e s p h e r e s
5
5'
I
Izl = 1 m a g E v
we c a n , by making use o f t h e
= const
which i s l o c a l l y f i n i t e .
, extract
a countable
We l e t ( p , - ) be a n
p a r t i t i o n of u n i t y s u b o r d i n a t e t o t h i s covering and we put
.
Since t h e terms of Q which a r e o f maximal degree 'k are i n v a r i a n t w i t h r e s p e c t t o t r a n s l a t i o n , there e x i s t s some
Bk =
€!
C > 0 such t h a t
5'
(SEC.
6)
D.O.'s WITH CONSTANT COEFFICIENTS
37
which when combined w i t h ( 6 . 2 . 3 ) g i v e s ( 6 . 2 . 1 ) .
We d e f i n e E by p u t t i n g , f o r
Cp
E Cz
:
I t can e a s i l y be shown t h a t E i s a d i s t r i b u t i o n by u s i n g ( 6 . 2 . 1 ) and t h e immediate i n e q u a l i t y
F i n a l l y , Cauchy's formula shows
The e x i s t e n c e o f a fundamental s o l u t i o n E shows t h a t any o p e r a t o r P(D)
#
0 with constant c o e f f i c i e n t s i s l o c a l l y solvable
i n d i s t r i b u t i o n space i . e . f o r any f E b' , t h e e q u a t i o n P(D)u = f admits a t l e a s t one s o l u t i o n
u
a
E
*
f
E B'
.
We can a l s o
38
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
-
investigate the global solvability of P(D), i.e. the surjectivity of the linear mappings 3
u
~ ( x )3
u
C"(X)
depending on the open subset X
c
IRn
.
PfD)u
E C"[X)
p(oju
E
a(x1
For these questions, the
treatises of RORMANDER [11 and TREVES [21 can be referred to.
6.3
HYPOELLIPTICITY OF OPERATORS WITH CONSTANT COEFFICIENTS: After the questions of existence, a further
important question concerns the investigation of the regularity of the solutions. For example, in which cases do the solutions of
P(x, D)u = f have no singularities other than those of f?
We
once again have the inclusion sing supp P ( x , Dju
DEFINITION 6.3:
C
sing supp u
, which leads us to:
A d i f f e r e n t i a l operator P(x, D) with coeff-
i c i e n t s i n C"(X) is said t o be h y p o e l l i p t i c i n the open subset X
if (6.3.1 .)
sing supp P ( x , D) u =
sing SUPP u
9
For operators with constant coefficients, we have:
PROPOSITION 6.4: i)
The following assertions are equivalent:
The operator P(D) is h y p o e l l i p t i c i n TRn
6)
(SEC.
The operator P(D) admits a fundamenta2 s o l u t i o n which
ii)
is
39
D.O.’s WITH CONSTAMT COEFFICIENTS
ern i n I R ~ -
COI.
The operator P(D) admits a parametrix ( i . e . a distrib-
iii)
ution F E d
such t h a t P ( D ’ J F
- do
E Cm(R”))
, which is
Cm
i n IR” - { o } .
PROOF:
i
The i m p l i c a t i o n s
iii
prove t h a t
=$
i
ii a iii
=$
are t r i v i a l .
We now
L e t t i n g F be a p a r a m e t r i x of P(D) , t o
,
8
within t h e m u l t i p l i c a t i o n of F by a f u n c t i o n from
t o 1 i n t h e neighbourhood of 0 , we may assume t h a t F
identical E
this
& I ;
allows us t o w r i t e
(6.4.1 .]
P(D),F
For any open s u b s e t X ion u E & ( X )
c
do
=
,
+
where
e E :C
,
B n , we need t o show t h a t i f a d i s t r i b u t -
E
P(D)u
satisfies
Cm(X)
then u
m
E
C (X).
For
t h i s , it i s s u f f i c i e n t t o show t h a t f o r any open s u b s e t X1 such that
yl c
c X,
we have u
m
E
equal t o 1 i n t h e neighbourhood o f (6.4
2.)
We have
X1.
g
g E C”(X,)
Suppose we have a
C (Xl).
=
n &‘(R”)
TI;we
P(Qu)
Co(X)
put
s
s i n c e a u = u on t h e open s u b s e t
By convolution o f e q u a t i o n (6.4.2)
The k e r n e l d i s t r i b u t i o n F(x
m
E
- y) of
w i t h F , we o b t a i n
t h e o p e r a t o r o f convolution
40
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
(h .3A) an& iii),
by F satisfies, f r o m
sing supp F(x
- y>
C
Diag(R" x
R"')
.
By applying Proposition 5.4, we can thus deduce that sing supp (F
*
g]
C
sing supp g m
The right-hand side of (6.4.3) is therefore C u
ently
. in X and consequ1
is also.
EXAMPLE 6.5: that the operators
The formulas (6.1.3), (6.1.4), (6.1.5)show
aar -
I
A
-A
a,
I
are hypoelliptic.
We now give an important class of hypoelliptic operators.
DEFINITION 6.6:
A d i f f e r e n t i a l operator P(x, D) with Cm
c o e f f i c i e n t s i n an open subset X X
c
IRn is said t o be e l l i p t i c i n
if i t s principal symbol does not vanish on
X x
R"
-
{ 0 ]
.
We shall prove in Chapter IV that elliptic operators are hypoelliptic; for the moment we prove in particular the following:
PROPOSITION 6.7:
If P(D) is an e l l i p t i c operator with
constcmt c o e f f i c i e n t s , then P(D) i s h y p o e l l i p t i c i n IRn
PROOF:
.
We shall now construct a parametrix F of P(D) which will
be Cm in lRn - 101.
If m
is the degree of P(D) , the assumption
of ellipticity is equivalent to
(SEC. 6 )
41
D.O.'s WITH CONSTANT COEFFICIENTS
By homogeneity, we deduce from t h i s t h a t
1
\P,({)
,
151"
c
2
and by choosing R s u f f i c i e n t l y l a r g e , we have t h e lower bound
[6.7.1.)
Ip(5)l
;151m
2
I f we have 6
W
E
C
, identical
5 R , t h e lower bound
, 151
5 E R"
for
151
t o 1 for
(6.7.1)a l l o w s
-1
2
2
.
R
and z e r o f o r
us t o d e f i n e
. This i s a p a r a m e t r i x of P(D), s i n c e we have
P(D)F
-
ho
=
- 1)
s'(e(5)
E 8
The r e g u l a r i t y o f F away from t h e o r i g i n w i l l f o l l o w from
LEMMA 6.8: xa F
For any multi-index
E c~(R")
PROOF OF THE LEMMA:
with
a
q = rn
L)e have
+ 1.1
-n- 1
.
The e q u a l i t y
(6.6.1 .) a e
shows t h a t i n t h e c a l c u l a t i o n of D (-),
S P
t h e t e r m s which i n v o l v e
the d e r i v a t i v e s of t h e function 8 provide a C a
x F.
m
contribution i n
It i s t h u s s u f f i c i e n t t o c o n s i d e r o n l y t h e d i f f e r e n t i a t -
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
42
i o n s which r e l a t e t o
-.1P
We can show by r e c u r r e n c e on la
1
that
f o r 151 2 R we have
=
Di (&)
a l i n e a r combination of terms of t h e form
(6.8 2.1
Furthermore, for any m u l t i - i n d e x
6 we have
c o n s e q u e n t l y , t a k i n g i n t o account (6.7.1) each t e r m on t h e r i g h t hand s i d e of ( 6 . 8 . 2 ) i s bounded above by
151
2 R.
These upper bounds imply t h a t t h e i n v e r s e F o u r i e r t r a -
nsforms a r e c e r t a i n l y i n C q ( I R n ) because r a b l e for
O(I{l-m-lal) for
151
2
I ~~q-m-~cul
R.
This lemma shows t h a t , f o r any i n t e g e r
E Cq(Rn)
1x12p F
F E Cm(Rn
-
{ 0
i s integ-
1)
with
q = rn
+ 2p
p
,
-n-1
;
hence
.
For o p e r a t o r s w i t h c o n s t a n t c o e f f i c i e n t s , H O W D E R h a s given a . c h a r a c t e r i s a t i o n o f t h e h y p o e l l i p t i c i t y of P(D) i n terms of t h e polynomial P ( 5 ) which i s e x p r e s s e d i n t h e f o l l o w i n g theorem:
(SEC.
6)
43
D.O.'s WITH CONSTANT COEFFICIENTS
The following a s s e r t i o n s are equivalent:
THEOkEM 6 . 9 :
The operator P(D) i s h y p o e l l i p t i c
i)
I f we have
ii)
iii)
N = {
6E
C"
There e x i s t s 6 > 0 and C
I
> 0
PROOF OF i)
+ ii):
If F
E
for
such t h a t f o r any multi-
5 E R"
for
, then
P(6) = 0 }
,
151
2
c
.
&' is a parametrix of P(D), and we
Put P(D) F
go
=
+
,
0
where
0
Eb
,
then by Fourier transformation this equality becomes
P[Cj.i(S)
(6.9.3.)
1 +
=
361
for
From the Paley-Wiener theorem, there exists C such that
(6.9.4.)
where A
= x
When 5
E
N, ( 6 . 9 . 3 )
SUP E SUPP
c m
1x1
0
be written
6 E c".
44
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
which, having r e g a r d t o ( 6 . 9 . 4 )
(1 and p r o v e s
+ 1 ~ 1 ~ 5)
c
,
implies
eA l I m
6\
6EN
for
(6.9.1).
The proof o f i i )
a , iii) i s b a s e d on p u r e l y a l g e b r a i c p r o p e r t -
i e s f o r which we r e f e r t h e r e a d e r t o HORMANDER [11 (Chapter IV).
i) i s proved by c o n s t r u c t -
F i n a l l y , t h e implication iii) i n g a p a r a m e t r i x F of P(D) i n t h e form showing t h a t t h i s i s CcD i n IRn
F
=
g’(#)
- (01, proceeding
and by
i n an e n t i r e l y
analogous manner t o t h a t o f t h e proof o f P r o p o s i t i o n 6 . 7 ;
this
i s p o s s i b l e due t o t h e upper bounds ( 6 . 9 . 2 ) .
I n t h e c a s e o f o p e r a t o r s P(D) which a r e n o t h y p o e l l i p t i c , we make use o f q u i t e g e n e r a l theorems t o d e s c r i b e t h e s i n g u l a r suppo r t of
C61
ANDER
7.
u
We r e f e r t o t h e monograph o f HORM-
f o r these questions.
OPERATORS AND DISTRIBUTIONS ON A MANIFOLD
7.1
x
from t h a t of Pu.
:
For
X
+
v
CHANGE OF VARIABLES I N DISTRIBUTIONS:
Let
Y be a diffeomorphism between two open s u b s e t s of I R n .
1 E LlDC(Y) , the
function
u = v
o
x
=
x*
V
1 E hoC(x)
9
(SEC. 7 )
OPERATORS ON A MANIFOLD
considered a s a d i s t r i b u t i o n , s a t i s f i e s f o r
Dx (F) = the
J(rl)
=
of v a r i a b l e s
‘y
E
+
, the
formula (7.1.1)d e f i n e s a d i s t r i b u t i o n
a(X)
denoted by
x x.
v and c a l l e d t h e i n v e r s e image o f If
an open s u b s e t Z o f IRn and i f show t h a t (X,
O
of d i r e c t image
X)* w
x+
REMARK 7.1: 0
J a c o b i a n determinant o f t h e change
)c
under t h e diffeomorphism
cp
:
x = x-’(y)
v E &(Y)
If now u
cp E C r ( X )
< u,cp > =
(7.1.1)
where
45
x-’
i s a diffeomorphism o f Y o n t o
w E &(Z)
* * X (X, w ) .
=
under
x1
x
v
, we
can immediately
We d e f i n e t h e o p e r a t i o n
a s b e i n g t h e i n v e r s e image under
.
x -1
The p r e s e n c e of t h e f a c t o r IJI a l o n g s i d e
i n (7.1.1)shows t h a t it i s n e c e s s a r y t o c o n s i d e r
the t e s t functions measure cp(x] dx
.
Q
n o t as f u n c t i o n s b u t as C
m
d e n s i t i e s of
This remark w i l l b e fundamental t o t h e d e f i n -
i t i o n o f d i s t r i b u t i o n s on a m a n i f o l d .
7.2 C
m
DIRAC MEASURE ON A HYPERSURFACE:
If
S
is a
h y p e r s u r f a c e o f an open s u b s e t X of Fin, t h e E u c l i d i a n s t r u c t -
ure of
mn
induces on S a Riemannian s t r u c t u r e , and we denote by
doS t h e a s s o c i a t e d volume element on S.
46
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
We d e s c r i b e a s t h e Dirac d e l t a f u n c t i o n on S t h e d i s t r i bution 6
S
&(X)
E
< tis, cp >
d e f i n e d by
Is
=
dOs(X1
Q(X)
m
More g e n e r a l l y , l e t f : X +IR be a C
grad f ( x )
S= { x E
x
=
x
..., af
(z(X]t
I
f[X)
=
0 ]
axn
.
putting
< b(f'l, cp >
=
m
n'
hypersurface of
b ( f ] E & (X) on
i,l*l
For example, i f f ( x ) = x
on t h e s e t
Then S i s a C
and w e d e f i n e t h e d i s t r i b u t i o n
8x1
mapping such t h a t
f 0
-(X))
Q E
9
d5&4
S by
cp E
9
w.
we once a g a i n have t h e d e f i n i t i o n of
t h e d i s t r i b u t i o n 6( x = 0). n
I f % i s a diffeomorphism Y
*
+
X such t h a t ( %
f ) ( y ) = yn,
t h e n it can e a s i l y be shown t h a t a d i s t r i b u t i o n % * ( S ( f ) ) i s precisely the distribution S(y = 0 ) .
n
It i s t o obtain t h i s
t r a n s f o r m a t i o n law t h a t w e need t o i n t r o d u c e t h e f a c t o r /grad f
1
i n t o t h e d e f i n i t i o n of 6 ( f ) .
For a thorough s t u d y of
t h i s t y p e o f d i s t r i b u t i o n , w e r e f e r t o GELFAND and. SHILOV c11.
7.3
CHANGE OF VARIABLES I N DIFFERENTIAL OPERATORS:
X be an open s u b s e t of Eln and P(x, D ) =
,@E
aa(x)
D@
Let be a
m
m
d i f f e r e n t i a l o p e r a t o r of degree m w i t h c o e f f i c i e n t s i n C The f o l l o w i n g i s a c h a r a c t e r i s a t i o n of t h e s e o p e r a t o r s :
(x).
7)
(SEC.
47
OPERATORS ON A MANIFOLD
Let P be a continuous Zinear mapping from
PROPOSITION 7 . 3
Then P is a d i f f e r e n t i a 2 operator of
Cz( X ) into Cm(X).
degree m if and on2y if f o r aZZ rea2-va2ued functions and any function a
.I
[7.3.1
E
C"(X) 0
R 3
+j
e-iTq ~ [ eiTp)[x) a
Im
with respect t o
The n e c e s s i t y i s obvious.
PROOF:
= x.%
with
5 E Rn
C"(X)
the expression c.--------*
is a poZynomiaZ of degree
cE
T
f o r every x i n X.
Conversely, l e t u s t a k e
and put
m e- i T x.5
(7.32.)
P(a eiTxgj
=
ak(a,
x , 51
,
Tk
k= 0
g j of
The c o e f f i c i e n t ok[a, x,
Tk
is a
m
c
f u n c t i o n o f x and E;
since t h e same i s a l s o t r u e o f t h e l e f t - h a n d s i d e .
5 and we can show (by r e p l a c i n g 5 by t
t
E
IR -
0
T
Moreover,
by t . r w i t h
1 ) t h a t ok i s a homogeneous f u n c t i o n of degree k w i t h
respect t o 5 ; consequently, it i s a homogeneous polynomial o f degree
k with respect t o
5 s i n c e it i s C
m
for
l e t K be a compact s u b s e t of X and we have a 1 i n t h e neighbourhood of K , for u
E
&,(X),
E
5
E
.$'(X)
lRn.
If we
identical t o
w e o b t a i n by
inverse Fourier transformation
By c o n s i d e r i n g t h i s l a t t e r i n t e g r a l as an i n t e g r a l w i t h r e s p e c t m
t o 5 w i t h v a l u e s i n C (X), t h e c o n t i n u i t y o f P. i m p l i e s t h a t
P.1
[XI
=
O
l,i.5 .
e-bsP(a e h c j
and by v i r t u e of ( 7 . 3 . 2 ) , we o b t a i n
c(5)
48
DISTRIBUTIONS AND OPERATORS
( C H A P . 1)
so that
n
E
%
T h e r e f o r e , P i s given by a d i f f e r e n t i a l o p e r a t o r of degree
m
=
(Pu)fx)
(7.3.3.)
ok[a, x, Dx)u
,
for
u
k=o
when it a c t s on f u n c t i o n s from .8 K ' i v e sequence of compact s u b s e t s K
j
By c o n s i d e r i n g an exhaustand a s s o c i a t e d f u n c t i o n s a
2'
t h e e q u a l i t y ( 7 . 3 . 3 ) shows t h a t t h e d i f f e r e n t i a l o p e r a t o r s
m Ok('j,
piece together into a d i f f e r e n t i a l
x, Ox)
k=o
m
o p e r a t o r of degree
w i t h c o e f f i c i e n t s i n C"(X); t h i s
concludes t h e proof o f t h e p r o p o s i t i o n .
The e x p r e s s i o n ( 7 . 3 . 1 ) p r o v i d e s a new means of c a l c u l a t i n g t h e p r i n c i p a l symbol o f P . t h a t t h e c o e f f i c i e n t of P,(x,
dcp(x)) a[x)
i f we p u t dq(x
0
.
T
m
I n f a c t , L e i b n i z ' formula shows i n ( 7 . 3 . 1 ) i s given by
I n p a r t i c u l a r , i f we have x
) = 5, and assume t h a t a ( x0 ) =
e f f i c i e n t o f rm i s given by P (x
m
0'
0
E
X, and
1, t h e n t h e co-
' 0 ) .
If % : X + Y i s a diffeomorphism between two open s u b s e t s of Bn, and P i s a d i f f e r e n t i a l o p e r a t o r of degree define the transported operator v
i.e.
E
C"(Y)
)& P o f P by
x:
m
on X , we
OPERATORS ON A MANIFOLD
(7.3.4.1
X*P
Proposition 7.3 shows immediately t h a t
m
operator of degree
e-iT
t(YOI[,
p)
T
m
=
(ei7e ) ( y o )
Ix,
where x
0
=
P is a differential
no)
E
Y x l R n , we c a l c u l a t e
i n t h e polynomial
where $ i s such t h a t d $ ( y o ) =
(73.5.)
x+
In order t o calculate i t s
on Y .
p r i n c i p a l symbol a t t h e p o i n t ( y o , the c o e f f i c i e n t o f
*
X * o P o X
=
49
P I m (yo'
e-iT
n0. TI,)
p(ei7
$(YO)
e
x I (x-'
0
(Yo)
We o b t a i n
t
=
TI,)
~ l ( ~ , )
x-1( y o ) .
We now g i v e a b r i e f o u t l i n e of t h e g e n e r a l i s a t i o n of t h e preceding d i s c u s s i o n t o t h e c a s e o f systems of d i f f e r e n t i a l operators.
$I d i f f e r e n t i a l systam ( q , r ) o f degree
m
consists
of an o p e r a t o r o f t h e form
where t h e c o e f f i c i e n t s a r e m a t r i c e s a,(x) m
C
dependence on x i n X.
.
C(Cq, C r )
with
It i s c l e a r t h a t such an o p e r a t o r
d e f i n e s a continuous l i n e a r mapping from
Cm(X; C r )
E
m
c0 (x;)'C
into
P r o p o s i t i o n 7.3 g e n e r a l i s e s immediately t o t h i s
case, it b e i n g s u f f i c i e n t t o t a k e t h e f u n c t i o n
a
in
Cz(X; C q ) and t o demand t h a t t h e e x p r e s s i o n (7.3.1)be a polynomial i n T w i t h c o e f f i c i e n t s i n . ' 6
We d e s c r i b e a s t h e p r i n c i p a l symbol o f P at a p o i n t
'1
DISTRIBUTIONS AND OPEmTORS
50
(x, 5 )
E
X
x
(CHAP. 1)
n
IR t h e l i n e a r mapping
The c o e f f i c i e n t o f .rm i n ( 7 . 3 . 1 ) i s e q u a l t o P,(x,
dq(x)).a(x),
which p r o v i d e s a new d e f i n i t i o n o f ? (x, 5 ) and a l l o w s us t o
m
prove t h a t ( 7 . 3 . 5 ) remains t r u e .
7.4 dimension C
m
DISTRIBUTION ON A MANIFOLD:
Let X be a manifold o f
n (we s h a l l assume i n t h i s book t h a t a manifold i s
and c o u n t a b l e a t i n f i n i t y ) .
-
We e q u i p C"(X) semi-norms '9
w i t h t h e t o p o l o g y d e f i n e d by t h e f a m i l y o f
p ( x , q),where cy
charts X : X 3 U
--L
U
x
d e s c r i b e s t h e f a m i l y of l o c a l
c R" o f X , and p m-)
of semi-norms d e f i n e d on C ( U ) .
describes t h e family
By u s i n g a p a r t i t i o n o f u n i t y ,
we can show t h a t it a c t u a l l y s u f f i c e s t o c o n s i d e r o n l y t h e semi-norms a s s o c i a t e d w i t h an a t l a s of X and we can show t h a t
C"(X)
is a Fr6chet s p a c e .
With an obvious n o t a t i o n , we d e f i n e as i n s e c t i o n 1 t h e t o p o l o g i c a l v e c t o r s p a c e s C;(X)
=DK(x),c:(x)
=D(x).
If
7
i s a complex v e c t o r bundle o f rank N on X, we denote by C"(X;
3 )' t h e
space of C
m
s e c t i o n s of
t o p o l o g y as above by r e d u c i n g t o of l o c a l t r i v i a l i s a t i o n s of 3;.
3
and we e q u i p it w i t h a
='* N C (U; C )
= (Cm(z))N
by means
(SEC. 7)
OPERATORS ON A MANIFOLD
R ( X ; 3)
We define s i m i l a r l y t h e spaces
ax; ?I
and
K
.
s)
q x ;
=
51
In o r d e r t o d e f i n e d i s t r i b u t i o n s on a manifold X , we s h a l l need f i r s t t o d e f i n e t h e bundle of d e n s i t i e s on X.
mapping from
A
n
E \ 0 i n t o Q: such t h a t
~ ( h w )= 111
('7A.l.I
,
W(W)
In p a r t i c u l a r , i f vl,..., v
1 E R\O
for
w €An E \ D
and
i s a b a s i s f o r E and i f A
n
we have
~(Av,, A
(7.4.2.)
...
A Av,)
More g e n e r a l l y , i f we have a order a on E i f
is a
n
A d e n s i t y w on a v e c t o r space E of dimension
(7.4.1)i s
= E
ldet A 1 w(v, A
R, we say t h a t
L)
...
E
GL(E),
A v ),
.
i s a density of
s a t i s f i e d when we r e p l a c e IhI by
j h l a ; we t h e n need t o use l d e t
A1
a
(7.4.2.).
in
The s e t o f d e n s i t i e s of o r d e r a on E c o n s t i t u t e s a complex vector space o f dimension 1 which we denote by /El
IRnI1
=
[ a
...
dxl
.
a'
dxn
I
a
E
C
.
f o r example
By m u l t i p l i c a t i o n , w e
define t h e b i l i n e a r mapping
[7.4.3.)
IEI,
X
and t h i s shows t h a t
]El, 3 IE
€3
i d e n t i f y t h e d u a l o f lEla w i
I f X i s a manifold of dimension
XI
spaces / T above x c 1
x
n , t h e family o f v e c t o r
arranges itself naturally into a vector
.
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
52
bundle o f rank 1, termed bundle of d e n s i t i e s of o r d e r
:
I n f a c t , i f X,,
X and w r i t t e n Q ( X ) . a
i s a c h a r t of X , it d e f i n e s a c h a r t AX and hence a b i j e c t i o n
$
:
X 3
U2 7
Ixl I a
5
Rn
C
x
1
3
u1
C
cl =mn
: An'TU1 P
ncy(ul)onto R,(cl]
of
on
a
u
,N
U1 x
An TG,
c.
If
i s another c h a r t o f X , formula
( 7 . 4 . 2 ) ( w i t h ldet A l a ) shows t h a t , i n t h e common open s u b s e t ,
151a
t h e coordinate t r a n s i t i o n function m u l t i p l i c a t i o n by Idet(X 1
o
'
()XI
la)-'
jJ@ .
o
Let p be a d e n s i t y o f o r d e r a on X , i . e . a C
Qu(X); i t s r e s t r i c t i o n c
of o r d e r a on U
.
j'
PjFj)
.
with a . ( ) 3
a "
E
p
I U:J i s
t r a n s p o r t e d by
we denote t h i s by p
*.(:.)
=
C (U. ) J
J
.
J
constitutes
I x.1 J
m
U
section of i n t o a density
and we put
j
ldz.1"
J
The t r a n s f o r m a t i o n l a w of t h e bundle
(x)
i m p l i e s t h a t i n t h e open s u b s e t U1 n Up, we have I -1 cy (7.4.4.) = ddX1 0 $
i2
a
-I
+,I
)I
Formula ( 7 . 4 . 4 ) shows t h a t we can speak o f real-valued d e n s i t i e s and of s t r i c t l y p o s i t i v e d e n s i t i e s .
To l e s s e n t h e n o t a t i o n a l
burden, we put
c"(x; nJx))
=
c"[lx
cy
I
.
1
c"(
Since t h e d e n s i t i e s o f o r d e r 1 a r e measures on X , we can i n t e g r a t e them; having r e g a r d t o ( 7 . 4 . 3 . ) , t h i s allows us t o d e f i n e t h e b i l i n e a r mapping
OPERATORS ON A MANIFOLD
53
r
when a
+ p
= 1. cy =
The p a r t i c u l a r case
0
,
p
= 1 m
define continuous l i n e a r forms on C o (
shows t h a t C"(X)
1x1)
functions
which l e a d s u s , i n
accordance w i t h Remark 7.1, t o s t a t e :
DEFINITION 7.5:
A d i s t r i b u t i o n on the manifold X i s a
continuous linear f o r m on the space C z ( 1 on X.
1x1 )
of d e n s i t i e s of order
&(x)
The space of d i s t r i b u t i o n s i s denoted by
More generally, i f
3 i s a comples vector bundle over X , we
describe as the space of d i s t r i b u t i o n a l sections of of continuous linear forms on
denote i t by
.
&[x;
.
f$(XI 8 3 1
C:(X;
S the space and we
( '3%denotes the dual bundle of
9). We have t h e i n j e c t i o n of
f(x;
9)
in
& ( x i s")
d e f i n e d by
the p a i r i n g
Taking i n t o account ( 7 . 4 . 3 ) and
(7.4.51,we
can show t h a t
the dual of t h e space C z ( X ) i s i d e n t i f i e d w i t h t h e space of d i s t r i b u t i o n a l s e c t i o n s o f Q,(X).
When X i s equipped w i t h a
s t r i c t l y p o s i t i v e d e n s i t y 1-1 (we can always c o n s t r u c t one o f t h e s e by using a p a r t i t i o n o f u n i t y f o r X ) we a r e a b l e t o i d e n k i f y t h e s e spaces, s i n c e we can use t h e t o p o l o g i c a l v e c t o r isomorphism
DISTRIBUTIONS AND OPERATORS
54
( C H A P . 1)
We now give a b r i e f o u t l i n e of t h e p r i n c i p a l p r o p e r t i e s of d i s t r i b u t i o n s on a manifold, l i m i t i n g o u r s e l v e s , f o r s i m p l i c i t y , t o t h e case
3=
c
.
a(X)
We have f o r t h e p a i r
,
PIX)
the
same t o p o l o g i c a l p r o p e r t i e s as t h o s e mentioned i n S e c t i o n 1, f o r t h e c a s e where X i s an open s u b s e t o f I R n .
.
P(X)
We have t h e continuous i n j e c t i o n of L1 ( X ) i n t o loc we l e t V be an open s u b s e t of X , a d i s t r i b u t i o n
u E S(X)
d e f i n e s by r e s t r i c t i o n t o C"( IVI) a d i s t r i b u t i o n
UIV
0
If
E P(V1
The same proof a s was used i n t h e case o f an open s u b s e t of lR
P[v)
shows t h a t t h e spaces
form a sheaf on X.
*
n
The concepts
of s u p p o r t and s i n g u l a r s u p p o r t g e n e r a l i s e i n t h e same way t o t h e c a s e of a manifold:
t h e space &'(X) of d i s t r i b u t i o n s w i t h compact m
support i s i d e n t i f i e d w i t h t h e d u a l of C (
1x1 ) .
--
We can a l s o g i v e a d e f i n i t i o n o f d i s t r i b u t i o n s w i t h t h e a i d of l o c a l c h a r t s .
:
Letting
be a c h a r t of X, t h e n i f
x 3 u
Ly
u E
b(x)
CR"
uLy
t h e diffeomorphism
p o r t s it i n t o a continuous l i n e a r form
w
XLy
[u\u)
x,
on
U i . e . i n t o a d i s t r i b u t i o n Z,on
r,
U,.
xB i s a n o t h e r c h a r t , w e have t h e c h a r t t r a n s i t i o n formula f o r t h e r e s t r i c t i o n s of 5,and
xLy[uen up)
and
ii t o t h e open s u b s e t s
%WLyn up'
B
trans-
If
7)
(SEC.
OPEMTORS ON A MANIFOLD
55
(7.5.2.) c
(x,)
Conversely, if
i s an a t l a s o f X , i n each open s u b s e t U, we N
u
take a d i s t r i b u t i o n
E &(x)
E
&(E@)
such t h a t (7.5.2) i s s a t i s f i e d
Then, t h e r e e x i s t s a unique d i s t r i b u t i o n
f o r a l l a,B. U
u *
such t h a t
N
=
Xcr(Ucy)
4
,
In fact, the distribut-
"@
ions
a
l i f t into distributions u
her i n t o a unique d i s t r i b u t i o n
cy
E &(UCY)
u E P[X)
which p i e c e t o g e t by v i r t u e of t h e
sheaf p r o p e r t y .
DIFFERENTIAL OPERATORS ON A MANIFOLD:
7.6
f o l d of dimension
n
, Proposition 7.3
DEFINITION 7 . 5 : cm(X);
degree
i f f o r a l l real-valued
a
E
l e a d s us t o s t a t e :
Let P be a continuous linear mapping from
~"(x) into 0 m
If X i s a mani-
we say t h a t t h i s i s a d i f f e r e n t i a l operator of rg E C"(X)
and alZ
c~(x), the f u z e t i o n
is a polynomial of degree
m w i t h respect t o
T
for a l l fixed x
i n X.
This d e f i n i t i o n shows t h a t t h e r e s t r i c t i o n P
IU
of P t o an open
subset U o f X i s a d i f f e r e n t i a l o p e r a t o r on U.
Let
X
:
X 3U
P
GCW"
be a c h a r t of X, and P a
56
DISTRIBUTIONS AND OPERATORS
( C H A P . 1)
m
m
l i n e a r mapping from C (X) i n t o C (X); we d e f i n e t h e l i n e a r oper0
ator P
m
X
,-
e
from C o ( U ) i n t o C (U) by p u t t i n g it
With t h i s n o t a t i o n we have:
PROPOSITION 7.7: C I ( X ) i n t o C"( X)
.
Let P be a continuous l i n e a r mapping from
Then, P i s a d i f f e r e n t i a l operator of degree m
i f and only i f f o r any chart P
X
x
from an a t l a s of X the operator
i s a d i f f e r e n t i a l operator of degree m.
The n e c e s s i t y f o l l o w s from P r o p o s i t i o n 7 . 3 .
PROOF:
The s u f f -
i c i e n c y i s immediate by u s i n g a p a r t i t i o n o f u n i t y s u b o r d i n a t e t o t h e c o v e r i n g o f X by c h a r t s from t h e given a t l a s .
We can d e f i n e t h e p r i n c i p a l symbol o f a d i f f e r e n t i a l o p e r a t o r P of degree
x
morphism
m
on X .
If
x
c
U
:
-+
U i s a c h a r t of X, t h e d i f f e o -
*
a l l o w s us t o d e f i n e on t h e c o t a n g e n t space T U a homo-
geneous f u n c t i o n Pu by l i f t i n g t h e p r i n c i p a l symbol o f
m
x
P.
The e q u a l i t y ( 7 . 3 . 5 ) shows t h a t t h e f u n c t i o n s P" combine t o g e t h e r
m
m
into a C
%
f u n c t i o n on T X.
T h i s f u n c t i o n i s denoted by
P m ( x , 5 ) and i s c a l l e d t h e p r i n c i p a l symbol of P ; geneous of degree
m
with respect t o
By c a l c u l a t i n g w i t h
6
E
T
*
X
it i s homo-
X.
a c h a r t , we can show t h a t t h e c o e f f -
i c i e n t o f -cm i n t h e polynomial
(7.6.1)i s
given by
(SEC. 7 )
57
OPERATORS ON A MANIFOLD
The concept o f a d i f f e r e n t i a l system g e n e r a l i s e s i n t o t h e concept of a d i f f e r e n t i a l o p e r a t o r a c t i n g upon t h e s e c t i o n s o f v e c t o r bundles. rank
Suppose
r
and
g
'? and
9
a r e complex v e c t o r bundles of
above X .
7.8: A continuous linear mapping
DEFINITION
P : c ~ ( x ,$1 +~ " ( x , Q) is a
cp E c"(x)
of degree m if for a l l real-valued a
E
is, f o r all x
x,
E
a polynomial of degree
with coefficients in the fibre
3
x
and a l l
F) the function
C:(X;
Let
differential operator
(F, Q)
: U -9
and
C,
i? CR"
f
uxcq
I
X
-
i?
u
with respect to
above x .
3
and g
f
s/
Iu
1
I
u
;
L
ExC'
y
E
-
Taking P t o be a continuous l i n e a r mapping from
C"[X, pf,g
6)
,
T
be a c h a r t above which t h e bundles
admit t r i v i a l i s a t i o n s
-
N
Sx of $
m
1
CI(x; b)
into
we d e f i n e by t r a n s p o r t a continuous l i n e a r mapping
~ l from C ~ ( Cq)
into
C""(r;
C')
by p u t t i n g :
58
( C H A P . 1)
DISTRIBUTIONS AND OPERATORS
17 .8.2.)
=
pf,g
PIUj
g*
0
f*
With t h i s n o t a t i o n , P r o p o s i t i o n 7.7 g e n e r a l i s e s immediately by replacing t h e operators of type P
X
t h e r e i n by o p e r a t o r s o f t h e
t y p e P f , g a s s o c i a t e d w i t h a covering of X by open c o o r d i n a t e
9 and 6 are t r i v i a l i s a b l e .
p a t c h e s above which t h e bundles
(s,$)
If P i s a
d i f f e r e n t i a l o p e r a t o r o f degree
$ and
by u t i l i s i n g l o c a l t r i v i a l i s a t i o n s of
which depends only on dq(x)
and a(x).
f u n c t i o n i s homogeneous of degree
5
dcp(x) =
we can show
c o e f f i c i e n t o f -rm i n ( 7 . 8 . 1 ) i s a f u n c t i o n
t h a t f o r x fixed,’the
iable
, then
m
m
Furthermore, t h i s
w i t h r e s p e c t t o t h e var-
and i s a l i n e a r mapping from
into X
QX
w i t h r e s p e c t t o t h e v a r i a b l e v = a ( x ), and we denote t h i s by Pm(x,
c)
t h i s i s by d e f i n i t i o n t h e v a l u e of t h e p r i n c i p a l
;
symbol o f P a t t h e p o i n t ( x , 6 )
EXAMPLE 7 . 9 : and n-1;
we c o n s i d e r t h e bundles
k *
,
exterior differential
C:(x,
(7.8.1)i s e- i T Q ( X )
k be an i n t e g e r c o n t a i n e d between 0
Let
S = (A T x j 8 c
from
T* X.
E
fl
Q
= (hk+’ T*X) e3
d
c
over X.
The
d e f i n e s a continuous l i n e a r mapping
6)
i n t o C”(X,
.
In t h i s case, the function
given by d(eiTrp.e)[x)
consequently
d
is a
=
(a,Q)
i 7
dcp(x)
A
a(x)
+
da(x)
9
d i f f e r e n t i a l o p e r a t o r o f degree
(SEC. 8 )
KERNEL DISTRIBUTIONS ON A MANIFOLD
1. The c o e f f i c i e n t o f a t t h e p o i n t (x, 5 ) (A
k
*
Tx
X) 2' 3
E
T
T
shows t h a t t h e p r i n c i p a l symbol of
*
Tx X ) 8 & d e f i n e d by l e f t e x t e r i o r
m u l t i p l i c a t i o n by t h e one-form
8.
d
X i s t h e l i n e a r mapping from
(Ak+'
into
C
*
59
it.
OPERATORS AND KERNEL DISTRIBUTIONS ON A MANIFOLD
L e t X , Y be manifolds equipped r e s p e c t i v e l y w i t h s t r i c t l y p o s i t i v e d e n s i t i e s y and v. t h e r e s u l t s of S e c t i o n s
4
With any d i s t r i b u t i o n continuous l i n e a r o p e r a t o r
We now show how t h e d e f i n i t i o n s and
and 5 g e n e r a l i s e t o t h i s s i t u a t i o n .
K E a ( X x Y) A :
, we
C:(Yl
can a s s o c i a t e a
J(X)
by p u t t i n g
Conversely, we have t h e k e r n e l s theorem:
THEOREM 8.1: A : C:(Y)
->
KA E P I X x Y)
PROOF:
P(X)
Any continuous l i n e a r operator i s defined by a unique kernel d i s t r i b u t i o n
such t h a t we have (8.1.1).
L e t (Ua) and ( V ) be a t l a s e s o f X and Y .
B
rjith t h e s e c h a r t s , Theorem
By t r a n s p o r t
4.4 shows t h a t f o r any a ,
f3 t h e r e
DISTRIBUTIONS AND OPERATORS
60
e x i s t s a unique d i s t r i b u t i o n
Ku,B E p ( u ,
(CHAP. 1)
such t h a t
X
Making use of t h e bundle p r o p e r t y , we deduce from t h i s t h a t t h e r e e x i s t s a unique d i s t r i b u t i o n KIUu x VB
Kac,B
=
K
f o r any a,@.
E
p [ X x Y)
such t h a t
By p a r t i t i o n of u n i t y , we t h e r e -
by deduce t h a t K s a t i s f i e s ( 8 . 1 . 1 ) .
REMARK
8.2:
I t i s c l e a r from ( 8 .1 .1) t h a t t h e k e r n e l KA a,
:
A
of an o p e r a t o r
Co(Y)
d e n s i t y p chosen on X.
x
" : X + X, 0
A :
C:(y]
does n o t depend on t h e
Furthermore, i f we have diffeomorphisms
Y
: Y -+ Y we d e f i n e t h e t r a n s p o r t e d o p e r a t o r
-
ICI
-J"X)
H
&();
o b t a i n e d from A by A = X,.A.e*. N
We s p e c i f y
N
a p o s i t i v e d e n s i t y v on Y ;
we can immediately show t h a t t h e
w
k e r n e l K- o f A s a t i s f i e s
A
c)
where K i s o b t a i n e d by t r a n s p o r t i n g K under t h e diffeomorphism A A Ie, X I : x x Y -X - x zY and where f i s t h e cm f u n c t i o n on Y d e f i n e d by 8+ v =
ry
f. v
.
I n p a r t i c u l a r , a change o f d e n s i t y on Y m a n i f e s t s i t s e l f on t h e k e r n e l K t h r o u g h m u l t i p l i c a t i o n by a s t r i c t l y p o s i t i v e C of t h e second v a r i a b l e .
m
function
(SEC.
8)
61
KERNEL DISTRIBUTIONS ON A MANIFOLD
F i n a l l y we n o t e t h a t i f t h e manifold Y i s n o t endowed w i t h a d e n s i t y , we s t i l l have a k e r n e l s theorem as l o n g a s w e seek K i n t h e space 8( (X x Y ;
no(xj
Q
me
Q, [Y) j
method of proof
used for Theorem 8 . 1 allows u s t o prove t h e f o l l o w i n g :
A continuous l i n e a r operator
THEOREM 8 . 3 :
A : C I I Y ' ) - . P ( x ) i s regularising i f and only i f
its kernel KA is
i n C"(X x Y). m
P(x) I
I f A i s a continuous l i n e a r o p e r a t o r from c 0 ( y ) i n t o
t h e d e n s i t i e s p and v a l l o w u s t o d e f i n e l i n e a r o p e r a t o r s 'A and
fl ( y ]
A* from C z ( X ) i n t o
REMARK
8.4:
, by
putting
P r o p o s i t i o n 4 . 5 g e n e r a l i s e s word f o r word t o
t h i s case.
THEOREM 8.5:
We assume t h a t the manifold Y i s compact.
Then i n t h i s case any regularising operator A :
c"(Y)
PROOF:
c"(x)
i s a compact operator.
Since A extends c o n t i n u o u s l y t o
f o r t i o r i i n t o a continuous o p e r a t o r from
$(yI
c0 ( Y )
8
into
it extends a m
c ( x ) , giving
62
DISTRIBUTIONS AND OPERATORS
(CHAP. 1)
t h e commutative diagram
I t , i s t h u s s u f f i c i e n t t o prove t h e f o l l o w i n g :
LEMMP,
j : c"(Y)
PROOF:
8.6:
->
I f Y is a compact manifozd, then t h e i n j e c t i o n
c"(Y) is compact.
We need t o show t h a t t h e r e e x i s t s a neighbourhood
Y
of 0 i n C"(Y) which is r e l a t i v e l y compact f o r t h e t o p o l o g y induced by Co(Y).
Let ( V . ) be a f i n i t e c o v e r i n g o f Y by open J
c o o r d i n a t e p a t c h e s . We d e f i n e
2/
t o be t h e s e t of f u n c t i o n s
m
v
E
C (Y) which t r a n s p o r t under each c h a r t t o f u n c t i o n s
having t h e i r f i r s t d e r i v a t i v e s bounded by 1.
By c o n s t r u c t i o n ,
m
i s a neighbourhood of 0 i n C ( Y ) composed o f e q u i c o n t i n u o u s functions;
t h u s it i s r e l a t i v e l y compact f o r t h e t o p o l o g y of
CO(Y).
9.
REGULAR OPEN SUBSETS OF IRn AND MANIFOLDS WITH BOUNDARY
We i n t r o d u c e i n t h i s s e c t i o n some c o n c e p t s which w i l l b e u s e f u l i n t h e s t u d y of boundary problems.
(SEC. 9 )
Let X be an open subset of IRn with clo-
DEFINITION 9.1 : sure
5 and
with boundary
ax
subset o f D n i f
ax.
E
ax,
We say t h a t X i s a regular open
i s a hypersurface o f D n and i f X i s locaZZy
situated on onZy one side of x
63
MANIFOLDS WITH BOUNDARY
ax.
This means t h a t f o r any
there e x i s t s an open neighbourhood U of x i n # c
diffeomorphism
x
:
U
+-
U of
and a
CI
u
onto an open subset U of IR" such
X'
ax
hr
u
X
n
X
In t h e preceding s i t u a t i o n we s a y t h a t
?
a t t h e boundary.
t h e hypersurface
ax
We n o t e t h a t
xlU
x
i s a l o c a l c h a r t of
ax
is a local chart of
a t x.
If X is an a r b i t r a r y open s u b s e t of Eln, we denote by
64
DISTRIBUTIONS AND OPERATORS
~ " ( 2t)h e
space of f u n c t i o n s cp
E ~"(ic) such
( C H A P . 1)
a"? E c0ljT)
that
Equipped w i t h t h e f a m i l y o f semi-norms
f o r any multi-index a .
I a@ q ( x ) 1 , where K is an a r b i t r a r y
com-
la/ 5 j p a c t s u b s e t of
?
If K i s a compact s u b s e t of
Frgchet space.
a,(?)=
and j is an a r b i t r a r y i n t e g e r 2 0 , C"(?)
2,
is a
we denote by
t h e (FrBchet) subspace o f C m ( z ) composed o f t h e
C",?)
f u n c t i o n s which a r e z e r o i n X \ K , and by C z ( ? ) =
a(?) the
inductive l i m i t of
8K ( 2 ) when K v a r i e s over t h e f a m i l y o f com-
2.
We h e n c e f o r t h assume t h a t t h e open s u b s e t
pact subsets of
X i s r e g u l a r ; we s h a l l s e e t h a t we can d e f i n e
C"(2)
n of t h e r e s t r i c t i o n s t o X o f Cm f u n c t i o n s inIR
.
a s t h e space More p r e c i s e l y ,
we have :
PROPOSITION 9 . 2 (SEELEY) :
-
There e x i s t s a continuous
Linear extension operator p : c m ( x )
C m o R n ) whose r e s t r i c t i o n
t o .8(X) i s a continuous Linear extension operator (again denoted by p ) p : S(X) +
PROOF:
x
R"+ = {
extension.
smn),
We begin w i t h t h e s p e c i a l c a s e
x
E a"
I
xn
>0 }
,
i n which we use a SEELEY
More p r e c i s e l y , f o r cp
-
E
m n C (IR+), we d e f i n e p ( Q ) by
(SEC. 9 )
x
where
65
W I F O L D S WITH BOUNDARY
i s e q u a l t o 1 i n t h e neighbourhood of 0 and
C;(B)
E
where (A ) i s a sequence o f r e a l numbers such t h a t
k
(9.2.1.)
j E KI
for a l l
>-:
Xk 2 k j
i s an a b s o l u t e l y
k z l convergent s e r i e s w i t h sum ( - 1)j ,
It i s immediately e v i d e n t t h a t t h e o p e r a t o r p s o cons t r u c t e d matches t h e r e q u i r e m e n t . I n t h e c a s e where
of lRn of
ax
charts
, we
c o n s i d e r a l o c a l l y f i n i t e c o v e r i n g of a neighbourhood
i n Bn by ( r e l a t i v e l y compact) open s u b s e t s U
xj
of
x a t t h e boundary.
Suppose we have 'p, Q ,
+
vj
=
j 21 Suppose $ E . j
supp
i s an a r b i t r a r y r e g u l a r open s u b s e t
X
cpj.
E
We p u t
Cm(Uo) and Cpj E
1
C:[uj)
m
c,(uj)
such t h a t
i n t h e neighbourhood of
i s such t h a t $
For c p ~C " ( ? ) ,
3
?.
= 1 in
rp = rp,
we have
Q
+
'pj rp
J We extend
i n t o @ = p ( ~ )E Cm(R")
, where
iP=@,+cPj
of l o c a l
j Uo = X \ (U Uj) . J
.
by p u t t i n g Q0
i s o b t a i n e d by e x t e n d i n g
j p ',
cp by 0 o u t s i d e o f U
0
and where
J
It is immediately obvious t h a t t h e o p e r a t o r p s o c o n s t r u c t e d meets t h e r e q u i r e m e n t . sequence
It remains t o prove t h e e x i s t e n c e of a
Xk s a t i s f y i n g ( 9 . 2 . 1 ) .
Cramer system
With N
E
N , we c o n s i d e r t h e
66
DISTRIBUTIONS AND OPERATORS
(CHAP. 1
A c a l c u l a t i o n o f Vandermonde determinants shows t h a t
2-j
Since t h e s e r i e s
converges, t h e r e e x i s t c o n s t a n t s C
j s l and c > 0 such t h a t
(1
+
2-3
5
c
J=1
,
and
We have t h e upper bounds
and s i n c e
Letting N
1 1 2a-l 2j - 1
-t m,
for R
2 1, we t h e r e b y deduce
we o b t a i n
Consequently ( 9 . 2 . 1 ) follows from ( 9 . 2 . 2 ) , ( 9 . 2 . 3 ) and ( 9 . 2 . 4 ) . We now consider t h e r e s t r i c t i o n o p e r a t o r
dR")-9 m)
It i s c l e a r l y continuous, and P r o p o s i t i o n 9.2 shows t h a t it is
(SEC. 9 )
By t r a n s p o s i t i o n , we o b t a i n an i n j e c t i o n o f t h e
surjective. dual
67
MANIFOLDS WITH BOUNDARY
.PR)
M) i n t o
of
We i d e n t i f y a continuous
ax)w i t h i t s image under t h i s i n j e c t i o n ,
l i n e a r form R on
with t h e d i s t r i b u t i o n
c
(92.5.)
.P[R").
u, ip
u
o n B n d e f i n e d by
>
=
a(~,,]
@ E
for a l l
i.e.
~(R"I,
We t h e n have:
PROPOSITION 9.3 :
By way of the i d e n t i f i c a t i o n ( 9 . 2 . 5 ) ,
the dual P ( y ) o f @) coincides with
J!+'($")
.
This e q u a l i t y
X
is algebraic m d topological, both of these two spaces being
r,,.
equipped e i t h e r with T s or with dual
&'R) of C"R]
coincides with
In the same manner, the
EL(R") X
PROOF: i f supp @
u i s d e f i n e d by ( 9 . 2 . 5 ) , we have < u, Q > =
If C
.
R"\
u E 8. (R")
and t h u s
iT
.
0
Conversely,
we have
(9.3.1.)
Suppose we have
LEMMA:
u E
Ir_(R")
and
ip
E 4R")
X
such t h a t @ = 0 i n X.
Then
< u, @ > = 0 .
This lemma shows t h a t if u E Ir_(R"),
L on
f l ) by
we d e f i n e a l i n e a r form
A
putting
a((p) = < u,
i s an a r b i t r a r y e x t e n s i o n of
cp E
9
>
.
Ed
.Pg) = E[R") X
(z) .
.
4R")
.@i)
Making use o f P r o p o s i t i o n 9 . 2 , w e have and t h u s .f,
@ E
where
a(cp)
=
<
U,
p(cp)
>
This proves t h e a l g e b r a i c e q u a l i t y The t o p o l o g i c a l e q u a l i t i e s f o l l o w from t h e
,
68
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
It now remains
c o n t i n u i t y of p and o f t h e r e s t r i c t i o n o p e r a t o r .
f o r us t o prove Lemma (9.3.1): through a p a r t i t i o n of u n i t y , we n can a s b e f o r e reduce t o t h e case X =lR+.
xn
@,(x) =
of
@(XI,
+
x, we have
< u,
>
0 since
< u,
@
=
REMARK 9.4
for all Q E
:
fl)
@
6
> ;.Q @e
E
> 0 , we put
i s z e r o i n t h e neighbourhood
; we t h e r e b y deduce t h a t
e
@
in
&(B(R")
when
E
-f
0.
We have t h e continuous i n j e c t i o n
,
with t h e function fo innn \
; since 0
e)
For
By means of ( 9 . 2 . 5 ) , f E
1 Lloc@in)
i s identified
d e f i n e d by f o = f i n
2, fo
= 0
X.
REMARK 9 . 5
:
&(XI
i s s e q u e n t i a l l y dense i n
& F] : by
a p a r t i t i o n of u n i t y we reduce t o t h e s e q u e n t i a l d e n s i t y of &(XI
in
&L'[R")
n i n t h e case X =B+. I n o r d e r t o e s t a b l i s h
X t h i s d e n s i t y , we proceed by r e g u l a r i s a t i o n ( s e e 1.1.1)by
n choosing p w i t h support i n B+.
REMARK 9.6 : manifold M ,
?
When i n D e f i n i t i o n 9.1 we r e p l a c e #
by a
i s c a l l e d a manifold w i t h boundary imbedded i n M.
A l l t h e preceding d e f i n i t i o n s and p r o p e r t i e s extend t o t h i s case
without d i f f i c u l t y .
(SEC. 9)
69
MANIFOLDS WITH BOUNDARY
We now g e n e r a l i s e t o t h i s s i t u a t i o n t h e r e l a t i o n s h i p s b e t -
x be
Let Y be a manifold and
ween k e r n e l s and o p e r a t o r s .
a
manifold with boundary imbedded i n M: we assume t h a t p o s i t i v e densities
and v a r e s p e c i f i e d on M and Y ; we n o t e t h a t
!J
i s a manifold w i t h boundary imbedded i n M
x Y
from Lemma 1 . 3
and P r o p o s i t i o n 9.2
fi
s e q u e n t i a l l y dense i n
a(iTx
X X Y
E ( ~ ( Y ) , i~ (X))
d
into
p)
is
We have
, afl)=
(M x Y’)
Y) =
and t h a t ,
f l ]8 &Y)
,
.
x Y)
x Y
P(M) and t h e space
iT
of continuous l i n e a r o p e r a t o r s from B(Y)
i s a subspace of
.
fi [M’))
.C(b(Y),
By v i r t u e of
Lemma (9.3.1),it can immediately be shown t h a t t h e o p e r a t i o n of passage t o t h e k e r n e l ( s e e Theorem 8.1) induces a b i j e c t i o n of
L[B[y),
t r u e f o r a l l cp
A
.&fl
f l [ x ) ) onto
E c(&fl),
E
G ),
.s([Y))
the kernel
JI E B(Y)
A
K Ea[Y x
.
NOW
0
suppose
J!(Y)
r : B(M)
; if
ion o p e r a t o r , we have
and t h a t formula (8.1.1) i s
x Y)
is the restrict-
, and
r € E(B[M), & [Y))
we c a l l
M) of A o r t h e k e r n e l of A ; we can show
a s above t h a t t h e o p e r a t i o n o f passage t o t h e k e r n e l i s a b i j e c t i o n of
e[fi),
.s([(v))
Y)
and t h a t it i s for all
x, 7 be two manifolds with boundary
imbedded
If E 4 Y )
Finally, l e t
b ( Y x
>
again c h a r a c t e r i s e d by:
cp€*),
onto
c
A?,
$V
>=c
K, $ v 63
q#h
.
i n manifolds M, N equipped w i t h p o s i t i v e d e n s i t i e s
u,
v.
70
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
A E L(fl),
Suppose we have
fl))
d
;
we d e f i n e t h e k e r n e l K
of A by c o n s i d e r i n g A a s a continuous l i n e a r o p e r a t o r from
m)
fi [M)
into
, or,
which amounts t o t h e same t h i n g , by
c o n s i d e r i n g t h e k e r n e l of t h e o p e r a t o r A
o
r E C[aN],
KEJJ XXTi
Pfl)) .
The above shows t h a t
( M X N ) ~ E [ M X N ) = P( M X N) X x N X V
By
x
u t i l i s i n g Lemma (9.3.1),we can a g a i n show t h a t t h e o p e r a t i o n of passage t o t h e k e r n e l is a b i j e c t i o n o f
i",
q#>
=
$
@
*v>
for all
H
cp E &M)
where
E B(N]
where
d[iT)) o n t o
and t h a t it i s c h a r a c t e r i s e d by
x N)
,(M X X Y
S(fi),
is an e x t e n s i o n of
Q
cp E
, or,
MI,
$ E
4%
f u r t h e r , by
i s an e x t e n s i o n of jl.
We do n o t use t h e n o t a t i o n
Pfl x y )
s i n c e we s h a l l
r e f r a i n from s y s t e m a t i c a l l y s t u d y i n g h e r e t h e % a n i f o l d w i t h corners" A E
x x y embedded i n M x N .
Z ( q y ) , d fl))
function i n in
xx
?'I.
I f t h e k e r n e l of t h e o p e r a t o r
i s t h e e x t e n s i o n by 0 o u t s i d e
x x Y, we s a y t h a t A i s
x x y of m
"an o p e r a t o r w i t h C
a C
m
kernel
I n c o n c l u s i o n , t h e k e r n e l s theorem i s s t i l l v a l i d
f o r manifolds w i t h boundary.
I
(SEC. 10)
Let
71
ADDITIONAL NOTES
7,
be two manifolds w i t h o r w i t h o u t boundary;
we
assume a s b e f o r e t h a t p o s i t i v e d e n s i t i e s l.~ and v a r e s p e c i f i e d . A continuous l i n e a r o p e r a t o r
A :
4 ~ ) c ~ ( v i) s
said t o
be r e g u l a r i s i n g i f it extends i n t o a continuous l i n e a r o p e r a t o r
-
from 'El(&
.
c"(Y)
equipped w i t h i t s s t r o n g d u a l topology
AS
i n Theorem
THEOREM 9 . 7 :
4.7, w e
An operator
, we
A :
ularising operator
Suppose
A
E
p (u)
b[*w)--9
ern
in
v x .W
A :
C(q.w),
is reg-
.
can prove:
Suppose t h a t W is compact.
THEOREM 9 . 8 :
-into
can prove:
uzarising if and only if i t s kernel is A s i n Theorem 8.5
Tb
4.w)9 ~ " ( y )
P(v])
j
Then. any reg-
is a compact operator.
a s i n (8.3.1), we d e f i n e t h e
transpose tA and t h e a d j o i n t A n of A by
W e can show t h a t t h e s t r o n g d u a l o f canonically w i t h
J(q
;
>(V)
i s identified
hence w e have a p r o p o s i t i o n analogous t o
Proposition 4 . 5 .
10.
ADDITIONAL NOTES AND EXERCISES 10.1
PALEY-WIENER-SCHWARTZ THEOREM:
of Theorem 2.9.
We use t h e n o t a t i o n
DISTRIBUTIONS AND OPERATORS
72
Let U be t h e F o u r i e r t r a n s f o r m o f a d i s t r i b u t i o n
a)
.
T E t?; t
5
(CHAP. 1)
Suppose 8
E
C m ( f ) i s such t h a t 8 ( t ) = 1 f o r
1/2, and such t h a t e(t) = 0 f o r t t 1.
- 'K[T))
.
v,[x)
=
U(C)
= < T(x), cp,[x)
e(x.T
> ,
We p u t
Show t h a t
and deduce from t h i s t h e upper bound
(2.9.1).
Conversely, l e t U be an e n t i r e f u n c t i o n i n C"
b)
T = rl[U
i n g (2.9.1). We p u t
sequence (see (l.l.l)),w i t h
)
I R" supp p
.
C
Let p
{ 1x1
p u t Tj = p j
*
T. Using Theorem 2 . 6 , show t h a t
c { x
I
dist.(x,
HOMOGENEOUS DISTRIBUTIONS:
t h e two open s e t s 7Rn o r E n \
let
T
x
morphism
E
<1 3 .
We
Deduce from t h i s t h a t
n3
1: 0 1.
For
We denote by R one o f
T E &[n)
and A > 0 ,
be given by t r a n s p o r t i n g T under t h e d i f f e o -
.&((n)
x
geneous o f degree
x
be a r e g u l a r i s i n g
J
Tj supp T c K .
10.2
.
K) 5 1 ]
j
satisfy-
c---)
m
(with m
.
E
x
E c
We say t h a t T i s homo-
) i n R i f TA = Am T f o r a l l
> 0.
a)
Show t h a t i f T i s homogeneous of degree
i s homogeneous of degree m - la1
Suppose
T
E S*[R")
, then
Da T
.
Show t h a t Da 6 i s homogeneous of degree
b)
m
-
n
-
la1 i n IR"
i s homogeous of degree
m.
I
Show
(SEC. 1 0 ) A
-
t h a t T i s homogeneous o f degree
c)
73
ADDITIONAL N O T E S
T E
Suppose we have
- m
n
(Rn)
A?'
i n IRn.
T
such t h a t
lRn\CO
m
is C
i
and homogeneous.
IR"\lo 3
is
Cm [for
T = UT
function Log 1x1. degree
-1
Suppose p v
-
E
X
1, and t h a t x pv
5 > 0 , and t o i w f o r 6
.I
( * ) 10.3
1 = X
identical t o 1 in the
(1
- U)T
]
i s t h e d eri va t iv e of t h e
1 is X
odd and homogeneous o f
Deduce from t h i s t h a t t h e
1.
I
f o r a > 0 we have
?U
U c--t Tu = e--CY x2/2
I
.
6U4'2Ie'Re
(r
2
0
,
CY
E
E S'(R1 Show t h a t
Show t h a t t h i s
f 0 , by
Deduce from t h i s t h a t for
2
X
for
We r e c a l l t h a t
> 0 and continuous f o r Re c1 2 0 .
e q u a l i t y remains t r u e f o r
a-'.
i T
--2 i1n (pv; 1 - in 6) .
2 FOURIER TRANSFORM OF e ihx '2.
c1
.
< 0 , and t h a t
F l y
holmorphic f o r Re
3
and t h a t
A is t h e f u n c t i o n e q u a l t o X
Show t h a t t h e mapping
main branch of
+
Jl(CR)
Show t h a t pv
Fourier t r a n s f o r m of pv
(10.2.1
C:(IRn)
c1 E
neighbourhood of 0 , w r i t e
d)
T E 8'[R")
Show t h a t
1R \ {O}, t h e F o u r i e r t r a n s f o r m of ei X x /2 is
taking the
is
74
(CF99. 1)
DISTRIBUTIONS AND OPERATORS
10.4
FUNDAMENTAL SOLUTION OF THE CAUCHY-RIEMA"
We c o n s i d e r i n IRd
t h e Cauchy-Riemann o p e r a t o r
2 Let Q be a r e g u l a r bounded open s u b s e t of IR
a)
OPERATOR:
.
By app-
l y i n g S t o k e s ' formula, show t h a t
for
u,
(+y
,
E C'(5)
an
z = x + iy,
b e i n g given i t s canon-
i c a l o r i e n t a t i o n as t h e boundary of a compact s e t .
b)
Suppose we have zo
u(z) =J& *
E
Q.
By a p p l y i n g (10.4.1)for
a f t e r having removed from Q a d i s k w i t h c e n t r e
zn and w i t h r a d i u s € , t h e n by l e t t i n g
E
t e n d t o 0 , show t h a t
dx dy
0
c)
Deduce from t h i s t h a t
of t h e operator
a az
.
-
1 is E = nz
a fundamental s o l u t i o n
6
d)
By o b s e r v i n g t h a t E i s homogeneous of degree
10.2.,b ) , show t h a t
10.5
-
1 (see
=
FUNDAMENTAL SOLUTION OF THE LAPLACE OPERATOR:
c o n s i d e r i n IRn t h e Laplace o p e r a t o r
A=k-a2
We
,
j=i ax:
a)
L e t R be a r e g u l a r open s u b s e t o f I R n .
Suppose we
ADDITIONAL NOTES
(SEC. 10)
have
u, v E C’[?i]
, at
75
l e a s t one o f t h e two b e i n g assumed t o
have compact s u p p o r t .
By applying S t o k e s ’ formula t o t h e ( n
-
1) - fo r m
show t h a t
Thus, deduce t h a t
n
=
grad u. grad v dx
an
where N i s t h e i n t e r i o r u n i t normal t o
and do i s t h e Riemannian
an (IRn b e i n g equipped w i t h i t s c a n o n i c a l E u c l i d i a n
density on structure).
Deduce from t h i s t h a t
b)
(10.5.1) f o r show t h a t
v
Suppose t h a t n 2 3 and
n= {
x E
R”
I
1x1
>
c]
E
4f]
,
.
By a p p l y i n g
and by p u t t i n g r = 1x1,
76
(CHAP. 1)
DISTRIBUTIONS AND OPERATORS
where un-l
is t h e a r e a o f t h e w i t sphere of l R n .
Deduce from t h i s t h a t
E = (2
- 1nlurr-l
~
x
l
- i s~ a
fundamental s o l u t i o n o f A when n 2 3.
c)
I n t h e c a s e n = 2 , proceed i n a similar manner by
-n+2 r e p l a c i n g 1x1 by Log 1x1 , and show t h a t
is a fundamental s o l u t i o n of A d)
Show t h a t , for n
2
Log 1x1
.
-
= - 1
3, we have
(use
I %I2
10.2, b )
10.6
1
E =
FUNDAMENTAL SOLUTION OF THE HEAT OPERATOR:
c o n s i d e r i n IR
n +1
We
the heat operator
W e put
a)
(with
T = 0
defines a distribution
+ iy, E E
y
<0
&"R"+')
fixed;
,
cp
E
f+l)
and t h a t t h i s d i s t r i b -
u t i o n does n o t depend on t h e number y < 0 chosen.
Show t h a t E i s a fundamental s o l u t i o n o f P , and t h a t
supp E
Let a
b)
transform o f
-
77
ADDITIONAL NOTES
(SEC. 10)
.
E
c { (t,
E €I"+'
I
t
C b e such t h a t Re a > 0.
e-at
t -y(t) 0
c)
x)
2
0 }
,
Show t h a t t h e F o u r i e r
is
1 a+io
Deduce from t h i s t h a t
and, by u s i n g 1 0 . 3 , t h a t
i . e . t h a t E is t h e ( l o c a l l y i n t e g r a b l e ) function
10.7
FUNDAMENTAL SOLUTION OF THE SCHRVDINGER OPERATOR.
'We c o n s i d e r i n En+'
t h e Schradinger operator
. a ) Show t h a t t h e formula
( where fixed;
cp E B(A"+')
?
=
Q
- iy
,y
c0
) defines a distribution E E S ( R w 1 )
and t h a t t h i s d i s t r i b u t i o n does n o t depend on t h e number y > 0
78
DISTRIBUTIONS AND OPERATORS
chosen.
Show t h a t E i s a fundamental s o l u t i o n o f P, and t h a t supp
b)
(CHAP. 1)
E c { ( t ,x ) E Rn+'
I
t
2
0 ]
.
By proceeding as i n t h e p r e c e d i n g e x e r c i s e , show t h a t
and t h a t
(it w i l l be noted t h a t t h e function
10.8
FUNDAMENTAL SOLUTION OF THE WAVE OPERATOR:
We
c o n s i d e r i n Bn+l t h e wave o p e r a t o r
a)
Show t h a t t h e formula
t h i s d i s t r i b u t i o n does n o t depend on t h e number
y > 0 chosen.
Show t h a t E i s a fundamental s o l u t i o n o f P and t h a t
SUPP
Et
where
c
E C
[t, x)
E R"
By o b s e r v i n g t h a t P(7,
b)
79
ADDITIONAL NOTES
(SEC. 10)
Show t h a t
"'w
=
Et(x)
t
2
0
3
.
5) =
*
E)(;l tn-?
=
I
f o r t > 0 , and t h e r e f o r e
that
A
Show t h a t E, e x t e n d s i n t o t h e e n t i r e f u n c t i o n
c )
Show t h a t I z
lU(c)I SUPP
5
El c
1 1
C 1.l
= 15 I and t h a t / B 1 5
elT1 and 5 13.
IqI.
Thus deduce t h a t
t h e n b y u s i n g 10.1, t h a t
DISTRIBUTIONS AND OPERATORS
80
(CHAP. 1)
By u s i n g (10.8.1),i n f e r t h a t
(10
a 2.1
Supp
E
{ (t, x )
c
I
E R"+'
1x1 I t
.
3
F u r t h e r p r o p e r t i e s of E w i l l be seen i n E x e r c i s e 1 0 . 3 o f Chapter
111.
10.9 f :
R"
-
R
u = f
*
We d e f i n e u = v
v E >([R") 0
# 0 when f ( x ) = 0.
, the
E
.
We s h a l l now
v
under
E C"(P( \ 0)
i n v e r s e image of f
f on t h e open s u b s e t
a l o c a l diffeomorphism
Zn,
v
be such t h a t
neighbourhood of a p o i n t xo
f(x) into
Let
be a Cm mapping such t h a t f ' ( x )
v E &'((R)
Let define
INVERSE IMAGE OF A DISTRIBUTION:
-1
( Rn
\
0)
.
f
.
In t h e
IRn such t h a t f ( x ) = 0 , we consider 0
5
x cc x
x(x)
transporting
of 33"
and we p u t
* 0
Show t h a t t h e d i s t r i b u t i o n s u, u
0
piece together i n t o a distribShow t h a t i f a sequence
u t i o n on l R n , which we denote by f* v.
v
J
E p ( R ) fl C"(R\O)
v -
J
0
in
cw((R\O)
,
m
: vJ-0
i s such t h a t then
f*
v)
=
v
J
-
in
0
&((R)
&(f)
in
Show t h a t (103.1
,I
SUPP(f*
f-l(s"pp
v)
,
, ,
(SEC. 1 0 )
In t h e c a s e where
ADDITIONAL NOTES
v
i s t h e D i r a c d i s t r i b u t i o n i n IR, show
t
that f
v coincides with t h e d i s t r i b u t i o n 6 ( f ) defined i n
Section 7 . 2 .
We s h a l l g i v e i n Chapter V I I I a more g e n e r a l d e f i n i t i o n o f the i n v e r s e image.
81
This Page Intentionally Left Blank
CHAPTER 2.
SOBOLEV SPACES AND A P P L I C A T I O N S
1.
DIRICHLET'S PRINCIPLE
Sobolev spaces a r e spaces of d i s t r i b u t i o n s corresponding
t o a c e r t a i n degree of r e g u l a r i t y and t h e y have, i n a d d i t i o n , a H i l b e r t space s t r u c t u r e .
They a r i s e i n a n a t u r a l manner i n
the study of boundary-value problems f o r d i f f e r e n t i a l o p e r a t o r s , as t h e simple example which f o l l o w s w i l l show.
We l e t X be a
r e g u l a r bounded open s u b s e t of IRn and we c o n s i d e r t h e f o l l o w i n g D i r i c h l e t problem:
Find
where
g
I;
such t h a t Au - u = 0 i n X ,
u / a x = in We can f o r example assume g
i s given.
begin by s e e k i n g u
E
C2(x).
E
C
2
(ax)
and
P h y s i c a l c o n s i d e r a t i o n s l e a d us t o
consider a q u a n t i t y which w e c a l l t h e energy o f u, d e f i n e d by
Then, we have
83
84
SOBOLEV SPACES AND APPLICATIONS
The above Dirichlet problem admits
DIRICHLET’S PRINCIPLE :
a t most one solution.
u i s a solution, i t minimises the
If
energy E over a l l the functions equal t o g
I n f a c t , suppose we have v Let h = v E(v)
-
- u.
i,
2 Re
E
2 C ( X ) such t h a t v
( g r a d u (x)
U(X)
From Green’s formula
-
(where
au denotes an
normal t o
ax),
and s i n c e h
I ax
I ax
= 0.
-
we s e e t h a t E ( v ) = 0.
m ) d x
I x A u . hdx
t h e d e r i v a t i v e of
-
u
+
E(h)
.
i, 2 -
+
- , h d x
along t h e e x t e r i o r u ni t
E(u) = E(h) since A u
-
u = 0
Now E ( h ) i s always p o s i t i v e o r z e r o , and
i s zero i f and only i f h
-
=
= g.
I ax
-.
+
g r a d u.grad h dx
on the boundary.
u is a solution:
We have, i f
=
E(u)
(CHAP. 2 )
Thus, E ( u )
i s c o n s t a n t , i . e . zero s i n c e
h
5 E ( v ) and t h e e q u a l i t y
i s equivalent
t o v = u.
O f c o u r s e , t h e p r e c e d i n g argument does not prove t h e e x i s t -
ence of a s o l u t i o n and i n f a c t t h e r e does n o t always e x i s t a solution
u
in
c2(?).
It i s t h e r e f o r e n e c e s s a r y t o seek 2 -
i n a space l a r g e r t h a n C ( X ) .
We observe t h a t E(u) i s t h e
norm a s s o c i a t e d w i t h t h e p r e h i l b e r t s t r u c t u r e on C 2 ( ? ) by t h e i n n e r product (u, v),
=
u
( (grad u
.v + grad
)u ;,
dx
defined
.
(SEC. 2 )
85
HS(IRn) AND H s o c ( X )
SPACES
The D i r i c h l e t p r i n c i p l e l e a d s us t o c o n s i d e r t h e completed
1 H i l b e r t s p a c e , which we denote by H ( X ) , and which, as we s h a l l
1-
see, c o i n c i d e s w i t h t h e Sobolev space H ( X )
H”(K~
=
{ u E L ~ ( x )I
aj
E L*(x)
u
1-s jr n
for
>
,
*
We s h a l l a l s o s e e t h a t we can d e f i n e i n a n a t u r a l manner t h e trace v
=
I ax
ax
of v on
{ v E H”(Y)
I
when v vlax
1-
H ( X ) , and t h a t t h e a f f i n e v a r i e t y
E
g
=
}
i s closed i n
$(XI.
nus,
by analogy w i t h t h e D i r i c h l e t p r i n c i p l e , we can c o n s i d e r t h e orthogonal p r o j e c t i o n
4
that
u
u of 0 onto
c;
we s h a l l show i n s e c t i o n
1i s i n f a c t t h e unique element of H ( X ) which i s =
au-u
0
in
u a x = 9
in
such t h a t
2.
THE SPACES
For
E
2
L
s
E
IR, we denote by HS (IRn ) the
u E S’(Rn)
space of d i s t r i b u t i o n s
I c12)s’z
.
HSDn)
DEFINITION 2.1 :
[I+
X
n
(R )
.
such t h a t
We equip H S ( E n ) w ith th e in n e r
product
and w i t h t h e associated norm
11
1 Is
defined by
86
SOBOLEV SPACES AND APPLICATIONS
-
We c o n s i d e r t h e t o p o l o g i c a l
We s h a l l f r e q u e n t l y p u t H S m n ) = H S .
: 88
v e c t o r isomorphism
(u, ")*
=
.
As u
81
d e f i n e d by we have
s
A
(CHAP. 2 )
v dx
from Theorem 2 . 5 of Chapter I .
HS i s t h u s t h e H i l b e r t space o b t a i n e d by t r a n s p o r t i n g t h e 2 n H i l b e r t space L (B ) by t h e isomorphism icular H
o
2 n = L (IR ) .
9 E IR
Let s,,
R S ;we
have i n p a r t -
be such t h a t s, 5
9 ;it
S
i s immediately c l e a r t h a t HS2 c H S
S
into H
H
and t h a t t h e i n j e c t i o n o f
i s continuous.
=
We put H+
fl
H'
,
s E R H4
=
u
,
8 c Hk
; we have
HS
and, from ( 2 . 8 . 1 ) o f
s E R Chapter I ,
.
&' c H-
upper bound
cy
15
I
I
i s continuous from H
S
From Theorem 2 . 1 0 o f Chapter I and t h e
I I I Icy/
,
a differentiation operator
i n t o Hs-lal.
Da
2 We have seen t h a t Ho = L ;
more g e n e r a l l y we have
For rn
PROPOSITION 2 . 2 :
tributions u
c
B([Rn) such
multi-index a s a t i s f y i n g
E
m y H" is
that
Icy1
D'
rn
S
the space of d i s -
u E L2(R")
,
for any
Moreover, the inner
product i n Hm is equivaZent t o the inner product
PROOF: we'have
-Da u
Suppose E
L
2
u E .B'
; from Theorem 2.10 o f Chapter I ,
i f and o n l y i f
ga ;I
E
L2
,
and i n t h i s
(SEC. 2 )
HS(Bn)
SPACES
AND HS (X) loc
87
e a s i l y be shown t h a t t h e r e e x i s t p o s i t i v e c o n s t a n t s C
such
C
1' 2
hence P r o p o s i t i o n 2 . 2 .
PROPOSITION 2.3
3
continuous;
HS; f o r h
E
E
HS
9
T~ u
4
u
I/f'-,
"11,
=
E
we denote by
IRn, x
-h
CX ,I
HS
In the ease where h = (0,
PROOF:
8
.. .
j '
h
0, h j , 0,
tends t o
-+
0
.. ., 0 ) a.
J
A
S
h
u the
Then
. the d i f f e r -
u in HS-l when
We now show t h a t t h e i n j e c t i o n o f
continuous; s i n c e onto
when
T h U - U
entiaZ quotient
T
I/ul/s
and in
is
c ,HS
the i n j e c t i o n of HS i n t o gr is
u under the t r a n s l a t i o n
image of h'
i s dense i n HS;
Suppose u
continuous.
8
The i n j e c t i o n
:
8 i n t o HS i s
is a t o p o l o g i c a l v e c t o r isomorphism o f
and o f HS o n t o Ho, it i s s u f f i c i e n t t o c o n s i d e r t h e
case s = 0 .
For
Cp
E8
we have f o r example r
6
88
(CHAP. 2)
SOBOLEV SPACES AND APPLICATIONS
so that
which proves t h e c o n t i n u i t y which we seek.
The i n j e c t i o n o f HS i n t o %' i s continuous; in f a c t , as p r e v i o u s l y we reduce t o c o n s i d e r i n g t h e c a s e s = 0 and we observe t h a t f o r u
I<
'1
P
>I
I'I,
5
Ho and
E
8
-D
HS
8
,we have
16 is dense i n H S
~ " ( 8 )= 8
i n L2, we s e e t h a t
3
E
IITllo
We now prove t h a t
t h e d e n s i t y of
Cp
in
s
Since
i s dense i n A
s
i s dense
-s 2
(L ) = HS
and t h e c o n t i n u i t y o f t h e i n j e c t i o n
t h e n imply t h e d e n s i t y o f
t o 0 when h+O
.
8
i n HS.
from t h e dominated convergence theorem.
Similarly, we obtain
;
SPACES
(SEC. 2 )
89
H S ( B n ) AND H S o c ( X )
Therefore t h e c o n c l u s i o n a g a i n f o l l o w s from t h e dominated convergence theorem.
F be a cZosed subset of Bn; we
Let
DEFINITION 2 . 4 :
Himn) the subspace of
denote by
HS formed by the elements with
support i n F. Since t h e i n j e c t i o n subspace o f H
S
Hs-&'
is c o n t i n u o u s ,
and we endow it w i t h t h e induced H i l b e r t s t r u c t u r e .
Suppose we have k
THEOREM 2 . 5 ( S o b o l e v ) : s
E
Himn) is a c l o s e d
IR such t h a t s
> n/2
+ k.
Then HS
k
c C (IR
n
E
2J
and
) , with continuous
injection.
By d i f f e r e n t i a t i o n , we reduce t o t h e c a s e where
PROOF: If u
k = 0.
E
HS, we have 6 = f g , where
f(p) = (I+ 1512)s'2 f
E
L
2
and g
E
:(I) , g(5) =
L2 when s
(I+ \~I')-"'fWe
n/2.
Schwartz's i n e q u a l i t y t h e n p r o v e s t h a t
with
2 cs
=
have
ds
.
Is[5)\ 8% = cs
Hence u
E
Co(Bn), w i t h
\\u\\s
9
SOBOLEV SPACES AND APPLICATIONS
90
THEOREM 2.6
The muZtipZication
:
bizinear and continuous from
PROOF: and t h a t
For
A Q U
Cp
E 8
S
and u
(Q, UJ
(CHAP. 2)
i n t o HS.
x
H5
E
HS, w e know t h a t -n
i s t h e Cm f u n c t i o n
(zR)
A
Cp
is
-'pu
* ;;
We have
Q u
.
S'
We t h u s
have
Now, we have: ( 2 . 6 . 2 . ) LEMMA ( P e e t r e ' s I n e q u a l i t y ) :
t h i s proves P e e t r e ' s i n e q u a l i t y f o r s t 0 , ( E b e i n g r e p l a c e d by
6 + TI.).
The c a s e s 5 0 can be deduced from t h i s by permuting
t h e r o l e s of
5,
rl.
( 2 . 6 . 2 . ) and ( 2 . 6 . 1 . ) show t h a t
SPACES H ' ( I R ~ ) and H'
(SEC. 2)
l o c (x)
91
The c o n c l u s i o n t h e n f o l l o w s from ( 2 . 1 . 2 ) of Chapter I .
We have seen t h a t
3 i s dense i n HS;
we s h a l l now make t h i s
statement more p r e c i s e by g i v i n g a c o n s t r u c t i v e procedure of approximation by t r u n c a t i o n and r e g u l a r i s a t i o n .
Let
THEOREM 2 . 7 : P ( x j dx
=
1
.
neighbourhood o f 0 .
Then
p
k
*
u and
xk
Let For
p E
x
E
k
u tend t o
c:(R") c:(IR
be such t h a t
1 be such t h a t
an i n t e g e r
u
> 0, we
i n HS when k
+
x
= 1 i n the
put:
+
m
.
SOBOLEV SPACES AND APPLICATIONS
92
(CHAP. 2)
from t h e dominated convergence theorem.
From Theorem 2.6, we have
and t h u s
For Cp 6
3
> 0 , t h e d e n s i t y of
E
i’iu
b such t h a t
we have Xk CP
=
- (PI\,
i n HS shows t h a t t h e r e e x i s t s
.
<
For
sufficiently large ,
k
ZP, and t h u s
We now proceed t o t h e s t u d y o f t h e d u a l o f HS.
S i n c e HS i s
a H i l b e r t s p a c e , we can i d e n t i f y it w i t h i t s d u a l by means o f t h e b i l i n e a r form on H for s on
#
s
x HS d e f i n e d by
HS x 8
8
d e f i n e d by u, v
-
H8 i s
-<
u, v
to 8
,
>
;
therefore, we
We have s e e n t h a t t h e i n j e c t -
o f dense image and i s continuous;
inuous l i n e a r form R on H
lls
But,
0 , t h i s b i l i n e a r form does n o t e x t e n d t h e b i l i n e a r form
s h a l l n o t use t h i s i d e n t i f i c a t i o n . ion
-v ) ~.
u, v -(u,
S
and we have
t h u s a cont-
i s c h a r a c t e r i s e d by i t s r e s t r i c t i o n
IS
E
S’
.
me i n j e c t i o n
(SEC. 2 )
SPACES H s ( l R n )
A-lls
o f (Hs)' i n t o 8'
a subspace of 8' exactly H-'.
allows us t o i d e n t i f y ( H s ) ' w i t h
; we s h a l l now s e e t h a t t h i s subspace i s
More p r e c i s e l y , we have:
The b i l i n e a r form on
THEOREM 2 . 8 : UP V
-<
u, v
on HS x H-'
u, v
u, v >
-1
HS, v
E
defined by
;(I G() 5 ) 85
, which
we
. I<
This bizinear form s a t i s f i e s E
8 x 8
> extends i n t o the continuous b i l i n e a r form
defined by
again denote by
u
93
and H Y o c ( X )
I llulls l \ v \ L s
u, v
for
and allows us t o i d e n t i f y the d u a l of HS with
H-',
the normed space H-'. PROOF:
If u
HS and v
E
E
H-',
we have
from Schwarz's i n e q u a l i t y .
Thus i f v
E
H-',
by A ( u ) =
<
U,
R
E
(H')';
we d e f i n e t h e continuous l i n e a r form R on HS v
> for
For u E 8
f
E
E
S
H ,
Conversely, suppose
from R i e s z ' s theorem, t h e r e e x i s t s a unique f
such t h a t k ( u ) = ( f ,
where v
all u
E
8'
. we
for all u
E
and we have l l k l l =
obtain
is d e f i n e d by
HS, we have v
E 8 ,
H-' w i t h
^v
= [I+
E
1512 15
?
;
since
H
S
11 f i l s
94
SOBOLEV SPACES AND APPLICATIONS
COROLLARY 2 . 9 :
For
m
E
,
N
HWm i s the space of d i s t -
r i b u t i o n s u of the f o m ( 2 .%I.j
The norm i n H-m i s equivaZent t o the norm
*
defined by the fomuZa 1u]-,2
F
Inf
Iul-,, , the
+--L
llfwlf
lower
Icy5 m
bomd being taken for a1 I possibZe decompositions ( 2 . 9 . 1 ) of u
.
This Zower bomd i s attained.
PROOF:
We e q u i p
H" w i t h
t h e inner product ( 2 . 2 . 1 ) .
H-m,
t h e norm
u
The d u a l of
Iu],
H"
that u
E
H-m
((V,
,
-m
equiv-
R i e s z ' s theorem shows t h e r e f o r e
i f and o n l y i f t h e r e e x i s t s v
< u , cp > =
(2.9.2.)
i s t h e n i d e n t i f i e d with u+-lu]
and d e f i n e s on H-m a norm denoted by
a l e n t t o t h e u s u a l norm o f H-m.
a s s o c i a t e d with
E
H"
v)Im f o r a l l
such t h a t
cp E 8
,
acy fa
I
and it a l s o shows t h a t we t h e n have
IuLm
(2.9.3.)
=
llvllm
see t h a t ( 2 . 9 . 2 ) i s equivalent t o u
f,
= (-1 j I
a
1 a@ -V E
E
F
la
with
m
L2 (Rn ), and t h a t ( 2 . 9 . 3 ) i s e q u i v a l e n t . t o
SPACES H S ( I R n ) AND H S o c ( X )
(SEC. 2 )
95
cp E 8
Conversely, i f (2.9.1) i s s a t i s f i e d , w e have, f o r
THEOREM 2 . 1 0 :
n
-
PROOF:
Let ( \ )
Suppose we have s , t
(Reillich).
s > t , and t h a t K is a compact subset of Bn s
H ~ ( R)
t
n
1is
.
E.
:
IR with
Then the i n j e c t i o n
compact.
be a sequence i n H
S
k
such t h a t l/uklls < 1.
We
now show t h a t we can e x t r a c t from t h i s a subsequence which i s convergent i n Ht.
Suppose we have
the neighbourhood of K .
u = TU
,
gu(
= (2~)’~
For u
E
*z ,
cp
E dRn)
equal t o 1 i n
S
HK, we have Da
= ( 2 ~-n) Du Cp A
*2 .
51
t o g e t h e r w i t h t h e e x i s t e n c e of a c o n s t a n t Ca such t h a t
As
(CHAP. 2)
SOBOLEV SPACES AND APPLICATIONS
96
We t h u s have:
(1
2 4 2
+ 151 )
IDcy
Uk[s)l
ccu
5
for a l l
k and a l l 5
E
IR",
a which shows t h a t t h e sequence (D uklk i s uniformly bounded on any compact s u b s e t o f l R n .
By a p p l y i n g t h i s r e s u l t f o r m u l t i -
i n d i c e s a such t h a t ( a (I 1, and by u s i n g A s c o l i ' s theorem, we s e e t h a t we can e x t r a c t from t h e sequence
%
a subsequence ( a l s o
denoted, w i t h an abuse o f n o t a t i o n , by uk ) such t h a t
(2.10.1)
\
converges uniformly on any compact s u b s e t of
We s h a l l now deduce from t h i s t h a t t h e subsequence u in H
t
.
Suppose R > 0 .
We have
where
f
Writing
we o b t a i n , s i n c e
t-s < 0
:
IIUk
-
2
Uallt
=
k
1,
W
n
.
converges
+
5
,
SPACES HS(iRn ) AND Hy,,(X)
(SEC. 2 )
97
so t h a t I2 can b e rendered a r b i t r a r i l y small i f R i s chosen s u f f iciently large.
For R f i x e d i n t h i s manner, I 1 can be rendered
a r b i t r a r i l y small i f k , R are s u f f i c i e n t l y l a r g e , by v i r t u e o f
(2.10.1).
Let sl, s, s2 be such t h a t s1 < s < s2.
PROPOSITION 2.11:
For any
E
> 0 , there e x i s t s
=
P
i . e . by p u t t i n g
a constant c E such t h a t
1
+ l5l2
, that
i.e. that
1 I S P
VS
+
-(Q-sl
) / ( ~ - 4s,-s
a
P
l/S2-S
By p u t t i n g
h = s
, this
i n e q u a l i t y can b e w r i t t e n
(CHAP. 2)
SOBOLEV SPACES AND APPLICATIONS
98
s -s
If ~p
2
1, ( 2 . 1 1 . 1 ) i s s a t i s f i e d s i n c e ( A p )
2 1 s -s
If Xp < 1, ( 2 . 1 1 . 1 ) i s s a t i s f i e d s i n c e ( X p )
2
1.
We now s t u d y d i s t r i b u t i o n s which behave l o c a l l y l i k e elements of
HS
. DEFINITION
suppose s u
E
HS
l 0 C
a'(X)
2.12 :
Let
x be an open stcbset
We denote by H ; o c ( X )
E 7R.
such t h a t
QU
E HS f o r a l l
of I R ~and
the space o f d i s t r i b u t i o n s Q
E
We equip
Cz(X),
( X ) with the topoZogy defined by the f a m i l y of semi-norms
u c ~ I ! ( p u ] /, ~ Q describing C E ( X )
.
I f U i s an open s u b s e t of X , t h e r e s t r i c t i o n o p e r a t o r i s e v i d e n t l y continuous from H
S
S
loc
(X) i n t o H l o c (U).
Furthermore,
we have :
PROPOSITION 2.13 :
( x ) into
i n j e c t i o n of
H~
s > n/2 + k
,
injection.
We have
loc
we have
i s a Frgchet space, and t h e
H;oc(X)
i s continuous.
P(X)
HS
loc
n
(X)
,
c Ck(X)
HYoC(X)
=
C"(X)
men with continuous
.
A differ-
s E R entiaZ operator of degree m with Cm c o e f f i c i e n t s i n
x
is
continuous from H'Joe (XI i n t o H~;:(X).
PROOF:
Let K . be an e x h a u s t i v e sequence of compact J
s u b s e t s of X and suppose we have Then t h e sequence o f semi-norms
cpj E u
c--,
f(X)
(IQj
equal t o 1 i n K u(ls s u f f i c e s t o
j'
99
SPACES HS(IRn) AND H!oc(X)
(SEC. 2 )
define t h e topology of H
S
loc(X): in fact, if
,
cp E c:(X)
we
have, f o r j s u f f i c i e n t l y l a r g e ,
wj
cp =
1k9.11,
and
=
lIwj ulIs
C
from Theorem 2 . 6 , C only depending on s and cp is metrizable.
E
all (Pu
k
f(X]
'p
E
, Wk
i s a Cauchy sequence i n H in H
IJ
S
,
%
.
has a l i m i t
and a f o r t i o r i i n
Thus u
E
HS
loc
u in
( X ) and
a', w i t h cpu
\
ulis
Thus H;oc(X)
be a Cauchy sequence i n H
(9
C:[x)
.
Let (\)
possesses a l i m i t
from t h i s t h a t
livj S
loc S
(X) : for
, thus
a' ; it
= u f o r all (9
S
+
follows
u i n Hl o c ( X ) .
The
other a s s e r t i o n s o f t h e p r o p o s i t i o n a r e immediate, by u t i l i s i n g i n p a r t i c u l a r Theorem 2.5.
DEFINITION 2.14 :
We denote by H:omp(X)
th e union of the
spaces H! when K describes the famiZy of compact subsets o f X , and we equip HEomp(X) w i t h t h e i n d u c t i v e l i m i t topology of th e topoZogies of th e HL.
THEOREM 2.15 :
The i n j e c t i o n s
c"(x)
C z ( X ) i s dense
-
in HS ( X ) and in H:omp(X). loc
H ; (X) ~ ~ and
-
C;(X)
Hs
(X)
COW
are continuous and of dense image and allow us t o i d e n t i f y th e duaZ of H Y o c ( X ) w i t h H-' ( X ) and the duaZ of HEOmp(X) w i t h c omp H;:c(X).
PROOF:
The c o n t i n u i t y of t h e i n j e c t i o n s f o l l o w s immed-
i a t e l y from t h e c o n t i n u i t y o f t h e i n j e c t i o n
R
- HS .
SOBOLEV SPACES AND APPLICATIONS
100
q?j a s i n t h e proof o f Theorem 2.13.
Suppose we have f u n c t i o n s If u
if p
S
Hloc(X),
E
it i s immediate t h a t
cp j u
--
i n HS ( X ) ; loc
u
i s a r e g u l a r i s i n g sequence, we have seen (Theorem 2 . 7 )
k
that
(CHAP. 2 )
pk
*
(cpj u)
-
i n H:oc(X).
t h e d e n s i t y o f C"(X) 0
-
P k * U
k
; hence
E
Similarly, i f
i n HS ( X ) when c omp
l-4
-+
k
i n HS when
cpj u
+=
S
HCD,,(X)
1
; hence t h e
4
i n HEomP(X).
d e n s i t y of C"(X) 0
We s h a l l now show t h a t each space HS ( X ) loc
, HLZmp(X)
is
i d e n t i f i e d w i t h t h e d u a l of t h e o t h e r by means of t h e b i l i n e a r form on HS
(X)
< h, v >
,
loc
x
HcZmp(X) where
x
E
d e f i n e d by
u, v-<
U, V
>
=I
i s an a r b i t r a r y f u n c t i o n having
C:(X)
t h e v a l u e 1 i n t h e neighbourhood o f supp v , and where t h e l a s t s n s e t of b r a c k e t s a r e t h o s e o f t h e p a i r i n g between H (B ) and F i r s t , it i s immediately c l e a r t h a t < u, v > depends
HdS(Bn).
This proves t h a t < u , v >
c o n t i n u o u s l y on u for v f i x e d . does n o t depend on t h e f u n c t i o n
x
chosen, and we t h e r e b y deduce
t h a t < u , v > depends c o n t i n u o u s l y on v f o r
u fixed.
Conversely, l e t R be a c o n t i n u o u s l i n e a r form on H and l e t
v
be t h e d i s t r i b u t i o n i n
< cp, v > = k(cp)
-
for all
cp E C"(X)
c'(X)d e f i n e d
ip
0
E
have
<
@,v >
= < Xip, v
=.
I
J(Xip)
.
x
m
E
C0 ( X )
Then
depends c o n t i n u o u s l y on
Cz(IRn) f o r t h e topology induced by H S @ i n ) .
v E H-'(R*)
by
; suppose we have
e q u a l t o 1 i n t h e neighbourhood of supp u.
S
loc ( X ) ,
In similar fashion,
We t h e r e f o r e
SPACES HS(IRn) AND H 7 0 c ( X )
(SEC. 2 )
i f R i s a continuous l i n e a r form on H
u E &[X)
distribution
jl
E
Cz(X).
If
rp E C(;X)
depends c o n t i n u o u s l y on by H-‘(IRn),
d e f i n e d by
thus
then @
E C:[R”]
p E HS(R”]
REMARK 2.16 :
S
c omp
101
( X ) , we c o n s i d e r t h e
u, jl > = a ( $ ) f o r a l l
@-<
rpu, @
> = A(cp5)
f o r t h e topology induced
and f i n a l l y
The preceding i d e n t i f i c a t i o n s a r e a l g e b r a i c .
We can show t h a t t h e y a r e a l s o t o p o l o g i c a l when we equip each dual w i t h t h e topology of uniform convergence on t h e bounded subdomains.
We can i n f e r from t h e t h e o r y of p s e u d o - d i f f e r e n t i a l ope r a t o r s t h e i n v a r i a n c e under diffeomorphism of t h e spaces
HYoc(X)
( s e e C o r o l l a r y 8.9 i n Chapter I V ; s e e a l s o E x e r c i s e
7.2 f o r a d i r e c t p r o o f ) :
I)
THEOREM 2.17 :
let
x:
Let U, U be two open subsets of IRn and
c
U
-t
U be a diffeomorphism.
Then
x+
d e f i n e s a topotN
ogicaZ vector isomorphism of HSl o c ( U ) onto HSl o c (U) as we21 as o f
This theorem allows us t o g e n e r a l i s e t h e d e f i n i t i o n of H Y o c ( X ) t o t h e c a s e where X i s a manifold:
DEFINITION 2.18 :
Let X be a mcmifold.
We denote by
SOBOLEV SPACES AND APPLICATIONS
102
HSoc(X) the space of d i s t r i b u t i o n s f o r any chart equip HS
loc
E
u
X :
(CHAP. 2 )
> (X)
such t h a t
-u
w
u
of X.
We
(X) with the topoZogy defined by the family of conc
tinuous semi-norms o f x w u i n H Y o c ( U ) ,
x
describing the famiZy
x.
of charts of
If U i s an open s u b s e t o f X , t h e o p e r a t o r of r e s t r i c t i o n t o U i s e v i d e n t l y continuous from H
S
loc
(x) i n t o H'l o c ( u ) .
Let X be a manifoZd.
THEOREM 2.19 :
( Sheaf property of HS (X)). loc
(i)
be such t h a t , f o r a22 x U of x in X w i t h
uIu
E
E
Let
HF(xJ
=
E >(X)
X, there e x i s t s a neighbourhood H:oc(U)
.
Then u
( i i ) Let K be a compact subset of X .
subspace
u
c u E H:~~(x) I
HS (X). loc Then the E
3 of IIuI(~,~
c K
supp u
S
Hloc(X) is hiZbertisabZe, and we denote by
an
I f the manifoZd X i s compact,
admissibZe norm in H;(X), S
we put H ~ ~ ~ = ( Hx'(x). ( i i i )Proposition 2.13
and Theorem 2.15 remain vaZid.
For the duaZity properties, we assume X t o be endowed with a positive density.
PROOF:
u E HYo,(Uj)
such t h a t indices J show t h a t
For ( i ) ,c o n s i d e r an a t l a s
.
X
U j'
-J J 5
U
of X
for any j belonging t o t h e s e t of
Let
x.: U
-
Xw u
E H:oc(U)
,
w
U
be an a r b i t r a r y c h a r t o f X ; we i . e . that
cp & u
E
H5
for
(SEC. 2 )
HSDn)
SPACES
me
r~ E C;[U~
n uj),
X(U
c o n s t i t u t e an open c o v e r i n g
c . )
of U; t h u s t h e r e e x i s t s a f i n i t e s u b s e t F o f J such t h a t
u
c
SUPP
x(u n u j ) ,
Let t h e r e be f u n c t i o n s
j E F
n uj))
E C:(X(U
'pj
.
j E F
i s transported i n t o a function $
j
, cpj cp E
X"
0
n UJ.I),
f c"(x.(U O
i s transported i n t o $
j
X
J
J*
y j y )&
F, hence
Q
x,
E HS
.
n
C:(X(U
U.
Ujj)
and
which belongs t o
u
Theorem 2.17 t h u s shows t h a t y . cp X*
HS by h y p o t h e s i s . E
in the
=
We have Q X, u
By t h e c o o r d i n a t e t r a n s f o r m a t i o n X j
for a l l j
1
j E F
neighbourhood of supp cp
yj CP X , u
=
'pj
such t h a t
J
E HS
LJ
Furthermore, we s e e
t h a t , f o r cp f i x e d , t h e r e e x i s t s a c o n s t a n t C such t h a t , f o r a l l
u
E
HS (X), we have loc
Thus
(2.19.1.) If
of
x.J
u i n HS
loc
(j E J)
Xj
i s an atZas of X , the semi-norms
(Uj) are s u f f i c i e n t t o define the topoZogy o f
' s
Hloc(X). W e now proceed t o t h e proof of ( i i )and ( i i i ) .
We can c l e a r l y S
choose J t o be countable i n (2.19.1),which proves t h a t Hloc(X)
i s metrizable.
Let ( \ )
be a Cauchy sequence i n H
u
X : U+U
any c h a r t s
-
(U) , therefore Hloc
of X ,
x \
x* \
S
loc( X ) ; for
i s a Cauchy sequence i n d
h a s a l i m i t u i n HS (U); i f X lot
x'
is
another c h a r t o f X, we can s e e by p r o c e e d i n g ' t o t h e l i m i t t h a t
ux, = [ X I
o
)(-I)* ux ; t h u s t h e r e e x i s t s u
E
8 (X) \
U
X
= X+ u for any c h a r t
x
of X.
We have
such t h a t
SOBOLEV SPACES AND APPLICATIONS
104 i n HS
Uk-
loc
Therefore HS
(X).
loc
U
-
cy
U
K c
(j E F) such t h a t
of X
'pj E C I ( U j )
For
i n t h e neighbourhood of K .
I=
j- E ~ ' j '
X j + ( ~ j u)-
loc ( X ) , and
u
Uj
.
vj
= 1
j E F
j
Let t h e r e be f u n c t i o n s
u =
S
L e t t h e r e be a f i n i t e f a m i l y o f c h a r t s
t h e r e f o r e complete. :
( X ) i s a Fr6chet s p a c e .
i s closed i n H
L e t K be a compact s u b s e t o f X ; H;(X)
xj
(CHAP. 2 )
and u
-f
0 in H
9
0
such t h a t
E HE(X),
u S
loc
we have
( X ) i f and only i f
i n HS f o r every j
F: t h i s i s c l e a r l y
E
n e c e s s a r y , and it i s s u f f i c i e n t from (2.19.1)by c o n s i d e r i n g
an a t l a s of X formed by t h e union of
x j and
complement of t h e neighbourhood of supp
an a t l a s o f t h e
qj.
Thus t h e topology
i n H s ( X ) can be d e f i n e d by t h e norm a s s o c i a t e d w i t h t h e i n n e r
K
product
We l e a v e t o t h e r e a d e r t h e t a s k o f g e n e r a l i s i n g t h e o t h e r p r o p e r t i e s s t a t e d i n P r o p o s i t i o n 2.13
3.
THE SPACES HS ( 2 )
and Theorem 2 . 1 5 .
AND H s o c ( ? ) . n
L e t X be a r e g u l a r open s u b s e t of IR Chapter I ) and l e t s E IR
H5(Rn)
-
&(X)
.
(see Definition 9.1,
The r e s t r i c t i o n o p e r a t o r
1":
d e f i n e d by r u = uIx, h a s f o r i t s k e r n e l
S
t h e c l o s e d subspace H @in) o f H S ( B n ) ; we t h u s have an isoIRL1\X morphism of HS / HS onto I m r . we equip H' / H' with the RT1\X Rn\X H i l b e r t - q u o t i e n t s t r u c t u r e , which t h e p r e c e d i n g isomorphism allows us t o t r a n s p o r t i n t o a H i l b e r t s t r u c t u r e on I m r .
This
leads t o :
Let X be a regular open subset of Bn
DEFINITION 3.1 : and l e t s it E
E
s'(x)
VQ denote by H s ( ? )
1R.
a h i t t i n g m extension
with the norn;
11~11,
il~ll,,
= Inf
the space of d i s t r i b u t i o n s
Z in
we equip
H'(IR~).
where
H'IX)
describes the famiZy
of extensions of u i n H ' (I$).
Tne decomposition i n t o an o r t h o g o n a l d i r e c t . sum
H'(R")
H~
=
(R")
8
R"\ X shows t h a t , for a l l
u E Hsfl],
e x t e n s i o n U o f u such t h a t
we have
I/uIls
=
II~ll,
U
'
[Hs (R") Rn\ X
)'
t h e r e e x i s t s a unique
(HS [I?"))' , and also t h a t Rn\X 2 We have H o ( ? ) = L ( X ) and, more
E
generally:
PROPOSITION 3.2 :
Suppose we have m
i s the space of d i s t r i b u t i o n s
The norm i n H-m(?) defined b y
u E P(X)
E
W.
Then H-m(Z)
of the form
i s equivalent t o the norm
u C-- I u
(CHAP. 2 )
SOBOLEV SPACES AND APPLICATIONS
106
where the lower bound i s taken f o r a l l the decompositions ( 3 . 2 . 1 ) of u.
-
PROOF:
Suppose we have
F
u
a*(fE)
E fi (X)
of t h e form ( 3 . 2 . 1 ) .
f:
E
obtainecl by e x t e n d i n g f a by 0 o u t s i d e o f
x.
u
We p u t
=
Icy
where
5.m
2 n L (R )
is
The d i s t r i b u t i o n
Y
u i s an e x t e n s i o n of u and, from C o r o l l a r y 2 . 9 . , w e have E- H-m(Rn)
with
Conversely, suppose u that
Fa
/\ulLm= \/Ul\-rn 2
E
r
E
H-m(%)
j
we c o n s i d e r
I,Fm IiF,lo .
H-m(IRn)
such
2
-
IUIErn
We s h a l l now s e e t h a t H1(?) eared i n the introduction.
PROPOSITION 3 . 3 :
Then, f o r
E
, From C o r o l l a r y 2 . 9 , t h e r e e x i s t s
n
L (IR ) such t h a t
compact.
U
~f
we p u t
is indeed t h e space which app-
More g e n e r a l l y :
We assume t h a t X = X?: rn E N
,
HmF)
d i s t r i b u t i o n s u E b ( X ) such t h a t
aty
or t h a t
ax
is
i s the space of 2 u E L (X) for lall m.
The inner product i n
e(?) i s equivaZent
t o the inner product
I
(3.3.1
.I
. . )
For t h e moment we s h a l l denote by
PROOF:
of t h e u
8‘(XI such
E
that
acy u
with t h e i n n e r p r o d u c t ( 3 . 3 . 1 ) .
8(?) t h e space
2
E L (x) f o r J c c 1s m , equipped From P r o p o s i t i o n 2 . 2 , we have
Hmg.‘) C PflT) w i t h c o n t i n u o u s i n j e c t i o n . Conversely, we Frnfl] C HmF)w i t h continuous i n j e c t i o n . can show t h a t This amounts t o p r o v i n g t h a t t h e r e e x i s t s a continuous l i n e a r m n e x t e n s i o n o p e r a t o r from Hm(x) i n t o H (B ) . II)
By l o c a l c h a r t s and
a p a r t i t i o n of u n i t y , it can e a s i l y be shown t h a t it i s s u f f n i c i e n t t o c o n s t r u c t such an o p e r a t o r i n t h e c a s e X =B+. We c o n s i d e r t h e continuous l i n e a r e x t e n s i o n o p e r a t o r
p :
a$) -B[R”)
c o n s t r u c t e d i n t h e proof of P r o p o s i t i o n
9 . 2 Chapter I , or even s i m p l e r , t h e e x t e n s i o n p c o r r e s p o n d i n g t o :
I t i s immediately e v i d e n t t h a t p i s continuous w i t h v a l u e s i n
?(Bn) for t h e t o p o l o g y induced by
Vn-n
H [R+) on
m+).
In
o r d e r t o conclude t h e p r o o f , it i s t h u s s u f f i c i e n t t o prove t h a t
m+)
i s dense i n
Tq)
,
which we s h a l l do by t r u n c a t i o n
Let t h e r e b e f u n c t i o n s
and r e g u l a r i s a t i o n .
xk
and pk as i n
t h e s t a t e m e n t of Theorem 2 . 7 , w i t h i n a d d i t i o n SUPP p
c {
X n I
0
]
.
For u
Epm:)
L e i b n i z ’ formula Shows
108
SOBOLEV S P A C E S AND A P P L I C A T I O € J S
a"[\
that
=
U)
\ a" u +
, where
v
(CHAP.
2)
i s a l i n e a r combinat-
v
ion of terms of t h e form
I
k-1"'
(a"' x) (x / k l a""
'pk = ( u
We now p u t of lcll
0
by 0 f o r xn < 0 .
u
*
p
> ,, k IR+
We have
+
"" =
where u
0
p I
"8
5
1)
.
i s t h e extension
vk E C " q )
and, for
s m;
i t s convolution w i t h p choice of p .
-
a' (U0 7
s i n c e , because
Since
k
a
E
Pk-(a' u
2
i n L (By)
Suppose
s E R.
where i : C;(Rn)
there-
Suppose we have
Then the i n j e c t i o n
Hs(T)
x is compact.
c ~ ( R " )i s
-
, and
, we
Then the i n j e c t i o n
The f i r s t i n j e c t i o n can be w r i t t e n r
c;(iT)-
3 ,
i n view of t h e i n L2(Fn)
u)"
a"
p-(,
1
xn = 0
pq) .
R with s > t .
i s compact when PROOF:
cy
c
i s supported by
i s continuous and of dense image.
CI@>-Hs(% s, t
*
u)"
in
P R O P O S I T I O N 3.4:
u)"
n
(a"
'pk-u
fore that
(a'
i s supported by { x I 0
can t h e r e b y deduce t h a t
p :
(a'
u
0
i
-Ht(T) 0
p , where
a continuous l i n e a r e x t e n s i o n o p e r a t o r ,
Hs(Rn)
i s t h e i n j e c t i o n , and where
-
r : Hs(Rn)
is the restriction.
H'fl')
Its continuity thus
follows from t h e c o n t i n u i t y o f i ( s e e P r o p o s i t i o n 2 . 3 ) .
r
second i n j e c t i o n can be w r i t t e n
ps
H'(KJ
:
-H'[R")
o p e r a t o r , where bourhood of
2,
x).
(K = supp
x
E
j
E
, where
S
0
and where J : H-:Ht
is the injection
I t s compactness t h u s f o l l o w s from t h e compactness F i n a l l y , suppose u
H S ( B n ) i s an e x t e n s i o n o f
5 i n HS(IRn ).
+
Xp
C"( Bin) i s a f u n c t i o n e q u a l t o 1 i n t h e neigh-
u
E
HS
(x)and suppose
; we know ( s e e P r o p o s i t i o n
2 . 3 ) t h a t t h e r e e x i s t s a sequence Cp, Cp,
D
i s a continuous l i n e a r e x t e n s i o n
of j ( s e e Theorem 2 . 1 0 ) .
%
D
E
Cz(lRn ) such t h a t
i n Hs(X), and hence t h e d e n s i t y o f C"j?)
in Hs(X).
The dua2 of H s ( x ) (resp: of
PROPOSITION 3.5:
h
Ci
> = < u,
v
>
for u
E
H'R) ,
v
the
,
X
.
We begin by showing t h a t < %, v > does n o t depend on t h e
extension
(3.5.1)
K
E H~'(R")
being an arbitrary extension of u i n t o HS ( Bn )
PROOF:
is
H-'(R")
Hs(T)) by
i d e n t i f i e d with the normed space HIS(Bn) ( r e s p : X pairing v
-
u
~p
We can t h e r e f o r e deduce t h a t
klX
<
The
chosen for
LEMMA:
Let
u
,
i . e . t h a t w e have:
U E H'IR"')
that: U = 0 i n X , v = 0 i n
nn\?.
,
v E
Then
H-S(Rn)
be such
< U, v > = 0
.
I n f a c t , by l o c a l c h a r t s and a p a r t i t i o n of u n i t y , we can
SOBOLEV SPACES AND APPLICATIONS
110
n reduce t o t h e case X = IR,,
(CHAP. 2 )
Suppose
U having compact s u p p o r t .
we have a r e g u l a r i s i n g sequence p k ( s e e Theorem 2 . 7 ) w i t h supp p
c
f
xn < 0
so t h a t
< U,
v >=lirn
ion
v
. < pk
pk
We know t h a t +6
U, v
> = 0
*
U
-
U in H
S
,
since the distribut-
i s z e r o i n t h e neighbourhood o f t h e s u p p o r t o f t h e t e s t
function pk " "
u , v > i s w e l l d e f i n e d and t h a t
This lemma shows t h a t
I<
u, v
I< u,
>)
v
lFl\, IlvIl-,
I 5
IN, ~vC, .
l i n e a r form on H s ( X ) .
r : HSIJ?")-HSfl]
E
so t h a t
Since t h e r e s t r i c t i o n o p e r a t o r
r i s a continuous l i n e a r
0
thus it i s i d e n t i f i e d with a d i s t r i b u t i o n
satisfying <
H-'(IR~)
,
S
Conversely, l e t R be a continuous
i s continuous, R
form on H S ( l R n ) ; v
of u i n H
for any e x t e n s i o n
s o t h a t v E H"(Rn)
\\v\Ls =
and
K
U, v > = A(U
1l .E
IX
) for a l l u
0
b\I
5
H'(IR~
E
1,
9
S i m i l a r l y l e t R be a continuous l i n e a r form on E l s ( I R n ) ;
from
X t h e Hahn-Banach theorem, it extends i n t o a continuous l i n e a r form ( w i t h t h e same norm) on H - ' ( D n ) a distribution U
u E HSR)
,
<
E
U,
H S ( B n ). v
>
=
which is i d e n t i f i e d w i t h
I f we put u = U
A(v)
for
I X Y we
have
v E HIs($)
and
X
I l ~ ~ I5, ll~li, = liali. DEFINITION 3.6:
Let
x be a manifold with boundary imbedded
i n a manifold M (see Remark 9 . 6 , Chapter 1) and Zet s
E
denote by H S o c ( X ) the space of d i s t r i b u t i o n s u E H ~ , , ( x ) N
t h a t cp
u
E
HSK)
f o r any chart X : U -U
of
?R
.
We
such
x at
the
(SEC. 3 )
SPACES HS (?)AND Hyoc
boundary m d for a l l q
E
(?)
111
We equip HSl o c (2) with
Cz(E).
topoZogy defined by the family of semi-norms
u
.
-
-,'?\?
llcp X* uijs
and semi-norms which are continuous i n HS l o c (X)
A s i n t h e proof of Theorem 2 . 1 9 , it can be shown t h a t it i s
a c t u a l l y s u f f i c i e n t t o c o n s i d e r an a t l a s of
?
We can t h e r e b y deduce t h a t H:oc(x)
cussion.
and t h a t i f K i s a compact s u b s e t of by elements of HS loc hilbertisable;
?,
i n t h e above d i s -
i s a Frgchet s p a c e ,
t h e subspace Hs(X)(formed
K
(x)which a r e z e r o i n X \ K) of
we denote by
Ibll s ,k
s > n/2
with c o n t i n u o u s i n j e c t i o n , when
n H;,,[R)
=
C"V)
;
loc
(x)i s
an a h i s s i b l e H i l b e r t norm
H;oc(Z)
I t can immediately be shown t h a t
in HE(?).
HS
+
Ck(y)
C
,
;
k
a d i f f e r e n t i a l o p e r a t o r of degree
m
S
w i t h c o e f f i c i e n t s i n C"(?)
H'l-o~(c?).
lo c (2
i s c o n t i n u o u s from HS
into
BY a p a r t i t i o n of u n i t y , we can c o n s t r u c t a c o n t i n u o u s
l i n e a r e x t e n s i o n o p e r a t o r from H
S
loc
(X)i n t o
(MI,and we can H' loc
t h e r e b y deduce t h e c o n t i n u i t y of t h e i n j e c t i o n
C"P)
-
HYocfl)
t o g e t h e r w i t h t h e d e n s i t y of C:(x)
in
We denote by HZ ( M ) t h e union o f t h e H s ( M ) when K X , comp K d e s c r i b e s t h e f a m i l y o f compact s u b s e t s of and we e q u i p
H:oc(?).
X
(M) w i t h t h e i n d u c t i v e l i m i t t o p o l o g y . When we assume X , comp t h a t M i s endowed w i t h a p o s i t i v e d e n s i t y , arguments analogous
:H
t o t h o s e of Theorem 2 . 1 5 S
d u a l of Hloc(X)
(resp:
and P r o p o s i t i o n 3.5 show t h a t t h e of HZs (M)) i s i d e n t i f i e d w i t h X,comp
SOBOLEV SPACES AND APPLICATIONS
112 H
< u, v > = < u, v > .v
E
u
where
HY~,~)
loc
3
-
HSR)
'pu E
u
REMARK 3 . 8 :
(x)i s t h e
H[MI I'
I
s' (X)
E
and a l s o t h a t
Let Y be a manifold ( w i t h o r without boundary)
D e f i n i t i o n s 2 . 1 8 and 3 . 6 :
9
N
9
by u s i n g t r i v i a l i s a t i o n s of
CN) = (HSm:))N.
Y.
above
We
) i n a manner analogous t o
HS(Rn;
c o o r d i n a t e p a t c h e s of Y , we a r r i v e a t
above
CN.) = (HS(Rn))N
We l e a v e t o t h e r e a d e r t h e t a s k
of g e n e r a l i s i n g t o t h e c a s e o f s p a c e s
of t h e s p a c e s
and where
I/'pu//, d e f i n e i t s t o p o l o g y .
be a complex v e c t o r bundle o f r a n k
HSm:;
u
space of t h e u
cp E f r X ]
for a l l
n a t u r a l l y d e f i n e t h e space HS ( Y ; loc
4.
E
v
I n t h e c a s e where X i s a r e g u l a r open s u b s e t
we can show t h a t HS
t h e semi-norms
or at
and
i s an a r b i t r a r y e x t e n s i o n of
Hs,,(M)
such t h a t
and l e t
E
x,conp
REMARK 3.7: o f E?
u
of
(CHAP. 2)
H;o,[Y;
3) t h e p r o p e r t i e s
n-1 E R
xn
HYoc(Y) g i v e n e a r l i e r .
TRACE THEOREMS, SPACES H:(X).
We pt
R"
= { x =
(XI,
xnj
I
XI
E R 3
.
We
s h a l l s e e t h a t we can d e f i n e i n a n a t u r a l manner t h e t r a c e on t h e hyperplane { xn = 0 condition s >
i.
1
o f an element
u E Hs(Rn.)
under t h e
-
TRACE THEOREMS , SPACES HE (X)
(SEC. 4)
THEOREM 4.1:
operator y : C:[Rn) ( y ~ ) ( x 'j
-
Let
s E
u ( x ' , 0) extends
B be such t h a t
C;(R"-']
s >
113
f.
The trace
defined by
in a unique manner i n t o a continuous
Zinear operator (again denoted by
y)
from Hs(IRn ) i n t o
HS-'(R"-'
We f u r t h e r observe t h a t t h e change of v a r i a b l e
By i n t e g r a t i n g b o t h s i d e s o f t h i s i n e q u a l i t y w i t h r e s p e c t
t o E ' , we o b t a i n
).
114
(CHAP. 2 )
SOBOLEV SPACES AND APPLICATIONS
$2
and hence Theorem 4 . 1 , s i n c e C " ( B n
trace operator y
ya = Y
0
T~
a
h = (0,
2 . 3 shows t h a t t h e mapping fixed ( s >
2 ) , continuous
, we
a
likewise define the
xn = a }
on t h e hyperplane
, where
8
) i s dense i n H S ( B n ) .
given i n W
For a
REMARK 4 . 2 :
Jplt
cs
llYUll+
..., 0, a )
CL
ya u
;
,
we have
*
thus Proposition
is, for u
from IR i n t o H
s-$ ( Bn-l
E
HS(IRn )
).
The d i s t -
r i b u t i o n a s s o c i a t e d w i t h t h i s f u n c t i o n ( s e e S e c t i o n 1 . 5 , Chapter
I ) coincides with
u
i s obviously t r u e for u
u
E
i n f a c t , t h e formula
;
E
C m ( W n ) , and t h e r e f o r e a l s o f o r a l l 0
HS s i n c e b o t h s i d e s of t h i s e q u a t i o n depend c o n t i n u o u s l y on
u ( t h e c o n t i n u i t y o f t h e right-hand s i d e f o l l o w s from t h e domina t e d convergence theorem by v i r t u e o f t h e upper bound
COROLLARY 4 . 2 :
s-j > Hs(IRn)
2.
L e t there by s
Then the operator
yj = y
If u
i n t o HS-j-'(Bn-').
E
E
IR and j
DJ
xn. Hs(Wn),
E
N such t h a t
i s continuous from then
il
fwzction of class Ck with respect t o xn with values i n Hs-k-i ( Bn-l
) for 0 < k
I
j.
is a
(SEC. 4)
TRACE THEOREMS, SPACES Hz( X )
PROOF: Dj
-
The f i r s t p a r t f o l l o w s from t h e c o n t i n u i t y o f
H"j(Rn)
: HS(Rn)
Xn
C1(R,
H3-17(R
HS, t h e f u n c t i o n
E
n-I
7
x -1 y n corollary.
X
-
The l a s t p a r t of P r o p o s i t i o n 2 . 3 shows
We assume j t 1. that, for u
and from Theorem 4.1.
x
n
i s in
yxn u
)) and a l s o t h a t i t s d e r i v a t i v e i s t h e f u n c t i o n
a
u
t h i s i m p l i e s t h e second p a r t o f t h e
;
Xn
(Trace on a h y p e r s u r f a c e ) . Let X be a mani-
COROLLARY 4.3:
fold, Y a hypersurface o f X, and suppose ator y : cz(x]-
s >
2.
Then the oper-
of r e s t r i c t i o n t o Y e x t e n d s wziqueZy
Cz(Y)
into a continuous l i n e a r o p e r a t o r (again denoted by y i from i n t o H:it(y),
H:oc(x)
-
ry
PROOF:
For xo
E
X : U
Y, l e t
U
be a diffeomorphism o f a
neighbourhood of xo i n X onto an open s u b s e t of IRn
such t h a t
r
X ( U ~ Y ) =
For
'PI,,=,
Exculxn=o};
, we
Q1-
Q,
E Co(Uo) =
(90 *
If u
c o n s i d e r cp E
C:(X),
From Theorem 4 . 1 , we have
weput
E
C"(E) 0
we have
]/yo X,(Yu)\Is+ 0
(9,
such t h a t
<(yu) I
Cs
= y[cp X, u) II(9
X., ulls
thus y i s continuous from C z ( X ) (equipped w i t h t h e topology induced by H s o c ( X ) )
i n t o Hs-'(Y). loc
Theorem 4 . 3 t h e n f o l l o w s
. ;
116
(CHAP. 2 )
SOBOLEV SPACES AND APPLICATIONS
from t h e d e n s i t y o f C z ( X ) i n H Y o c ( X ) .
COROLLARY 4 . 4 :
Y : Cp;)
-
Suppose s
1.
>
The trace operator
defined by
C:[R*')
(yu)(x') = u(x',
0)
extends miqueZy i n t o a continuous Zinear operator y from i n t o Hs-$ ( Bn-l 1.
)Hs( :B
PROOF:
For
u E
HsE] ,
r a r y extension of
u
we p u t yu =
1.
into HS(Bn
yG where
"u i s an a r b i t -
The f a c t t h a t y"u does not
i f o l l o w s f o r example from Remark depend on t h e e x t e n s i o n ?chosen cv
4.2.
\lyulIs4
We have
llyu\lsa
5
cs
?iE Cz(En ) ;
11U11,
5
.
therefore
for a l l 5 , thus
Cs lluilg
When
u
E
w
yu = u
we can choose
C:(Ti;)
I xn-0
m u
1 Xn=o
t h u s proved t h e e x i s t e n c e of an e x t e n s i o n y ; -n s -n f o l l o w s from t h e d e n s i t y o f Co(m+) i n H (B+) m
,
We have
i t s uniqueness
.
From C o r o l l a r y 4 . 4 , we proceed e x a c t l y as i n t h e proof of Corollary 4 . 3 i n order t o e s t a b l i s h t h e following:
COROLLARY 4 . 5 :
-
(Trace a t t h e boundary)
fold w i t h boundary m d suppose s y :
C;R)
>
d.
Let
x be a mani-
Then the operator
c"(ax> of r e s t r i c t i o n t o ax extends uniquely i n t o 0
a continuous l i n e a r operator y from HS (x)i n t o H s ~ ! ( ax). loc We conclude t h i s s e c t i o n by s t u d y i n g t h e c l o s u r e of C"(X) 0 i n HS loc
(x). To do t h i s ,
we s h a l l need t h e f o l l o w i n g :
(SEC. 4)
TRACE THEOREMS
PROPOSITION 4.6:
IR
, and
v
0
Zet j
Let
and t
E
E
IR
Then, f o r
v
DJ 6 is in Ht ( I R " ) if m d onZy i f t + j < -
1.
v E H t+''(Rn-l
PROOF:
We put
1
non-zero,
and if
We then have
IIv € 3 D j bitt
(4.6.1)
117
be t he Dirac measure a t 0 i n
6
.
SPACES Ht (X)
=
c t , j l l v l l t + j ~' f
u = v @ Dj
6
;
we have
.=
;(I1) 5; .
By u s i n g , as i n (4.1.2)¶t h e change of v a r i a b l e
+ 15t\2)4 qn , we
gn = (1
thus obtain:
hence t h e c o n c l u s i o n .
REWRK 4 . 7 :
1 -2
t
-
Proposition
t h e operator
v 8
v
Ht+2 ( IRn -1) i n t o H t ( B n ) . putting
- t
Hs-3( Bn-l
cp
E
4.6 shows i n p a r t i c u l a r t h a t , for
6 is
c o n t i n u o u s from
By t r a n s p o s i t i o n we o b t a i n , by
= s , a c o n t i n u o u s o p e r a t o r A from H'(B
n
and it is immediately obvious t h a t Acp
Cmo(IRn ) .
) into
=
y
cp
for
We t h u s o b t a i n a f r e s h p r o o f o f t h e Trace
Theorem 4 . 1 .
PROPOSITION 4.8:
We put X = IR:
(Rn 1 f r e s p : utions
u of the form
.
t H,,(Bn))
u =
For
is th e space of d i s t r i b vj @D' 6
osje-ts.
,
v.
with
J
E Ht +* (R n-I ) COW
-4 ,
( for t
2
PROOF:
If
u
(resp : v j
we have
t
t
{ xn
1). t
i s o f t h e above form, t h e n
hn>. Since u = 0 3 ,we know
HaX,cornp
plane
E Ht+j+&(Rn-l)) = { 0
HaX(Rn)
t n ( r e s p : u € HaX(!? )) from P r o p o s i t i o n
'€
(CHAP. 2)
SOBOLEV SPACES AND APPLICATIONS
118
E
4.6.
Hax,comp (Rn) Conversely, suppose
6(IRn) i s supported by t h e hyper-
(SCHWARTZ L11) t h a t
u
(which we
can be w r i t t e n uniquely i n t h e form
assume non-zero) N
c
<
V 8 0' 6 , w i t h v Ei[Rn-') and vN # 0 . By j J j=o 2 ' again u s i n g t h e change o f v a r i a b l e Y we = (1 + 1g'I TIn
=
U
obtain, since
2
I\ui/t =
;({)
ii
(1 +
=
$
)1cj(fi)
:
j=o
7;lt
I"
h
N
N IE 11 + j=o
\ t3i'1
*, V,(rlj
T;1f
B S ~dqn
c Thus t h e i n t e g r a l w i t h r e s p e c t t o
ri
n
i s convergent f o r almost n
a l l 5 ' ; by choosing such a 5 ' w i t h deduce t h a t
i
(I+
dy, <
+ a .
vN(gt)
+ o,
#
0
, we
therefore
so t h a t N < - t -
1 2
.
I n t h e same manner, t h e i n t e g r a l w i t h r e s p e c t t o 5 ' i s convergent
for almost a l l q n ;
by g i v i n g ( N
+ 1) such v a l u e s t o n,,
we see
by s o l v i n g a Cramer system t h a t t h e N + 1 f u n c t i o n s 2 @+$: (I + 1 5 1 1 1 2 vJ a r e i n L ~ ( I R ~ - ' ) , i . e . v E j 0 j 5 N. F i n a l l y suppose u E H:x[Rn) Since
H ~ + ~f oG r
.
U
E
a(Rn)
i s supported by t h e hyperplane
know (SCKWARTZ c11 t h a t
u
{ xn = 0 ] , we
can be w r i t t e n u n i q u e l y i n t h e form
u
TRACE THEOREMS , SPACES H : (X)
4)
(SEC.
vj
o
@
, where
Dj
js t r u n c a t i o n ( s e e Theorem
119
t h e sum i s l o c a l l y f i n i t e .
By
2.7), u i s t h e l i m i t o f a sequence
hn) and we have seen t h a t u [kj E Hax,cornp t
< D 6, rir > = 0
for R
thus v . = 0 if j J
‘
- t -
2
t
(Rn)
HaXJcornp
THEOREM
For
-.L2
r+
aa’
E C,(R
n-I
)
we have
We t h e n proceed e x a c t l y a s in t h e
t o prove t h a t
v.
J
E
.
Ht+M(R”-’l
Then
Let X be a regular open subset o f Rn.
4.9:
the closure H’
# j.
0 ,loc
(x) (resp: H:(x))
Hs(X)) is the space o f the t h a t y aa u = 0 for a l l
IJ
ci
of c;(x)
in
HS~~(X) (resp:
(resp: Hs(X)) such 1 s a t i s f y i n g I a1 < s-2 ( y denoting
E
H:ocfl)
ax).
I n p a r t i c u l a r , cz(x) i s dense 1 i n HSoc(X) ( r e s p : H s ( X ) ) i f and only if s I 5. the trace operator on
PROOF:
S
We can c o n s i d e r t h e space H 0 , l O C (X) as b e i n g t h e b i m
S Co(X) in Hloc(X). Now, a c c o r d i n g t o t h e end o f
p o l a r space o f
S e c t i o n 3 , t h e space o r t h o g o n a l t o C:(X) space of t h e v
E
H (R”) I’such
that
~
Cp
u
E
E
C;(X),
XI comp i . e . w i t h t h e space
--s
i s i d e n t i f i e d with t h e
< v,
HW,cow
i f and o n l y i f H’loc (X)i s i n H’(x) 0
Cp
[$)
> = 0 for
.
Thus
all
SOBOLEV SPACES AND APPLICATIONS
120
-=
(4.9.1,)
U,
v
>
=
0
(CHAP. 2 )
v f H-'
for all
bx, cornp
.
(R")
By p a r t i t i o n of w i t y , we s e e t h a t , i n ( 4 . 9 . 1 ) , we can f o r c e t o have s u p p o r t i n an open c o o r d i n a t e p a t c h of
2
v
a t t h e boundary,
n and t h i s allows us t o reduce t o t h e case X =IR+.
P r o p o s i t i o n 4 . 8 t h e n shows t h a t ( 4 . 9 . 1 ) i s e q u i v a l e n t to 9
i.e. t o v
J
E ,< yj u,
&<s, E H-~+H (~""1 cornp
0< j < s Hsoc
(2).
- 51
.
,
(- I)J v . > =
J i.e. finally t o
o y
for a l l
J
u =
o
for
Theorem 4 . 9 i s t h u s proved i n t h e case of
I n t h e case o f Hs(?) we proceed i n t h e same manner.
S
Let us c o n s i d e r more p a r t i c u l a r l y t h e space H ( X ) ( i . e . 0
t h e c l o s u r e of C z ( X ) i n H s ( X ) )
and l e t us e q u i p it w i t h t h e m
Since t h e i n j e c t i o n o f C (X) i n t o
topology induced by Hs(?).
0
H Z ( X ) i s continuous and of dense image, t h e d u a l of H:(X) n a t u r a l l y i d e n t i f i e d w i t h a subspace o f &'(X)
is
and w e s h a l l
l i m i t o u r s e l v e s h e r e t o s t u d y i n g t h e v e r y simple case where s
E
2i ( s e e LIONS-MAGENES [11 for t h e c a s e s
For m
THEOREM 4.10 :
topologically v i t h H
PROOF:
-m
E
E
IR).
IN, t h e duaZ of H:(X)
is i d e n t i f i e d
-
(X).
Let R be a continuous l i n e a r form on H m ( X ) ; we
i d e n t i f y t h i s with t h e d i s t r i b u t i o n u
0
E
&'(X)
d e f i n e d by
(SEC.
4)
< u, cp >
TRACE THEOREMS, SPACES H:(X)
=
a[Q) for
all
cp E f ( X )
,
From t h e Hahn-Banach
Theorem and P r o p o s i t i o n 3 . 5 , t h e r e e x i s t s
< u, cp > = < U, rp > f o r
such t h a t
We t h u s have u = U
E
IX
U
E
HZm(R"]
.
X all
cp E C z ( x j
Conversely, suppose u
H-m(?).
u i s of t h e form
From P r o p o s i t i o n 3 . 2 . ,
121
u =
F
a'
E
H-m(X).
fa
,
la5 m with fa
E
i s t h e e x t e n s i o n of f we have Cp
u,
'p
o
a a ( f o j , where fa
We p u t U =
L2(X),
a
by 0 o u t s i d e of
,= < U,
Q
>
,
?.
For
E
2 n L (R )
E f(Xj
,
which depends c o n t i n u o u s l y on
for t h e topology induced by H m ( X ) , i n view o f P r o p o s i t i o n
3.5 s i n c e U
HIm(IRn).
This proves t h a t t h e d u a l of H:(X) is X -m a l g e b r a i c a l l y i d e n t i f i e d with H ( X ) ; t h e t opol ogi c al i dent i E
f i c a t i o n i s o b t a i n e d i n s i m i l a r manner.
REMARK 4.11 Propositions
:
Suppose s
E
R i s such t h a t
s >.
i.
4.6 and 4.8 show t h a t t h e l i n e a r mapping
i s c o n t i n u o u s , i n j e c t i v e and o f c l o s e d image.
I t s t r a n s p o s e , which i s a c t u a l l y t h e r e s t r i c t i o n mapping
122
(CHAP. 2 )
SOBOLEV SPACES AND APPLICATIONS
i s thus surjective.
By p a r t i t i o n of u n i t y and l o c a l c h a r t s a t t h e boundary,
w e can show t h a t t h i s s u r j e c t i v i t y r e s u l t remains v a l i d when
?,
En i s r e p l a c e d by
IRn-l
by
ax,
and y
j
u by
y (+
k)J
,
u
where X i s a r e g u l a r open s u b s e t o f I?? w i t h compact boundary n and where v i s a f i e l d o f v e c t o r s o n B t r a n s v e r s a l t o
ax.
L e t # be a complex H i l b e r t s p a c e .
R E M K 4.12 :
ax,
t h a t t h e p r o p e r t i e s of F o u r i e r t r a n s f o r m a t i o n g e n e r a l i s e
W e know
without
d i f f i c u l t y t o t h e case o f f u n c t i o n s or d i s t r i b u t i o n s w i t h v a l u e s i n 3$ ; f o r example, w e have P a r s e v a l ’ s formula
for
u, v E g($; #)
.
This allows us t o g e n e r a l i s e word f o r
word a l l t h e preceding results and p r o o f s t o t h e c a s e o f Sobolev spaces o f d i s t r i b u t i o n s w i t h v a l u e s i n
5.
3.
APPLICATION TO THE DIRICHLET PROBLEM
We r e t u r n t o t h e example i n t h e i n t r o d u c t i o n , i n a r a t h e r more g e n e r a l form. compact boundary operator P =
-
Let X be a r e g u l a r open s u b s e t o f En w i t h
ax
A +
and l e t P be t h e e l l i p t i c d i f f e r e n t i a l
V(x), where V
E
C”(?)
is such t h a t t h e r e
(SEC.
5)
DIRICHLET PROBLEM
123
e x i s t c o n s t a n t s c and C s a t i s f y i n g 0 < c 5 V[x) IC
for a l l
x.
X E
THEOREM 5 . 1 :
E
ti4 [ax)
Suppose we have f
E
H
-1 -
(X) m d
Then there e x i s t s a ttvlique u E H 1 R) such
,
that
u, v
for
E HIR]
,
From t h e assumptions made concerning
V ( x ) , ( ( u , v ) ) i s an i n n e r product e q u i v a l e n t t o t h e i n n e r product of H1(
2).
We f i r s t prove Theorem 5 . 1 in t h e p a r t i c u l a r c a s e f = 0. Since C ” ( X ) 0
“u, we put
1 i s dense i n H o ( X ) ,
.I)
= =
0
( 5 . 1 . 3 ) i s then equivalent t o
for all
c u EH’(K)
I yu
4 . 5 , 4.9 and Remark 4 . 1 1 show t h a t v a r i e t y i n H1(X)
1 parallel t o Ho(X),
v
=
E HA[X) g
1
.
Theorems
c i s a closed a f f i n e and t h e r e f o r e t h a t
SOBOLEV SPACES AND APPLICATIONS
124
u
E
(CHAP. 2 )
u is the
H1(%) s a t i s f i e s (5.1.1),(5.1.2) i f and o n l y i f
o r t h o g o n a l p r o j e c t i o n ( r e l a t i v e t o t h e i n n e r product ( ( u , v ) ) ) of 0 onto
c,
i s t h e unique element
u
i . e . i f and only i f
which minimises t h e energy E(u) = ( ( u , u)).
of
It remains f o r us t o prove Theorem 5 . 1 i n t h e c a s e g = 0 .
From Theorem 4.9, c o n d i t i o n s (5.1.1), ( 5 . 1 . 2 ) a r e t h e n equivalent t o =
((u, Since f
E
<
f, cp
>
E c:[x)
f o r a l l cp
-1 -
H
( X ) , Theorem 4.10 shows t h a t I
a c o n t i n u o u s l i n e a r form on H (X). 0
q))
=
<
f, (p
for all
.
u E H;(XI
i s i d e n t i f i e d with
f
This means, from R i e s z ' s
t h e o r e m , t h a t t h e r e e x i s t s a unique u [(u,
and
E
1
H (X) such t h a t 0
cp E C:[X)
t h i s concludes
j
t h e proof of weorem 5.1. The method used i n t h e p r e c e d i n g p r o o f , c a l l e d t h e v a r i a t i o n a l method, allows c o n s i d e r a b l e g e n e r a l i s a t i o n s ( s e e LIONS c11).
For example, suppose X i s a r e g u l a r bounded open s u b s e t ofIR
n
and P ( x , Dx) i s a d i f f e r e n t i a l o p e r a t o r o f degree 2m w i t h c o e f f i c i e n t s i n C"(%), form
Pu =
which we can always assume t o be of t h e
De(a
aB
DCYu )
with
We assume t h a t P i s s t r o n g l y e l l i p t i c i n e x i s t s c > 0 such t h a t
a
cr8
C C"R)
x, i . e .
,
that there
5)
(SEC.
DIRICHLET PROBLEM
125
Let us consider t h e s e s q u i l i n e a r form w i t h r e s p e c t t o
u, v
E c;(x)
u, v
m -t
:
-
-
v >
=
D a u , DB v
< a
>
Ictk ms5 l s m af3
-
.
This extends i n t o a continuous s e s q u i l i n e a r form
u, v
on
a(u, v]
Hm(X) x H ~ ( X ) .
G%rding's i n e q u a l i t y ( s e e Theorem 8.6, Chapter IV) i m p l i e s that there exists A
(5.1.4.)
0
such t h a t
Re a ( . ,
2 Ilull,
C
5
- A,
Iblj,2
Hf(x)*
for
We can then s t a t e t h e following:
Suppose we have
THEOREM 5 . 2 :
gj
E: H W H ( W )
such t h a t Re A
for j = O , . . . , A
2
0,
f
E
H-,flT)
and
Then f o r A
m-1.
there e x i s t s a zcnique u
E
E
C given
?(?) which is a
solution of the Dirichlet problem:
1
(P
+
A).
=
'jU
where we have p u t
( i n t h e s e n s e of d i s t r i b u t i o n s i n X )
f
[j = 0,
gj y
J
-1)
j
u
=
-
vectors o n D n transversal t o
The uniqueness of
...*
Y(T 2) ay
,
,
y being a f i e l d of
ax.
u c l e a r l y follows from ( 5 . 1 . 4 ) .
not give i n d e t a i l t h e proof o f t h e e x i s t e n c e of
u
We do
(see
Lax-Milgram theorem, E x e r c i s e 7 . 4 ) , as we s h a l l be e s t a b l i s h i n g some more g e n e r a l r e s u l t s i n Chapter V I .
SOBOLEV SPACES AND APPLICATIONS
126
6.
(CHAP. 2)
SOBOLEV SPACES AND REGULARISATION
O c c a s i o n a l l y we need t o prpve a r e g u l a r i t y r e s u l t f o r s o l u t i o n s of an e q u a t i o n Pu = f from an a p r i o r i i n e q u a l i t y o f t h e
for a l l v
€aK, with
appropriate indices s
S i n c e we cannot
j'
apply t h i s i n e q u a l i t y d i r e c t l y t o a d i s t r i b u t i o n led t o regularise
u
u
, we p,
and t o e s t i m a t e Sobolev norms of
and of t h e commutator
[P
.
, p, * ]u
are
*U
For t e c h n i c a l r e a s o n s , which we s h a l l d i s c u s s i n Chapter VII, s e c t i o n 6, we b e g i n by i n t r o d u c i n g a f a m i l y of e q u i v a l e n t norms on H
S
which depend on a parameter 0 < 6 S 1.
We f o l l o w c l o s e l y t h e methods o f HORMKNDER c11, c21.
DEFINITION 6.1
:
If w e have 0
<
6 5 1, w e p ut f o r
:
E H'(R")
It i s c l e a r t h a t f o r 6 f i x e d , t h i s i s a norm which i s equivalent t o
lhf
llu'Is, 6
and f u r t h e r m o r e , we have :
/ 6-0
IIU
Ils+l
for
u
E'+'H
0
(SEC. 6 )
SOBOLEV SPACES AND REGULARISATION
127
The u s e f u l n e s s o f t h e s e norms i s t o show t h a t i f a d i s t r i b u t i o n
u
E
HS s a t i s f i e s
Ilulls, 6
C
S
w i t h C independent of 6, t h e n
s+l t h i s distribution is in H and
I1ulls+l
S
C
.
We g i v e below a n o t h e r e x p r e s s i o n f o r t h i s norm
If we have s
PROPOSITION 6.2 :
f o r 0 < 6 S l and
u t i l i s e the'decomposition u = u 1
2
1/6
.
1
6
and
for
(1 +
+
1851
S
, we
+ u" where t h e F o u r i e r and t h a t o f u" i s z e r o
Then, we can show t h a t
by o b s e r v i n g t h a t for
+I
If we have u E H
151 < 1 / 6
t r a n s f o r m o f u' i s z e r o f o r
IS1
B, there e x i s t C1 and
.
uEHS
PROOF of t h e 2nd i n e q u a l i t y .
for
E
ki2iS 5
1651
2
1 :
(1 +
id2Is+l
(1 + 1 6 5 12
I- 1
<1 :
Id 21s+l
2 s+l
2 (1 +
151
I
(1 + 160121-1
.
I n o r d e r t o prove t h e f i r s t i n e q u a l i t y i n ( 6 . 2 . 1 ) , it i s s u f f i c i e n t t o n o t e t h a t we always have
(CHAP.,2 )
SOBOLEV SPACES AND APPLICATIONS
128
T h i s e q u i v a l e n t e x p r e s s i o n for t h e norm
I'
'Is,6
w i l l allow
us t o prove t h e f o l l o w i n g :
PROPOSITION 6.3: a)
Let P(x,D) be a d i f f e r e n t i a l operator of degree m
with Cm c o e f f i c i e n t s which are constant outside a compact domain.
Then for a l l s
Let
b) ill
and
alZ s
i12 E ?R,
x
E
IR there e x i s t s
C
such t h a t
be a diffeornorphisrn between two open subsets
of Bn, and l e t K be a compact subset of i12; then for
there e x i s t s
PROOF OE
a).
c
such t h a t
If u = u' + u" i s a decomposition of
u E Hs+m, from (6.2.1) we have
By u t i l i s i n g t h e c o n t i n u i t y o f P i n Sobolev s p a c e s , we t h e r e b y deduce t h a t t h e r e e x i s t s a c o n s t a n t C which depends o n l y on s
and P,such t h a t
(SEC. 6)
SOBOLEV SPACES AND REGULARISATION
129
which i m p l i e s ( 6 . 3 . 1 ) .
The proof of b ) is i d e n t i c a l and i s b a s e d on t h e i n v a r -
I/ (Is
i a n c e of t h e norms
under diffeomorphism.
We now g i v e an e x p r e s s i o n f o r
I/ulls
using regularJ
isations of
u.
If we have p
rn
C ( I R n ) , we assume t h a t f o r an i n t e g e r
E
0
h 2 0 , we have A
P(53
(6.3.3)
= 0
(151hl
5'0
#
We assume a l s o t h a t : 6
p(t5) = 0 for a l l t
(6.3.4)
EP
implies
5 =
0 if
5 Em".
We can e a s i l y c o n s t r u c t such f u n c t i o n s , by p u t t i n g p = A h 6 where
m
€3
E
g(0) 4 0
C and 0
THEOREM
with h > s+l
6.4
.
.
Suppose we have s
:
E
IR and a function p
Then, there e d s t C2 and C > 0 such t h a t 2
f o r u E HS and 0 < 6
S
I , We have put
p,(x]
=
C-n
(The second i n e q u a l i t y i n (6.4.1)does not make use o f asswrrption ( 6.3.4)I
PROOF:
.
We have
p(x/e)
.
130
SOBOLEV SPACES AND APPLICATIONS
(CHAP. 2 )
with F(5,6)
(6 -4.2)
=
J:
I^p(€g)12
C-2s-2
(1
+
62/e2)-1
$
.
Since h > s+1, F extends i n t o a continuous f u n c t i o n on ?Rnx CO,ll, t h u s , i t i s s u f f i c i e n t t o show t h a t t h e r e e x i s t . c o n s t a n t s
we o b t a i n
We t h e r e f o r e deduce
A
and s i n c e p € 8
, we
F(t,6)
5
t h u s o b t a i n t h e second i n e q u a l i t y i n
(6.4.3).
I n o r d e r t o prove t h e f i r s t i n e q u a l i t y i n (6.4.31, we observe t h a t
and we t a k e
7)
(SEC.
131
ADDITIONAL NOTES
We have c1 > 0 , s i n c e t h e i n t e g r a l depends c o n t i n u o u s l y on TI and
6
satisfies t h e condition ( 6 . 3 . 4 ) .
The f o l l o w i n g theorem w i l l be proved i n Chapter I V i n t h e more g e n e r a l case where
THEOREM 6.5 :
degree
m with
compact domain.
Let
P
i s a pseudo-differential operator.
be a d i f f e r e n t i a l operator of
P(x,D)
Coo c o e f f i c i e n t s which are constant outside a
I f we have s
REMARK 6 . 6 :
B , then there e x i s t s C such t h a t
E
The upper bound
(6.5.1) i s s t i l l v a l i d
if
t h e o p e r a t o r P depends c o n t i n u o u s l y on a parameter v a r y i n g w i t h i n a compact space.
7.
ADDITIONAL NOTES AND EXERCISES.
7.1
POINCARE'S INEQUALITY.
a)
Suppose suppcp
[express
9
cp
00
E c,,(#]
c { x
I
i s such t h a t
a S x s ~ + L *] 1
and m a j o r i s e t h e i n t e g r a l
as t h e i n t e g r a l of
1
132
SOBOLEV SPACES AND APPLICATIONS
u s i n g t h e Cauchy-Schwarz Let
b)
m
inequality]
(CHAP. 2)
I
be an i n t e g e r t 1.
Deduce from t h e
preceding question t h a t t h e r e e x i s t s a function
] 0, + O Q [
3 Ld
c(L)
+ O0 [
[&Y,
tending t o 0 with
L and such t h a t
Let X be a r e g u l a r open s u b s e t o f IRn such t h a t
c)
u
E
\to"
H m ( X ) and show t h a t 0
IIuII,,,
Thus deduce t h a t i n Theorem 5 . l w e can r e p l a c e t h e
0 < c 5 V(x)
assumption
a)
i s a norm e q u i v a l e n t t o
H~(x) .
on d)
7.2
ull,
,
I C
by 0 5 V(x) 2 C.
INVARIANCE OF HS UNDER DIFFEOMORPHISM
s E ] 0, 1 [
With
P a r s e v a l ' s e q u a l i t y t o prove
J.,
x
Rn
'
v[x
- v ( y ] 12
1. -
yln
where we have p u t
b)
*
Fs({)
=
Show t h a t Fs i s a homogeneous f u n c t i o n of degree 2s
with r e s p e c t t o
6.
Deduce from t h i s t h a t t h e f u n c t i o n d e f i n e d by
is e q u i v a l e n t t o t h e norm
l\Vl\s
on H S ( B n ) .
(SEC. 7 )
Use t h e e x p r e s s i o n [ v
c)
133
ADDITIONAL NOTES
0 5 s < 1.
Is
t o prove Theorem 2.17 when
Deduce from t h i s t h e case s 2 0 , t h e n f i n a l l y t h e
case s < 0 by making use of t h e d u a l i t y of Sobolev s p a c e s .
7.3
Show t h a t t h e f o l l o w i n g i n c l u s i o n s a r e s t r i c t :
8 C H+m
H-=
C
C
8'
2 -1 and 6 1 . [use f o r example t h e d i s t r i b u t i o n s (1 + x )
7.4
LAX-MILGRAM THEORFM
Let V be a H i l b e r t space and l e t
We assume
q u i l i n e a r form.
a
a
t o be c o e r c i v e ; t h i s means t h a t
t h e r e e x i s t s a > 0 such t h a t la(u, u ) l
2 CY llul(V2for a l l
v.
U
Prove t h a t f o r any continuous a n t i l i n e a r form T on V ,
a)
t h e r e e x i s t s a unique u
(7.4.1 .)
v such t h a t
E
a(u, v)
A E c(V,
[Let
be a continuous ses-
f o r a l l u,v.
V)
=
T(v)
for a l l
v
Ev,
be t h e o p e r a t o r such t h a t a ( u , v ) = (Au, v ) ~
Show t h a t A i s i n j e c t i v e and t h a t i t s image 'is
b o t h dense and c l o s e d .
If
f
i s t h e element o f V such t h a t
T ( v ) = ( f , v ) ~ t,h e n t h e s o l u t i o n o f ( 7 . 4 . 1 ) i s given by
u = A
. fl.
-1
b)
Apply t h i s r e s u l t w i t h V = H m ( X ) t o prove Theorem 4.2, 0
a f t e r f i r s t r e d u c i n g t o t h e c a s e where t h e g c) that
I f we f u r t h e r assume t h a t
a
j
a r e zero.
i s Hermitian, t h e n prove
u i s a s o l u t i o n of (7.4.1)i f and only i f
u minimises
SOBOLEV SPACES AND APPLICATIONS
134
-
(CHAP. 2 )
the functional
v E V
-
a(v, v)
7.5
E
2 Re T(v)
.
R
INEQUALITIES
HORMANDER'S
m with
Let P = p ( x , D ) be a d i f f e r e n t i a l o p e r a t o r o f degree c o e f f i c i e n t s which are C
a)
We p u t
p1 =
CC
&
(x,
D)
,
By u s i n g formula ( 1 . 1 . 3 ) o f Chapter I , show t h a t
P, cp = i p(xl c p ) b)
c)
i xl Rp
=
-f
I (XI
.
cp E C;(~J
SUP
x E SUPP 'p By u t i l i s i n g P o i d c a r g ' s i n e q u a l i t y ( E x e r c i s e
t h a t t h e r e e x i s t s a function L
for
Show t h a t
.t,
where
-
.
n i n a neighbourhood 0, of 0 i n R
L'--C(L)
7.1),prove
t e n d i n g t o 0 when
0 , such t h a t
Wll
cpf
5 C(L1
*
Q = [P , P1l
where d)
2 lcp!!*l
( n o t e t h a t Q is o f degree 2m
m[;
-
Deduce from t h i s t h a t t h e r e e x i s t L
two f u n c t i o n s : LI--C[L]
LO, +
for
,L
t e n d i n g t o 0 when L
$(L)
-+
0
-
> 0 , C 2 0 and
from 10, LJ
0 , such t h a t
2).
into
(SEC.
7)
135
ADDITIONAL NOTES
e ) 'In t h e c a s e where P has c o n s t a n t c o e f f i c i e n t s and where 52 i s a bounded open s u b s e t of R n , prove t h a t
llpll,
=
l)p*
pun,
and t h a t t h e r e e x i s t s C t 0 such t h a t
Thus deduce t h a t i f P is non-zero,
7.6
t h e r e e x i s t s C such t h a t
LOCAL SOLVABILITY FOR DIFFERENTIAL OPERATORS OF PRINCIPAL TYPE WITH REAL PRINCIPAL SYMBOL
n Suppose Q i s an open s u b s e t of IR ; l e t P = P ( x , D ) be a d i f f e r e n t i a l o p e r a t o r o f degree
m
w i t h Cm c o e f f i c i e n t s i n R.
We suppose t h a t P is o f p r i n c i p a l t y p e , i . e . t h a t :
dt P m b l
51 f
where P
i s t h e p r i n c i p a l symbol (homogeneous o f degree
m
0
for
x
~
n ~ E R " \ O
m ) o f P.
With a E Q, we denote by B ( a , L ) t h e open b a l l w i t h c e n t r e a
and r a d i u s L .
a)
Prove t h a t t h e r e e x i s t s c > 0 such t h a t for
5 E R"
Deduce from t h e p r e v i o u s q u e s t i o n and f?om s e c t i o n
,
7.1 t h a t
136
(CHAP. 2)
SOBOLEV SPACES AND APPLICATIONS
t h e r e e x i s t s C t 0 such t h a t
Deduce from t h i s , again by u s i n g s e c t i o n 7 . 1 , t h a t t h e r e e x i s t
C ,C 1
2 0 such t h a t :
We s h a l l a c t u a l l y g i v e l a t e r ( s e e s e c t i o n 1 1 . 3 o f Chapter IV) a simpler proof of ( 7 . 6 . 1 ) . Prove, w i t h t h e a i d of
c)
exists a function L
t o 0 when L
+
2 lly\lwl
(7.6.2.) d)
+-
M ( L ) from
(7.5.1) and ( 7 . 6 . 1 ) t h a t t h e r e
10, L ~ ]i n t o LO,
+
4,t e n d i n g
0 , such t h a t :
5
M(L)
(i/b/\E +
lip*
(pll?)
for tP
c:(e(a~
Ll).
Henceforth we make t h e supplementary assumption t h a t
m are r e a l - v a l u e d .
t h e c o e f f i c i e n t s of t h e polynomial p Prove t h a t t h e r e e x i s t s M
for
Q-
e)
E :C ( s ( a ,
2
0 such t h a t
L)
Deduce from ( 7 . 6 . 2 ) and ( 7 . 6 . 3 ) t h a t t h e r e e x i s t L > 0
and M t 0 such t h a t
(SEC. 7 )
137
ADDITIONAL NOTES
(7.6.4.1 f) With t h e number L chosen i n t h i s manner, suppose f
E
H-(mll(flJ
.
E = { P* cp
We p u t
I
cp € Cz(B(a,
L)]).
Deduce from ( 7 . 6 . 4 ) t h a t
L : E - C
*
g = p cp
*
(f, CPJ
i s a well-defined a n t i l i n e a r form on E , and i s continuous when we e q u i p E w i t h t h e topology induced by L from t h i s t h a t t h e r e e x i s t s
Pu = f i n B ( a , L ) .
2
(B(a,L)).
2 u E L [B(a, L)) such t h a t
Infer
This Page Intentionally Left Blank
CHAPTER 3
SYMBOLS, OSCILLATORY
INTEGRALS
AND S T A T I O N A R Y - P H A S E THEOREMS
1.
INTRODUCTION.
Let P ( x , D ) be a d i f f e r e n t i a l o p e r a t o r of degree m
C
n c o e f f i c i e n t s i n an open s u b s e t X of IR
.
m with
By F o u r i e r t r a n s -
formation, we can e x p r e s s it i n t h e form:
If w e assume t h a t
P = P(D)
i s e l l i p t i c with constant coeff-
i c i e n t s , we have s e e n ( s e e Chapter I , P r o p o s i t i o n 6.7 ) t h a t we can d e f i n e a p a r a m e t r i x F of P i n t h e form F = (5”)
(#).
L e t t i n g Q be t h e o p e r a t o r o f convolution w i t h F , we l i k e w i s e say t h a t t h i s i s a p a r a m e t r i x of P because t h e o p e r a t o r s P o Q and Q o P - I a r e r e g u l a r k i n g .
-
I
The o p e r a t o r Q i s e x p r e s s e d by
f
and we have proved t h a t t h e f u n c t i o n q = l i k e a homogeneous f u n c t i o n of degree -m; any multi-index a , we have
139
e
behaves f o r 151
.+
more p r e c i s e l y , f o r
+
m
(CHAP. 3)
SYMBOLS AND OSCILLATORY INTEGRALS
140
I;?cy
S(5)1
=
d(l+ \I\) -m-l I. )
When P(x, D ) i s e l l i p t i c w i t h v a r i a b l e c o e f f i c i e n t s , we a r e t h u s l e d t o t r y t o c o n s t r u c t a p a r a m e t r i x Q of P i n t h e form
(12.)
(Qu)(x)
=
where q ( x , 5 ) behaves f o r
+
+ = l i k e t h e function c j ( S ) .
For example, as a f i r s t approximation, we may use t h e f u n c t i o n
We a r e t h u s l e d t o d e f i n e and s t u d y a c l a s s o f such f u n c t i o n s q , termed symbols; t h e o p e r a t o r s a s s o c i a t e d w i t h t h e formula
( 1 . 2 ) w i l l be c a l l e d p s e u d o - d i f f e r e n t i a l o p e r a t o r s .
2.
SYMBOLS Let X be an open s u b s e t of Wn ( n 2 0 ) and l e t N be an
integer degree m
1.
2 E
We d e f i n e t h e space Sm(X
x
N
W ) of symbols of
IR.
DEFINITION 2.1 :
We denote by Sm(X x W N ) the s e t of
complex-valued functions a E
~ " ( x x RN ) such t h a t , f o r any
compact subset K of X and any multi-indices C such t h a t (2.1.1.1
laxP a,a
a(x,
ell
5
cIi +
a, 6
there e x i s t s
I.
w1
for x E K , ~
Ve say t h a t 8 i s the frequency variable.
E
.
R
~
(SEC. 2 )
141
SYMBOLS
EXAMPLE 2 . 2
A polynomial
:
5)
p(x,
F
=
ICY
coefficients a a
For m
E
c"(x)
E
9
with
is in S ~ ( X x B").
+
IR, t h e f u n c t i o n (x, 5 )
(1
+ 1%) 2 )m/2
is in
mn).
Sm(X x
I n o r d e r t o g e n e r a l i s e D e f i n i t i o n 2 . 1 t o c e r t a i n open sub-
-
s e t s of X
m u l t i p l i c a t i v e group R (x, 83
The
we n e x t i n t r o d u c e a number of concepts.
X B N ,
+
N
a c t s on X xIR
N
tel
(x,
,
t E $'
by:
A s u b s e t o f X xIR
is s a i d t o
be conic i f it i s i n v a r i a n t under t h i s a c t i o n .
We d e f i n e t h e conic s u p p o r t o f a f u n c t i o n a
E C"(X
N
x p(
o u t s i d e of which
r\o rc =
)
t o be t h e s m a l l e s t c l o s e d conic s u b s e t
i s zero.
a
If
r
cx
{ ( x , e l c r 1 8 4 0 ] ,and We (x, tell (x, el E r, t 5 I 3.
RN
x
,
we p u t
=
c
Let
DEFINITION 2 . 3 : N X xB
.
We denote by
r
N
a l s o p u t I R ~=IR 0
\ 0.
be an open conic subset of
s m ( r )the s e t of complex-valued functions
a E Cm(r) such t h a t , f o r any compact subset K of
r and f o r a n y
multi-indices a , 8, there e x i s t s C such t h a t : (2.3.1.
we p u t
)
\a! a:
s+"(rI
a(x, =
ell
u $(rI m
I CCI
+
, s-(rI
p \ IwI.1 =
for ( x ,
n s"'(rI m
e) E
K'
(CHAP. 3)
SYMBOLS AND OSCILLATORY INTEGRALS
142
m
Sm(I') i s obviously a v e c t o r sub-space of C (I');
a
E
Cm(I')
which i s homogeneous of degree
a function
m with r es pe ct t o 0 ,
( i . e . s a t i s f y i n g a ( x , t e ) = tma ( x , e l f o r t > 0, (x, 0 ) A function a
Sm(r).
E
l a r g e i s i n Sum(r).
E
r)
is in
Cm(r) which i s zero f o r 101 s u f f i c i e n t l y
We can immediately show t h e f o l l o w i n g :
c S"(r] c S"'(r) c s+"(r)
We have S'"(r]
PROPOSITION 2 . 4 :
for m < m'. Suppose
a,B
3 e,
6
Suppose a
, b E swl~l(r)
E S"(r)
a
E
cm(r); then
E
E Sm(r \ 0)
a
.
+m
In particular, S
.
(r)
-
b
E
S-OP(I').
then ab E S""'[r)
;
sm(r) if
-
b when a , b
E
I
and only if
i s a commutative a l g e b r a , and S-m(I')
i d e a l of t h i s ; we writ e a
a
a E
S"'(r]
+m
S
i s an
( r ) , with
Being a symbol i s a l o c a l p r o p e r t y i n t h e
s e n s e o f open c o n i c s e t s :
Let
PROPOSITION 2.5 :
x
x
N IR
(xol
.
e0)
Suppose E
r\
PROOF: we can assume
r;
0
a
E
c"(r)
, there
r
be an open conic subset
,
I f , f o r m y point
e x i s t s a conic neighbourhood
According t o t h e f i n a l p a r t o f P r o p o s i t i o n 2 . 4 ,
r
we put e =
=
r\
0
el
L e t K be a compact s u b s e t of
101 ; we have
Inf (XI
,
E K
E
> 0.
By h y p o t h e s i s ,
(SEC. 2)
f o r any
r
(xo, go) E 8 ) in
of ( x
0 )
143
SYlmOLS
I
0
t h e r e e x i s t s a c o n i c neighbnurhood V
such t h a t t h e upper bounds ( 2 . 3 . 1 ) a r e
satisfied for (x, 8 )
191
V,
E
By c o v e r i n g K by a f i n i t e
we s e e t h a t we o b t a i n t h e
0'
upper bounds ( 2 . 3 . 1 ) f o r (x, 8)
,
s
2
number of such neighbowhoods V
KC.
E
m
We s h a l l now s e e t h a t symbols of degree
PROPOSITION 2 . 6
N1 R ~ X R , ,J
,
Let f
T,
:
b
PROOF: C
-
__+
el
E srn(F2)I
Let K
1
r. 3
be an open conic sinbset of
,
( j = 1 , 2 ]
(XI
If
Let
:
r2 (Y,
be
wl
and homogeneous of
Cm
degree 2 ( i . e . commuting with the action ofm')
a = b
then
o
E
f
be a compact s u b s e t of
C
1
= (y(X,B),h(x,e))
r e s p e c t t o 8 , and
h
;
r 1; we
N2 , t h e r e 0
exist c, c' > 0
clel I Ih(x, 811 5 cllB1 f o r (x, 8) E K:
thus obtain (x, 9')
la(x, 911 2 C(1
E KY
( 2 . 3 . 1 ) f o r (x, 8 )
.
+
.
We
Ih(x, 911 1" I - C ' ( l + l B l ) m
We now e s t a b l i s h t h e upper bounds C
E
We p u t
i s homogeneous of degree 1 w i t h r e s p e c t
1
for
r2.
have
g i s homogeneous o f degree 0 w i t h
t o 8 ; s i n c e g(K ) i s compact inIR such t h a t
.
srn(r,] ,
f(K1) = K2, w i t h K2 = f ( K ) a compact s u b s e t of f(X,B)
a r e preserved
More p r e c i s e l y :
under c e r t a i n changes o f v a r i a b l e .
n,
0
K1 by r e c u r r e n c e on
/a1
+ 161.
We assume
144
(CHAP. 3 )
SYMBOLS AND OSCILUTORY INTEGRALS
them t o be s a t i s f i e d ( f o r any 101
+
.
I p-1
ax j
with Since
,
f
o
3% ( 2 . 3 . 1 ) for
Icy1
m) when
and any
Now, we have
so(rlJ ,
E
b
0
a ‘k
+ 181 5
axj ahk
E
,satisfy
f
p-1
aej
sl(r~)
E
.
s-’(rlI
upper bounds o f t h e t y p e
( f o r degrees r e s p e c t i v e l y
m, m-l), L e i b n i z ’ s formula proves t h a t t h e same i s a l s o t r u e o f
axj
(for degrees r e s p e c t i v e l y m, m - 1 ) .
aej a
thus obtain f o r
We
t h e upper bounds ( 2 . 3 . 1 ) when
The f o l l o w i n g i s a u s e f u l s p e c i a l c a s e of t h i s p r o p o s i t i o n . Suppose
r1 is
an open conic s u b s e t of X
x
IRN and suppose 0
f
is
t h e mapping c o r r e s p o n d i n g t o passage i n t o p o l a r c o o r d i n a t e s i n
: ! R
x
:R 3
e l +->f
(x,
Then i n t h i s c a s e
rl
degree 1,.of
r2
= f
(rl 1
[2.6.1.)
C
a E
f
(x,
ms e IeI) E x
x
sN-,
x
R+
i s a diffeomorphism, homogeneous of
o n t o t h e open conic s u b s e t
X x SN,l
Sm[rlJ
x R+
, and
P r o p o s i t i o n 2 . 6 shows t h a t :
if and o n l y i f a
0
f-”
E
sm(r2)
The two p r e c e d i n g p r o p o s i t i o n s a l l o w us t o g e n e r a l i s e t h e
.
145
SYMBOLS
(SEC. 2 )
d e f i n i t i o n of Sm( r )
.
DEFINITION 2 . 7
manifold
We define a conic m m i f o l d t o be a
:
equipped with a Cw action of the m u l t i p l i c a t i v e
M
-k
group IR such t h a t : (2.7.1 .]
For any h
E
M, there e x i s t s a diffeomorphism
homogeneous of degree 1, of an open conic neighbourhood of A i n M onto an open conic subset
DEFINITION 2.8 :
m
E
t h a t , f o r any h
Suppose we have a conic manifold M and
E
EXAMPLE 2 . 9 :
E
fibre.
B
If
E
a
D
f'
x
satisfying
.
E Sm(r)
i s a v e c t o r bundle on a C
W
manifold,
E \ 0 , t h e bundle E deprived o f i t s 0 - s e c t i o n .
we denote by
+
C w ( M ) such
E
M, there e x i s t s a diffeomorphism
the conditions of (2.7.1)w i t h
For t
r ofnn x IRNa .
Sm(M) denotes the space of f m c t i o n s a
R.
x, u
, we
t
t a k e as a c t i o n m u l t i p l i c a t i o n by
i n each
Then E \ 0 , and any open conic s u b s e t of E \ 0 , i s a
conic manifold.
If M i s a conic manifold, any C
homogeneous of degree
m
W
function,
i n M, i s i n Sm(M).
PROPOSITION 2 . 1 0 :
i) Let M1, M2 be two conic manifolds- and l e t f : MI
_+
then
k$ a = b
be a o
f
C"
mapping of degree 1.
E Sm(M1)
.
If
b E Sm(t$)
,
146
(CHAP. 3)
SYMBOLS AND OSCILLATORY INTEGRALS
i i ) Let M be a conic manifoZd, and Zet
and m' with
aj €
S"'(U~)
a
5
m
%
J
(mod. Sm'(Uj E
PROOF:
n uk, we
n U,)
have
f o r aZZ j, k
E
J.
Then
Sm(M) unique modulo Sm'(M) such t h a t
a .(mod. Sm'(U
J
Iuj
. Let there be
5 m' < m
such t h a t , i n u .
there e x i s t s a a
-=
)J € 3
j
Suppose we have
be a covering of M by upen conic s e t s . r n E B
(U
.))
J
f o r a22 j
E
J.
P a r t ( i ) i s an immediate consequence o f PropI n o r d e r t o e s t a b l i s h ( i i ) ,we f i r s t observe t h a t
o s i t i o n 2.6.
+
t h e q u o t i e n t space M' = M / B
i s a manifold and t h a t t h e
u is J j -1 c o n i c , we have U = p (U!); s i n c e t h e U! form an open c o v e r i n g j J J of M', we know t h a t we can t h e r e f o r e f i n d a l o c a l l y f i n i t e sub-
p r o j e c t i o n p: M
-+
M I i s crn; we p u t U! = p ( u . 1 ; s i n c e J
c o v e r i n g , and hence by i n v e r s e image under p a sub-covering of t h e c o v e r i n g of M by t h e U . which i s l o c a l l y f i n i t e i n t h e J c o n i c s e n s e ( i . e . such t h a t f o r any compact K o f M y t h e r e i s o n l y a f i n i t e number of i n d i c e s c o r r e s p o n d i n g t o a non-empty i n t e r s e c t i o n withIR
4-
K).
It w i l l t h e r e f o r e be s u f f i c i e n t t o
prove ( i i )when t h e c o v e r i n g i s of t h i s form. of u n i t y , we know t h a t t h e r e e x i s t a ' j
SUPP a'. c
E
ui
C"(Ml)
By a p a r t i t i o n such t h a t
I C Y ~ = 1 i n M'; by p u t t i n g CY j = CY' j 0 P J J we o b t a i n f u n c t i o n s a j E f ( M ] homogeneous of degree 0 ,
,
w i t h Supp a . c U j J
,
aj = 1 i n M.
3
I
(SEC. 2 )
We p u t
147
SYMBOLS
a =
aj
cyj
(where
c1
j by 0 o u t s i d e U
mjlu
of
.aj]
3 k
E
J we have:
a
=
We have a E Sm(M), a n d , for
,
c:[m
denotes t h e extension
j “j
E m j l u k ak
=
a.1
j J luk j m’ (mod S”’(u,]) and a a %(mod S (Uk)); hence t h e I’k existence of a . The uniqueness o f a modulo Sm’(M) i s
Iuk j therefore
obvious.
We now c o n s i d e r a c o n i c manifold N , and a c o n i c submanifold M o f N ( i . e . a sub-manifold M of t h e manifold N which 4-
i s s t a b l e under t h e a c t i o n o f IR
in M ) .
of t h e p r e c e d i n g p r o p o s i t i o n shows t h a t
If a
&IM
E
Sm(N) , p a r t ( i )
E Sm(M)
.
Conversely , we have :
Let M be a ZocaZZy cZosed conic sub-
PROPOSITION 2.11 :
manifoZd of a conic mrmifoZd N; for a
ii E
S”’[N)
PROOF:
extending
E
Sm(M), there e x i s t s
a.
We f i r s t show t h a t t h e p r o p o s i t i o n i s t r u e when we
r e s t r i c t o u r s e l v e s t o a s u f f i c i e n t l y s m a l l conic neighbourhood ( i n N) o f any p o i n t of M.
By u s i n g t h e diffeomorphism
x
of
(2.7,1),t h e n by t a k i n g p o l a r c o o r d i n a t e s inIRN ( s e e (2.6.1)), we can reduce t o N = Y xIR
+ , where
0
Y i s a manifold.
The pro-
j e c t i o n on Y of t h e sub-manifold M of N i s a l o c a l l y c l o s e d sub-manifold X of Y and we have M = X x lR
+
since M i s conic.
148
(CHAP. 3 )
SYMBOLS AND OSCILLATORY INTEGRALS
By taking local coordinates in Y we reduce to the case where N = (U x V ) x
R',
{ 0
M = (U x
R ' neighbourhoods of 0 in IRp, I "
we define a = a(y, z ; t )
1) X ' R
.
where U, V are open a = a(y ; t)
If
E
by
N
a(y,
z ;
E
Sm(M)
,
t ) = a(y, t)
In order to prove that the proposition is true globally, we of N which j are locally finite (in the conic sense) and, as in the proof consider a covering of M by open conic subsets W
of Proposition 2.10, we construct functions
CY
aj = 1
and
j
pj E c ~ ( N )
,
c supp B J. c W jl Bj'l J in the neighbourhood of M. We
homogeneous of degree 0, with supp in supp
cyjl
cy.
; now, we have seen that there with a j cuj*a exist a E Sm(Wj) such that EL -- a j j\wj n M jlWj nM * H H If we put a = p j aj then E Sm(N] and
have a
aj
I
N
a d
,
=c .
N
alM
[B,.ZjIIM
-
aj = a
In applications, we encounter symbols which possess an expansion into homogeneous components in the following sense:
DEFINITION 2.12 : a
E
Sm(M) i s a c l a s s i c a l symbol of degree
f o r any j
E
IN a f w l c t i o n
degree m-j, such t h a t
any i n t e g e r k
2
1.
symbols of degree
a,
(a
-
*
E Cm(M), k-I j=o
aj-
We denote by S:l(M) m.
We say that
Let M be a conic manifold.
m i f there e x i s t s
homogeneous of
) E SWk[~)
for
the space of c l a s s i c a l
(SEC. 3 )
149
ELLIPTIC SYMBOLS
I n t h e c a s e of t h e space X
a E C”(X
t h e space of t h e
X I R
x RN)
It i s c l e a r t h a t t h e f u n c t i o n s a
N
, we
m
denote by Scl(X
x
#)
such t h a t
j
a r e uniquely determined f o r
; t h e f u n c t i o n a0 i s ‘ c a l l e d t h e p r i n c i p a l p a r t ,
a E Szl(Uj
m, of
homogeneous of degree
EXAMPLE 2.13 :
a.
We can show t h a t t h e symbols c o n s i d e r e d
i n Example 2 . 2 a r e c l a s s i c a l , and t h a t t h e y admit r e s p e c t i v e l y a s p r i n c i p a l p a r t of degree m:
3.
ELLIPTIC SYMBOLS DEFINITION 3 . 1 :
rn
E
IR.
Suppose M
We say t h a t a symbol a
i f there e x i s t s b
E
S-”(M)
E
-3
a con:> manifold and
Sm(M) i s e l l i p t i c of degree m
with d e b
E
1
(modulo
s-’(M))
Note t h a t t h i s d e f i n i t i o n only depends on t h e c l a s s o f
S~(M)/S“+’ (MI
a
in
.
From P r o p o s i t i o n 2 . 1 0 , e l l i p t i c i t y i s a l o c a l p r o p e r t y i n t h e s e n s e o f conic neighbourhoods; i n o r d e r t o s t u d y t h i s , we can t h e r e f o r e reduce t o t h e c a s e where M i s an open conic s u b s e t
of
x
XIRE.
(CHAP. 3)
SYMBOLS AND OSCILLATORY INTEGRALS
150
X x Bt; a symbol a
r
Let
PROPOSITION 3 . 2 :
be an open conic subset of
s m ( r )i s e l l i p t i c of degree
E
only i f , f o r any compact subset K of
, if and
m
r, there e x i s t s
c > 0 such
that
el\
la(x,
(3.2.1.)
\el"
c
2
(x,
for
PROOF: K1
e] E
We assume t h a t ( 3 . 2 . 1 ) i s s a t i s f i e d ;
c$c
s u b s e t s of
... c r
K.
J
c
...
\el
KC and
2
E E
(u
c2
2
...
2
2
lR+ and t h a t F i s a s u b s e t of
(K5+1<j
)
since
*
be an exhaustive sequence of compact
and we l e t c 1
j 21
C
we l e t
c
2
j
..
be a sequence of
a s s o c i a t e d c o n s t a n t s , i n accordance w i t h ( 3 . 2 . 1 ) . that
-1
K.
J
c c Kj+l
r,
We suppose
and we p u t
.
It can e a s i l y b e shown t h a t A i s c l o s e d , and we t a k e a f u n c t i o n p E C m ( T ) e q u a l t o 1 on A and w i t h support i n B.
n o t v a n i s h on B , we d e f i n e b t o prove t h a t
(x, 0 )
E
r E S-"(r)
I! E c"(r) .
, we
Since
does
We p u t r = ab
-
1;
observe t h a t any p o i n t
T admits a conic neighbourhood To such t h a t f o r E
a
s u f f i c i e n t l y s m a l l , we have To c A , and s i n c e
r
E
> 0,
i s zero i n t h e
neighbourhood of A it s u f f i c e s t o apply P r o p o s i t i o n 2 . 5 .
Like-
(SEC. 3 )
wise, i n o r d e r t o prove t h a t b o r i z e t h e d e r i v a t i v e s of cy
axl 0
151
ELLIPTIC SYMBOLS
b =
a:,(:)
E
S-m(r) it i s s u f f i c i e n t t o m a j -
on A .
b
On A we have
and we have a l r e a d y observed i n Chapter I
( 6 . 8 . 2 ) t h a t t h i s i s a f i n i t e l i n e a r combination of terms of t h e
form
a$1
a
.... aB .
J.a
a
lBJ +
with
j.1-1
....+ le,I
bl
=
;
a
we t h e r e f o r e immediately deduce t h e d e s i r e d upper bound o f 3 . b x,e i n any domain ( K C ) € , w i t h K a compact s u b s e t o f
Conversely, i f w e t a k e a compact s u b s e t K of since
Ir[x,
, there
r E S-I(r)
Q)\ 5
Ib(x, 0 ) \
$
5 1(1
c la.bl 2 1 i n (Kc)' 2
for
exists
+ lQ\)-"
for (x,e)
and f i n a l l y
a
i n ( K C ) € which proves t h a t
DEFINITION 3 .3 :
l(1 E
and p u t r = ab
We say t h a t a symbol a
E
S
1,
and
+
Consequently we have
KC.
E
18/)'m
1a(x,Q)1
2
21
s a t i s f i e s (3.2.1).
Suppose M i s a conic manifold, and m +m
-
> 0 such t h a t
E
E (KC)e
(x, 0)
r
r0 .
(M) i s e l l i p t i c of degree
m
E
IR
at a
point of M i f it i s an e l l i p t i c symbol of degree m in an open conic neighbourhood of t h i s p o i n t .
of p o i n t s of M a t which
a
We denote by E l l ( a ) the s e t
i s elliptic;
t h i s i s an open conic
s e t whose complement i s a closed conic s e t called the characteri s t i c s e t of
a
and denoted by c h a r ( a )
.
.
(CHAP. 3)
SYI’BOLS AND OSCILLATORY INTEGRALS
I n t h e c a s e of a symbol
a
#)
Sm(X x
f
, we
define t h e e l l i p t -
i c i t y , E l l ( a ) and c h a r ( a ) by c o n s i d e r i n g a
EXAMPLE 3.4: part a
4.
0’
i s a c l a s s i c a l symbol w i t h p r i n c i p a l
a
If
homogeneous o f degree
, we
m
-1
have c h a r ( a ) = a.
(0).
ASYMPTOTIC EXPANSIONS OF SYMBOLS
We can meaningfully c o n s i d e r a formal s e r i e s o f symbols ( a . ) J with degrees m
j
tending t o
-
,
m
when M i s a c o n i c manifold.
We s t a t e :
DEFINITION
4.1:
Let ( a . ) J
rn.
symbols
a:
be a c o w t a b l e family of j
,
~
E S J [ ~ ) ; we assume t h a t for any r e a l r , there
J
i s only a f i n i t e number of i n d i c e s say t h a t a f u n c t i o n a
E
Cm(M)
j such t h a t m .
satisfies a
-
J
aj
2
r.
We
i f there
J
e x i s t s a sequence of r e a l numbers (JJ) tending t o k f o r a l l k:
such t h a t
For example, w i t h t h e n o t a t i o n o f D e f i n i t i o n 2.12, we have
& - r _ 7 a j . j
2 0
THEOREM 4 . 2 :
Let
(aj)j
be a f a m i Z y of symbols as in
(SEC.
4)
D e f i n i t i o n 4.1.
Then there e x i s t s a
s-"(M), such t h a t a PROOF:
15 3
ASYMPTOTIC EXPANSIONS
-
The uniqueness o f
a
the existence of
, we
, with
aj
a
Sm(M) unique modulo
E
m = max m
j
j.
modulo SVm i s immediate.
To prove
w i l l not r e s t r i c t t h e g e n e r a l i t y , bear-
i n g i n mind P r o p o s i t i o n 2.10 ( T i ) , by assuming t h a t M = open c o n i c s u b s e t o f X x 1R".
r
i s an
We b e g i n w i t h t h e p a r t i c u l a r c a s e
where J = W and where t h e sequence (m.) i s s t r i c t l y d e c r e a s i n g J (and tends t o
-
x(0) = 0 if
m).
S
Suppose we have
X E C=(R~]
1/2, and x(0) = 1 i f 101 2 1.
e x h a u s t i v e sequence o f compact s u b s e t s o f
r.
such t h a t Let (K.) be an J
We can f i n d an
i n c r e a s i n g sequence ( t. ) such t h a t ; J
(we can e a s i l y c o n s t r u c t t h e t that for
j
,
j f i x e d , t h e r e . i s o n l y a f i n i t e number of upper bounds
( 4 . 2 . 1 ) t o b e r e a l i s e d , and t h a t
if 18 I
i n s t e p w i s e f a s h i o n by o b s e r v i n g
)
+
.
(
l i8
i)mj-l rn - 0
We t h e n p u t
+ m
X ( e ) aj(Xt 8 ) which i s c l e a r l y i n C m ( r ) s i n c e j=o j t h e sum i s l o c a l l y f i n i t e ; t h e convergence of t h e s e r i e s B(x, 95
2-j k- 1
a
=
a l l o w s us t o deduce from ( 4 . 2 . 1 ) t h a t
- j=o aj
rnk
E S
(r) for any
i n t e g e r k t 1.
1 54
We now pass on t o t h e e x i s t e n c e o f
p o s s i b l e values of m
bk =
-
rn
j
k bk
i n the general case;
j
j ranges over J .
when
n k E S (r) , and we c o n s i d e r a
aj
=n
a
we
t h e s t r i c t l y d e c r e a s i n g sequence of t h e
denote by ( n k l k
a
(CHAP. 3)
SD3OLS AND OSCILLATORY INTEGRALS
.
Then f o r a l l R
E
We p u t m
E
C
(r)
such t h a t
IN w e have:
k E M
mined by
n
-1
2
r
n
t h e requirements s i n c e
REMARK 4.3:
.
> nr+l
F i n a l l y , t h e symbol
a -+
r+l
'
m
We can confirm d i r e c t l y from t h e d e f i n i t i o n s Let ( J h ) h for h
be an a r b i t r a r y p a r t i t i o n of t h e s e t o f i n d i c e s J ;
%
(note t h a t
-
%
.
a j E J h E
S %(M),
a
Then we have
j mh=
where
m
max j E H
REMARK 4 . 4 :
J
-
]
E
H,
ah h E H
.
I n t h e case where J = TN and where t h e sequence
a
( m . ) i s d e c r e a s i n g , we have J any i n t e g e r k, we have
c a s e where M =
meets
.
- 0
t h a t we have t h e f o l l o w i n g a s s o c i a t i v i t y p r o p e r t y .
suppose
a
r,
[a
-
a
LV
k
J Z O
a.)
J
E
5
i f and o n l y i f f o r
Srnktl(M)
.
In the
j=o t h i s condition i s equivalent t o t h e following:
for any i n t e g e r k , and f o r any compact s u b s e t K o f r , t h e r e e x i s t s k "k+l- 1 c such t h a t la: aB, (a-( )1a j I ( x , I ~ ( +1 \ e l ) j-o
1 55
TOPOLOGY
1.1
f o r (x, 9 ) E KC and
+
k
I n f a c t , i n o r d e r t o majorise t h e d e r i v a t i v e s of o r d e r k' > k , it suffices t o write
k
k'
k'
and t o n o t e t h a t
k'
m
j=k+l
5.
TOPOLOGY ON THE SYMBOL SPACES
Let
r
be an open conic s u b s e t o f X x IR
subset K c
r
N
.
With any compact
j we a s s o c i a t e a semi-norm on
and any i n t e g e r
Sm(r ) by p u t t i n g
m( r ) a metrizable topological vector
t h e s e semi-norms d e f i n e on S
space s t r u c t u r e , because it s u f f i c e s t o consider an exhaustive
(5).
sequence of compact s u b s e t s
We have t h e approximation r e s u l t :
PROPOSITION 5.1:
If a
E
Sm(r), we take
p E
t o 1 i n the neighbourhood of 0 rmd we put p,[ Then
%=
pk.a
E S-(r)
and f o r a l l rn'
>
9)
Cz(lRN -i
p
identical
(j).
m, the sequence %
156
(CHAP. 3 )
SYMBOLS AND OSCILLATORY INTEGiiALS
converges t o
PROOF:
f o r the topology of
a
If CK
i s a semi-norm on
,j,m'
sm'(r). sm' ( r ),
since a
E
sm(r)
t h e r e e x i s t s C such t h a t
For'cc' = 0 , t h e v a l u e of t h e l e f t - h a n d s i d e i s O(km-m') because 18
I
2
k for 8
m-m'
a g a i n O(k
e
E
supp. ( 1
- pk)
) because w e have
-
E supp $(I pk)
.
;
i f a ' # 0 the left-hand side i s
k 5
\el
5 2k for
Consequently, by u t i l i s i n g L e i b n i z ' s
formula, we deduce t h a t
which proves t h e p r o p o s i t i o n . We can t h u s immediately deduce t h e f o l l o w i n g r e s u l t concerning e x t e n s i o n by c o n t i n u i t y :
PROPOSITION 5 . 2 :
space of
Let R be a linear mapping defined on the sub-
s-"(r) , formed
by the functions which vanish f o r
Ie I
large, and with values i n a Frechdt space F. We assume t h a t f o r all m
E
m,
2
i s continuous f o r the topoZogy induced by
Then there e x i s t s a unique extension of
sm(r).
l, : S+"(r] -. F which i s
(SEC. 6 )
conkinuous on each
6.
sm(r ) .
VARIOUS GENERALISATIONS
(6.1.) Up in C
157
GENERALISATIONS
.
t o now, we have considered symbols with v a l u e s
We a r e now going t o d e f i n e t h e symbol-sections
of a
Complex v e c t o r bundle above a conic manifold, under s u i t a b l e assumptions.
Let
M be a conic manifold and l e t m
on M; we s p e c i f y a C
a c t i o n ofIR
+
F be a v e c t o r bundle
on F which commutes, v i a t h e
p r o j e c t i o n of F onto M y with t h a t ofIR
+
in M
We can show t h a t t h e r e e x i s t s , above a conic neighbourhood of any p o i n t of M y a t r i v i a l i s a t i o n of F commuting w i t h t h e
+
allowing a r e d u c t i o n t o M = T ( a n open conic
actions o f B s u b s e t of Rn
N
x
Do) , i . e . F
I
r
d
x C ; we p u t i n t h i s case
Sm(M, F ) = ( S m ( r ) ) d , hence t h e g e n e r a l d e f i n i t i o n of Sm(M, F ) by r e s t r i c t i o n and t r a n s p o r t . If F
a
z(F,,
F),
where F
F
1’ 2
t h e preceding t y p e , we d e f i n e Sm(Y;
a r e two bundles on M of
z(F,,
F2)] accordingly.
158
SYMBOLS AND OSCILLATORY INTEGRALS
We say t h a t
E Sm(M;
a
(CHAP. 3 )
i s r i g h t e l l i p t i c of
s(F1, F,))
degree m ( r e s p : l e f t e l l i p t i c ) i f t h e r e e x i s t s b
E S-”(M; s(FZ,
(resp : ba
-
IF
F,))
such t h a t ab
E
g(F1, F,))
S-’(M;
a
t h e c a s e of a c l a s s i c a l symbol part a
0
- 1F2 E S”(U:
.
e(F2, F,))
We can show t h a t , i n
w i t h homogeneous p r i n c i p a l
of degree m y t h e r i g h t e l l i p t i c i t y ( r e s p : l e f t ) of
a
is e q u i v a l e n t t o t h e r i g h t i n v e r t i b i l i t y ( r e s p : l e f t ) o f ao.
(6.2.)
I n f a c t , t h e r e i s not j u s t one s i n g l e t y p e of
symbols b u t r a t h e r we can s a y t h a t t h e r e a r e almost as many o f them as t h e r e a r e t y p e s of problems, and we need t o be a b l e t o a d a p t one t o t h e o t h e r .
For example, i f P(D) i s a h y p o e l l i p t i c
operator then i n order t o include functions o f t h e type
48
t h e upper bounds ( 6 . 9 . 2 ) i n Chapter I l e a d t o t h e
m
.
( r) With 0 < p < 1 m we d e f i n e t h e space S ( r ) o f symbols o f degree m and of t y p e p P f o l l o w i n g g e n e r a l i s a t i o n of t h e s p a c e s S
by r e p l a c i n g t h e upper bounds ( 2 . 3 . 1 ) by
\a, B a,W
a(x,
where 6 = 1 - p .
ell
I
c
(1
+
\ e l lm-P I 4 + s l e l
We can prove t h a t a l l t h e p r e c e d i n g r e s u l t s
i n t h i s c h a p t e r g e n e r a l i s e , w i t h o n l y minor m o d i f i c a t i o n s , t o symbols of t y p e p ( s e e HORMANDER C31, C51 f o r symbols which a r e
s t i l l more g e n e r a l ) .
(SEC.
7.
7)
OSCILLATORY INTEGRALS
159
OSCILLATORY INTEGRALS
7.1.
INTRODUCTION:
When
2 (5)
i s defined e x p l i c i t l y ,
e x p r e s s i o n (1.1)i s w r i t t e n f o r m a l l y as
however t h i s i n t e g r a l i s n o t a b s o l u t e l y convergent w i t h r e s p e c t to
5.
The o b j e c t i v e of t h i s s e c t i o n i s t o prove t h a t we can i n f a c t work w i t h ” i n t e g r a l s ” of t h e t y p e (7.1.1)and w i t h i n t e g r a l s which are even more g e n e r a l , as i f t h e y were a b s o l u t e l y convergent.
We s h a l l e x p l a i n t h e i d e a u n d e r l y i n g t h e method
i n t h e p a r t i c u l a r c a s e where P ( x , D ) i s t h e i d e n t i t y o p e r a t o r i n one dimension.
I n o r d e r t o g i v e a meaning t o t h e i n t e g r a l
we can c l e a r l y w r i t e it as an i t e r a t e d i n t e g r a l
however t h i s method i s n o t v e r y e a s y t o h a n d l e .
Since t h e
d i f f i c u l t y e x i s t s f o r 151 l a r g e , it i s s u f f i c i e n t , a f t e r trunc a t i o n with r e s p e ct t o i n t e g r a t e for 151 t 1.
6, t o o n l y c o n s i d e r t h e case where we We n o t e t h a t :
160
SYMBOLS
AND OSCILLATORY INTEGRALS
(CHAP. 3 )
which l e a d s u s , a f t e r formal i n t e g r a t i o n by p a r t s w i t h r e s p e c t to
y , t o put
Y € B
and t o d e f i n e t h e l e f t - h a n d s i d e by t h e a b s o l u t e l y convergent i n t e g r a l appearing i n t h e right-hand s i d e , t h i s being l e g i t i m a t e because it can e a s i l y be shown t h a t t h e r i g h t - h a n d s i d e o f
(7.1.2.)c o i n c i d e s w i t h t h e i t e r a t e d i n t e g r a l
T h i s t y p e of g e n e r a l i s e d i n t e g r a l h a s been s t u d i e d systema t i c a l l y by H@"DER
C51
under t h e name o f o s c i l l a t o r y
integral.
7.2.
DEFINITION OF OSCILLATORY INTEGRALS:
subset. of.IRn, ( n
2
0), l e t
cp E C"(x
x
#\
0)
Let X be an open be a r e a l - v a l u e d
7)
(SEC.
161
OSCILLATORY INTEGRALS
f u n c t i o n which i s homogeneous o f degree one w i t h r e s p e c t t o
e E R N \ o
and l e t
a E Sm(X x # ) be a symbol; we
consider t h e i n t e g r a l
It i s c l e a r t h a t t h i s i n t e g r a l i s a b s o l u t e l y convergent when
m
-
N , and i n t h i s case it depends continuously on a
E
Sm ;.
w e s h a l l prove t h a t we c a n , u s i n g i n t e g r a t i o n s by p a r t s , make it meaningful f o r a r b i t r a r y
m, i f
Q
We observe t h a t t h e mapping
(7.2
2.)
a E
s-"(x
x RN)
has no c r i t i c a l p o i n t s .
9, :
c
=
a(a)
1~(8u) E
c
i s a l i n e a r form; i n o r d e r t o demonstrate t h e c o n t i n u i t y of R i n t h e sense o f P r o p o s i t i o n 5 . 2 , we e s t a b l i s h t h e f o l l o w i n g :
PROPOSITION 7.3 :
f o r (x, 8) E X x
F?
If we assume t h a t
d x,
e
cp(x, 0)
f
\ 0 , then there e x i s t s a first-order
d i f f e r e n t i a I operator
PROOF:
For an o p e r a t o r o f t h e form (7.3.1), we have
We make t h e right-hand s i d e i d e n t i c a l t o 1 by p u t t i n g
o
p
where
Since
Q
- p)
(1
(CHAP. 3 )
SYMBOLS AND OSCILLATORY INTEGRALS
162
E
i s i d e n t i c a l t o 1 f o r 10 I 5 1 and where
CIIRN)
h a s no c r i t i c a l p o i n t s , it i s c l e a r t h a t
-2
D E S
(X x
N
R )
and c o n s e q u e n t l y t h e c o e f f i c i e n t s
a j y bj, c are i n t h e s p a c e s i n d i c a t e d .
It i s u s e f u l f o r l a t e r work t o n o t e t h a t t h e t r a n s p o s e tL
b J l c ' i n t h e same symbol s p a c e s .
o f L h a s c o e f f i c i e n t s a:,
This lemma l e a d s us t o s t a t e :
DEFINITION 7 . 4 :
x
x(#
\ 0)
r is an open conic subset of
If
rp E
we say t h a t a function
c"(r)
is a
phase function if it is homogeneous of degree 1 with respect t o with r e a l values and such t h a t d
8,
xl
PROPOSITION 7 . 5 : X x[RN
\ 0)
,
If
Q
e
Q(X,
9)
f
0 on
r.
is a phase function on
then:
i ) the linear form 11 in ( 7 . 2 . 2 ) extends uniquely i n t o a
linear form 8 on -s+"(X
x
#) , which is continuous
on any Sm.
This extension is termed an o s c i l l a t o r y integraZ and i s denoted by (7.5.1
.)
A(,)
=
i(f'cx'
" a(x,
0) u(x) dx d 6
.
(SEC.
7)
For a
E
163
OSCILLATORY INTEGFULLS
m
, we
S
have the expression
when k > m + N, where L i s taken as i n 7 . 3 . ii) I f
pk
i s defined as i n Proposition 5.1, then f r
iii) For
a IJ
-
f i x e d i n Sm, the mapping
1 p 1 E
E cp1
C
i s a d i s t r i b u t i o n of order k (for k > m+N) which we denote by
(7.5.4
e
i
5
eiQ(x$ ,.(a)'
0) dQ
and which we caZZ the i n t e g r a l d i s t r i b u t i o n associated with the phase
tp
cmd the amplitude
PROOF OF i ) :
7.3, r e p l a c e e*
If
a
a
E
by Lk(e*)
S-m, we c a n , by v i r t u e of Lemma
i n (7.2.1), and by i n t e g r a t i o n by
p a r t s we o b t a i n
ela(x)ldx
de.
Furthermore, t h e p r o p e r t i e s o f t h e c o e f f i c i e n t s of tL imply t h a t
t~ maps SP c o n t i n u o u s l y i n t o sP-' Consequently, i f t h e i n t e g r a l which
a
E
f o r any r e a l v a l u e p .
Sm and i f we choose k such t h a t k > m+N,
OCCUTS
depends c o n t i n u o u s l y on
on t h e r i g h t - h a n d s i d e of
a
E
(7.5.5)
Sm; t h u s , it s u f f i c e s t o a p p l y
164
(CHAP. 3)
SYMBOLS AND OSCILLATORY INTEGRALS
Proposition 5.2.
I n o r d e r t o prove i i ) , we u t i l i s e t h e c o n t i n u i t y of R on
m'
any subspace S
and we conclude by u s i n g P r o p o s i t i o n 5.1.
To prove i i i ) , we n o t e t h a t t h e r e p r e s e n t a t i o n
7.5.2)
allows us t o m a j o r i s e I ( a u ) by 0
when k > m+N.
O s c i l l a t o r y i n t e g r a l s which depend on one parameter behave l i k e o r d i n a r y i n t e g r a l s , i n t h e s e n s e t h a t we c a n , under c e r t a i n assumptions, d i f f e r e n t i a t e or i n t e g r a t e under t h e i n t e g r a l s i g n
ii* PROPOSITION 7 . 6 : Q
Let Y be an open subset of BP and Zet
be a phase f u n c t i o n on
dx,e Q
(x, y ,
u E C ~ ( Xx Y )
o
0)
,
X x Y x
.
we p u t f o r
If
a y
E
(
E
\~0) ~ such t h a t
S ~ ( Xx Y x
R ~ ) and
Y
cc
Then i ) We have
F E f(y)
and we can d i f f e r e n t i a t e m d e r
(SEC. 7 )
165
OSCILLATORY INTEGRALS .c
A s i n L e m a 7.3 we can again undertake w i t h t h e
PROOF OF ti): parameter
t h e c o n s t r u c t i o n of an o p e r a t o r
y
L[x, y, e, a, a,
J
E
ax)
So[X x Y x
with c o e f f i c i e n t s
#I,b,,J
c
E
S-'(X
x Y x
#)
.
If k > m+N,
then ( 7 . 5 . 2 ) shows t h a t w e have
=
F(Yl
/,/e%
(tlkl[au 1 dx d 0
and s i n c e t h i s i n t e g r a l i s
a b s o l u t e l y convergent , t h e u s u a l Fubini theorem allows us t o d r a w t h e r e q u i r e d conclusion.
PROOF OF i ) : symbol ition
a
By t r u n c a t i o n , we can reduce t o t h e case where t h e
i s zero f o r 18 I 2 1.
If we have p k ( 8 ) as i n Propos-
5 . 1 , we denote by F k ( y ) t h e f u n c t i o n o b t a i n e d when a
replaced by
%
is
= a.pk.
The o p e r a t o r L above allows us t o adapt t h e proof of P r o p o s i t i o n
7.5
i ) and i i ) t o t h e case where we have a parameter, i n o r d e r =
lim Fk(y) uniformly w i t h r e s p e c t t o y. k'w Fk E C y ( Y ] ; if a i s a multiFurthermore, it i s c l e a r t h a t
t o prove t h a t
F[y]
index, we immediately o b t a i n t h e expression
166
SYMBOLS AND OSCILLATORY INTEGRALS
(CHAP.
3)
with
Consequently, t h e argument used f o r Fk can s t i l l be used t o prove that
uniformly w i t h r e s p e c t t o y .
-
Henceforth, we s h a l l omit t h e s i g n
on an o s c i l l a t o r y i n t e g r a l
s i n c e P r o p o s i t i o n 7.6 allows us t o manipulate it l i k e an o r d i n a r y integral.
We can g i v e an i n c l u s i o n f o r t h e s i n g u l a r support of a d i s t r i b u t -
i o n of t h e t y p e ( 7 . 5 . 4 ) .
PROPOSITION 7.7:
We p u t
')a(x,
all
8
Suppose we have
E RoN
;
.
\ 0
X x RN
0
then the d i s t r i b u t i o n
c {x E X I there exists
PROOF:
J?
8)dB s a t i s f i e s
s i n g SUPP A
(7.7.1.)
=
If cp i s a phase f m c t i o n on
and if we have a E sm(x x RN) A = /eiv(xs
R",
x
0
E
X
8f 0
such t h a t
,
Q ~ ( x ,B)
e) f
Q ~ ( X ~ ,
t h e homogeneity and t h e c o n t i n u i t y of
t h a t t h e r e e x i s t s an open neighbourhood V o f xo such t h a t
o Q
I
03 for show
(SEC. 8)
9)
cpL(X,
167
INTEGRAL OPERATORS
-
f 0
f o r (x,
a) E
x
V
N
Ro
.
Furthermore, P r o p o s i t -
7.6 shows t h a t t h e mapping
ion
V 3 x
=
A(x)
eiY(”,
‘)a[x,
9)d8
E C
i s Cm and s i n c e we have
we t h e r e f o r e deduce t h a t inclusion
8.
IV
E
, which
C”[V]
demonstrates t h e
(7.7.1).
INTEGRAL OPERATORS ASSOCIATED WITH A PHASE AND AN AMPLITUDE
CONSTRUCTION 8.1: Q
a
A
E
Let X , Y b e open s u b s e t s o f lRn
be’ a phase f u n c t i o n i n Sm[X x Y x
.K E J1(X
x Y)
N
R )
.
X x ’f x P N
\
0)
Bp ;
let
and l e t
We c o n s i d e r t h e i n t e g r a l d i s t r i b u t i o n
a s s o c i a t e d w i t h t h e phase rg and t h e amplitude
a
i.e. l8.1.1.)
Let
< K, w > =
A : f(Y)
For w E f l y )
icp(x’yse)
J’[X)
, Av
&(x, y,
0 ) w[x, y) dx dy d9,
be t h e o p e r a t o r w i t h k e r n e l K .
i s t h u s t h e d i s t r i b u t i o n i n X d e f i n e d by
168
(CHAP. 3)
SYMBOLS AND OSCILLATORY INTEGRALS
(8.1.2.)
< Av, u z
e+(x,y,el
a(x, y,
e)
u(x) ~ ( y )dx dy d o ,
and A i s termed t h e i n t e g r a l o p e r a t o r a s s o c i a t e d w i t h t h e phase and t h e amplitude
Q
a
i ) If K i s the kernel of A, we have
PROPOSITION 8 . 2 : t8.2.1.)
s i n g supp K
e c
iii) If
d .
.
there e x i s t s
c {(x, y) E X x Y I
~t with
cp(x, y , 0)
Ti(.,
f
0
y,
el
for
=
.
o 3
N
,
(x, Y, 8 ) E X x Y x 3,
x, e then A extends i n t o an operator which i s continuous from E ~ ( Y )into
P(x]
.
PROOF: i) f o l l o w s by a p p l i c a t i o n of ( 7 . 7 . 1 ) t o K . For i i ) , P r o p o s i t i o n 7.6 and ( 8 . 1 . 2 ) show t h a t i f m
t h e n Av i s t h e C
(63.2.)
,
v E f(Y)
f u n c t i o n i n X d e f i n e d by
(Av)(x)
ei~(x9y,91 , . ( a
=
y,
e)
v(y) dy d e
.
The c o n t i n u i t y of A i s o b t a i n e d by m a j o r i s i n g a semi-norm
'K, j
(A) a f t e r having r e p l a c e d ( 8 . 2 . 2 ) by an i n t e g r a l o f t h e t y p e
(7.5.2) with
k
> m+N+j
.
(SEC. 8 )
1.69
INTEGRAL OPERATORS
For i i i ) , we n o t e t h a t t h e o p e r a t o r tA i s o b t a i n e d by permuting t h e r o l e s of x , y i n (8.1.2),b u t i ) shows t h a t tA i s continuous from
c:(x]
into
c"(Y)
and consequently P r o p o s i t i o n 4 . 5 of
Chapter I shows t h a t A e x t e n d s i n t o an o p e r a t o r which i s c o n t i n -
&'(Y)
uous from
into
J~(x)
.
This formula is deduced from t h e F o u r i e r i n v e r s i o n formula by u t i l i s i n g t h e passage t o t h e l i m i t ( 7 . 5 . 3 )
EXAMPLE 8 . 4 :
m
Let P(x, D ) b e a d i f f e r e n t i a l o p e r a t o r w i t h C
c o e f f i c i e n t s on X .
By d i f f e r e n t i a t i n g i n t h e o s c i l l a t i n g i n t e g r a l
(8.3.1 we o b t a i n
(8.4.1
E W L E
8.5:
Composition w i t h a C
We d e f i n e t h e o p e r a t o r
f* : C"(Y]
, (8.3.1)
shows t h a t
For
x E X
m
mapping
+
c"(X)
f
from X i n t o Y .
by f*v = v
o
f.
170
SYMBOLS AND OSCILLATORY INTEGRALS
EXAMPLE 8.6:
Trace Operator.
Suppose we have
and suppose Y i s an open s u b s e t o f T i r . of TRn
(CHAP. 3)
'A
=
R"
X
n'PR
L e t X be t h e open s u b s e t
d e f i n e d by
-
Let y be t h e t r a c e o p e r a t o r on X:
c;(Y)
3
v
yv
E Cp)
,
t h e n we have y = f * w i t h f ( x ) = (x, 0 ) .
4%01
IW1IXl =
i
Therefore Example 8.5
shows t h a t
E W L E 8.7:
Pseudo-differential operators.
These are t h e
o p e r a t o r s P d e f i n e d by ( 8 . 4 . 1 ) when we r e p l a c e t h e polynomial with respect t o p(xl y ,
5)
5, P(x, 5),
E
Sm(X X
by an a r b i t r a r y amplitude
x
X
R")
.
These w i l l be s t u d i e d i n d e t a i l
i n Chapter I V .
8.8
Cp
GENERALISATION TO THE CASE WHERE THE PHASE FUNCTION
I S DEFINED IN AN OPEN C O N I C SET.
set of
X
xPN\
0)
Let
r
be an dpen c o n i c sub-
and l e t Q be a phase f u n c t i o n i n
r.
The
methods of S e c t i o n 7 a l l o w u s , w i t h minor m o d i f i c a t i o n s , t o d e f i n e and study t h e o s c i l l a t i n g i n t e g r a l
ii
eirp(x'
when
a E Sm(X x RN)
a[.,
and when
0 ) u ( x ) dx d e a
satisfies
,
u
E cz(X]
171
STATIONARY PHASE
i s a phase f u n c t i o n i n an open conic s e t
If cp
r
cx
X
Y x(R
N
conic supp a
\
0 ) and a E Sm(X x Y x RN) i s such t h a t
c r u (x
+
o p e r a t o r A : C:(Y) t h e amplitude
y x
{o})
B'(X)
, then,
we define t h e i n t e g r a l
q,
a s s o c i a t e d w i t h t h e phase
and
as i n S e c t i o n 8.1, and we o b t a i n analogous
a
properties.
GENERALISATION TO THE CASE OF SYMBOLS OF TYPE
8.9
(see Section
6).
p >
0
All t h e p r e c e d i n g c o n s t r u c t i o n s and p r o p e r t i e s
remain v a l i d , w i t h minor m o d i f i c a t i o n s , when t h e amplitudes
a
a r e symbols from S:.
9.
STATIONARY PHASE THEOREM
9.1
CASE WHERE THE PHASE IS NEVER STATIONARY. Rp, Rq
be open s u b s e t s o f t h e behaviour f o r
I(xy
(9.1.1.1
where
T
cp E
c"(x
X
Y)
+
7)
+
m
;
Let X, Y
we o f t e n have occasion t o s t u d y
of i n t e g r a l s of t h e t y p e
.iT(x'
a(x, y, 71 dy
i s r e a l - v a l u e d and where a E Sm(X x Y x
I n o r d e r t o avoid any convergence problems, we assume t h a t zero f o r
y
o u t s i d e a compact s e t K c Y.
a
R +). is
172
SYMBOLS AND OSCILLATORY INTEGRAL3
I f the function
cp
(CHAP. 3 )
h a s no c r i t i c a l p o i n t s w i t h r e s p e c t t o t h e i n t -
e g r a t i o n v a r i a b l e y , then I i s of r ap i d decrease with res p ec t t o T;
more p r e c i s e l y , we have:
PROPOSITION 9 . 2 :
We asswne t h a t
(p(x,
Y
+ , where
x R )
E S-”(X
and i n t h i s case I(x,
d
y) f 0 on X x Y , T
is t h e frequency
variab t e .
PROOF:
The f i r s t - o r d e r o p e r a t o r
S u b s t i t u t i n g back i n t o ( 9 . 1 . 1 ) we o b t a i n , a f t e r i n t e g r a t i o n by parts ,
I(x,
7)
Since any
k
i.i.p
=
(?Ik
El
E
.
( t L ) k a dy
Swk(X
x Y x
and f o r any compact s e t
R+) H
, we cx
x
t h e n deduce t h a t f o r
R+ w e
have t h e upper
bound
The d e r i v a t i v e s of I are d e f i n e d by i n t e g r a l s of t h e same t y p e as I ;
t h u s we m a j o r i s e them i n t h e same manner, and t h i s proves
the proposition.
9)
(SEC.
9.3
STATIONARY PHASE
CASE WHERE
173
HAS A QUADRATIC CRITICAL POINT.
The
above p r o p o s i t i o n l e a d s us t o s t u d y t h e behaviour o f I when possesses a c r i t i c a l p o i n t . t h a t i n which
We b e g i n w i t h t h e s i m p l e s t c a s e , i . e .
i s a q u a d r a t i c form which is non-degenerate
g,
More p r e c i s e l y , w e assume t h a t
with r e s p e c t t o y .
Y1 where
=
1/2
< Q(x)Y,
Y t
Q ( x ) i s an i n v e r t i b l e symmetric m a t r i x which h a s
ence on x
E
X , and we t a k e Y = Bq
.
.)
I ( x , 71
E swq/2 (x
a
x
R+)
and t h i s symboZ admits an asymptotic expansion
(93.4.) where R(xl
aY )
=
-2i
< Q-’[x)
C
m
depend-
I n t h i s c a s e , we have:
satisfies (9.3.2
cf~
ay, ay >
and s i g n Q ( x ) = ( n m b e r of p o s i t i v e eigenvaZues of
Q(X
(number o f negative eigenvazues of Q ( x We begin w i t h a F o u r i e r t r a n s f o r m c a l c u l a t i o n .
174
SYMBOLS AND OSCILLATORY INTEGRALS
LEMMA 9.4:
(CHAP. 3)
Let Q be an invertibZe symmetric matrix in IRq i
Then the Fourier transform of the f u n c t i o n exp (5 < Qy, y
.
is
given by the f u n c t i o n
PROOF OF THE L E W :
Let M b e an o r t h o g o n a l m a t r i x such t h a t
tM Q M i s d i a g o n a l ;
t h e change o f v a r i a b l e y = Mx i n t h e c a l c u l -
a t i o n o f t h e F o u r i e r t r a n s f o r m t a k e s us t o t-h e c a s e where Q i s 9 2 diagonal. I f we t h u s p u t < Q Y , Y > = h j YJ , we have
i
~ X P ( Z< Q Y , Y
>I
=
We a r e t h u s l e d t o c a l c u l a t e
fi
1
2
exp($ h j y j )
j - 1 S(e
i/2
A j !)
and we have seen
i n E x e r c i s e 10.3 of Chapter I t h a t w e o b t a i n t h e f u n c t i o n
We hence o b t a i n t h e lemma, by o b s e r v i n g t h a t
We now r e t u r n t o t h e proof o f t h e theorem.
We a p p l y P a r s e v a l ' s
formula i n t h e i n t e g r a l (9.3.1),and we o b t a i n
with
(SEC. 9 )
175
STATIONARY PHASE
(where * i n d i c a t e s t h a t we have t a k e n t h e F o u r i e r t r a n s f o r m w i t h respect t o the variable y ) .
The advantage of e x p r e s s i o n
(9.4.2)
l i e s i n t h e f a c t t h a t T appears i n t h e e x p o n e n t i a l v i a t h e f u n c t ion
71.
;
t h u s , we can o b t a i n t h e asymptotic expansion of J by
u t i l i s i n g t h e Taylor expansion o f t h e e x p o n e n t i a l f u n c t i o n .
LEMMA 9.5:
For any i n t e g e r N
2
1, we have
.)
(9.5.1
where the remainder rN s a t i s f i e s the upper bounds
\(z) d j
(9.5.2.)
PROOF OF THE LEMMA:
For j = 0 , t h e upper bound ( 9 . 5 . 2 ) i s i n
f a c t j u s t t h e u s u a l upper bound f o r t h e remainder i n t h e Taylor formula.
We o b t a i n t h e upper bounds ( 9 . 5 . 2 ) f o r N
2
j > 0 from
t h e c a s e j = 0 by d i f f e r e n t i a t i n g t h e e q u a l i t y ( 9 . 5 . 1 )
9.6
CONCLUSION OF THE PROOF OF THE THEOREM:
t h i s l e m a t o the exponential i n the integral s
by--
where
27
< Q-’(x)q,
>
.
We o b t a i n ,
j times.
We a p p l y
(9.4.2),r e p l a c i n g
(CHAP. 3)
SYMBOLS AND OSCILLATORY INTEGRALS
176
and
<
=
RN(X,
J (x,
We n o t e t h a t Remark 4 . 4 ,
k
E s
7 )
Q-”(XI~,
m-k
(x
X
R+)
7 >I a(x,
?,,TI TI
9
; t h u s , i n view of
it s u f f i c e s t o show t h a t t h e remainder RN s a t i s f i e s
t h e upper bounds
uniformly f o r
1.1
+
w i t h i n a compact s u b s e t K c X and f o r
x
5 N
We b e g i n by p r o v i n g t h i s w i t h ci = B = 0.
,
U t i l i s i n g ( 9 . 5 . 2 ) , we o b t a i n :
(9.6.4.)
IRN(x,
Furthermore, s i n c e
a
from ( 9 . 6 . 4 )
E
K.
C T
i s a symbol o f degree
support with respect t o
uniformly f o r x
I
711
y
, we
m
w i t h compact
have f o r any i n t e g e r LI
2
0
By t a k i n g II s u f f i c i e n t l y l a r g e , we deduce
, and ( 9 . 6 . 5 )
the inequality
(SEC. 9 )
STATIONARY PHASE
177
I n t h e g e n e r a l c a s e , L e i b n i z ' s formula shows us t h a t
a:
3;
i s a f i n i t e sum o f terms o f t h e form
R"XI
r
with (y
= a'
+
(y"
1
fi = fi' i-
We can show by r e c u r r e n c e on
$1'
Iff' 1 +
,
\fi'
1
aa' ax8 ' rN( ...)
that
is
a f i n i t e sum of terms of t h e form
By combining Lemma 9.5 and t h e immediate Gpper bound
we e a s i l y o b t a i n t h e upper bound ( 9 . 6 . 3 ) .
EXAMPLE 9 . 7 :
Suppose we have X
n
c
I
Y =
R
n
X
W
n
;
a p o i n t of Y i s denoted by y = ( z , < ) and we d e f i n e Q(x) = Q by
< Qyp y > = < z I 5 > R(x1
i
a,)
e i z~. 5
.
We have d e t Q = 1, s i g n Q = 0 and
n
=
a2 3] azj a6j 1
I n t h i s c a s e Theorem 9 . 3 g i v e s
n
a(x, z ,
TI&
dc
I:(
-k k s o
R~ a \Z=<=O
178 9.8
E c"(x
, we
x Y)
say t h a t (xo, yo)
with res pe ct t o
cp
y
if
= 0 and i f t h e Hessian m a t r i x
yo)
cp(Xo,
cp
c r i t i c a l p o i n t of
2
(ayjYk
C R I T I C A L POINT W I T H
HAS A NON-DEGENERATE
I f we have
i s a non-degenerate 'p(Xo,
cp
CASE WHERE
RESPECT TO y :
a,
(CHAP. 3)
SYMBOLS AND OSCILLATORY INTEGRALS
yo)),
= Q
i s invertible.
Jk
I n t h i s c a s e , t h e i m p l i c i t f u n c t i o n theorem shows t h a t t h e r e 0
e x i s t neighbourhoods U , V of x t h e r e e x i s t s a unique y ( x ) 0
and y ( x ) = y
0
a.2
, we
V satisfying
can reduce
, this
(p(x, y ) l y , y ( x )
YY m a t r i x when U i s s m a l l .
y
s u c h , t h a t f o r any x
.
Q(x) =
We p u t
E
, y0
aY
E
U, 0
=
~ ( x ,y ( x ) )
b e i n g an i n v e r t i b l e
By a change o f v a r i a b l e s w i t h r e s p e c t t o
t o t h e form (9.3.1),t h i s b e i n g e x p r e s s e d
Q
i n t h e f o l l o w i n g theorem:
THEOREM 9.9:
t h a t (xo, y o ) i s a non-degenerate c r i t i c a l point of respect t o y
.
such t h a t i f we put
.I
z = h(x, y),
cp(x, Y ) for
Moreover
h
Q
with
Then, a f t e r appropriately shrinking the open
s e t s U, V, there e x i s t s a Cm function
(9.9 .I
We assume
(or Morse's lemma w i t h p a r a m e t e r ) .
- cp(x,
(x,
Yl
E
h
: U x V
we have
3 t~.Q(x).z
YIxI) = u
x
v
.
i s a diffeomorphism with respect t o y f o r
f i x e d , and s a t i s f i e s
R9
x
(SEC, 9 )
( h is termed the "Morse diffeomorphism")
PROOF:
179
STATIONARY PHASE
.
T a y l o r ' s formula w i t h i n t e g r a l remainder allows us t o
write
Let us for t h e moment assume t h e e x i s t e n c e of a C
u
x
"
3
I.,
Y1 I
R
R[x, Y )
function
e(Rq, A')
f
Then, t h e f u n c t i o n h ( x , y ) = R ( x , y ) . ( y
m
-
such t h a t
y ( x ) ) satisfies the
c o n d i t i o n s o f t h e theorem, a f t e r a p p r o p r i a t e l y s h r i n k i n g U and V
s o as t o have a diffeomorphism w i t h r e s p e c t t o
y
for
x
fixed.
I n o r d e r t o prove t h e e x i s t e n c e o f t h e f u n c t i o n R , we cons i d e r t h e mapping @,
dependent on t h e parameter x
E
U , defined
by
8 3 R
Q".
+(RI
=
t~ ~ ( x R)
d e n o t e s t h e space of symmetric
where
x = x
A
0
, the
g x g
matrices.
For
d e r i v a t i v e of @ a t t h e p o i n t R = I d i s given by t h e
l i n e a r mapping
180
(CHAP. 3)
SYMBOLS AND OSCILLATORY INTEGRALS
this linear mapping is surjective, and since every the image of
’
5 Q”
A
A
E
S
is
E Q-’ 8 , it is therefore bijective.
Consequently, @ is a local diffeomorphism in the neighbourhood of the identity operator, and the function -1 R[x, y ) = @ (Q(x, y ) ) is defined for (x, y )
E
U x V and satis-
fies ( 9 . 9 . 3 ) when U and V are small enough.
We are now in a position to state the theorem of stationary phase with parameter.
THEOREM 9.10:
We adopt the assumptions of Theorem 9 . 9 , and
we a s s m e t h a t the symboZ a E Sm(X x Y x R+) i s zero f o r (x, y)
u x v.
outside a compact subset of 2
Q(x)=
k
ayy C P~ ~,
we put
and l we denote ~ by
: U x W 3 (x, 2 ) W
y = k(x, z)
E
~ V
~ the inverse
of the Morse diffeomorphism f o r x f i x e d i n U. Then the integraZ (9.1.1) satisfies
ii] This symbol a h i t s an asymptotic expansion i n U x B’ of the form
~
(SEC. 9 )
STATIONARY PHASE
181
iii) I n p a r t i cu l a r , the leading term i n t h i s expan-
sion is given by
PROOF:
The assumption c o n c e r n i n g t h e s u p p o r t o f
i n t h e i n t e g r a l (9.1.1)i n t h e
t o w r i t e t h e phase f u n c t i o n form ( 9 . 9 . 1 ) ; a n d
we have
a l l o w s us
a
I ( x , 7)
=
Then, we use t h e Morse diffeomorphism i n o r d e r t o e f f e c t t h e change of v a r i a b l e s
y --+
z
=
h ( x , y)
,
and we f i n d
and c o n s e q u e n t l y Theorem 9.10 f o l l o w s from Theorem 9 . 3 .
9.11
VARIOUS GENERALISATIONS:
word f o r word when
a
All t h e r e s u l t s g e n e r a l i z e
is a symbol w i t h v a l u e s i n a f i n i t e -
dimensional v e c t o r space F :
a € Sm(X x Y x R+ ; F)
.
All t h e
(CHAP. 3 )
SYMBOLS AND OSCILLATORY I N T E G R A L S
182
r e s u l t s l i k e w i s e g e n e r a l i s e , w i t h minor m o d i f i c a t i o n s , t o t h e case
a
where
i s a symbol o f t y p e p w i t h p > 112.
A D D I T I O N A L NOTES AND EXERCISES
10.
( * ) 10.1
SYMBOLS W I T H
UNIFORM U P P E R BOUNDS W . R . T .
We say t h a t a E BSrn(R" x F'?)
a
a
E
Srn (R" x
RN)
and if
s a t i s f i e s ( 2 . 1 . 1 ) w i t h C independent of t h e compact s e t K .
We p u t
BS+ =
a)
Show t h a t BS*
b)
Supposing a
a.b-1
E
BS
-1
We say t h a t
,
BS'"
ffi-=Qffim
i s a graded sub-algebra of S
+m
BSm, prove t h a t t h e r e e x i s t s b
a
ell
2
c(l
+
lei,"
for
i s e l l i p t i c of d e g r e e ' rn
E
.
. E
BS-m such t h a t
m BS j and m j \a
.
-
2
~
Prove t h a t t h e r e e x i s t s a
unique modulo BS--, such t h a t for any k we have (8
101
rn
k
ad)
E BS
k+'
Under t h e s e c o n d i t i o n s we w r i t e
.
a1 .
.
Suppose we have a sequence of symbols ( a j l j
c)
j
E
M
i f and only i f t h e r e e x i s t s c > 0 such t h a t
la(x,,
a
if
x I N Bn:
.
with
E BS%
,
(SEC. 1 0 ) 10.2.
OSCILLATORY INTEGRALS AND DECOMPOSITION I N T O PLANE WAVES :
b E Sm(R x [O,
Suppose we have a symbol
a)
183
ADDITIONAL NOTES
v E P(R)
and c o n s i d e r t h e d i s t r i b u t i o n
+
0
[)
d e f i n e d by t h e
0s c i l l a t i n g i n t e g r a l
~ ( y )=
['
m
.
eiY5b(y,t)d5
JO
Let t h e r e be f ( x ) = 0.
such t h a t f'(x)
p
0 when
Consider t h e d i s t r i b u t i o n u = f * v
E
#(IRn)
f : R"
-t
R
i n E x e r c i s e 10.9 of Chapter I.
u(x) =
j'
defined
Prove t h a t
.
, 5)df
b(f(x) 0
[Approximate b by symbols w i t h compact s u p p o r t i n 51 b)
= cp(x
Let a = a ( x , 0 ) be a symbol inIR
, 6)
be a phase f u n c t i o n i n
Consider t h e d i s t r i b u t i o n
and assume t h a t (p'(x X
, 6)f
0
u
n
x I R N and l e t
R" x (RN\O)
E &(R")
when
cp(x
d e f i n e d by
, 6)
u
0
Prove (by changing i n t o s p h e r i c a l c o o r d i n a t e s ) t h a t
i s t h e unit sphere of EN and where dw i s t h e where S N- 1 Riemannian d e n s i t y on S Euclidian structure].
N- 1
[EquipIR
N with i t s canonical
184
(CHAP. 3 )
SYMBOLS AND OSCILLATORY INTEGRALS
c)
By a p p l y i n g t h e above i n t h e c a s e
.ix.9
s i D”-’(
do ,
show t h a t
-Y1 - i n 6 ( y ) )
pv
(cf. 10.2.1,L)
Deduce from t h i s t h e f o l l o w i n g decomposition i n t o p l a n e waves o f t h e Dirac d i s t r i b u t i o n inIR
6(x)
6(x)
=
i(21~)’”
= n(*)-”
DW’(pv
Js
):
(D”‘S)(x.w)
n
1YX’ .
(n t 2) :
dw
if
n
i s even
w
dw
if
n
i s odd
.
*I
10.3
LACUNAS OF THE FUNDAMENTU SOLUTION OF THE WAVE
OPERATOR : We a g a i n adopt t h e n o t a t i o n of E x e r c i s e 1 0 . 8 i n Chapter I , and we suppose t h a t n b 2 .
E, E B (R”)
We r e c a l l t h a t t h e d i s t r i b u t i o n
i s d e f i n e d by
(SEC. 1 0 )
a)
185
ADDITIONAL NOTES
Reasoning a s i n t h e above e x e r c i s e , e s t a b l i s h t h e decomp-
o s i t i o n i n t o p l a n e waves:
where
1 = D**(Pv- Y
V(Y)
- i n 6(y))
E &(R)
,
Deduce from t h i s t h a t
(10.3.1) w E
where
(R)
is d e f i n e d by r c 2 1
(10.3.2)
r(Y> =
D
(10.3.3)
w[y) =
-in D rr-2 b ( y )
b)
i f n is even,
Pv;
i f n is odd.
From (10.3.1), (10.3.3) and from (10.9.1)of Chapter I ,
deduce t h a t when
{x E R"
1
1x1
n
i s odd, we have E
< l},
Chapter I ) t h a t
1
= 0 in
and t h e r e f o r e ( s e e s e c t i o n 10. 8 of
supp E l
P
{x E R"
t h i s ( s e e (10.8.1),Chapter I )
I
1x1 = I]
.
I n f e r from
t h a t t h e s u p p o r t of t h e funda-
mental s o l u t i o n E of t h e wave o p e r a t o r i s e q u a l , when odd and 2 3, t o t h e cone
c)
Suppose
for x
E
E?
:
{ ( t , x ) E R*'
1t
f : R-C
i s continuous.
= 1x13
n
is
. Prove t h a t we have,
186
where u
S =
o f En-', i s t h e area of t h e u n i t sphere S n-2
n-2
u0 = 2
d)
(CHAP. 3 )
SYMBOLS AND OSCILLATORY INTEGRALS
sn-l
[Cut
with
up i n t o zones x.o = c o n s t a n t , and p u t
X ,.,I.
W ' e s h a l l h e n c e f o r t h assume t h a t
t o s t u d y t h e r e s t r i c t i o n of E
1
i s even, and we propose
n
.
c 11
t o (1x1
Deduce from (10.3.1), ( 1 0 . 3 . 2 ) and ( 1 0 . 3 . 4 ) t h a t t h i s r e striction is the C
m
function
where, for 0 5 r < 1, we have p u t
n-3
cn
J'
- (2in)-"
-1 (&)I
F""1 and Cn =
=
(1
+
s/r)l-"(l-s2)
on-2
ds
I
.
Prove, by i n t e g r a t i o n by p a r t s , t h a t n
Deduce from t h i s t h a t
a2k-2
t Re c a l l t h a t
'2k4 Calculate d i r e c t l y t h e i n t e g r a l
j
IR2 , 1x1 < 1.
@GI
when x
E
X.D+
1
dm
Deduce from t h i s t h a t
F2(r) =
1
(1
- r)-3
and t h u s , by u t i l i s i n g (10.3.5), t h a t t h e r e s t r i c t i o n of El t o (x
E R"
11x1
<
11
i s equal t o
ADDITIONAL NOTES
(SEC. 1 0 )
with
I n f e r from t h i s t h a t , when
is even, t h e r e s t r i c t i
n
f
t h e fundamental s o l u t i o n E of t h e wave o p e r a t o r ( s e e (10.8.1) Chapter I ) t o t h e cone
--
< t') is
{(t,x)ER#Ilxl
given by
n-1
Kn(t
2
- bI2) 2
10.4
FUNDAMENTAL SOLUTION OF THE WAVE OPERATOR FOR SPACE DIMENSION EQUAL TO 1, 2 OR 3 :
We once more adopt t h e n o t a t i o n of E x e r c i s e 10.8. o f Chapter I.
a)
I n t h e case n = 1, prove t h a t E
1 is the characteristic
f u n c t i o n of t h e i n t e r v a l [-1, 11, and t h u s t h a t E i s t h e c h a r a c t e r i s t i c f u n c t i o n of b)
{(t,x)
E R2
1
1x1
t
3
-
I n t h e c a s e n = 3, c o n s i d e r ( c f . s e c t i o n 7.2, Chapter I )
t h e d i s t r i b u t i o n 6(1
- 1x1 2 ) .
Calculate i t s Fourier transform
by u t i l i s i n g s e c t i o n 7.2 of Chapter I and ( f o r example) t h e formula ( 1 0 . 3 . 4 ) .
Thus deduce t h a t
188
c)
(CHAP. 3 )
SYMBOLS AND OSCILLATORY INTEGRALS
I n t h e c a s e n = 2 , we e x p r e s s E on t h e b a s i s of t h e c a s e
n = 3 by u s i n g t h e "method o f d e s c e n t " .
r(t, x1 , "2 , 5 )
Let
denote t h e fundamental s o l u t i o n
I
c o r r e s p o n d i n g t o t h e c a s e n = 3.
d e f i n e s a d i s t r i b u t i o n on IR
3
Prove t h a t t h e mapping
w i t h s u p p o r t i n { t t 01, which i s a
a2
fundamental s o l u t i o n of
.
Prove t h a t
t h i s d i s t r i b u t i o n c o i n c i d e s w i t h t h e fundamental s o l u t i o n E.
Thus deduce, by u t i l i s i n g t h e l a s t p a r t of b ) , t h a t E i s t h e function
10.5
1
5 Y(t]
y(t
-
- 1x1 1- 112
1x1) (t2
2
a
STATIONARY PHASE THEOREM FOR A SURFACE INTEGRAL.
Let S be a h y p e r s u r f a c e of Bn and suppose a
m
e q u i p S w i t h t h e Riemannian s t r u c t u r e induced byIR denote t h e a s s o c i a t e d d e n s i t y by da.
C (S).
E
0
n
We
and we
We assume t h a t t h e t o t a l
c u r v a t u r e K ( y ) o f S does n o t v a n i s h a t any p o i n t y of t h e s u p p o r t of
a.
Supposing
x
E R"\O
,
l e t y . ( x ) denote t h e p o i n t s J
y E S n supp a where t h e normal t o S i s p a r a l l e l t o x ; t h i s case t h e i n t e g r a l
in
(SEC. 10)
189
ADDITIONm N O T E S
e x h i b i t s t h e f o l l o w i n g behaviour a s
T +
+
I(~ = ,x)
(10.5.1)
(&)? 7
a(Yj(xll
1
4 eit"/4)0j,,
0
j
i.rx.y
,(XI
K(Yj(X11l
- -W1 +
where
:
w
2 l
0 (7
is e q u a l t o t h e number of p o s i t i v e p r i n c i p a l c u r v a t u r e s
minus t h e number of n e g a t i v e p r i n c i p a l c u r v a t u r e s when t h e normal t o S a t y . ( x ) i s o r i e n t a t e d p o s i t i v e l y i n t h e d i r e c t i o n J of x.
a)
Prove t h a t we can reduce t o t h e c a s e where
a
has i t s
s u p p o r t i n t h e domain of a l o c a l c h a r t c o n t a i n i n g a t most one p o i n t y . ( x ) and of t h e form J
y'
Show t h a t i n such a c h a r t we have
b)
E R"-'
-
(y',
f(y'))
E
S
.
I(?,x) =
Deduce (10.5.1)from t h e s t a t i o n a r y phase theorem by ob-
s e r v i n g t h a t t h e modulus of t h e t o t a l c u r v a t u r e i s e x p r e s s e d i n t h i s c h a r t by
( s e e Dieudonn6 C11, Chapter XX, Problem 1 4 . 1 ) .
This Page Intentionally Left Blank
CHAPTER 4
PSEUDO D I F F E R E N T I A L OPERATORS
1.
DEFINITION
1.1 INTRODUCTION :
In order t o invert e l l i p t i c operators,
we s a w i n ( 1 . 2 ) of Chapter I11 t h a t we were l e d t o g e n e r a l i s e t h e d i f f e r e n t i a l o p e r a t o r s by o p e r a t o r s of t h e form
m n where p i s i n - a space S ( X x IR )
.
Writing out t h e F o u r i e r
t r a n s f o r m , we o b t a i n an e x p r e s s i o n u s i n g an o s c i l l a t o r y i n t e g r a l depending on t h e parameter x E X :
(1.1.2)
Moreover, we a l s o want t h e o p e r a t o r s of t h e form (1.1.2) t o possess t h e s t a b i l i t y p ro p e r t i e s of t h e d i f f e r e n t i a l operators f o r t h e o p e r a t i o n s of a d d i t i o n , o f composition, o f transposition,
....
For t x a m p l e , i f we compose P w i t h t h e o p e r a t o r of m u l t i p l i c a t i o n
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
192 m
by g
C ( X ) , we o b t a i n f o r t h e composition t h e e x p r e s s i o n
E
where t h e amplitude p ( x , S ) g ( y ) i s a n element of Sm(X x X
x
Bn).
This l e a d s us t o s t a t e t h e f o l l o w i n g :
DEFINITION 1.2 m
E R U
( p . d.0.
)
.
{-=I
:
Let X be an open subset of lRn, and l e t
We define a pseudo-differential operator
of degree
m
in
x
t o be an i n t e g r a l operator P
associated with the phase function V(X,Y,b)
(x,y,s)
= (x-Yl.5
EX x X x
R"
and with an amplitude p(x,y,g)
E Sm(X x X x
R") ,
( 5 i s t h e frequency v a r i a b l e ) .
we denote by L m ( X ) the space of p . d . 0 . 's o f degree
By d e f i n i t i o n , a p . d . 0 .
i n X.
P is written:
I f we c a r r y out t h e i n t e g r a t i o n w i t h r e s p e c t t o formally obtain-
m
5 i n (1.1.2)we
(SEC. 1)
DEFINITION
193
I n t h e p a r t i c u l a r case where p i s homogeneous w i t h r e s p e c t t o
5,
t h e k e r n e l K i s a d i s t r i b u t i o n which fs homogeneous w i t h r e s p e c t W
t o z and which h a s C dependenceon x.
I t w a s i n t h e form (1.2.2),termed a s i n g u l a r i n t e g r a l , t h a t CALDERON-ZYGMUND [I1 s t u d i e d t h e s e o p e r a t o r s and developed t h e Subsequently , numerous o t h e r a u t h o r s
f i r s t symbolic c a l c u l u s .
expanded and g e n e r a l i s e d t h e t h e o r y , r e p l a c i n g (1.2.2)by t h e more t r a c t a b l e e x p r e s s i o n ( 1 . 2 . 1 ) ; we c i t e t h e names of KUMANO-GO,
UNTERBERGER-BOKOBZA,
KOHN-NIRENBERG,
s h a l l p r i m a r i l y f o l l o w t h e approach o f H O W D E R
HORMANDER.
We
[5], but
p u t t i n g t h e emphasis more on t h e r o l e of t h e s t a t i o n a r y phase theorem.
1.3
IMMEDIATE PROPERTIES.
From P r o p o s i t i o n 8 . 2 of
Chapter I11 , we know t h a t any p s e u d o - d i f f e r e n t i a l o p e r a t o r P
i s continuous from C"(X)
i n t o C"(X)
0
o p e r a t o r from
g(X)i n t o
B1(X)
and extends i n t o a continuous Suppose K
P
i s the kernel dist-
r i b u t i o n o f P ; P r o p o s i t i o n 7.7 of Chapter 111, shows t h a t
(1.3.1)
s i n g supp
%
consequently P r o p o s i t i o n
(1.3.2)
s i n g supp
c Diag(X x X)
,
5 . 4 o f Chapter I shows t h a t w e have
Pu c
s i n g SUPP u
#
IJ
E
C"X)
I n t h e p a r t i c u l a r c a s e where p does n o t depend on y we have t h e r e p r e s e n t a t i o n (1.1.1)of Pu f o r u deduce by c o n t i n u i t y t h a t
m
E
C ( X ) , and we t h e r e f o r e 0
0
194
PSEUDO-DIFFERENTIAL OPERATORS
< Pu
(1.3.3) for u
E
,v
&'(X), v
f d\eix5p(x,t]
> =
1
c(E)
(CHAP.
4)
p ( x , g ) v ( x ) dx
(/eix5
C z ( X ) , since t h e function
E
i s of r a p i d decrease.
v(x)dx
The r e g u l a r i s i n g o p e r a t o r s a r e p s e u d o - d i f f e r e n t i a l o p e r a t o r s :
PROPOSITION 1 . 4 :
equal t o L-"(X)
The s e t of regularising operators i s
.
fl Lm(X)
which i s i t s e l f equaZ t o
rn
PROOF:
P E
Supposing
there exists
a
E
m
S
n Lm(X) ,
t h e n for a l l m
E
IR
such t h a t P can be w r i t t e n i n t h e form
(1.2.1). Its kernel distribution (1.4.1)
=
Kp(X,Y)
k i s in C (X
x
X ) when m <
-
i e
n
-
k.
Consequently K
P
E
C"(X
m
x
We now assume t h a t Kp i s C ; we can w r i t e P i n t h e form
(1.2.1)by t a k i n g P(X,Y,51 = and
/p(5)
&
=-i(x-y)5
= 1
.
p ( 5 ) Kp(x,y)
We t h u s have p
F i n a l l y , it i s c l e a r t h a t L-"(X)
with E
p
E CI(R")
S-m and P E L-m.
c n Lm(X).
m
X).
(SEC. 1)
195
DEFINITION
We can always reduce t o t h e c a s e o f a proper p.d.0.
if
we argue modulo a r e g u l a r i s i n g o p e r a t o r ; t h i s g i v e s us
PROPOSITION
1.5
If we have
:
,
E Lm(X)
P
then we Can
write
PnP'+
(1.5.11
R
where R i s a regularising operator and
PI
E
Lm(X)
is a
proper operator.
PROOF:
We prove t h a t t h e r e e x i s t s a f u n c t i o n
p
m
E c (x
x
x)
such t h a t t h e p r o j e c t i o n s
are proper.
Let
be a l o c a l l y f i n i t e c o v e r i n g of
(Vj) j € J
X by r e l a t i v e l y compact open s u b s e t s ; t h e n
i s a neighbourhood of a r e proper.
A =
Diag ( X x X )
U =
u Fj x F j
such t h a t p
llu
If V i s a neighourhood of A such t h a t
t h e n it s u f f i c e s t o t a k e f o r p a C"(X
x
j
and
5
cU
,
X ) function identical t o
1 on V and w i t h s u p p o r t i n U.
We d e f i n e t h e o p e r a t o r s P' and R by t h e i r k e r n e l s
196
PSEUDO-DIFFERENTIAL OPERATORS
4)
(CHAP.
By c o n s t r u c t i o n t h e o p e r a t o r P' i s proper.and t h e i n c l u s i o n
( 1 . 3 . 1 ) shows t h a t R i s r e g u l a r i s i n g .
F i n a l l y , P'
L"'(X),
E
s i n c e i f p i s an amplitude f o r P , t h e n p p i s an amplitude f o r
P'
. REMARK 1 . 6 :
P r o p o s i t i o n 5 . 3 of Chapter I shows t h a t i f t h e n it i s a c o n t i n u o u s o p e r a t o r on each of
P i s a p r o p e r p.d.0. t h e following spaces:
A CHARACTERISATION OF P.D.O.'s.
2.
We a r e now going t o g e n e r a l i s e t o p.d.0.'~t h e c h a r a c t e r i s a t i o n ( s e e P r o p o s i t i o n 7.3 of Chapter I ) of d i f f e r e n t i a l operators.
The r e s u l t i n q u e s t i o n i s somewhat t e c h n i c a l , b u t
it c o n t a i n s p r a c t i c a l l y a l l t h e a n a l y s i s n e c e s s a r y t o t h e t h e o r y of p . d . 0 . ' s ; i n p a r t i c u l a r , t h e symbolic c a l c u l u s and i n v a r i a n c e under diffeomorphism a r e e a s i l y deduced from it.
THEOREM 2 . 1 : a
E
Sq(X x
Suppose P E ~"'(x] ;
N
we take a symbol
R ) and a real-valued f u n c t i o n Jr E
C"(X
N
x
RD)
such
a
and
e *
that dx$(x,8)
4
0 for ( x , e )
E
c o n i c supp o f
$(x,e) i s homogeneous o f degree 1 w i t h respect t o
8
0,
197
CHARACTERISAT1ON
(SEC. 2 )
We assume t h a t P is proper o r t h a t
a
has compact support
with respect t o y, and we p u t b(x,6) = e b E SW[X
i)
Then :
6) P(a B i t ) ( x , 6) x
N
Ro]
i i ) we have an expression for the asymptotic expcmsion
We s h a l l s e e t h a t t h e g e n e r a l term of (2.1.4) i s i n
w -
k -/el 2l
S
,
and consequently t h i s asymptotic sum i s
meaningful.
PROOF:
From P r o p o s i t i o n 2 . 5 o f Chapter 111, it s u f f i c e s
t o prove t h e theorem i n a conic neighbourhood of a p o i n t (Xo
, e0)
E X x RoN
.
By r e p l a c i n g
p r e s s i o n (1.2.1) for P , we o b t a i n
(2.1.6)
b(x,6) = e- i * [ x , 01
u by
a ei'
i n t h e ex-
198
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
where by h y p o t h e s i s we can reduce t o t h e c a s e where t h e f u n c t i o n y
4
p(x,y,{)a(y,0)
h a s s u p p o r t w i t h i n a compact s u b s e t K
X.
c
We u t i l i s e t h e e q u i v a l e n c e ( 2 . 6 . 1 ) o f Chapter 111, and we
put
T
= 101
,
variables 5 =
(D
TQ,
.
0
=
A f t e r having made t h e change o f
we o b t a i n
We d e f i n e t h e number
r =
inf c o n i c supp a
(Y,W)E
Y E K assumption ( 2 . 1 . 1 ) shows t h a t r > 0. be a f u n c t i o n i d e n t i c a l t o 1 for zero f o r
171 2 Er
Let
171 <
.
'i
~ ( 7 f) C:(R") and e q u a l t o
We b e g i n by proving t h a t t h e i n t e g r a l
-"' (*).
is in
x SW1 x
S-"(X
L(CU,Y,'I),Dj,) satisfies
u,
l/eiT@ p.a.p d y d7)
= e
I(X,CU,T~
=
-1 a&[e i 7
R+)
.
The f i r s t - o r d e r o p e r a t o r
- TI-*
I d y *(Y,cu)
1=
T@
akei T @
.
< dy
$(Y,d-4
9
Dy
By v i r t u e of t h e choice of
t h e r e e x i s t s f o r any i n t e g e r k and any compact s u b s e t
H c X x SW1 some C such t h a t
'
for 7 2 1
(SEC. 2)
199
CHARACTERISATION
We t h e r e b y deduce t h a t I i s of r a p i d d e c r e a s e w i t h r e s p e c t t o
T
and t h e same proof shows t h a t t h i s i s a l s o t r u e of i t s d e r i v a t i v e s .
We now p l a n t o a p p l y t h e s t a t i o n a r y phase theorem t o t h e remaining integral
J(x,w,T) = b(x,cu,'r)
dy
We have
@
I
d y $(Y,w)
-
'.
I(x,m,T)
-7 ,
d
7
@ = x-y
;
consequently 0
i s s t a t i o n a r y a t t h e p o i n t ( y , n ) d e f i n e d by y = x and
7
.
= d y $(x,u))
If we p u t yo
I
A t t h i s p o i n t , t h e Hessian m a t r i x i s given by
xo,
I
dy *(xo
I
wo)
, Theorem 9.9
of Chapter
I11 shows t h a t t h e r e e x i s t neighbourhoods, U , Q , W of X o,
mo
, ( y o , a) i n which
we can u t i l i s e a Morse diffeomorphism;
a f t e r a p p r o p r i a t e l y s h r h k i n g U x R , we can assume t h a t W i s a neighbowhood of t h e s e t
Wo p(y,q)
C(X
9
dy t(x,wI)
1
x
Eu
I
UI E
n]
.
If we l e t
be a C z ( W ) f u n c t i o n , i d e n t i c a l t o 1 on Wo, we can a p p l y
( * ) Transkrtor's Note:
"sign" is h e r e an a b b r e v i a t i o n f o r " s i g n a t u r e " .
(CHAP.
PSEUDO-DIFFERENTIAL OPERATORS
200
4)
t h e s t a t i o n a r y phase theorem t o t h e i n t e g r a l
J1(x,w,7)
with
T
= e-"$
(&)npiT@ p.a.p
as t h e frequency v a r i a b l e . J 1 E Sw
(2.1.8)
(U x
J1(x,~7)
n
dy d 7
We f i n d and
x R')
N
-k
%
(1-p)
[
Skow,Y,T,Dy,ql
P(X,Y,~T)k4Y,Td
k S
51 =
d y h d
where Sk i s a d i f f e r e n t i a l o p e r a t o r of degree 5 2k which depends o n l y on k and $. bourhood o f
7
I t should b e n o t e d t h a t p i s z e r o i n t h e neigh-
= dy$(x,ru)
f o r (x, w ) e conic supp a and x
F i n a l l y , we prove t h a t we have
J2 = J
- J1 E
S-"(U
x
nx
E
K.
R']
.
To do t h i s we c o n s i d e r t h e f i r s t - o r d e r o p e r a t o r
L = D(< dy$(Y,col
- 17
where we have p u t D = ( Idy$(y,ru)
(1
- P ( Y , T ) ) ( ~- P(T))D(x,u,Y,T)
t h e frequency v a r i a b l e q .
-
1 -(I p ) ( 1 7
- p)L
9
> + < X-Y,Dy>)
Dy
- 712 + -2
Es
(u
IX-YI
x
2 -1
)
nx x
x
so that
R")
for
By c o n s t r u c t i o n we have
-
(eiT%)= (I p ) ( 1
L e i b n i z ' s formula shows t h a t
(
p a
+)k
- p)
eiT@
.
i s a sum of terms
(SEC. 2)
CHARACTERISATION
and s i n c e
171
above by
C
2
r 7
T
~
201
on t h e s u p p o r t o f 1 - P, t h i s i s a l s o bounded
(I ~+
lT1)”k ~
These upper bounds imply t h a t J2 i s o f r a p i d d e c r e a s e w i t h r e s p e c t t o T and we proceed i n t h e same way f o r i t s d e r i v a t i v e s . F i n a l l y , we have shown t h a t b
Q
J1, which proves (2.1.3), and it
remains t o g i v e t h e asymptotic expansion of J1. by
&
By r e p l a c i n g
T
i n (2.1.8), we o b t a i n
(2.1.9)
b(x,@I
-
bk(x,e)
k S bk
where
(2.1.101
E
Sw-k
c
RN)
i s given by
bk(x, 9)
= IQ+Y and
(X x
k, Q,
B, Y
>1
(x , e)
(x,
‘k,a,p,y
ID^a DyP(x,x,dxt(x, $
S2k i s a homogeneous f u n c t i o n of degree la1
w i t h r e s p e c t t o 8 , which depends n e i t h e r on a o n l y on $. above by
91)a,Y a( x,e >
nor on
p
-
k
but
The degree of t h e g e n e r a l t e r m of ( 2 . 1 . 1 0 ) is bounded m
+ q - E1 .1 + B +
yl
; c o n s e q u e n t l y it t e n d s t o
-
m,
PSEUDO-DIFFERENTIAL OPERATORS
202
+
when l a / or 181 or I y I t e n d s t o
m y
(CHAP.
4)
which allows u s t o apply
Remark 4 . 3 of Chapter I11 and t h e r e f o r e c o n s i d e r ( 2 . 1 . 9 ) as an asymptotic sum w i t h r e s p e c t t o ( a , 6 ) .
-
b(x,ej
12.1.11)
Dca Dy8 p ( x , x , d x ~ ( x , e ) ~ . d , , 8 C x , e ~ a 1
1
q +
where
a,B
a,B,J, and
a
.
independent of
where
q
B 1
+I-
2lel
E S
d
We f i n d
(X x
RN)
and depends only on
We t a k e advantage o f t h e f a c t t h a t d
a,B
t o determine it by t a k i n g
p
p
i s a polynomial w i t h r e s p e c t t o 5 and g
is
i n the particular
m
E
C (X).
A s we
observed a t t h e beginning of S e c t i o n 1, t h i s means t h a t P i s t h e d i f f e r e n t i a l o p e r a t o r Q(x,D)
=
b(x,B)
g
.
We can t h u s d i r e c t l y c a l c u l a t e
e-i*(x'e) Q(x,D) (g.a e i*)
by a p p l y i n g t h e Leibniz-Hormander formula (1.1.3)o f Chapter I , and w e o b t a i n
If we p u t g ( y ,
e)
=
*(.,el +
c d g(x,e) Y
,y - x >+ r
, formula
(1.1.4) of Chapter I shows t h a t
e-i$
a$8
Q ( ~ , D ~ Ie (i 9~
=
> - a,:*' a!
Q ( x , d X f ( x , B ) ) D ~ ( ae * )
CY
and s u b s t i t u t i n g i n t o (2.1.12),we o b t a i n
(2.1.13)
b =
cm 1
CYlB
( D ~as p ) ( x , x , d x t ( x , 8 ) ) . D ~ ( a eir) lY=x'
(SEC. 2 )
203
CHARACTEEiISATION
By i d e n t i f i c a t i o n w i t h (2.1.11)ywe deduce from t h i s t h e which proves t h e formula ( 2 . 1 . 4 ) .
e x p r e s s i o n f o r t h e terms d aa
A s an a p p l i c a t i o n of Theorem 2 . 1 , we give below a c h a r a c t e r i s a t i o n of p s e u d o - d i f f e r e n t i a l o p e r a t o r s on an open s u b s e t
x
c ?Rn.
PROPOSITION 2.2:
Let P be a oontinuous lin e a r operator
from C I ( X ) i n t o C m ( X ) . Then, P e-ixs P(a
(2.2.11
eixsl
E
E
S ~ ( Xx
Lm(x)'if and onZy if we have
R") for a l l
a
E
c:(x).
If we f u r t h e r assume t h a t P is proper, i t s u f f i c e s t o have (2.2.1) with a = 1.
PROOF:
I t remains t o prove t h a t t h e c o n d i t i o n i s s u f f i c i e n t - .
Let ( p . ) be a p a r t i t i o n of u n i t y on X w i t h f u n c t i o n s p J j and l e t
a J.
on supp p j ;
For u
m
be f u n c t i o n s i n C 0 ( X ) , w i t h
w e assume t h a t t h e supp
Rj j
E
Cz(X)
m
E
Co(X) identical t o 1
are also locally f i n i t e .
W
E
C (X),by F o u r i e r i n v e r s i o n we have: 0
J
J
I n t e r p r e t i n g t h i s l a s t i n t e g r a l a s an i n t e g r a l w i t h r e s p e c t t o ,€ with v a l u e s i n C w ( X ) , t h e c o n t i n u i t y o f P allows us t o w r i t e 0
(CHAP.
PSEUDO-DIFFERENTIAL OPERATORS
204
Since
u =
, we
pju
4)
t h e r e f o r e deduce t h a t
J
where t h e amplitude P(X,Y,e;I
-c
J i s c l e a r l y an element of
Pj(xlh)
Pj(Y1
Sm(X x X x
l o c a l l y f i n i t e with respect t o
R")
s i n c e t h i s sum i s
.
y
Since P i s p r o p e r , it d e f i n e s a continuous o p e r a t o r P
Ca(Rn)
* C"(X)
C"(X)
where t h e f i r s t arrow denotes t h e r e s t r i c t i o n t o X.
If we apply
t h i s operator t o the integral
m n c o n s i d e r e d a s an i n t e g r a l w i t h v a l u e s i n C (IR ) , we o b t a i n
r
with
(2.2.4)
p(x1sl =
=-ix[:
,,(,ixl:
) E Sm(X x R")
.
SYMBOL OF A P.D.O.
205
This l e a d s us t o :
I f we Let P be a proper operator taken
PROPOSITION 2 . 3 : from L m ( X ) ,
then there e x i s t s a unique
p E Sm(X x
R")
such
Moreover, i f p ( x , y , S ) is an amplitude f o r
that we have ( 2 . 2 . 3 ) .
P, we have (2.2.5) @
We say t h a t p ( x , S ) i s the complete symbol of P and we w rite P = p(x,D).
PROOF:
We have j u s t proved t h e e.xistence o f
ness a r i s e s o u t o f t h e d e n s i t y o f
&(I?"]
p
.
;
t h e unique-
The expansion
( 2 . 2 . 5 ) f o l l o w s from a p p l i c a t i o n of ( 2 . 1 . 4 ) t o t h e p a r t i c u l a r case ( 2 . 2 . 4 ) .
3.
SYMBOL OF A PSEUDO-DIFFERENTIAL OPERATOR
3.1
COMPLETE SYMBOL:
We s h a l l now e x t e n d t h e n o t i o n o f
a complete symbol t o an a r b i t r a r y o p e r a t o r P i n g P r o p o s i t i o n 1.5, w e can w r i t e P = P'
E
Lm(X).
By u t i l i s -
+ R' and we denote by
p ' ( x , S ) t h e complete symbol of t h e proper o p e r a t o r PI. consider a d i f f e r e n t decomposition o f t h i s t y p e , P = P" the difference p' symbols i s i n
-
S-o(X
If we
+ R",
p" between t h e corresponding complete x
R")
s i n c e t h e o p e r a t o r P'
- P
= R"
- R'
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
206
If we denote by o(P)(x,c) the
is both proper and in L--(X).
equivalence class of p' in Sm / S--, we have thus defined a linear mapping
Lm[X) 3 P ->
o(P) E Sm(X
x
R") /
This is clearly surjective, and Proposition
S-"(X
x
.
R")
1.4 shows that its
kernel is L-m( X); therefore we have an isomorphism (3.1.1)
Lm /
1-OD
->
0.
Sm
/
S-"
called the isomorphism of the complete symbol.
REMARK 3.1.2.
:
The above shows that any operator be-
m longing to L (x) can be written, to within a regularising operator, in the form p(x,D) with
3.2. PRINCIPAL SYMBOL.
e Sm(x x
p
If P
E
R")
.
Lm(X), it is clear that we
have the equivalence p
E LW1(X)
>-(
a(P] E
s-'
/ s-=
.
Consequently, we deduce from (3.1.1),by passage to the quotient, a simpler isomorphism:
(3.2.1)
Lm
/
Om
LW1
sm / Sm-l
->
s
called the isomorphism of the principal symbol of degree m.
If P = P(x,D) is a differential operator, then om(P) admits a canonical representative in Sm( X part P,(x,E)
x
Bn ) , namely the principal
of the polynomial P ( x , S ) .
(SEC.
4)
ALGEBRA
& SYMBOLIC CALCULUS
207
Lm(X), we s h a l l o f t e n commit t h e n o t a t i o n a l m abuse o f i d e n t i f y i n g u (P) w i t h a r e p r e s e n t a t i v e i n S , and w e m Given P
E
s h a l l do t h e same for c r ( P ) .
4.
ALGEBRA AND SYMBOLIC CALCULUS OF P.D.O. ' s I n Chapter I we encountered t h e d e f i n i t i o n of t h e t r a n s -
pose tP and t h a t of t h e a d j o i n t P* o f an o p e r a t o r P on an open s u b s e t of lRn.
For p.d.o.'s we have:
THEOREM
4.1
:
If P
E
Lm(X) , then
t
P and P* are i n Lm(X)
and the complete symbols admit the following expansions: (4.1.1)
(4.1.2)
a PROOF: m
with C
Since t h e t r a n s p o s e or t h e a d j o i n t of an o p e r a t o r
k e r n e l i s an o p e r a t o r w i t h C
m
k e r n e l , we w i l l n o t
r e s t r i c t t h e g e n e r a l i t y by assuming t h a t P i s p r o p e r .
If we p u t p ( x , h )
t
o(P)[x,S)
, then
P c a n be w r i t t e n
208
(CHAP. 4)
PSEUDO-DIFFERZNTIAL OPEFATORS
By n o t i n g t h a t we have
v>
,
<W,tP">
Q
v
E
,
CIfx)
we o b t a i n
We t h e r e f o r e deduce t h a t tP E Lm(X), and t h e expansion (4.1.1) t h e n f o l l o w s from ( 2 . 2 . 5 )
t
The P
c a s e can be deduced from t h i s s i n c e P
*
=
t-
P.
F o r composition, we have
THEOREM 4 . 2 :
Suppose we have P
E
Lm(X) and Q
E
Lm' (X);
we assume t h a t a t l e a s t one of t h e s e is a proper operator. Then, (4.2.1 )
R
,
P
E LWm' (X)
and for t h e complete symbol
PROOF:
We s h a l l n o t r e s t r i c t t h e g e n e r a l i t y by assuming
t h a t both of these operators a r e proper.
(4.2.3)
e-ixt(Q
p)(eiRs)
e-iXt
Q
rei"!
= eixzQ(u(P)eixz)
We have
e-ixI ~ ( ~ ~ ~ f ) ]
.
Consequently ( 4 . 2 . 1 ) can be deduced from ( 2 . 1 . 3 ) and from Proposition 2.2.
The expansion ( 4 . 2 . 2 ) f o l l o w s from
4)
(SEC.
ALGEBRA & SYMBOLIC CALCULUS
209
a p p l i c a t i o n of ( 2 . 1 . 4 ) t o ( 4 . 2 . 3 ) .
REMARK 4.3 :
The composition o f p r o p e r p . d . 0 . ' ~ i n d u c e s ,
v i a t h e isomorphism of t h e complete symbol, an a s s o c i a t i v e
l a w of composition on
/ s-".
S+w
This l a w i s denoted as
=#
and can be d e f i n e d d i r e c t l y by
(4.3.1)
-
(qSfd(x,g)
1 7 :a
a!
q(x,g)
.
:D P(X,E)
F i n a l l y , we have i n v a r i a n c e o f t h e p.d.0.'~ under diffeomorphism.
THEOREM
4.4
subsets X , Y o f B n .
be a diffeomorphism between two open
Suppose P
operator transported by (4.4.1
x
Let
:
I
x
x*P
E
L ~ ( x )and l e t X*P denote the Then
onto Y.
E
Lrn(Y)
and f o r the complete symbol we have
-
~(GPI(Y,TIcy
(4.4.21
PROOF:
1
7 a;
t
~ ( P H X , x'(X).T)DF(eir)
( Z P X
We w i l l n o t reduce t h e g e n e r a l i t y by assuming t h a t
P i s p r o p e r ; i n t h i s case
x*P i s a l s o proper and we u t i l i s e t h e
c h a r a c t e r i s a t i o n i n Proposition 2.2.
We have
consequently, ( 2 . 1 . 3 ) of Theorem 2 . 1 shows t h a t t h e l e f t - h a n d s i d e of ( 4 . 4 . 3 ) i s i n Sm(Y x B n ) . equal t o taking
g ( ~ P ) ( y , ~ ); in (2.1.4)
:
This l e f t - h a n d s i d e i s a l s o
hence t h e expansion
(4.4.2)by
PSEUDO-DIFFERENTIAL OPERATORS
210
a = 1
4.5
,
,
= u(P)(x,Z>
P(X,Y,$)
(CHAP.
4)
= X(X).T.
*(x,Tl)
For p r i n c i p a l symbols, it
REDUCED SYMBOLIC CALCULUS,
f o l l o w s from t h e e x p r e s s i o n g i v i n g t h e f i r s t t e r m of t h e expan-
(4.1.1.), (4.1.2),(4.2.2),(4.4.2),
sions
t
a,(
(4.5.11
(4.5.31
o,~Pl~x,- 51
=
p)rx,c)
OrrHm~( Q o
(4.2.2),we
A s an a p p l i c a t i o n of
(Q)(x,!I
0 ',
om(%Pl ( Y , 7) = O,(PI
(4.5.4)
t h a t we have
(x,
a,(P)(x,!)
t
X' (XI .Ill w i t h y = X(X]
.
deduce (under t h e c o n d i t i o n s of
Theorem 3 . 2 ) t h a t t h e commutator
(4.5.5)
[P
, Q]
Q
= P
-Q
where t h e Poisson b r a c k e t {
is in
P
L~
ml-I
(XI
and
} i s d e f i n e d by
n
REMARK 4.6
:
Case of vector-valued p . d . 0 . ' ~ .
L e t E , F be f i n i t e dimensional v e c t o r spaces on C
.
We
denote by Lm(X; E , F) t h e space of p . d . 0 . ' ~ of t h e form (1.2.1) when t h e amplitude
p
E Sm(X x R" ; L(E, F)) ca
.
Then, such
m
o p e r a t o r s a c t c o n t i n u o u s l y from C (X; E) i n t o C (X; F ) and 0
p o s s e s s a l l t h e p r o p e r t i e s of s c a l a r - v a l u e d p . d . o . ' s e x c e p t i o n of
(4.5.5) and (4.5.6). For
with t h e
example, Theorem 2.1
f o l l o w s from Remark 9.11 o f Chapter 111; it i s of c o u r s e
P.D.O.'s
a
necessary t o t a k e
< SW(X
x
RN ; F),
P E L ~ ( X ;
E
, F)
b
E Sm(X x
a,(P)
E Sq(X
211
ON MANIFOLDS
x
RN ; E)
and we t h e n f i n d
The p r i n c i p a l symbol of
i s c l e a r l y an element
R" ; f(E, F)) / Sm-'.
For t h e composition,we assume
Q , P E Lmm' (X ; E
Q E L m' ( X ; F, G ) and we f i n d t h a t
, G)
.
F i n a l l y , i f we t a k e i n n e r p r o d u c t s ( r e s p . Hermitian produ c t s ) on E and F, t h e t r a n s p o s e ( r e s p . t h e a d j o i n t ) of
P E Lm(X ; E
, F)
i s an o p e r a t o r tP ( r e s p . P*) belonging t o
L m ( X ; F, E ) .
5.
P.D.O.'s
ON MANIFOLDS
I n t h i s s e c t i o n X denotes a manifold of dimension
n.
Following t h e d e f i n i t i o n of d i f f e r e n t i a l o p e r a t o r s ( D e f i n i t i o n
7.6 o f Chapter I ) and from t h e c h a r a c t e r i s a t i o n o f p . d . o . ' s d e s c r i b e d i n S e c t i o n 2 , it i s n a t u r a l t o s t a t e :
DEFINITION 5 . 1 :
Suppose
m
ER U
;
{-o]
P : C:(X)
L m ( X ) the s e t of continuous operators
-
we denote by C"(X)
which s a t i s f y the following condition: f o r any real-valued f u n c t i o n
*(x,8)
E
C"(X
x
:R 1,
homogeneous of degree 1 with respect t o 6, and f o r any amplitude a
E
Sq(X x
dx$(x,8)
RN )
with compact support i n x such t h a t
f 0 f o r (x,6) E
conic supp a , we have
212
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
F i r s t l y , Theorem 2 . 1 and P r o p o s i t i o n 2.2 show t h a t t h i s d e f i n i t i o n c o i n c i d e s w i t h t h e former d e f i n i t i o n when X i s an If X i s a manifold and we have P
open s u b s e t o f IRn,
E
Lm(X),
and i f we l e t U be an open s u b s e t of X , we d e f i n e t h e o p e r a t o r PIu as b e i n g t h e compose
c"(x) 0 ->
->
CpI)
c"(x) ->
c"(u)
where t h e f i r s t arrow i s t h e n a t u r a l i n j e c t i o n and l a s t arrow denotes t h e r e s t r i c t i o n t o U ; D e f i n i t i o n 5 . 1 shows t h a t PIu
E
Lm(u).
D e f i n i t i o n 5.1 i s obviously i n v a r i a n t under diffeomorphism; consequently i f
->
-
%(PI,,)
EL
x :X
transported operator
U
3
s t r i c t l y positive density. P
E
Rn
m N
(u)
.
i s a c h a r t of X , t h e
We assume t h a t X i s equipped w ith a
PROPOSITION 5 . 2 :
of a p . d . 0 .
U c
Then t h e kernel d i s t r i b u t i o n
Lm(X) is Cm outside of Diag ( X
x
X).
= L-"(x)
=
Further-
more, we have the following e q u a l i t i e s : (5.2.1.)
Operators w i t h kerne l s E
PROOF:
Suppose (x
0
neighbourhoods o f x
0'
yo
c"[x
, yo) 8
x
x)]
Diag(X
n P(x) . m
x
X ) and l e t U , V be open
chosen s u f f i c i e n t l y small f o r them t o
be a l s o d i s j o i n t c o o r d i n a t e p a t c h e s .
Consequently i n t h e open
(SEC.
5)
P.D.O.'s
set W = U u V
213
ON MANIFOLDS
c R"
X :W
we can d e f i n e a c h a r t
.
Furthermore, we know ( s e e ( 8 . 2 . 1 ) , C h a p t e r I ) t h a t t h e r e s t r i c t ion o f P t o W
-
W i s t r a n s p o r t e d by
x
(x,x)
into a distribution
K such t h a t N
(5.2-2)
K=KNg
P where
g
is a C
, Since K
m
N
f u n c t i o n and P = x*P.
i s Cm o u t s i d e of t h e d i a g o n a l o f
? X F,
we deduce
P The same method allows
t h a t t h e k e r n e l of P is Cm i n U x V .
us t o prove t h e i n c l u s i o n
n Lrn(X)
c { o p e r a t o r s w i t h k e r n e l i n C"(X x X ) l
rn
.
A proof analogous t o t h a t o f P r o p o s i t i o n 9 . 2 of Chapter I11 m
proves t h e i n c l u s i o n : { o p e r a t o r s w i t h C F i n a l l y , we obviously have
L-=(X)
c
k e r n e l s ) c L-m(X).
n Lrn(X) ,
rn
and t h i s
concludes t h e proof of ( 5 . 2 . 1 ) .
The q u o t i e n t L" / L-m i s a s h e a f
PROPOSITION 5.3 :
s e t s (U ); ci
p
= lUiY
Let there be a covering of X by open
we s p e c i f y operators
Then, there e x i s t s P
pw
E
on X ; t h i s i s
Pcu E
L ~ ( U @ ) such t h a t
L m ( X ) unique modulo L W m ( X ) such t h a t
( t h e sign E s i g n i f i e s moduZo
.
214
PSEUDO-DIFFERENTIAL OPERATORS
PROOF:
(CHAP.
4)
Taking a f i n e r covering where a p p r o p r i a t e , we
w i l l n o t r e s t r i c t t h e g e n e r a l i t y by assuming t h a t t h e Ua are r e l a t i v e l y compact and l o c a l l y f i n i t e .
Suppose ( p a ) i s a
s u b o r d i n a t e d p a r t i t i o n o f u n i t y with f u n c t i o n s p
CY
E
and suppose we have f u n c t i o n s pa
W
Co(Ua)with
f CI(ue)
identical t o 1
c1
on SUPP P,.
We d e f i n e P by
(5.3.1
1
p u
b,
=
pa P&
cz It i s c l e a r t h a t P i s a continuous o p e r a t o r from C"(X)
into
0
s i n c e t h e sum ( 5 . 3 . 1 ) i s f i n i t e f o r a l l u
C"(X) Taking
a
E
C"(X). 0
and $ a s i n D e f i n i t i o n 5 . 1 , we have
(5.3.2) N t h e right-hand s i d e i s a f i n i t e sum o f symbols i n Sm+q(X x iE ) ;
consequently we c l e a r l y have P
We now show t h a t
p
rPP
Lm(X).
E
Since supp p
'
cy
rl
Up
c U, n U
8'
we have (5.3.31
R
where
us
'a p ~ IU l B
x u
=
(pa
- 1)
PB p,
=
a ' (
p
Pp
'8
p~)lUg p,
+ R
aB
i s an o p e r a t o r w i t h C
B r n
S u b s t i t u t i n g (5.3.3) i n t o (5,3.1),
we o b t a i n
m
kernel i n
215
P.D.O.'s ON MANIFOLDS
P
REMARK 5 . 4
P*
The above p r o p o s i t i o n and i t s proof remain
:
v a l i d i f we everywhere r e p l a c e t h e e q u a l i t i e s modulo L
m'
e q u a l i t i e s modulo L
This sheaf
-m
by
w i t h m' < m .
p r o p e r t y a l l o w s us t o g i v e an e q u i v a l e n t form,
which i s p o s s i b l y more n a t u r a l , f o r t h e d e f i n i t i o n of p . d . o . ' s on X.
PROPOSITION 5.5 :
C:(X)
4
x :u
C"( X)
. Then
H
4
u c R"
PROOF: necessary.
, the
Let P be a continuous Linear operator
P E Lm(X)
operator X * ( P
i f and onZy i f f o r any chart mc
IU
i s i n L (u)
We saT7 f o l l o x i n g D e f i n i t i o n
.
5 . 1 t h a t t h i s was
I n o r d e r t o show t h a t it i s a l s o s u f f i c i e n t , it
s u f f i c e s t o apply P r o p o s i t i o n 5 . 3 t a k i n g a c o v e r i n g of X by open c o o r d i n a t e p a t c h e s .
F i n a l l y , we n o t e t h a t P r o p o s i t i o n 1 . 5 and i t s proof remain v a l i d when X i s a m a n i f o l d .
216
(CHAP.
PSEUDO-DIFFERENTIAL OPERATORS
6.
4)
SYMBOLIC CALCULUS OF P.D.O,’s ON A MANIFOLD
6.1.
DEFINITION OF THE PRINCIPAL SYMBOL OF AN OPERATOR P E Lm(X).
x
With each c h a r t
lu)
-
sm(r x R”) / SW1. -1 allows diffeomorphism x
U,,,(&P
the
:U
/
U
cR”
us t o l i f t it t o an element denoted
S”l(T*U
\ 0) independently
x
choice of t h e c h a r t diffeomorphism same r e a s o n we have
we a s s o c i a t e t h e symbol
Formula ( 4 . 5 . 4 ) shows t h a t
E
E Sm(T*U\ 0)
by q,(P)
N
uu(p) = av(P)
of th e
d e f i n e d on U. in
T*(U
For t h e
n V) \
0
I
thus
t h e combination p r o p e r t y of t h e symbols ( s e e P r o p o s i t i o n 2 . 1 0 o f Chapter 111) proves t h e e x i s t e n c e of a unique element
cr € Sm(T*X \ 0)
/
rn
for any open sub-
This symbol u i s c a l l e d t h e p r i n c i p a l symbol
set of c h a r t s . of degree
alu = uu
Sm-’ such t h a t
of P and i s denoted by crm(P).
The l i n e a r mapping p E Lm(X)
_I_*
/
E S”(T*X\O)
o,(P)
S-’
is s u r j e c t i v e ; Remark 5 . 4 allows us t o reduce t o c o n s i d r i n open s u b s e t s of IRn i n o r d e r t o prove t h i s . P r o p o s i t i o n 5 . 5 shows t h a t cr,(P) P
E
m- 1 L (X).
sm-l
i f and only i f
Hence w e have t h e isomorphism o f the p r i n c i p a l
symbol of degree
(6. 1. 1)
E sm-I /
Furthermore,
m
Lrn(X)
/ Lnc1[X)
For composition we have:
Om
*
Sm(T*x \O)
/
s”’(T*X
\O)
(SEC. 6 )
SYMBOLIC CALCULUS ON A MANIFOLD
P E Lm(X)
Suppose t h a t
THEOREM 6 . 2 :
217
Q E Lm’ (X)
where one a t l e a s t i s a proper operator; then Q
PROOF:
,P
E LWm1(X)
From D e f i n i t i o n 5 . 1 , we can immediately show t h a t Q
P E Lmcm’(X)
.
The d e f i n i t i o n of t h e p r i n c i p a l symbol shows t h a t i f it is s u f f i c i e n t t o v e r i f y ( 6 . 2 . 1 ) i n an open c o o r d i n a t e patch,U. Suppose we have
cp
bourhood of supp $.
$ Q P
,0
E Cz(U)
w i t h q = 1 i n t h e neigh-
We have
3
$ Q q P
e
6 Q cp P
+
+Q(I-(P)P
since
$ Q(I
- cp)
E L-”(x)
.
m
which concludes t h e proof s i n c e $ i s a r b i t r a r y i n C (U). 0
THEOREM 6 . 3 :
Assume t h a t X i s endowed with a s t r i c t z y
218
(CHAP. 4 )
PSEUDO-DIFFERENTIAL OPERATORS
p o s i t i v e density.
If P
E
(6.3.1)
tP
(6.3.2)
P+ E P ( X )
PROOF:
E
P(X)
L ~ ( x ,) we then have I
q t P 1 (x, 51 =
;
O,(P*I(X,51
=
+I
(X,-C)
m m
I
L e t t i n g U be an open s u b s e t of X , we c l e a r l y have
c o n s e q u e n t l y ( 6 . 3 . 1 ) and ( 6 . 3 . 2 ) f o l l o w from meorem 4 . 1 and from P r o p o s i t i o n 5 . 5 .
PROPOSITION 6 . 4 :
Any p . d . 0 . P
E
L m ( X ) extends contin-
uousLy i n t o an operator from &'(x) i n t o .I?'(x).
PROOF:
We have P = t ( t P ) ; now Theorem 6 . 3 i m p l i e s t h a t
t~ i s continuous from
m
c o (XI i n t o c
m
( X I ; consequently P r o p o s i t i o n
6 . 4 f o l l o w s from Remark 8.4 of Chapter I .
DEFINITION 6.5 :
i f f o r any chart
the operator h ( P
U
lu
A p.d.0.
P
c x &cc
E
R"
L m ( X ) i s called c l a s s i c a l ,
,
the complete symbol of
u ) E L m (U) is c l a s s i c a l .
We denote by
Lm(X) the subspace of c l a s s i c a l p . d . 0 . ' s of degree m. C
Formulas ( 4 . 4 . 2 ) , ( 4 . 1 . 1 ) ,
( 4 . 1 . 2 ) and ( 4 . 2 . 2 ) show t h a t t h i s
c l a s s i s s t a b l e under diffeomorphism, under t r a n s p o s i t i o n , under passage t o t h e a d j o i n t , under r e s t r i c t i o n and under composition, by u s i n g t h e proof of Theorem 6.2.
6)
(SEC.
SYMBOLIC CALCULUS ON A MANIFOLD
219
The p r i n c i p a l symbol o f an o p e r a t o r P E Lm admits a
c
CI(T*X
c a n o n i c a l r e p r e s e n t a t i v e u ( P ) which i s a
rn
f u n c t i o n and i s homogeneous of degree
PROPOSITION
E
(x0,&,)
6.6
T*X \
Suppose P
:
01 ,
with respect t o 5
m
s i n c e t h i s i s t r u e i n any l o c a l c h a r t .
\o)
Moreover, we have
.
E Lz(X)
Then f o r
we have
U
7-
where:
of
a E C:(X)
xo
$ E CO1(X)
PROOF: x
and has the value 1 i n the neighbourhood
,
dJl(xo) =
to
f
d$
on SUPP a.
0
Let U be an open c o o r d i n a t e p a t c h
containing
OD
0,
and suppose we have a , @E Co(U) w i t h a = 1 i n t h e
neighbourhood of supp B and B(x ) = 1. 0
A t the point x = x
0,
we have
e-iT* p(a
eiT9‘)
e-i~* p plCra .i~$>
+
e - i ~ tB P(I-U] iaei T * ) . -m
The second t e r m on t h e r i g h t - h a n d s i d e i s i n S c a l c u l a t e t h e f i r s t t e r m we can r e p l a c e P by P
(B+)and t o
IU
and conse-
q u e n t l y reduce t o t h e c a s e where X’is an open s u b s e t o f IR”. I n t h i s c a s e we can r e p l a c e P by u p ( x , D ) and a p p l y Theorem 2 . 1 ; we o b t a i n
e-iT*p(aeiT’)(xo)
= a,(P)(x,
, T d)(xo))a(xO)
md S”’.(R+)
2 20
(6.6.1).
which proves
REMARK 6.7: bundles.
Case of p . d . o . ' s
f ,9
Let
o p e r a t i n g on s e c t i o n s of
as,$$) t h e vector
denote by
C ~ ( X; 9)
.
9
3; i n t o
C"(X
4
F on C.
and
X
bundle above
rn
f
and
g
9
8 and
of
P :
, $1
if
above a l o c a l
x -
U of X we have ( w i t h t h e n o t a t i o n of ( 7 . 8 . 2 ) o f
-+
E
Lm(i;
E
, F)
We denote by vX t h e c a n o n i c a l p r o j e c t i o n denote by
* nx f
* %g 3
i n v e r s e images o f
modulo Sm-',
(3
U
E
t h e b u n d l e s on and
9
under nX.
. T*X
Sm(T*IJ\o
of t h e choice of
combine i n t o an element o f
\0
* T X\
and
X
which a r e
0
g
.
Sm(T*X \ 0 ; L(n>
independent, These elements
, r;
g e n e r a l i s a t i o n of t h e r e s u l t s of S e c t i o n s 5 and
Definition
,g)
6
5
U
9 ) ) / Sm-'
c a l l e d t h e p r i n c i p a l symbol of P and denoted by u,(P).
Lm(X ; 9
and We
which can be l i f t e d
* , '9))
; $(nu% f
4
The p r i n c i p a l symbol of
S"(c x R" ; L(F,G))
i s an element of
i n t o an element
type
We
of t h e morphisms
i s an element of L (X ; 9
; 9)
Chapter I ) :
Pf,g
E
, having
We s a y t h a t a continuous o p e r a t o r
f o r any t r i v i a l i s a t i o n s chart U
X
be v e c t o r bundles above
as f i b r e f i n i t e - d i m e n s i o n a l v e c t o r s p a c e s
of
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
The
t o p . d . 0 . ' ~ of
p r e s e n t s no new d i f f i c u l t i e s .
Likewise,
6 . 5 and t h e p r o p e r t i e s of c l a s s i c a l p . d . o . ' s g e n e r a l i s e
t o t h e spaces
L:(X
;9
, Y)
.
7)
(SEC.
7.
ELLIPTIC P.D.O.'s
221
ELLIPTIC PSEUDO-DIFFERENTIAL OPERATORS
I n t h i s s e c t i o n , X d e n o t e s a manifold. To t h e concept of an e l l i p t i c symbol t h e r e corresponds t h e concept o f an e l l i p t i c o p e r a t o r .
DEFINITION
of degree
7.1:
If P
E
L m ( X ) , we say t h a t P i s e l l i p t i c
i f i t s principaZ symboZ o,(P)
m
symboZ of degree m i n
i s an e l z i p t i c
.
Sm(T*X\ 0)
If P i s a d i f f e r e n t i a l o p e r a t o r o r more
EXAMPLE 7 . 2 :
generally a c l a s s i c a l p.d.0.
of degree
m
, t h e n it i s
i f and o n l y i f t h e homogeneous f u n c t i o n a,(P)
elliptic
does n o t v a n i s h on
T* X \ 0.
I n f a c t , e l l i p t i c o p e r a t o r s a r e t h o s e which a r e i n v e r t i b l e i n t h e s e n s e of t h e f o l l o w i n g theorem:
The operator P
THEOREM 7 . 2 :
m
E
Lm(X) i s e l l i p t i c of degree
i f and only i f there e x i s t s a proper operator Q
that P.Q.
1
Identity.
Q.P. E I d e n t i t y ;
E
L-,(X)
such
Such an operator Q then s a t i s f i e s
it i s unique modulo
and i s termed a para-
metrix of P. PROOF:
The e x i s t e n c e o f Q c l e a r l y i m p l i e s t h e e l l i p t i c i t y of
om(P) s i n c e we have
222
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
. g,(Q)
o,(P)
-
1 .E
*
-1
S
(T X l 0)
I n o r d e r t o prove t h e e x i s t e n c e o f Q, w e make use o f an approximation
procedure which i s important i n i t s own r i g h t .
By h y p o t h e s i s , we know t h a t t h e r e e x i s t s q
0
E S'"(T*X
\ 0)
such t h a t
(7.2.1) Let
orn(P).qo
Qo
E L'm(X)
-1
-1
E S
*
.
(T X \ O )
be proper such t h a t
,
orn(QO)= q,
then
( 7.2.1) implies
(7.2.2)
P.
- I = A,
0,
.
E L-'(X)
We now prove t h e f o l l o w i n g :
m LEMMA 7.3 :
such t h a t rnk
We take a sequence of p.d.o.'s -0.
k=
i ) There e x i s t s
S
.
Then:
f
L "(X)
Rk E L k ( X )
rn
proper, such t h a t for any
integer h
(s
(7.3.1)
-
h
E
Rk)
Lmh+'(x)
Such an operator S i s m i q u e modulo L--(X)
ii) If
R E L-'(X)
.
and we w r i t e
i s proper, and if
T E Lo(X)
is proper such t h a t (7.3.2)
- &
(-llkRk
then we have
(7.3.3)
(1+ R).T
P
T.(I
+ R)
I
Id
.
7)
(SEC.
ELLIPTIC P.D.O.'s
PROOF:
u
If
x u
223
i s a l o c a l c h a r t of X , we can
-D
rn define
Su E L '(U)
by c o n s i d e r i n g t h e asymptotic sum of t h e CI
complete symbols i n U o f t h e o p e r a t o r s X ( R
*
k1U)
( t h i s being Then, we
p o s s i b l e i n view of Theorem 4.2 of Chapter 111). d e f i n e S s a t i s f y i n g ( 7 . 3 . 1 ) by combining t h e S
U by v i r t u e of
--m
The uniqueness modulo L
Proposition 5.3.
(7.3.1).
i s obvious from
The v e r i f i c a t i o n of ( 7 . 3 . 3 ) i s immediate from t h e -
d e f i n i t i o n of T.
We r e t u r n now t o t h e proof of t h e theorem.
We t a k e
Q = Q0 .T where T is d e f i n e d by ( 7 . 3 . 2 ) and R i s a proper op-0
erator equal t o R
0
modulo L P . Q
(7.3.4)
I
; then Q
Id
E
L-m(X)
i s proper and
.
The same method w i l l a l l o w us t o c o n s t r u c t a proper such t h a t
Q'
(7.3.51
.
P a I d
GI' E )X("'L
.
M u l t i p l y i n g ( 7 . 3 . 5 ) on t h e r i g h t by Q, ( 7 . 3 . 4 ) i m p l i e s t h a t Q e Q '
which proves t h a t Q i s a p a r a m e t r i x o f P and t h a t a p a r a m e t r i x
i s unique modulo L-O0'
REMARK 7 . 4 : bundles
Case of p . d . o . ' s
o p e r a t i n g on s e c t i o n s of
.
We once more adopt t h e n o t a t i o n o f Remark
P
E Lm(X ; 3
,3)
e x i s t s a symbol
;
6.7.
Suppose
we say t h a t P i s r i g h t e l l i p t i c i f t h e r e
PSEUDO-DIFFERENTIAL OPERATORS
224
q,
E
S-m(T*X \ 0
.
o,(P)
q,
- Id
IT?))
; L(?T:$,
E
(CHAP.
such t h a t
*
*
; L(nxY, nx9))
S-'(T*X\O
4)
.
Theproof of Theorem 7.2 shows t h a t P i s r i g h t e l l i p t i c i f and only i f t h e r e e x i s t s
Q
E
; 3, 9 ) such t h a t P . Q E I ;
L-"(X
we say t h a t Q i s a r i g h t parametrix of P.
We have t h e analogue of t h i s "on t h e l e f t " , and we have t h e concept of a two-sided
parametrix when 3 and
9
a r e of t h e
same rank.
For o p e r a t o r s of t h e c l a s s i c a l t y p e , we have a simple c r i t e r i o n of e l l i p t i c i t y ; t h i s i s
PROPOSITION 7.5
P E
Supposing
:
L:(X
;9
,'3) ,
then
P is l e f t e l l i p t i c ( r i g h t e l l i p t i c ) if and only if for any
E
(x,5)
T*X \ O
the linear mapping
i s injective (surjective). and
9
a,,,(~)(x,5l
E 4sx
Moreover, if the bundles
9,) 3
are of the same rmk, any Zeft ( r i g h t ) parametrix of P
i s a two-sided parametrix.
To prove t h a t P admits a l e f t p a r a m e t r i x , we s h a l l
PROOF:
t
now d e f i n e t h e a d j o i n t P
of P.
To do t h i s , we endow X w i t h a
s t r i c t l y p o s i t i v e d e n s i t y and we d e f i n e a Hermitian s t r u c t u r e on
3; - a n d on
7
(we can do t h i s l o c a l l y by means of a
(SEC. 7 )
ELLIPTIC P.D.O.'s
225
t r i v i a l i s a t i o n and we re-combine by p a r t i t i o n o f u n i t y ) .
and C i ( X ; 3) a r e each equipped w i t h an
Ci(X,S)
t h e spaces
.
, lg (p u
9
L:(X
=
v$
(u
; 3 , 9)
*
p
9
c;(x,sJ ,
u E
Vl9
whose p r i n c i p a l symbol s a t i s f i e s
operator equivalent t o P
*
and
X
,
.
3 X
i s a proper
*
-*
The p r i n c i p a l symbol P P i s for a l l
*
rn
gEx ;
d*
If P
it t h u s has t h e same p r i n c i p a l symbol
t h e i n j e c t i v e l i n e a r mapping (0 ( P ) ( x , C ) )
Fx i n t o
v € CO"(X,Y)
6 . 7 ) t h a t t h i s i s an o p e r a t o r belonging
f o r t h e Hermitian s t r u c t u r e s on 9
a s P*.
and
The a d j o i n t P* o f P i s d e f i n e d by t h e e q u a l i t y
and we know ( s e e Remark to
, l9
(
i n n e r product denoted r e s p e c t i v e l y as
(
Thus
it i s t h e r e f o r e i n v e r t i b l e .
-*
.
(x,c) E T X \ O
a,(P)(x,S)
Consequently t h e Let Q E L4m(X
o p e r a t o r P P i s e l l i p t i c of degree 2m.
from
;9
, 9)
be a p a r a m e t r i x of t h i s o p e r a t o r ; we have 4
Q. P
(3.5.1) The e q u a l i t y
.P
=
I
+
A
with
R €L--(X;9,9)
(7.5.1) shows t h a t t h e o p e r a t o r Q?* E L-,(X
;9 ,9)
i s a l e f t p a r a m e t r i x of P.
The o t h e r a s s e r t i o n s of t h e p r o p o s i t i o n are immediately obvious,
.
226
PSEUDO-DIFFERENTIAL OPERATORS
7.6
(CHAP.
4)
CASE OF SYSTEMS WHICH ARE ELLIPTIC I N THE SENSE OF DOUGLIS-NIRENBERG
. P =
We s a y t h a t a system
...
where
(‘jk) j=l, J k=Ii*.o9K
rn pJk
E
i s of t h e t y p e ( s , t )
Lcjk (X)
E
IRJ x IRK i f
f o r any ( j , k ) E J x K . We d e s i g n a t e a s t h e mjk = tk - j p r i n c i p a l symbol of P of t y p e ( s , t ) t h e f u n c t i o n from Cm(T*X
\0
;
46
cJ))
d e f i n e d by t h e m a t r i x
A system P of t y p e ( s , t ) i s s a i d t o be l e f t ( r i g h t ) e l l i p t i c i n
t h e sense of Douglis-Nirenberg, if f o r any t h e m a t r i x CT
s, t
(P) ( x ,
THEOREM 7.7 :
5) is
(xi{)
E
injective (surjective).
T*X\ 0
We have :
A system P of type ( s , t ) i s l e f t ( r i g h t )
e l l i p t i c i f and only i f there e x i s t s a system Q of type ( t , s ) which i s proper and such t h a t
Q
.
P t
I
(P
.
Q E
.
I)
Moreover, when K = J , one-sided e l l i p t i c i t y i s equivalent t o the existence of a two-sided parametrix 9.
PROOF:
We equip X w i t h a Riemannian m e t r i c s o a s t o be
a b l e t o d e f i n e t h e l e n g t h 151 of a cotangent v e c t o r .
r
E
With
IR, we denote by A r a proper o p e r a t o r from Lz(X) having 151
f o r i t s p r i n c i p a l symbol.
we denote by A
The o p e r a t o r A
-1 a p a r a n e t r i x of A r r
.
r
r
i s t h u s e l l i p t i c and
With t h e v e c t o r s
E
J B ,
(SEC. 8 )
P.D.O.'s
&
SOBOLEV SPACES
we a s s o c i a t e t h e d i a g o n a l system of t y p e (s,O) matrix A
t
E
S
(ASJjk = As
with
if
227
d e f i n e d by t h e
j = k , and 0 o t h e r w i s e .
For
J
K -1 IR we s i m i l a r l y d e f i n e a d i a g o n a l system A t o f t y p e ( O , t ) .
Then , t h e formula
p l a c e s systems P of t y p e ( s , t ) i n b i j e c t i o n w i t h t h e o r d i n a r y
8 , GJ)
E Lo(X ;
systems
.
Using t h e symbols it can be
immediately shown t h a t P i s l e f t ( r i g h t ) e l l i p t i c i n t h e s e n s e c
( s , t ) i f and o n l y i f P i s l e f t ( r i g h t ) e l l i p t i c .
Consequently,
t o prove Theorem 7.7 it s u f f i c e s t o use P r o p o s i t i o n 7.5 and t o c
CI
observe t h a t i f Q i s a l e f t ( r i g h t ) p a r a m e t r i x o f P , t h e n
-1
Q = At
. -Q .
As
REMARK 7.8 :
i s a l e f t ( r i g h t ) p a r a m e t r i x of P.
Tne d e f i n i t i o n s and r e s u l t s concerning
systems of t y p e ( s , t ) g e n e r a l i s e immediately t o t h e c a s e where
d(
and
GJ
9 and 9 above X which
a r e r e p l a c e d by bundles
decompose i n t o d i r e c t sums o f b u n d l e s :
J
K
9 = 0 Sk k= 1
8.
P.D.O.'s
59.; 0 j=1
AND SOBOLEV SPACES
Let X be an open s u b s e t of IRn. the p.d.o.'s,
gj
To prove t h e c o n t i n u i t y o f
we b e g i n by showing t h a t a p o s i t i v e e l l i p t i c
228
PSEUDO-DIFFERENTIAL OPERATORS
(CHAP.
4)
o p e r a t o r admits a k i n d o f approximate square r o o t .
Let C
LEMMA 8.1 :
E
L o ( X ) be such t h a t
i t s principal
symbol u 0 ( C ) is r e a l and s a t i s f i e s for any compact subset K
Then, there e x i s t s a proper (8.1.2)
C
PROOF:
=
B*B
B
+
R
E
c
X
L o ( X ) such t h a t
with
R E L-"(X)
.
The assumption (8.1.1)means t h a t f o r any compact
s u b s e t K c X , t h e r e e x i s t s c > 0 such t h a t
Thus d ( C ) i s an e l l i p t i c symbol and we can c o n s t r u c t a f u n c t i o n 0
p
E
c"(X
x
R")
of t h e same t y p e a s t h a t u t i l i s e d during t h e
proof o f P r o p o s i t i o n 3 . 2 i n Chapter 111.
bo = p j4 -
E So(X x
R")
Then
and bo i s e l l i p t i c .
i s a proper o p e r a t o r w i t h p r i n c i p a l symbol b
it s a t i s f i e s
c
- Bo* . Bo = R-,
ELqX)
0'
If Bo
Lo(X)
E
by c o n s t r u c t i o n
.
We can t h e n c o n s t r u c t by r e c u r r e n c e proper o p e r a t o r s B
j
E
L-J(X)
such t h a t f o r any i n t e g e r j 2 0 , we have
Knowing t h e B for k < j, t h e c o n d i t i o n (8.1.3) means t h a t B k j must s a t i s f y
(SEC. 8 )
P .D.O.' s & SOBOLEV SPACES
229
(8.1.4)w i l l be
Noting t h a t u- ( R - j ) is r e a l , t h e c o n d i t i o n
j
s a t i s f i e d i f t h e p r i n c i p a l symbol b 2 bo
t h i s i s possible since b
0
of B
u-j(R_J3
=
bj
j
j
satisfies
i
is elliptic.
i n order t o
It now only remains t o t a k e
obtain ( 8 . 1 . 2 ) .
REMARK 8.2
The same proof a l l o w s us t o c o n s t r u c t an
:
approximate square r o o t of C , i . e . a proper B
L o ( X ) such t h a t
E
2
C - B .
REMARK 8 . 3 :
case where C
lgl-m
E
This lemma g e n e r a l i s e s immediately t o t h e
Lm(X).
We need of course t o r e p l a c e u 0 ( C ) by
i n (8.1,1),and we f i n d B i n Lm'2(X).
u,(c)(X,5)
As an a p p l i c a t i o n of t h i s lemma, we have
THEOREM
8.4
:
Let P
E
Lo(X) be proper, such t h a t there
e x i s t s M such t h a t f o r my compact subset K
c
Then, there e x i s t s a proper and seLf-adjoint.R
X
E
L-O0(X) such
that (8.402)
lpu 1,2
sM
IIUIC
+
(R u
,u)
for
u E c:(x).
2 30
PSEUDO-DIFFERENTIAL OPERATORS
I n particular, any p.d.0. into
'Lcow (XI
PROOF: C =
8-
(CHAP.
4)
beZonging t o L o ( X ) i s continuous from
2
Lloc(x1
a
The assumption (8.1.1) i m p l i e s t h a t t h e o p e r a t o r
*
P P s a t i s f i e s t h e assumptions of Lemma 8.'l.
sequently, t h e r e e x i s t s a proper B
E
0
L (X) and R
Con-m
E
L
(X) such
that
( 8.4.3
)
P*P
+
B*B
=
h?
+R
a
The e q u a l i t y ( 8 . 4 . 3 ) shows t h a t R i s s e l f - a d j o i n t and a l l o w s us
where (
,
) denotes t h e i n n e r product i n LL(X); t h i s proves
(8.4.2).
THEOREM 8.5 :
If P
E
L m ( X ) , then f o r any s E D , P i s
I f i n addition P i s continuous from H'c omp (XI i n t o H:,:(x). S proper, it i s continuous from H c omp (X) i n t o HE;Zp(X) and from HSl o c(XI i n t o H;,?X).
PROOF:
Since t h e theorem i s t r i v i a l f o r a r e g u l a r i s i n g
o p e r a t o r , we w i l l n o t r e s t r i c t t h e g e n e r a l i t y by assuming t h a t P i s proper.
We r e c a l l t h a t t h e norm HS of u
E
C:?Bn)
is
d e f i n e d by
1M1,
=
IIAsulb
o p e r a t o r whose complete symbol i s
where
(1
As
h e r e denotes t h e
+ 1512 ) 4
2
L e t A:
(SEC. 8 )
P.D.O.’s
& SOBOLEV SPACES
be a proper o p e r a t o r e q u i v a l e n t t o A
: A’-:
A:
metrix of
IIPu/Is-m = ll~s,mPuli,
but
E
A:-’
P,A ,s
A:
= I
5
IlA&
Lo(X)
S
+ Rs.
and l e t A;
For u
”A:
A u;
E
I, p Rs
I
231
-1
C”(X), 0
+
be a parawe o b t a i n
I l ~ ~ m p ~ s u /I( o
E L - W
.
Consequently Theorem 8.5 follows from Theorem 8 . 4 .
We now g i v e a r e s u l t which i s known by t h e name of
Garding’s i n e q u a l i t y .
THEOREM 8 . 6 :
If P
E
Lm(X) , we assume t h a t there e x i s t s
a > 0 such t h a t f o r any compact subset K c X
Then, f o r any s
m
5,
E
>
0 and any compact subset K, and for aZZ
there e x i s t s c such t h a t
PROOF:
We d e f i n e t h e o p e r a t o r
t h e assumption ( 8 . 6 . 1 ) allows us t o a p p l y Lemma 8 . 1 (or more p r e c i s e l y Remark 8 . 3 ) t o t h i s o p e r a t o r ; t h u s . t h e r e e x i s t s a proper B
E
Lm’*(X)
and R
E
L-“(X)
such t h a t
PSEUDO-DIFFERENTIAL OPERATORS
232
1
* )U
~ ( +p P
= (a
- 2) Am A,, E
*
s z
u
+ 6*B u + R u
(CHAP.
with
4)
u E CI(X)
Taking t h e i n n e r product w i t h u , we o b t a i n
F i n a l l y , f o r any K , t h e r e e x i s t s C2 such t h a t for
u
E B,
,
by s u b s t i t u t i n g t h e hence we deduce ( 8 . 6 . 2 ) w i t h C = C C 12' upper bound f o r
1 (R
REMARK 8.7 :
u, u) I i n t o ( 8 . 6 . 3 ) .
G e n e r a l i s a t i o n s t o t h e c a s e of systems.
Let E,F be Hermitian v e c t o r spaces of f i n i t e dimension. We now g i v e a b r i e f i n d i c a t i o n o f how t h e r e s u l t s of s e c t i o n
8 g e n e r a l i s e t o t h e c a s e where P i s a p . d . 0 . b e l o n g i n g t o Lm(X ; E , F ) .
By s e l e c t i n g b a s e s of E and F , we can reduce t o
t h e s c a l a r c a s e i n o r d e r t o prove t h e c o n t i n u i t y of P from HS (X c omp
j
E ) i n t o HS-m(X loc
j
F).
For Ggrding's i n e q u a l i t y , we
(SEC. 8)
ON SOBOLEV SPACES
P.D.O.'s
assume t h a t
;E
P E L:(X
, E) and we
233
replace the inequality
( 8.6.1) by t h e i n e q u a l i t y
where t h e s i g n 2 i s t a k e n i n t h e s e n s e of t h e o r d e r r e s p e c t i n g t h e Hermitian o p e r a t o r s . Next we n o t e t h a t w e do n o t need t h e f u l l p r e c i s i o n of Lemma I
it s u f f i c e s t o prove t h e
8 . 1 t o prove Ggrding's i n e q u a l i t y ; e x i s t e n c e of a proper
*
C
8 . 8
+
B E Lmj2(X ; C
,
R,
with
, E)
E
RIEL
such t h a t m-I
.
(X;E,E)
To do t h i s , we t a k e for B a p r o p e r o p e r a t o r w i t h homogeneous
Jo,(C)(x,~f
p r i n c i p a l symbol e q u a l t o
(here
J
denotes t h e
p o s i t i v e square r o o t i n t h e p o s i t i v e Hermitian o p e r a t o r s ) .
The
remainder of t h e proof o f ( 8 . 6 . 2 ) i s p r a c t i c a l l y unchanged.
The c o n t i n u i t y o f t h e p . d . o . ' s
allows us t o g i v e a new def-
i n i t i o n of Sobolev spaces :
PROPOSITION 8.8:
u E lfoc(X)@
(8.8.1)
If s
I
E
u E ,f)"X), proper,
Moreover, the semi-norms proper P
E
IR, we have the equivalence P
and for a l l we have
1 1 P~ ullo ,
2
E LIDc(xl
p u
for
E L"(X)
cp
E
Ls ( X ) , also define the topology of H S o c ( X ) .
Cz(X) and
.
234
PSEUDO-DIFFERENTIAL OPERATORS
PROOF:
If
u
E
S
Theorem 8.5 i m p l i e s t h a t P u
Hloc(X),
cp
and t h a t f o r e v e r y
E
C"(X)
t h e r e e x i s t JI f o r every u
Conversely, suppose P
let Q
E
L-'(X)
Thus i f
+
R u
,
and t h i s a l s o shows t h a t f o r
H
S
loc ( X )
L:oc(X)
and C such
C:(X)
For u
E
R E L-"(X)
where
, we
P u E LFo,(X)
E
E
E
4)
.
L s ( X ) i s e l l i p t i c and p r o p e r , and
E
be a p a r a m e t r i x o f Q.
u = QPU
(CHAP.
have u
E
cp E Co(X)
H
S
loc
8 ( X ) , we i s proper
have:
.
( X ) from Theorem 8 . 5 ,
, l l c p ~ I ( ~i s
bounded
I\$,P u \ b + \I$, R u \Io , w i t h
above by a q u a n t i t y of t h e form
From t h i s w e deduce t h e i n v a r i a n c e of Sobolev spaces under d i f f eomorphism.
COROLLARY 8.9 :
Let
x
be a diffeomorphism between two open
subsets X , Y of B n . Suppose u
E
a' ( x ) ;
then
(8.9.1)
moreover,
x
*
defines a homeomo2yphism between these two SoboZev
spaces.
The proof i s immediate i n view of P r o p o s i t i o n 8.8 and t h e i n v a r i a n c e under diffeomorphism o f spaces of proper pseuao-diffe r e n t i a l o p e r a t o r s of g i v e n degree. A s r e g a r d s m a n i f o l d s , w e have
(SEC. 8 )
P.D.O.'s
If X is
THEOREM 8.10:
& SOBOLEV SPACES
a
235
,
of dimension n
Cm manifoZd
then Theorem 8 . 5 generalises word f o r word t o t h i s case.
Let K be a compact s u b s e t o f X and l e t
PROOF:
( Uj)
jtl,.
..,
be a f i n i t e c o v e r i n g o f K by open c o o r d i n a t e p a t c h e s U. of c h a r t s J N xj : u 4 U j of X . We p u t u o = x \ uj. I n view o f N
u
J
i
Theorem 2.19 then Pu
E
o f Chapter 11, we have t o show t h a t i f u
HS-m(U ) and depends c o n t i n u o u s l y on u ( 0 S j loc j
Suppose w e have
E
'pj
C,"(Uj)
the neighbourhood o f supp
rpjPu
=
Since +,u
cpj
pj
'pj
=
o
+
P 4 . u
J
and
R~
P jl. u f o r 1 S j J
Xj
*
(p
lu -1.) -
.
and
Q
j '
R.u
J
S
N.
Co(Uj)
I
N)
.
equal t o 1 i n
We have where
-a,
EL
' m
qj E
H:(X)
E
, we .(XI
Rjt=qjP(1-.qj)
.
are l e d t o study
We have
Consequently Theorem 8.5 t o g e t h e r w i t h
D e f i n i t i o n 2.18 o f Chapter 11, show t h a t
rp P q j u E H T i 3 X )
5
.
The c o n t i n u i t y i s e s t a b l i s h e d i n t h e same manner.
E l l i p t i c p.d.o.'s
THEOREM 8.11:
manifoZd and l e t P
are hypoelliptic;
more p r e c i s e l y we have
(On e l l i p t i c r e g u l a r i t y ) . E
L m ( X ) be a p.d.0.
which i s e Z l i p t i c and
proper. i)
If u
E
Let X be a Cm
&'(X) , we have the equivalences:
236
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
more generally
s i n g sup2
I f we have s, t
ii)
E
=
PU
.
s i n g supp u
IR , then for any compact subset K
c
x,
there e x i s t s c such t h a t
PROOF:
Q E Lem(X)
Letting
be a p a r a m e t r i x o f P , t h e proof i s
based on t h e e q u a l i t y
(8.11.4)
Thus
u = Q
.PU +
Ru
,
where
u Eb(X)
and R E L-OJ(X).
(8.11.1.)~ ( 8 . 1 1 . 2 ) and (8.11.3)f o l l o w from t h e c o n t i n u i t y
of t h e p . d . 0 . ' ~ . (8.Il.S)
We always have t h e i n c l u s i o n ( s e e ( 1 . 3 . 2 ) )
P u c s i n g supp u
s i n g supp
and t h e i n v e r s e i n c l u s i o n f o l l o w s from inclusion
(8.11.4)and from t h e
(8.11.5)for Q.
REMABK 8.12:
Let
F
of f i n i t e dimension over X.
and
9
b e complex v e c t o r bundles
Taking
p
E
L ~ ( X;
F
, 3) , we
l e a v e t o t h e r e a d e r t h e t a s k of showing t h a t a proof s i m i l a r t o t h a t of Theorem 8.10 allows us to prove t h a t P i s continuous from
(SEC. 8 )
I-?
a P
P.D.O.'s
( X ; 3) i n t o HSm '
loc
237
& SOBOLEV SPACES
( X ; 3) ,
Furthermore, if P i s proper
and l e f t e l l i p t i c , Theorem 8 .1 1 and i t s proof g e n e r a l i s e immediately.
I n t h e c a s e o f c l a s s i c a l p . d . 0 . ' s , we have a
converse t o t h e i n e q u a l i t y (8.11,3),as f o l l o w s :
PROPOSITION 8.13:
If P E LE(X ; 9
as s m e t h a t f o r any compact s e t K
c
, 9) , and t
< m,
we
X, t h e re e z i s t s C such t h a t
Then, P i s l e f t e l l i p t i c . From P r o p o s i t i o n 7 . 5 , it s u f f i c e s t o prove t h a t for a l l
PROOF: (xo
, so)
from
E T*X \ 0
sx i n t o
, the
gx.
of p.d.0.'~ i n I R n .
,5 )
l i n e a r mapping cr,(P)(X
We b e g i n by reducing t o t h e c a s e of a system Let U b e a neighbourhood o f x
0
which i s an
open c o o r d i n a t e p a t c h o f X o v e r which t h e bundles M N a r e isomorphic t o U x 6 and U x C , Suppose we have i d e n t i c a l t o 1 i n t h e neighbourhood of x t o 1 i n t h e neighbourhood o f supp and i f u
is i n j e c t i v e
9
0
,
and $
E
3; and
cp
9
m
E
Co(U)
Cz(U) i d e n t i c a l
If we f i x K = supp Q ,
m
E
C K ( X ) we have
Pu = P cpu = +$P ( p u Since ( i - $ ) P q
+
(1-JI)Pcpu
,
i s r e g u l a r i s i n g , we can r e p l a c e P by $ P cp
in
t h e i n e q u a l i t y (8.13.1)and t h i s allows us t o work w i t h i n t h e open s e t U.
X=R",
Consequently, w e w i l l n o t reduce t h e g e n e r a l i t y by t a k i n g
3 = X x $ ,
% = X X $
.
238
PSEUDO-DIFFERENTIAL OPERATORS
Suppose we have
E C;(R"
v
; C)'
(CHAP. 4)
and let u (x) = v ( x ) e i7X.go 7
f o r T > 0. We study the behaviour, as T
-+
+
m,
of both sides of
(8.13.1) when we take u = uT.
If we let Am be the p.d.0. with complete symbol (1
, we
+ 1012)rn/2
11
'7
have
Ilm =
11
'7
\b
consequently we obtain
Likewise, we obtain
112
By combining (8.13.1) and these two latter inequalities, we obtain
and as v
is arbitrary, this implies that
I+) 1 and consequently
c
l~,(P)(xtr,~
u (P)(x rn
,
5),
.
V(X)
I
is injective.
We now assume that X is a compact manifold endowed with a
(SEC. 8 )
P.D.O.'s
239
& SOBOLEV SPACES
s t r i c t l y p o s i t i v e d e n s i t y and we l e t 3 be a Hermitian v e c t o r bundle of f i n i t e rank over X. d e f i n e s a continuous o p e r a t o r
]p
in
P from
C"(X
, 9)
it
; 9) i n t o
we can a l s o c o n s i d e r it a s an unbounded o p e r a t o r
; 9) ;
c"(X
P E L"'(x ; 9
If
L2(X ; 9)
of dense domain
D
P
.
Hm(X ; 9)
Under t h e s e c o n d i t i o n s we have
Suppose t h a t P is e l l i p t i c .
THEOREM 8.14:
Then,
i ) The a d j o i n t of t h e unbowzded operator (ByD) i s the unbowzded
operator (P*,D*) associated w i t h t h e e l l i p t i c p . d . 0 .
P*E Lm(X;S)
with
i i ) The operator
dim Ker P
<+
D* = D = f l ( X ; 5 )
.
P i s o f Fredholm t yp e , i.e . :
,
Im
P i s a closed s e t of f i n i t e codimension.
Moreover, Ker B = Ker 6 c C"(X L2(X
(8.14.1)
, S] = Ker
; S]
*
6@Im P
,
i i i ) P i s s e l f- ad j o i n t i f and only i f P = P*, and i n t h i s case we
have (8.14.2)
L'(X
, F) =
(8.143)
C"(X
; 9) = Ker P O P(C"(X
PROOF OF i ) : Since
C"(X
Ker P 0 I m IP
; 9) i s dense i n
i
9))
.
D = Hm(X
, 9)
240
PSEUDO-DIFFERENTIAL OPERATORS
(CHAP.
4)
The mapping v E L2(X
D* =
(8.14.2)'
, 9)
E Cm(X,%) + ( h , v ) E b ; i s continuous f o r t h e
( t h e l a s t b r a c k e t i s t a k e n i n t h e s e n s e of t h e p a i r i n g between Cm(x,%)
and
).
&(X,9)
S u b s t i t u t i n g i n t o ( 8 . 1 4 . 2 ) ' , we
obtain
D* ={v
EL2(X,9)/
P*v E L z ( X , 5 ) )
.
The e l l i p t i c r e g u l a r i t y p r o p e r t y ( s e e Remark 8 . 1 2 ) i m p l i e s t h a t
D
c
*=
D
.
#(X;5)
P*v = P*v f o r v
E
D , s i n c e B* i s d e f i n e d by t h e e q u a l i t y
(RJ,v) = PROOF OF i i ) :
The e q u a l i t y ( 8 . 1 4 . 3 ) ' shows t h a t
, P*v) ,
(u
u , v
ED
.
The e l l i p t i c r e g u l a r i t y r e s u l t shows a l s o t h a t
Ker 1p = Ker 6 c C"(X
;S;>
.
If w e l e t Q b e a l e f t p a r a m e t r i x of
P, w e have
(8.14.4)
Q.P = I
+ R1 ,
This e q u a l i t y shows t h a t I
a compact operatclr from
with
R1 E
L'"(X;
+ R1 = 0 on Ker P , C"(X
;F )
5)
.
and s i n c e R1 i s
(see Theorem 8.5 of Chapter I )
(SEC. 8 )
241
P.D.O.'s & SOBOLEV SPACES
this implies that dim(Ker k) < +
.
eo
If we let Q' be a right parametrix of P, we have
(a.
14.5)
I+
=
P.Q'
This equality shows that
% , with
R2 E L'"(X;
9)
.
Irn 63 3 Irn(PQ') = I m ( I + R2)
since R 2 is a compact operator from C"(X;
9)
, we
and
know that
Im(1 + R 2 ) is a closed subset of finite codimension;
is therefore also true for Im 63
.
the same
We note that if we multiply
( 8 . 1 4 . 4 ) on the right by Q', we thereby deduce that Q is equivalent to Q' and it is thus a both-sided parametrix. It can immediately be shown that
(Irn F*jL = Ker P
;
furthermore the above shows that P*is also of Fredholm type; I
consequently I m p * is closed and equal to (Ker Ip)
, hence ( 8 . 1 4 . 1 ) .
PROOF OF iii): Following i) and ii), it only remains for us to prove ( 8 . 1 4 . 3 ) .
If we have
u
E
C"(X
U,
E
;S)
, the
decomposition
( 8 . 1 4 . 2 ) allows us to write u = u
However, u2
,+
t
u
u2
,
- u,
E
with
C"(X
; S)
;
Ker 63
, u2
E frn P
consequently the elliptic
regularity result implies that u2 is the image under element of Cm(X,S)
REMARK 8.15:
integer defined by
.
P of an
and this proves ( 8 . 1 4 . 3 ) .
The index of a Fredholm operator P i s the
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
242
index
(IF) = d i m Ker p - c D d i m
Imp
.
With t h e assumptions of Theorem 8.14, we have
(Irn R)" = Ker P* ;
consequently i n t h i s c a s e t h e i n d e x can a l s o
be e x p r e s s e d by Index
(P)
dim Ker P
=
-
*
d i m Ker P
=
index
I f we assume f u r t h e r m o r e t h a t P i s a c l a s s i c a l p . d . o . ,
(6)
.
t h e theorem
o f ATIYAH-SINGER [11 shows t h a t t h e i n d e x o f P can be e x p r e s s e d using topological inva ria nts associated with X ,
S and om(P) ,
.
( s e e a l s o ATIYAH-BOTT-PATODI [11)
We conclude t h i s s e c t i o n w i t h t h e f o l l o w i n g :
REMARK
8.16:
p.d.o.'s
p
Suppose
1
E 1 1 I 11
o f t y p e p and o f degree
m
;
t h e space Lm(X) of
P
can be o b t a i n e d by r e p l a c i n g
t h e s p a c e s o f o r d i n a r y symbols by symbols o f t y p e p .
The exten-
s i o n of t h e r e s u l t s o f Chapter 111 t o symbols o f t y p e p a l l o w s us
t o generalise
, with
some minor m o d i f i c a t i o n s , t h e d e f i n i t i o n s and
r e s u l t s of t h i s chapter t o p . d . o . ' s
of type p .
7
For t h e c a s e
p =
$,
we r e f e r t o CALDERON-VAILLANCOURT [11,
MELIN-SJOSTRAND [11, and for o t h e r g e n e r a l i s a t i o n s we r e f e r t o BEALS-FEFFERMAN [11 , ( s e e a l s o E x e r c i s e 1 1 . 7 ) .
(SEC. 9 )
9.
ELLIPTIC COMPLEXES
243
ELLIPTIC COMPLEXES AND HODGE THEORY
Let X be a manifold and l e t Eo,
...,E R
be complex v e c t o r
bundles o f f i n i t e dimension over X. I n d i f f e r e n t i a l geometry we f r e q u e n t l y encounter t h e following s i t u a t i o n :
DEFINITION
9.1:
operators P = (P,) same degree,
We describe as a
differenti.aZ
C O ~ ~ E ?o Xf
the coZlection of d i f f e r e n t i a l operators of the say,
m
Pk : C"(X
; Ek)
4 C=(X
8
Ek+lI
k = O,.**,l-I
9
such t h a t
If we denote by E k ( x ) t h e f i b r e o f Ek above x
( x , f ) E T*X
, the
0
p r i n c i p a l symbol of degree
a,(Pk)(x,f)
l i n e a r mapping
E d[Ek(x]
E~+'J(X~)
E
m
.
X, for of Pk i s a We t h e n
d e f i n e a sequence of l i n e a r mappings
,which a l s o c o n s t i t u t e a complex by v i r t u e of ( 9 . 1 . 1 ) .
DEFINITION 9 . 2 :
f o r a22 ( x , f )
The complex (P,)
* E T # \ O , the
REMARK 9.3:
i s said t o be e l z i p t i c i f
sequence ( 9 . 1 . 2 ) i s exact.
A complex reduced to a s i n g l e o p e r a t o r P = Po
244
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
i s e l l i p t i c i f and o n l y i f t h e sequence OJPI 0% f )
0
Eo(x)
_I*
_1_3
om(P)(x,<)
i s e x a c t , i n o t h e r words i f
0
E1(x)
i s a n isomorphism.
We
a r e c l e a r l y e n c o u n t e r i n g once more t h e concept of an e l l i p t i c operator.
EXAMPLE 9 . 4 : k = D,..,n
k * We p u t Ek = A (T X) 8 6
(De Rham complex) m
= dim X .
The C
s e c t i o n s of t h i s bundle c o n s t i t -
k Ute t h e space R (X) of complex-valued d i f f e r e n t i a l k-forms. e x t e r i o r d i f f e r e n t i a l of degree
nk(X)
dk
C
nk+'(x)
k
The
d e f i n e s a l i n e a r mapping
and we have
d
We saw i n Example 7.9 of Chapter I t h a t
k+l
$
O
dk
is a differential
o p e r a t o r of degree 1 whose p r i n c i p a l symbol a t a p o i n t
(x,5) U)
E
*
-
T X \ 0 is t h e l i n e a r mapping
E (A'TSX)
C3 C
ol(dk)(x,5)w
= igA
U)
E (A
k+l
*
Tx X I
@C.
We now show t h a t i n t h i s c a s e t h e sequence (9.1.2)i s e x a c t . Let
(el, . . , . . , O n ) k *
U)
E ( A TxX) 8 U)
be a b a s i s o f XT:
,
such t h a t
el
= 5.
If
it decomposes i n t h e form
=
0
~1
al<.*mc@k The e x a c t n e s s t o degree
IY
9
Am..A
0
%
with
E6
u ) ~
k a r i s e s from t h e f o l l o w i n g e q u i v a l e n c e s :
ELLIPTIC COMPLEXES
0
u)
cy
= 0
when
cyl
245
4
1
We can always assume t h a t X i s endowed w i t h a s t r i c t l y p o s i t i v e d e n s i t y p and t h a t each ian structure.
* Pk
i s endowed w i t h a Hermit-
Under t h e s e c o n d i t i o n s , we d e f i n e a p r e h i l b e r t
, )k
structure (
Ek
bundle
; Ek)
on each C"(X
space and t h e a d j o i n t
of P i s a d i f f e r e n t i a l o p e r a t o r of degree k
*
Pk : C"(X
; Ek+l)
+ C"(X
; Ek)
m
k = Ol,..,a
This a l l o w s us t o d e f i n e s e l f - a d j o i n t d i f f e r e n t i a l o p e r a t o r s of degree 2m
PROPOSITION
operators Ak
PROOF:
,
Suppose
t h e mapping
9.5:
I f t h e complex (Pk) i s e l l i p t i c , then t h e
k = 0, ...,a,
are e l l i p t i c .
(x,p) E T*X \ 0
.
It s u f f i c e s t o prove t h a t
246
PSEUDO-DIFFERENTIAL OPERATORS
is injective.
u,(pk)(~l~) = ak
We put
(CHAP.
4)
t h e n t h e formulas of
j
t h e symbolic c a l c u l u s show t h a t
X be a v e c t o r from Ek(x) such t h a t 6kh = 0 , t h e n , u t i l -
If we l e t
i s i n g t h e m e t r i c of Ek (x), we o b t a i n
n consequently a X = 0 and Y - ~X k
= 0.
However, s i n c e t h e sequence
( 9 . 1 . 2 ) i s e x a c t , we t h e r e b y deduce t h a t
h C I m ak,l
and t h i s
implies t h a t X = 0.
PROPOSITION 9.6:
I f (P,)
i s an e l l i p t i c complex, then there
e x i s t proper pseudo-differential operators
(9.6.1) Plk,
.
Qk
E
We say t h a t t h e
L’m(X
i Ek+l
1
+ Qk
Pk = I
+%
Ek)
(%) c o n s t i t u t e
Qk
such that k = O,...,A
.
where Rk i s regularising.8
a p a r m e t r i x o f th e complex
.I,@ PROOF:
Let
Tk
E LmZrn(X ; Ek
e l l i p t i c o p e r a t o r Ak. (9.6.2)
Tk+l Pk
Ek)
be a parametrix of t h e
We can convince o u r s e l v e s t h a t Pk Tk
(modulo a r e g u l a r i s i n g o p e r a t o r )
(SEC. 9)
247
E L L I P T I C COMPLEXES
by m u l t i p l y i n g on t h e l e f t by Tk+l
and on t h e r i g h t by Tk i n t h e
immediate e q u a l i t y Pk
We put
'k-1
*
GIk = Tk
Qk-l
=
Ak Pk
E
Pk
, Ek) , which
; Ek+l
L"m(X
=
'k
+
Ak+l
'k-1
which, h e a r i n g i n mind ( 9 . 6 . 2 1 ,
Tk-1. P*k-1
+
gives Tk PE Pk
is e q u i v a l e n t t o
Id
I
.
We have t h e g e n e r a l i s e d Hodge theorem: THEOREM 9 . 7 :
Suppose
P
si
('k)k=O,.
complex on a compact manifold X. i)
The space Im Pk-l
..,
A
i s an e l l i p t i c
Then:
i s a closed subspace, of f i n i t e codimension, Ek).
The
i s thus o f f i n i t e dimension;
it
of Ker Pk equipped with the topology induced by C " ( X ,
quotient space Ker Pk/Im Pk-l i s denoted by $ ( P )
and i s called the kth cohomology space of the
complex ( P , ) . i i ) We put
#(PI
(9.7.11
Moreover, dim $(P)
= Ker Ak
$c <+
P)
, and
we obtain
*
3
Ker Pk fl Ker Pkml
.
rmd we have the orthogonu2 d i r e c t
sun decomposition ( f o r the structure (
,
)k
I
248
PSEUDO-DIFFERENTIAL OPERATORS
C"(X
(9.7.2)
, E ~ )=
i i i ) The canonicaZ mapping I m Pk-l i n the equaZity
PROOF OF i ) :
*
-
Q Im pk
pk
4)
.
o f passage t o the quotient under
j
(9.7.1)induces a linear isomorphism
:
$(PI
(9.7.31
$(PI Q Im
(CHAP.
Hk(P)
The e q u a l i t y ( 9 . 6 . 1 ) i n d u c e s on t h e subspace = I + Rk;
Ker P
k t h e e q u a l i t y Pk-l.%-l
c o n s e q u e n t l y t h e proof
o f i ) i s analogous t o t h a t o f i i ) i n Theorem 8.14.
PROOF OF i i ) :
The i n c l u s i o n
, utilising
cp E Ker Ak
C
k
3 (P)
follows
To demonstrate t h e i n v e r s e i n c l u s i o n ,
from t h e d e f i n i t i o n of Ak. we t a k e
*
Ker Pk fl Ker Pk,,
t h e s c a l a r product (
, )k, and
we o b t a i n =
(ky
= ('k(P thus
cp E Ker Pk
9
n Ker
*
) k = ('k'kcp
'kv]
+ ('k,l'k,l'p
*
-
'k-lv)
S
v)
I
.
The a p p l i c a t i o n of Theorem 8.14 operator A
9
*
*
+ ('k,l'P
Pk
*
k shows t h a t dim $(P)
decomposition
The assumption (9.1.1)shows t h a t
C
+
iii) t o t h e s e l f - a d j o i n t w
and g i v e s t h e o r t h o g o n a l
(SEC. 9 )
249
ELLIPTIC COMPLEXES
consequently, we have
*
Im Ak = Im P* 'k-1 k-1
0 Im PkPk
so t h a t
Im Ak
(9.7.6)
c Im
.
*
0 I m Pk
PkWl
E q u a l i t i e s ( 9 . 7 . 1 ) and ( 9 . 7 . 5 ) show t h a t t h e subspaces $(P)
,
, Im
I m P,k,
+6
taken two a t a t i m e , a r e o r t h o g o n a l ;
Pk,
consequently ( 9 . 7 . 2 ) follows from ( 9 . 7 . 4 ) and ( 9 . 7 . 6 ) .
PROOF OF i i i ) : have (p
cp E Ker A k such t h a t
*
E
Let us f i r s t show t h a t
Ker Pk-l
and
ep E Im
t h e c l a s s of an element ii,
Pk,
E
is injective;
i f we
j ( q ) = 0 , we t h u s have b o t h which i m p l i e s t h a t
j is surjective;
now show t h a t
j
suppose
Ker Pk.
$
E
cp =
0.
Hk(P); t h i s i s
The decomposition ( 9 . 7 . 2 )
allows us t o w r i t e
Applying Pk t o ( 9 . 7 . 7 ) , we o b t a i n
and having r e g a r d t o ( 9 . 7 . 7 ) and (g.l.l), we a r e l e f t w i t h
0
c
*
PkPk$2
so t h a t
We
P+k Jr,
= 0
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
250
Consequently,
$ =
JI, + Pk-,$,
, and
t h i s proves t h a t
j(+) = j ( $ o ) .
EXAMPLE 9 . 4 : space $(PI manifold X.
(continued).
In t h i s particular case, the
i s c a l i e d t h e kth De Rham cohomology group o f t h e When X i s compact, Theorem
9.7
shows i n p a r t i c u l a r
t h a t a l l t h e s e cohomology groups a r e of f i n i t e dimension and t h a t each De Rham cohomology c l a s s can b e r e p r e s e n t e d uniquely using
a harmonic form (Hodge t h e o r e m ) .
For more d e t a i l s concerning t h e m a t e r i a l i n t h i s s e c t i o n , . we r e f e r t h e r e a d e r t o t h e o r i g i n a l a r t i c l e by ATIYAH-BOTT
c11 and
t o t h e monograph of GILKEY c11.
FRIEDRICH'S LEMMA AND GENERALISATIONS
10.
We now concern o u r s e l v e s w i t h s t u d y i n g t h e behaviour o f t h e commutator o f a p . d . 0 .
P(x,D)
with a r e g u l a r i s a t i o n
;
p,*
t h i s s t u d y i s based on t h e f o l l o w i n g theorem which g i v e s a s o r t of l i m i t e d expansion w i t h r e s p e c t t o
THEOREM 10.1:
(pV)(x)
of
= jeix5p[x,{);(5)&
x for
p.d.0.
Suppose we have
,
o f [P, p E *I when E
E
-f
0.
P E L m (Rn ) of the form
where the symboZ p
x outside a compact subset K.
associated with the symboZ D:p(x,E)
is independent
We denote by Pa the and we denote by
p:
FRIEDRICH'S LEMMA
(SEC. 10)
the f u n c t i o n ~p E C;(R")
.
E-"
pa(
2)
I f we have s
with E
251
p@(x) = x"p(x)
IR and
0
2 0,
and
then for a22 N, there
e x i s t s C such t h a t
for u PROOF:
and
C
>0
.
If P i s an o p e r a t o r w i t h c o n s t a n t c o e f f i c i e n t s t h e
commutator [P, p,*]
i s i d e n t i c a l l y z e r o , and we can reduce t o
c o n s i d e r i n g t h e case where t h e symbol of K .
E H'~
p
is zero for x o u t s i d e
Using F o u r i e r t r a n s f o r m a t i o n , we can w r i t e
hence, for u
E 8 , we
obtain
I n similar f a s h i o n , we o b t a i n
We u t i l i s e t h e Taylor formula up t o o r d e r N t o expand w i t h r e s p e c t to
E
t h e difference
PSEUDO-DIFFERENTIAL OPERATORS
252
( C W .
4)
By s u b s t i t u t i n g (10.1.3) and ( l 0 , l . b ) i n t o t h e l e f t - h a n d s i d e of
(10.1.1)and t a k i n g account of (10.1.51,we o b t a i n
The i n t e g r a l e x p r e s s i o n for t h e remainder rN shows immediately t h a t , for any k
0, t h e r e e x i s t s C such t h a t
2
(10.1.6)
I n a d d i t i o n , t h e assumption concerning t h e s u p p o r t of p ( x , E ) i m p l i e s t h a t for a l l h
2
0 , t h e r e e x i s t s C such t h a t
A (10.1.7)
I P ( M
8
511
To m a j o r i s e t h e norm
c
(1
(IR (e)ull
N
+ 151," (1 +
s+a
, we
l%51)-h
specify
v E 8 and we
c o n s i d e r t h e i n n e r product
Taking account of (10.1.7)and of P e e t r e ' s i n e q u a l i t y , t h e r e
follows
FRIEDRICH'S LEMMA
(SEC. 1 0 )
253
t h e right-hand s i d e of (10.1.8) i s bounded above by
Consequently, by t a k i n g
h
s u f f i c i e n t l y l a r g e , we t h u s deduce
which p r o v e s (10.1.2).
A s a p a r t i c u l a r c a s e , we have F r i e d r i c h ' s Lemma
COROLLARY 1 0 . 2 :
We adopt the same asswnptions as i n the
preceding theorem, together with the f u r t h e r condition P(0) = 1. Then for aZZ s (10.2.1
1
E
B, there e x i s t s C such t h a t
IICP,
P,*3lIS
for
Moreover, PROOF:
[P
, p,*3
5
cIlulls+ncl
u EH~++"
0
and e
in HS when u
>o E
H
s+m-1
E - 0
The i n e q u a l i t y (10.2.1) i s o b t a i n e d by s e t t i n g
254
PSEUDO-DIFFERENTIAL OPERATORS
N = u = 1 and by r e p l a c i n g
If u
E
S
s
i s dense i n H
s +m- 1
-
4)
by s - 1 i n t h e i n e q u a l i t y ( 1 0 . 1 . 2 ) .
u
p,*u
we know t h a t
(CHAP.
in
s
and s i n c e g
O'E
t h e i n e q u a l i t y ( 1 0 . 2 . 1 ) a l l o w s us t o e x t e n d
t h e conclusion t o a l l u
E
H
s +m- 1
We now g i v e t h e proof o f Theorem 6 . 5 o f Chapter I1 w i t h i n a more t h i s i s as f o l l o w s :
general s e t t i n g ;
Let P be a p . d . 0 . a s i n Theorem 10.1; l e t
COROLLARY 1 0 . 3 :
there e x i s t s
E
IR
and
p
s a t i s f y i n g the supplementary condition
( 6 . 3 . 3 ) of Chapter 11, w i t h h > s + l .
Then, there e x i s t s C such
that
PROOF:
We r e p l a c e
[P
we m a j o r i s e each term.
6.4 of Chapter
, p,*Ju For
by i t s e x p r e s s i o n (10.1.1)and
0
<
< N , we
u t i l i s e Theorem
I1
A
and s i n c e Pa i s an o p e r a t o r of degree
m
, P r o p o s i t i o n 6.3
of
Chapter I1 ( o r r a t h e r i t s immediate g e n e r a l i s a t i o n t o p . d . 0 . I s ) shows t h a t
(SEC. 10)
since
jell
FRIEDRICH'S LEMMA
255
t 1.
To m a j o r i s e t h e remainder, we u t i l i s e (10.1.2)w i t h
r e p l a c e d by s-1 and
c1
we o b t a i n
= 1-s;
and t a k i n g N t 2 , we have an upper bound given by
REMARK 1 0 . 4 :
s
2 CVllul\s+m-l
A l l t h e above r e s u l t s a r e s t i l l v a l i d when
t h e symbol of P depends continuously on one parameter varying w i t h i n a compact s u b s e t ;
i n f a c t , it i s s u f f i c i e n t t o n o t e
t h a t we can choose t h e c o n s t a n t C i n (10.1.7)independent of t h i s parameter.
A s an a p p l i c a t i o n of C o r o l l a r y 1 0 . 2 , we s h a l l prove t h e
i d e n t i t y between a weak s o l u t i o n ( i. e . i n t h e d i s t r i b u t i o n a l s e n s e ) and a s t r o n g s o l u t i o n ;
COROLLARY 1 0 . 5 :
more p r e c i s e l y we have
Let P(x,D) be a p . d . 0 . s a t i s f y i n g the
assumptions of Theorem 10.1, and of degree u
E
with
Hw'[R") f
m
.
We assume t h a t
s a t i s f i e s , i n the d i s t r i b u t i o n a l sense, Pu = f
E L2(Rn)
.
Then, there e;cists a sequence
256
U
PSEUDO-DIFFERENTIAL OPERATORS
J E
n Hml
Cm(R"l L*
J
u
5
u
J
Hm-1,
4)
md
u
jm
u i s a strong solution.
uj = u
It s u f f i c e s t o put Hml*
u
say t h a t
we
I%. 4 f j
PROOF:
such t h a t
(CHAP.
*
P,/j
*
We know t h a t
and C o r o l l a r y 1 0 . 2 shows t h a t
p u
We conclude w i t h t h e f o l l o w i n g :
REMARK 10.6:
All t h e above d i s c u s s i o n g e n e r a l i s e s word
for word t o t h e c a s e where P i s a matrix-valued p . d . 0 .
11.
A D D I T I O N A L NOTES AND EXERCISES
11.1
WITH UNIFORM SYMBOLS :
P.D.O.'s
g l o b a l problems i n IRn
( s e e Chapter V I ) it i s convenient t o make
use o f p . d . 0 . ' ~ whose symbols a r e i n BS Chapter I ) .
For c e r t a i n
-I-=
(see S e c t i o n 1 0 . 1 of
More s p e c i f i c a l l y , we denote by BLm t h e subspace o f
L m ( I R n ) composed of t h e p . d . 0 . ' ~ of t h e form ( 1 . 2 . 1 ) where t h e amplitude
a)
p
I f we have
i s i n BSm(IR2nx E n ) . P E
o p e r a t o r with k e r n e l
Q
BLm
,
t h i s i s therefore a regularising
.
K E CcD(IRn xm")
Prove t h a t f o r any m u l t i - i n d i c e s
a,
8 , t h e function
(SEC. 11)
(x
(llml*l)
u - Y ) B caxlY
K(x,Y)
[Use t h e e x p r e s s i o n (1.4.1)f o r p
i s bounded i n E
b)
Deduce from t h i s t h e e q u a l i t i e s
BL-
=
n
257
ADDITIONAL NOTES
BSm w i t h
m"
m C- n
xR"
-
.
lel 1
s e t of o p e r a t o r s w i t h k e r n e l
km=
m
s a t i s f y i n g (11.1.1)
.
Prove t h a t an o p e r a t o r belonging t o BL-m i s continuous from
c)
8
into d
,
from
8'
8 ' and from H-- i n t o H
into
+oo
.
C U t i l i s e t h e e x p r e s s i o n f o r P, i n terms of i t s k e r n e l ; . A continuous l i n e a r o p e r a t o r w i t h k e r n e l d i s t r i b u t i o n K i s
d)
s a i d t o be B-proper i f t h e r e e x i s t s C such t h a t
With P E BLm., prove t h a t t h e r e e x i s t s a B-proper P '
R
E
BL--
such t h a t P = P'
function ~ ( x - y ) where
x
E
+ R.
E
[Multiply t h e k e r n e l o f P by a
C z ( I R n ) i s e q u a l t o 1 f o r 1x1 2 con-
s t a n t 1.
e)
Given t h a t P i s a B-proper o p e r a t o r , prove t h a t
op(x
(11.1.2)
i f and o n l y i f P
E
BLm.
BLm and
,y )
=
cix5 Pie- ) E Em
I n t h i s c a s e , we have
258
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
[Put b(x
,
, 5)
z
= p(x, x + z
, c),
where p is an amplitude for
P, and prove that f o r all a , 6 and k
there exists C such
that
(
~
~
4la: 1a$(x
, t , r l l * c ( l + ~sl)*lel
(1
+ 1cl1-k ;
thus deduce an upper bound f o r
by splitting this latter integral into two parts corresponding to
Is1 *
f)
By utilising the decomposition in d) together with the
2
expression (11.1.3)and c), deduce that an operator P acts continuously from 8'into 8 and from 8 ' into 8'
E
m BL
, and
that
(11.1.2), (11.1.3)remain valid; u P is called the complete
symbol of P. Let P
g)
E
BLm be a B-proper operator.
Apply the Taylor
formula with integral remainder to the function A
b(x
, 6 ,5 +
k
0, we have
2
5 ) between 5 and 6 + 5 to show that for any
Thus deduce, by utilising (11.1.4)and Peetre's inequality
(SEC. 11)
ADDITIONAL NOTES
259
t h a t we have
Generalise t h i s r e s u l t t o t h e case P
h)
Suppose P
BLm.
E
i s t h e c l a s s of u
BLm.
The p r i n c i p a l symbol o f degree
i n BSm
P
E
/
BSm-l; we denote t h i s by
0
m
m of P (P) ;
show t h a t we have t h e l i n e a r isomorphism
i)
If we have P
E
BLm, t h e n
t P, P" a r e i n BLm and
[If p ( x , y , 5 ) i s an amplitude for P , t h e n p ( y , x , - 5 ) i s an amplitude for
t
p and we o b t a i n
*
S i m i l a r l y , for P
j)
If P
E
(11.1.6) by using (11.1.5)#
.I
BLm, prove t h a t t h e r e e x i s t s b ( y , 5 )
E
BLm w i t h
such t h a t
(11.1.8) J
[We express tP by means o f an e x p r e s s i o n o f t h e t y p e (11.1.3),
260
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
i . e . we w r i t e
t
P = a,
Then, o b s e r v i n g t h a t P =
(x, D).
t t
( P),
LP we t h e r e b y deduce t h a t P i s d e f i n e d by t h e amplitude
, Y , fl
P(X
=
Otp(Y
9
4) =
b(Y
I
fl
3
Given Q E BLm' and P E BLm, prove t h a t
k)
mm'
and t h a t we have
L P E B L
(11.1.9) [Express P by means of (11.1.8) and w r i t e Q i n t h e form
(11.1.3).1
e)
,
riu-ml- 1 E @L
1.
Show t h a t P i s e l l i p t i c i f and o n l y i f t h e r e e x i s t s
E
BL-m such t h a t PQ =
m)
[p
We s a y t h a t P E BLm i s e l l i p t i c i f u m ( P ) i s e l l i p t i c i n
BSm.
Q
Thus deduce t h a t
Let C
E
(11.1.10>
I+R
,
QP
= I+R'
with
R
, R'
E BL"
,
BLo; we assume t h a t t h e r e e x i s t s c > 0 such t h a t
, fl
O,(Cl(X
2
c
for
2
I/C
.
Prove t h a t t h e r e e x i s t s B E BLo such t h a t c = B*B+R w i t h [Use t h e proof of Lemma 8.1 w i t h t h e s i m p l i f i c a t i o n
R E BL-=.
which f o l l o w s from t h e f a c t t h a t t h e i n e q u a l i t y (11.1.10)i s uniform w i t h r e s p e c t t o
If P S
from H
E
x.1
BLm, deduce from t h i s t h a t P o p e r a t e s c o n t i n u o u s l y
i n t o HS-m.
[Proceed as i n Theorem 8.5, o b s e r v i n g t h a t
(SEC. 11 )
n)
Let P
(11.1.11)
261
ADDITIONAL NOTES
E
BLm; we assume t h a t t h e r e e x i s t s
Re Pm(x
, 5)
Prove t h a t for any
2
E
a151"
151
for
> 0 and any s <
a > 0
2'
2
such t h a t
;1;
.
there exists C
such t h a t we have Ggrding's i n e q u a l i t y :
0 )
G e n e r a l i s e t h i s e x e r c i s e t o t h e c a s e of matrix-valued
operators.
Note:
t h e method of proof i n d i c a t e d i n t h i s e x e r c i s e a l s o
.
m
provides a n o t h e r means o f s t u d y i n g p . d . 0 . ' s belonging t o L (X) ( s e e HORMANDER C41
f o r a p r e s e n t a t i o n of t h i s . )
This method
i s a l s o s i m p l e r a s l o n g as w e are n o t concerned w i t h q u e s t i o n s of change of v a r i a b l e , and it allows a number of important g e n e r a l i s a t i o n s ( s e e BEALS [11).
(*)
11.2
P . D . O . ' s WITH UNIFORM SYMBOLS AND DEPENDENT ON
ONE PARAMETER I n Chapter V I I we s h a l l make use of p . d . 0 . ' ~ which depend on a parameter v a r y i n g w i t h i n an i n t e r v a l of .the form [yo,
N We denote by PSm(lRn x l R ) t h e s e t o f f u n c t i o n s
+
a[.
262
PSEUDO-DIFFERENTIAL OPERATORS
, 5 , Y)
p(x
E C"
mn x I R x~ [yo
> 0 and such t h a t
I
axU arP
p(x
, + II
for a certain
for a l l ( a , 8 ) t h e r e e x i s t s C such t h a t
, 5 , YI I
c ( 151 +
y ~ w l ~f olr a l l
g
x
Y 2 Yo
and
We denote by PLm t h e space of p . d . o . ' s
where t h e amplitude p
4)
(CHAP.
E
of t h e form
P(')
PSm(lR2n x W").
The r e s u l t s and p r o o f s concerning t h e p.d.o.'s belonging t o
BL
m
m
.
B
,
extend almost word f o r word t o p . d . o . ' s belonging t o PL
We s h a l l i n d i c a t e o n l y t h e main m o d i f i c a t i o n s .
I n a ) it i s n e c e s s a r y t o r e p l a c e (x
-
y)
B
k
by y (x
-
y)
where k i s an a r b i t r a r y p o s i t i v e i n t e g e r .
m
I n c o n d i t i o n (11.1.2),it i s n e c e s s a r y t o r e p l a c e BS PSm and t o change (1 + 161) i n t o ( y +
ltl)
by
in.tl1.1.b).
I n g ) , P e e t r e ' s i n e q u a l i t y must be used i n t h e form
(Y +
Ir + t61IS *
CS(Y
+
1~11"(1+ t
lcl)lsl ,
y 2 1
A l l t h e asymptotic expansions of symbols a r e s t i l l v a l i d as l o n g as we c o n s i d e r them i n t h e graded a l g e b r a PS
+co
=
u m
In particular if P ( ' )
E
PLm and Q ( ' )
E
PLm' t h e n
PSm.
.
ADDITIONAL NOTES
(SEC. 11)
e ) we
In PSm.
say t h a t P(')
Let y
E
is elliptic i f
+ 151
(Y'
For example
26 3
) "2
U
m
(P) i s e l l i p t i c i n
i s e l l i p t i c of degree s .
IR \ 0 ; we denote by HS t h e space HS endowea w i t h Y
t h e e q u i v a l e n t norm
(') Supposing P
PLm and s
E
E
R , prove t h a t t h e r e e x i s t s C
independent of y such t h a t
\I1&)u
c Ill u 111,
5
9
u E 8
'1's-m; y iY I n c o n d i t i o n (11.1.11)we need t o r e p l a c e lSlm by (y +
. /51)m
and we t h e n o b t a i n Ggrding's i n e q u a l i t y ( 1 1 . 1 . 1 2 ) w i t h norms
5,
y ) be
, y)
f 0
Finally, l e t p(x,
5 ERN,
Y 2 0
,
(5
a Cm f u n c t i o n for
x ER"
,
We assume t h a t p i s homo-
geneous o f degree m w i t h r e s p e c t t o ( 5 , y ) and t h a t a l l i t s d e r i v a t i v e s a r e bounded f o r Show t h a t p
E
151 + Y =
1
N
BSmDn xIR ) and t h u s deduce t h e concept of an
o p e r a t o r "with homogeneous p r i n c i p a l symbol" t o g e t h e r w i t h t h e concept o f a " c l a s s i c a l " o p e r a t o r i n PL
+co
.
2 64
PSEUDO-DIFFERENTIAL OPERATORS
(CHAP.
4)
SECOND PROOF OF THE ESTIMATE ( 7 . 6 . 1 ) OF CHAPTER 2.
11.3
Let P = p ( x , D ) be a d i f f e r e n t i a l o p e r a t o r of p r i n c i p a l t y p e of degree
m
n i n an open s u b s e t Sl o f IR
d i f f e r e n t i a l o p e r a t o r Q : c"(0)
.
Show t h a t t h e
dl) d e f i n e d
C"(S2,
4
by
By u t i l i s i n g t h e e l l i p t i c i n e q u a l i t y ( s e e (8.11.3)and Remark
8.12) and t h e Poincard i n e q u a l i t y ( s e e E x e r c i s e 7 . 1 of Chapter 11), deduce t h e i n e q u a l i t y
11.4 Let T
(7.6.1)o f
Chapter 11.
CONVOLUTION WITH CERTAIN HOMOGENEOUS DISTRIBUTIONS. E
8 (Bn) be
a d i s t r i b u t i o n which i s homogeneous of
degree m ( s e e S e c t i o n 1 0 . 2 o f Chapter I ) . Assume f u r t h e r t h a t T i s Coo i n lRn \ 0 .
Show t h a t t h e o p e r a t o r
of convolution w i t h T i s i n L-m-n(E?).
11.5
ENDOMORPHISMS OF COMPLEXES AND TRACES
L e t ( E j , P j ) be an e l l i p t i c complex above a compact mani f o l d X endowed w i t h a d e n s i t y dx.
We t a k e an endomorphism
(Tj) o f t h i s complex, i . e . a f a m i l y of endomorphisms T Cm(X;Ej),
a)
j
which f o r a l l j s a t i s f y :
Show t h a t each T . induces an endomorphism, denoted by J
of
(SEC. 11 )
265
AEDITIONAL NOTES
H ( T j ) , of t h e space H J ( P ) .
b)
Assume t h a t t h e T. a r e of f i n i t e r a n k ; prove t h a t we have J
t h e a b s t r a c t L e f c h e t z formula
C U t i l i s e t h e e x a c t sequences 0 and 0
.-(
Irn p
J-1
0
of t h e t y p e
-
A 0 C
A
I=
Ker P
Hj(p,]
4
4
G
) = Trace (T
Trace (T
c)
Ker p j
4
.-(
4
4
j
Irn P
j
4
0 t o reduce t o s i t u a t i o n s for which we have
0
1A
; Ej)
C"(X
+
T r a c e (T
IC
]
.
We d e f i n e t h e k e r n e l d i s t r i b u t i o n
Kj € . B ' ( X x X ; E
EJ)
j
of T . by J
= for u E C"(X , Ej) and J j ' * We r e c a l l t h a t t h e s i g n Pp d e n o t e s t h e o u t e r v E C"(X , Ej)
.
tensor product.
K.
We assume t h a t
J
E
C"(X
x X
; Ej
):E
;
prove
t h a t we a g a i n have t h e i n e q u a l i t y (11.5.1)when we p u t
Tj
Trace [approximate t h e T
j
= Jx[tmce KJ. ( x , x ) ) dx
by o p e r a t o r s of f i n i t e rank i n t h e s e n s e of
t h e convergence of t h e k e r n e l s i n
C"(X
x X ;
EjP Ef;] 3
0
266
11.6
4)
(CHAP.
PSEUDO-DIFFERENTIAL OPERATORS
COTLAR-KNAPP-STEIN LEMMA
Let X be a measured s e t whose measure i s denoted by dC. We t a k e a measurable mapping f E X
a Hilbert space.
h: X
x
L'(X)
II A&,
4
9
E k[#)
We suppose t h a t t h e r e e x i s t s a f u n c t i o n
X +lR which i s t h e k e r n e l of an o p e r a t o r H , bounded i n and such t h a t
II
5
h2(5,
71
I\ 9" 11
s h2(5,7
I
for
J%d{
Then, we propose t o show t h a t t h e i n t e g r a l
weakly and d e f i n e s a bounded o p e r a t o r A such t h a t C =
with
a)
11 H 11
.
by a c o n s t a n t M;
TI E x
x X.
converges
11 A 11
s
c
Letting
H2n-1
, use
11 Ap I(
i s uniformly bounded
we s h a l l assume t h i s i n t h e f o l l o w i n g t e x t .
Show t h a t f o r a l l n
[Use t h e e q u a l i t y
c)
(I,
Show t h a t it i s s u f f i c i e n t t o e s t a b l i s h t h i s r e s u l t when
X i s of f i n i t e measure m and when
b)
is
where
E
11 A 112
Hhl(fl
=
IN, we have
11 A*A 11
, f 2n )
1
1)
.
A
=
* n ll(A A]
11
Thus deduce t h a t
b e t h e k e r n e l of t h e o p e r a t o r
(11.6.1) t o show t h a t we have
(SEC. 11 )
ADDITIONAL NOTES
Thus deduce t h a t tend t o
+
m
11.7
show
11 A 112" 5 that 11 A 11
M ?rr-lm 5
CALDERON-VAILLANCOURT
We t a k e a r e a l number p a ( x , y, 5 )
E
Cm(Rn XRn
XW")
E
j
t h e n by l e t t i n g
n
c
THEOREM.
l[ an8 an amplitude
LO,
which s a t i s f i e s f o r any m u l t i -
index a , 6
where we have put
= (1
+
.
I{])'
L e t A be t h e o p e r a t o r
d e f i n e d by
h ( x ) = JJe i(x-y)f
(11.7.1)
a(x
I
Y
I
5)
u(y)dy
&
for u E c:(R") 2 n We now have t o show t h a t A i s continuous i n L &7 1; we s h a l l
.
follow t h e approach of a method d e s c r i b e d i n BOUTET DE MONVEL
C31.
a)
Give a meaning t o t h e o s c i l l a t o r y i n t e g r a l (11.7.1) by u s i n g
an i n t e g r a t i o n by p a r t s w i t h t h e a i d o f t h e d i f f e r e n t i a l operator
where
x
E C"(w")
i s zero i n t h e neighbourhood of 0 and
2 68
15 1
equal t o 1 f o r
b)
(CHAP. 4)
PSEUDO-DIFFERENTIAL OPERATORS
large.
Consider t h e d i f f e r e n t i a l o p e r a t o r
= (1
AT
Ix
- y12N)-1 (1 + i(x
i(X-YI5
Prove t h a t
s a ti s f i es
+
hoe
s"
- Ylg
[ i P 1 ( 1+
= 0
%Y b
q) .
(-
and t h a t b = t La
S
-
1x
f o r any m u l t i - i n d i c e s a , B .
For 5
C)
n
IR we c o n s i d e r t h e o p e r a t o r A
E
=
(%.)(XI
sBi[x-
%u(x)
By observing t h a t
d e f i n e d by
, y , 5)
%(x
1
ei(x-y)c
t
5
b ( x ,, Y
LdYIdY 9
51
s
U(Y1 dY
prove t h a t t h e k e r n e l of A is bounded above i n modulus by C(l
+
Ix
-~
the operators A
d)
C*,
.)
1
5
~
2
5
n z, that
n
are uniformly bounded i n L @ ) ,
Prove t h a t t h e k e r n e l
,,b, Y l
~
Thus deduce by choosing N >
ei(x*s-yo')
c%,,,of
qq
i s given by
, Z , g) c( y , z , T)
leiz('K1b(x
Let M be t h e d i f f e r e n t i a l o p e r a t o r d e f i n e d by Z
MZ
- TI
= ( 1 + (+ + %)-21C
Show t h a t f o r any k
E
2 -1
El -
($ + 5,,)-2AZ
IN we have
(tNk(b(x
9
, CF(Y
9
z
9
711
5
3
.
dz,
ADDITIONAL NOTES
(SEC. 11 )
269
and deduce an analogous upper bound for
e)
Show t h a t t h e i n t e g r a l
s
h(g
, 'I])dl)
i s a bounded
Thus deduce t h a t t h e o p e r a t o r H w i t h k e r n e l h i s bounded i n L'. I n accordance w i t h t h e Cotlar-Knapp-Stein the operator
Show t h a t
a
a , bounded i n L2,
Lemma, we c o n s i d e r
d e f i n e d by
extends t h e o p e r a t o r A .
ff =
This Page Intentionally Left Blank
CHAPTER 5
E L L I P T I C BOUNDARY-VALUE
1.
PROBLEMS
INTRODUCTION Let Cl b e a r e g u l a r open s u b s e t o f IRn, w i t h boundary
an;
we t a k e , i n a neighbourhood o f a l l i n Bn, a f i e l d o f v e c t o r s v transversal t o
an,
y . of o r d e r j on J
and we d e f i n e f o r j E IN t h e t r a c e o p e r a t o r
by y
j
v ] ' u]
=
u
,an-
Suppose we
n
have P ( x , D ) , an e l l i p t i c d i f f e r e n t i a l o p e r a t o r i n IR X
of degree
m, and c o n s i d e r p r o p e r p s e u d o - d i f f e r e n t i a l o p e r a t o r s d .-k 'j,k
(30) f o r j = 0,.. . ,
p-1
. . , m-1
and k = 0,.
( w i t h p a p o s i t i v e i n t e g e r and d . given r e a l s ) . J
We propose t o
study t h e e x i s t e n c e , uniqueness and r e g u l a r i t y of s o l u t i o n s of t h e boundary v a l u e problem
and where yu i s t h e where B i s t h e m a t r i x (B j , k ) j = O,..,p-l k = O,..,m-l C0lumi-l
m a t r i x (Yj
dj = o,.
. , ym-l, 271
u
(CHAP. 5)
ELLIPTIC BOUNDARY PROBLEMS
272
For example, t h e D i r i c h l e t problem corresponds t o
and t h e Neumann problem corresponds t o
t h e inward unit normal t o
an
.
There a r e b r o a d l y two t y p e s of method f o r s t u d y i n g (1.1). One o f t h e s e c o n s i s t s of m a j o r i s i n g a norm of
u
by s u i t a b l e
norms o f t h e d a t a f , g , and t h e n concluding by employing f u n c t i o n a l a n a l y s i s arguments.
The o t h e r t y p e o f method
c o n s i s t s of reducing (1.1)to a problem on t h e boundary 3 0 , t h e n s t u d y i n g t h i s problem by means of t h e t h e o r y of systems of e l l i p t i c p s e u d o - d i f f e r e n t i a l e q u a t i o n s on a manifold w i t h o u t boundary.
D e s c r i p t i o n s o f t h e f i r s t t y p e o f method a r e g i v e n ,
for example, i n AGMON - DOUGLIS - NIRENBERG 111, LIONS - MAGENES Here we s h a l l d e s c r i b e t h e second t y p e of method,
[11.
f o l l o w i n g t h e work of HORMANDER c21.
For c l a r i t y , we s h a l l s t a r t w i t h a formal d e s c r i p t i o n of t h e c a s e i n which P has c o n s t a n t c o e f f i c i e n t s and R i s bounded.
If
R.
(p
E Cm@)
,
Then, for u
we denote by
E
Cam)
cpD
, P(uo) -
i t s e x t e n s i o n by 0 o u t s i d e
( P U ) ~is a d i s t r i b u t i o n
(SEC. 1)
c a r r i e d by
INTRODUCTION
an,
273
which depends only on P and on yu; we put
(1.3.)
P(u0)
E:
(Pup
+
p" YU
c
I n f a c t , P yu i s a m u l t i p l e l a y e r of o r d e r m-1:
i n a neighbourhood of a p o i n t
t r a n s p o r t by a l o c a l c h a r t of IRn of
an
an
i n t o {xn = 0 } and v i n t o
after
which transforms fi i n t o { x = ( x g l xn) E R
n
1
xn
> 0
7 ,
( i n what f o l l o w s , such a l o c a l axn
c h a r t w i l l be c a l l e d a l o c a l c h a r t of
y)
)
, P i s written
and we have t h e jump formula ( c f (l.b.l), Chapter 1):
A l l t h e above c l e a r l y h o l d s f o r an a r b i t r a r y r e g u l a r open
s e t fi and f o r an a r b i t r a r y d i f f e r e n t i a l o p e r a t o r P.
Carrying o u t a l e f t convolution of b o t h s i d e s o f (1.3) with a fundamental s o l u t i o n E of P = P ( D ) we o b t a i n , p u t t i n g P u = f : (1.6.)
u
=
(E * f o )
In
+
(E * F ~ U ) , ~
This formula e x p r e s s e s u as t h e sm of a volume p o t e n t i a l and a s u r f a c e p o t e n t i a l ; we s h a l l show t h a t t h e s e two p o t e n t i a l s a r e in C m ( E ) , so that taking traces i n ( 1 . 6 ) :
274
(CHAP.
ELLIPTIC BOUNDARY PROBLEMS
5)
(1.7.)
In p a r t i c u l a r , i f
u
C : C"(aL2,
o b t a i n yu = C ( y u ) , where
i s d e f i n e d by
c
v
y
=
E c"(E)
[(E
*
and u
c"(an, c")
C")
Cm(ahl,
J
].
~ v ) , ~
C")
It can e a s i l y be
From (1.7),i f u
shown t h a t C i s a p r o j e c t o r .
(l.l), then yu E
s a t i s f i e s PU = 0 , we
E
Cm(?2)
satisfies
satisfies
i s given by ( 1 . 6 ) .
Conversely, i f
and i f we p u t
u E Cm(z)
v E
u = (E
*
cm(an, c"')
follhl
satisfies
+ (E * pt) In '
then
and s a t i s f i e s (1.1).
We t h u s see t h a t (1.1)reduces t o i n v e s t i g a t i n g t h e system
I n what f o l l o w s , we show t h a t t h e p o t e n t i a l s
(SEC. 2 )
275
REGULARITY OF BOUNDARY POTENTIAL
have t h e s t a t e d r e g u l a r i t y a t t h e boundary, and t h e n go on t o i n v e s t i g a t e t h e o p e r a t o r C ( c a l l e d t h e CalderBn p r o j e c t o r ) . F i n a l l y , we apply t h e r e s u l t s o b t a i n e d t o t h e i n v e s t i g a t i o n o f
(1.1)under c e r t a i n e l l i l s t i c i t y assumptions.
2.
REGULARITY OF THE POTENTIALS AT THE BOUNDARY We use t h e n o t a t i o n d e f i n e d a t t h e s t a r t of t h e I n t r o -
duction.
u E J'(L-2).
Let
t r a c e s up t o o r d e r k ( k
x
:
u
v
cnn
of
u has s e c t i o n a l
We say t h a t E
IN) on 30. i f , f o r any l o c a l c h a r t
(fi,
V ) and f o r any cp
t r a n s p o r t e d i n t o a f u n c t i o n o f c l a s s Ck i n x s u f f i c i e n t l y s m a l l ) with values i n J ' [ R X ~ ' ) naturally define the traces
y J. u = y 0
[(l
">j u ]
If u
Ik
E
ulan
('i), then
on ail up t o o r d e r k f o r k < s
for 0 S j
u =
-
a,
(for x
.
We t h e n
u has s e c t i o n a l t r a c e s
and
yj u E Hs- j-$? loc
(an)
has a s e c t i o n a l t r a c e of o r d e r 0 on a R , we
j
E
define t h e l a y e r v
a.
Let
(1,v ) , we
IN ; by means of l o c a l c h a r t s of
8 Dj 6
s e c t i o n a l t r a c e s up t o o r d e r
( 1 . 3 ) , (1.5).
t 0
( s e e Chapter 1 1 ) .
&'(a)
&'(an) and.
n
is
1
n a t u r a l l y d e f i n e i t s e x t e n s i o n u o by 0 o u t s i d e v
TU
as d i s t r i b u t i o n s on as2.
(0 5 j 5 k )
u E Hsoc
For example, i f
yo
n
E c;(u),
E 5'an(R")
.
If u
E
B'(C2)has
m y we a g a i n have t h e jump-formula
276
ELLIPTIC BOUNDARY PROBLEMS
A
Now l e t operator.
< L:(Rn)
(CHAP. 5 )
be a c l a s s i c a l p s e u d o - d i f f e r e n t i a l
We i n t r o d u c e t h e c o n d i t i o n s A i s proper, and each term of the complete symbol
(2.1.)
1 aj(xI j s
of A i s a r a t i o n a l f r a c t i o n i n 5.
5;)
Using t h e formulas of t h e symbolic c a l c u l u s for p . d . o . ' s ,
it can be shown t h a t t h e c o n d i t i o n s ( 2 . 1 ) a r e p r e s e r v e d under diffeomorphism, t r a n s p o s i t i o n , composition, and passage t o t h e parametrix i n t h e e l l i p t i c c a s e . t h u s s a t i s f i e d when
A
In p a r t i c u l a r , they are
i s t h e proper parametrix of an e l l i p t i c
d i f f e r e n t i a l operator.
Suppose we have
THEOREM 2 . 2 : (2.1) and
u
E
B'(Rn]
zero i n a.
A E LE(Rn)
Then (Au)
In
satisfying
has sectional
traces of any order on an.
PROOF:
By p a r t i t i o n of u n i t y and l o c a l c h a r t s we can
n reduce t o t h e c a s e 0 =IR+.
We can assume
u t o have compact
s u p p o r t ; we t h e n know t h a t t h e r e e x i s t c and R such t h a t
1;(5)\ that
5 c[l A'u
A' E LP'(R")
+
151) a for 5
k n C [R )
.
E
IRn, and it can e a s i l y be shown
when k
- -n-1
<
and
Hence A'u can be taken t o be as r e g u l a r a s one
wishes inlRn when lt' i s s u f f i c i e n t l y s m a l l , and it i s s u f f i c i e n t t o e s t a b l i s h t h e theorem i n t h e c a s e
a(x, 5) & ~ ( 5 a(x, ) 5)
with
a
A =
g[x1 D)
,
homogeneous of degree p w i t h
277
REGULARITY OF BOUNDARY POTENTIAL
(SEC. 2 )
r e s p e c t t o 5 , where ~ ( 5 =) 0 f o r 5 n e a r 0 and ~ ( 5 =) 1 f o r
Is(
2
For
1.
@> =
E f(R"),
@
\
G((l)
85
we have ( c f . (1.3.3), Chapter IV)
\
eiX' ~ [ x ,5 ) q x )dx
+ < Ru, @ >
15/21
where R i s t h e r e g u l a r i s i n g o p e r a t o r with k e r n e l
Putting B = A
-
R , we t h u s have
5 , so
By h y p o t h e s i s , a ( x , 5 ) i s a r a t i o n a l f r a c t i o n i n
t h a t it f o l l o w s t h a t t h e p o l e s 5, subset of
C. when x d e s c r i b e s a compact s u b s e t of Rn and 5 '
n-1 d e s c r i b e s a compact s u b s e t of IR
For 5 ' f i x e d , F[ (l', f o r 15,
I
$),
F($
=
i
e*S
.
i s t h u s holomorphic i n
sufficiently large.
g
remain w i t h i n a compact
En
Since
[- Dx)Q
(4%g)
@ I x ) ) dx
s
we see t h a t F(5) i s of r a p i d decrease with r e s p e c t t o 5 f o r
278
5'
ELLIPTIC BOUNDARY PROBLEMS
E
Bn-l and I m En 2 0 when
@ c
supp
{ xn > 0
a ( 5 ) is slowly i n c r e a s i n g w i t h r e s p e c t t o 5 Im
and
R
(CHAP.
.
I
for
s1
5)
Since
E
we t h e r e f o r e o b t a i n by c o n t o u r deformation
5 0
a t 5, i n ( 2 . 2 . 1 ) :
where
(1
r5'
- 1%'
2 112
1 )
i n t h e c a s e 15'
3
{ Im
of t h e h a l f - p l a n e 'n
[
completed by t h e segment
of a ( x
j
E l ,
5
0
E n ) such t h a t I m
I
1
- ( 1 - 15'12 1112
I
< 1, i s a p o s i t i v e contour
enclosing t h e poles
5
> 0 ( f o r x describing
supp 0 )
I t m a y be n o t e d , i n view of t h e homogeneity of can choose
rs
contour o f
{ Im
for
&! E r
5'
I
'
=
IS'\r
for
> 0 1.
5' E R"
1
~
~
2
11 ,
We t h e n have
.
\ $1
where
a , t h a t we
r
is a f i x e d
5 c(l
+ 15'1 )
REGULARITY OF BOUNDARY POTENTIAL
(SEC. 2 )
We t a k e i n p a r t i c u l a r SUpp $
C
{ xn > 0 1
.
@ (x)
3
cp(xl) $(xn)
279 with
From ( 2 . 2 . 3 ) , ( 2 . 2 . 2 ) , we o b t a i n
It can be shown t h a t t h e o r d e r o f i n t e g r a t i o n can be changed, s o t h a t we can w r i t e :
where
( B y i n t e g r a t i n g by p a r t s w i t h r e s p e c t t o x ' , we s e e t h a t G(xn,
6') i s o f r a p i d d e c r e a s e w i t h r e s p e c t t o C ' , and hence The
t h a t t h e right-hand s i d e of ( 2 . 2 . 4 ) i s m e a n i n g f u l ) . e q u a l i t y (2.2.4)i s c l e a r l y t r u e f o r deduced f o r imating
u E &'(R")
1
and i s
by proceeding t o t h e l i m i t , approx-
u by i t s r e g u l a r i s e d forms pk
support i n { x s 0 n
,
u E C ;(R')
*
u
, with
p
having
( s e e (l.l.l),Chapter I).
F i n a l l y , ( 2 . 2 . 4 ) and ( 2 . 2 . 5 ) show t h a t , i n
Cx,'OI,
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
280
ca
Bu c o i n c i d e s w i t h t h e f u n c t i o n from C
xn C->
(2.2.6.)
y,(Au)
into
&'(an).
P ( R-1 ) )
d e f i n e d by
Tne above proof shows t h a t t h e mapping
i s continuous from ~u
In
I
(cp
REMARK 2.3 : u -->
-+ (R
E
D'(B") I
u =
o
QI
in
THEOREM 2 . 4 : ( R e g u l a r i t y of t h e s u r f a c e p o t e n t i a l s a t t h e boundary )
Suppose we have
A
i ) The operator
E L:(R")
satisfying ( 2 . 1 ) . A(v,@ 6 )
K : v >-t
3,
Then
i s contin-
uous from c"(~Q) i n t o crn(E).
If v E
p(an)
Kv has sectional. traces of a n y order on
I
i i ) The operator
v -->
A f t e r transport b y a local chart of symbol. i s a
0
(XI'
5 ' ) ->
\r
i s the principal. s y m b o l of A, and
the poZes En of
ao(x', 0 ;
iii) For all. s
E
5'' %)
(n,
0 ;
' . ( , a
r
LF+'(an)
i s in
yO(Kv)
an
.
i t s principal.
y),
s', $1 8% ,
where
i s a contour enclosing such t h a t I m 5, > 0 .
IR, K is continuous from H '
loc
(an) i n t o
2 81
REGULARITY OF BOUNDARY POTENTIAL
(SEC. 2 )
For ( i ) ,we can r e p l a c e A by t h e o p e r a t o r B = A - R
PROOF:
used i n t h e previous proof;
u = v(x')
i c u l a r case
taking (2.2.4), (2.2.5) i n the part-
,v
@ b[xn)
€ C:[R*'),
we s e e t h a t f o r
xn > 0 :
(2.4.1.)'
=
(Bu)(x)
k(x,
5') ; ( I 1 )$5'
where
so t h a t ( i ) t h e n f o l l o w s as t h e right-hand s i d e o f ( 2 . 4 . 1 ) i s Cw with r e s p e c t t o x E 7 R n .
For ( i i ) ,w e reduce t o ~ ' i 8 v ti) 1.1' <
where .'l(xl,
-
E
c0(wn]
C?
for v
1, and t h e n t h a t A"
E Lp'+'(Rn)
5')-
-
Ir.
= l R:
a'(xl,
and we n o t e f i r s t t h a t
,
E C;(R*'I
A'(v@
b)(X',
0)
A'
E
=
A"(xl,
, Dx,)v
h a s a s complete symbol 0 ;
again r e p l a c e A by B as above.
s', G) d #
.
We can t h e r e f o r e
Using ( 2 . 4 . 1 ) , ( 2 . 4 . 2 ) , w e o b t a i n
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
282
For ( i i i ) ,we reduce t o t h e o p e r a t o r B by n o t i n g t h a t f o r A'
E LP'(Rn)
v
I C' l\vl\s if
E CCD(R"") 0
and
-
<
S-P+P'
1/2
cp E C:(Rn)
, we
and P' I
( ( 4 . 6 . 1 ! , Chap. 21,
have:
We s h a l l t h u s show t h a t t h e r e exists a c o n s t a n t C such t h a t ~ \ ( ~ ~ ) , ~ \ ~Is c~ 1-1v11l, / f~o r a l l
i s g i v e n by ( 2 . 4 . 1 1 , ( 2 . 4 . 2 ) ,
v
, where
E CL(R*')
assuming t h a t
a(x,E)
(Bu) In
has
compact s u p p o r t i n x ,
For a l l multi-indices
LEMMA 2.4.3: constant 1xp D;
a,B,
there e x i s t s a
such that
C
k(x,
5'11
c
(1
+ 15'1) P+l+cun+n
for
x
n
2
o
.
I n f a c t , from ( 2 . 4 . 2 ) and from L e i b n i z ' s formula, D Z k is a f i n i t e l i n e a r combination o f terms o f t h e form
we t h e n deduce t h a t By i n t e g r a t i n g by p a r t s w i t h r e s p e c t t o 5 ny xB'D" k i s a f i n i t e l i n e a r combination o f terms o f t h e form X
with
b
=
i s of degree
$ a) .
Elh($ fin
p
+
p
- fin
5
We t h e n have ( 2 . 4 . 3 ) s i n c e p
+ CY, - 8,
b
.
We now e x t e n d k(x, 5 ' ) f o r xn 5 0 such t h a t t h e e x t e n s i o n ,
(SEC. 2 )
REGULARITY OF BOUNDARY POTENTIAL
again denoted by
k
, still
283
x and sat-
has compact s u p p o r t i n
i s f i e s t h e upper bounds i n Lemma ( 2 . 4 . 3 ) f o r a l l x
E
IRn
.
We
can, f o r example, use a SEELEY e x t e n s i o n by p u t t i n g f o r xn 5 0 :
where (A ) i s a sequence as i n ( 9 . 2 . 1 1 , Chapter I .
P
We p u t h
5'1
Y
LEMMA 2.4.4
:
-ix e
C
For a l l q , r
E
=
k(x,
5')
dx
IT, there e x i s t s a constant
C such that
From 2 . 4 . 3 we have, f o r given p
ID;
k(x,
5'11
5
CJl
+ 1 5 ' p
+I-cu
n (1
E
+
l'J and f o r any multi-index
IX'11-q
(1
t
\""I
s o t h a t , choosing p > 1 and supposing q > n-1:
which r e a d i l y g i v e s Lemma 2 . 4 . 4 .
Since U extends (Bu)
In'
it w i l l s u f f i c e t o show t h a t
(1
+
ci
\s'\))-P
284
ELLIPTIC BOUNDARY PROBLEMS
( C W . 5)
i.e. that
( r and. q w i l l be chosen l a t e r ) . To conclude, it w i l l be s u f f i c i e n t , t h e r e f o r e , t o show t h a t
The change of v a r i a b l e s that
7'
v
=
T'
Tn =
(1
+
17 1 )yn
shows
(SEC. 2)
REGULARITY OF BOUNDARY POTENTIAL
285
Now Peetre's inequality (see Chapter 11) shows that
so that, from (2.4.7), (2.4.9) we have:
From (2,4.11), we have (cf. (2.1.2), Chapter I)
Finally, we choose q
Z
n
- 1, q
Then, from (2.4.12) we have:
7
s
-
+
n-l/2, r
2
\p-s\ + q
,
ELLIPTIC BOUNDARY PROBLEMS
286
and h e n c e , from ( 2 . 4 . 1 0 ) , ( 2 . 4 . 1 3 ) 3-
which e s t a b l i s h e s ( 2 . 4 . 8 ) and completes t h e proof o f Theorem 2 . 4 .
THEOREM 2 . 5 ( R e g u l a r i t y of t h e volume p o t e n t i a l a t t h e boundary) :
Suppose we have A E L:(R~)
1 A(fo)]
operator f -->
Then t h e
satisfying ( 2 . 1 ) .
~ ~ (into 5 ) cmm).
is continuous from L.
PROOF:
Suppose
f
+
f i s an e x t e n s i o n o p e r a t o r which i s con-
t i n u o u s from C m ( n ) i n t o C m ( J R n ) ; we have
i s t h e e x t e n s i o n by 0 i n t o
n of
s h a l l t h e r e f o r e show t h a t g into C"(5);
-+
g =
flnl,
- u , where = R" \7i .
= f
fO
&
01
(Au) i s continuous from la
C"(nl
u We
)
f o r t h i s , we reduce t o t h e o p e r a t o r B used i n t h e
proof of Theorem 2 . 2 , and we use ( 2 . 2 . 3 ) .
By i n t e g r a t i n g by
p a r t s w i t h r e s p e c t t o x', we s e e t h a t f o r any R t h e r e e x i s t s a c o n s t a n t C such t h a t and
I m
$
t
0
h a v e , f o r xn > 0 :
;
(C(5)l 5
C (1
+ \<'I)-'
for 5 '
E
Bn-l
( 2 . 2 . 3 ) and ( 2 . 2 . 2 ) t h u s show t h a t we
(SEC. 2 )
.)
(2.5.1
so t h a t Bu
E
ID
R M K 2.6:
c"(E]
,
Under t h e c o n d i t i o n s of t h e preceding theorem,
we can show t h a t H;OC
287
REGULARITY OF BOUNDARY POTENTIAL
?>
f
( E ) t o Hii!(E)
[
when s >
]
extends c o n t i n u o u s l y from ID l/2 ( s e e BOUTET DE MONVEL [11).
A(fo)
-
By analogy w i t h t h e o p e r a t o r s which occur i n t h e s o l u t i o n of t h e D i r i c h l e t problem, t h e o p e r a t o r s K i n Theorem 2 . 4 a r e c a l l e d Poisson o p e r a t o r s ;
by analogy w i t h t h e c a s e i n which A is d i f f -
e r e n t i a l , the operators operators.
f -7
[
y,
A(f0lln
]
are called trace
A l l t h e previous r e s u l t s and t h e d e f i n i t i o n s of t h e
Poisson o p e r a t o r s and t h e t r a c e o p e r a t o r s a r e i n f a c t v a l i d under a weaker assumption t h a n ( 2 . 1 ) :
t h e t r a n s m i s s i o n assumption
f o r A ( s e e BOUTET DE MONVEL C21).
PROPOSITION 2 . 7 :
(2.1).
K
Suppose we have
A
E L!(R"]
satisfying
Then the transpose o f the Poisson operator :
&*(an]
>
>-
v
&'mI ([ A ( v @ 6lll"Io
i s the operator
T'
:
Cmm
1
i
> c"(an1
f
t
7
Y , ( [ ~ A ( ~ ~ ~ I , ~ ,I )
288
ELLIPTIC BOUNDARY PROBLEMS
We need t o prove t h a t < f , Kv > = < T' f,
PROOF: f
Cim)
E
.
= Bn \
where
,v
E C;(an)
n
,
We reduce t o t h e c a s e R = IR,
t h e e q u a l i t y i s immediate when f
E
C"(E), 0
-EI
f E
.
C:(n)
;
When
we c o n s i d e r a r e g u l a r i s i n g sequence X
pe(xn) = ] , (p
n
with
Supp p
lim
To c o n c l u d e , we show t h a t i n C"(Q).
v > for
c
xn > 0 ]
yo t Atp,
*
, We have
f o )= y o ( [ t A ( f o ) ] , n r )
e -0 This i s obvious i f A i s a p s e u d o - d i f f e r e n t i a l
o p e r a t o r o f degree l e s s t h a n -n,
so t h a t the required equality
f o l l o w s by a s t r a i g h t f o r w a r d passage t o t h e l i m i t i n (2.5.1).
lFnder the conditions of Proposition 2.7,
COROLLARY 2.8:
the operator T' extends continuouszy from
d (5)t o
ti'(an).
Xhe same is true for the t r a c e operator
T :
f
->
Yo-([
A(fo]]
) when the degree u of A i s l e s s than
In
- 1/2. PROOF: i n t o -C:tc)
Since
, TI
TI
=
+,
K and s i n c e K i s c o n t i n u o u s from C:(an)
e x t e n d s c o n t i n u o u s l y from
8 (z)t o a' (an).
Since T'f
289
REGULARITY OF BOUNDARY POTENTIAL
(SEC. 2)
,
2 (f) Lloc
f O
= Tf
<
p
if
- 1/2
.
,
H;&[R"]
is i n
A(fo)
and t h u s
We conclude t h i s s e c t i o n w i t h a p r o p e r t y r e l a t i n g t o t h e e x i s t ence of s e c t i o n a l t r a c e s .
Suppose P ( x , D , )
THEOREM 2 . 9 : operator of degree
m
i s an e z l i p t i c d i f f e r e n t i a l
i n R", and suppose t h a t u
E
i s ext-
&'(ill
endable t o IR". (i) I f
Pu E cm(z)
,
has sectional, traces of any order
u
on an. (ii)
If
H;oc(E) ,
Pu E
k on a R i f
order
PROOF:
k
For - ( i )l,e t
put Pu = f and PE
-
u = (Q(fo)
we have
c s +m
has sectional traces up t o
u
-
cEp(f)
f o = g;
.
1/2
be an e x t e n s i o n of u;
we
l e t Q be a proper p a r a m e t r i x o f P ;
+ Qg + RC)
In
R E L-m(Rn)
with
;
we
then have ( i )from Theorems 2 . 2 and 2 . 5 , s i n c e Q s a t i s f i e s ( 2 . 1 ) and s i n c e
g
vanishes i n R.
For ( i i ) ,we p u t Pu = f and we c o n s i d e r extending Pv E
f
; let
Cm(E),and
REMARK 2.10:
v = u
-
[QF),,,
hence ( i i ) s i n c e
Suppose M c
;
we have
6 E HYz(R")
,
&'(a) i s a subspace of extendable
d i s t r i b u t i o n s , endowed w i t h a topologq such t h a t t h e r e e x i s t s a
290
(CHAP. 5)
ELLIPTIC BOUNDARY PROBLEMS
a continuous e x t e n s i o n o p e r a t o r : M
+
Remark ( 2 . 3 ) and
#(lRn).
t h e p r e c e d i n g proof show t h a t , i n Theorem 2 . 9 , t h e t r a c e s of
u
depend c o n t i n u o u s l y ( a s d i s t r i b u t i o n s on 3 0 ) on
By u s i n g t h e r e s u l t of E x e r c i s e 6 . 3 , given
REMARK 2 . 1 1 :
l a t e r i n t h i s c h a p t e r , Theorem 2 . 9 and i t s proof may r e a d i l y be g e n e r a l i s e d t o t h e c a s e i n which t h e assumption of e l l i p t i c i t y
an i s
of P i s r e p l a c e d by t h e assumption t h a t
noncharacteristic
f o r P.
3.
THE CUDERON PROJECTOR Let Q
L-m(Bn) be a p r o p e r p a r a m e t r i x of t h e e l l i p t i c
E
C
d i f f e r e n t i a l o p e r a t o r P o f degree m.
= 1 (2.1).
+ R'
with
Suppose u
R , R ' E L'm,
E
Cmm
1,
We have
PQ
I
I
+ R,
We r e c a l l t h a t Q s a t i s f i e s Applying Q on t h e l e f t t o b o t h
s i d e s of (1.3), we o b t a i n , p u t t i n g Pu = f :
so t h a t , taking t h e t r a c e s : yu
=
y
-
(P
DEFINITION 3 . 1 :
ct"Can,
G ~ )
P" yu) I"
+
Y(e(foII,n
-
Y
R'(UO)
.
The operator C :
cmc3n, PI
defined by
cv = y
[(Q P v) ,"]
(SEC. 3)
CALDERON PROJECTOR
( s e e ( 1 . 3 ) , (l,k)¶ (1.5) and Theorem 2.4) i s termed the CaZderdn projector associated with P, Q, we have u E
yu = C(yu)
,
c"@)
+ Y(Q(fo)),n-
y.
By construction,
for
?p'(uo]
,
f = Pu
PROPOSITION 3.2 :
i ) The operator
v
+-+ (Q k v)
,
i s , f o r aZZ u
E
By
continuous from i i ) Flt? have 'j1A
E L*-a(ao)l iii)
proper.
c i s a projector i n the sense t h a t
PROOF:
C2
By t r a n s p o r t u s i n g a l o c a l c h a r t o f
- c E L-"(an) . (52,
v),
(1.4)
and (1.5) show t h a t
Since t h e o p e r a t o r s
Q
P
Mk+l
Dkxn
(2.1)and a r e o f degree - ~ - l
'
Di n
j-a-1
'Q+k+l
gk
xn
satisfy
respectively,
( i )and ( i i )follow from Theorem 2 . 4 ; t h i s a l s o shows t h a t C has a s p r i n c i p a l symbol ( o f degree j-t):
j
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
292
where
r
5, of
p(x',
i s a contour o f
0;
[ Irn
5'' k) such
#
e n c l o s i n g t h e zeros
>0 ]
that Im
5,
To e s t a b l i s h ( i i i ) ,we c o n s i d e r
> 0.
v
E
C"(af2,
by c o n s t r u c t i o n of C , we have
put u = [Q r v ) , , ;
,
C")
and
yu = Cv
;
N
s i n c e PU = (R P v ) , ~ we a l s o have
n
s o t h a t C - 'C
Thus
c2
can)
from
= r, w i t h
-c
E L-"(asl)
to
c"(an)
s i n c e r extends continuously
,
Let c be t h e p r i n c i p a l symbol of C , i . e . t h e m a t r i x
I
I
(cj,,tj l , t = o l . . l rrri
degree j - 2 ) of C c(x',
5')
where c
For
j ,R'
i s t h e p r i n c i p a l symbol ( o f
j ,R
(XI1
i s a p r o j e c t o r from !t
m
5')
, the
E T*(an)
\
0
image of which we s h a l l
make e x p l i c i t ; following t r a n s p o r t by a l o c a l c h a r t of denote by
P(X',
denote by
s+(x',
Xn;
5')
5',
In)
1
m,
v) we
t h e p r i n c i p a l symbol o f P ; we
(resp : S'[xl,
5'))
t h e space of
(SEC. 3 )
CALDERON PROJECTOR
xn
s o l u t i o n s U bounded f o r
;r 0
uu
=
(Dj
5', D
p(x', 0;
xn
of t h e
)a =
.
0
~ ~ ~ l ~ j s o , ~ * ,and w~ 5')
Df(x',
{ yLl
=
U
E
From t h e e l l i p t i c i t y of P , t h e e q u a t i o n has no r e a l r o o t 5
)
xn 5 0
(resp :
ordinary d i f f e r e n t i a l equation We p u t
293
n'
and hence Cm =
furthermore, t h e number o f r o o t s 5
n'
5'))
S'(x',
5', 5 )=
p(x', 0 ;
(Ds(x',
5')
@ D-(x',
Irn5
f o r which
. 0
5')
> 0 ,
i s l o c a l l y c o n s t a n t and t h e r e f o r e independent o f (XI,
5')
\
E T*(R+')
when n t 3; s i n c e t h i s i s
0
m when 5 ' i s r e p l a c e d by - E ' ,
changed i n t o i t s complement i n
we s e e t h a t t h i s number i s e q u a l t o m/2 when n t 3.
In the
case n = 2 , we hy-pothesise t h a t t h i s number i s e q u a l t o mj2. We t h u s have
5')
dim D'(x',
5')
PROOF:
= D+(X',
Let
u
m/2.
Using the preceding notation, we have
PROPOSITION 3.3 :
Irn c ( x ' ,
I
5')
I
E S+[xl,
and thus dim Di( X I ' 5 ' ) = rn/2.
%I)
and
U O
be i t s e x t e n s i o n by
0 f o r x < 0 ; as i n ( 1 . 3 ) , ( l a b ) , (la?),we have t h e jump n formula:
PIX', 0;
5 ' ' DXn )U"
=
-i1
is t h e p r i n c i p a l symbol o f P * s i n c e j' j we t h e r e b y deduce t h a t
where p
Uo
E
s'(R) ,
294
ELLIPTIC BOUNDARY PROBLEMS
(CHAP.
5)
and, by a s t r a i g h t f o r w a r d contour d e f o r m a t i o n , we have
x
for
r
where
i s t a k e n as i n ( 3 . 2 . 2 ) .
2
n
0
We o b t a i n
r
therefore
yU
E Im
5'1
yLl = c ( x ' ,
i . e . from ( 3 . 2 . 2 1 , we have
5')
c(x',
For xn
E
E
C
and
.
m Conversely, l e t w
,
YU
be such t h a t
w E
Im
c(x',
5')
W, we d e f i n e U ( x n ) by t h e e q u a l i t y (3.3.1) w i t h y R U
r e p l a c e d by w
y~ = c(x',
Then U E S+[x',
R'
I*)W
= w,
REMARK 3.4 :
5')
and
so t h a t w E D+(x',s')
,
The above p r o p o s i t i o n shows t h a t t h e rank of
c(x', 5 ' ) remains c o n s t a n t , and t h e r e f o r e t h a t t h e subspaces I m c(x',
.
5')
a r e o r g a n i s e d i n t o a v e c t o r sub-bundle of
rank m/2 of t h e trivial bundle Cm x
T*(an) \
0
above
(SEC.
4)
APPLICATION
T*(an) \
4.
0
,
295
We denote t h i s sub-bundle by
I m c.
APPLICATION TO ELLIPTIC BOUNDARY-VALUE PROBLEMS Suppose we have
We denote by b t h e p r i n c i p a l symbol of B, i . e . t h e m a t r i x ( b j ,k B
) where b
j,k'
i s t h e p r i n c i p a l symbol ( o f degree d - k ) of
j ,k
j
5.9 E
For ( x ' ,
l i n e a r mapping of C
F :Im
c
.
5'9
b(x',
+
Cp x T*(an)\
:
defines a
Im
s')
C(X',
We denote by t h e c o r r e s p o n d i n g morphism of
0
Assume t h a t b" i s s u r j e c t i v e .
admits a r i g h t parametrix
&(('i) x &'[an,
5')
T*(an)\O.
v e c t o r b u n d l e s on
THEOREM 4.1
b(x',
m i n t o Cy whose r e s t r i c t i o n t o
H
we denote by
T*(an)\O,
CP))
a,
Then
P
The orthogonal of Im p ( i n
i s contained i n C"m]
x C"(aC2,
Cp9
.
If moreover fi i s bounded, then Im p i s closed and of f i n i t e codimension.
PROOF: (BC)j,k
BC i s a y x m m a t r i x o f o p e r a t o r s
d .-k E LcJ
(an)
,
and t h e s u r j e c t i v i t y o f i t s p r i n c i p a l
symbol i s e q u i v a l e n t t o t h a t o f
2.
BC i s t h e r e f o r e
296
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
r i g h t - e l l i p t i c i n t h e sense of Agmon-Douglis-Nirenberg,
3
~"(5) x c"(an,
i s continuous from
CF)
into
C m ( E ) i n view of Theorems 2 . 4 , 2 . 5 , and it can e a s i l y be shown
PB
that
=
I
+Q ,
f ->
Now t h e o p e r a t o r from
Y'
0
with
RICE) t o B'(an, C") Q i s of degree
Y[Q(f0)],
extends continuously
i n view of C o r o l l a r y 2.8, s i n c e
- 1 a t most f o r 0 5
,
j 5 m-1
f o r e i t s l e f t composition with r B i s continuous from into
C"(an,
Ch)
and i t s l e f t composition w i t h R P A B to
has i t s k e r n e l i n Coo(;
X
cmmn). E)
The o p e r a t o r f
tQ
+
tQ
,
with
( f , g ) i s i n t h e orthogonal of
(f, g)
t
-
t
- > [ ~ ( f o ) ] , ~
R
i s therefore
The orthogonal of I m 6 c o i n c i d e s w i t h Ker
tp = I
f?(f, g)
E
is
and t h e o t h e r o p e r a t o r s which
appear i n ( 4 . 1 . 2 ) a r e c l e a r l y r e g u l a r i s i n g .
now
#(E)
w
continuous from
regularising.
There-
C"m)
t 6 ;
t
f ? r e g u l a r i s i n g ; hence i f
Im p , we have
x C"(a0,
Cu).
F i n a l l y , i f 52 i s
(SEC. 4)
APPLICATION
bounded, we know t h a t
~"(5) x c"(an,
R is
.
CP)
%us
297
a compact o p e r a t o r from
Im(1
+ e) i s
c l o s e d and of f i n i t e
codimension, and t h e same a p p l i e s f o r Irn p which c o n t a i n s
+ e)
Irn (I
.
formula f o r t~ :
tP(f")
=
( t P f ) O
+ ;;(Yf)
and we t h e r e f o r e deduce t h a t
i s i n Ker
t 33 i f and o n l y i f
We now proceed t o t h e c a s e of Sobolev s p a c e s .
We denote
by m ' t h e s m a l l e s t i n t e g e r 5 m such t h a t B j = 0,
..., p-1.
= 0 f o r k 2 m', j ,k Then B y u i s w e l l d e f i n e d when u has s e c t -
i o n a l t r a c e s on a R up t o o r d e r m'
-
1; from Theorem 2.9, t h i s
i s t h e c a s e , i n p a r t i c u l a r , when u i s an e x t e n d a b l e d i s t r i b u t i o n
Pu
E
Hs(,E)
Let s , u
E
IR w i t h
such t h a t
with
s > rn'
s > rn'
- rn -
o p e r a t o r ( n o t n e c e s s a r i l y bounded)
- rn - 1/2 1/2
.
. We c o n s i d e r t h e
29 8
This domain is dense i n H
U
loc
(5)
; u s i n g Theorem 2 . 9
is c l o s e d .
and Remark 2 . 1 0 , it can be shown t h a t ps
IU
Suppose t h a t
THEOREM 4.3 : s,a
(CHm. 5 )
ELLIPTIC BOUNDARY PROBLEMS
E R
0s,0
a r i g h t quasi-inverse
Q
=lo
,
s > rn'-m-1/2
with
'
:
Hloc
(z)
F is
surjective.
a 5 s+m.
Then Ps 1u a d n i t s
-
i n the sense t h a t Hr61/2 loc
(an)
is continuous, with P s , o ~ s , o = I + l?s,a~ from H ( ,n ;l
orthogonaZ of
r61/2[an)
into
x Hloc
Im ps
,a
(in
H Z
c"(;)
($1
n, cornp
i s the same as the orthogonal of Irn P bozmded, then I m ps (equal t o t h a t of I m
PROOF:
Let
c o n t i n u o u s from H
f S
loc
.
x
c"(ahl,
x
Hemp
HYoc
(-3
continuous
Cp)
.
The
(an)I
If, furthermore, R i s
i s cZosed and has f i n i t e codimension
SU
P
Let
).
7
be an e x t e n s i o n o p e r a t o r ,
( 5 ) i n t o 'H
loc
Dn)
; we d e f i n e
Q
=,a
( f , g]
(SEC. 4 )
APPLICATION
299
as b e i n g t h e r i g h t - h a n d s i d e of formula ( 4 . 1 . 1 ) d e f i n i n g Q ( f , g ) ,
a
c
b u t w i t h f o r e p l a c e d by f .
The c o n t i n u i t y of
SIU
f o l l o w s from P r o p o s i t i o n 3 . 2 and from t h e assumption
05
It can be shown t h a t
S+m
=
I
+
i s d e f i n e d by t h e r i g h t - h a n d s i d e of
fs,u(f, 9)
where
Q s,u N
,;'("C
viously has values i n
f[aQ
x
belongs t o t h e dornaizi
t
thRt
Ker Ker
(F, G) =
c C:fi) t p
SlU
BS,,
t
p
.
t p ( ~ G, )
x C:(aQ
= Ker
Of
Cb)
, which
i s bounded,
es,o i s
HS(E)x H**1/2(a*l
it follo*rs i n m e d i a t e l y
%if,
Since
, we
c B;,,
obtain
g i v e s t h e s t a t e d e q u a l i t y between
L i e o r t h o g o n a l of t h e images o f
R
If (F,G)
Cp)
t
; fs,u ob-
f
in t h i s formula r e p l a c e d by
( 4 . 1 . 2 ) w i t h f0
%U
,p
.
F i n a l l y , when
a compact o p e r a t o r from
,
and t h e r e f o r e I m ps
?U
i s closed,
with f i n i t e c o d h x n s i o n .
Suppose t h a t b
s , E~ R
with
,
u
E p (0)
be extendabZe t o IRn such t h a t
i i ) Suppose
u' < u
injectiue. i ) Let
-
.,
THEOREM 4 . 4 ( E l l i p t i c R e g u l a r i t y ) : Let
s > m1-wI/2
i S
0 5 s+m
and l e t K be a compact subset of
; then there e x i s t s a constant C such t h a t
.
300
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
f o r aZZ u with support i n K s a t i s f y i n g the assumptions of ( i ) . iii) We have iv)
K e r ps
If, furthermore,
ta 51
= Ker P
c
C-m]
is bomded, then
,
Ker bJ
st0
= K e r 63
i s of f i n i t e dimension.
For ( i ) ,we p u t Pu = f and
PROOF: we have Pv =
7 , with
N
cp = -(Rf
,
1,
,
v
I
u
- (Gjf),, H
;
I
I n view o f Theorem 2 . 9 ,
v h a s s e c t i o n a l t r a c e s of a l l o r d e r s on a R , and we have P ( v o ) = yo
+ g(yv),
Applying Q on t h e l e f t t o t h i s e q u a l i t y ,
we o b t a i n (4.4.2.
1
(I
')
YV
E c"(an,
C")
@
0 - 6 1/2 Hloc (an1
.
Now t h e i n j e c t i v i t y of t h e p r i n c i p a l symbol o f
i s equivalent t o t h a t of
left-elliptic that
F; t h e r e f o r e
i n t h e s e n s e of Agmon-Douglis-Nirenberg, so
(SEC. 4 )
APPLICATION
301
From ( 4 . 4 . 2 ) and from P r o p o s i t i o n 3 . 2 , we deduce t h a t
E
w
for
, which
HYocfi) u
and y
j
then gives t h e s t a t e d r e g u l a r i t y p r o p er t i es
u
u since
-v
=
($9
E)if:H::
,
We prove ( i i ) i n t h e same way by u s i n g t h e e l l i p t i c i n e q u a l i t y
bounded,
(4.4.1)shows
by H'(E)
and H''(E)
4.5
t h a t t h e t o p o l o g i e s induced on Ker 63
a r e c o i n c i d e n t , which gives ( i v ) .
LOPATINSKI'S DETERMINANT AND CONDITION:
The boundary-
w
value problem (1.1)i s s a i d t o be e l l i p t i c i f b i s b i j e c t i v e . me number
u
of s c a l a r boundary c o n d i t i o n s must t h e n be e q u a l t o
m/2, and t h e preceding theorems show t h a t 63 o p e r a t o r s ( w i t h t h e same i n d e x ) when
n is
63s,a
I
bounded.
..
are indexed We conclude
t h i s s e c t i o n by g i v i n g i n v e r t i b i l i t y c r i t e r i a f o r b i n t h e case I.I
= m/2.
Following t r a n s p o r t by a l o c a l c h a r t of
note by p ( x , 5 1 , b j Y k ( x ' ,
(E,v), we
5') t h e p r i n c i p a l symbols of P, B
de-
j ,k
r e s p e c t i v e l y , and we p u t
p+-(x*;
+-
Irn
% ) a r e polynomials i n En whose ze,ros
5 1 ,
>
o
; pf i s o f degree m/2 i n L1.-
1
5')
6
5,.
5n a r e such t h a t
Let denote t h e remainder
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
302
of t h e E u c l i d i a n d i v i s i o n of
$1
5''
bj(x';
by
pi(x';
$,I
9. .
E l ) i s i n v e r t i b l e i f and o n l y i f t h e
From P r o p o s i t i o n 3 . 3 ¶ b(x:
o n l y bounded s o l u t i o n U , for x 2 0 , o f n
i s U = 0. S i n c e t h e bounded s o l u t i o n s U , f o r x t 0 o f t h e e q u a t i o n n
5''
p(x1, 0;
tion
+
p
=
0
a r e t h e s o l u t i o n s U o f t h e equa-
U(xn)
=
0
b!(x';
s',
Dxn) U(xn)
[XI;
5''
)
Dx
, and
since
n
b
5',
.(XI;
3
Dxn)U
-p +( x *;6')5'' i sDx b(x',
(b;, k
=
)U = 0
J
, we
Dx,]U
when
s e e t h a t t h e i n v e r t i b i l i t y of
n
e q u i v a l e n t t o t h a t of t h e m a t r i x
1
5'
1) jyk=o, ~, I p-
1
.
The d e t e r m i n a n t o f t h i s m a t r i x
i s c a l l e d t h e L o p a t i n s k i determinant o f (1.1)r e l a t i v e t o t h e l o c a l chart considered. Lopatinski condition i n
We s a y t h a t (1.1)s a t i s f i e s t h e
(x', 5')
-
i f t h i s d e t e r m i n a n t i s non-zero;
t h i s condition i s equivalent t o t h e i n v e r t i b i l i t y of b ( x ' ,
5.
6').
EXAMPLES The D i r i c h l e t problem i n
o p e r a t o r P of degree
m
is
for the elliptic differential
303
EXAMPLES
We t h u s have i n t h i s c a s e p = m/2
P
!?he corresponding operators 5
Bj,k
= gj,k
I
,
m' = m/2 ,
The D i r i c h l e t problem ( 5 . 1 ) i s e l l i p t i c .
THEOREM 5.2:
u
,
Ps,o
(with s
>
m+l
- -,2
s+m) are therefore indexed ( w i t h the same index) when Q i s
t h i s index is zero when P is formazly s e l f - a d j o i n t or
bounded;
strongZy e l l i p t i c .
PROOF:
Using t h e n o t a t i o n o f
b.(x', 5)
J
=
4
=
p = mj2); t h e r e f o r e
0,
j,k
.,., p-1
,
view of Remark 4 . 2 ,
(f,
g)
(5.2.1
E
c"fi]
x
bv.(xl,
J b1
51
(xl,
4 . 5 , we have for
I*) = BjIk
j,k and t h i s shows t h a t
Ker
c"(an,
t P
cb)
j = 0
,..., G-I( p u t t i n g for
(5.1) is elliptic.
i s t h e s e t of t h e such t h a t
.)
Now, we have l o c a l l y
that (5.2.1) is equivalent t o
In
304
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
=
tPf
,
0
yj
=
f
0
- c(yf)
g . b e i n g e q u a l t o t h e c o e f f i c i e n t of D j 6 i n J t h e r e f o r e follows t h a t
cf
K e r tp
(5.2.2.)
E
, yj
o
=
{
E
f
=
Crn(qP*f
t
,
yj
=
f
if P = P
+
~ ( u ,u)
5
G
We t h e r e f o r e deduce t h a t K e r p' = { 0
Ker
t
p'
=
Let s >
{ 0 }
nH1 -2
+ 1.
,
}
E
hence t h e i n d e x o f P'
and u = s+m; t h e n
ps 9 0
REMARK 5 . 3 :
IR such t h a t
E
%(nl
0
6' i s assoc-
I n view o f ( 5 . 2 . 2 ) , we s i m i l a r l y have
, and
Index
Garding's
u
where
G's,o. 1
Index
i s t h e operator
. =
pi
Index p'
9U
I n t h e c a s e i n which P =
we have Green's formula
i s equal t o zero.
PS,,
-
(If, 0) A s t h i s o p e r a t o r i s compact, we have:
p =
63 t h e n
JIUII~
f->
Index
3
2
for all
i a t e d with P' = P
3
We now suppose
i n e q u a l i t y e n s u r e s t h e e x i s t e n c e o f c > 0 and A Re(Pu, u)
that
i s bounded.
t h a t P i s s t r o n g l y e l l i p t i c and t h a t
o
O 5 j 5 p-I
, so
h a s z e r o i n d e x i n t h e c a s e where $2 i s bounded.
(5.2.3.)
f =
for
0
*
Ker p
p
Ker
In particular
0
and it
0 5 j 5 1.1-1
for x
,
Osjsp-I
for
-
=
0
.
A and f2 i s bounded,
-
G&rding's i n e q u a l i t y ( 5 . 2 . 3 )
where N i s t h e inward u n i t normal. t h u s h o l d s f o r A = 0.
P
:
co.rnl
Therefore
cmfi] (-
and
H'(5)
c
AU,
yo u )
1----------b (-
but
Hr1/*(an)
x
Yo u5
w i t h continuous i n v e r s e ( f o r
is bijective
,
u(s+~!)
Now l e t P be e l l i p t i c of degree 2 ;
we c o n s i d e r t h e problem
Pu
=
f
in
0
y, u
=
g
in
an
(5.4.1
p = m/2 = 1
I n t h i s case of
i s an isomorphism,
Hsm>
u
~ > - 3 / 2
c"(an)
-
u *-I
: Bs,o
x
, ,
4.5 , b(x', 5 ) = 5,
m'
= m
b'(x',
.
and, u s i n g t h e n o t a t i o n
5')
=
i-
<(XI,
Cn(x', 6') denotes t h e r o o t 6, of p ( x ' , 0;
5')
s', &) =
Hence ( 5 . 4 ) i s e l l i p t i c ;
p o s i t i v e imaginary p a r t .
where
0
i n the
p a r t i c u l a r c a s e of t h e Neumann problem, i . e . o f problem P = -
a
,
L,
= N
, Green's
with
formula ( 5 . 3 . 1 ) shows t h a t
(5.4)w i t h Ker p
i s c o n s t i t u t e d by l o c a l l y c o n s t a n t f u n c t i o n s i n a , and a l s o t h a t
Ker tp that
i s c o n s t i t u t e d by t h e
A f = O
,
~
~
i s locally constant i n
f
-
n ,
[f, g) E
0 g =, - yo f
g =
CG : ) x C:(an) , i ; e . such t h a t
- f Ian
such f
.
We now r e t u r n t o t h e D i r i c h l e t problem for an e l l i p t i c o p e r a t o r
306
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
P of degree m; suppose we have
n
D = Hm(D)
where t h e domain D of IF i s
H[(Cl]
I.
with
p = m/2.
For t h i s boundary-value problem, we have t h e analogue
of Theorem
8.14, Chapter
IV:
Suppose t h a t R is bowzded.
THEOREM 5 . 5 :
( i ) IP is a closed operator with dense domain.
*
a d j o i n t IP has the same domain D as Y
1p
Its
and is defined by
u f o r u E D. IP i s s e l f - a d j o i n t if and only if P i s
p* u = P
forma1ly se 2f-ad.joint.
(ii)
is an indexed operator.
Ip
PROOF:
(4.4,1),t h e r e exists
From
C such t h a t
We t h e r e f o r e deduce, u s i n g Remark 2.10, t h a t IP i s c l o s e d . Theorems 4.3,
4 . 4 show t h a t Ker IP i s o f f i n i t e dimension and
Im
P = (Ker p*)'
*
v
E
D
u
E
D.
w
, IP
E
*
*
.
i s t h e s e t of t h e v uously on u
We t h u s have
and t h a t I m I P i s c l o s e d .
contained i n C m ( l ) ,
We now i n v e s t i g a t e p ; i t s domain D E
L2 ( Q )
such t h a t ( P u , v ) depends c o n t i n -
2
D f o r t h e topology induced by L ( R ) and, f o r
v i s c h a r a c t e r i s e d b y : (Pu, v)
Taking, i n p a r t i c u l a r , u E C"[D), 0
=
(u, p*
V)
for
it f o l l o w s t h a t ,
(SEC.
6)
for v
E
307
ADDITIONAL NOTES
D
* , we
*
P
t h e n have
*
2
,
p v E L (sl)
v =
and t h e r e -
f o r e , from Theorem 2 . 9 , t h a t v h a s s e c t i o n a l t r a c e s on 30 up t o o r d e r m-1;
it can e a s i l y be shown t h a t we s t i l l have t h e jump
formula
P*(Vo)
=
(p*
u)" + g("fV)
Furthermore, for u E H m ( n ) and v
(pu, v)
=
( u , P*(vo))
,
for t h e o p e r a t o r P
E
L
2
*.
( a ) , we have
where t h e b r a c k e t on t h e right-hand
s i d e i s t h a t of t h e d u a l i t y between p ( Q ) and HIm@in).
Thus,
n
u
for
E Hm(n) (pus
and v
.I
=
E
D*,
(.,
w e have:
P*V)
2 which shows t h a t v E L (n)
,
+ (u,
;(Yv))
,
(W))
(u
I
0 for u
t
E
D.
It can be shown t h a t t h i s l a s t c o n d i t i o n i s e q u i v a l e n t t o y; v = 0 f o r 0 S
0
6.
5
j
5
p-1.
j
] =D
I p-1,
and hence t h a t
i n view o f Theorem
4.4.
ADDITIONAL N O T E S AND EXERCISES
6 . 1 EXAMPLE OF A PSEUDO-DIFFERENTIAL OPERATOR WHICH DOES NOT SATISFY THE REGULARITY PROPERTY OF THEOREM 2 . 5 Let
c1 >
-
a 1; l e t x+ denote t h e f u n c t i o n o n B which is
a equal t o x f o r x > 0 and e q u a l t o 0 for a
x
s 'i n' L o p e r a t o r A of convolution w i t h x+ i ) (see Exercise
1 1 . 4 , Chapter I V ) .
6
0.
Show t h a t t h e
w i t h 1-1 = -1-a
308
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS Let
a
3
such t h a t f ( x ) = 1 f o r 0 5 x 2 1.
E CO(R )
f
Calculate t h e r e s t r i c t i o n of A(fo)
deduce from t h i s t h a t
6.2
A(fD)
to
f Ca(R3)
30 , 1 [ , when
cy
.
$N
and
CAUCHY'S FORMULA AND THE HILBERT OPERATOR "
a)
a
Let
be t h e open u n i t d i s c i n B ' ,
and l e t
be t h e Cauchy-Riemann. o p e r a t o r i n
p=t(&+i$
m 1
E =
We r e c a l l ( s e e 1 0 . 4 , Chapter I ) t h a t
z = x + i y ) i s a fundamental s o l u t i o n of P .
IR2
.
(with
We adopt t h e
n o t a t i o n of S e c t i o n 1.
Let v E C"(
an).
V
Show t h a t t h e f u n c t i o n
(E * P v ) in
i s given by
v(2) L e t zo
E
=
aa.
- 5, ALl 1
2in
t
-2
dt
(2
E
01
Show t h a t
dt.
9
where C E i s obtazhed by e x c l u d i n g from
an
t h e a r c with c en tr e
z and w i t h l e n g t h 2 ~ . Deduce t h a t t h e Calder6n p r o j e c t o r 0
c
:
c"(an)
-
c"(M)
(SEC. 6 )
i s given by
ADDITIONAL NOTES
C
3
c
(I +
H)
,
309
with
where we have p u t
Deduce t h a t t h e boundary v a l u e problem
has a s o l u t i o n
1
u
u E
c"0
u
holomorphic i n
n
"Ian = i f and only i f t h e d a t a v s a t i s f i e s t h e
equation v = Hv, and t h a t
u
i s t h e n unique and given by t h e
Cauchy formula
t-z m e operator
H : c"(M)
dt
(z
c"(an)
-.)
Hilbert operator ( f o r the unit c i r c l e ) . Show t h a t when we s e t z = eig(z
E
E n)
,
is termed t h e Show t h a t H'
,8
E R),
= I.
H becomes
i d e n t i c a l t o t h e o p e r a t o r o f convolution ( i n t h e sense of p e r i o d i c d i s t r i b u t i o n s w i t h p e r i o d 2 i ~ )w i t h t h e d i s t r i b u t i o n
I!
P
1 --2rr
1
2in
coefficientsof in L
2
(an).
p.v.
cot
59
Calculate t h e Fourier
and show t h a t H extends t o a u n i t a r y o p e r a t o r
310
(CHAP.
ELLIPTIC BOUNDARY PROBLEMS
5)
WEAKENING OF ASSUMPTION ( 2 . 1 )
6.3
By a p p r o p r i a t e choice of t h e contour
r5'
i n the equality
(2.2.3), show t h a t a l l t h e r e s u l t s of S e c t i o n 2 s t i l l hold i f assumption ( 2 . 1 ) i s r e p l a c e d by: A i s p r o p e r , and each term
of t h e complete symbol
, 5)
a.(x
j20 J
function with respect t o
of A i s a r a t i o n a l
5 i n a symmetric conic neighbourhood
( i . e . i n v a r i a n t under homothetic t r a n s f o r m a t i o n s w i t h non-zero
r e a l r a t i o ) of t h e f i b r e d space conormal t o an.
6.4
LOCAL REGULARITY I N THE NEIGHBOURHOOD OF A POINT
OF an. We adopt t h e n o t a t i o n o f S e c t i o n
E
f*
0
pu
q,(E)
E
neighbourhood of x Pu
HS' and
E
that u
with s 0
Byu
E
> rn'
8 ( n ),
E
for
H
Cl'
O
6.5
and. t h a t S
j
< sw-3
an.
f'4
yj u E
u*
with
.
,f"+
e x t e n d a b l e , such
and i n
r e p e a t i n g t h e proof o f p a r t ( i ) of Theorem u
E
- rn - A. Assume
(respectively i n
E
L e t xo
i s i n j e c t i v e for a l l
, and
( m )1 {a]
T*,
that
, f')
c(x
Suppose t h a t
4.
an)
s' 4- m
t h a t i n some we have
.
By
4 . 4 , prove t h a t
i n some neighbourhood of x
0
GLOBAL REGULARITY
We adopt t h e n o t a t i o n o f S e c t i o n bounded, and a i l bounded.
4, assuming
t o be un-
Suppose t h a t t h e d i f f e r e n t i a l
(SEC. 6 )
ADDITIONAL NOTES
operator P i s e l l i p t i c i n
BLm(R")
311
.
Show ( c f S e c t i o n 11.1,
Chapter I V ) t h a t t h i s i s e q u i v a l e n t t o t h e f o l l o w i n g c o n d i t i o n s : the c o e f f i c i e n t s o f P, t o g e t h e r w i t h a l l t h e i r d e r i v a t i v e s , a r e bounded i n IRn; moreover, t h e r e exists c > 0 such t h a t 1P,(%rl
2
c
C: f R"
for x
151 = 1
where p
m
i s t h e homogeneous p r i n c i p a l symbol o f degree m o f P .
" Suppose t h a t b i s i n j e c t i v e , and t h a t we have such t h a t S
>
m'-m-&
P u
,
E H'(E) 0 5
s
,
E H O I . (an) ~ ~
BY u
.
+m
4.4.1) i s
fying t h e above assumptions.
5 . 5 remains t r u e f o r
6.6
with
Using Theorem 4 . 4 and a
parametrix of P i n BL-mfJRn), show t h a t u p r i o r i e s t i m a t e of t h e t y p e
u f) :(aH
E
Ha(;)
and t h a t an a
v a l i d for a l l
u
satis-
Deduce t h a t p a r t ( i ) o f Theorem
P and
CONVERSE OF THE REGULUITY THEOREM
Assume t h a t t h e boundary-value problem (1.1)s a t i s f i e s p a r t ( i i )
o f t h e s t a t e m e n t o f Theorem 4 . 4 ( a p r i o r i e s t i m a t e ) .
By
applying t h e i n e q u a l i t y ( 4 . 4 . 1 ) f o r K c Cl and by u s i n g Propo s i t i o n 8.13, Chapter I V Y show t h a t P i s e l l i p t i c i n
By a p p l y i n g t h e i n e q u a l i t y where
v
E &,,(an
, (3 m
E)
,
(4.4.1) f o r
show t h a t :
;
u = (QPv)
.,
ELLIPTIC BOUNDARY PROBLEMS
312 for all
v E
JK,(an
,
.
E)
(CHAP.
Then deduce t h a t t h e o p e r a t o r
injective.
6.7
CASE OF SYSTEMS WHICH ARE ELLIPTIC I N THE SENSE OF DOUGLIS-NIRENBERG
Suppose
5
i s a manifold w i t h boundary, imbedded i n a man-
i f o l d M ( s e e Remark 9 . 6 , Chapter I ) .
J E
L:
0 Ej jt 1
I
,
F =
, where
Fi
@
Let
tne E
i= 1
j'
F. are vectorial 1
bundles above M.
Suppose we have an e l l i p t i c d i f f e r e n t i a l o p e r a t o r P:
, t) E I
(. i.e.
zJ
,'
( s e e S e c t i o n 7.6, Chapter I V ) an o p e r a t o r P o f t h e form
(pi j)i=1,. j-1,
.....,
P i,j
where
,I
Cw(M
Let
G
d 0 Gj
P
j=l
an.
Ej)
Y
* CaD(M, Fi)
J
i s a d i f f e r e n t i a l o p e r a t o r of degree t
above
x
Let
,
j
- si '
with
where G . a r e v e c t o r i a l bundles J
8 : Cm(M
e n t i a l o p e r a t o r of t y p e ( r
, E)
4
, t ) E IRd
Cm(M x
'
IRJ.
, G)
be a d i f f e r -
5)
(SEC.
6)
ADDITIONAL NOTES
313
Consider t h e boundary-value problem
I n o r d e r t o a l l o w t h i s problem t o be s t u d i e d by means of a formalism analogous t o t h a t i n Chapter V , it i s convenient t o introduce a r e a l f i e l d v t r a n s v e r s a l t o
an,
and t o p u t
, f E) ,
where
m
i s an i n t e g e r l a r g e enough t o e n s u r e t h a t t h e f o l l o w i n g two conditions a r e s a t i s f i e d :
(6.7.1.)
t h e boundary c o n d i t i o n
i n t h e form
B(w) = g W
, w i t h B = (Bo,B1
Bk : C"(m,E) * C (W,G) i s (r
...B,-i,
i s expressed where
a d i f f e r e n t i a l o p e r a t o r of t y p e
..., m-1
t-k) for k = 0,
(&)lang
(we p u t ( t - k )
j~
= t.-k for
j = ly...yJ).
(6.7.2.) y j t$
The jump formula for P does n o t i n v o l v e any terms
such t h a t j 2 m.
We can t h e r e f o r e w r i t e t h i s formula i n
the form
~(u") = (pu]
O
+
F(yu )
for
u E
cW(E , E)
,
#u
where P(yu) i s a d i s t r i b u t i o n c a r r i e d by
an
which depends o n l y
on P and YU.
It i s t h e n a simple m a t t e r t o a d a p t S e c t i o n s 3 and taking account of t h e f o l l o w i n g remarks:
4,
314
ELLIPTIC BOUNDARY PROBLEMS
(CHAP.
5)
) i s of type We t a k e a parametrix Q of P such t h a t Q = (Q j ,k
a)
( t , s ) and each o p e r a t o r c o n s t i t u t i n g Q
satisfies (2.1).
j ,k
The e x i s t e n c e of such a parametrix i s immediate: we t a k e Qo of t y p e ( t , s ) having as p r i n c i p a l symbol t h e i n v e r s e matrix of 0
( P ) ; p u t t i n g , f o r example, PQo = I
s ,t
Q
degree -1, and we t a k e
b)
-
R , we have R of
.
Q,Rk
N
We d e f i n e t h e Calder6n p r o j e c t o r C = y Q
, Qm E) * C"(W
C : C"(S
Q
m
,
E)
C
p
p s e u d o - d i f f e r e n t i a l o p e r a t o r of t y p e ( t
o,[c)
Let
[uwSt-a
s
be t h e p r i n c i p a l symbol of
C
('k,
.
F.
(%,A)
-
9
k, t
A3'k1 Iro,
E, F
U
and t h a t t h e mapping onto t h e image of
0
0
,
+.
that
yU
; 5'
... I
m-1
(E,
V)
and
i s a projector,
+
, where S+ ( X I , 5')
(C)(x', 5 ' )
,0
s,t (p)(X'
- a).
i s an isomorphism of S (x', 5 ' )
n
U
, 5')
uo(X'
space o f s o l u t i o n s U , bounded f o r x system
a
Ck,j
It can be shown, following
t r a n s p o r t by a l o c a l c h a r t t o t h e boundary of t r i v i a l i s a t i o n s of
We have
,D
i s the
2 0, of the differential
) U(x,)
= 0,
We then deduce
x"
t h a t t h e image of uo(c)(x'
5')
does not depend on t h e choice
( t o w i t h i n an isomorphism) of t h e i n t e g e r m y and a l s o ( s e e f o r example I N C E [11 ) t h a t t h e rank of a o ( C ) ( x ' , 5 ' ) i s e q u a l t o one h a l f of t h e degree i n 5, i.e. t o
3
(g J
tj
-
dim Ej
of
- j'
det
0
s,t
(P)(x',
dim F . )
J
I
0 ; 5'
, 5") ,
with, i n t h e
c a s e n=2, a r e s t r i c t i o n analogous t o t h a t i n d i c a t e d p r i o r t o t h e statement o f P r o p o s i t i o n 3.3.
(SEC. 6 )
c)
ADDITIONAL NOTES
315
Consider t h e p r i n c i p a l symbol b of B:
b =
- k (Bk))k,o
I...I
and t h e r e s t r i c t i o n
I
i; of
b
"
t o Image
0
0
Thus b i s a morphism o f Image u ( C ) i n t o G.
(C).
0
N
I t can be shown t h a t t h e p r o p e r t y of b b e i n g i n j e c t i v e o r s u r j e c t i v e depends n e i t h e r on t h e i n t e g e r m n o r on t h e f i e l d v chosen,
d)
.. lan
For j = 1,. ,J l e t m! denote t h e s m a l l e s t i n t e g e r f o r
which
bu]
m'
can be e x p r e s s e d by means o f t h e t r a c e s
..
j = 1,. ,J and 0
y u . such t h a t
k J assumption
J
-4
-
-
Under t h e
1.
,...,I
,...,J ,
and j = 1
4 . 1 t o t h e c a s e of t h e o p e r a t o r
g e n e r a l i s e Theorem
Suppose t h a t m! J
- si
m' j for i = 1 S k S
4
< t
j
f o r j = 1,...,J.
E
Hs,(E,
If
F)
u E B 1 ( n I E)
,
i s extendable and s a t i s f i e s
P u
(Bu) lan i s meaningful.
p S 0, consider t h e operator
For
show t h a t
316
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
G e n e r a l i s e Theorem 4 . 4 s i m i l a r l y ( i n p a r t i c u l a r , show t h a t i f
u
f)
Ef(n
, E)
is e x t e n d a b l e and s a t i s f i e s
Give p r o p e r t i e s analogous t o t h o s e e s t a b l i s h e d i n Exercises
6.4, 6.5, 6.6.
6.8
THE OBLIQUE DERIVATIVE PROBLEM
We c o n s i d e r a r e g u l a r open domain 51 of
We p r e s c r i b e on ail a r e a l
N t h e inward u n i t normal t o 351.
b v ( X ' )
vector f i e l d
XI
v ( x ~ )f 0
f o r a l l x'
E
n
ER
,
an.
We p u t
+ X(K1)N , where
v(x*) = ~ ( x ' )
dx')
t i a l component o f v ( x ' ) and A(x') of v ( x ' ) .
IRn , and we denote by
E
cu
of c l a s s C
E Txl(&)
,
such t h a t
i s t h e tangen-
IR i s t h e normal component
We s t u d y t h e o b l i q u e d e r i v a t i v e problem
6)
(SEC.
ADDITIONAL NOTES
Let xh
a)
XI+
=
y'
E
9
Y,)
Suppose we have a l o c a l c h a r t from
x(X')
x(xh) = 0. Y = (Y'
an.
317
an
t o t h e neighbourhood o f xh, w i t h
Show t h a t t h e mapping (--t
x =
X'
+ y,
"x')
where x' =
x-1 y ' ) ,
i s a diffeomorphism of a neighbourhood of 0 i n t o I R n , and t h a t i t s inverse i s a l o c a l chart into
A ay"
i n t o xh t r a n s p o r t i n g N
from
Show t h a t t h e p r i n c i p a l symbols of t h e t r a n s p o r t -
a t i o n s under
of t h e o p e r a t o r s A and
respectively
1tX'(X')q'12
T' >
+
X(x')
.
\
and
are
< X'(X') T[x')
Show t h a t t h e L o p a t i n s k i d e t e r m i n a n t of
(6.8.1) i n t h e chart
b)
+{
i a v
i s equal t o
Deduce from t h i s t h a t t h e boundary-value problem (6.8.1)
is a l v a y s e l l i p t i c when n = 2 , and t h a t for n
2
3 it i s
e l l i p t i c i f and o n l y i f t h e v e c t o r v ( x f ) i s nowhere t a n g e n t i a l to
c)
an.
i s bounded; l e t u = Kh denote t h e unique
Suppose t h a t
s o l u t i o n o f t h e D i r i c h l e t problem
Show t h a t
u
i s a s o l u t i o n of t h e oblique d e r i v a t i v e
318
(CHAP. 5 )
ELLIPTIC BOUNDARY PROBLEMS
problem
i f and only i f
T
where
=
1i A av
O
K
i s a ( s c a l a r ) p . d . 0 . of degree 1 i n
an, of which t h e p r i n c i p a l symbol, following t r a n s p o r t by a l o c a l chart
an, i s
I n v e s t i g a t e t h e problem ( 6 . 8 . 1 ) i n t h e case n = 2 ,
d)
Q = ' ~ =z x p
of
xl,&_yv
E
+ iy E
\z\< I 2
2
( i d e n t i f y IR
and C ) .
1
v[z> =
zp f o r z
E
an, where
More p r e c i s e l y , c o n s i d e r t h e
d e f i n e d by
Suppose we have
u E
C"(3)
~ = b ; = ~ () 2. - i 3U
'g i f and only i f we have
with r e a l values.
Put
Show t h a t u i s i n t h e k e r n e l of
(SEC. 6 )
319
ADDITIONAL NOTES
u E cmc2 u
holonorphic i n
R e (zPU(z)) = 0
Deduce from t h i s t h a t
dim Ker p = 2
+ 2i
zP~(z) = a-P
<1
,
+-
saj
h = 0 , and
r e a l values.
G) ; E( f i
-
2p
be t h e
Show t h a t
+
a.z* J
Im
h(z)
121
< 1, h ( 0 ) = 0.
u i s i n t h e k e r n e l of p i f and o n l y i f
i s p u r e l y imaginary.
-
J
w i t h h holomorphic f o r
Deduce from t h i s t h a t
dim Ker p = 2
when p > 0 ,
when p = 0 .
e n t i r e - s e r i e s e x m n s i o n of U.
121
= 1
d i m Ker p = 1
I n t h e c a s e p < 0 , l e t U( z ) =
for 0 <
1.
for
when p < 0.
Conclude t h a t
Now l e t
g
E C”(W)
with
We know t h a t t h e r e e x i s t s a unique f u n c t i o n holomorphic i n
with G(0) E W ( i n f a c t , i f
g(eie)
such t h a t Re G = g i n
=
+-
ane in’ n = - o
an,
is the
F o u r i e r s e r i e s expansion of g , we have
Show t h a t t h e c o n d i t i o n s Au = 0 i n
an a r e e q u i v a l e n t t o
a,
3V
= g
in
ELLIPTIC BOUNDARY PROBLEMS
320
( CH AP . 5 )
u E cmm Re (z'U(Z) Deduce from t h i s t h a t
p
-
~ G ( z ) )P 0
zpU(z) = 2G(z) f o r IzI < 1
g
P
empty)
Let P1,
= 1 ,
When p > 0,
U holomorphic i n R ,
Deduce from t h i s t h a t t h e image 1.
Conclude t h a t t h e index of t h e
i s i n a l l cases equal t o 2 ( 1
Suppose Rl, R
=
-
121
-
p)
.
ELLIPTIC BOUNDARY-VALUE PROBLEMS WITH INTERFACES:
6.9
an,
.
i s o f codimension 2p
operator
for
i s s u r j e c t i v e when p 5 0.
show t h a t ( 6 . 8 . 2 ) i s e q u i v a l e n t t o :
of
n
U holomorphic i n
(6.82)
r ,
2
are two r e g u l a r open domains i n IRn such t h a t
3% =
r u r' , w i t h r , r '
d i s j o i n t (and
r'
possibly
.
P
degrees m
2
be two e l l i p t i c d i f f e r e n t i a l o p e r a t o r s i n I R n
m2 r e s p e c t i v e l y .
, of
Suppose t h a t i n t h e neighbourhood
o f 8 R 2 t h e r e is a f i e l d of v e c t o r s v t r a n s v e r s a l t o
an2.
For
(SEC.
6)
321
ADDITIONAL NOTES
with
Cons i d e r t h e b o m dary-value problem
1
PIUl
=
fl
in
nl
P2u2
=
f2
in
%
B(l)y(l)u1 B'y'u2
=
+ g'
B(2)y(2)u2
in
r'
g
in
r
.
We adopt t h e f o l l o w i n g assumption w i t h r e g a r d t o t h e degrees
of t h e o p e r a t o r s :
a)
Suppose we have, f o r R = 1,2, a p a r a m e t r i x Q o f P , R R'
We
(CHAP. 5 )
ELLIPTIC BOUNDmY PROBLEMS
322
c o n s i d e r t h e t h r e e Caldero'n p r o j e c t o r s C ( l ) , C ( 2 ) ,
C ' defined
by
Let cl, c 2 , c ' denote t h e i r p r i n c i p a l symbols, and l e t bl, b 2 , b '
B'.
denote t h e p r i n c i p a l symbols of B ( l ) ,
By a g a i n u s i n g
t h e method o f r e d u c t i o n t o t h e boundary, we a r e l e d t o c o n s i d e r i n g t h e morphism
Show t h a t Theorem 4 . 1 remains v a l i d when
b)
Cm&)
by
(3 Ce&)
s
cm(an
and t h e assumption t h a t R bounded.
s
$1
by
Cm(E)is
replaced
C"(T,dL) t4 Cm(%C?'),
i s bounded i s r e p l a c e d by:
al,
Q2
[ I n p a r t i c u l a r , it may be noted t h a t t h e m a t r i x of
operators ( B ( l ) C ( l ) ,
B(2)C(2)) is right-elliptic
i n t h e sense of
Douglis-Nirenberg when i t s rows a r e given t h e i n d i c e s
ml
- dil)
=
%
- d[.')J
, the
(j = 0,0.01p-11
B(l) C ( l ) a r e g i v e n t h e i n d i c e s ml
-
columns of
k (k = 0 , .
. ., m1 - 1) and ~
t h e columns o f B ( 2 ) C ( 2 ) t h e i n d i c e s m2
C)
for
- k (k =
m i denote t h e s m a l l e s t i n t e g e r 5 m such k 2 . 4 , j = 0,0..,pI . We d e f i n e m i Let
O,..,,
m
2
- 1)l.
t h a t B(l) = 0
j ,k
and m' s i m i l a r l y .
(SEC.
6)
Suppose
323
ADDITIONAL NOTES
s1
, u1 , S2 , o2
ER
a r e such t h a t sl
Adapt Theorem 4 .3 a s b e f o r e .
Adapt Theorem
of Douglis-Nirenberg,
- -3 ml
Replace
4 . 4 i n similar f a s h i o n .
t h e i n j e c t i v i t y of
> mi
It may b e n o t e d t h a t
implies t h e l e f t - e l l i p t i c i t y ,
i n t h e sense
of t h e matrix o f o p e r a t o r s
when t h e f i r s t b l o c k o f columns i s g i v e n t h e i n d i e
ml
-
% ml
"2 ml d)
k(kd,o.,,m,=l~
t h e second b l o c k o f columns
- k[ktO,...,~-l) , nil-?), j (J=OV...,
-3 - ii("P
(j=O,..,,~-l)
d
-dc2)
' 9 3
t h e f i r s t b l o c k of rows t h e i n d i c e s t h e second t h e i n d i c e s
,and t h e
t h i r d t h e indices
(~O,-O.,P-~)
Give p r o p e r t i e s analogous t o t h o s e e s t a b l i s h e d i n E x e r c i s e s
,
ELLIPTIC BOUNDARY PROBLEMS
324
6.4, 6.5, 6.6.
(CHAP. 5 )
Generalise everything done earlier in the con-
text of manifolds with boundary,
bundles,
and systems which
are elliptic in the sense of Douglis-Nirenberg (see Exercise 6.7)
e)
Consider problem (6.9.1)in the case
r 1 4 N
Uz(X1
,
Put u
$1
*
uz(X'9
I
(ul
,
N
u2)
, with
By this means reduce the problem
to one which is elliptic in the sense of Douglis-Nirenberg (see Exercise 6.7) for u in the open domain
all'
CHAPTER 6
EVOLUTION EQUATIONS
1,
THE CAUCHY-KOVALESKI AND HOLMGREN -THEOREMS
1.1
INTRODUCTION:
Suppose P(x, D ) i s a d i f f e r e n t i a l X
W
o p e r a t o r of degree
m
with C
coefficients i n IRn.
One n a t u r a l i d e a for a t t e m p t i n g t o f i n d a s o l u t i o n
u
in
t h e neighbourhood o f 0 o f t h e Cauchy problem (1.I .I
.I
P(x, Dx]u
=
f(X)
c o n s i s t s of c a l c u l a t i n g t h e d e r i v a t i v e s ~ ( ~ ' ( from 0 ) the equations. CI
If an < m y we immediately o b t a i n t h e u (0)by u s i n g ( 1 . 1 . 2 ) .
Then, t o c a l c u l a t e :D
u ( 0 ) from ( l . l . l )we , assume t h a t t h i s
equation can be w r i t t e n i n t h e form
32 5
(CHAP. 6 )
EVOLUTION EQUATIONS
326
f
a,(O)
with
.
0
I n t h i s c a s e , it can be shown t h a t
t h e r e e x i s t s a unique formal s o l u t i o n of (1.1.1)and ( 1 . 1 . 2 ) ; furthermore, when t h e d a t a aa, f , gj a r e a n a l y t i c i n a neighbow-
u i s i t s e l f analytic.
hood of 0 , t h e s o l u t i o n
W e now give a somewhat more g e n e r a l formulation of t h i s result. n
THEOREM 1 . 2 :
~ ( a )=
suppose
cj aj i s a linear j-1
form on IRn with c o e f f i c i e n t s c
j
Consider an equation of
2 0.
the form
(1 2.1
.I
=
DY u
aQ(x) D'
u
+
f(x)
(Y
with aa =
o
i f
1.1
>
Iyl
or
~ ( a )2 ~ ( y )
.
we impose the following supplementary conditions:
Then, given
f,
a,&' 8
a n a l y t i o i n the neighbow-
hood of 0 , there e x i s t s a unique f u n c t i o n u which i s analytic i n the neighbowhood of 0 and which s a t i s f i e s (1.2.1)and (1.2.2).
For example, i f we t a k e ! 2 ( a ) = an and
,..., 0, m)
Y = (0
, we
o b t a i n t h e Cauchy problem (1.1.3), ( 1 . 1 . 2 ) .
PROOF OF UNIQUENESS:
Suppose t h a t
cp i s e q u a l t o z e r o ;
then
(SEC. 1)
THEOREMS
327
condition (1.2.2)i m p l i e s :
'D
(1.2.3.)
=
u(0)
0 f o r any multi-index
j
p j < yj f o r a t l e a s t one index
u
such t h a t
.
For i n d i c e s o f t h e form y + 6 w i t h 6 2 0 , equation (1.2.1) and L e i b n i z ' s formula give
b
(1,2.4.~Dy+b ~ ( 0 ) ~ )
" If
f
q(B )De
cu+'u(0) *
aty(0) 0
+ 'D
f(0)
W
.
i s z e r o , it can be shown by r e c u r r e n c e over t h e s e t of
*+'D
numbers l l ( y + 6 ) t h a t
u(0)
=
o
for a l l 6
2 0.
In f a c t ,
t h e m u l t i - i n d i c e s which appear i n t h e right-hand s i d e of ( 1 . 2 . 4 ) s a t i s f y t h e r e c u r r e n c e assumption s i n c e
In o r d e r t o prove t h e e x i s t e n c e , we d e f i n e s e r i e s expansion about t h e o r i g i n .
of
, so
Cp = 0
we o b t a i n t h e case
u
by i t s Taylor
Replacing
u
by u
- cp
,
t h a t we f o r c e t h e d e r i v a t i v e s
u t o s a t i s f y (1.2.3).
Next, we s u c c e s s i v e l y determine t h e d e r i v a t i v e s of o r d e r g r e a t e r t h a n y by imposing t h e e q u a l i t y ( 1 . 2 . 4 ) . prove t h e a n a l y t i c i t y o f
u
i n t h e neighbourhood o f 0 , we s h a l l
show t h a t t h e r e e x i s t c o n s t a n t s Mo
M
(M,,
..., Mn)
E
I n order t o
such t h a t
E
IR
+
and'
328
EVOLUTION EQUATIONS
(1.2.6.)
ID"
u(O)] ff
where
Mff = MI
1
x
I
...
Iff]!
Mo Mff ff
x Mn
n
(CHAP.
f o r any multi-index
2
6) 0,
-
We know, by h y p o t h e s i s , t h a t t h e r e e x i s t c o n s t a n t s L and Lo such t h a t
By means o f a s u i t a b l e choice of t h e M by r e c u r r e n c e over t h e numbers k(a).
f o r a l l t h e a = y+p such t h a t
it i s s a t i s f i e d f o r
c1
j'
we prove (1.2.6)
We assume t h a t it i s t r u e
A(yS1.1) C k?(y+6)
= y+& ,
and we show t h a t
S u b s t i t u t i n g t h e upper bounds
(1.2.7)i n t o ( 1 . 2 . 4 ) we o b t a i n , t a k i n g account o f ( 1 . 2 . 5 ) :
( :) ( i: )
The combinatorial i n t e r p r e t a t i o n o f t h e binomial c o e f f i c i e n t s shows t h a t
5
( ) 'w
5
imp1ies t h a t
and consequently
1
.
Using t h i s i n e q u a l i t y , ( 1 . 2 . 8 )
(SEC. 1)
32 9
THEOREMS
Without r e d u c i n g g e n e r a l i t y , we can assume t h a t L 2 1 ; we p u t
M
j
= 2 L Aj '
j = 1,.
..,
n
w i t h A 2 1.
Then t h e b r a c k e t i n
t h e right-hand s i d e of ( 1 . 2 . 9 ) i s bounded above by t h e q u a n t i t y
F.
L.2"
LO + -MO
j*
It i s s u f f i c i e n t t o t a k e A and Mo s u f f i c i e n t l y l a r g e t h a t t h e quantity (1.2.10) i s smaller than u n i t y ;
t h i s t h e n a l l o w s us
t o argue by r e c u r r e n c e and cdncludes t h e proof of t h e thoerem.
REMARK 1 . 3 :
The statemerit and t h e proof of Theorem 1 . 2
remain v a l i d word f o r word when t h e c o e f f i c i e n t s a (x) a r e a N x N m a t r i c e s and t h e f u n c t i o n s
REMARK
1.4:
cp
, f,
u
have v a l u e s i n C".
Expression (1.2.10)shows t h a t t h e r a d i u s o f
convergence of t h e Taylor s e r i e s expansion f o r
u
i s bounded
below by a number > 0 which depends o n l y on t h e c o n s t a n t L a p p e a r i n g i n t h e upper bounds (1,2.7),i . e . on t h e s m a l l e s t r a d i u s o f convergence p o f t h e Taylor s e r i e s expansions about 0 of t h e f u n c t i o n s a a , f ,
cp
and on t h e
SUP
f E C" 121 IP/2
lacr(.1I
330
(CHAP.
EVOLUTION EQUATIONS
6)
As a special case of this theoren we deduce:
THEOREM 1.5 (Cauchy-Kovaleski):
Suppose P(x, Dx) i s an
operator with c o e f f i c i e n t s anaZytic i n some neighbourhood of a point xo
E
IRn.
We take f u n c t i o n s f
in the neighbourhood of xo.
cp
$ which are anaZytic
Suppose tha-b $ i s rea2-valued and
satisfies
We denote by S the hypersurface defined i n the neighbourhood of
$[XI = $[xo)
0
x by the equation
,
Xhen i n some neighbow-
ho'od of xo there e x i s t s a unique a n a l y t i c soZution of the Cauchy prob Zem
PROOF:
Suppose for example that
x ( x o )
0
;
then the
axn change of variables
(1.5.3.1
Yn =
$(XI
Yj = x
- .O(x0)
- xoj j
j
o
1,
..., n-I
0
is an analytic diffeomorphism of a neighbourhood of x onto a neighbourhood of y = 0; it transforms problem (1.5.2) into
331
THEOREMS
(SEC. 1)
Formula (7.3.5) i n Chapter I f o r t h e t r a n s f o r m a t i o n o f t h e
..., 0,
N
p r i n c i p a l p a r t s shows t h a t
P,[O,
f 0
1)
e n t l y w e can i s o l a t e t h e t e r m i n Dm i n ( 1 . 5 . b ) ,
n
.
Consequ-
and we t h e r e f o r e
come down t o t h e p a r t i c u l a r c a s e of t h e example given a f t e r t h e statement of Theorem 1 . 2 .
REMARK
1.6:
I n t h e c a s e where P i s an N x N system,
assumption (1.5.1)has t o be r e p l a c e d by t h e c o n d i t i o n
f
d e t Pm(XO' d $ ( X 0 l )
0 *
This assumption l e a d s us t o i n t r o d u c e :
DEFINITION 1 . 7 :
A hypersurface S i n IRn
, of
c l a s s C1,
is
said t o be c h a r a c t e r i s t i c a t xo for the operator P(x, Dx) of degree
m
i f
Pm(xo, d$(xo))
0
=
, where
$(x) = 0 i s an
equation of S i n the neighbowhood of xo. (P i s an N x N system, t h i s condition must obviously be replaced b y d e t Pm(xoI dr)(xo))
=
0 )
.
For example, t h e hyperplane xn = 0 i s c h a r a c t e r i s t i c a t 0 i f and only i f t h e c o e f f i c i e n t of DZ i n P ( x , Dx) v a n i s h e s a t 0.
REMARK 1.8:
I n t h e c a s e o f o p e r a t o r s P(D) w i t h c o n s t a n t
c o e f f i c i e n t s , t h e uniqueness of t h e s o l u t i o n of t h e Cauchy
332
(CHAP.
EVOLUTION EQUATIONS
6)
problem p(oju (I .I .?,
=
o
1 j = OJ...,
j xn=0 = 0
rn-I
m
with
=
degree of P(D)
i s d i r e c t l y r e l a t e d t o t h e f a c t t h a t t h e hyperplane xn = 0 i s noncharacteristic. I n f a c t , suppose t h a t
P,(O,
homogeneous of degree
m
..., , the
0, 1)
-
function
zero a n a l y t i c s o l u t i o n of problem ( 1 . 1 . 7 ) .
0
.
Then, i f P i s
u(x) = xn nH1
i s a non-
More g e n e r a l l y , i f
we drop t h e assumption of homogeneity, we can f i n d i n HORMANDER m
El], Chapter V , t h e c o n s t r u c t i o n of a non-zero C
s o l u t i o n of
(1.1.7).
Nonetheless, i n c e r t a i n c a s e s t h e r e i s a s u b s t i t u t e theorem f o r l o c a l e x i s t e n c e and uniqueness w i t h r e s p e c t t o a c h a r a c t e r i s t i c surface;
t h i s i s as f o l l o w s :
THEOREM 1 . 9 (Goursat-Beudon):
with the conditions
Consider the equation
333
THEOREMS
(SEC. 1)
Suppose t h a t the functions aa, f , cp
are a n a l y t i c in the neigh-
bourhood of 0 ; then there e x i s t s a unique s o l u t i o n u o f (1.9.1), ( 1 , 9 . 2 ) , which is a n a l y t i c
in the neighbourhood of 0 .
To prove t h i s , it i s s u f f i c i e n t t o apply Theorem 1 . 2 with y = (I,
0,
..., 0,
-1)
~ ( a =) a,,
and
+m
cyl
There i s , i n f a c t , no need t o assume t h e a n a l y t i c i t y of t h e s o l u t ion i n o r d e r t o have l o c a l uniqueness i n t h e Cauchy problem (1.5.2).
We have:
THEOREM 1.10 (Holmgren):
operator of degree hood V of xo.
Suppose P ( x , Dx) is a d i f f e r e n t i a l
with c o e f f i c i e n t s anuZytic in a neighbour-
m
Suppose t h a t
S
is a hypersurface containing xo We put
and t h a t S is defined by an a n a l y t i c equation $ ( x ) = 0.
v+
=
{
X
E V
1 '#(XI 2 0
istic for P a t xo.
E
s is not character-
which s a t i s f i e s
C?(V')
P u
=
0
in
V+
DCYujS
=
0
in
vns
.)
is necessarily equal t o zero i n
PROOF:
Suppose t h a t
Then there e x i s t s a neighbourhood W of xo
such t h a t any f u n c t i o n u
(1.10.1
.
}
W+ =
{x E w I
]CY/SWI
2
o
}.
To prove t h i s uniqueness r e s u l t we proceed by d u a l i t y ,
using t h e e x i s t e n c e of a s o l u t i o n of a t r a n s p o s e d problem.
After
c a r r y i n g out t h e a n a l y t i c change of v a r i a b l e s ( 1 . 5 . 3 ) , we can
334
( c w . 6)
EVOLUTION EQUATIONS
assume t h a t S i s d e f i n e d by t h e e q u a t i o n y
n = 0.
Next, we
a p p l y t h e a n a l y t i c change o f v a r i a b l e s
(1.10.2.) XJ
=
j = 1,
Yj
which t r a n s f o r m s t h e hyperplane y,
2 x = x n 1
with e q u a t i o n
P+
=
-
]
(x
xn
2
,..
+ XI2
+
xkl.2
P
We p u t
xn-l
... +
n-1
= 0 i n t o t h e paraboloid
2 +
...*
]
i?+
=
P+
n 7,
I
where V corresponds t o t h e open s e t V a f t e r t h e change of
.. v a r i a b l e s , and. we denote by P t h e o p e r a t o r o b t a i n e d by t r a n s p o r t of
w
.
P
We now have t o show t h a t t h e r e e x i s t s a neighbourhood
of 0 such t h a t i f
E
Cm(h+) s a t i s f i e s
(1.10.3.)
Suppose r > 0 ;
we d e f i n e t h e s e t s
~ ~ = { x ~ ~ ~ ~ I m fa i x } ; j < x = ~{ xl ~ ~ + i x ~ I r } ; j f o r r s u f f i c i e n t l y small, we have t h e i n c l u s i o n s
p;CB r C V . The Cauchy-Kovaleski Theorem 1 . 5 and Remark 1 . 4 show t h a t
it i s p o s s i b l e t o f i n d an r 0 s u f f i c i e n t l y small t h a t f o r any h
E ]
0, ro
[
Cauchy problem
and for any polynomial
q(xl,,.,,
x
n-1
1
the
(SEC. 1)
(1,10.4.)
THEOREMS
?(x,
DX)v
=
0
admits an a n a l y t i c s o l u t i o n
v
335
in
Br
i n Br.
X
T
L
we a p p l y Green's formula i n t h e compact domai
n
6;:
I
then t a k i n g account of (1.10.3)and (l.lO.b)y we o b t a i n
where C ( x ' ) i s a f u n c t i o n which depends o n l y on t h e c o e f f i c i e n t s of P and which does n o t v a n i s h .
Equations (1.10.3) and (1.lO.b) imply t h a t t h e l e f t - h a n d s i d e of (1.10.5)i s e q u a l t o z e r o ; for
lx']
4 r h
since
q
consequently u ( x ' , h ) = 0
i s an a r b i t r a r y polynomial;
this
, It i s t h e r e f o r e s u f f i c i e n t t o t a k e
shows t h a t u = 0 i n '.6 rO
W = B r . 0
REMARK 1.11:
The d e t e r m i n a t i o n of ro shows t h a t t h e open
s e t W depends o n l y on t h e open s e t Y , on t h e r a d i u s of converg-
336
(CHAP. 6 )
EVOLUTION EQUATIONS
ence of t h e Taylor s e r i e s expansions a t x
0
of t h e c o e f f i c i e n t s
of P and o f J, and on t h e modulus of t h e c o e f f i c i e n t s as i n Remark 1 . 4 .
REMARK 1 . 1 2 :
I n a l l t h e c a s e s f o r which t h e r e i s an e x i s t -
ence theorem f o r problem (1.10.4),t h e above proof shows t h a t t h e r e i s a l s o a uniqueness theorem (1.10.1).
I n t h e c a s e where P has c o n s t a n t c o e f f i c i e n t s and S i s a Theorem 1.10 and Remark 1 . 8 show t h a t i n
hyperplane,Holmgren's
o r d e r t o have l o c a l uniqueness, it i s n e c e s s a r y and s u f f i c i e n t t h a t S be n o n - c h a r a c t e r i s t i c .
We t h e n a l s o have g l o b a l unique-
ness :
Suppose P(D) is a d i f f e r e n t i a 2 operator
PROPOSITION 1.13:
with constant c o e f f i c i e n t s ;
is qon-characteristic for P. and supp u c
Pu = 0 i n
PROOF: I ) with
suppose t h a t the hype2rpZane xn = 0
Let
8" +
.
u
E .P(R"]
be such t h a t
Then u = 0.
Let p k be a r e g u l a r i s i n g sequence ( s e e (l.l.l), Chapter
p
,
E):R(:C -n
w i t h support i n IR,,
u
The f u n c t i o n
and s a t i s f i e s
P
\
k
= u
*
= 0 in
Pk
.
co
is C
,
A s P has
c o n s t a n t c o e f f i c i e n t s , Holmgren's Theorem 1.10 and Remark 1.11 imply t h a t
\
= 0 and consequently t h a t
u
=
lim k m
uk
=
0
,
337
NECESSARY C O N D I T I O N FOR WELL-POSEDNESS
(SEC, 2 )
NECESSARY C O N D I T I O N FOR THE CAUCHY PROBLEM TO BE WELL POSED
2.
The Cauchy-Kovaleski and Holmgren theorems a r e e s s e n t i a l l y l o c a l r e s u l t s ; i n t h i s s e c t i o n we give a necessary c o n d i t i o n f o r the Cauchy problem t o be w e l l posed g l o b a l l y .
x = [xo,
Let
XI
,...,
xn)
;
E R"+'
we p u t x = (xo, X I ) .
We denote by P ( x , D ) a d i f f e r e n t i a l o p e r a t o r of degree m w i t h X
coefficients i n C ~ ( I R ~ + ' ) .
DEFINITION 2 . 1 :
Let T E D ; we say t h a t th e Cauchy prob-
lem i s we ll posed for P i n t h e half-space xo
,
f E C"(R"+')
a wique u
.I
(2.1 .I
E
gj
E Cm[R")
j
P
0,
..,
m-I
2
T, i f f o r any
,
there e r k t s
Crn(Dn'l) such t h a t ~
u
=
f or
x
for
x
0
0
> T
,
= T
and
O;U=
g3
j = 0,
...,
m-I
.
we say t h a t t h e Cauchy probZern i s w e ll posed i n t h e d irection x
2
N = (1,O
..., 0)
if it i s we12 posed in each h a lf space
T.
It i s p o s s i b l e t o o b t a i n another formulation by r e p l a c i n g t h e i n i t i a l c o n d i t i o n s by a c o n d i t i o n on t h e s u p p o r t . s i m p l i f y t h e n o t a t i o n we t a k e T = 0 .
To
338
(CHAP. 6)
EVOLUTION EQUATIONS
Suppose t h a t the hyperplane x 0 = 0 i s
PROPOSITION 2.2 :
noncharacteristic for P.
Then the Cauchy probZem i s well posed
for P i n the half space x 0
2
f o E C”(Rw’)
with
e x i s t s a unique u
0
(2.2 2.)
supp f o c
E C”(R”+’)
P u*
PROOF:
0 i f and only i f for any
-’R+
= (x
1
x
0
2
o
}
there
c
y’.
such t h a t =
and
fo
supp uo
The proof of necessity is immediate, since all the
derivatives of the solution u of (2.1.1)with g = 0, j m-1 are equal to zero on the noncharacteristic j = 0,
...,
hyperplane x0 = 0.
Conversely, Borel’s Theorem allows us to construct a function v E C”(A”+’)
D’ v = 0
g
j
of which the traces on x0 = 0 satisfy
for j = O,..., m-1 and f = Pv 0
-
f flat on xo = 0 .
Then if uo is the solution of (2.2.2), the function uo + v is a solution of (2.1.1).
The problem of giving a general characterisation of operators for which the Cauchy problem is well posed remains open; nonetheless, a large number of partial results exist.
We give
below a necessary condition on the principal part of P.
THEOREM 2.3 : (Lax-Mizohata)
Suppose t h a t for any T
E
IR,
NECESSARY C O N D I T I O N FOR WELL-POSEDNESS
(SEC. 2)
339
the hyperplane xo = T i s noncharacteristic f o r P , and t h a t the
. . , 0).
Cauchy probZem i s well posed i n the d i r e c t i o n N = (1,0,. We t h e n have t h e i m p l i c a t i o n
We s t a r t by g i v i n g t h e f o l l o w i n g lemma:
n+l
Suppose K i s a compact subset of lR
LEMMA 2 . 4 :
; then
there e x i s t s a number C and an i n t e g e r k 2 0 such t h a t
.I
(2.4.1
lul,
with
=
‘v13
5
c IPul,
I
IDcy
v(x)I
sup
for a l l
u E
cK OD
la1 5 j x 5 0 0
m
PROOF:
L e t f E CK ; t h e S e e l e y e x t e n s i o n ( P r o p o s i t i o n
9.2, Chapter I ) of
C;(R”+’)
f- = flRn+,
-
which w e denote by Sf-.
t o B n + l i s a function i n
Suppose V i s a compact neigh-
bourhood of K ; we can f o r c e Sf- t o have s u p p o r t i n V f o r any f . The c o n t i n u i t y of t h e Seeley e x t e n s i o n shows t h a t f o r any int e g e r k t 0 t h e r e e x i s t C and q such t h a t
I~fl,
(2.4.2.)
For u
m
E
5
c 1f-I;
,
aD
f o r any
f
E cK
C K , we p u t f = Pu, g = Sf- and we d e f i n e , w-ith
.
EVOLUTION EQUATIONS
340 E
(CHAP.
6)
> 0, t h e function
fe
ee
=
f
+
-
(I
E
ee)g
c;
9
X
Oe(x) = e(f)
where x
0
9 E Cm(R)
and
t 1 and e q u a l t o u n i t y f o r xo
Let uE E Cm(TRn+’) =
P U
(where
,
f
E
E
inf
=
a
i s e q u a l t o zero f o r
0.
be t h e unique s o l u t i o n of
c
suppue
lxD.a}
(x
=
H
a
.
xg)
X E V
Since f
n+l = f onB, we have u
E
E
= u onIRy+l and
(2.4.3.)
By h y p o t h e s i s , P i s a continuous b i j e c t i o n of t h e Fr6chet space E
{u I
=
u E
c”(R”+’),
supp u
c H~
1
; consequently
t h e r e e x i s t C , an i n t e g e r k and a compact s e t K’ such t h a t
(2.4.4.)
CY
1u(x)1 S
sup
1.1
xEK
C
( D Pu(x)i
,
f o r any
u E E,
5 k x E K’
Applying ( 2 . 4 . 4 ) t o uE and t a k i n g account of ( 2 . 4 . 3 )
, we
obtain
]uI,
(2.4.5.1
I
c
Furthermore, we have function
(7
c
- g)
ee(f
-
m
Ifelk
,
Ifelk
-
for any u
Igik
E cK ,
e > 0.
since the e‘o g) has support i n t h e s t r i p
(SEC. 2 )
0 5
x0
341
NECESSARY C O N D I T I O N FOR WELL-POSEDNESS
is flat for x
5 E and
0
= 0.
We t h e r e f o r e deduce from
(2.4.5) that
14;
c
5
Id,
8
and t a k i n g account of ( 2 . 4 . 2 ) we f i n a l l y o b t a i n
I n o r d e r t o prove t h e theorem, we c o n s i d e r t h e p o i n t x ? 0 and we use a method adapted from I V R I I HORMANDER
-
and
PETKOV [ll
C71.
Let Co
E
C be such t h a t
If I m 5 ,
We denote i t s m u l t i p l i c i t y by r .
#
0 , we can
assume for example t h a t I m 5, < 0 , s o t h a t we n e c e s s a r i l y have
5 ' # 0.
The v e c t o r
plane x
0
(Re Go,
5')
= 0 ; hence we can apply a l i n e a r change of v a r i a b l e s
which p r e s e r v e s t h e v a r i a b l e x
0
into (0,
..., 0 ,
5 ' = en = ( 0 , (2.4.6.)
i s t r a n s v e r s a l t o t h e hyper-
1).
..., 0 , P,,,(o;
and which transforms (Re
We t h e r e b y reduce t o 5 , =
5 0' 5 ' )
- i,
1) and we have
D,,
0
,..., D,,)
where Q i s homogeneous of degree m
=
-
[ D + ~ i 0")'
Q ( D ~ , D,,]
r and Q ( - i , 1) # 0.
I n o r d e r t o l o c a l i s e t h e o p e r a t o r a t x = 0 and t o g e t r i d
o f terms o f lower o r d e r , I v r i i and Petkov i n t r o d u c e t h e change
(CHAP. 6)
EVOLUTION EQUATIONS
342
of v a r i a b l e s Yj
j = O
= x j P' j
,.,., n
,
with
p > O
,
sj>0
,
The o p e r a t o r P(x, D ) is transformed i n t o X
The i n e q u a l i t y ( 2 . 4 . 1 ) i m p l i e s t h a t f o r any compact s e t
K there exists
po
1
where S = max s
j
such t h a t
j'
The i d e a behind t h e proof c o n s i s t s of c o n s t r u c t i n g an asymptotic s o l u t i o n u
P
of
pP "p
=
o(p- ")
i n a neighbourhood
of 0 s o as t o c o n t r a d i c t t h e i n e q u a l i t y ( 2 . 4 . 7 ) .
s = 2r
3
if
j = I,..., n-I
and
so = sn = 4 r
then
We seek an asymptotic s o l u t i o n i n t h e form
w i t h v o ( 0 ) = 1.
We p u t
;
(SEC , 2 )
343
NECESSARY C O N D I T I O N FOR WELL-POSEDNESS
Expression ( 2 . 4 . 6 ) l e a d s u s t o t a k e f o r t h e phase f u n c t i o n cp(y) =
(o0 + ID,^
-
- iyo
+
, since
y,
O0cp, on v'l
0
Cp
this satisfies
f
=
; moreover
0
With t h i s choice o f c p t h e asymptotic expansion of P
-
s t a r t s w i t h a term of degree m
q(Do
l e a d s t o t h e equation sufficient t o take v
E
Cm
O
of 0.
+
the
P
r, and s e t t i n g t h i s t o zero
c On)
r
vo(y)
=
0
*It is
e q u a l t o u n i t y i n t h e neighbourhood
K1
The equations which f o l l o w a r e of t h e form
r
q ( D o + i On)
(2.4.8.) where F
j
E
=
Vj(Y)
K1 i s determined from t h e v o ,
Cm
Fj(Y)
. . , vj-1'
We s o l v e ( 2 . 4 . 8 ) i n t h e neighbourhood of 0 by convolution with t h e fundamental s o l u t i o n ( 6 . 1 . 3 ) of Chapter I of t h e Cauchy-Riemann o p e r a t o r
i(D 2
0
+ i On)
.
We can t h u s f i n d an
i n t e g e r J such t h a t t h e p a r t i a l sum
J
J
juo
satisfies
f o r p s u f f i c i e n t l y l a r g e ; t h i s c o n t r a d i c t s ( 2 . 4 . 7 ) and concludes t h e proof of t h e theorem.
REMARK 2.5
:
A completely analogous theorem e x i s t s f o r
344
EVOLUTION EQUATIONS
(CHAP. 6 )
d i f f e r e n t i a l systems, it b e i n g s u f f i c i e n t t o r e p l a c e P by t h e m determinant of t h e p r i n c i p a l p a r t i n c o n d i t i o n ( 2 . 3 . 1 ) ( s e e IVRII-PETKOV c11).
REMARK 2 . 6 :
Apart from t h e c a s e of o p e r a t o r s w i t h con-
s t a n t c o e f f i c i e n t s , a c h a r a c t e r i s a t i o n of o p e r a t o r s f o r which t h e Cauchy problem i s w e l l posed i s known o n l y i n a number of special cases.
For example, i f we assume t h a t t h e r o o t s a t
5, of ( 2 . 3 . 1 ) a r e of c o n s t a n t m u l t i p l i c i t y , we have a c o n d i t i o n which i s n e c e s s a r y (FLASCHKA-STRANG c11) and s u f f i c i e n t (CHAZARAIN [I]) and which i n v o l v e s lower-order terms of P , i . e .
t h e Levi c o n d i t i o n .
If we suppose t h a t t h e r o o t s a t 5
0
a r e of
m u l t i p l i c i t y l e s s t h a n o r e q u a l t o 2 , a g a i n we have a number of v e r y complete r e s u l t s ( s e e IVRII-PETKOV [ L ] , HORMANDER
[TI).
When t h e c o e f f i c i e n t s are a n a l y t i c , BONY and SCHAPIRA 111 have shown t h a t c o n d i t i o n
( 2 . 3 . 1 ) i s s u f f i c i e n t f o r problem ( 2 . 1 . 1 )
t o admit a unique a n a l y t i c s o l u t i o n when t h e d a t a a r e a n a l y t i c .
I n S e c t i o n 4 , we s h a l l show i n p a r t i c u l a r t h a t i f t h e r o o t s
a t 5, are simple, t h e n c o n d i t i o n ( 2 . 3 . 1 ) i s n e c e s s a r y and sufficient.
3.
HYPERBOLIC OPERATORS WITH CONSTANT COEFFICIENTS I n t h e c a s e where P h a s c o n s t a n t c o e f f i c i e n t s , Ggrding
(SEC. 3 )
HYPERBOLIC OPERATORS
345
has given an a l g e b r a i c c h a r a c t e r i s a t i o n of t h e o p e r a t o r s P(D)
for which t h e Cauchy problem i s w e l l posed.
The s u b j e c t i s
d i s c u s s e d i n depth i n HORMANDER [11; however, for completeness, we s h a l l reproduce a p a r t of t h i s d i s c u s s i o n h e r e , s i n c e c e r t a i n
4 i n connection with quest-
r e s u l t s w i l l be needed i n S e c t i o n
i o n s r e l a t i n g t o speed o f p r o p a g a t i o n .
Throughout t h i s s e c t i o n , we s h a l l be i n En.
Assume t h a t the Cauchy problem i s
PROPOSITION 3 . 1 :
well posed f o r the operator P(D) i n the d i r e c t i o n N Then there e x i s t s a r e a l number y
+
[3.1.1.) P ( S
7
PROOF:
#
0 for
En \ 0.
such t h a t
5 E R",
T
E C and I m
7
< Yo.
The g e n e r a l i t y w i l l n o t be r e s t r i c t e d i f we t a k e
.., 0 ,
N = (0,
N]
0
E
1).
The argument used i n proving (2.4-.4)
a p p l i e s p a r t i c u l a r l y i n t h i s c a s e ; consequently, for any compact s e t K t h e r e e x i s t C , an i n t e g e r k and a campact s e t K' such that (3.1.2.)
lU(xj1 I
sup
x
EK
c
ID~(P(D)
sup
I4 s x
E K'
and
Let
T E
put 5 = ( 5 '
,
C be such t h a t P(S', T)
E C~(R"I
~(X)II
k
7)= 10 with
supp u
5'
E
C
En-',
; we s h a l l show t h a t t h e r e e x i s t s yo
xy and
.
346
EVOLUTION EQUATIONS
independent o f 5 ' such t h a t suppose t h a t I m T 5 0.
Im
T
x
n
E
Cm(R)
We can t h e r e f o r e
R:,)
t o t h e function
etx,')
i s e q u a l t o u n i t y for
Since P(D) u(X) = 0 f o r
2 0.
E
e i(x-a).6
u(x)
0
6)
To show t h i s , we apply t h e i n e q u a l i t y
( 3 . 1 . 2 ) ( w i t h K reduced t o a p o i n t a
where
.
Yo
2
(CHAP.
x
n
x
,
an
and zero for
2~
2 -an
and u ( a ) = 1,
2
we o b t a i n
1
c
(1 +
1611k+m e& a,,.Im
T I
or, w i t h r e a l c o n s t a n t s b and c , (3,1,.3.)
b Log(1
+ /GI) + c
Im
T
,
when
P(5',
T)
= 0,
However, a r e s u l t due t o Seidenberg-Tarski ( s e e HORMANDER
[11, Appendix) shows t h a t t h e l o g a r i t h m i n ( 3 . 1 . 3 ) may be replaced by a c o n s t a n t , and t h i s proves t h e p r o p o s i t i o n .
This p r o p o s i t i o n l e a d s t o
DEFINITION 3.2 :
An operator with constant c o e f f i c i e n t s is
said t o be correct i n the sense o f Petrovski i n the direction N = (0,
i/
..., 1) i f
the c o e f f i c i e n t of the term of highest degree i n En i n
P ( 5 ) i s independent o f 5 '
ii/ condition (3.1.1) i s s a t i s f i e d for a certain y o .
(SEC. 3 )
347
HYPERBOLIC OPERATORS
EXAMPLES :
The h e a t o p e r a t o r
- Ax'
3%
The SchrBdinger o p e r a t o r
the direction N.
i s correct i n Q
Xn
-A
X'
i s c o r r e c t i n t h e d i r e c t i o n s N and -N.
Remark 1 . 8 shows t h a t i n g e n e r a l we cannot expect t o have uniqueness o f t h e Cauchy problem for a c o r r e c t o p e r a t o r ; nonet h e l e s s w e have
PROPOSITION 3.3 :
Suppose P is a correct operator in the
d i r e c t i o n N; then it adnits a fundamentaZ s o l u t i o n E with
-
support i n R:
.
PROOF:
By h y p o t h e s i s , t h e polynomial P ( E ' , T ) f a c t o r i s e s P Const. XI (7 Aj(I')) , consequently t h e r e e x i s t s
into
-
j=1
c > 0 such t h a t
(3.3.1.)1
~P(T,Y I I
2
c IIm
7
- Yojp
I
for and
7
5' E '-?I
-< yo
.
This i n e q u a l i t y allows us t o d e f i n e a d i s t r i b u t i o n F E
a'
348
EVOLUTION EQUATIONS
6)
.
1
if E = F
and t h e r e f o r e P(D) E =
(CHAP.
I n o r d e r t o show t h a t Cp E CR (:]: Cp
in (3.3.2).
supp
F
C
- ,
Rn
we t a k e
The Paley-Wiener theorem a p p l i e d t o
i n c o n j u n c t i o n with t h e Cauchy theorem, shows t h a t expression
( 3 . 3 . 2 ) i s independent of y f o r y < y
0
and t h a t t h e r e e x i s t s a
c o n s t a n t C such t h a t
(3.3.3.) where
.1
F, 'P =
5
inf
..
x t SUPP cp upper bound ( 3 . 3 . 3 ) shows t h a t
-
uently t h a t
supp E
REMARK 3.4 :
C
< F,
~p
> = 0
and conseq-
.
We say t h a t P(D) i s an e v o l u t i o n o p e r a t o r i n
t h e d i r e c t i o n N i f it admits a fundamental s o l u t i o n with n support inIR+.
An o p e r a t o r which i s c o r r e c t i n t h e sense o f
P e t r o v s k i i s an e v o l u t i o n o p e r a t o r ; t h e converse does not h o l d , however, and we r e f e r t o HORMANDER
C41
f o r a c h a r a c t e r i s a t i o n of
evolution operators.
Remark 1 . 8 and P r o p o s i t i o n 3 . 1 show t h a t a n e c e s s a r y c o n d i t i o n f o r t h e Cauchy problem t o be w e l l posed f o r P(D) i n t h e d i r e c t i o n N i s t h a t P be h y p e r b o l i c i n t h e sense of t h e following d e f i n i t i o n :
(SEC. 3 )
349
HYPERBOLIC OPERATORS
DEFINITION 3.5
We say t h a t P(D) i s hyperbolic i n the
:
direction N i f
(i)
Pm(N) # 0
(ii)
The condition (3.1.1) i s s a t i s f i e d f o r some yo
.
The l a s t p a r t of t h i s s e c t i o n w i l l be devoted t o proving t h a t t h i s c o n d i t i o n i s s u f f i c i e n t ; t o t h i s end, we r e q u i r e c e r t a i n a l g e b r a i c r e s u l t s r e l a t i n g t o hyperbolic o p e r a t o r s .
F i r s t , we have:
PROPOSITION 3.6 :
Suppose P(D) i s a hyperbolic operator
i n the d i r e c t i o n N; then i t is hyperbolio i n the d i r e c t i o n
-
N
and its principal p a r t Pm (D) is hyperboZio i n the d i r e c t i o n N.
PROOF: P(5 +
Since Pm (N) # 0, t h e sum of t h e r o o t s i n
-T
of
N) i s an a f f i n e f u n c t i o n of 5 of which t h e imaginary
T
p a r t i s bounded above by m y
*
0,
it i s t h e r e f o r e c o n s t a n t .
Consequently, t h e r e e x i s t s some y such t h a t P(S+ T
N) f 0
for
(5,
7)
E R“
x C
and ]Im 71
Y
and t h i s proves t h e f i r s t a s s e r t i o n .
+x
-
PI(, + T N) ; h e + = now t h e r o o t s i n T of t h e l e f t - h a n d s i d e a r e l o c a t e d w i t h i n a
We have
strip
1Im TI
I””
3
T
N)
; therefore the roots i n T
of t h e l i m i t
(CHAP. 6)
EVOLUTION EQUATIONS
350
are real.
rn of P i s h y p e r b o l i c , t h i s does not
If the principal part P
i n g e n e r a l imply t h a t P i s i t s e l f h y p e r b o l i c ( s e e SVENSSON [11); however, we have
Suppose t h a t a12 the r o o t s of the
PROPOSITION 3.7 :
equation i n (3.7.1
.]
:
T
Pm(l+
T
N)
=
0
are r e a l and d i s t i n c t for 5 n o t proportiona2 t o N. P(D) is hyperbolic
Then
We say t h a t it i s
in the d i r e c t i o n N .
s t r i c t Zy hyperbo l i c .
a2 xn
For example, t h e wave o p e r a t o r hyperbolic i n t h e d i r e c t i o n ( 0 , proposition, we t a k e N = ( 0 ,
LEMMA 3.8
:
If
P
,
. ..,O,
PROOF:
.)I
2
c IIm
Let X h ( E ’ )
factorising P
m’
we o b t a i n
is strictly To prove t h e
1).
1);we s t a r t by f i r s t proving
is s t r i c t l y hyperbozic in the
d i r e c t i o n N , then there e x i s t s c (3.8.1.) I P ( S * ,
. .. , 0 ,
Ax,
TI
(
> 0
such t h a t
ISI +
be t h e r o o t s i n
171)”+’
T
of
m (s,,
P
7) =
0 ;
HYPERBOLIC OPERATORS
(SEC. 3 )
Gh(E1,
where
7)
n
=
(7
- Xj(c'))
351
a
and
i s a non-zero
j# h
constant. m
IGhCI1l
The f u n c t i o n
T)I2
does n o t vanish f o r
h=l
( I 1 ,7) E
(R"" x
c )\
0 and i s homogeneous of degree 2(m - 1);
t h e r e t h e r e f o r e e x i s t s c > 0 such t h a t
We now r e t u r n t o t h e proof of t h e p r o p o s i t i o n . P(g',
7) =
bound
pm(!',
lQ(sl,.)I
( 3 . 8 . 1 ) shows t h a t
+ Q(E', 7) Since we C [ltll + lT1)w' I the
7)
I
P[%',
7) f Owhen
1 I m 71
We put
have an upper inequality 2
y
for y
sufficiently large.
N a t u r a l l y , P r o p o s i t i o n 3 . 3 a p p l i e s f o r hyperbolic opera t o r s ; we s h a l l show f u r t h e r t h a t E has support i n a cone of
Q J E o 1
.
(CHAP. 6)
EVOLUTION EQUATIONS
352
Suppose P is a hyperboZic operator i n the
DEFINITION 3.9:
we denote by r(P, N) the connected component o f N
domain N ; the open s e t
c
5
1
5 E R"
and P,(S)
f
]
0
in
.
We s h a l l now show t h a t P i s a l s o h y p e r b o l i c i n each d i r e c t i o n of We s t a r t w i t h t h e following lemma:
r(P, N).
LEMMA 3.10:
r(P, N) i s equaZ t o the s e t o f the 6
such t h a t the poZynomiaZ Pm(5 + roots in
T.
PROOF:
Let
r'
and by d e f i n i t i o n
I n o r d e r t o show t h a t
r'
i s connected. T/ =
then
is+p
r'
C
t h i s s e t i s c l e a r l y open
5 E
for P
,
r1
and f u r t h e r , N
numbers which sum t o u n i t y .
E
and
A
p
5E
are positive real
By h y p o t h e s i s
we have t h e
factorisation
and consequently t h e p o l y n o d a l
P,(v
+
T
N)
= P,(ht
admits t h e r o o t s
h T
+PN +T J
-)z
r N)
< 0
.
= P,(N)
II (p + 1
T
-h
Tj]
F i n a l l y , i f 5 belongs t o t h e boundary of TI, t h e r o o t s of P (5 + m .
T
N) a r e n e g a t i v e or zero and t h e r e e x i s t s a t l e a s t one
zero r o o t s i n c e 5
4 rl.
.
r'
it remains for us t o prove t h a t
More p r e c i s e l y , we s h a l l show t h a t i f
N E ri where
En
N) has onZy s t r i c t l y negative
T
be t h i s l a t t e r s e t ;
Pm(c) 4 0
E
Hence P,(5)
P
0
which proves
re,
(SEC. 3 )
HYPERBOLIC OPERATORS
353
t h a t 5 a l s o belongs t o t h e boundary o f r ( P , N ) and consequently
rl
6
.
q p , N)
The cone r(P, N) depends only on P,;
however we have
Suppose P i s hyperboZic i n the direction N.
THEOREM 3.11:
~ e t 9 E q p , N) ; -then (3.11.1.)
P(5+ T
N
+ P?)
f 0
andImb S PROOF: i e n t of
is
0
7
< yo
.
, ZER"
= 0, t h i s i s condition (3.1.1).
For Im
urn
Im
for
f
P,(n)
0 ;
The c o e f f i c -
consequently t h e number of zeros
p such t h a t I m y < 0 i s c o n s t a n t when t h e parameter T remains i n
t h e h a l f plane I m T equation
P(5 +
(3.11 2.)
7
N
However, f o r
yo.
+ PI)
T - ~P[{
which converges t o P,(N
0
+ +
T(N
A?)'=
Lemma 3.10 shows t h a t
P(,N
'I
l a r g e and p = AT, t h e
i s equivalent t o
+ An))
= 0
0 vrhen Im
T +
+
0
I?)
f
-
m
.
f o r Re h > 0 ,
consequently, t h e same a p p l i e s f o r (3.11.2)when I m
'I
< Y o , and
t h i s proves t h e theorem. A s an a p p l i c a t i o n , we deduce
THEOREM 3 . 1 2 :
I f P i s hyperbolic i n the direction N, it
i s hyperbolic i n my direction of r(P, N) and, moreover, t h i s
(CHAP. 6 )
EVOLUTION EQUATIONS
354 cone is convex.
q E
Let
PROOF:
r ( P , N)
hyperbolic i n t h e d i r e c t i o n
;
we s t a r t by showing t h a t P i s N with
q -k E
E
In fact,
> 0.
Theorem 3.11 i m p l i e s
~ ( +5 ~ ( +7
#
e N))
o
and Lemma 3.10 shows t h a t ing
1 = (51
by t a k i n g
E
-e
+
N)
e
N
Irn T c inf(0, y o
-58
for P,(q
, we
+
e N)
#
0
for
t h e n deduce t h a t
> 0 small enough t o ensure t h a t r-
-
E
E
, 5 > 0.
E Writ-
‘Q € r ( p , N) N remains i n
t h e open s e t r(P, N ) . To prove t h a t t h e cone r(P, N ) i s convex we show t h a t t h e
segment j o i n i n g two of i t s p o i n t s qI and q2 remains i n i t s i n t erior.
I n f a c t we showed d u r i n g t h e proof of Lemma 3.10 t h a t
t h i s segment is i n s i d e t h e cone
r[p,
7,)
= r ( P , N)
An analogous r e s u l t holds for s t r i c t l y hyperbolic o p e r a t o r s ; this is: THEOREM 3.13: N , then
I f P is s t r i c t l y hyperbozic i n the direction
it is s t r i c t l y hyperbolic in any direotion o f the erne
r(P, N).
PROOF:
For 5
f a c t o r i s a t ion
(3.13.1 .)
E
IRn and not p r o p o r t i o n a l t o N , we have t h e
(SEC. 3 )
where t h e
HYPERBOLIC OPERATORS
a r e Cm f u n c t i o n s which a r e mutually d i s t i n c t f o r
T.(*)
g E R" \
355
J
(mN)
the single root
#
, Since P,"] T =
-
h when
, the
0
5 t e n d s towards
J
E r(p,
N)
, not
J
consequently,
h N;
.
t h e f u n c t i o n s T . (5) a r e continuous on lRn Suppose we have
r o o t s T . ( C ) tend t o
p r o p o r t i o n a l t o N , and i-1
E
IR ;
we deduce from ( 3 . 1 3 . 1 ) :
(3.13.2 .) We now show t h a t
The homogeneity and t h e c o n t i n u i t y of t h e f u n c t i o n s
T
j
show t h a t
which proves ( 3 . 1 3 . 3 ) s i n c e Lemma 3.10 t e l l s us t h a t t h e
T
. (TI)
J
are s t r i c t l y negative. The a s s e r t i o n ( 3 . 1 3 . 3 ) and t h e c o n t i n u i t y of t h e f o r any j t h e r e e x i s t s p
j
( 3 . 1 3 . 2 ) shows t h a t t h e p
j
a r e r o o t s of
now only remains t o show t h a t t h e y
5 i s not proportional t o p
j
= pk = y with
Tj(h
X
E
+ pll) IR
.
=
~
J
q.
j
P,(5
imply t h a t
j
+ pJ. 7)
T.(S
such t h a t
T
= 0
+ p7)
, and
= 0
.
It
a r e d i s t i n c t i n p a i r s when
In f a c t , i f there existed
j and k d i s t i n c t , we would have
~ + p7) ( 5= 0
Then t h e e q u a l i t y
shows t h a t A = 0 , 2 . e . t h a t
and consequently
P,[s
s + pl"rl
+ pl"rl) 0
=
'5 +
Pl"rl = h N w i t h
0 = Am p,")
, which
i s a contrad-
(CHAP. 6)
EVOLUTION EQUATIONS
356
iction. We a r e now i n a p o s i t i o n t o prove t h e p r i n c i p a l r e s u l t of t h i s this is
section;
THEOREM 3.14:
direction N ;
Suppose P i s a hyperbolic operator i n the
then P admits a unique fimdamentaZ s o l u t i o n w i t h
support i n the dual cone
r*p, N) PROOF:
=
c x E R" I
x.5
2
o for
rp,
~ Z Z5 E
N)
3
Since P i s , i n p a r t i c u l a r , c o r r e c t i n t h e sense of
P e t r o v s k i i n t h e d i r e c t i o n N , P r o p o s i t i o n 3.3 shows t h a t i t
-
admits a fundamental s o l u t i o n E w i t h support i n By and defined
bY
with
Re y
< y,
.
Theorem 3.11 i m p l i e s t h a t t h e f u n c t i o n
i s holomorphic i n t h e open s e t the fact6risation
shows t h a t
Rn
- i r(P,
N)
.
Furthermore,
(SEC. 3 )
HYPERBOLIC OPERATORS
357
(3.14.2.) when
Re y
.
< y,
The Paley-Wiener theorem a p p l i e d t o cp
implies t h e e x i s t e n c e of C such t h a t exp HJYN
IG(g + iy
(3.14.3.)
,
5 E R"
for
X
+ i ~ 7 1 15
N
and y r e a l .
c
+ 17)
(151 + I Y l +
I~lIW1
We deduce from ( 3 . 1 4 . 2 ) ,
( 3 . 1 4 . 3 ) t h a t we can apply a t r a n s l a t i o n of t h e contour of i n t e g r a t i o n i n ( 3 . 1 4 . 1 ) and o b t a i n
y
for a l l
,
h < 0 and
.
E r [ P , N)
To show t h a t E has support i n T*(P, N), we t a k e
cp
E :C
such
that
CC
sup, cp
7 E
where
.
TIP, N)
( 3 . 1 4 . 3 ) show t h a t supp
E
C
cx I
supp
E c
r*p,
E R"
Wneri X V
< E, cp >
X.?
NI
x
2
.
0
1
x.1
-+
0
-
5
my
.
11 , and
0
I
t h e i n e q u a l i t i e s (3.14.2) Consequently, t h i s shows t h a t
Since r ( P , N ) c o n t a i n s a neighbowhood of N , t h e d u a l cone T*(P, N ) i n i n c l u d e d i n
$U { 0 I .
From P r o p o s i t i o n 2.2
of Chapter I , we can t h e r e f o r e perform a convolution of a d i s t n r i b u t i o n w i t h support i n R, and a d i s t r i b u t i o n with support i n I'*(P, N ) ;
t h i s immediately i m p l i e s t h e uniqueness of a fundam-
e n t a l s o l u t i o n w i t h support
n i n W, ( t h i s a l s o follows from Prop-
358
(CHAP. 6)
EVOLUTION EQUATIONS
F i n a l l y , P r o p o s i t i o n 3.1, Remark 1 . 8 and
o s i t i o n 1.13).
Theorem 3.14 show t h a t we have
The following assertions are equivalent:
THEOREM 3.15: i/
the Cauchy problem is well posed f o r P(D) i n the direction
N ii/ P is hyperbolic in the d i r e c t i o n N iii/ P admits a fundamental s o l u t i o n with support in a cone C
such t h a t
cn
{ x
I
S 0 ]
x.N
=
.
{ 0
I n t h e case where P(D) i s an r x r system, we have
The fozzowing assertions are equivalent:
PROPOSITION 3.16: i/
Thg system
P
(Pij(D))ij
I
=I. . r
a d n i t s a fundamental
s o l u t i o n with support in a cone C such t h a t
c n ii/
C
x
1
X.N
2
o 1=
C o 3.
The scalar operator p = det(Pij(D])
i s hyperbolic in the
.
direction N
When t h i s is s a t i s f i e d , we say t h a t P is hyperbolic i n the d i r e c t i o n N.
PROOF :
i/
j
ii/
.
Let E = (Ehk) be t h e fundamental
s o l u t i o n of P w i t h support i n C ;
(Pij(D)bo)*(Ehk)
P
bO.I
.
we have
Taking t h e determinant i n t h e sense
of t h e convolution t h i s e q u a l i t y shows t h a t
4)
(SEC.
pg0 Since we have
*
=
det(Ehk)
bo
i s hyperbolic i n t h e d i r e c t i o n
p
I n o r d e r t o prove t h e i m p l i c a t i o n
uce t h e matrix
Let Eo be t h e fundamental s o l u t i o n o f p ;
Q(D)
=
E = Q(D)
shows t h a t P admits
introd-
the equality
p(D)
1
. I)
(Eo
as a fundamental
.
R+n U { 0 ]
s o l u t i o n w i t h s u p p o r t i n a cone o f
4.
, we
i/
ii/-
of t h e c ofac t ors of P i j ( D ) ) .
Q = (Qs,,(D))
F"D)
, Theorem
c supp E c C
supp ( d e t ( E h k ) )
3.15 i m p l i e s t h a t t h e o p e r a t o r N.
359
HYPERBOLIC CAUCHY PROBLEMS
HYPERBOLIC CAUCHY PROBLEMS WITH VARIABLE COEFFICIENTS
4.1
INTRODUCTION:
We c o n s i d e r t h e Cauchy problem f o r
f i r s t - o r d e r systems o f t h e form
P = Dt
- A(t,
X,
where A
Dx)
i s an N x N matrix of p s e u d o - d i f f e r e n t i a l o p e r a t o r s o f degree one .in x
E
X and w i t h C
co
dependence on t h e " t h e " parameter
.
t
We now propose t o prove, under c e r t a i n assumptions, t h e
u
e x i s t e n c e and uniqueness of
Dt u
- A(t,
X,
DX]u
such t h a t
=
f
R x X
in
(4.1 .I.] U
It=o =
The i n v e s t i g a t i o n may be c a r r i e d o u t i n analogous f a s h i o n when X n
is a compact manifold without boundary o r when X = IR
.
We
360
(CHAP. 6 )
EVOLUTION EQUATIONS
s h a l l r e s t r i c t our a t t e n t i o n t o t h e l a t t e r c a s e s i n c e it r e q u i r e s c e r t a i n a d d i t i o n a l assumptions on t h e behaviour of t h e symbol of A f o r l a r g e 1x1
.
More p r e c i s e l y , we s h a l l assume h e n c e f o r t h
t h a t t h e complete symbol a ( t , x, 5 ) of A i s i n t h e space
BS1(RW1 x R" ; L(CN, CN )) ( s e e 1 0 . 1 , Chapter 111) and t h a t it admits a p r i n c i p a l p a r t a l ( t , x, 5 ) which i s homogeneous of degree one i n 5 f o r 161 2 1.
u
Then, f o r
E
C"(l
0 T]
; 8)
,
we have :
(4-1.2.)(Au)(t,
x)=
i)dcc
a(t, x, 5) u(t,
eixE
C"{[
0 T] ; S),
I n t h e s p e c i a l c a s e where A i s a d i f f e r e n t i a l o p e r a t o r with
DJ
$Ad
c o n s t a n t c o e f f i c i e n t s of t h e form
, Proposition
3.16 shows t h a t t h e Cauchy problem (4.1.1)i s w e l l posed i f and only i f t h e e q u a t i o n dat (T
-
n A
has only r e a l r o o t s i n T for 5
=
5 )
j j E
Bn
.
o For example, numerous
systems o f p h y s i c a l o r i g i n l e a d t o t h e c a s e where t h e matrices A . a r e hermitian. J
I n t h e g e n e r a l case of o p e r a t o r s of t h e form (4.1.2), we s h a l l prove i n t h i s s e c t i o n t h a t t h e Cauchy problem i s w e l l posed f o r o p e r a t o r s which a r e symmetrisable i n t h e sense of t h e following definition : DEFINITION 4 . 2 :
The operator Dt - A is s a i d t o be s y m e t -
risabZe i f there e x i s t s a
Cm
mapping:
(SEC. 4)
361
HYPERBOLIC CAUCHY PROBLEMS
51 E R"+'
(t, x,
x (R" \
with values i n N
x
of degree 0 i n
5 E R" \ 0
0)
- ro(t,
x,
5)
N hermitian matrices, which i s homogeneous and which i s bounded together
with a l l i t s d e r i v a t i v e s f o r 161 = 1 and such t h a t (4.2.1 ,)
r o ( t , x, 5 )
(4.2.2 ,)
r o ( t , x,
with c
.
5)
(x,
i s hermitian
5)
c 1
2
independent of
> 0
x,
a,(t,
5 ) E R"
x R" \ 0
and of t i n a
compact subset of 13.
We s t a r t by e s t a b l i s h i n g an energy i n e q u a l i t y .
THEOREM 4.3 :
Let P = Dt
Then, for a12
erator.
s
E
-
A be a symetrisabZe op-
IR and T > 0 , there e x i s t s a
C
such that
for
t
E
LO
TI and
u
E Go([ 0 T
Furthermore, we can rep2ace u ( 0 ,
3;
.)
H*')
n C'([
by u(T,
.)
0T
3 i Hs) .
i n the r i g h t -
hand s i d e of ( 4 . 3 . l I s .
PROOF:
We s h a l l show t h a t , under t h e c o n d i t i o n s o f t h e
theorem, t h e r e e x i s t s a C ' such t h a t f o r
T'
E
r 0, T ]
we
EVOLUTION EQUATIONS
362
(CHAP.
We s h a l l s e e t h a t ( 4 . 3 . 2 ) s i m p l i e s ( 4 . 3 . 1 ) s ;
J/Is
]IPu(t',
in
We f i r s t n o t e t h a t it i s s u f f i c i e n t t o
establish (4.3.2)
0
s i n c e by a p p l y i n g ( 4 . 3 . 2 ) 0 t o v = A
(where A s is t h e p . d . 0 . w i t h symbol t a i n (4.3.2)
( 4 . 3 . 2 ) s has
O(//u[t',
t h e advantage o f a l l o w i n g e r r o r s of o r d e r
6)
(1
2 42)
+ Igi )
S
u
we ob-
by o b s e r v i n g t h a t from S e c t i o n 11.1 o f Chapter
S
IV we have
In o r d e r t o prove ( 4 . 3 . 2 ) s , we i n t r o d u c e a s e l f - a d j o i n t R ( t , xI Dx]
p.d.0.
+
E
w i t h Cm dependence on t . d
F(t) = z ( R u ( t ,
w i t h p r i n c i p a l symbol r ( t , x, 5 )
BLo
0
The i d e a i s t o c a l c u l a t e (we denote by (
.), u ( t , .))
2 n p r o d u c t i n L (B ) and by
1 1
,
) the inner
t h e a s s o c i a t e d norm).
We o b t a i n
F(t)
=
(R 3, u,
+ (Ru,
at
u)
+
(R;
U,
U)
where t h e symbol of t h e o p e r a t o r R ' i s o b t a i n e d by d i f f e r e n t i a -
t
Or, p u t t i n g Pu = f . t. i ( R u , Au)
t i n g t h e symbol of R w i t h r e s p e c t t o
F(t)
(4.3.3.)
=
i ( R 4 u, u)
+
i ( R f , u)
-
+
(Ru, i f )
+
( R i u,
u)
The assumption ( 4 . 2 . 1 ) i m p l i e s t h a t t h e o p e r a t o r
RA
-
#-
A R i s of degree 0 , and c o n s e q u e n t l y E x e r c i s e 11.1 i n
Chapter IV shows t h a t
(SEC.
4)
HYPERBOLIC CAUCHY PROBLEMS
363
Substituting in (4.3.3) and integrating between 0 and T' we obtain, taking account of the L
with T'
E
[O TI
2
continuity of the operator
,
Furthermore, assumption ( 4.2.2) allows G&ding 's inequality to be used (see (11.1.12),Chapter IV): there exist
Replacing R by R + C A-2, .with C = 0.
we can assume that (4.3.5) holds
This gives (4.3.2) by substituting into (4.3.4). S
In order to show that (4.3.2) implies (4.3.1) S
S Y
it is
'2
sufficient to apply the following lemma to y ( t ) = Ilu(t, and
2
I P U h
LEMMA
Jlls 4.4
:
=
cp(t1
*
Suppose y and cp
are p o s i t i v e continuous
fwrctions i n CO. TI which, w i t h constants a , C the i n e q u a l i t y
2
0, s a t i s f y
364
(CHAP. 6)
EVOLUTION EQUATIONS
.
The right-hand s i d e o f ( 4 . 4 . 1 ) i s t h e s o l u t i o n
PROOF: of
2'
5
cz
+
z(t)
CZQ
=
, c
Z(0) =
[
(Y
+
m.
,/I
We have
[z(t')
and consequently t h e f u n c t i o n p = y P(t)
Putting
z
+ cp(t'))
dt'
- z satisfies
1
s c $(t) =
i:
p(t')
dt'
, we
obtain:
F i n a l l y , r e p l a c i n g t by - t , we s e e t h a t we can r e p l a c e u(0, . ) by u ( T , . ) i n ( 4 . 3 ; 1 ) s .
By u s i n g a f u n c t i o n a l a n a l y s i s argument, we shall now deduce from t h e i n e q u a l i t i e s ( 4 . 3 . 1 ) ence, uniqueness and r e g u l a r i t y .
S
a theorem concerning e x i s t -
4)
(SEC.
365
HYPERBOLIC CAUCHY PROBLEMS
Suppose P = D t - A i s symetrisabZe. Let Then there e x i s t s a IF?,f E L2 ([ 0 T ] ; Hs) and g E HS. THEOREM 4.5 :
s
E
u E c"([ 0 T ] ; Hs)
unique
( in the sense o j
Pu = f (4.5.1
.>
.I
u[o,
=
such t h a t
If f E C"([
3
u ECm([ O T
PROOF:
;
3;
x
$1
H+")
and g
E
H*
S
remains v a l i d for
then
The proof of uniqueness i s immediate; i n f a c t i f satisfies D
0T
E C1([
[
H*)
1; HS) 0 T 3 ; H*')
u E C"([ U
0 T
T
9
Moreover, the energy inequaZity ( 4 . 3 . 1 ) u.
a(]o
t
u = Au, we deduce t h a t
and t h e i n e q u a l i t y (4.3.1),,
t h e n shows
t h a t u = 0.
I n o r d e r t o prove t h e e x i s t e n c e , we i n t r o d u c e t h e space
E =
f
cp E Car[ 0 T
3;
H-)
I rp(T)
= 0 'J
.
We s h a l l now show
*
t h a t it i s p o s s i b l e t o d e f i n e a l i n e a r form R on P E by
*
For t h i s we n o t e t h a t t h e energy i n e q u a l i t y ( 4 . 3 . 1 ) f o r t h e Cauchy problem r e l a t i n g t o t h e o p e r a t o r P
Ilu(T,
.)I],
on t h e right-hand s i d e o f ( 4 . 3 . 1 ) - s
i s symmetrisable by r
-1) 0
.
We t h u s h a v e , f o r cp
*
-S
holds
with
*
(note t h a t P E
E:
366
(CHAP. 6 )
EVOLUTION EQUATIONS
so t h a t
This shows t h a t R i s w e l l d e f i n e d and i s , moreover, continuous
.x
2 L ([ 0 T
f o r t h e topology induced on P E by
.
3 ; H-')
Consequently t h e Hahn-Banach theorem i m p l i e s t h e e x i s t e n c e of an element
u E
(L2([ 0 T
3;
=
H-'))l
0T
L2([
1; Hs)
such t h a t
[
*
(4.5.2.) (u, P 9) = rp
for all
( 4 . 5 . 2 ) shows t h a t Pu = f i n
ot u
u
E
=
Co[[
AU
+fE
0T
L2(r
1; H*l]
E
f E C"([ u
o
T
3;
H*')
];
E C"([ 0 T
(u(0,
s t e p by s t e p t h a t
3
x
T
.
R")
(9, d o 1
*I)
c
,
x
R")
Thus
so t h a t
I
.), rp(Ol .)]
and g E H
H+")
3;
o
cp E I(:C
&([ 0 T
E and consequently u( 0 ,
0T
d t , -11 d t + T1
. A f t e r i n t e g r a t i n g by p a r t s w i t h r e s p e c t
t o t , (4.5.2) implies
all
.)I
In p a r t i c u l a r , if
E.
E
(f(t,
Hs")
+m
.)
= g.
, the
and s i n c e D
u E Ck([
0T
1t.now remains t o show t h a t
3; u
t
= (9, cp(Ol
.)I
for
If
above argument shows t h a t
u = Au + f we then deduce
H*)
for a l l k
E c"([ 0 T
t h a t it s a t i s f i e s t h e energy i n e q u a l i t y
1; HS)
(4.3.1) . S
2
0.
and
(SEC. 4)
HYPERBOLIC CAUCHY PROBLEMS
367
Suppose we h a w sequences (f.) , ( g . ) such that J
fJ
E c;([ o
and that g
E C"([
5
T
3
x
R")
and f
E CI(R") and
0T
j
f
gJ + g
J
2
in L ([ in HS.
o T 3 ; )'H
,
Let
] i H*)
be the solution of Pu = f uj(O, j j' The inequality (4.3.1) applied to u - \ shows that u S j j Hs) so that u Cauchy sequence in Co([ 0 T j in Co([ 0 T In the limit, we have Pc = f, Hs) z(0, ,) = g ; consequently, the uniqueness shows that
u
j
.
3;
. ) = gj.
is a
-
1;
H
u
= u.
Furthermore, the inequality (4.3.1)s for u is obtained by proceeding to the limit in this inequality applied to the u j'
We now give some examples of symmetrisable operators.
EXAMPLE 4.6
:
When the principal symbol a of A is 1
hermitian, the operator is obviously symmetrisable.
For
example, the Maxwell equations in a vacuum are written classically as
- CWlE 4 where E and H are functions in C3, defined overIR . a,
E =
metric system
H
3, H =
(CHAP. 6)
EVOLUTION EQUATIONS
368
0 0 0 0 0 1 ' 0 -1 0
To
ary-lo
0 0 00-1
O
& I 0
EXAMPLE 4.7
:
The o p e r a t o r P = Dt
- A is
s a i d t o be
s t r i c t l y hyperbolic i f t h e eigenvalues of a ( t , x , 6 ) a r e r e a l
1
and d i s t i n c t when
(t, x,
5) E Rwl
x (R" \ 0)
.
Suppose
f u r t h e r t h a t al i s independent of ( t , x ) f o r 1x1 s u f f i c i e n t l y large.
The p r o j e c t o r o n t o t h e eigen-subspace
associated with t h e eigenvalue X . ( t , J
(4.7.1-Ipj[t,
X,
of a l ( t , x,
5)
x , 5 ) i s g i v e n by
-
5) = 21n
XI
51 I s
(t, x) E RW1 k f h
(4.7.1)shows
i s independent of ( t , x) j f o r 1x1 s u f f i c i e n t l y l a r g e , and i s homogeneous of degree 0 i n Expression
5 E
]Rn \ 0
For
Y E
.
that p
The same t h e r e f o r e a p p l i e s f o r t h e f u n c t i o n
CN, we have
4)
(SEC.
N
N
Thus, r a i s h e r m i t i a n . 0 1
F i n a l l y , r o ( t , x, 5 ) i s
v
E cN \ o
0
.
EXAMPLE 4.8 :
c
=
p o s i t i v e - d e f i n i t e s i n c e (r ( t , x, ~ > v v ,] for
369
HYPERBOLIC CAUCHY PROBLEMS
lipj v f
o
>
J
S t r i c t l y hyperbolic s c a l a r operator of
o r d e r m.
Suppose we have
m-I
(4.8.1 .)
where
P ( t , x, Dt'
AWj
Dxl
= a w j (t, x, D,)
suppose t h a t a
=
- j=o
D;
with e
m-j
:D
€ss"s(R"l
x
R"]
0
5 E R" \ 0
and which
i s independent of ( t, x) for 1x1 s u f f i c i e n t l y l a r g e .
The Cauchy problem f o r P i s w r i t t e n f o r m a l l y as f o l l o w s :
(43.2.) PU = f
,
k Dt ulk0
=
k
gk
01.m.s
-1
We s h a l l now reduce t h i s t o a f i r s t - o r d e r system; t o do t h i s , we p u t u
= D5-1 A,,+~ u and j Then, ( 4 . 8 . 2 ) i s e q u i v a l e n t t o
u
;
a (t, x, 5) m-J
admits a p r i n c i p a l p a r t
m- j which i s homogeneous of degree m-j f o r
Am-j
=
(ul,
..., ),u
370 l4.8.3.
EVOLUTION EQUATIONS
)
DtU-AU
=
F
(CHAP.
1
r
U t=o
=
6)
G
where
with
A
B = j
F = G = Consequently B . i s a p . d . 0 . .3
a It, X I *j+l
o f degree 1 w i t h symbol
14(wj) and
11(1 + 151
A h a s as homogeneous
p r i n c i p a l symbol o f degree 1,
It can e a s i l y be shown t h a t p,(t,
x, T,
5)
where p
m
det(23
- Al(t,
X,
5))
=
is t h e p r i n c i p a l symbol of P.
The above a l l o w s us t o s t a t e t h e f o l l o w i n g :
THEOREM AND DEFINITION
4.9
:
The operator P defined i n
(4.8.1) i s said t o be s t r i c t l y hyperbolic i f for a l l (t, x, EJ E '"R
x
[R"
\
03
the zeros i n
of i t s principal
(SEC. 4)
symbol
371
HYPERBOLIC CAUCHY PROBLEMS
x,
p,(t,
~j =
T,
-
7’
0
aWJ [t, x,
5)
are
7’
J=O
real and d i s t i n c t .
We assume i n addition t h a t p is m independent of ( t , x) f o r 1x1 s u f f i c i e n t l y large. Let
,
sE R
f E
Lz([ 0 T
1 ; H’),
j = 1,
g j E H’+*j
..., m.
Then there e x i s t s a unique m-I
n
E
k = o
ck([ o
T
3;
H
such t h a t Pu = f i n & ( ] 0 T [ x R”) Furthemnore, there e x i s t s C
~
,
1
D$’
~
~
~ ( 0 .) ,
=
gd
such t h a t
IT
for a l l
s a t i s f y i n g the above assumptions and
f,gj
.
t E [ O T ]
FinaZZy, i f
have
u E C”([
PROOF:
E C”([
f
0T
3;
Hs”)
0T
.
3;
Hs”)
and
g5
E H*
we
It i s s u f f i c i e n t t o a p p l y Theorem 4.5 t o t h e
system ( 4 . 8 . 3 ) .
The s t a n d a r d example o f a s t r i c t l y h y p e r b o l i c o p e r a t o r i s t h e wave o p e r a t o r
P
~i
2 Dt
-c
2
(X)
4(
where
c
is a strictly
m
positive C large).
f u n c t i o n (which i n t h i s c a s e i s c o n s t a n t f o r 1x1
P h y s i c a l l y , c ( x ) i s i n t e r p r e t e d as b e i n g t h e wave
propogation speed.
We s h a l l show more g e n e r a l l y t h a t we
.
~
372
EVOLUTION EQUATIONS
(CHAP.
6)
have t h e n o t i o n o f a propagation speed when P i s a s t r i c t l y hyperbolic d i f f e r e n t i a l o p e r a t o r .
Consider a d i f f e r e n t i a l o p e r a t o r of degree m, D ) , with C
P = P ( t , x, Dt, of IRn+l.
m
c o e f f i c i e n t s i n an open s u b s e t S2
X
Let p
m
be i t s p r i n c i p a l p a r t ; saying t h a t P i s
s t r i c t l y hyperbolic w i t h r e s p e c t t o s a y i n g t h a t f o r a l l ( t , x) P ~ , ~ ( T ,=~ p], ( t , t o N = (1; 0).
x, 7 ,
E
(see Definition 3.9).
i s equivalent t o
S2 t h e polynomial
5) is
We denote by
t
s t r i c t l y hyperbolic with respect
r(t,
x ) t h e cone T(Pt,xl
N]
We s t a r t by g i v i n g a d e f i n i t i o n of t h e
hypersurfaces which w i l l p l a y a r o l e analogous t o t h a t o f t h e hyperplane t = 0 .
DEFINITION 4.10
:
1 in IRn+l A hypersurface S of c l a s s C
i s said t o be spacelike f o r P a t a p o i n t ( t , x) covectors normal t o
s
a t ( t , x) are i n
r(t,
X)
E S
if the
u (- r(t,
x))
I t i s said t o be spacelike f o r P i f t h i s i s true a t a l l points.
,
(SEC. 4)
HYPERBOLIC CAUCHY PROBLEMS
373
We n o t e t h a t w i t h t h i s d e f i n i t i o n , Theorem 3.13 shows that p
t YX
i s s t r i c t l y h y p e r b o l i c w i t h r e s p e c t t o any non-zero
conormal o f S a t ( t x),
This w i l l form t h e b a s i s of t h e
f o l l o w i n g l o c a l uniqueness r e s u l t .
DEFINITION 4.11 : of degree
m
Suppose P i s a d i f f e r e n t i a l Operator
with Cm c o e f f i c i e n t s which i s s t r i c t l y hyperbolic 0
in a neighbowhood V of a p o i n t ( t o y x 1. surface of '?.
L et S be a Cm hyper-
Suppose t h a t s is spacelike for P a t ( t o y xo).
Then there e x i s t s a neighbowhood W of ( t o yxo) such t h a t i f u E we
P[v)s a t i s f i e s
have u = 0 i n
PROOF:
Let
bourhood of (to,
~u =
o
in
v
,
D~
uIs =
o
Icy]
Imi
w.
x X
0
be a diffeomorphism d e f i n e d i n t h e neigh-
1 which
transports S into
,
374
(CHAP. 6)
EVOLUTION EQUATIONS
L1
3 I r =0 1.
('E;
S = {
r )
and (to, xo) into 0.
Let P be the op-
erator obtained by transporting P. Now ( 7 . 3 . 5 ) , Chap. I, shows
(7, 3
that
(7, 5 ) ;
= Pt
0'0
0'
0
consequently
is strictly 0'0
~
-
hyperbolic with respect to N = (1; 0). ument shows immediately that
p-t,x
c
c
c
A transversality arg-
is still strictly hyperI)
bolic with respect to N' when (t, x) and N' are respectively .*
sufficiently near 0 and N.
The above allows us to prove the
existence of a neighbourhood W in which uniqueness applies, by proceeding as in the proof of Theorem 1.10 (see Remark 1 . 1 2 ) . We are in fact then led to using an existence theorem for a Cauchy problem of the type (1.10.4). For this, it is suffitcient to apply Theorem 4.9 to P after extending it to the whole of IRn+l.
This is possible by virtue of:
LEMMA 4.12 :
Let Q = Q(t, x, Dt, Dx) be a d i f f e r e n t i a ? ,
operator ( o r pseudo-differential operator w i t h respect t o x) of degree m, with Cm c o e f f i c i e n t s , which i s s t r i c t l y hyperbolic i n a b a l l Br of radius r a t 0.
Then there e x i s t s a s t r i c t l y
c
hyperboZic operator Q of degree m which coincides w i t h Q i n the ball B
r/2
and which has constant c o e f f i c i e n t s outside the b a l l
PROOF: z e r o for s
2
Let
@(s]
E C"(R)
be equal to unity for s
r , and lying between 0 and 1.
The mapping
5
r
-2'
(SEC. 4)
375
HYPERBOLIC CAUCHY PROBLEMS
and i s z e r o o u t c o i n c i d e s w i t h t h e i d e n t i t y map i n t h e b a l l B r/2 s i d e t h e b a l l Br.
x),
Then t h e o p e r a t o r Q w i t h complete symbol Q ( p ( t ,
7,
g)
meets t h e r e q u i r e m e n t s .
This l o c a l uniqueness r e s u l t can be g l o b a l i s e d .
For ex-
t h e f o l l o w i n g theorem e s t a b l i s h e s t h e e x i s t e n c e of a
ample,
f i n i t e wave p r o p a g a t i o n speed for s t r i c t l y h y p e r b o l i c d i f f e r e n t i a l operators :
Suppose P i s a d i f f e r e n t i a l operator of
THEOREM 4.13 :
degree m, s t r i c t l y hyperbolic i n
[ 0, T ]
constant c o e f f i c i e n t s f o r 1x1 large.
u p
x
R"
and with
We put
SUP
t€[ O,T1,xERn
151 = where
T
j
let Let
DJ u t
c s0
E
(to,xo>
= { (t, x ]
=
c
u E C"([ jt=o
j
I
are the zeros with respect t o
For cone
1
=
t
0, T
E
I -
(0, x) ~x 0T
o
I
[
in
s0
1; b )
2
x
R"
O, to [ xo/
'I
of
p,.
we define the backwards
,
< v to
1x
- xoI < v ( t o - tl 3
3 denote
i t s trace on t = 0 .
be such -that Pu = 0 i n C and
f o r j = 0,
..., m-1.
;
Then u = 0 i n
c.
PROOF: For e
C
u E Cm[[
F i r s t we suppose t h a t
E ] 0,
, we
to
tE E
put
t
0
-
0
0
,
5X<
1
h3 \ x
- xo\2 -
the part S
corresponding t o
Y
2
v [t
IX
-
0,
x
T
R")
.
Consider, f o r
of t h e hyperboloid
- x tEI2+ x2 t62 v2[I- 1)
xoI
.
and we denote by
E
t h e backward cone i s s u i n g from ( t E ,x )
E
(CHAP. 6)
EVOLUTION EQUATIONS
376
tE and 0 4
=
0
.
t s 1 tE
The h y p e r s u r f a c e s S a r e s p a c e l i k e
Y
et
t0
f o r P , and we have -
-.
I
.+
kJ % o
ce =
put A =
[ X E[
= 0 for
1.1
We
'
0, 1 [ I 'D
ulSx
s; m-I 3 ,
This i s c l e a r l y a c l o s e d domain
xb
c o n t a i n i n g 0 , and P r o p o s i t i o n 4 . 1 1 shows t h a t it i s open. We deduce froni t h i s t h a t
A =
[ 0,
1
[I
,
and hence u = 0 i n
CE*
S i n c e C i s t h e union o f t h e CE, w e o b t a i n t h e c o n c l u s i o n o f Theorem 4.13 when
0, T]
u E C"(E
X
.
R")
I n t h e g e n e r a l c a s e , we can assume ( b y t r u n c a t i n g t h e neighbowhood of C ) t h a t
u
E
cm([
0 T ] ];'H
u
in
f o r some
p5
for j = 1
... m.
By h y p o t h e s i s , we have f = 0 i n C and each g
= 0 i n So.
Let
u
E
IR.
We p u t Pu
f
,
$-I
ulb0
=
j
(SEC. 4)
p
k
377
H Y P E R B O L I C CAUCHY PROBLEMS
n (x) be a r e g u l a r i s i n g sequence i n IR ; t h e n t h e s o l u t i o n
\
of
t h e Cauchy problem P U k
=
f+Pk
c"(: o T
is i n
3
.
R")
x
We again denote by C € t h e back-
-
ward cone i s s u i n g f r o m
(to E, xo]
and by S
t = 0.
For k s u f f i c i e n t l y l a r g e , we have
CE and
gj
*
pk
=
in S
0
energy i n e q u a l i t y ( 4 . 9 . 1 ) \(t,
S
.
E'
consequently
with s = u
. ) t e n d s towards u ( t , . ) i n H
obtain u = 0 i n C
0
-
E
=
0
in
= 0 i n CE.
The
f W
\
i t s t r a c e on
pk
m + 1 shows t h a t
when k
-+
+
We t h u s
which e s t a b l i s h e s t h e theorem.
E'
We t h e r e f o r e deduce t h a t t h e Cauchy problem i s w e l l posed for P.
THEOREM 4 . 1 4 :
i n Theorem 4.13. gj
E C"(R")
(4.14.1.)
The asswrptions on P are the same as those
Then, f o r
j = 0,
Pu = f
..., ,
m-I
f
0T
1
X
En]
-
the Cauchy problem
t uIt=o
0'
a h i t s a unique s o l u t i o n i n
PROOF:
E C"([
j = 0,
'j
o T 3 x R")
..., a
The uniqueness follows from Theorem 4.13.
s h a l l now prove t h e e x i s t e n c e . i d e n t i c a l t o u n i t y for I s 1 s 1.
Suppose We put
wl
6(s] E CZ(R]
We is
378
( c w . 6)
EVOLUTION EQUATIONS
Theorems 4.9 and 4.13 show t h a t t h e r e e x i s t s a unique s o l u t i o n
uk E C z ( [ 0 T I x
I?")
of
(4.14.1) for the data
f k and g
j,k'
Theorem 4.13 shows t h a t % ( t ,x) = % , ( t ,x) for k' t k and
1x1
-
5 k
u E C"([
vT. 0
T
Consequently, t h e f u n c t i o n s
2
R"]
x
REMARK 4 . 1 5 : data f
E
u E Cmc 0 T
REMARK
3, & J , t h e
4.16
satisfies
a limit
(4.14.1).
The same method a l l o w s us t o show t h a t f o r
E &*
,there
s o l u t i o n of
(4.14.1).
0 T ] ; &),
C"([
u
and
B admit
gj
e x i s t s a unique
I n t h e c a s e where t h e o p e r a t o r P i s no l o n -
:
g e r d i f f e r e n t i a l , t h e r e i s i n g e n e r a l no n o t i o n of a propag a t i o n speed ( s e e E x e r c i s e 7 . 1 ) .
5.
PAMBOLIC CAUCHY PROBLEMS I n t h i s s e c t i o n , we c o n s i d e r o p e r a t o r s P o f t h e form
P = Dt
-
A where A ( t , x, Dx) is a p . d . 0 .
of degree m > o i n
x , whose complete symbol a ( t , x, 5 ) i s i n BLm(IR"+'
x IR" ; qEN,
p))
and admits a p r i n c i p a l p a r t
which is homogeneous of degree m i n
DEFINITION
5.1
i f there e x i s t s a
:
3 E IRn\ 0
.
am
We say t h a t P i s paraboZic f o r t f
ern mapping:
r o ( t , x, 5 ) w i t h values i n N
(t, x, 5) E I R ~ 'x (R"\ 0 ) --> x
N hermitian matrices, which i s
(SEC. 5 )
PARABOLIC CAUCHY PROBLEMS
homogeneous of degree
in 5
o
I
with a22 i t s d e r i v a t i v e s f o r
E
5
Dn
I
1
379
and bowzded together
0
= 1, and which i s such t h a t
where c i s a constant,^ > o independent of (x, 5 ) and of t i n a compact subset of IR ,
We s h a l l now p r o v e , a n a l o g o u s l y t o Theorem 4.3 , t h e f o l l o w i n g energy i n e q u a l i t y :
THEOREM 5 . 2 :
t
.
Let P = Dt - A be a paraboZic operator f o r
Then, f o r aZZ s
E
and
R
T >
0,
there e x k t s C
such t h a t r T
( w i t h m' = 3m) for u E C1([oT]
PROOF:
t E COT]
; Hs) n C " ( [ o T ]
I
and
, H-')
,
We s t a r t by showing t h a t , under t h e above cond-
i t i o n s , t h e r e e x i s t s C' such t h a t for t
E
COT] we have
380
EVOLUTION EQUATIONS
(CHAP.
6)
t h e l e f t - h a n d s i d e a l l o w s u s , v i a an argument analogous t o t h a t used for Lemma 4.4, t o show t h a t i n f a c t we have
We t h e n deduce, by a g a i n u s i n g (5.2.2) t h a t , an upper s'
bound for
loT Ilu(t',
of.(5.2.3)s.
d t ' i s given by t h e r i g h t - h a n d s i d e
The i n e q u a l i t y ( 5 . 2 . 2 ) s has t h e advantage of be-
i n g p r e s e r v e d i f P i s p e r t u r b e d by an o p e r a t o r of degree m
-
i n x.
I n f a c t , P r o p o s i t i o n 2 . 1 1 of Chapter I1 i m p l i e s t h a t
for a l l
E
Ildt
1
> o we have t h e upper bound
9
lII,,',
A s i n t h e proof o f Theorem 4.3, we s e e t h a t it i s s u f f icient t o establish
5.2.~)~.
Using n o t a t i o n analogous t o t h a t used i n t h e proof of Theorem 4.3, we s t a r t from t h e e q u a l i t y (4.3.3).
By v i r t u e of
assumption (5.1.1), G h i n g ' s i n e q u a l i t y shows t h a t t h e r e e x i s t
(SEC.
5)
c > o
PARABOLIC CAUCHY PROBLEMS
381
and C such t h a t
S u b s t i t u t i n g i n ( 4 . 3 . 3 ) , we o b t a i n for t F ( t ) I-c Ilu(t,.)llz+C
10 TI
E
. ) I f + Ilu(t,-)lk-
[llu(t
2
'lf(t I 2
2
from which we deduce t h a t f o r t
The upper bound ( 5 . 2 . 2 )
0
E
[o
TI we have
follows from ( 5 . 2 . 4 ) by pro-
ceeding as for ( 4 . 3 . 4 ) and ( 4 . 3 . 5 ) .
Using f u n c t i o n a l a n a l y s i s arguments analogous t o t h o s e used )t
i s parabolic
i n t h e proof of Theorem 4.3, and n o t i n g t h a t P for t \
, we
deduce:
Suppose P = Dt - A i s parabolic f o r t f .
THEOREM 5 . 3 :
Let
s EIR
I
T> o
I
E L2 ([o T ] ; H-'),
f
there e x i s t s a unique ( i n &(lo u
E
L2[[o
(5.2.1ls. u
E
)
T[xW") T] ;
If
u E Go( [o T ] ; HS)
and u ( o
,
g
E
HS
.
Then
such t h a t Pu = f
.
Furthermore,
and s a t i s f i e s the energy i n e q u a l i t y
Hm')
f E Cm ( [ o T ] ;
Cm([o T ] ; H+>
.) =
g
H+>
and g
E
H
+m
, then
.
We s h a l l now show t h a t t h e c l a s s i c a l p a r a b o l i c o p e r a t o r s
382
EVOLUTION EQUATIONS
(CHAP.
6)
a r e i n f a c t p a r a b o l i c i n t h e above s e n s e .
DEFINITION AND THEOREM 5 . 4 : P = D
The operator
t - A ( t , x, DX ) i s said t o be parabolic i n the sense of
Petrovski f o r t
Im X 2 c
I
5
?
if'there exists c
> o
such t h a t
1"
Assume
f o r any eigenvalue A of am (t, x, 5 ) . f u r t h e r t h a t a m ( t , x , 5 ) i s independent of ( t , x) f o r
It I + 1x1 s u f f i c i e n t z y large; then such an operator P is
parabolic i n the sense o f D e f i n i t i o n 5.1.
The proof i s based on t h e f o l l o w i n g lemma:
L E W 5.5 :
Let M be an N
x
N matrix whose e i g e n v a h e s
have s t r i c t l y p o s i t i v e imaginary parts.
Then there e x i s t s a
h e m i t i a n matrix H such t h a t H > o and I m ( H . M ) >
PROOF:
We know t h a t t h e r e e x i s t s a Jordan b a s i s
..., e N ) ; l e t
(el,
0.
T denote t h e change-of-basis
matrix.
We
have :
T
with
M T-'
e~
j
ImX.
J
Let
E
>
0;
replacing t h e basis (el,
=
o
or 1
> o
..., eN) by
the basis
5)
(SEC.
383
PARABOLIC CAUCHY PROBLEMS
( e l , Ee2, ,
..,
N-1 E
e,),
we can assume t h a t a
= o or
j
E.
We o b t a i n
Furthermore, we can p u t T i n t h e form T = U
P-'
H M
I
U
We o b t a i n
u n i t a r y and P s t r i c t l y p o s i t i v e h e r m i t i a n .
M =
. P with
+ E N) U P ; p u t t i n g H = P2, we have L e t w E CN; we f i n d t h a t U*(D + E N) U p
U*(D
P
h (H M w
.
, W)
E
Im
(D U
ID P W1l2
1 c with c = i n f
P w
, U P w) +
E
-
wll
(Im X.) > o J
j
E
c jlu P
and
C
PII
h
(N U P
.
11
N
, U P W)
w
Consequently, t h e matrix H meets o u r requirements when we
This lemma shows t h a t f o r ( t o ,x there exists H
0
> o such t h a t
by c o n t i n u i t y we a l s o have
( t , x , 5 ) n e a r ( t o ,xo, 5 ' ) .
Ln
0
,
5')
E
[o
TI x I R n x Sn-l,
(Ho am(to, xo, go) )
fm (Ho a,(t,
x,
5) ] >
o
> o for
By p a r t i t i o n o f u n i t y , t h i s
;
384
EVOLUTION EQUATIONS
(CHAP.
6)
allows us t o c o n s t r u c t r ( t , x , 6 ) s a t i s f y i n g ( 5 . 1 . 1 ) for 0
1
6
I
= 1, and we next extend ro i n t o a homogeneous f u n c t i o n
of degree o i n 5.
EXAMPLE 5 . 6 :
L e t Q ( x , D ) be a s t r o n g l y e l l i p t i c X
s c a l a r d i f f e r e n t i a l o p e r a t o r of degree 2m w i t h c o e f f i c i e n t s
I
independent of x for
at +
(x
x
1
large.
Then t h e o p e r a t o r
Dx) i s p a r a b o l i c i n t h e s e n s e of P e t r o v s k i for t f ;
t h i s i s t h e case with t h e h e a t operator.
6.
SEMIGROUPS OF OPERATORS AND APPLICATIONS
6 . 1 INTRODUCTION
:
If we assume i n Theorems 4.3 and
5 . 3 t h a t t h e o p e r a t o r s A ( t , x , Dx) a r e independent o f t , t h e n t h e corresponding Cauchy problems w i t h f = 0 can be w r i t t e n i n t h e form o f an a b s t r a c t d i f f e r e n t i a l e q u a t i o n du = AU 1 ~ ( 0 )P (6.1 .I.)
dt
u
0
where A i s a l i n e a r o p e r a t o r w i t h domain D = H1(R";
A(x, Dx)
CN) (D = Hm(Rn; CN) i n t h e Banach space
f o r 5 . 3 ) which r e p r e s e n t s
E = L2(Rn; C N j
.
I n t h e s p e c i a l c a s e where A i s continuous from E i n t o E , t h e s o l u t i o n of ( 6 . 1 . 1 ) i s given by u ( t ) = G u w i t h t o
l z kl tk Ak
Gt. = - k > o
=
etA
,
In applications, A i s not i n
(SEC. 6 )
385
SEMIGROUPS
R+
g e n e r a l continuous; however t h e s o l u t i o n
3 t --u(t)
of
(6.1.1)i s continuous f o r s u i t a b l e s p a c e s ; t h i s l e a d s t o :
DEFINITION 6.2
We designate as a strongly continuous
:
semigroup (abbreviated t o S.G.) defined on a Banuch space E, a mapping t E
R+
(6.2.1.)
.
-
Gt
Gtl+%
E =
GtlO x E
for a l l
l6.2.2.)
E) such t h a t
J(E,
G4!
,
E
f o r tl,
x(.)
L 6E. 3 :
~
E E
E =
We t a k e
we p u t ( G t x)(s) = x ( s
-
z 0
I
* I
Go
the f u n c t i o n i s continuous.
tEE+-GtxEE
E
5
?([ 0 ,
+=
[I
,
and f o r
t).
We can d e f i n e , i n a c e r t a i n s e n s e , t h e d e r i v a t i v e of G
t
at
t = 0.
DEFINITION 6.4 erator of a S.G.
o
=
:
(Gt)
C
x
We designate as the i n f i n i t e s i m a l genthe operator A with domain
E E
I 5
defined by
Ax =
lim
-
1 ; ( G
lim
~x
- x)
exists
o+
-1s (Gs x - x)
s '0 +
Thus, i n Example 6.3, w e have and
A =
d
,
D =
HI([
0,
+
0)
[)
1
386
(CHAP.
EVOLUTION EQUATIONS
6)
An S.G. has a t most e x p o n e n t i a l growth; t h a t i s
PROPOSITION 6 . 5 :
real numbers
w
(6.5.1.)
liGt\/
Let (Gt ) be an S.G.;
then there e x i s t
and M such that S
M eWt
for
9
t
[I
We can endow E with an equivalent norm
which G s a t i s f i e s ( 6 . 5 . 1 ) with M = 1 and the same t
PROOF:
[ 0,
t
1
Since t h e o p e r a t o r s G
1, t h e
t
0
.
111
for
2
w.
are simply bounded f o r
Banach-Steinhauss theorem i m p l i e s t h e e x i s t e n c e
o f M such t h a t
\\Gt\l
S
for
M
t € [O I ]
a
Any r e a l t t 0 may be w r i t t e n t = p + r w i t h p
r
E
[
E
Bi and
P
0, 1 [ ; t h i s allows us t o w r i t e Gt = (G1)
Gr.
We t h e r e -
f o r e deduce t h a t
MW1
IIGt\l
-
I
M ep Lg
M
em w i t h
u)
=
Lg M
which proves ( 6 . 5 . 1 ) .
UlxlIi
We p u t
=
sup
l[e'"
Gt x\I
; t h i s i s an equiva-
t r o l e n t norm s i n c e t h e upper bound i n ( 6 . 5 . 1 ) i m p l i e s
l1\~1]l
M
11~\1
and we c l e a r l y have
l\x\l 9 lilx\lf
.
Finally,
(SEC. 6 )
387
SEMIGROUPS
(s+t)
-u)
=
/je
sup
G*t
cut XI1
S T D
hl
REMARK 6.6
If we p u t
:
. y ,
S.G.
lllGtll\
which s a t i s f i e s
Gt S 1
=
.
Gt
eWt
,t h e n
&
(Gt)
i s an
We s a y t h a t it i s a
contraction S.G.; P r o p o s i t i o n 6 . 5 shows t h a t i t i s always possi b l e t o reduce t o t h i s c a s e .
PROPOSITION 6.7 :
The infinitesirnaz generator (A, D) of
a semigroup possesses the foZZowing properties: For aZZ x
a/
and
E
D , we have Gt x
E
t Ax = A Gt x
D, G
x = Gt Ax.
GI
t
b/
The domain D i s dense i n E.
c/
The operator A i s closed; i n other words i t s graph
1
{ ( x , Ax)
PROOF OF a/
For A
x
E
=
e
E0
3
is a cZosed subspace of E
x
E.
> 0 , we p u t
-1e ( G e - I)
Then we have t h e e q u a l i t i e s
Gtx
A e
=
GtA
X
c
=
,
-1( G t + e ~ - G G t ~ )
x E E
0
We t h u s have a / by t a k i n g x E D and making
E
t e n d t o 0.
,
388
6)
(CHAP.
EVOLUTION EQUATIONS
For s > 0 , we p u t
PROOF OF b /
M
It i s c l e a r t h a t
5
-
x
.
x
o+
s
It can e a s i l y be shown from t h e d e f i n i t i o n s t h a t f o r x
E
E,
w e have t h e e q u a l i t i e s
(6.7.1 .] making
A
Msx E
=
AsM
=
x
MsA
E
E D
and
AMSx
=
A
S
x
D , t h e second e q u a l i t y i n As x
(6.7.2.)
PROOF OF Ax-,
y
n m we have A x S
obtain M
s
=
that
,
MS Ax
=
and
x
n - xn
,
Furthermore, i f
x E
for
Suppose we have a sequence x n
c/
i n E.
E
D
.
D such t h a t
From
(6.7.2),
m
= Ms Axn ; t h e n making
n
t e n d towards
y = As x ; it t h e n f o l l o w s t h a t x
REMARK 6.8 : uxlb
(6.7.1)shows
x E E
for
which p r o v e s , i n p a r t i c u l a r , t h e d e n s i t y of D . E
x
t e n d t o 0 , t h e f i r s t e q u a l i t y shows t h a t
E
Msx
x
U
E
+ -,'we
D and Ax = y .
By v i r t u e o f c / , t h e norm o f t h e graph
1}x11 + llAxll endows D w i t h a Banach space s t r u c t u r e and,
t a k i n g account o f a / , t h e S.G.
(Gt)
induces an S.G. on D .
We now r e t u r n t o t h e a b s t r a c t Cauchy problem.
(SEC.
6)
389
SEMIGROLJPS
THEOREM 6 . 9 :
Let
A be a Zinear operator with domain D
i n E.
Suppose t h a t A i s the i n f i n i t e s i m a l generator of an S.G.
(Gt).
Let
a unique
E C"([
0 T
C1([ 0 T
3;
f
u E
(6.9.1 .)
1; 0 ) n
E)
= Au(t)
u'(t)
for
and uo
C"([
+
D ; then there e x i s t s
E
0 T
1;
D) such t h a t
f(t)
,
t 2 0
u(0)
.
uo
=
We aZso have the same concZusion with the assumption f E
C1([ 0 T
3 ; E)
We s h a l l now prove uniqueness. u
0
= 0 , f = 0.
. Let
The f u n c t i o n v(s) = G
t-s
u
be a s o l u t i o n w i t h
.u(s) (0 5 s
t) is
S
d i f f e r e n t i a b l e and has t h e d e r i v a t i v e
Thus v ( s ) = v(0) = 0 , s o t h a t u(t) = v ( t ) = 0 .
I n o r d e r t o prove e x i s t e n c e , we can suppose t h a t u
0
s i n c e G u i s t h e s o l u t i o n of (6.9.1) with f = 0. t o
= 0
We s h a l l
now show t h a t
0
me
i s a solution.
mapping
01s 4 t
-
G G S f(s)
E
is c o n t i n u o u s , i n view o f Remark 6.8; consequently
u E Go([
0T
];
D)
, We now c a l c u l a t e t h e d e r i v a t i v e of u;
t a k i n g h > 0 , we have:
D
(CHAP. 6)
EVOLUTION EQUATIONS
390
Since A i s a c l o s e d o p e r a t o r , t h e assumption on f shows t h a t for h
+
0 , t h e right-hand s i d e p o s s e s s e s a l i m i t e q u a l t o
A u(t) + f(t).
u i s a s o l u t i o n o f (6.9.1)
This shows t h a t
and i s i n t h e space Ci([
1; E).
0 T
I n t h e c a s e o f t h e assumption f
E
1
C ( [ 0 T I ; E) we use t h e
same t y p e o f argument, t a k i n g
u(t)
=
Gs f(t-s)
,
ds
.
e x e r c i s e 7.3.)
(cf.
I n o r d e r t o apply t h i s theorem, w e f i r s t need a c h a r a c t e r I n view of Remark
i s a t i o n of t h e g e n e r a t o r s of S.G.'s.
6 . 6 , we
s h a l l c o n f i n e our a t t e n t i o n t o t h e c a s e o f c o n t r a c t i o n S.G.'s.
PROPOSITION 6.10 :
Suppose ( Gt
i s a contraction S. G.;
then we have the following properties:
a/
The mapping
h E C
-
+a,
Rh
Gt d t E S(E,
= 0
i s defined and hoZomorphic f o r Re 1 > 0 IIRX// b/
Reh
, and
satisfies
1
*
The operator RA is the resolvent of A i n the open s e t
Re X > 0 ; i . e .
E)
(SEC. 6 )
SEMIGROLJPS
(1-A)
Rk
0
4
55
9
5
D
391
=
(1-A)
5
The proof of a/ follows immediately from the upper bound for
1 IGt[ I .
In order to prove b/, we start by establishing the
following lemma :
6.11
LEMMA
A
:
R1 x
E
We have
E
PROOF:
4
,
XRh-x
x E E
o+
.
We have by definition.
-At
(eke
- I) G t x d t - -1
e
I"
€
e- h t Gt x d t .
0
It is therefore sufficient to apply the Dominated Convergence Theorem in order to obtain the lemma.
From this lemma, we deduce that x A
Rh
x
=
1 R1 x
Furthermore, A
E
-x
and R
E
E implies RX x
; in other words (1
x
- A].RX
=
E
D and
IE ,
commute; consequently, we obtain in the
limit ARX x = RA Ax for x
E
D and this concludes the proof of the
proposition.
This proposition has the following important converse:
THEOREM 6.12 (Hille and Yosida) :
Suppose A is a linear
(CHAP. 6 )
EVOLUTION EQUATIONS
392
operator defined in a domain D which is dense in t h a t there e x i s t s A.
such t h a t its reso2vent
Suppose
E.
RA = (A
- A)”
e x i s t s and s a t i s f i e s
lPAIl
(6.12.1 .]
71
5
for
A
A,
2
Then, (A, D) is the infinitesirnu2 generator of a contraction S. G.
The proof w i l l be performed i n a number of s t e p s .
-.
First s t e p A RA x
We show t h a t f o r x
x
E we have
E
From t h e d e f i n i t i o n of R
A+,
A R
A x = R A Ax + x for x
A RA x
-
x
for x
E
E
1
we have
D , and t h e r e f o r e (6.12.1) i m p l i e s
Since t h e o p e r a t o r s A R
D.
A are
equicontinuous and D is dense, t h e convergence holds f o r a l l x
E
E.
Second s t e p
Construction of a semigroup Gt .
Az Rh
BA = A A RA i s e q u a l t o
The o p e r a t o r
-A
; it i s t h e r e f o r e t Ex continuous and we can a s s o c i a t e with it t h e group e ‘‘ This
.
-
group w i l l s e r v e t o approximate ( G ) s i n c e Bh x
=
A Rh Ax
t
Ax
when x
A + -
commute, t h e same i s t r u e of BA and %
*
U’
E
0.
Since R
A
and R
P
t h i s allows us t o w r i t e
6)
(SEC.
SEMIGROUPS
393
u s i n g t h e upper bound.
l:BXll
=
=- t X
t h2 FIX lle
11
e-tX e
5
I ,
I
we t h e n deduce t h a t
This i n e q u a l i t y shows t h a t f o r a l l x
e
BA
x converge uniformly Vhen
s u b s e t of R+ . We denote by
h
-+
D t h e functions
f o r t i n a compact
w
Gt x t h e l i m i t f u n c t i o n ; t h i s i s
t h e r e f o r e continuous and s a t i s f i e s
llGt
XI\
F i n a l l y , s i n c e D i s dense, we can extend G S.G.
E
t
5
1 for x
E D.
i n t o a contraction
on t h e whole o f E .
Third step
W e now show t h a t
g e n e r a t o r of '(Gt].
Let
be i t s domain; l e t t i n g h e
Bh x - x
A
is t h e i n f i n i t e s i m a l
be t h e g e n e r a t o r of (Gt)
-+
, the
w
and l e t
equality
Bh x ds
=
implies t h a t It G t x - x
p
Gt A x ds
(x
E D)
.
394
(CHAP. 6 )
EVOLUTION EQUATIONS c
We then deduce t h a t A extends A and s i n c e (Ao
-
A ) i s by
e
-
h y p o t h e s i s a b i j e c t i o n of D onto E , we must have D = D and t h u s A = A.
The f o l l o w i n g c o r o l l a r y i s o f t e n used i n a p p l i c a t i o n s :
Let A be a l i n e a r operator with domain
COROLLARY 6.13 :
Suppose t h a t there e x i s t A
D dense i n E .
the resolvent RA = (A - A ) - l
and w such t h a t
e x i s t s and s a t i s f i e s
A - w 1
-<
(6.13.1 .)
0
Then A i s the i n f i n i t e s i m a l generator of an S.G.
( G t ) such
wt
that
llGt\j I e
PROOF:
We immediately g e t back t o t h e p r e v i o u s theorem
by c o n s i d e r i n g t h e o p e r a t o r B = A
REMARK 6.14
:
-
w.
Feller-Miyerdera and P h i l l i p s have given a
c h a r a c t e r i s a t i o n o f S.G. g e n e r a t o r s without making any a s s m p t i o n s on t h e growth o f
(lGt(l
, It is n e c e s s a r y t o r e p l a c e
( 6 . 1 3 . 1 ) by t h e following c o n d i t i o n : t h e r e e x i s t l o , w and M such t h a t
llql
M
(x - w)n
for
A S A0
and
n E N
.
(SEC.
6)
6.15
395
SEMI GROUPS
APPLICATION TO THE HEAT EQUATION WITH DIRICHLET CONDITIONS
Let
an,
be a bounded open s u b s e t o f Bn w i t h r e g u l a r boundary =
and l e t Q(x, Dx)
qw(x) Dw
be a d i f f e r e n t i a l
Ial s 2 m We suppose t h a t t h e r e
operator with c o e f f i c i e n t s i n C a ( E ) . e x i s t s c > 0 such t h a t Q
(6.15.1 .)
5
QX)
for
For given f and u0 ’ we s e e k
( (at (6.15.2
-
Q(X,
D x j ) u(t,
1
5) E
(x,
x
R”
u satisfying
XI
u(0, X I U(t,
= Id2”
2
XI
=
f ( t , XI
=
.,b)
I
0
IR+ x an
in
R+
x
(Dirichlet condition)
( f o r m = 1, t h i s i s an e q u a t i o n of t h e h e a t t y p e ) .
We t r a n s f o r m t h i s problem i n t o an a b s t r a c t d i f f e r e n t i a l e q u a t i o n by i n t r o d u c i n g t h e l i n e a r o p e r a t o r A w i t h domain
D =
l-?”(n) n HZ(n)
Au = Qu.
i n t h e space E = L
2
( n ) , d e f i n e d by
We t h e n have:
THEOREM
6.16
:
The operator ( A , D ) ‘is the generator of a
strongly continuous semigroup i n E .
n
39 6
(CHAP. 6 )
EVOLUTION EQUATIONS
PROOF:
The Girding i n e q u a l i t y ( 8 . 6 . 2 )
t h a t t h e r e e x i s t c > 0 and A
such t h a t
0
Re [- Q [ x , D)u, u>
(6.16.1.)
+
~ [ u ,u) u
for
0
.
2
ED
llulf,
c
1
and
2
.
A,
- A i s i n j e c t i v e from D i n t o E
Consequently, t h e o p e r a t o r A for A 2 A
, Chapter IV, shows
Furthermore, we showed i n Theorem 5 . 2 , Chapter V,
t h a t i t s index i s z e r o ; it i s t h u s b i j e c t i v e .
I n order t o
a t t a c h an upper bound t o t h e r e s o l v e n t of A, we minorise t h e norm
- A)u[f
=
h2
llulf +
Using (6.16.1) w i t h A = A o ,
II(A
-
z
~ ) u f
- 21
(A'
- 21 Re(Au,
(\Au(r
u>
,
u E
D
I
D ,
we o b t a i n A ~ ) / / U ~ C
for
u E
for
A
from which we deduce
- A)-'\l
!\(A
(6.16.2.)
I
1
p
.
X,
hO
This i n e q u a l i t y and C o r o l l a r y 6.13 imply Theorem 6.16,
2 s i n c e i n t h i s case t h e space D i s obviously dense i n L ( a ) .
From Theorem 6.9 we deduce t h a t f o r and
u
0
E
$"(EJ n
H:[n)
, the
f
E
C1([
0T
3;
L2(fi))
problem ( 6 . 1 5 . 2 ) admits
a unique s o l u t i o n
Ed[[
o
T
I; ~ ~ ( nn ) co([ )
o
T
3;
~ " ( E In
H;(SI>)
We can i n f a c t show t h a t it i s s u f f i c i e n t t o assume t h a t f E Cot[
0T
3;
L2(Cl])
.
.
(SEC. 6 )
397
SEMIGROUPS
6.17
REMARK
If w e r e p l a c e S2 byIRn and i f we assume t h a t
:
the coefficients q
a r e bounded t o g e t h e r w i t h a l l t h e i r deriva-
CY
t i v e s , t h e n Theorem 6.16
remains v a l i d word f o r word.
In fact,
m t h e t h e o r y of pseudo d i f f e r e n t i a l o p e r a t o r s i n BL ( s e e Chapter I V , S e c t i o n 11.1) allows us t o prove t h e i n e q u a l i t y
(6.16.1) , and t h e Lax-Milgram theorem ( s e e S e c t i o n 7 . 4 , Chapter 11) shows t h a t A
6.18
-
A is bijective for A
A
0
.
APPLICATION TO THE WAVE EQUATION WITH DIRICHLET CONDITIONS
We make t h e assumptions on 0 and Q which were made i n S e c t i o n 6.15; i n a d d i t i o n , we t a k e Q i n t h e form
a.
q.
1,J
cients
.[XI a, +
(terms of degree 5 1) w i t h r e a l c o e f f i -
lJ
qij
.
For g i v e n f , u
0,
u
1,
we seek
u satisfying
( D i r i c h l e t boundary c o n d i t i o n )
.
(CHAP. 6 )
EVOLUTION EQUATIONS
398
I n o r d e r t o reduce t o a n a b s t r a c t d i f f e r e n t i a l e q u a t i o n , we s t a r t by r e p l a c i n g (6.18.1) by an e q u i v a l e n t system of f i r s t
a,u-au u(0)
(6.18.2.1
B U
IR+ x an
=
F
=
uo
=
o
and we denote by A t h e o p e r a t o r d e f i n e d by
THEOREM 6.19 :
S.G.
a
on D.
We t h e n have
The operator (A, D ) i s t h e generator of a
in t h e space E .
PROOF:
I n o r d e r t o i n v e s t i g a t e t h e e x i s t e n c e of t h e
(SEC. 6)
399
SEMIGROUPS
E
D
satisfying the
g i v e n i n E.
This
e q u a t i o n i s e q u i v a l e n t t o t h e system
I
v = A u - h
(A‘
- Q)u
=
+ Ah
k
However, from t h e r e s u l t s proved i n connection w i t h Theorem
6.16, t h e r e e x i s t s X
0
1x1
such t h a t f o r =
admits a unique s o l u t i o n u E i-?[m 2
i n L (Q).
1x1
2
h,
2
Ao, t h e e q u a t i o n
h
17 Hi[Q) for any g i v e n h
This i m p l i e s t h a t t h e r e s o l v e n t R o f A e x i s t s when
.
X
I n o r d e r t o a t t a c h an upper bound t o i t s norm, it i s
convenient t o endow t h e H i l b e r t space E w i t h a new e q u i v a l e n t H i l b e r t norm.
We d e f i n e a new i n n e r product ( (
,
) ) by
putting
(lu, U’))-
(:)
Isij a,
U,
i, with
U =
,
U1
=
have t h e G&rding i n e q u a l i t y
aj
us)
(1:)
+
C(U,
, where
u‘)
+ CV, vll
C i s such t h a t we
(CHAP. 6 )
EVOLUTION EQUATIONS
400
Thus t h e a s s o c i a t e d norm
111
. 111
i s equivalent t o the
o l d norm on E .
The upper bound f o r t h e r e s o l v e n t w i l l f o l l o w from
LEMMA 6.20 :
There e x i s t s
C'
such t h a t
n (6.20.2.)((AU,
aj
(Sij
U))=
+
c(v~
i,J= 1
[u
-
(pu, v).
1 Since C " ( Q ) i s dense i n H ( a ) , we have t h e e q u a l i t y o n 0 U, a . V) + [LU, v) f o r all gU, v j = csi j J i,J= 1
a,
E
H
2 -
( a ) and v
first order.
E
1
Ho(R),
and L i s a d i f f e r e n t i a l o p e r a t o r of
S u b s t i t u t i n g t h i s e q u a l i t y i n t o ( 6 . 2 0 . 2 ) ~we
obtain Re((AU, U))
=
C Re(v, u)
+
*[Lu,
v)
*
(SEC. 6 )
SEMIGROUPS
401
This proves t h e lemma.
F i n a l l y , proceeding e x a c t l y as for ( 6 . 1 6 . 2 ) , we deduce from (6.20.1) t h a t
- As-91
5
1x1
for
+o
z
lo
I
and t h i s proves t h e theorem.
(Using E x e r c i s e 7 . 2 below, it can be shown t h a t A i s t h e g e n e r a t o r of a s t r o n g l y continuous g r o u p ) .
A s an a p p l i c a t i o n , we deduce t h a t f o r
problem (6.18.1) admits a unique s o l u t i o n
REMARK 6 . 2 1 :
I f we r e p l a c e s2 bylR
n
, we
have t h e analogue
o f Remark 6.17.
REMARK 6.22 :
I n ( 6 . 1 8 . 1 ) , we can r e p l a c e t h e D i r i c h l e t
c o n d i t i o n by t h e Neumann c o n d i t i o n
ac
P
o b t a i n analogous r e s u l t s ( s e e E x e r c i s e 7.5 below).
0
j
we t h e n
EVOLUTION EQUATIONS
402
(CHAP. 6)
ADDITIONAL NOTES AND EXERCISES
7.
7.1
COUNTER-EXAMPLE TO THE PROPERTY OF FINITE PROPAGATION SPEED IN THE NON-DIFFERENTIAL CASE
Let P = D
t
- A(Dx),
where A
E
1
L (B)i s t h e pseudo-
d i f f e r e n t i a l o p e r a t o r w i t h symbol 161.
The o p e r a t o r P i s t h u s
s t r i c t l y h y p e r b o l i c w i t h c o n s t a n t c o e f f i c i e n t s and of degree one. Show t h a t t h e s o l u t i o n of t h e Cauchy problem
i s given by
U(t
, f)
Show t h a t f o r t
5 5
-
e-it!c\
#
= e-it
Is 1
0 , t h e i n v e r s e F o u r i e r t r a n s f o r m of
does not have compact support ( n o t e t h a t
i s not
c").
7.2 GENERATOR OF A STRONGLY CONTINUOUS GROUP Let E be a Banach space; we have t h e n o t i o n o f a s t r o n g l y
-
-+
continuous group i f we r e p l a c e lR a)
group. (Gt)t
b)
t ER
Let
We denote by (A ~
o;
G
t
E
, D)
by B
i n D e f i n i t i o n 6.2.
C(E, E ) b e a s t r o n g l y continuous
t h e g e n e r a t o r of t h e semigroup
show t h a t (-A, D) g e n e r a t e s a semigroup.
Conversely, l e t (A, D ) be an o p e r a t o r such t h a t (A, D ) and
(-A, D ) g e n e r a t e semigroups; t h e n show t h a t t h e semigroup
7)
(SEC.
ADDITIONAL NOTES
403
g e n e r a t e d by A extends onW i n t o a group.
CLet G l ( t ) -A;
and G 2 ( t ) be t h e semigroups g e n e r a t e d by A and
w e d e f i n e G ( t ) by G ( t ) f o r t t 0 and by G 2 ( t ) f o r t 5 0. 1
Show t h a t f o r x
E
D t h e f u n c t i o n u ( t ) = G ( t ) x is i n
1
C ( B Y E ) n Co(WyD) and s a t i s f i e s u ’ ( t ) = A u ( t ) f o r t
E
IR; deduce
from t h i s t h a t G ( t ) i s a g r o u p l .
FURTHER DISCUSSION OF THEOREM 6.9
7.3
L e t E be a Banach space and suppose ( A , D ) g e n e r a t e s a semigroup ( G ) i n E .
Consider t h e d i f f e r e n t i a l e q u a t i o n
t
(7.3.1)
u’(t)
a)
Take f
of
(7.3.1):
E
Co(%+,
D ) and u
0
for
E
t 2 0
D and l e t
and u(0)
u
E C 1(R+
I1 u ’ ( t ) II s C (
[use t h e expression
Let
f
uo
ED
.
be t h e s o l u t i o n
11
Auo
, E), +
we haye
f ( 0 ) 11
+
(I f ’ ( s ) 11 ds) for
b)
-
Show t h a t f o r a l l T t h e r e e x i s t s C such t h a t
If i n a d d i t i o n f
(7.3.3)
+ f(t)
= Au(t)
E C’(E+
u(t)
E)
f E ( t ) = M E f ( t ) where M
E
P
Gt uo
.
+
; we define f o r
E
t E
3
YO, T
f(t-s)ds
1
,
> 0 t h e f u n c t i o n f c by
i s as d e f i n e d i n t h e proof of
404
(CHAP. 6)
EVOLUTION EQUATIONS
Proposition 6.7,
Show that
and that f converges to f in E
fe
E C'(E+ ; E) n Co(F+ ; D)
C'm+
; E)
.
1 Let f E C E+ ; E ) and u0 E D; prove that (7.3.1)
c)
unique solution u
c
C
1
E+ ; E ) n Corn+
admits a
; D) and that this s o l -
ution satisfies the inequalities (7.3.2) and ( 7 . 3 . 3 ) .
[Let uE be the solution corresponding to fE '* use the inequalities (7.3.2) and (7.3.3) to show that uE is a Cauchy sequence].
Let k
d) Dk
=
E
lN ; we define by recurrence ?Be spaces Dk b4-
E Dkm1
{U
1A
and we put D0 = E.
u E Dk1,3
Show that
Dk is stable under G and is dense in E, and show that it is a t Banach space for the norm
Let k
2
2; take
E c ~ ( E )n Cka(Dl)
f
Dk; show that the solution u
and uo
E
space
c~(E)
n c~-'(D,) n
888
n C'(D,J
n
...n
C'(D~-,)
of (7.3.1) is in the
.
7.4 ENERGY INEQUALITY FOR THE WAVE EQUATION We adopt the conditions of Theorem 6.19. that there exists C such that if u corresponding to data f
E
Let T > 0; show
is the solution of (6.18.1)
C1(CO,TI; L2(n)),
ug E
!-?(E) n Ho(n), 1
7)
(SEC.
405
ADDITIONAL NOTES
[ u s e t h e i n e q u a l i t y ( 7 . 3 . 2 ) 1.
7.5
WAVE EQUATION WITH NEUMA" BOUNDARY CONDITIONS.
We adopt t h e assumptions of problem
( 6 .18.1) and we assume
t h a t t h e c o e f f i c i e n t s of t h e o p e r a t o r Q(x, Dx) a r e r e a l .
Ln
what f o l l o w s we s h a l l c o n s i d e r o n l y r e a l - v a l u e d f u n c t i o n s I n (6.18.1) we r e p l a c e t h e D i r i c h l e t c o n d i t i o n s by t h e Neumann condition :
where
-
d d e n o t e s d i f f e r e n t i a t i o n i n t h e d i r e c t i o n conorrnal t o Q, dn
J
We wish t o e x t e n d t h e proof o f Theorem 6.19 t o t h i s s i t u a t i o n .
a)
We d e f i n e on E = H1(C)
x
L 2 ( a ) t h e symmetric b i l i n e a r form
where U = (u,v) and U ' = (u',v') a r e i n E . p o s s i b l e t o choose a s s o c i a t e d with ( ( space E .
c
,
such t h a t t h e norm
Show t h a t it i s 0
11
) ) d e f i n e s t h e topology of t h e product
406
EVOLUTION EQUATIONS
6)
2 2 Let H N ( Q ) be t h e c l o s e d subspace o f H ( Q ) c o n s t i t u t e d by t h e
b)
s e t of the u
E
H
2
(Q) which satisfy ( 7 . 5 . 1 ) .
2 product space D = H N ( R )
A
(CHAP.
=(:
1
x
We d e f i n e t h e
1
H ( Q ) and t h e o p e r a t o r
)from D i n t o E .
Show t h a t t h e r e e x i s t s C such
that
be
c) Ihl
((AU
, U))!
/If
s Clll U
Prove t h a t t h e o p e r a t o r ( A
-
> C ; deduce from t h i s t h a t ( A ,
for a l l
A): D
-t
U ED
.
E i s bijective for
D) g e n e r a t e s a s t r o n g l y
continuous group i n E .
e)
Use t h e i n e q u a l i t y (7.3.3) i n o r d e r t o prove t h a t t h e sol-
u t i o n o f d ) s a t i s f i e s t h e f o l l o w i n g energy i n e q u a l i t y :
(SEC. 7 )
for t
E
407
ADDITIONAL NOTES
LO,TA, w i t h C independent of t h e s o l u t i o n u.
(It
should be p o i n t e d out t h a t t h e method o f semigroups can a l s o be a p p l i e d t o mixed problems of o r d e r g r e a t e r t h a n 2; s e e MIZHOHATA [ll )
.
EQUATIONS OF THE SCHRGDINGER TYPE
7.6
Suppose Q(x, D ) i s a formally s e l f - a d j o i n t d i f f e r e n t i a l X
o p e r a t o r of degree 2m inIRn.
Suppose t h a t i t s c o e f f i c i e n t s a r e
bounded, t o g e t h e r w i t h a l l t h e i r d e r i v a t i v e s , and t h a t t h e r e exists c
Let A
a)
u
0
E
> o such t h a t
E
jE \ 0 and w
H2m such t h a t ( A
-
E
L2; prove t h a t t h e r e e x i s t s a unique
i Q ) u = w.
[Use t h e r e s u l t s of E x e r c i s e
11.1 o f Chapter IVY and t h e Lax-Milgram theorem ( s e e S e c t i o n
7.4, Chapter I I ) ] .
b)
Deduce from t h i s t h a t t h e o p e r a t o r Q, considered as an
unbounded o p e r a t o r i n L2 w i t h domain D = H 2m , g e n e r a t e s a s t r o n g l y continuous group of u n i t a r y t r a n s f o r m a t i o n s .
408 c)
EVOLUTION EQUATIONS Apply t h i s r e s u l t t o t h e s t u d y of t h e e q u a t i o n of
Schradinger t y p e :
(3,
- i Q)u =
f , u ( 0 ) = uo.
(CHAP. 6 )
CHAPTER 7 MIXED HYPERBOLIC PROBLEMS
1.
INTRODUCTION
D ) i s a d i f f e r e n t i a l o p e r a t o r w i t h Cm
Suppose P ( t , x, Dt, c o e f f i c i e n t s inIRn+'(t
X
E
IR, x
E
I R n ) , which i s s t r i c t l y hyperbolic
with r e s p e c t t o t h e "time v a r i a b l e "
t
( s e e D e f i n i t i o n 4.9,
Chapter V I ) .
Let R be a r e g u l a r open s u b s e t ofIR
n
and T > 0 ; p u r e l y w i t h
a view t o s i m p l i f y i n g t h e n o t a t i o n i n what f o l l o w s , we s h a l l assume
an
t o be connected.
The mixed Cauchy-Dirichlet
problem
f o r t h e wave e q u a t i o n ( s e e S e c t i o n 6.18, Chapter V I ) can be g e n e r a l i s e d i n t o a mixed problem of t h e following form: given
f,
g, h =
boy. . . I
-
hm-l ) , f i n d
u satisoing
(Pu)(t, x) f(t, X I (Bu)(t, x ' ) = g(t, x ' )
in in
(1.11
]O,T[ x Cl ]O,T[ x &2
-
[boundary c o n d i t i o n s ) k
Dtu(O, x)
5
hk(x)
in
Cl
for
k
0,
...
,-I
( i n i t i a l conditions) More p r e c i s e l y , we s h a l l , i n t h i s c h a p t e r , , d i s t i n g u i s h two t y p e s of problem (1.1):t h e systems c a s e and t h e s c a l a r c a s e . I n t h e systems c a s e , P i s of degree m = 1, b u t i t s c o e f f i c i e n t s
410
(CHAP. 7 )
MIXZD HYPERBOLIC PROBLEMS
are N
x
N matrices; B(t,
Cm for (t, X I )
E
W
x
XI)
aR.
is a given d
x
N matrix of class
In the scalar case, P is of degree m
with scalar coefficients, and
Bu
=
u)
(Bj j=l,...,d
,
with
m
is a given differential operator of degree r. - k+l with J coefficients which are scalar and C in IR x an; we have put 0)
real-valued vectors transversal to
as2
.
Throughout the subsequent discussion we shall use the following assumptions: (1.2.)
7R x
an is non-characteristic for P.
(1.3.)
n an is compact, o r n is a half spaceIR+
(1.4.)
P and B have constant coefficients outside a compact set.
Then, under a supplementary condition known as the u n i f o m
Lopatinski condition, we shall give existence and uniqueness results for the solutions of (1.1).
These results are based
essentially on an a priori estimate f o r the problem without initial conditions P u - f
in
R x R
Bu = g
in
Rxan
411
SPACES AND OPERATORS
(SEC. 2 )
i n t h e spaces eyt L2 when t h e parameter y i s s u f f i c i e n t l y l a r g e . This a p r i o r i e s t i m a t e w a s e s t a b l i s h e d by KREISS c1.1i n t h e case of systems and by SAKAMOTO c11 i n t h e s c a l a r c a s e .
Here we
s h a l l adopt t h e method of K r e i s s , and we s h a l l show how t h i s a l s o allows t h e a p r i o r i e s t i m a t e t o b e o b t a i n e d f o r t h e s c a l a r
n=
,
R:
y
b
by u s i n g a r e d u c t i o n t o f i r s t n o r d e r with t h e a i d of s u i t a b l e p s e u d o - d i f f e r e n t i a l o p e r a t o r s c a s e when
P
i n t a n g e n t i a l v a r i a b l e s t , x ’ ( s e e AGRANOVITCH [11
.
We s h a l l s e e i n p a r t i c u l a r t h a t t h e uniform Lopatinski c o n d i t i o n i s s a t i s f i e d i n t h e c a s e of a s c a l a r o p e r a t o r of
m
order
w i t h D i r i c h l e t boundary d a t a ; t h i s forms a general-
i s a t i o n of problem (6.18.1) of Chapter V I .
On t h e o t h e r
hand, t h i s c o n d i t i o n i s not s a t i s f i e d i n t h e c a s e of Neumann boundary d a t a ( s e e Remark 6 . 2 2 i n Chapter V I , and E x e r c i s e (9.4).
2.
OPERATORS AND SPACES USED 2.1
Background and n o t a t i o n .
If R is a r e g u l a r open s u b s e t of IRn and i f y
e”L2(R
by
IR
x C2
x 0)
such t h a t
product ((u,vll,;y
-
and w i t h t h e norm
E
IR, we denote
t h e H i l b e r t space of f u n c t i o n s
E L2[R
e’*u
“=”u
I
llull
0 ;Y
x
e-”vll
n)
, endowed =
d e f i n e d by
u in
with t h e inner
BRxn
e-2*G
dt dx
,
(CHAP.
MIXED HYPERBOLIC PROBLEMS
412
7)
We d e f i n e i n similar f a s h i o n t h e H i l b e r t s p a c e
e*L2(R
x
bn)
(u,vIOiy
endowed w i t h t h e i n n e r p r o d u c t :
I
=
, e-*v]
(e-*u
and w i t h t h e norm
lUl20 ;Y
-
j
=
e-**uv'
,
d t dx'
Rxan
d e f i n e d by
1e-*u1
j
5
luI2 d t dx'
e-**
P
Rxbn
where dx' denotes t h e measure induced by t h e Riemannian s t r u c t u r e of
an.
We s h a l l d e s c r i b e t h e o p e r a t o r s which w i l l be used i n t h e c a s e where
R
n i s t h e h a l f spaceIR+.
x &I=
R" =
1
{(t, X I )
F i r s t we c o n s i d e r
t
E R
, x'
E
R"-' 1
and we denote by u , 5 ' t h e d u a l v a r i a b l e s o f t , x'.
, We r e f e r
t o S e c t i o n 1 1 . 2 , Chapter I V Y f o r t h e d e f i n i t i o n o f t h e symbols
E
PSm(R" x
y E
Lye, +
A
0
R"
x
[ ,
L t , + 4)
which depend on a parameter
as w e l l as f o r t h e p r o p e r t i e s of t h e
pseudo-differential operators
We d e f i n e t h e o p e r a t o r A
dyl = A ( t ,
x'
; Dt,
Dxl
, y)
.
.
A u = eyt A(y1 e-w u Y Thus, f o r u E C z i R n ) and by p u t t i n g T = u iy, we have:
Y
by
i t ?+ i 2 3 1 A ( t ,
-
x,
i
U'
S ' , Yl
!'I& &'.
(SEC. 2 )
413
SPACES AND OPERATORS
i s convenient t o use t h e n o t a t i o n Ay
X'
When A ( t , x ' ; u, i n T, 5' w i t h f o r a l l y.
T
i
c',
Dt + i Y
8
.
Y)
Dx,,
y ) = P ( t , x ' , T,
= u - i y , we s e e t h a t A
5 ' ) i s a polynomial
= P ( t , x', Dt
Y
We c l e a r l y have for t h e o p e r a t o r s A
analogous t o t h o s e of t h e o p e r a t o r s A (Y)
Y
.
Y
Dx')
properties
In particular, if A
i s of degree 0 , t h e r e e x i s t s a c o n s t a n t C such t h a t
for a l l u
E
e Y tL2 (R n)
.
p a r t i c u l a r t h e symbols operators A
s
Y
.
y > -
yo
and for a l l
I n t h e f o l l o w i n g , we s h a l l use i n
A" =
( 3+ o2 +
15'
I
)
and t h e
We n o t e t h a t if A i s o f degree -1, t h e r e e x i s t s
a c o n s t a n t C such t h a t
.
(2. I I1
C
for all and f o r a l l
In f a c t , since t h e operator A
Y
A
Y
u E
B*
Y2K L2(Rn)
.
i s of degree 0 , w e have
which g i v e s ( 2 . 1 . 1 ) s i n c e we have
We now c o n s i d e r t h e h a l f space
We h e r e use t a n g e n t i a l o p e r a t o r s which depend on t h e
414
( CH AP . 7 )
MIXED HYPERBOLIC PROBLEMS
normal v a r i a b l e x : i f n
A = A(t,
x ;
Q
,tg,
y)
i s a symbol
in
-
-
we a s s o c i a t e with i t , f o r y Av : C:(R=f')
C"(R;+l)
2
70 , t h e
operator
defined by
If A i s of degree 0 , we have an e s t i m a t e of t h e form
Reduction 2.2 We r e t u r n t o problem (1.5) i n t h e case where P i s s c a l a r of o r d e r m ,
,
n = R:
a
V =
Gn
We a l s o assume t h a t t h e c o e f f i c i e n t of Dm i n P does not X n vanish, s o t h a t we can reduce t o t h e c a s e where P i s of t h e form
with P . of degree j. J
U = (u~,...,urn).
u
= Am-J DJ-l j Y x,, Then ( 1 . 5 ) can be w r i t t e n
As i n 4.8, Chapter V I , we put
I
and
415
SPACES AND OPERATORS
(SEC. 2 )
(2.2.11
where
t
By
'Bj,k,Y
1j=4,...,d k=l
...
, ,m
We see t h a t A i s of degree 1, B
i s of degree 0 , and t h a t
t h e p r i n c i p a l p a r t (homogeneous o f degree 1) of A has t h e c h a r a c t e r i s t i c polynomial
5"
p i s t h e p r i n c i p a l symbol of P .
p(t,
x ;u
- iy , 5 )
,
where
We thus come down t o a s i t -
u a t i o n analogous t o t h a t of problem (1.5) i n t h e case o f systems, b u t with pseudo-differential operators i n t a n g e n t i a l v a r i a b l e s .
416
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
In order to encompass both types of problem (when n=R?, n+l
-
in R+
=
V
a
b;; ) ,
we are led to consider the problem
n
:
ProbZem 2 . 3
I
Y
- gA
Dxnu By" =
=
I -+I A E PS (R+ x R" x
where
a ( t , x;
form with principal part degree 1, and
Eo, + m[)
is of N
x
N matrix
u., g 1 ; Y) homogeneous of
B E PSo(R" x R" x
matrix form with principal part
[To b(t,
+ m[) XI
is of d
{',
; cr,
y
1
x
N
homo-
geneous of degree @.
We put
5")
A(X,
x =
= det(lnI
(t, x ;
-
a(X))
u
{I,
.
yj
and
We adopt the following "strict
hyperbolicity I ' assW p t ion :
(2.3.1)
When we put
T
= u - iy, then A ( X , 5 )is a polynomial n
of degree N in T,
(T,
5,)
which possesses N zeros in
which are real and simple when
Furthermore, we have A #
uER\O
.
O when
5 E R" \ 0 3
0
,
.
y = 0
,
We note that assumption (2.3.1) is satisfied in the case of
417
SPACES AND OPERATORS
(SEC. 2 )
problem (1.1)i n view of t h e s t r i c t h y p e r b o l i c i t y of P.
We
a l s o n o t e t h a t assumption ( 2 . 3 . 1 ) i m p l i e s
(2.3.2)
For y > 0 , a ( X ) h a s no r e a l e i g e n v a l u e s .
(2.3.3)
For y = 0 , t h e eigen-subspaces a s s o c i a t e d w i t h t h e r e a l e i g e n v a l u e s of a ( X ) a r e of dimension 1.
To prove ( 2 . 3 . 3 ) , we f i x t , x, o , 5 ' and p u t a ( X ) = a ( y ) ,
0,)
A(X,
= A(y
and e l , . . . , e
N
, 0),
.
Let A
be a r e a l e i g e n v a l u e of a ( 0 )
a b a s i s of CN such t h a t el i s an e i g e n v e c t o r o f We denote by C,(y)
a ( 0 ) associated with A . columns of t h e matrix A1
-
,.. .,C,(y)
a(y) i n t h i s basis.
the
We t h u s have
C ( 0 )= 0 , and assumption ( 2 . 3 . 1 ) i m p l i e s t h a t
1
M 5 (0,
1) = det ( c ; ( o )
Thus t h e columns C 2 ( 0 ) , proves ( 2 . 3 ; 3)
...,
, c2Col,
...,
C"O1)
4
0
CN(0) a r e independent, and t h i s
.
We f i n a l l y make t h e f o l l o w i n g assumption which, i n t h e c a s e o f problem (l.l), f o l l o w s from
(1.4)
a f t e r using a l o c a l
c h a r t of t h e boundary.
(2.3.4)
The symbols a and b a r e c o n s t a n t for 1x1 s u f f i c i e n t l y l a r g e and a r e independent of t f o r It1 s u f f i c i e n t l y large.
We conclude t h i s s e c t i o n w i t h two r e s u l t s which will be
418
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
used i n S e c t i o n 6.
These a r e i n f a c t very simple s p e c i a l c a s e s
of g e n e r a l p r o p e r t i e s of p a r t i a l h y p o e l l i p t i c i t y a t t h e boundary
For s
(HORMANDER C21).
= L2@+
Ho9'(Y)
E
IR, we p u t
i s t h e r e f o r e t h e space of t h e
D"
u
t,X'
E
.
, Hs(R")) u
f o r la1
L2(R:+')
1
E
2 n+l L [R+ )
5 k
.
u
for
xn)
x',
, where
E )+C $'(::
Em(T' x R")
= A ( t , x',
1
(R+
such t h a t
We denote by
A : C,(R+'n+l )
m 'n+l BL (IR+ ) t h e space o f t h e o p e r a t o r s defined by (Au)(t,
H ~k, n+l
For k E ? N ,
xn, Dt,
-O
QD 'n+l C (R+
)
Dx,)u(.,.,xn)
A i s a symbol from t h e space
d e f i n e d i n 11.1, Chapter I V .
o p e r a t o r i s c l e a r l y continuous from
-n+l
Such an
into
HO''(IR+
H~ ,s-m -n+l OR+ ) .
Let
PROPOSITION 2.4:
P = Dx
- A.
n such t h a t
PROOF: for a
n
For k an i n t e g e r Pu
E
BL1 (K"+1)
let Then
.
We r e q u i r e t o prove t h a t
We a l s o have
S k
.
Da' D
,
+
we put H ~ k, 'n+l (R+
2 1,
Hk-I (R+ -"+I )
a = (a',an), la1 = 0.
E
A
u
E
Db:
I
Hk(E:+)'
E
.
be
2 n+l L (R+ )
This i s t r u e by hypothesis when
a
a I
Da'D nAu
+
Da8D npu :
t h i s g i v e s t h e r e q u i r e d conclusion by r e c u r r e n c e over an s i n c e Au E H0,k-I (R+ '"+ )I Pu E Hk-'(T'')
,
THEOREM 2. 5 :
l e t P(x, Dx) be an N
.
Let X be a regular open subset of x
IRn and
N d i f f e r e n t i a l system of degree 1 w i t h
coefficients i n f o r P.
419
SPACES AND OPERATORS
(SEC. 2)
cm(nn). Suppose E = {u E
Let
L:oc(F)
t h a t ax i s non-characteristic 2 P u E Lloc(X)] I endowed
1
with the seminorms of (u, Pu) i n L2
loc
Then C : ( x )
(2) x
LFOc(x)
.
i s dense i n E and the trace operator y :
Cp)
> c:(ax)
"lax
>-
extends uniquely i n t o a continuous operator ( a l s o denoted by yi
from E i n t o H-112(ax). loc
PROOF:
When C:(z)
is endowed with the topology induced
y : C:(g)
by E, the continuity of
+
H-1'2loc
(ax)
immediate consequence, after using local charts of
is an
x at the
boundary, of the following lemma: 12.5.1
3
LEMMA
We adopt the conditions of Theorem 2.5,
n t h a t the c o e f f i c i e n t s of P are constant assuming t h a t X =B+,
outside a compact domain and t h a t the c o e f f i c i e n t of Dx i s the n identity. Then there e x i s t s a constant C such t h a t
In order to establish this lemma, we consider the extension 0
u of u by 0 for xn < 0. P(u0)
I
(Pu)"
+ T1 w
The jump formula
8 6(x,)
shows that
420
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
which g i v e s Lemma ( 2 . 5 . 1 ) s i n c e , i n view of P r o p o s i t i o n 4.6, Chapter 11, t h e r e e x i s t s a c o n s t a n t c > 0 such t h a t -1
I n o r d e r t o conclude t h e proof of Theorem 2 . 5 , it now remains t o prove t h a t
C"(y) 0
By t r u n c a t i o n and
i s dense i n E .
p a r t i t i o n of u n i t y , we s e e t h a t it i s s u f f i c i e n t t o approximate by elements u
j
from
Cr(y) a given element u
support i n t h e case X =By;l e t p
j
I
(u"
* pj)iX
;
E w i t h compact
be a r e g u l a r i s i n g sequence
supp p j C {x,
( s e e (l.l.l), Chapter I ) w i t h u
j
E
we know t h a t u
j
-t
5
01
;
2
we put Further-
u in L (X).
more, we have
where p
*
j
i s t h e o p e r a t o r of convolution w i t h
*
NOW
j'
*
i s supported by {x = 01; t h u s ( p P(us)lx = n J 2 which t e n d s towards Pu i n L (X) Finally, (Pu)")
P ( u o ) -(PU)'
(p j
p
.
lX
F r i e d r i c h ' s lemma ( s e e S e c t i o n 10, Chapter IV) shows t h a t
pj
[P,
u j
-+
-
*3"
0
2 n i n L (JR )
.
We conclude from t h i s t h a t
u i n E.
REMARK 2.6 :
Under t h e c o n d i t i o n s of Lemma 2.5.1,
the
above argument shows t h a t Theorem 2.5 remains v a l i d when we
(SEC. 3 )
UNIFORM LOPATINSKI CONDITION
REMARK 2.7 :
421
Under the conditions of Theorem 2.5, we see
immediately (by continuity and density of C " ( x ) 0
(2.7.1)
Y($u)
=
yS.W
u E
for
E
and
in E) that:
t E C"(F]
.
This allows us to localise in the neighbourhood of a point
The trace commutes with transport by diffeomorphism ( global o r local).
n In the case X =lR+, the jump formula P(uD)
3.
P
( P L J ) ~+
1
@6
remains valid for
u E E
THE UNIFORM LOPATINSKI CONDITION
3.1 Introduction We consider problem (1.5) in the systems case, with R =lR:,
P homogeneous with constant coefficients, B constant, f = 0:
.D
n
u
-
A ( D ~ , D.,)u
= o
I., 3, 0)
(3.1 .I] Bu(t, x ' ,
0) =
g(t, x')
By partial Fourier-Laplace transformation relative to the tangential variables (t, X I ) ,
we formally obtain the ordinary
.
MIXED HYPERBOLIC PROBLEMS
422
(CHAP.
7)
d i f f e r e n t i a l system w i t h i n i t i a l c o n d i t i o n s , and parameters (T,
€,I),
where 5'
E
IRn-l,
T
=u
- iy
,
u ER
,
y
>0 :
Before s t u d y i n g ( 3 . 1 . 2 ) it w i l l be u s e f u l t o g i v e some p r o p e r t i e s of t h e s o l u t i o n s v of t h e d i f f e r e n t i a l system
D
(3.1.31
v xn
- A(T,
f8)v = 0
(Xn
E Rl
a
The s o l u t i o n v o f (3.1.3) such t h a t v ( 0 ) = w is
5')
iXnA(T,
.
(of order m 3' j t h e eigenvalues of t h e m a t r i x A(7, 9 ' )
vIxnl = e j P l,...,L)
i . e . t h e zeros a t ,€
n
w
We denote by A
of t h e e q u a t i o n
-
where P(Dt, D X t , D ) = D P
xn n' shows t h a t A . 4 IR.when 5 ' J
A(Dt, E
d e t P(7, f ' ,
0),
= 0
, ,
Dx,) , The h y p e r b o l i c i t y of
Bn-l and I m T = y > 0.
We con-
s i d e r t h e decomposition
a
-
m
5
of GN j=l i n t o a d i r e c t sum of t h e g e n e r a l i s e d eigen-subspaces of
GN
A(T,
c
@ Ker [(AjI
A(T,
3'))
€,I).
L
Denoting by of w , we o b t a i n
w =
)1w j 51'
t h e corresponding decomposition
(SEC. 3 )
42 3
UNIFORM L O P A T I N S K I CONDITION
and we t h e n r e a d i l y deduce t h a t v remains bounded f o r % 2 0 i f and only i f w
j
= 0 when Im A
j
A s i n t h e case of e l l i p t i c
< 0.
o p e r a t o r s ( s e e S e c t i o n 3 of Chapter V ) , t h e space of s o l u t i o n s
v of ( 3 . 1 . 3 ) i s t h e r e f o r e w r i t t e n where
5')
S*(T,
It, y)
E-(u,
{I,
+
E'(ol
cN =
!I,
,
f')
@ S-(T,
i s t h e subspace of t h e s o l u t i o n s v of
( 3 . 1 . 3 ) which remain bounded f o r k x
E*(a,
5')
S+(T,
.
0
2
n
We denote by
t h e subspace of CN d e f i n e d by
Y) = {v(O)
=
Y>
Iv
@ f I m h > O j
8 E-(o,g1,Y)
E+(o,g',Y)
E S*(T,
.
!I)]
-
Ker "hjI
We t h u s have in A(T,
and t h e dimension
S'1
5
p =
I
m
Imh > O
j
of
3
+ E (a, E l , aER
9
a, $',
y) is independent of
Y>OI
Y
( for
E Rwl,
0
We denote by C a contour of C e n c l o s i n g t h e eigenvalues
hj(j = 1, plane
...,
1) of A ( T ,
{* I m !,>
o ]
c'),
and by
enclosing
k C a contour of t h e h a l f -
{hj
1
* Im 1.J > o ] .
We know t h a t
and t h e r e f o r e t h e s o l u t i o n v of (3.1.3) such t h a t v(0) = w i s given by
MIXED HYPERBOLIC PROBLEMS
424
+
It i s obvious t h a t
+
N
+(u, 0 1 ,
onto E
f
E s
.
+
P-w E E-(o, g ' , Y)
and
c
V
(7, { I )
Thus P
y) p a r a l l e l t o
+
+
7)
that
i s t h e p r o j e c t o r from
E-(D,
y) = Image P
i c u l a r we have E (0, $ 1 ,
, so
(CHAP.
5 1 ,
Y), and i n p a r t -
+,
It i s c l e a r t h a t t h e
We now r e t u r n t o problem ( 3 . 1 . 2 ) . problem
h a s , for a l l w
E
C a y a unique bounded s o l u t i o n v for xn 5 0
+
i f and only i f t h e r e s t r i c t i o n B ( u , E l , y ) of B t o
+
E (a, E',y)
+
i s an isomorphism of E (n,
homogeneity i n ( a ,
c',
c',
E
that
J7-l
u2+
-
E
.
By
and f o r a l l
if and only i f it i s r e a l i s e d for a l l u ,
l2 +
d
y ) , we s e e t h a t t h i s c o n d i t i o n i s r e a l -
ised for all y sufficiently large, for all u 5 1
y ) onto C
I
,
y
>0
L o p a t i n s k i c o n d i t i o n is s a t i s f i e d .
,
E l ,
y such
We t h e n say t h a t t h e
This allows us t o g i v e some
e x i s t e n c e and uniqueness r e s u l t s f o r (3.1.1) ( s e e HERSH c11, CHAZARAIN-PIRIOU [l], SAKAMOTO
C11).
However, t h i s c o n d i t i o n
(SEC. 3 )
UNIFORM
425
LOPATINSKI CONDITION
i s u n s t a b l e and t h e r e f o r e not s u i t a b l e f o r g e n e r a l problems I n f a c t , we s h a l l show l a t e r
with v a r i a b l e c o e f f i c i e n t s . t h a t t h e subspace (a,
el,) #
+ E (n,
y ) has a l i m i t when y
e l ,
-+
O+ w i t h
The uniform Lopatinski c o n d i t i o n f o r ( 3 . 1 . 1 )
0.
+
c',
c o n s i s t s of r e q u i r i n g B ( a ,
y) t o be an isomorphism not only
f o r y > 0, but a l s o i n t h e l i m i t f o r y = O+, (a, 5 ' ) # 0.
We now r e t u r n t o t h e g e n e r a l problem (1.1)i n t h e c a s e of systems.
Suppose
uo, 5~ E T*(R x an)
(to, x;,
and
yo > 0 ; we s h a l l now g i v e a n a t u r a l d e f i n i t i o n of a subspace
+
E (t,, x:, uo, gi, yo) : we c o n s i d e r a l o c a l c h a r t a t x;, and t h e corresponding local c h a r t s o f T*(R")
, an , T*((an)
we use t h e n o t a t i o n
;
x
of
fi
t o signify
t r a n s p o r t by t h e s e c h a r t s , and we denote by p t h e p r i n c i p a l symbol of P.
We have
The subspace
PROPOSITION 3.2 :
ponding t o the operator the l o c a l chart
PROOF: n formIR+.
x
N
H
P(to,
X:
+
N
E (uo, ti, yo)
corres-
i s independent of
; Dt, DJ X
used.
We denote t h i s by
We can obviously assume t h a t f2 i s a l r e a d y of t h e
We c o n s i d e r a c o o r d i n a t e change
1
N
N
xs = x'(x',
xnl
wxn
Xn)
--
X"(X"
H
x = X(X)
9
and p u t
426
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
With t h e n o t a t i o n used i n t h e I n t r o d u c t i o n 3.1,
+
E (o0, 5:, (Dx
n
-+ E
yo) corresponds t o t h e equation
- Abo'
x;,
To'
0;))
= 0
.(X,)
whereas
I
H
0;s
(Go,
(a,D
Yo) +
X
bI
corresponds t o t h e equation
- A(to,
x;,
T ~ ,0;))
yGn) =
Now we can reduce t h i s eauation to-the by successively p u t t i n g w(xnl =
0
n
4bX"
e
v(xn)
y Fn)
(a)
preceding equation
X
= w
and
; s i n c e both t h e s e transformations keep
--+ t h e s o l u t i o n s bounded i n W , t o g e t h e r with t h e i r i n i t i a l v a l u e s ,
( 3 . 2 ) i s proved.
It should be pointed out t h a t t h e p r o p e r t i e s
of invariance of t h e subspaces E
+ will
be i n v e s t i g a t e d more
s y s t e m a t i c a l l y i n Section 8.
I n order t o b r i n g t o g e t h e r t h e case of systems and t h e s c a l a r case for problem (l.l),we g i v e :
DEFINITION 3.3 :
We consider the s i t u a t i o n o f Problem
(SEC. 3 )
2 . 3 , and we put x
ER:,
For
X =
,
o E R
y > 0,
we put
{x
fl
.
ERW1
E+(X) =
,
,
Y 2 0
0 [(Ker Irn I . > 0 J
J
We denote by
1
= (t, x ; u, f ' , Y)
>
1.1 =
,
t E R
(Q,f',Y)O?.
- a(X)) m 9 ,
XjI
where the A . are the eigenvalues (of order m
o f a(X)
427
UNIFORM LOPATINSKI CONDITION
j
j
..
j = 1,. , a )
the dimension
mj
Im I . > 0 J
(independent o f X) o f E+(X).
+
We obviously have properties for the subspaces E (X) analogous to those given in Section 3.1 for the subspaces E+(U,
.
f ' , Y)
+
+
In particular, we again have E (X) =
Image P (X), where
P+(X)
5
2irr
+ since P (X) is
Cm with respect to X and remains of constant
+
rank v , the subspaces E (X) are organised into a sub-bundle
E+ of class Cm of the trivial bundle CN over
xn
+
{y
We shall now investigate the behaviour of E (X) when For
y > 0, we
put
A(x, 5,) 1
A (x, 5,) such that h =-A'A
= det
I , ! (
- a(x)l
= II (5,%tIml.>O ,J
.
hjl
=
y +. 0.
rn k n (En - lj) J ,
j=1 j
We consider the matrices
a The decomposition
> 05
CN =
@
j=l
m.
Ker [(a(X)
- njI) 9
.
428
MIXED HYPERBOLIC PROBLEMS
shows t h a t
k
E ( X ) = Ker
i
,
Q (X)
7)
(CHAP.
.
= dim E'(X)
Rank $(X)
+
Furthermore, t h e Cayley-Hamilton theorem shows t h a t Q ( X ) Q-(X)
,
= E+(X)
Q+(X) C Ker Q-(X)
= A ( X , a ( X ) ) = 0 , s o t h a t Image and consequently we o b t a i n
E+(x)
(3.3.2)
=
y >
Q+(x> f o r
Imge
+
The c o e f f i c i e n t s of t h e polynomials ( i n En) have C"
tn)
A-(X,
f,]
A'(X,
dependence on X f o r y > 0 , s i n c e
.
o
i s the
+
c h a r a c t e r i s t i c polynomial o f t h e r e s t r i c t i o n of a ( X ) t o E-(X).
5
A (X, f,)
We s h a l l now show t h a t t h e c o e f f i c i e n t s of continuously t o
E
suppose X,
:
A . ( o f o r d e r m ; j = 1, . . . , a ) J j
xn
extend
{y = 01, and denote by
t h e eigenvalues of a ( X ) .
Let
0
.
D1,.. ,DR be d i s c s which a r e mutually d i s j o i n t c e n t r e d We know t h a t t h e r e e x i s t s a neigh-
r e s p e c t i v e l y at A l,...,Aa. bourhood V of X
0
in X
such t h a t , f o r X
E
V, t h e eigenvalues
(counted according t o t h e i r o r d e r ) o f a ( X ) a r e d i s t r i b u t e d as
m1 eigenvalues i n Dl,..., m i n Da.
Let 1.1 denote t h e number of R j eigenvalues of a ( X ) i n D . which have p o s i t i v e imaginary p a r t
when
X
f V n {y > 01 1
+
A (Xo,
S,)
=
.
-
3
II (tn Ij)'j
We p u t
A 8
Ago$,,
f,)
0
II (f,
m .-p
- "1
J
j
j=l j=l and it can r e a d i l y be shown, u s i n g t h e r e l a t i o n s h i p s between
c o e f f i c i e n t s and r o o t s , t h a t t h e c o e f f i c i e n t s of t h e polynomials
A
f
[Xi
1n )
t h u s extended t o
x
.
a r e continuous i n
formulas ( 3 . 3 . 1 ) and ( 3 . 3 . 2 ) l e a d us t o extend t h e subspace E+(X) t o
by p u t t i n g
Q+(X) = A*(X,
a(X))
,
and
The
(SEC. 3)
429
UNIFORM LOPATINSKI CONDITION
+
E + ( X ) ='Image Q ( X ) f o r a l l x E
2
.
We know t h a t Q + ( X ) i s of
rank p f o r y > 0 , and we s h a l l show t h a t t h e same a p p l i e s f o r y = 0.
F i r s t , we s h a l l g i v e a reduced form o f a(X) i n t h e neigh-
X
bourhood o f an a r b i t r a r y p o i n t Xo of
PROPOSITION 3.4 :
For
Xo
:
E X , we denote by {A
,,...,hk]
the (possibly empty) s e t cf t?ze reaZ eigenvalues of a ( X , ) ,
.. .
and
by m . the m l t i p Z i e i t y of We ¬e by h .( j=l, ,k) J J k rn the nwnber (possibly zero) o f eigenvalues (counted according t o t h e i r m u l t i p l i c i t y {k
Im A > O]
.
)
of
a(Xo)
located i n the half plane
Then, there e x i s t s a change of b a s i s of GN ,
W
w i t h C dependence on X i n the neighbourhood o f Xo i n X , which p u t s a(X) i n t o block-diagonal form, having blocks dimension
"1,. . .%, m+ , m-
where the eigenvalues of
:
a'(X]
are i n the h a l f plane
.
{* Im 1 > 01 and where, for j = 1,. .k, we have
Of
430
( c w . 7)
MIXED HYPERBOLIC PROBLEMS
(3.4.2)
PROOF :
Let C.(j = ly...yk)be a small contour of C enJ
closing only A [k
Let Ck be a contour of the half plane
j’
Im X > 0)
enclosing only the eigenvalues of a(X 0 ) which
are located in this half plane.
For
€ 2 sufficiently near
X
Xo, consider the projectors
These have Cm dependence on X, and we know that: Rank Pj(X)
-
rn
j
(j = I,...,k)
,
Rank Pk(X)
-
m
f
,
The image under P.(X) of a basis of Image P.(Xo) is for X J
J
sufficiently near Xo, a basis of Image P.(X) having Cm dependence J
on X.
We construct similarly a basis of Image Pk(X)
, which
(SEC. 3)
431
UNIFORM L O P A T I N S K I C O N D I T I O N N
gives a basis of C
p u t t i n g a ( X ) i n t o a form (3.4.1) i n which &
t h e eigenvalues of a (X) a r e i n t h e h a l f p l a n e and, for j = l , . . . , k ,
01
t h e m a t r i x a . ( X ) has only t h e eigenvalue J
A . (of o r d e r m . ) . J J
{* IRIA >
Om.
I n each subspace C
...,k ) ,
(j = 1,
we apply
a change of basis which p u t s a . ( X o ) i n Jordan form; we t h u s J
o b t a i n ( 3 . 4 . 2 ) w i t h ones everywhere on t h e diagonal immediately It may be
above t h e p r i n c i p a l d i a g o n a l , i n view o f ( 2 . 3 . 3 ) . n o t e d , from ( 2 . 3 . 2 ) t h a t t h e b l o c k s a . ( X o ) ( 3 . 4 . 1 ) when
XD
E x n {y, 01
THEOREM 3.5 :
sub-bundte o f c t a s s
X
n
{y
> 01
E+(of rank
PROOF:
do n o t appear i n
+
The subspaces E ( X ) are organised i n t o a Cm o f
, which
p) of
.
J
the t r i v i a l sub-bundle CN over
extends i n t o a continuous sub-bundte
the t r i v i a l bundle CN over
x
,
I n view of t h e arguments p u t forward following
t h e statement of D e f i n i t i o n 3.3, we r e q u i r e t o prove t h a t
+
Rank Q (XD) = p f o r Xo
E
x
I?
{y = 01. By P r o p o s i t i o n 3 . 4 , we
may assume t h a t a ( X o ) i s of t h e form ( 3 . 4 . 1 ) with X = Xo, and t h a t (3.4.2) i s s a t i s f i e d .
We t h e n o b t a i n
432
MIXED HYPERBOLIC PROBLEMS
-
with
Qk = h ( X o
A-(x0,
for j = l,...,k,
Qj = A (Xo, aj(Xo))
C,)
k a (xo])
, k
NOW
- Aj)mj-PJ A_({")
ll (5 n
=
.
j=1
, where
and
A-
i s the
We thus o b t a i n Q- = 0 , mj-P j with T
c h a r a c t e r i s t i c polynomial of a - ( X _ ) . " Qt i n v e r t i b l e ,
and
-
Qj = (aj(Xo) AjI] Tj j i n v e r t i b l e f o r j = 1,. ,k. Now, from (3.4.2) , we s e e t h a t m .-li j i s g e n e r a t e d by t h e f i r s t 1-1 t h e image of ( a . ( X ) - A . 1 ) j J O m. c a n o n i c a l b a s i s v e c t o r s of C J , s o t h a t we have
..
'
+
k
GI (Xo) =
Rank
pj
+ m+=
.
p
1'5 REMARK 3.6 :
t h a t , f o r Xo E 2
The conclusion o f t h e above proof shows
n
{y = 03
, and
f o l l o w i n g a change of basis t
o f CN o f t h e t y p e i n d i c a t e d i n P r o p o s i t i o n 3.4, E ( X o ) i s
k i d e n t i f i e d w i t h t h e subspace k m
-
&@
6c j 6 j=1 by t h e f i r s t 1-1
j
Cm
Q
P C j6
+
cm
of
j=1 1-1
,
where C j i s t h e subspace g e n e r a t e d m c a n o n i c a l b a s i s v e c t o r s of C j.
Looking ahead t o S e c t i o n 5 , it w i l l be u s e f u l a t t h i s p o i n t t o determine t h e number p
Using the n o t a t i o n of Proposition 3.4
PROPOSITION 3.7 :
i n the ease where
Xo
a, Dj(X) =
(X)
j'
,
E
n
,
and denote by a
{ y = 01
we p u t
j
t h e lower l e f t element
(SEC. 3 )
433
UNIFORM LOPATINSKI CONDITION
o f the matrix
is given by:
-
when
p j = mj/2
=
a.
J
i s real and non-zero, and v j
is even
rn
j
f l
m
clj
Then
D j (Xo),
when rn
2
j
?a. > Q
i s odd and
J
,
For 6, E D \ 0, assumption (2.3.1) shows that N
(3.7.1)
A(7,
5,)
n
= c
(7
- ~~(5,)) ,
ER
with ?,({,)
bl and c
#o
,
Furthermore, A ( T , 6 ) is of the form n
(3.7.2)
A(7,
f,)
N
= f,
+ c1(T)6r1 +
om.
+
CN(T),
By equating the terms of highest degree in 6, in (3.7.1) and (3.7.2)we obtain c A(T,
5,)
and
1
~ bA (
E
7
D and we then deduce from (3.7.1)that ,
f,)
are real for
It then follows that the zeros 5, of conjugate pairs when zeros 5
n
of
Aj(7,
5),
y
A(T,
y = 0
and {
n
ER
.
f ) form imaginary n.
= 0; the same therefore applies for the
and of
N
A(T, S,)
; thus
A
J
(7, gn)
and
434
MIXED HYPERBOLIC PROBLEYS
N
f,)
A(?,
5 ER
a r e r e a l f o r y = 0 and
A = Al
... A$
n
,
is r e a l ; since
.
Since
aA
.
we t h e n deduce t h a t
a A (oo, hj)
(Go,
i s non-zero by v i r t u e of
aA . assumption (2.3.1), we s i m i l a r l y s e e t h a t
ab T1 #ao,
We have t h u s proved t h a t
hj)
Y
.
(oo ; lj)f 0
i s r e a l and non-zero.
Furthermore a s t r a i g h t f o r w a r d c a l c u l a t i o n g i v e s
since
x.1
J
-
a.(x J
O
-
) =
hence
ER \O
CY.
J
.
0 0--0 F i n a l l y , we c o n s i d e r t h e polynomial e q u a t i o n
A(oo
- iY,
f,)
r o o t of o r d e r m Thus, when y t h e equation
= 0 *
j’
+ 0,
.
for 5
For y = 0 , t h e value
= A
n
j’
6, =
A.
J
is a
t h e v a l u e y = 0 i s a simple r o o t . ( i . e . t h e r o o t s 5, of j = 0) a r e , i n a c e r t a i n s e n s e ,
t h e r o o t s 5, n e a r A
A.(o
- iy, 6,)
3 0 near t o t h e r o o t s 5 of t h e equation n
m
aA -$ (uo, hj) *
icr More p r e c i s e l y , t h e t h e o r y j‘ o f Puiseux expansions ( s e e LEFCHETZ c11) shows t h a t t h e s e can
with
p =
be w r i t t e n , f o r y 2 0 s u f f i c i e n t l y s m a l l , in t h e form:
where t h e
(icy.) Jr
denote t h e m . p o s s i b l e e v a l u a t i o n s o f J
(SEC. 3 )
.
(icu.) J
435
UNIFORM L O P A T I N S K I C O N D I T I O N
It t h e n follows t h a t p
j
i s equal t o t h e number
of t h e s e e v a l u a t i o n s w i t h p o s i t i v e imaginary p a r t ; t h i s t h e n allows us t o conclude by means of an elementary c a l c u l a t i o n .
We can now g i v e t h e f o l l o w i n g d e f i n i t i o n
We say t h a t problem ( 2 . 3 ) s a t i s f i e s the
D E F I N I T I O N 3.8 :
uniform Lopatinski condition if the r e s t r i c t i o n b+(t,
x'
; o,
f',
E'ft,
x'
; U,
f ' , Y) i s an isomorphism f o r a l l t
x' E (0,
R-'
,
f', Yl f 0
Y)
of
,
oER
b ( t , x'
; o,
O'ER')"
,
O ' , Y) t o
Y 2 0
E
such t h a t
0
We r e f e r t o E x e r c i s e 9.5 for t h e s t u d y of an example.
The uniform L o p a t i n s k i c o n d i t i o n i m p l i e s t h a t t h e number d of s c a l a r boundary c o n d i t i o n s i s e q u a l t o t h e rank p of E
+.
P r o p o s i t i o n 3.2 allows us t o g i v e a n a t u r a l d e f i n i t i o n of t h e uniform L o p a t i n s k i c o n d i t i o n f o r problem (1.1)i n t h e case of systems.
I n t h e s c a l a r v e r s i o n of problem (l.l), and under t h e
c o n d i t i o n s of t h e r e d u c t i o n 2.2, we reduced problem (1.5) t o problem ( 2 . 2 . 1 ) .
We shall now g i v e an i n t e r p r e t a t i o n of t h e
uniform L o p a t i n s k i c o n d i t i o n for t h i s problem. t h a t p = p ( t , x,
'I, 5 )
of P we d e f i n e , for
X
Remembering
i s t h e p r i n c i p a l symbol ( o f degree m )
E x n {y > O}
,
+
a subspace D (X)
436
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEE
+
analogous t o t h e subspace D
(XI,
5') a p p e a r i n g i n P r o p o s i t i o n
3 . 3 , Chapter V : t h i s i s t h e space o f t h e w of t h e form w
j
E(w~,...,W~)
c"
E
.
= Dj-'v(CI)
f o r j = 1,. . , m y where v i s a s o l u t i o n ,
bounded f o r x t 0 , of t h e d i f f e r e n t i a l e q u a t i o n n (3.8.1)
p(t,
XI
,
Q
- iy , 51,
D~ n
1 v(x,)
=
o
.
+
It can e a s i l y be shown t h a t t h e subspace E (X) c o r r e s -
ponding t o t h e reduced problem ( 2 . 2 . 1 ) i s given, for y > 0 , by:
This e q u a l i t y shows, i n p a r t i c u l a r , t h a t t h e subspaces
D+(X) a r e o r g a n i s e d i n t o a bundle of rank
p over
p r o p e r t i e s analogous t o t h o s e o f t h e bundle E
+
x
which has
( s e e Theorem 3 . 5 ) .
Returning t o ( 2 . 2 . 1 ) , we n o t e t h a t t h e p r i n c i p a l p a r t of
IB
i s t h e matrix
denotes t h e p r i n c i p a l symbol of degree r - k + 1 of j,k j o r more p r e c i s e l y , i t s v a l u e a t (t, XI, t~ iy, ~ 9 )
where b B
j,k'
From
-
3 . 5 . 2 ) , we t h e r e f o r e s e e t h a t t h e uniform L o p a t i n s k i
UNIFORM LOPATINSKI CONDITION
(SEC. 3 )
437
c o n d i t i o n for t h e reduced problem ( 2 . 2 . 1 ) i s expressed by t h e condition (3.8.3.)
If we p u t b ( X ) = ( b j t k ( t ,
X I ,Q
- iy, 5 ’ ) ) j=l,...)d k-1
+
...
,
m
+
then t h e r e s t f i c t i o n b (X) of b ( X ) t o D (X) is an X E
isomorphism for aZZ
x Ix,=o
-
I n t h e g e n e r a l case of t h e s c a l a r problem (l.l),by l o c a l c h a r t s of t h e boundary t r a n s p o r t i n g t h e f i e l d v i n t o
a
7 n ’
and by a r g u i n g as for P r o p o s i t i o n 3.2, we d e f i n e t h e bundle D
+
over
{(t,
XI,
U, g I , Y ) J ( t ,
0 , 511
XI,
E
T*(R x
W,Y
2
o,(u, P , Y I + ~ ’ J
w i t h t h e uniform L o p a t i n s k i c o n d i t i o n d e f i n e d by t a k i n g ( 3 . 8 . 3 ) .
REMARK 3.9
:
L o p a t i n s k i determinant i n t h e s c a l a r c a s e .
We r e t u r n t o problem (1.1)i n t h e s c a l a r case w i t h Q =Bt, V
=
a ‘-qand d = p.
We denote by p t h e p r i n c i p a l symbol
..
of P , and we p u t
p(t,
XI,
u
- iy, 51, 5,) = P ( X ,
.
5,)
As i n S e c t i o n 4.5, Chapter V, we c o n s i d e r , f o r y > 0 , a f a c t o r isation
P[x,
E,)
-
P+(x,
!,)
P-(x,
),!
s
where
pf(x, ),!
i s a polynomial i n 5 whose zeros a r e i n t h e h a l f p l a n e n
{* I m f,, > O’J
.
The s o l u t i o n s , bounded for xn 2 0 , of
e q u a t i o n ( 3 . 8 . 1 ) a r e t h e s o l u t i o n s of t h e d i f f e r e n t i a l
438
( c w . 7)
MIXED HYPERBOLIC PROBLEMS
+
x- ) v ( xn ) = 0.
equation p ( X , D
By s u c c e s s i v e d i f f e r e n t i a t i o n s ,
+
I,
we t h e r e f o r e deduce t h a t we can p a r a m e t r i s e t h e bundle D
, over
where t h e elements of t h e m a t r i x M ( X ) a r e expressed i n terms
+
of t h e c o e f f i c i e n t s of p ( X , 5 ) by means o f f i x e d polynomials. n
+
Now we s a w p r e v i o u s l y t h a t t h e c o e f f i c i e n t s o f p ( X , 5,) continuously t o
2
extend
; t h e same t h e r e f o r e a p p l i e s t o t h e m a t r i x
+
M, and ( 3 . 9 . 1 ) t h u s f u r n i s h e s a g l o b a l frame f o r t h e bundle D
over
X
.
We denote by
t h e p r i n c i p a l symbol o f t h e o p e r a t o r B
j'
and we denote by
t h e remainder of t h e E u c l i d i a n d i v i s i o n of b .J( X , 5 n ) by
+
p (X,
cn).
Then we have
I n f a c t it i s s u f f i c i e n t , by c o n t i n u i t y , t o show t h a t t h i s e q u a l i t y holds f o r y > 0.
k = 1,
..., m,
+
w i t h p (X,
Then we have w
= Dk-'v(O)
for
xn
Dx ) v ( xn ) = 0. n
The l e f t - h a n d s i d e
(SEC.
439
UNIFORM LOPATINSKI C O N D I T I O N
3)
t h u s g i v e n by
gives
{n)
bj(X,
since
E
+
bj(X,
{,,I
modulo p+(X,
E n ) , which
(3.9.2).
If we p a r a m e t r i s e D
+
by wl,
...')w !J
anduse
(3.9.21,we a r e
l e d t o consider t h e (Lopatinski) determinant
and t o e x p r e s s t h e uniform L o p a t i n s k i c o n d i t i o n
EXAMPLE
3.10
:-
Cauchy-Dirichlet problem for a s c a l a r We a r e h e r e concerned w i t h problem (1.1)
o p e r a t o r o f o r d e r m.
i n t h e s c a l a r c a s e w i t h d = p y B.u = J t h e n o t a t i o n of Remark
+
(3.8.3)by
u (j = l,...,~). Using
j-1
3.9, we have b = b+ =' :5 J J
for
...,p.
Therefore
LoppyB = 1, and t h e -miform L o p a t i n s k i c o n d i t i o n
(3.9.3) is
j = l,...,p,
sat i s f i e d
and b
j,k
= 6 . . for j, k = 1, ~ , k
.
COUNTER-EXAMPLE 3.11 : Cauchy-Neumann problem. We t a k e as an example t h e s c a l a r problem (1.1)i n t h e c a s e
Using t h e n o t a t i o n o f Remark
p
+
(x, 6),
= {n
- +/(o-iV)'-
3.9, we have
lC1I2
,
d =
v
where
= 1, bl = En, I-
J
denotes
440
MIXED HYPERBOLIC PROBLEMS
7)
(CHAP.
t h e r o o t with p o s i t i v e imaginary p a r t when y > 0 , or i t s limit when y + O+.
We t h u s have
+
b,(X,
sn) = +./i0-iVl2 -
It1l2
I
+
hence
t h i s quantity vanishing f o r
,
y = 0
u2 =
.
\!*I2
me
uniform Lopatinski c o n d i t i o n ( 3 . 9 . 3 ) i s t h e r e f o r e not s a t i s f i e d .
4.
ENERGY INEQUALITIES
We s t a r t by e s t a b l i s h i n g an a p r i o r i e s t i m a t e f o r t h e symmetrisable o p e r a t o r s i n t h e u n r e s t r i c t e d spaceIR u
E CCD(Rn+')
THEOREM
n+l
.
For
and y > 0 , we d e f i n e t h e norm
4.1
:
Suppose P =
-
Dt
A ( t , x, D ) i s a f i r s t X
order operator which i s symmetrisable i n the sense of D e f i n i t i o n 4 . 2 , Chapter V I . c > 0 appearing i n ( 4 . 2 . 2 )
, Chapter
independently of ( t , x, 5 ) . constants C and y
PROOF:
0
We a l s o assume t h a t the constant VI,
can be chosen
Then there e x i s t p o s i t i v e
such t h a t
This i s analogous t o t h e proof o f Theorem 4.3,
Chapter V I , t h e n o t a t i o n of which we u s e h e r e .
We c a l c u l a t e
4)
(SEC.
441
ENERGY INEQUALITIES
G(t) = bdt' 2 y t ( R u , I G(t) =
(4.1.2)
.
u)
P u t t i n g Pu = f , we o b t a i n
-
y e-2Yt(Ru,
u)
+
e-'2YtIrn(Ru,
f)
-
4
+
u)
e-2YtIrn(RAu,
u)
e-'*(R;u,
I n t e g r a t i n g both s i d e s of t h i s e q u a l i t y with respect t o t from
-m
to
Yj(R
+coy
e'*u
we o b t a i n
, e'*u]dt
= -JIrn(RA
e'%u
e-*u>
dt
As i n t h e proof o f Theorem 4.3, Chapter VI, we t h e n deduce by u s i n g t h e Garding and Bchwarz i n e q u a l i t i e s t h a t
with We t h u s have, f o r C'Y
lu11,2;y
s
SY
E
c'
0
> 0:
Ilull~iy +
4=Y CJ
llf1I20 ;Y
from which we deduce (4.1.1)by choosing
E
< C'
*
C' IluIlo;y 2
and yo
sufficiently large.
We s h a l l now g i v e a p r i o r i e s t i m a t e s f D r problems 2 . 3 w i t h boundary c o n d i t i o n s .
( 1 . 5 ) and
We r e f e r t o S e c t i o n 2 . 1 for t h e
d e f i n i t i o n o f t h e s p a c e s eYtL2 and t h e c o r r e s p o n d i n g norms.
MIXED HYPERBOLIC PROBLEMS
442
fomn Lopatinski condition (see 3.8)
yo
Y
7)
We consider probZem 2 . 3 , assuming the uni-
THEOREM 4.2 :
put P - Dxn - A
(CHAP.
.
t o be s a t i s f i e d , and we
Then there e x i s t p o s i t i v e constants C and
such t h a t
for a l l y
2
y
0
and f o r a22
-
.
-n+l u E Co(R+ )
T h i s theorem forms one of t h e fundamental r e s u l t s of t h i s
c h a p t e r , and it w i l l be e s t a b l i s h e d l a t e r on.
For t h e moment,
we s h a l l show how we can deduce from it energy i n e q u a l i t i e s for problem (1.1).
COROLLARY 4 . 3 :
We consider problem (1.1)i n the ease of
systems, assuming the uniform Lopatinski condition t o be satisfied.
Then there e x i s t p o s i t i v e constants C and y 0
such t h a t
f o r aZ1
y 2 Yo
PROOF: and B case
Y
and f o r a l l
u
E
C:(R
x
3)
.
Since P and B a r e d i f f e r e n t i a l , w e have P
Y
= P .
= B f o r a l l y , and t h u s ( 4 . 3 . 1 ) follows from (4.2.1) i n t h e
n where fi =It?+.
I n t h e case where fi
i s a r e g u l a r open
4)
(SEC.
subset of IRn with
an
an
compact, we consider a f i n i t e covering of
by open c o o r d i n a t e patches
Uj(l
5
boundary; suppose we have f u n c t i o n s p ',
E
443
ENERGY INEQUALITIES
c:(n)
of
OD
cp. E Co(uj)
J
= I
such t h a t
J]
j
in
t o the and From Theorem
,
4 . 1 , we have
Now
P('pou) = 'Po Pu
of degree 0.
2
YIl'po~//o;y
+ [SP, 'Po3P
with
'Pol
[P,
We t h u s deduce an upper bound of t h e form
2 -CY' llpullo;y +
7C" l I ~ l 2/ o ; y
for
.
Y 2 Yo
...,J
S i m i l a r l y from ( 4 . 2 . 1 ) it follows t h a t f o r j = 1,
and
y sufficiently large:
Adding t o g e t h e r t h e i n e q u a l i t i e s obtained, we have
which g i v e s ( 4 . 3 . 1 ) f o r y s u f f i c i e n t l y l a r g e .
By reducing t h e s c a l a r case t o t h e case of systems of f i r s t o r d e r ( s e e Example 4.8 of Chapter V I , and 2.2),
Theorems 4 . 1 and
4.2 allow an energy i n e q u a l i t y t o be s t a t e d f o r problem
(1.5)
444
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
i n t h e s c a l a r case.
For
s E
IR and y E R \ 0
,
we d e f i n e
x ‘i)by r e s t r i c t i o n of t h e elements of t h e H i l b e r t space $(R Y For u E eYtH”yR x H;(Rn+l) t o R x n ( c f . Exercise 9 . 1 ) .
z)),
llutls;y
We denote by
e’*u
t h e norm of
H;(R
in
x
’i)
I n p a r t i c u l a r , it can b e shown ( s e e E x e r c i s e 9 . 1 l a t e r on i n t h i s chapter) t h a t , f o r k
u
E eY Lt 2 (R
x
n)
E N , e*Hk,(R
x
‘i) i s
E L2(R
e-*D*u
such t h a t
t h e space of t h e x
n)
cy ( o f d i f f e r e n t i a t i o n w i t h r e s p e c t t o
multi-index
for any
t , x) of leng-
t h k, and t h a t we have t h e uniform norm equivalence w i t h r e s p e c t to
y :
H:(R
We d e f i n e t h e H i l b e r t space HS(lRn),
Y
x
an)
from
by f i x i n g once and f o r a l l a f i n i t e a t l a s of as2 and a
p a r t i t i o n of u n i t y .
For
E ewH;(R
V
t h e norm of e-Ytv i n HS(R x Y
x
an).
h)
we denote by
We t h e n o b t a i n ( s e e
E x e r c i s e 9 . 2 ) t h e following theorem:
THEOREM
4.4
:
We consider problem (1.1)i n the scalar
case (P o f order m), assuming t h e uniform Lopatinski condition t o be s a t i s f i e d .
for a l l .
2
y
0
Then there e x i s t C , yo > 0 such t h a t :
and f o r a l l
u
E C:(R
x
’i)
(SEC.
4)
445
ENERGY INEQUALITIES
We now return to the proof of the fundamental theorem 4.2. We shall achieve this using a procedure analogous to that which allowed Theorem 4.1 to be proved by symmetrisation using the operator R.
In fact, in the next section we shall establish
the following fundamental result:
THEOREM 4.5 :
=
We put
xn
{?
+ O2 + lC11*
= 13
and we consider problem 2.3, asswning the uniform Lopatinski condition t o be s a t i s f i e d .
xo
= (to, x o i Do'
5;'
Yo)
Then, for a l l
c
z
3
there e x i s t matrices
r(X), T(X) of class Cm i n a neighbourhood V of Xo i n
such t h a t : r(X) i s hermitian, T(X) i s i n v e r t i b l e and, p u t t i n g H
T-l a T = a with c
> 0
,
bT =
F ,
there e x i s t constants C and
c
and:
matrix (possibly with just a s i n g l e block) which s a t i s f i e s : (4.5.1)
hO is Y
14.5.2)
e(X) 2
Cm up
t o the boundary, for y
2
0 , hfzl 2 c I Y
CI
We shall now show how Theorem 4.2 is deduced from Theorem
4.5.
By pseudo-differential partition of unity, we see that
446
MIXED HYPERBOLIC PROBLEMS
(CHAP. 7 )
it i s s u f f i c i e n t t o e s t a b l i s h t h e energy i n e q u a l i t y ( 4 . 2 . 1 ) with I ‘Eyu , where t h e symbol @ i s t h e extension Y t o a homogeneous f u n c t i o n of degree 0 ( w i t h r e s p e c t t o
u r e p l a c e d by u
u, y,
of a f u n c t i o n cp E Cz(V)
!I)
.
I n f a c t it can be s e e n ,
as i n t h e proof of C o r o l l a r y 4 . 3 , t h a t t h e e r r o r s due t o t h e commutators
[Py,
@$
@J
[By,
and
a r e absorbed by t h e
when y i s s u f f i c i e n t l y l a r g e y l l ~ 1 1 ~ and 10;y 0 ;Y E c:(v) i s equal t o (see i n particular (2.1.1)). Suppose
l’
terms
u n i t y i n t h e neighbourhood of supp cp e x t e n s i o n of
0
, and
let Y
denote t h e
t o a homogeneous f u n c t i o n of degree 0.
Extend-
i n g T i n similar f a s h i o n , we s e e t h a t it i s s u f f i c i e n t t o e s t a b l i s h t h e energy i n e q u a l i t y ( f o r u ) w i t h A r e p l a c e d by Y Y
(IT-’), AY[YT],
and B
Y
f a c t , when we p u t estimate
lglo;y
( ~ ) y ( y T ~ ‘ ) y @I y
g
I
(YT-’)yf
.
Clflo;y
%
By(”)y
r e p l a c e d by
,
.
In
we c l e a r l y have t h e
Furthermore, we have
modulo an o p e r a t o r of degree -1, and
t h i s allows us t o o b t a i n e s t i m a t e s i n t h e o t h e r d i r e c t i o n . F i n a l l y , using an argument based on e x t e n s i o n of t h e symbols (more p r e c i s e l y , t h e i r r e s t r i c t i o n t o
)I )
t o symbols
which a r e c o n s t a n t o u t s i d e a compact neighbourhood of supp (p ( c f . t h e proof of L e m a 4.12, Chapter V I ) , we reduce t o t h e c a s e where, i n t h e s t a t e m e n t of Theorem 4 . 5 , r e x i s t s g l o b a l l y and
N
a = a
,
H
b = b
.
We t h e n c o n s i d e r a symbol R with
p r i n c i p a l p a r t r , and we p u t
K(xnl =
myu, dxn d
s
where t h e i n n e r product i s t h a t of e y t L 2 (lRn ) f o r xn’ 0 f i x e d .
(SEC. 4)
-
Dx
and suppressing the index uY V Y = n for the operators and the functions, we obtain (cf. (4.1.2)):
Putting y
447
ENERGY INEQUALITIES
5
= -2 Im(RAu, u ) ~ + ; ~2 Im(Ru, f)
K(x,)
0;Y
+
(R;
u).0;Y
U,
n
Integrating both sides of this equality, with respect to
x from 0 to n
+coy
we obtain
((Ru,
(4.5.4)
LJllo;y
=
2s-
Im(RAu, u) o;y . dxn
0
-
J+k;;,
ulo;y
dxn
0
0
Now we have, with a constant c' > 0 :
(4.5.5)
Im(RAu, u) for
y
.
sufficiently large
In fact, this follows from (4.5.1), (4.5.2):
for the
contribution of the symbol h(X) we apply GBrding's inequality, as a symbol of degree 0. For the Y contribution of the symbol e(X) we apply Gbrding's inequality, considering
considering
A4
e(x)
as a symbol of degree 0, and
conclude by means of the obvious lower bound
inequality, we obtain
for
y
sufficiently large
.
.
448
MIXED HYPERBOLIC PROBLEMS
Using
( 4 . 5 . 4 1 , ( 4 . 5 . 5 1 , (4.5.61, we o b t a i n :
and a r g u i n g as a t t h e end o f Theorem 4 . 1 , we t h e n deduce t h e inequality (4.2.1) for y sufficiently large.
5.
CONSTRUCTION OF THE SYMMETRISER T h i s s e c t i o n i s devoted t o t h e p r o o f of Theorem 4.5.
t h u s c o n s i d e r t h e neighbourhood o f a p o i n t x
0
E
C
.
We
We
s t a r t w i t h a s i t u a t i o n analogous t o t h a t o f e l l i p t i c boundaryv a l u e problems: t h i s corresponds t o t h e c a s e where t h e m a t r i x a ( X ) has no r e a l e i g e n v a l u e s . 0
From ( 2 . 3 . 2 ) , a p o i n t Xo such t h a t yo > 0 i s ,
e l l i p t i c point. i n particular
5.1
,
We t h e n s a y t h a t Xo is an
elliptic.
r
C o n s t r u c t i o n of e l l i p t i c point X
0
i n t h e neighbowhood o f an
.
I n view of P r o p o s i t i o n 3.4, a change of b a s i s o f C m
C
lil
of c l a s s
i n t h e neighbourhood of Xo a l l o w s us t o reduce t h e problem t o
t h e c a s e where
where t h e eigenvalues
+
of a ( X ) ( r e s p . of a - ( X ) )
have p o s i t i v e imaginary p a r t
m
(resp. negative).
Let
CN = c
N c o r r e s p o n d i n g decomposition of C
m +
.
@ C
-
denote t h e
+
The subspace E (X)
(SEC. 5 )
449
CONSTFlUCTION OF SYMMETRISER
+
111
corresponds t o t h e subspace C
t h r o u g h t h e change o f b a s i s .
Using a procedure analogous t o t h a t u s e d for Lemma 5 . 5 , Chapter
VI, changes o f b a s i s i n C
m+
m-
a l l o w us t o reduce t h e
and C
* Im a*
problem t o t h e c a s e where
(X ) 2 c ' I
f o r X n e a r Xo,
We s e e k r ( X ) i n t h e form
where c ' i s a p o s i t i v e c o n s t a n t .
t h e n have I m r ( X ) a ( X ) = e ( X ) , with
s o t h a t ( 4 . 5 . 2 ) i s s a t i s f i e d ( h e r e t h e r e i s no c o n d i t i o n
(4.5.1) t o satiso).
We s h a l l show t h a t ( 4 . 5 . 3 ) i s s a t i s f i e d
when we choose p s u f f i c i e n t l y l a r g e .
N
b(Xo)w
-E cN =
w
For
++
= b w
shows t h a t
+ b
+
m
c
+ G?
m
c-
.
- c
b w
,
we p u t w = ( w
+ , w-I
I
The uniform L o p a t i n s k i c o n d i t i o n
i s an isomorphism.
There t h e r e f o r e e x i s t s a
c o n s t a n t C ' such t h a t
]w+I2
(5.1.1)
C'( 1w'I2
+
lz(Xo)w12)
= -lw+12
+
p1w-1~ =
5
for a l l
w
E cN
Now we have -(r(xo)w,
w]
and t h e r e f o r e , u s i n g (5.1.1):
1w+12
+
p1w-1~
- 21w+12
450
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
2
5
+ (p
Iw+12
- X ' l lw-I2 - X ' li;(x0lw12
and (4.5.3) is then deduced by choosing
p
such that
p
-
2C'
> 0.
We now proceed to the fundamental part of this section; this consists of the construction of r a non-elliptic point X
in the neighbourhood of
(which is such that y
0
0
= 0).
This
construction will prove to be relatively complicated, except in the case where the real eigenvalues of a(X ) are all simple. 0
We start by discussing this case, and we in fact assume in order to simplify the notation, that a(Xo) has no eigenvalues which are not real (the contribution of such eigenvalues would in fact give r i s e to arguments analogous to those for the elliptic case 5.1, and, furthermore, will be treated fully in the general case).
We then say that X
0
is a strictly
hyperbolic point.
5.2
Construction of r in the neighbourhood of a strictlx hyperbolic point X
0
.
In accordance with the notation used for Proposition 3.4, we denote by X a,(X),
1'
..., %(X)
... , AN
and by
the eigenvalues of a(X) in the neighbourhood
of Xo (with a.(Xo) = Aj). J
the eigenvalues of a ( X o )
The a.(X) are real for J
y
= 0, and a
change. of basis of CN of class C" in the neighbourhood of Xo
5)
(SEC.
451
CONSTRUCTION OF SYMMETRISER
allows a r e d u c t i o n t o t h e diagonal form
I n accordance w i t h t h e n o t a t i o n i n t r o d u c e d i n P r o p o s i t i o n 1 ba and put u = 1 o r p = 0 3.7, we p u t aj = (Xo)
T $
,
j
j
depending on whether I m a . ( X ) > 0 o r I m a . (X) < 0 f o r y > 0 J 3 s u f f i c i e n t l y small. From P r o p o s i t i o n 3.7, o r p r e f e r a b l y by a s t r a i g h t f o r w a r d d i r e c t v e r i f i c a t i o n , we can s e e t h a t
a. E R \ O J
or
,
'j
= 1 o r 0 depending on whether
CL
j
> 0
4-
CL
j
x
< 0 , and E ( X ) corresponds i n t h e new b a s i s t o 0
= (t, x ; u ;
H
X = (t, x ;
u,
S' , Y) ,
st,
0)
we p u t T a y l o r ' s formula w i t h r e s p e c t t o
;
t h e v a r i a b l e y shows t h a t
(5.2.1)
aj(x)
+ iY and where H
j
Dj(y)
i s Cm.
We seek r ( X ) i n t h e diagonal form
+
Aj(x)
452
(CHAP. 7)
MIXED HYPERBOLIC PROBLEMS I
where t h e r
Since a . ( X ) i s r e a l , (5.2.1)
a r e r e a l constants.
j shows t h a t I m r . a . ( X ) = y r
.
J J J J t h a t c o n d i t i o n s of t h e t y p e ( 4 . 5 . 1 )
>
rj cyj
for
0
J
c
2
Re D.(X) + y . r . Irn H.( X ) and t h e r e f o r e
j = 1,
Suppose we choose, f o r example r r
= -p
j
w
E
J
J
a r e s a t i s f i e d when
j
...,
.
N
= 1 for a
j
> 0 , and
< 0 , where p i s a p o s i t i v e c o n s t a n t . For j + -+ CN, we denote by w i t s p r o j e c t i o n onto E ( X o ) , and we p u t
w = w
for a
+
-.
+ w
Then -(r(Xo)w,
=
w)
- lw+I2 + PI~'[~
, and
the
uniform L o p a t i n s k i c o n d i t i o n t h e n allows us t o deduce, e x a c t l y as f o r t h e e l l i p t i c c a s e 5 . 1 , t h a t c o n d i t i o n ( 4 . 8 . 3 ) i s s a t i s -
f i e d when p i s s u f f i c i e n t l y l a r g e .
We now go on t o t h e g e n e r a l c o n s t r u c t i o n of bourhood of a p o i n t X
0
r
i n t h e neigh-
P r o p o s i t i o n 3.4
such t h a t yo = 0.
and t h e arguments employed i n t h e e l l i p t i c case 5 . 1 allow us t o reduce t o t h e c a s e where
%(X)
h a s , i n a neighbourhood of
Xo, t h e b l o c k d i a g o n a l form ( 3 . 4 . 1 ) w i t h , furthermore, f
* (X)
Im a
positive definite.
We now i n t r o d u c e , o r r e c a l l ,
t h e following notation:
NOTATION 5.3 For X = (t, x ; For j = 1,
.. . , k ,
u, f ' , y )
, we
N
put X = (t, x ;
u,
c o n s i d e r t h e m a t r i x w i t h dimensions m
E', j
0).
CONSTRUCTION OF SYMMETRISER
A
j
453
=
We p u t
We t h u s have H
Ly
(5.3.1)
aj(X)
= hjl + A j
+
Bj(X) m
where t h e m a t r i c e s Bj, Dj, Hj a r e C
+
iy Dj(X]
, B.(Xo) J
+
& . ( XJ I
= 0, and where t h e
l e f t lower element cy. o f D.(Xo) i s r e a l and non-zero ( s e e J J Proposition 3.7). Furthermore, we can assume t h a t B.(X) has r*
J
r e a l c o e f f i c i e n t s i n view of t h e f o l l o w i n g lemma, which i s due t o RALSTON [ll.
LEMMA 5 . 4
In Proposition 3.4, it is p o ssib le t o choose
:
H
th e change of b a s i s such t h a t t h e matrices B.(X) have real. c o e f f i c i e n t s (j = 1,
PROOF:
J
..., k).
In t h e s i t u a t i o n of P r o p o s i t i o n 3.4, we s h a l l show
.. .
N
t h a t t h e r e e x i s t s , f o r j = 1, , k an isomorphism U.(X) o f J m C j, w i t h Cm dependence on X i n t h e neighbourhqod o f X LI
0
such t h a t (5.4.1)
-
us-1 (X)
(Aj
+
Bj(2))
H Uj(X]
=
Aj
+ Bj[X)
N
454
MIXED HYPERBOLIC PROBLF,MS
(CHAP.
7)
" Suppressing w i t h a m a t r i x B!(X)which has r e a l c o e f f i c i e n t s . J t h e index j i n o r d e r t o s h o r t e n t h e n o t a t i o n , we denote by el,
..., em t h e
.,
111
c a n o n i c a l b a s i s of f o r R = 0,
c"
= c j.
..., m-1
and s i n c e B . ( X ) = 0, J
- ! L
t h e v e c t o r s ( A . + B .( X ) ) e c o n s t i t u t e , J J m II
R = O,...,m-1 and X s u f f i c i e n t l y n e a r X
Since
0
.
We d e f i n e t h e iso-
d. ,
"
O
a b a s i s f o r Cm f o r
R R = 0, morphism U . ( X ) by U . ( X ) A e = (Aj + B j ( X ) ) ' e my J J j m We t h e n have ( 4 . 5 . 1 ) w i t h a m a t r i x B ! ( X ) of t h e form 9. ,
-
..., m-1.
J
Now it i s c l e a r t h a t , f o r i = 1,
. .., m
&
, bi(X)
from t h e s i g n , e q u a l t o t h e c o e f f i c i e n t of gm-l n N
det(A.
J
+ B?(X) J
- !,I4)
N
= det(a5(X)
- {"I) ,
is, a p a r t
in
and we s a w during
t h e proof of P r o p o s i t i o n 3.7 t h a t t h e s e c o e f f i c i e n t s a r e a c t u a l l y r e a l s i n c e i n t h i s case y = 0.
Taking account of t h i s lermna,
we seek r ( X ) i n block diagonal form
( rl
(5.4.2)
0
I -PI
0
w i t h r .(X) h e m i t i a n (j = 1,
J
.. ., k )
and p > 0.
Conditions of
t h e t y p e ( 4 . 5 . 2 ) a r e t h e n c l e a r l y s a t i s f i e d by t h e l a s t two
5)
(SEC.
455
CONSTRUCTION OF SYMMETRISER
We shall construct the r . (X) such that conditions of J TO do the type (4.5.1) are satisfied by the first k blocks.
blocks.
this, we seek r . ( X ) in the form J
N
(5.4.31
-
= Ej
rj(X)
+ F j (X) + iy
G
j m
F.(X) are real and symmetric, F. E C , F.(Xo) = 0 , j’ J J J and where G is real and anti-symmetric. Under these conj ditions the matrix r . ( X ) is hermitian and in view of (5.3.1) J we have :
where E
Im r a j j
-
Im [(Ej
where M = M(X)
+ Fj) ( A j + B j ) l + Y R e ( E f j
+G A ) j j
D(Y) + O(IX-’j;l)
is Cm and of the form
+W
.
In order to meet the conditions (4.5.1), we therefore impose: v
(5.4.41
(Ej
N
+ Fj(X))
(Aj
Re (E.D.(X
(5.4.51
J J
We note that condition H
for X = X
0
.
0
)
+
+ Bj(X))
Gj~j)
symmetric
positive definite
(5.4.4) reduces to
E . A . symmetric J J
Now the following lemma can be proved in ele-
mentary fashion:
LEMMA 5.5
The synunetric matrices ,E such t h a t j E . A . is symmetric, are the matrices o f the form J J
:
.
MIXED HYPERBOLIC PROBLEMS
456
e / I /
(505.1
1
E =
J
/
We t h e r e f o r e t a k e E
j
/
I
i n t h e form ( 5 . 5 . 1 ) .
Then c o n d i t i o n
( 5 . 4 . 4 ) reduces t o
+ Bj(T))
Fj(x)(Aj
(5.502)
H
+ EjBj(X)
symmetric
.
&
This p r o p e r t y i s embodied i n t h e c o e f f i c i e n t s of F.(X)
J
- 1)/2 e q u a t i o n s .
through a l i n e a r system of m , ( m
J
,w
We seek
j
Fj(X) i n t h e form
Fj(X) = H
( "p 1 )
,
N
- 1. Then j t h e l i n e a r system i s s q u a r e , and t h e a s s o c i a t e d homogeneous
where @.(X) is a symmetric m a t r i x w i t h dimensions m
J
I.
t o F.(x~)A.symmetric, f o r X = Xo, or, J J We t h u s s e e t h a t ( 5 . 5 . 2 ) i n view of L e m a 5.5, t o F.(Xo) = 0 . system i s e q u i v a l e n t
J
(SEC. 5 )
457
CONSTRUCTION OF SYMMETRISER c
i s expressed i n terms o f a Cramer system f o r X n e a r X 0' and c
t h e r e f o r e admits a unique s o l u t i o n FI(X) which i s o f c l a s s Cm J
H
With t h i s choice of
and which i s e q u a l t o z e r o f o r X = Xo. .4
We s h a l l s e e t h a t c o n d i t i o n F . ( X ) we have c o n d i t i o n ( 5 . 4 . 4 ) . J f i r s t , we ( 5 . 4 . 5 ) i s s a t i s f i e d w i t h a s u i t a b l e choice of G recall that a
j
j' denotes t h e lower l e f t element of D.(Xo); t h e r e J
t h e r e f o r e e x i s t s a c o n s t a n t C which depends only upon t h e c o e f f i c i e n t s of E
Re(E.D.(X J J
for a l l w
j
>w,
(see 5.5.1) w)
- (Iwll2
2
elajlwlI
+ Clw112
0
ma E C j,
w ' = (w2
with
We impose upon E
(5e5.3
2
such t h a t
j
1
,...,
t h e condition
elaj
2
3
Then ( 5 . 4 . 5 ) i s an obvious consequence of
LEMMA 5.7 :
symetrCc matrix
For any constant G:
Re( GjAjw,
PROOF:
w)
We t a k e
there e x i s t s a real, anti-
such that
J
(5,7.1)
C,
2
- Iwll 2
+ Clw'I2
for a22
Gj
a t r i d i a g o n a l antisymmetric m a t r i x .
Then
m-2
and we l e a v e it t o t h e r e a d e r t o verif'y t h a t
(5.7.1) i s
w E
.
GmJ
458
MIXED HYPERBOLIC PROBLEMS
s a t i s f i e d when t h e c o e f f i c i e n t s
g i ,i-1
(CHAP.
7)
a r e chosen successively
t o be s u f f i c i e n t l y l a r g e . To summarise, we have shown t h a t i f t h e matrix E
j
i s chosen
t o be of t h e form ( 5 . 5 . 1 ) , and s a t i s f i e s ( 5 . 5 . 3 ) , then we can construct
r . ( X ) i n t h e form (5.4.3) such t h a t conditions of the J
type ( 4 . 5 . l ) a r e s a t i s f i e d f o r h ( x ) = r.(X) a . ( X ) . J
J
To conclude
t h e proof of Theorem 4.5, it now remains t o satisfy condition (4.5.3).
By c o n t i n u i t y , it i s s u f f i c i e n t t o satisfy t h i s
where we r e c a l l t h a t
+
We a l s o r e c a l l ( s e e Remark 3 . 6 ) t h a t t h e subspace E (X-) correeponds, i n t h e new b a s i s , t o
w
E
(X,>
k
* @ j=1
For w
E
-+
CN, we p u t w = w
w on E ( X o ) .
+
-
+w
, where
+ w
d'
@
a+"
6
i s t h e p r o j e c t i o n of
Using t h e uniform Lopatinski condition and
arguing e x a c t l y as at t h e end of t h e c o n s t r u c t i o n 5.1, we see t h a t i n order t o e s t a b l i s h ( 5 . 7 . 2 ) and t o conclude t h e proof,
(SEC. 5 )
459
CONSTRUCTION OF' SYMMETRISER
it i s s u f f i c i e n t , t o apply t h e f o l l o w i n g lemma:
LEMMA 5.8 :
E
j
For any number
p > 0,
there e x i s t matrices
o f . t h e form (5.5.11, s a t i s f y i n g (5.5.31, such t h a t the
matrix r(Xo) defined by ( 5 . 7 . 3 ) s a t i s f i e s th e following property: there e x i s t s a constant -(r(Xo]w,
(5.8.1)
w)
with
C"
C"
2
( - l ~ + +1 ~Plw'1*)
CN.
E
w
for
It i s obviously s u f f i c i e n t t o satisfy t h e in-
PROOF:
m equality (5.8.1) f o r a l l w and w -I- ( r e s p . w-)
E C
j when r(Xo) i s r e p l a c e d by E
j'
i s r e p l a c e d by t h e p r o j e c t i o n of w i n C?'j
m.-p.
(resp. i n c (5.8.2)
J
-(Ejw,
J ) , i.e. w)
+
Cqt(-lw+12
2
- 12)
plw
for a l l
w
m
EGj
where
= - aR f o r
We have, p u t t i n g e
i+A = m . + l J
R = l,...'rn
j
(cf. ('5.5.1)):
i+L = m +2
J
Let us f i r s t assume t h a t m . is even; from P r o p o s i t i o n J
3.7 we t h e n have (5.5.3),
Q
p
5
u s-3'
1 J
= m 1 2 ; we f i x u j 1
E
IR s a t i s f y i n g condition
=md it can e a s i l y be seen t h a t ( 5 . 8 . 2 ) i s
s a t i s f i e d when we t a k e u
3
= u
5
=
... = 0 ,
and when t h e p o s i t i v e
460
MIXED HYPERBOLIC.PROBLEMS
u4,
(CHAP.
... a r e
chosen i n t u r n t o be s u f f i c i e n t l y l a r g e . m.+l t h e c a s e where m i s odd and CI > 0 , we have p = j j j 2 ' 02,
we f i x u 1 < 0 such t h a t
when t h e p o s i t i v e numbers o be s u f f i c i e n t l y l a r g e .
m
3'
O5
c2 = a4 = *.=
... a r e
F i n a l l y , when m
6.
0
and
chosen i n t u r n t o
j
i s odd and a
0 , we
j
- 1
; we f i x 51 > 0 such t h a t o a 2-3' j 2 l j c o n t i n u e as i n t h e p r e c e d i n g c a s e .
have 1-1
In
and it can be shown
c y j $ 3I
t h a t ( 5 . 8 . 2 ) i s s a t i s f i e d when we t a k e
7)
=
and we
SOLUTION OF THE PROBLEM WITHOUT I N I T I A L CONDITIONS Before s o l v i n g t h e mixed problem p r o p e r , we s h a l l f i r s t
i n v e s t i g a t e i n t h e s p a c e s eyt Hk t h e boundary-value ( w i t h o u t i n i t i a l c o n d i t i o n s ) Pu = f , Bu = g .
problem
We s h a l l e s t -
a b l i s h e x i s t e n c e and uniqueness r e s u l t s u s i n g f u n c t i o n a l a n a l y s i s methods, s t a r t i n g from t h e energy i n e q u a l i t y ( 4 . 3 . 1 ) and t h e analogous i n e q u a l i t y f o r an a d j o i n t problem.
I n order
t o s h o r t e n t h e d e s c r i p t i o n , we c o n s i d e r t h e c a s e o f f i r s t - o r d e r systems.
6.1
We s t a r t by d e f i n i n g t h e concept of an a d j o i n t problem.
NOTATION AND ASSUMPTIONS
We c o n s i d e r inIRn+'
an N x N matrix d i f f e r e n t i a l o p e r a t o r
of first-order
a n open s u b s e t
of IRn, and, f o r ( t , x ' ) E IR x
an,
a surjective
(SEC.
6)
461
PROBLEM WITHOUT INITIAL CONDITIONS
M -C
d
matrix B(t,x') : C
.
We adopt the following assumptions: m
(6.1.1.) The coefficients of P and B have C
(t,x)E R"+'
dependence on
m a n , are
and(t,x')E
constant for 1x1
sufficiently large, and are independent of t for 1 t 1 sufficiently large. (6.1.2.) The hypersurface IR
x as2
is non-characteristic for P.
We endowIRn with its canonical Euclidian structure and
W
aR with the corresponding Riemannian density.
x
where ( ( Lz(R
,
) ) and ( ,) respectively denote the inner product of
x n; CN)
v =(vl,
and of L2(R xan; CN) and where we have put
M(t,x')
=
...,v n )
denoting the inward unit normal to an.
v
j=l
J ( X I )Mj(t,x')
I
assumption (6.1.2) signified that M(t,
N
x
Stokes'
N matrix for all (t,x')EIR
x
XI)
The
is an invertible
an.
We now propose to transform the boundary term ( M q ,$) in the equality (6.1.3) so that it contains B Q
.
This leads
us to pose the following question.
c
Question 6.2;
x(cN,cN-d)
Find matrices
and -i(t,x')EL(C
(t,x')E R
N,c d )
B(t,x')
, A(t,x')
depending on
as in (6.1.1), with, at each point
in
MIXED HYPERBOLIC PROBLEMS
462
..
(6.2.1) for all
=
<McY,'B>
c a ,$ f C N ,
c
d+
N-d
C
where t h e b r a c k e t s denote r e s p e c t i v e l y
N d N-d t h e c a n o n i c a l h e r m i t i a n s t r u c t u r e s of C , C , C
.
Let us assume f o r t h e moment t h a t t h i s q u e s t i o n has been resolved.
If
B f C N and t h u s
Q
f Ker A n KerB
c1
=
and c o n s e q u e n t l y A
0;
,we have
t h i s shows t h a t
2 = '1Ker
s i n c e d i m K e r B = N-d
B:
< Ma ,B> = o
for a l l
KerAnKerB+j
Ker B + CN-d
i s an isomorphism
i n view of t h e s u r j e c t i v i t y of B.
t h e n deduce t h a t A i s s u r j e c t i v e and t h a t We t h u s have c a n o n i c a l isomorphisms
We
CN= Ker ABKerB,
CN4 H(CN/KerA)=
Ker
,
Consequently, Ker B , c o n s i d e r e d as a v e c t o r bundle over
a R , must be t r i v i a l i s a b l e s i n c e it i s isomorphic t o t h e N-d t r i v i a l bundle ( R x a n ) x C , I n o r d e r t o r e s o l v e Q u e s t i o n 6 . 2 ,
R
x
we a r e t h e r e f o r e l e d t o make t h e f o l l o w i n g assumption: (6.2.2.)
Assume t h a t t h e r e e x i s t s a b a s i s
e,(t,x'),ooo,e in
N-d
(t;~') of Ker B ( t , x ' ) ~depending a s
( 6 . 1 . 1 ) on (t;x')f
Rxan
Under t h e s e c o n d i t i o n s , we have
PROPOSITION 6 . 3 :
We adopt t h e asswnption ( 6 . 2 . 2 ) .
(6.3.1,) There e m k t s A ( t , x ' ) depending on ( t , x f ) c IR (6.1.1)
such t h a t
CN=
K e r A@KerB*
x
Then:
an as in
(SEC. 6 )
PROBLEM WITHOW I N I T I A L CONDITIONS
463 L
.
-
Having chosen such an A , there e x i s t unique A , B such
(6.3.2.)
c
d
t h a t B, A , A s a t i s f y Question 6.2. w
Moreover, B i s s u r j e c t i v e , and we have
(6.3.3. )
CI)
K e r , B = (M.Ker B)
.
I
I n o r d e r t o e s t a b l i s h (6.3.1)
PROOF:
t e n d i n g e l , . . . ,eN-d i n t o a b a s i s e l , . . .
, we
s t a r t by ex-
,eN f o r CN, a g a i n sat-
i s f y i n g c o n d i t i o n s of t h e t y p e ( 6 . 1 . 1 ) : f o r example, we n o t e
CN= Irn[B*)$KerB,
that e N-d+l”
*
and we t a k e f o r t h e v e c t o r s
t h e column v e c t o r s of B*.
’eN
t o take for A ( t ,
Then, it i s s u f f i c i e n t
t h e m a t r i x which a s s o c i a t e s w i t h an element
X I )
o f CN i t s f i r s t N-d components i n t h e b a s i s e 1( t , x ’ ) ,
To e s t a b l i s h ( 6 . 3 . 2 ) , we decompose an element
K e r A@ K e r B
into
ci
=
c1
1
+
ci
2
ci
...,e N ( t , x ’ ) . E
CN =
and we s e e t h a t ( 6 . 2 . 1 ) i s
equivalent t o
,, c
(6.3.4)
<
(6.3.5)
< Ma2 ,B>
Mal;B>
N
=
< Bal, A B >
=
< Au 2,
c
f o r a l l o!,E K e r A ; B E C
H
We p u t M2
c
= MIKer B
: Ker B
BB5eN4 f o r allo!$
+.
C
N
,
Since A
Ker B
;B
N
N
€@
2 = A/Ker B
i s an isomorphism, ( 6 . 3 . 5 ) i s e q u i v a l e n t t o w
< ~ V $ A ; ’ ~ , B > ~ C < ~ ; B ~ > ~ f~ o r a l 1 y c C i.e.
c
c
N-d
N , f 3 ~ C ,
.
464
MIXED HYPERBOLIC PROBLEMS
(6.3.6) c
A = (MI B;’Y
Similarly, (6.3.4) is equivalent to
M
1
where
and B denote the restrictions of M and B to Ker A. 1
Definition (6.3.6) shows that N
B = rank (h$
rank
GI)= dim I(M,
Ker B
) = d i m Ker B = N - d ,
hence the surjectivity of B.
Finally, it can be shown that we have the equalities H
-1
KerB = Ker(
)* = ( I r n M 2 $ -1 )A ={Im
DEFINITION 6.4 :
MI Ker
d’-
(.Mbke B
)f
Let Q, P, B be as i n 6.1, and suppose
the t r i v i a Z i t y assumption (6.2.2)appZies f o r Ker B.
Consider
the associated boundary-value prob lem:
(6.4.1)
Pu=f II.
in R x n a n d B u = g in
Rxan ,
H
Suppose B, A, A s a t i s f y Q u e s t i o n 6.2.
The boundary-value
c.
prob Zem associated w i t h of
a, Pw, B
i s caZZed an a d j o i n t probZem
(6.4.1). d..
We note that we do not have uniqueness of B y but that (6.3.3) still applies.
From (6.1.3) and
(6.2.11,we have
6
(SEC.
465
PROBLEM WITHOUT I N I T I A L C O N D I T I O N S
We note that problem (6.4.1)is an adjoint of the problem c
defined by R , P*, B
and that the arguments presented after the
statement of Question 6.2 show that H
(6.4.31
REMARK
x a n i c N ) ; ~ g 0)=
JIE C:(R 6.5
:
= C ~ ( Axan;c$)
n In the case where R =B+,the triviality
assumption (6.2.2) is satisfied automatically since
IR
x
aR = Bn is contractile.
In the general case, this
supplementary assumption has to be made for B in order to have the notion of an adjoint problem with a boundary condition of the same type as that of the original problem.
This
supplementary assumption can be avoided by generalising the boundary condition of (6.4.1)as follows: we replace C
G
complex vector bundle
of rank d onL?
jective isomorphism of CN onto
& .
x
d
by a
an, and B by a
We make for f
SUT-
and for
this isomorphism assumptions analogous to those appearing in
(6.1.1)for 1x1 or It1 large.
The whole of the above argument
adapts readily to this situation, which is locally isomorphic to the previous situation.
PROPOSITION
6.6
:
We consider the s i t u a t i o n 6.1, asswning
t h a t P i s s t r i c t l y hyperbolic with respect t o t and t h a t problem (6.4.1)s a t i s f i e s the uniform Lopatinski condition. Then, m y problem which i s a d j o i n t t o (6.4.1) s a t i s f i e s the backward uniform Lopatinski condition l i . e . the condition by
466
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
changing y
2
o into y
5
o i n the d e f i n i t i o n s i n Section 31.
PROOF: Throughout t h e f o l l o w i n g , we omit t h e v a r i a b l e ( t , x ' ) E IR
x
aR and we i n d i c a t e by t h e s i g n
a t i n g t o a problem which i s a d j o i n t t o
I
t h e notation rel-
(6.4.1).
We s t a r t by showing t h a t it i s s u f f i c i e n t t o e s t a b l i s h t h a t
(6.6.1)
*(o,
5'
-Y) = ( M ~ ~ c T , % ' , Y ) y f o r Y z o
I
(0 ,5'
,'Y)
40;
I n f a c t , t a k i n g account of ( 6 . 3 . 3 ) , we t h e n have N
I
U
KerBnE+(a,
, S t , -y)
= (MkerB)
I
~(ME+(Q;~';Y))
= ( M K e r B + ME+ (Q
,%' ;Y))
which proves P r o p o s i t i o n 6.6 s i n c e we have
I n order t o establish
(6.6.1),we
c o n f i n e our a t t e n t i o n t o y >
0.
I
= (MCN]'=
{o]
E=N-b =N-d
c a n , by c o n t i n u i t y ,
By l o c a l c h a r t s , we reduce t h e
n problem t o R =B+. The p r i n c i p a l symbol of P i s P(T r
,5 ) =
*
n
T I ~ + fj F Mj + 1 6J. MtJ,snE@ , T =
and t h a t of P
We n o t e t h a t f o r
~ ( 7 ~=57 )I d
Q-
N
-
P h , 5' I En) = P*17,
Since M
n
is
iv; a € R ,Y>
we have
0,
5' I 5 , L
i s i n v e r t i b l e , it can e a s i l y be seen t h a t it i s
sufficient t o establish
( 6 . 6 . 1 ) i n t h e c a s e where Mn = I d .
s a w i n S e c t i o n 3 t h a t t h e subspaces
+
E-(7;5')
We
a r e t h e images
(SEC.
6)
467
PROBLEM WITHOUT I N I T I A L CONDITIONS
+ where C- e n c l o s e s t h e z e r o s at 5
kIm5
that
n
70.
n
of
det p(7
S i m i l a r l y , t h e subspace
;S1,sn)
such
P(0;S'l-y)
is
t h e image of t h e p r o j e c t o r
Thus f i n a l l y w e have
The concept o f an a d j o i n t problem allows us t o g i v e an existence r e s u l t .
In Proposition
6.7 and 6.8 below, we con-
s i d e r problem (1.1)i n t h e c a s e of systems, assuming t h e uniform L o p a t i n s k i c o n d i t i o n t o be s a t i s f i e d .
PROPOSITION 6.7 :
YsY,;f€
exists
There e x i s t s
eYtL2(Rxn)
u € eYt L2 (A xn)
PROOF:
and
y,
> o such t h a t
g € eY 9I2 ( R x a n ) ,
for there
such that Pu = f and Bu = g .
F i r s t , we n o t e t h a t t h e boundary c o n d i t i o n Bu = g
i s meaningful, s i n c e Theorem 2 . 5 shows t h a t i f u and Pu a r e 'in eYt L2 (R x
n 3,
t h e n u admits a t r a c e u
E x3R)
H*oc(R
xan),
468
(CHAP. 7 )
MIXFD HYPERBOLIC PROBLEMS
To prove the existence of u, we proceed as in the proof Of .u
Theorem 4.5, Chapter VI.
Suppose (P*,B) is an adjoint problem;
%$=
E =($ECz(RxxiCN)i
we define.the space
; we
0)
shall show that it is possible to define a linear form R on P*E by putting J ( W ~i
I
=
((f ;$1)
- i(s,X$1
We have the obvious upper bounds
and
We then deduce from the enera inequality (4.3.1) applied to the backward adjoint problem (for y
This shows that, for y
2 yo
1:
is well defined and is,
2 y,,R
furthermore, continuous for the topology induced on P*E by
e-yt L2@1
x
R).
Consequently, the Hahn-Banach theorem implies
the existence of an element u
E
eyt L2(B
x C2)
such that
H
(6.7.1)
‘((u ,.P$)? = ( ( f , $ ) ) -, Ai$() Q (IR
Taking il, in C:
(6.7.2)
Pu-f
x
f o r all
$6E
N
R;C ) , (6.7.1)shows that
.
Theorem 2.5 and t.he equality (6.4.2) show that we again have
(by density and continuity):
(SEC. 6)
469
PROBLEM WITHOUT INITIAL CONDITIONS
(607.3)
((Pu;JI!g=
+ i(Bu,i$)+
((u;F$)) for a l l
JI E:C
Combining t h i s e q u a l i t y w i t h
i(Aus$)
;cN>.
R XT
( 6 . 7 . 1 ) and ( 6 . 7 . 2 ) , we o b t a i n
CI
(Bu-g,A*)=
o
Q EE ,
for all
and c o n s e q u e n t l y Bu = g , i n view o f ( 6 . 4 . 3 ) .
I n o r d e r t o i n v e s t i g a t e t h e r e g u l a r i t y of t h e s o l u t i o n , we s h a l l now e s t a b l i s h
PROPOSITION 6.8
integer k u
2
We p u t
:
and B u C eYt
E
lac
tf” (aC) Since M
where M, = I d .
(Y)
,
We t a k e an
= P
,
eYt &+I (c)
u
0
. i s i n v e r t i b l e , we can reduce t o - t h e c a s e
If we p u t v = e-Yt u, t h e n e-yt
- i
$(ax)
E eyt
lac
t h e n we have u E eYt $+I (C)
eYt &+I (ax)
PROOF:
with P
r=c
-1; then there e x i s t s yk > 0 such t h a t if
E eYt &(c) n eY t L2(c) , P u E
and u
Rx
Pu = P
(VlV
y and t h e energy i n e q u a l i t y ( 4 . 3 . 1 ) i s
written
for
1
y r y , and cpcH (C)
T h i s means t h a t we have t o show t h a t i f V E $+’(C)
,
pCY3 v
E $+”(c)
and
v
lac
$(ac> and ”\& E !$+’(ax). E
k = -1, i t is s u f f i c i e n t t o show t h a t
B
~
v
2 L (C)
n $(c) ,
tP+’(a~) E
We n o t e t h a t f o r v
E L2(ac). lac
then
470
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
S i n c e 3 R i s bounded (we s h a l l c o n f i n e
OUT
a t t e n t i o n t o t h i s case;
t h e method g i v e n below may be a d a p t e d t o t h e c a s e does i n f a c t t h e n become s i m p l e r ) we can cover
-
n =By and by a f i n i t e
u
'j j =o,...,~, Ujj such t h a t f o r j # o U . i s bounded and i?.C and f o r J J j = o we have U 0 c n and Xo = I d e n t i t y . Let
number of open c o o r d i n a t e p a t c h e s
b e a p a r t i t i o n of u n i t y s u b o r d i n a t e t o t h i s
cr(uj)
qjE
covering w i t h
#
for j
0.
We r e q u i r e t o show
t h a t we have:
that is
I n o r d e r t o a v o i d having t o d i s t i n g u i s h t h e c a s e j = o i n t h e f o l l o w i n g we s h a l l readIR
n+l
,
-n+l instead o f B + i n the
s t a t e m e n t s concerned w i t h t h i s c a s e .
and Bj
=xj+ o 8 O X *
J-
.
Since Q
j
i s a d i f f e r e n t i a l operator of
o r d e r 1, t h e commutator which a p p e a r s i n t h e e q u a l i t y Qj(J'j~j)
= $ (Q w.) j j J
+
[ QJ. , $ . ] w J
i s a n o p e r a t o r o f m u l t i p l i c a t i o n by a f u n c t i o n
for
5 #
o and
e,
~
ej E
czc.) J
i s bounded t o g e t h e r w i t h a l l i t s d e r i v a t i v e s ;
(SEC.6 )
471
PROBLEM WITHOUT I N I T I A L CONDITIONS
t h e assumptions on v and Pv imply t h a t f o r a l l j
Consequently, P r o p o s i t i o n 2 . 4 , shows t h a t it i s s u f f i c i e n t t o prove t h a t f o r a l l j :
1
-I--
2 \~jwj(.rxn)\k+l
(6.8.2.)
'j
E
l$fk+l(#:+l)
dxn
< +=
w
j
,
,
i.e.
( w i t h our convention,
t h i s i n t e g r a l i s t a k e n between
--m
and
+co
for j =
0)
For t h i s , we s h a l l u s e t h e technique i n t r o d u c e d i n S e c t i o n
6 , Chapter 11.
where
p
E
For
cp E #(I?),
we put
s a t i s f i e s t h e assumptions of Theorem 6.4, Chapter 11,
with h > k + 1.
With t h i s n o t a t i o n , c o n d i t i o n ( 6 . 8 . 3 ) i s
e q u i v a l e n t t o t h e e x i s t e n c e of a c o n s t a n t C such t h a t
i n t h i s case c o n d i t i o n ( 6 . 8 . 2 ) i s e q u i v a l e n t t o t h e e x i s t e n c e of a c o n s t a n t C such t h a t
(6.8.5)
N6($jWj)
5
c
for all
O C b r l
and a l l j.
472
MIXED HYPERBOLIC PROBLEMS
7)
(CHAP.
To o b t a i n t h e s e upper bounds, we a p p l y t h e i n e q u a l i t y
(6.8.1)t r a n s p o r t e d with respect t o ( t ,
YllP, +
2
*'JrjWj\IO
by
x
X I ) ) ;
to
we o b t a i n for y
+ 1 P&*'JrjWjIO
+
\ k P E *l,QjJ$jwj\\E
pc * ~ $ ~ w( * I~ d e n o t e s convolution
\\P&
2 ye
,
2 5 ~c [ / / P & * ~ $ j Q j w j / / o2
*I Bjwj\\:
1+
C[
\ PE
*I
Jr .B.w
12 +
J J d
1~ P e " . B j N j " j l ~
,
2 -1 We m u l t i p l y t h i s i n e q u a l i t y by and t h e n we i n t e g r a t e w i t h r e s p e c t t o
&-2(k+11(1+$) E
u s i n g Theorems
10,lI; a f t e r changing
E
t h e order of i n t e g r a t i o n with respect t o
9
E
and x
n
and a f t e r
6.4 and 6 . 5 o f Chapter I1 and Remark 6.6 of
Chapter I1 ( w i t h xn as p a r a m e t e r ) ; we o b t a i n (6.8.6)j
y'Nb($jWj)
+ Sb($jwj]
5
%*
~\*jQjWj(.'"n)\~,b
+ C"\fjBjWj
We have immediately t h e f o l l o w i n g upper bounds:
dxn
(SEC. 6 )
473
PROBLEM WITHOUT INITIAL CONDITIONS
and by a p p r o p r i a t e l y augmenting y , we can absorb t h e t e r m C N (Q.w.)
6
J J
i n t o t h e l e f t - h a n d s i d e of
(6.8.6)
j'
I n order t o
we u s e a n i d e a have a n analogous t r e a t m e n t f o r t h e term i n 0 . w J j' due t o TARTAKOFF [l]. of t h e c h a r t s
x.J
This c o n s i s t s of s p e c i s t i n g t h e c h o i c e
for j
#
We a g a i n denote by v a v e c t o r
0.
f i e l d d e f i n e d i n t h e neighbourhood o f a i l and t r a n s v e r s a l t o a i l , and w e t a k e f o r
xj
Chapter V, S e c t i o n 1.
(E,
ch a rt s adapted t o
v ) i n t h e s e n s e of
With t h i s c h o i c e , t h e c o o r d i n a t e
t r a n s i t i o n s a r e o f t h e form w
x = (x',x,)€
w i t h y' = y '
( X I )
uj
x
-x(x)
N
= (Y',Yn)E
ui
and P r o p o s i t i o n 6 . 3 of Chapter and yn = x n'
I1 shows t h a t t h e r e e x i s t s C such t h a t
for and f o r a l l
(i[ 5
Hk( ci) n L2 ( zi)
o
w i t h support i n - a
f i x e d compact domain.
We now r e t u r n t o t h e t e r m i n e . w ~j
ctp,= 1, 'J
in
(6.8.6)j; since
we have
(6.8.9) and from ( 6 . 8 . 8 ) , we deduce
Adding, and t a k i n g account of P r o p o s i t i o n 6.3 o f
474
MIXED HYPERBOLIC PROBLEMS
Lxapter 11, we o b t a i n :
and t h i s a l l o w s t h i s term t o be t r e a t e d l i k e (6.8.7) a f t e r
(6.8.6) . for j =
adding t o g e t h e r t h e i n e q u a l i t i e s
J
0,.
. . ,J.
F i n a l l y , f o r y 2 y, s u f f i c i e n t l y l a r g e , we o b t a i n
+ ItvH;]+c[\BV\*
k+l
+
IVlZ]
which proves (6.8.4) and ( 6 . 8 . 5 ) , and t h e r e f o r e P r o p o s i t i o n
6.8.
COROLLARY 6.9 :
e x i s t s yk > u EeYt u
E
0,
Corzsider an i n t e g e r k
such t h a t if
L ~ ( c ) ,P u 6 and
eYt
u
eyt
lac
y 2 y
I-F(z), E
eYt
k'
Bu
then there
and if u s a t i s f i e s :
E
eYt
$(ax),
then
I-F(ac) u
We s t a r t by p r o v i n g t h a t
PROOF:
2 0;
lac -~
E e Y t Ho(aC)
.
To do t h i s we a p p l y P r o p o s i t i o n 6.8 w i t h k = -1, n o t i n g t h a t Remark 2.6 e n s u r e s t h a t
ulac
E
eYt H-'(aC).
a p p l i c a t i o n of P r o p o s i t i o n 6.8 shows t h a t and t h a t u
lac
E
. .,k.
eYt Hj(3-z) f o r j = 1,.
Then, r e p e a t e d
u E eYt
d (C)
6)
(SEC.
PROBLEM WITHOUT I N I T I A L C O N D I T I O N S
475
F i n a l l y , we have THEOREM 6.10
(Existence, uniqueness and r e g u l a r i t y )
We consider problem (1.1)i n the case of systems, asswning Then
t h a t the uniform Lopatinski condition i s s a t i s f i e d . there e x i s t y o and C such t h a t : for y g
E
yo
an) there e x i s t s a unique
x
.*L2(R
2
u
,f
E
eytL2(R
E e*
L'(R
in
Rxan
x
xn)
n)
which s a t i s f i e s
(&lo.l)
Pu= f
in
Rxhl
,Bu=
g
.
Furthermore, we have (6.10.2)
For any i n t e g e r k that: for y 2 y
kJ
soZution u
u
E
PROOF:
2 0,
for f
E
there e x i s t s y ( w i t h yk " y o ) such k
eYtI$(R
xa), gEeytI$(Rxan),
of (6.10.1) s a t i s f i e s : eYt
@ (R~T), UIRxan E
eYt # ( R
xan) ,
I n view of Proposition 6.7 and Corollary 6.9, it
only remains t o e s t a b l i s h t h e uniqueness of (6.10.2) and ( 6 . 1 0 . 3 ) . t h e d e n s i t y of C:(B seen t h a t
the
x
u, t o g e t h e r with
F i r s t , using Corollary 4.3 and from
E)
i n eYt H1 ( R x r ) it can e a s i l y be
476
MIXED HYPERBOLIC PROBLEMS
In particular, Corollary 6.9 shows that the inequality
(6.10.4)holds for all Pu =
0,
u
E eYt L 2 ( W x n )
satisming
Bu = o when y is sufficiently large; this gives the
uniqueness of u.
In order to prove (6.10.2) and ( 6 . ~ 1 . 3 )we ~ consider sequences
, PL*,
f j E C ~ ( xRA)
pectively to f, g in
gj
E C ~ ( Wx an)
converging res-
For y sufficiently large,
we know that there exists u E eYt HI( R x n ) such that j The inequality (6.10.4)applied to the Pu, j = fj , Buj = gj. differences u
-%
in eYt L2( R x n )
shows that u u jy j/,
x
an
are Cauchy sequences
, ey t L 2 ( R x a n ] , and therefore that they tend
respectively to limits v , w in these spaces.
Pu. + Pv in 8 ' ( R x n ) , J
In particular,
and therefore Pv = f.
,
Since u j Theorem 2.5 shows that
v
-t
and Pu + P v in eYt L2(R x n) j 1 in H-' (lR x an), and therefore that 'j\Rxan+ "IRxan loc x
an
= w and Bv = g.
Thus v is actually the (unique)
solution u of (6.10.1)~ and (6.10.3) is obtained by proceeding to the limit in the energy inequality for the u j
THEOREM 6.11 (Support
E
eyt H1(B
Suppose t o € R ; then in t h e
x
E).
(SEC. 6 )
477
PROBLEM WITHOUT I N I T I A L CONDITIONS
statement of Theorem 6.10, we can choose y o i n such a way t h a t we also have the following property: if f and g are zero
.
for a l l t < t, , then the s o l u t i o n u is zero f o r t < t o
By t r a n s l a t i o n , it i s s u f f i c i e n t t o t a k e t, = 0.
PROOF:
We u s e t h e following lemma:
LEMMA 6.12 for t <
:
then
0;
Suppose y
tfdl,;,,
5
>
l{f\lo;y
f o r y'
is a sequence tending towards +
if yj
Yt
fE e
o and l e t
Y t 2 f E f! e j L (C) i s such t h a t 'J f = o for t < 0 .
m
2
E
2 L (Z]
be zero
Conversely,
y.
and if
sup \\flO,, .I
5
e+
The f i r s t p a r t of t h i s lemma i s obvious. converse, we t a k e
-
> o and we c o n s i d e r (p
then
m
For t h e
co (1 0
0
,-EL
X
n);
we n o t e t h a t
We now r e t u r n t o t h e proof o f Theorem 6.11. rc)
y,
2
Y o , where y is c.l
let y 2
t
.
, f E eyt
t h e number appearing i n Theorem 6.10;
Lz(C) an2
g
E
eYt
L2(aZ) be z e r o f c r
From Theorem 6.10, f o r any i n t e g e r j E N t h e r e e x i s t s
u.E eY J* t L2 (C) such t h a t Pu =
f , Bu = g w i t h J j j = y + j and by v i r t u e o f (6.10.3),we have
a unique
j '
Yo
Suppose
MIXED HYPERBOLIC PROBLEMS
yo i s
We s h a l l now show t h a t i f
chosen from t h e beginning
t o be s u f f i c i e n t l y l a r g e , t h e n uj+l = u f o r a l l j , and t h i s j w i l l prove Theorem 6.11 s i n c e a p p l i c a t i o n of Lemma 6.12 shows t h a t u. i s zero f o r t < J
v
= u
j
j+l
e(t)
- u j y and we t a k e
= 1 f o r t S o and
8 v:
that
e
o = QPvi=
Y.t e J
(P
To do t h i s , we p u t
0.
e(t)
L2 (C),
9 E f(R) such t h a t 8 > 0 , -t = e f o r t 2 1. It i s c l e a r and we have
+ i #)(e")
and o = g B v = B ( e v )
By a p p l y i n g t h e uniqueness r e s u l t of Theorem 6.10 t o t h e
-
problem corresponding t o
+
P
i
and B y we o b t a i n v . = o
e
J
when y o i s chosen t o be s u f f i c i e n t l y l a r g e .
7.
SOLUTION OF THE MIXED PROBLEM Suppose R
number >
0.
i s a r e g u l a r open s u b s e t of IRn and l e t T b e a
If
u
P u e L 2 ( l o , n xn),
u
1-
(2.7.3))
%z,(O], P(U0)
E
L2(]o,n
xn)
satisfies
Theorem 2 . 5 and (2.7.1) a l l o w us t o
and we have t h e jump formula ( s e e
-- (Pup
-
where u0 denotes t h e e x t e n s i o n of
u
by o f o r t .<
0.
(SEC. 7 )
479
SOLUTION OF MIXED PROBLEM
I n t h e c a s e of i n i t i a l d a t a which i s e q u a l t o z e r o , we have
THEOREM
We
7.1 ( E x i s t e n c e , uniqueness and r e g u l a r i t y ) .
consider probZem (1.1)i n the systems ease, assuming the uniform Lopatinski condition t o be s a t i s f i e d . g E $(Lo
f EHk([o T I x a )
traces ~j f
TJ
xan)
IN I
be such t h a t the
are zero for 0
and D j g
S j 5 k-1.
Itzo
I t = o
Then there e z i s t s a unique
PU
(7.1 1)
k E
Let
u
E
#(Lo
BU = g l u
= f
such t h a t
T I x?i)
- 0
To prove t h e e x i s t e n c e o f u, we consider
PROOF: d
"
?E
e x t e n s i o n s f , g o f f , g such t h a t
'i; E.Hk(ac)
a r e zero for t <
0,
I$@)
,
t h i s b e i n g p o s s i b l e by We
v i r t u e o f - t h e assumptions made f o r t h e t r a c e s of f and g. A. ,
L.
n o t e t h a t f and g a r e i n t h e spaces eyt Hk for a l l y >
0;
by
t a k i n g y s u f f i c i e n t l y l a r g e , Theorem 6.10 shows t h a t t h e r e exists
'ii E eYt I$@) such t h a t
N
N
N
Pu = f
..
Theorem 6 . 1 1 shows t h a t u i s z e r o for t <
N
B u = f.
0;
consequently
d .,
N
u =
";)O
=
and t h e jump formula i m p l i e s t h a t u
It = #
only remains t o put
"110
(7.1.1)i n $(Lo
TI
x
i).
n xn
0.
It
0
t o obtain a solution of
480
MIXED HYPERBOLIC PROBLEMS It i s s u f f i c i e n t t o prove t h e uniqueness when k = u
x
~'(10
n)
be Such t h a t pu = o ,
BU
=
0,
n
=
u It =
and B ( u o )
be such t h a t o
e(t) =
that y >
0,
<
3
T[xan
euoE
B(e u o ) € eYt L2(aC)
suppe
eYt L2(C],
0;
0
0
in
Let t o m
8 E c ( R ) such
t o< T, and t a k e a f u n c t i o n
1 f o r t < t o and
we have
1-0,
o in
Let
0.
t h e jump formula i m p l i e s t h a t uo a g a i n s a t i s f i e s P(uo) =
]-m,T[x
7)
(CHAP.
.
C]-e,fl
Then, f o r
P ( Q u o ) ( eY t L2(C) ,
Since P(8
uo)i
and B ( 8 u o ] a r e z e r o f o r
t < to,Theorem 6.11 i m p l i e s t h a t 8 u o = o f o r t < t o, s o t h a t u = o f o r t < t o, and f i n a l l y u =
The c o n d i t i o n s imposed i n
0.
(7.1)on t h e t r a c e s of t h e d a t a
f and g a t t h e i n s t a n t t = o a r e i n f a c t t o o s t r o n g .
It i s
s u f f i c i e n t t h a t t h e y s a t i s f y c o m p a t i b i l i t y c o n d i t i o n s ; we s h a l l s t a t e what t h e s e a r e i n t h e c a s e k = +
m.
For k =
0,
a theorem w i t h non-zero i n i t i a l d a t a i s g i v e n i n M U C H 111.
7.2 C o m p a t i b i l i t y r e l a t i o n s f o r t h e d a t a We s t i l l c o n s i d e r t h e c a s e o f t h e m a t r i x problem (1.1) with
P = Dt
-
A(t,x,Dx]
By s u c c e s s i v e d i f f e r e n t i a t i o n s w i t h r e s p e c t t o t , we obtain
D~ = A
+ P , D:
= [A(')+
A,A)+
(A+DJ
p
,...,
(SEC. 7 )
481
SOLUTION OF MIXED PROBLEM
(7-2.1.)
+ Ck(t,x,Dt,Dx)
Dtk =A,(t,X,DX)
with
%+I = % where (4,)'l)
A
D
+ (%)('I ,
D
P
$+, 4, + =
Dt
D
Ck
i s obtained by applying t h e o p e r a t o r Dt t o t h e
4, .
c o e f f i c i e n t s of
u
Let
,
Pu=f, B u = ~ u
lo a .Ti) ,
E
we put
= h a
R R We have Dt g ( o , x ' ) = (Dt Bu)
(0,
x') f o r a l l R E T J , x'
E
3R;
u s i n g (7.2.1)¶we t h e n deduce t h a t t h e d e r i v a t i v e s of f , g , h on t h e edge conditions
{ t=o
, x'Ean] ,
must s a t i s Q t h e following
which a r e known a s compatibility
rezations:
R
Conversely, we have
THEOREM 7 . 3 :
Consider probZem (1.1)i n the systems
case, assuming the uniform Lopatinski condition t o be s a t i s f i e d and assuming 0 t o be bounded.
Suppose we have
I
f
E
C"([o
a x 5) , h
relations (7.2.2). u E C"([o
71 x 5 )
s a t i s f y i n g the compatibizity
C"($)
Then, there e x i s t s a unique such t h a t
P u = f ,
B u = g ,
u
= h .
482
MIXED HYPERBOLIC PROBUMS H
Let
PROOF:
77 x
E c""o
f
R")
and
(CHAP.
7)
i;E c ~ ( R " )
be e x t e n s i o n s of f , g; w e know from Theorem 4 . 5 of Chapter VI c.
that there exists
N
Pv= f
Cauchy problem H
where
w-
w
xn
\[or]
f ! x R")
EC'([o
V
,
Since
0 , I 3 i g - B ~
v
a s o l u t i o n of t h e
P u t t i n g U = u-V,
O
t h e problem b e i n g s t u d i e d i s t r a n s -
formed i n t o PU = F, BU = G , U F = f-PV=
.
= h L
,
,
4=
= H , where we have put 0
, H=h-v
E Cm([o r]x a),
1
= o m t-0
t h e functions
PV
, 8V v L
satisfy t h e c o m p a t i b i l i t y r e l a t i o n s
, and
same a p p l i e s f o r t h e f u n c t i o n s F, G , H .
by l i n e a r i t y , t h e Thus, it i s
s u f f i c i e n t t o e s t a b l i s h t h e e x i s t e n c e of u i n t h e p a r t i c u l a r cas'e where f =
0,
h =
0.
The c o m p a t i b i l i t y r e l a t i o n s ( 7 . 2 . 2 ) t h e n i n d i c a t e t h a t g
i s flat for t =
0,
and s i n c e R
i s bounded we can use Theorem
7 . 1 t o o b t a i n t h e e x i s t e n c e of u i n
n #(to7~ xi'i>= c"(t0 ~3
kzo
n)
F i n a l l y , t h e uniqueness follows from Theorem 7 . 1
RFMARK 7.4 :
.
The p r o p e r t y o f f i n i t e propagation speed
which we s h a l l e s t a b l i s h i n S e c t i o n 8 ( s e e Theorem 8.5) shows t h a t t h e assumption t h a t R i s bounded can b e dropped from t h e s t a t e m e n t of Theorem 7.3.
I n f a c t , it i s s u f f i c i e n t t o
O
(SEC. 8 )
483
FINITE PROPAGATION SPEED
proceed as i n t h e proof of Theorem
4.14 of Chapter V I t o
o b t a i n t h e e x i s t e n c e of a Cm s o l u t i o n .
REMARK 7 . 5 :
I n t h e s c a l a r c a s e , t h e same t y p e of
argument allows us t o prove t h e analogue of Theorem 7.3. a l s o r e f e r t h e r e a d e r t o BALABAN J . KATO
We
f11,SAKAMOTO 111 and
C11.
REMARK 7.6 :
For a p p l i c a t i o n t o c e r t a i n e q u a t i o n s of
P h y s i c s , t h e assumptions on ( P , B, Q) need t o be g e n e r a l i s e d i n several directions.
For example, when t h e Lopatinski
c o n d i t i o n is no l o n g e r uniform, o r when P is n o t s t r i c t l y h y p e r b o l i c , o r when t h e boundary contains corners.
an i s
c h a r a c t e r i s t i c or
For a number of r e s u l t s i n t h e s e d i r e c t i o n s
we r e f e r f o r example t o MAJDA-OSHER C11, OHKUBO-SHIROTA 111, SARASON [ll, and t o t h e i r b i b l i o g r a p h i e s .
See a l s o
Exercise 9.4.
8.
FINITE PROPAGATION SPEED A s f o r t h e Cauchy problem, we s h a l l o b t a i n a f i n i t e
propagation speed p r o p e r t y f o r t h e mixed problem by s u i t a b l y deforming t h e h y p e r s u r f a c e { t = o
1.
I n t h i s c o n t e x t , it w i l l
be u s e f u l t o p r e s e n t t h e uniform L o p a t i n s k i c o n d i t i o n i n a more i n t r i n s i c fashion.
484
MIXED HYPERBOLIC PROBLEMS
8.1 Construction We t a k e F , p , V ,
7)
(CHm.
v as f o l l o w s :
F is a
real-valued v e c t o r space of dimension n + l ; p i s a polynomial which i s homogeneous of degree 1 i n F , w i t h v a l u e s i n t h e space of N x N m a t r i c e s w i t h complex c o e f f i c i e n t s ( i n t h e f o l l o w i n g , F w i l l be t h e f i b r e of T)ain+l a t a p o i n t of IR
x
an, and p will
be t h e r e s t r i c t i o n t o F of t h e p r i n c i p a l symbol of t h e o p e r a t o r P ) ; V and v a r e two independent elements from t h e space F; we suppose t h a t p i s s t r i c t l y hyperbolic w i t h r e s p e c t t o V ( s e e
#
P r o p o s i t i o n 3.7, Chapter V I ) and t h a t p ( v )
TIo€>
Then l e t
and y,2 0 be such t h a t
Rv
,...,en-1
such t h a t (V,e
el,..
denote by ( a ,
F; we put 6 ' =
'
3
is a b a s i s of F, and we
t h e corresponding c o o r d i n a t e s i n and we denote by -n = ~ ( a c', , 6 ) n
(El, ...,En-l)
t h e e x p r e s s i o n for t h e polynomial p , and by o r d i n a t e s of
T,;
l e t E+(IJ,
f o .
of C
,... , en-1 , v )
cn-l ,c n )
( 7 , ,y o )
N as f o l l o w s : we choose
We d e f i n e t h e subspace E+(To,Y,) el
0.
, g;
(0,
,%:)
t h e co-
, Y o ) be t h e subspace c o r r e s -
ponding t o t h e o p e r a t o r v ( Dt , Dxl Dx ) . n p a r t i c u l a r , for Y o > o E+(o,, g:, Yo)
[See S e c t i o n 3; i n
,
i s t h e space of t h e
v ( o ) when v d e s c r i b e s t h e space of bounded s o l u t i o n s for x
n
of t h e d i f f e r e n t i a l e q u a t i o n
rr(
, 5; ,D, n) v(x,]
0, -iY,
e x a c t l y as i n t h e proof o f P r o p o s i t i o n 3.2, t h a t t h e subspace of t h e v e c t o r s when
v
el
Ei(o,
, 6; , Y o )
,. ..,
en-l
i s r e p l a c e d by
Xv
2
-0
0
1;
it can be shown
is independent of t h e choice
and t h a t it remains unchanged with
A >
0
.
We denote t h i s
(SEC.
a)
485
FINITE PROPAGATION SPEED
subspace by E+(7\,,yO]
. N
We now c o n s i d e r a l i n e a r mapping b : C
-t
8.
We s a y t h a t
F , p , b , V , v s a t i s f y t h e uniform L o p a t i n s k i c o n d i t i o n i f , f o r
all
F
and f o r a l l y t o such t h a t
(n,
y)
+
BV
s t r i c t i o n of b t o E ( q , y ) i s an isomorphism.
#
0,
t h e re-
By t h e homo-
+
g e n e i t y of E (rl, y ) i n ( q , y ) , we s e e t h a t t h i s c o n d i t i o n i s not modified i f V i s r e p l a c e d by XV w i t h
A >
0.
ASSUMPTION 8.2 We now l e t M b e a manifold of dimension n + l , and l e t P b e m
a d i f f e r e n t i a l o p e r a t o r of d e g r e e 1 i n M , w i t h C
N
x
N matrix
Suppose y o E M and l e t S, Y b e two germs a t
coefficients.
y o of h y p e r s u r f a c e of M.
We suppose t h a t P i s s t r i c t l y hyper-
b o l i c i n t h e d i r e c t i o n conormal t o S a t y o , and t h a t Y i s noncharacteristic f o r P at y o .
We f i x a connected component S component Y y
0
+
+
of M \ S and a connected
of M \ Y , i . e . o r i e n t a t i o n s a l o n g t h e conormals a t
t o S and Y .
Let B : Y +
N
d
m
S(C ,6: ) be a C mapping.
We
assume t h a t t h e uniform L o p a t i n s k i c o n d i t i o n i s s a t i s f i e d , i . e . : i f V, v a r e v e c t o r s conormal t o S , Y a t y o , w i t h p o s i t i v e
o r i e n t a t i o n , t h e n t h e uniform L o p a t i n s k i c o n d i t i o n i s s a t i s f i e d by F, p , b , V, v where F = T* M, and where p i s t h e r e s t r i c t i o n Yo
t o F o f t h e p r i n c i p a l symbol of P, and where b = B(yo)..
As
486
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
we have seen, this condition is independent of the choice of the conormal vectors V, v .
With these assumptions, we have
PROPOSITION 8.3 (Local uniqueness)
Let W be a neighbourhood of y o i n M. a neighbourhood
Then there e x i s t s
W, o f y o i n M such t h a t : i f
u E Lloc(W,CN) 2
satisfies
in in
pu= o B U = O
S
then u = o i n
PROOF: S and
y0=o
in
u = o
w n s+nP
wn S + ~ Y wnsn~+
w,nS+nY+
By transport by a local chart of M transforming
Y into hyperplanes, the problem reduces to M = Rw'
, s = { t = o ] , y={xn=03 ,
s+={t>03
,
Y+={xn>o].
As in the proof of Holmgren's Theorem (see 1.10, Chapter VI) , the diffeomorphism
x* x,
*
t + 1x1
tt+
the problem to be reduced to the case
then allows
M=RrH1,y o = o ,
487
FINITE PROPAGATION SPEED
(SEC. 8)
I
I I
For (t, x) =
0,
the operator P(t, x, Dt, DX ) is strictly
hyperbolic with respect to t; the same therefore applies when (t, x) is fixed in a sufficiently small neighbourhood of For (t, x') =
0,
the restriction of B(t, x') to is an isomorphism for all
E+(t,x';a,!',y) y t
o such that
0.
2
Q
+Is'\'2 + y 2= 1
applies when (t, X I )
(0,
5')
E
n IR ,
By continuity, the same
is sufficiently near
An argument based
0.
on extending B and the coefficients of P, analogous to that used during the proof of Lemma 4.12 of Chapter VI, thus allows the problem to be reduced to the case where: P, B are defined inIRn+l, Bn the coefficients of P and B are constant outside a compact set, P is strictly hyperbolic with respect to t, and Then let
the uniform Lopatinski condition is satisfied.
r o > o be sufficiently small that
We shall conclude by showing that u = o in Let
uoE L 2 ( R r 1
s+nv+n{t
n
{t
1 W S+n@n{t
S+ n { t
.
be defined by u 0 = u . in
and by u o = o otherwise.
Since the trace
of u on S is zero, the jump formula (see ( 2 . 7 . 2 ) ,
( 2 . 7 . 3 ) ) shows
488
MIXED HYPERBOLIC PROBLEMS
P(uo)=o
that
all that
t
(CHAP.
B(ue)=o
We o b v i o u s l y have
and
The uniqueness p r o p e r t y i n Theorem
0.
7)
uo=o for
7.1 t h e n
shows
u o = 0, and t h i s concludes t h e p r o o f .
PROPOSITION 8.4 :
We consider problem (1.1)i n t h e
systems case, assuming the uniform Lopatinski condition t o be satisfied.
Then there e x i s t s
(t,x')E R x a n ,
p >
such t h a t : f o r a l l
o
the uniform Lopatinski condition i s s a t i s f i e d MI
by F, p , b y V , v w i t h
)
F S = ~ , ( ~R'
of the p r i n c ip a l symbol o f P , b = B ( t , vector t o B t h e cone
PROOF
x
a R a t ( t,
[(a,S)E
:
X I )
,V
,
p = restriction to F v =
X I ) ,
inward conormal
= a ve c t o r from F belonging t o
.
Rx R"1 151 c p o l
By u s i n g l o c a l c h a r t s of
2?
at t h e boundary, we
n s e e t h a t it i s s u f f i c i e n t t o s t u d y t h e c a s e i n which R =B+. We r e c a l l ( s e e Theorem 3.13, Chapter V I ) t h a t t h e r e e x i s t s
v
> o such t h a t f o r a l l ( t , X I )
E
W
an,
x
p i s s t r i c t l y hyper-
b o l i c w i t h r e s p e c t t o any V from t h e cone
1 r=ccQ,61\\$1 c T;.d for
( C ~ $ ' ~ Y ) R"+' E
we p u t X = ( t , x!;u,
.
For V
such t h a t
E
r
y20,
such t h a t IVI = 1 and
2 + \ % ' \2 + Y 2 = I
Q
i-
E l , y ) and we denote by E (X) t h e subspace
V
i-
E ( u , E ' , y ) corresponding t o F, p , V, v.
To s t u d y t h i s sub-
s p a c e , we a g a i n use C o n s t r u c t i o n 8.1; we s e e t h a t , i n t h i s , we can choose f o r t h e v e c t o r s e l , . . . , e
n-1
n- 1 , and t h a t we t h e n have v e c t o r s of B
t h e canonical b a s i s
8)
(SEC.
489
FINITE PROPAGATION SPEED
We can clearly suppose that the coefficient of 5, in p is the identity matrix so that the matrix aV(X) is ~ ( o - ~ ~ , ~ ' , S U ) = ~ ~ I - ~ , , ( X where ) , Cm
Once again we employ the same
with respect to (V, X).
arguments for the matrix a as those used after the statement
V
of Definition 3.3: we factorise the characteristic polynomial
A
V
of a
V
*
+ -
into AV = AV Av, the coefficients of the polynomials
Av being continuous with respect to [V,X) $(X)
= A,(X,a,(X),
; if we put
$(XI,
we have $ ( X ) = I m a g e
and the
uniform Lopatinski condition is expressed by the fact that the matrix B(t,
i
X I )
&v(X) is of maximal rank d =
p,
i.e. by the
condition F(V,X)
(8.4.1)
>
for all X
o
where we have put F(V,X)
=
M
a
x
1 ,et(cj
6j,<. .<j di N
and where we have denoted by C1,
(v, X I I
...,CN
,cj
CVI XI) d
the columns of the
matrix B
The function F is continuous, and the uniform Lopatinski condition for problem (1.1)indicates that (8.4.1)is satisfied for V=V,
, where we have put
R x Rn
V,=(l,o]~
From the compactness in (V, cry
c',
y)
.
and from properties
of the type ( 2 . 3 . 4 ) which are valid for It I large or IxlI large,
490
(CHAP. 7 )
MIXED HYPERBOLIC PROBLEMS
we s e e t h a t Inf F(V,, X) > o and t h a t F i s uniformly continuous. X Thus (8.4.1) i s s a t i s f i e d when V i s s u f f i c i e n t l y n e a r V, and t h i s concludes t h e p r o o f .
From P r o p o s i t i o n s 8.3 and 8.4
, and
by u s i n g t h e same t y p e
of "sweeping" argument as t h a t used f o r Theorem 4.13 i n Chapter V I , we o b t a i n
THEOREM 8.5 (Propagation Speed) We consider probZem (1.11 i n the systems ease, assuming the uniform Lopatinski condition to be s a t i s f i e d . p >
o as i n Proposition 8.4.
For
(t,,x,)E
Suppose
I o , X~R I
we consider the backward cone
Let
u
I
E
n ( I o ,x~R ) ) o i n c n (JO,T[:
be such t h a t
L2(C
PU=
xn)
BU= o
in
cn
(-Jo,T[ x m )
u = o
in
cn
({o} x n )
men u = o i n
c
n
(30,nx n) .
9.
ADDITIONAL NOTES AND EXERCISES
9.1 SPACES eyt H‘ Y a/ We p u t Rn+l = IR
x
Bn,w i t h g e n e r i c p o i n t ( t,x), and we
denote t h e d u a l v a r i a b l e by ( a , E ) . F o r y
E
IR\O,s
E
R,we consider
d e f i n e d i n S e c t i o n 1 1 . 2 , Chapter I V .
t h e H i l b e r t space H S ( B n + l )
Y
I I\u/]g;,
We r e c a l l t h a t t h e norm
i n t h i s space i s
d e f i n e d by:
We c o n s i d e r t h e space eyt HS(Rn+’)
Y
such t h a t
b/
-Yt
E
8
Let k
E
B.
5 n+l Hy(R ]
of t h e
, endowed w i t h
u
E &(R“+’)
t h e norm
Show t h a t t h e f o l l o w i n g t h r e e q u a n t i t i e s
are equivalent: 9
1-0
By n o t i n g t h a t
R
E
I V and f o r
e*
u
(90101.)
u
(0
-
E .P(RW’)
E s*(Rw’) eyt H:(Rn+l)
,
iY]’
for such t h a t
show t h a t :
i s t h e space o f t h e
u E e*
L2(Rw1]
u E L2 (Rn+l ) f o r [ a / = k , and e‘ Y t D a t,x we have t h e f o l l o w i n g e q u i v a l e n c e , uniform w i t h such t h a t
respect t o y:
.
c/ with y s E
(CHm. 7 )
MIXED HYPERBOLIC PROBLEMS
492
0
.
A
Consider a symbol
>0
.
E
PSm[Rwl
X
[ yo, + 01 []
Deduce from S e c t i o n 1 1 . 2 , Chapter I V , t h a t for
IR t h e r e e x i s t s C
0 such t h a t :
2
and d/
R“+’
X
Y
2
Yo
Deduce from (g.l.l), ( 9 . 1 . 2 ) a p r o p e r t y of
Let k E N .
i n v a r i a n c e of t h e spaces
n el
eyt $(Rw’) Y
under diffeomorphism
w i t h r e s p e c t t o x.
G e n e r a l i s e t h i s p r o p e r t y t o t h e c a s e of
s wl ) s p a c e s evt HV(R
n
when s
E
IR ( u s e t h e symbolic c a l c u l u s
for t h e o p e r a t o r s A which i s deduced immediately from S e c t i o n Y’ 1 1 . 2 , Chapter I V )
e/
.
Suppose R i s a r e g u l a r open s e t i n l R
n
.
A s i n t h e case
o f o r d i n a r y Sobolev s p a c e s , we d e f i n e t h e H i l b e r t space eyt H;@i
x
n) by
restriction.
of (9.1.1)when s = k
f/
spaces
E
Show t h a t we have t h e analogue
B.
n When R =lR+ o r when 30 i s compact, we d e f i n e t h e H i l b e r t eYt H:(R
x
an)
: when a R i s compact, we proceed as i n
t h e c a s e of o r d i n a r y Sobolev s p a c e s by choosing once and for a l l
a f i n i t e a t l a s o f ah and a p a r t i t i o n o f u n i t y ( d i f f e r e n t choices l e a d t o e q u i v a l e n t norms on
eYt H;(R
x
an)
b e i n g uniform w i t h r e s p e c t t o y f o r Iyl 2 y o Y t h e norm of u i n e yt H;
denote by IuI S;Y
(P, x
,
t h e equivalence
yo
an).
0).
We
(SEC. 9 )
g/
The g e n e r a l p o i n t o f
with x'
49 3
ADDITIONAL NOTES
E
IRn-l,
AS = [y2
We a g a i n u s e t h e t a n g e n t i a l symbol
x t 0. n
+ a2 + 15'1
i s denoted by x = ( x ' , x n ) ,
+
2 4 2
)
.
By u s i n g (9.1.1),show t h a t
we have, f o r k €IN, t h e f o l l o w i n g e q u i v a l e n c e o f norms, uniform with respect t o y :
9 . 2 . ENERGY INEQUaLITY IN THE SCALAR CASE. The n o t a t i o n is t h a t used i n t h e s t a t e m e n t of Theorem
4.4.
We a g a i n u s e t h e r e d u c t i o n of t h e s c a l a r o p e r a t o r P o f
a/
o r d e r m t o a system of f i r s t o r d e r , as i n Example 4.8 o f Chapter V I .
We t h u s have, f o r
u
E Cm(Rwl] 0
Deduce from t h i s , by u s i n g (g.l.l), t h a t t h e r e e x i s t s
By u s i n g Theorem 4.1, and y
0
i n f e r from t h i s t h a t t h e r e e x i s t
c
2
0
> 0 such t h a t :
b/
Suppose t h a t !2 =R:and
(9.1.3), show t h a t t h e r e e x i s t s c
apply t h e reduction 2.2 > 0 such t h a t :
Using Theorem 4 . 2 , deduce from t h i s t h a t t h e r e e x i s t
Using
494
MIXED HYPERBOLIC PROBLEMS
c/
By p a r t i t i o n of u n i t y , deduce Theorem
4.4
from
(9.2.1ly
(9.2.2).
ALGEBRAIC PRFLIMINARY TO EXERCISE
9.3
N
Suppose F i s a v e c t o r subspace of C Hermitian m a t r i x . N
s t r u c t u r e on C
We denote by (
9.4
and M is an N x N
) t h e c a n o n i c a l hermitian
and by L t h e a s s o c i a t e d o r t h o g o n a l i t y r e l a t i o n .
We suppose t h a t t h e subspace F i s maximal amongst t h o s e f o r which M i s p o s i t i v e , i . e . which s a t i s 0 (Mu, u) 2 0 for a l l
u
E
F.
a/
We p u t G = (M.F)'.
Show t h a t -M i s p o s i t i v e on G. [take v
b/
0
E G,
and s t u d y t h e s i g n of M on t h e subspace
Prove t h e i n c l u s i o n
Ker M
C
F
n
G
Sthe i n c l u s i o n Ker M C G follows from t h e f a c t t h a t M
i s h e r m i t i a n ; for t h e i n c l u s i o n Ker M C F , c o n s i d e r a v e c t o r Vg
E Ker M
and study t h e s i g n of
M on t h e subspace
[V
) + F],
0
9)
(SEC.
Use t h e i n c l u s i o n ' F
c/
exists a basis (el,.
-!-
(-:
form
495
ADDITIONAL NOTES
.., e N ) of
-
(Ker M)'
c
C" i n which M i s w r i t t e n i n t h e
where MI i s an i n v e r t i b l e p
( p = rank M ) , and F i s w r i t t e n F = { u
( r = codim F)).
t o show t h a t t h e r e
E
N
-- ... ur
Iul
C
p matrix
x
= 03
A l s o prove an analogous r e s u l t r e l a t i n g t o
M and G.
9.4
POSITIVE SYMMETRIC SYSTEMS
Let s2 b e a r e g u l a r bounded open s u b s e t of Bn. t h e matrix d i f f e r e n t i a l operator P = where
U(X) =
where x
E
$
m
dependence on x
on t h e subspace B
X
aj +
C(x)
E
5.
We suppose t h a t
i s o f c o n s t a n t rank p , v,(x]]
*denotes t h e
We t a k e a complex v e c t o r bundle
an, and we denote
suppose t h a t for a l l x
E
..,,
(v,(x],
outward unit normal t o 30. o f rank r over
Aj(x)
Vj(X]
~(x)
8 0 and
A,(.]
N m a t r i x and t h e A.(X) a r e N x N J
x
hermitian matrices with C t h e matrix
c
We c o n s i d e r
J=l
i s an N
C(x]
a, +
n
by B
X
i t s f i b r e over x.
We
an t h e h e r m i t i a n m a t r i x M(x) i s p o s i t i v e
and t h a t B
X
f o r which M(x) i s p o s i t i v e .
i s maximal amongst t h e subspaces
We now propose t o i n v e s t i g a t e t h e
mixed problem:
(9.4.1.1
i
Pu u(t,
f
e
XIlan
u(o,.x)
=
E
8,
ug
by showing t h a t t h e o p e r a t o r
- Au
+. x . nI
[o,
in in
Lo, + W E
in
n
I
-
Aj(x]
a J.
x
u
an
- C(x) u
496
MIXXD HYPERBOLIC PROBLEMS
i s a semigroup g e n e r a t o r .
I n t h e f o l l o w i n g , a l l f u n c t i o n s and d i s t r i b u t i o n s have values i n C
a/
vA'{
I\
.
Show t h a t C m ( E ) is dense i n t h e space
u E L2 (R)IAu E L2 (0)
Show t h a t t h e mapping
1
normed by
-
3 u
C"m)
extends c o n t i n u o u s l y i n t o a mapping o f from t h i s t h a t t h e Green's formula
(Au,
(9~2.1
42 L
remains v a l i d f o r u
A*
=
E
-A+
* .I
b,
=
V and v
A
(C(x)
A E
+
H 1( $-2 ) ; we have p u t
+ C*(X)
aA - E 2, I.)) 1 j
[ p r o c e e d e x a c t l y as i n Theorem 2.51. b/ 2?
Let xo
E an;
show t h a t t h e r e e x i s t s a l o c a l c h a r t of
t o t h e boundary and a t r i v i a l i s a t i o n o f B over t h i s c h a r t
such t h a t , a f t e r t r a n s p o r t , t h e matrix M(x) and t h e bundle Bx
N
k
and
Bx
=
{ u E c Iu,
P
...
i n v e r t i b l e p x p m a t r i x and N
I
- r
%-r
I
0
where MI(x) i s an
5 p [use Exercise
Deduce from t h i s t h a t i n t h e neighbourhood o f x
0
9.31.
there exists
(SEC. 9 )
an N
N m a t r i x B(x) such t h a t u
x
497
ADDITIONAL NOTES
B(x] M(x)u
3
0
.
E
Bx i f and o n l y i f
Use t h i s f o r m u l a t i o n of t h e boundary con-
d i t i o n and t h e f i r s t p a r t o f a / t o prove t h a t it i s p o s s i b l e t o d e f i n e a c l o s e d subspace o f VA by
c/ i n D,
Use b / t o prove t h a t t h e space D
0
=D
n H1m]
i s dense
T h i s a l s o e n a b l e s D t o be c o n s i d e r e d as t h e c l o s u r e of
n p a r t i t i o n of u n i t y , reduce t o t h e c a s e of Fi+ E
n
> 0 and
> 0 c o n s t r u c t approximations
u
El7
where: u1 and ulI denote t h e components i n Cp and CN-P
of u
E
D,
p
E
i s a r e g u l a r i s i n g sequence i n t h e v a r i a b l e s
(xl,.. . , xn-1 ) and t h e 0 rl a r e a r e g u l a r i s i n g bequence i n t h e v a r i a b l e x and w i t h s u p p o r t i n x n n
d/
A
2.
c
0'
.
0.
Show t h a t
Using ( 9 . 4 . 2 ) , show t h a t t h e r e e x i s t s c0 such t h a t
deduce from t h i s t h a t :
49 8
Note t h a t t h e maximality of t h e B
X
e/ v
(CHAP. 7)
MIXED HYPERBOLIC PROBLEMS
2
L
E
Show t h a t t h e space ( A
,
( X + A*] v = D
that
+ A ) D i s dense i n L 2 .
+ A)u, v ) =
be such t h a t ((A
i s n o t used h e r e .
0 for all u
v € V , = { u E L 2
I
E
[Let
Do, and show
A*uEL2]
and
A
r
I
w
Use E x e r c i s e 9 . 3 t o show t h a t t h e s p a c e s B
=(M(x) Bx) X d e f i n e a v e c t o r bundle o v e r which -M is p o s i t i v e and show t h a t
F i n a l l y , deduce from t h i s t h a t v = 0 by a y p l y i n g d / t o (A
* -
, D).
f/ Deduce t h a t (-A, t i n u o u s semigroup i n L'(il)
D ) i s a g e n e r a t o r o f a s t r o n g l y conand a p p l y t h i s t o t h e s o l u t i o n of t h e
mixed problem ( 9 . 4 . 1 ) .
Show t h a t t h e r e s u l t s of t h i s e x e r c i s e can be a p p l i e d
g/
t o t h e c a s e of t h e Maxwell e q u a t i o n s (Example
4 . 6 i n Chapter VI
i n an open s e t R , by adding as boundary c o n d i t i o n t h e c o n d i t i o n
A
V(X)
E(x)
o
CY
U[X]
A
H(x)
for a l l
x
E an
where a i s a given r e a l number and where A denotes t h e v e c t o r product i n m
h/
3
.
Show t h a t t h e r e s u l t s of t h i s e x e r c i s e can a l s o be
(SEC. 9 )
499
ADDITIONAL NOTES
a p p l i e d t o t h e c a s e o f t h e wave e q u a t i o n w i t h Neumann boundary
U i s a s o l u t i o n of t h e symmetric system
,
0
a, u
- ax,
=
,
-ax,
- a,
.a.
n
I
a a
a
-ax
9.5
9
n
EXAMPLE OF A BOUNDARY-VALUE PROBLEM FOR A FIRST-
ORDER DIFFERENTIAL SYSTEM SATISFYING THE UNIFORM LOPATINSKI C O N D I T I O N . I n IR x B3, we c o n s i d e r t h e ( 2 x 2 m a t r i x ) d i f f e r e n t i a l o p e r a t o r P = p(Dt y
= Dt-
D z ) d e f i n e d by
Y'
(;-:) (: Dx-
We denote by Show t h a t
Dxy D
7, f ,
det p(7, 5,
T, C
7, 61
:)
DY
-
0 (_i
t h e d u a l v a r i a b l e s of
.i" -
- l'f - f
%
o) t,x,y,Z.
,
and
t h e r e f o r e t h a t P i s s t r i c t l y hyperbolic with r es pe ct t o t .
Let
!,In
[ (x,
v a l u e problem
y, z )
E R3
I
x
> 0
Consider t h e boundary
500
MIXED HYPERBOLIC PROBLEMS
Show t h a t f o r y i n Section 3 i s
where
z
=
7+
>0
{ (a,
16
,
7
E
E+(u, 7, C, v) Z a t
t h e subspace
ef U
E
C2
- iy
I
(CHAP.
-*
8 5 - 7
, and where +
square r o o t w i t h p o s i t i v e imaginary p a r t .
defined
denotes t h e
Deduce t h a t
problem ( 9 . 5 . 1 ) s a t i s f i e s t h e uniform L o p a t i n s k i c o n d i t i o n .
7)
CHAPTER 8
MICROLOCALISATION
1.
GENERAL PROPERTIES OF THE SINGULAR SPECTRUM (WFL Let X be an open s e t i n Rn and suppose u
d e f i n i t i o n o f t h e s i n g u l a r spectrum of
u
a' (x).
E
The
(which we denote by
WFu; s e e D e f i n i t i o n 3.1, Chapter I ) can be formulated a s f o l l o w s , remembering t h a t a subset when
6
Let
1.1
DEFINITION
(xo
w
and a conic neighbowhood IY
E
Cr(V)
t h a t for a l l N
1
5
C(l
E
W
Ex
x
(R" \
.
O)
we say
+
,
of 5, i n Rn \ 0 such t h a t , for any
a u has rapid decrease i n
IR, there e x i s t s
la\)*
(R" \ 0) , and
s u b s e t o f X.
, 5,)
A
when 5
The s i n g u l a r spectrum of
X x
E
WFu i f there e x i s t s a neighbourhood V of xo i n X
t h a t (x0 ,f 0 )
I&{)
of B n \ 0 i s c o n i c a l i f t 5
and t > 0 ( s e e Chapter I , S e c t i o n 1):
E W
function
W
E
C 2 0
w
l i . e . . such
with
W).
u is a conic c l o s e d s u b s e t of
we have WF(u y )
ES
(WFu)
I Y
when Y i s an open
We have a l s o seen ((3.1.3), Chapter I ) :
PROPOSITION 1.2:
Let
nx : X . x (R"
\
0 ) -X
be
(CHAP. 8 )
MICROLOCALISATION
502
the projection n,(WFu)
P
(x,!)
.
-x
supp sing u
.
For u
E
. 8 ' ( x ) , we have
We can a l s o o b t a i n p a r t i a l r e g u l a r i t y p r o p e r t i e s :
PROPOSITION 1 . 3 :
IRn i s denoted by x =
put 5 = ( 5 '
, En).
1
{(O,O,O,sn)
Let u (XI
E
8' ( P ; the generic point of
, x,) with x'
E
Dn-l ; we similarly
we a s s m e t h a t m u does not i n t e r s e c t
5, E R \ O}
Then, f o r
,
cy m
support s u f f i c i e n t l y near (0,0 ) , au is
C
E C:(Rn)
with
i n xn with values
which are x' d i s t r i b u t i o n s (see Section 1.5, Chapter 1).
PROOF:
A
and C > 0 such t h a t ciu i s r a p i d l y d e c r e a s i n g i n
i n IR"
{c a.
By h y p o t h e s i s , t h e r e e x i s t a neighbourhood V o f (0,0 )
E Rn
I
15'
1
5
Cl{,l]
for
ci E
We f i x such a f u n c t i o n
Cz(V).
A
Since au i s slowly growing ( s e e ( 2 . 8 . 1 ) , Chapter I ) , it can
be shown t h a t t h e r e e x i s t s 1-1
e x i s t s CN w i t h
Ia(s)jS
5 = [If, E Rn t u r n t h e two Cases
E
IR such t h a t , f o r a l l N, t h e r e
$(I i
lsn1]4
(1 -1-
lS'I)p"+N
for
( t h i s f o l l o w s immediately by examining in
[5*1
I
Clh]
,
ItJl >
Cl%l
)
.
Thus
c1u
503
SINGULAR SPECTRUM
(SEC. 1)
is i d e n t i f i e d w i t h t h e f u n c t i o n
d e f i n e d , f o r xn
E
u
E
m
Co(R
, &'(R -1
))
IR , by
We s h a l l now g i v e a v a r i a n t o f D e f i n i t i o n 1.1.
For t h i s
p u r p o s e , we s t a r t w i t h t h e f o l l o w i n g p r o p o s i t i o n :
PROPOSITION 1.4:
Suppose we have v
E
c ( X ) and
l e t W be
A
an open cone i n I R n \
0
Then, f o r f E c"(x) W' i n I R n \
PROOF: have
with
0 such that
such that v is rapidly decreasing i n W.
,
A
fv c
is rapidly decreasing i n any cone W
.
C l e a r l y , we can assume t h a t
504
(CHAP. 8 )
MICROLOCALISATION
By h y p o t h e s i s , t h e r e e x i s t s c > 0 such t h a t
15-
'I)\ 2 c(
1c1 +
r a p i d decrease of exists
u
E
4
W.
and t h e slow growth of
2,
when
?
6
E
W ' and rl
I n view o f t h e we see t h a t t h e r e
IR and, f o r a l l N, a c o n s t a n t CN such t h a t , for 6
E
W':
and hence t h e r a p i d decrease o f I1 i n W'.
08
F i n a l l y , since f
i s bounded, we have:
COROLLARY 1.5: (x,
, 5,) E x
x
Let u
(R"\ O)
.
E
&'(x)
and
I n order t h a t
, 5,) k WFU , i t
(x,
i s necessary and s u f f i c i e n t t h a t there e x i s t a a t xo, and a conic neighbowlhood i s rapidZy decreasing i n PROOF:
w
E
Cz(x), non-zero
of 5, i n Dn \ 0 such t h a t
w.
The c o n d i t i o n i s c l e a r l y n e c e s s a r y ;
t o s e e t h a t it is
s u f f i c i e n t , we c o n s i d e r a neighbowhood V o f x
0
i n X such t h a t
# 0 for x
a(x) f
5
505
SINGULAR SPECTRUM
(SEC. 1)
,
E Cr(X)
p/a
if B
V;
E
c z ( V ) , we have Bu = f v w i t h
E
.
v = au
The c o n c l u s i o n t h u s f o l l o w s
from P r o p o s i t i o n 1 . 4 . COROLLARY
1.6:
f E C:(X),
Let
closed cone i n E n \ 0 such t h a t
and l e t W be a
u E &(X)
(Supp f x W)
n WFu
=
.
b
h
[Then f u i s rapidly decreasing i n W.
PROOF:
Let
such t h a t
f be
a c l o s e d c o n i c neighbourhood o f W i n IR" \ 0
(Supp f x
9 n WFu
n
.
d
For
xo E Supp u
,
A
t h e r e e x i s t s a neighbourhood V of xo i n X such t h a t au i s r a p i d l y when
decreasing i n
n Sn-l).
of
ci E
C"(V) ( t h i s f o l l o w s from t h e compactness 0
Let t h e r e b e a f i n i t e c o v e r i n g o f supp f by V
aj E C,"(Vj)
of t h i s t y p e , and l e t t h e r e b e f u n c t i o n s
a
j
= 1 i n supp f .
C
fu =
We have
f(aju)
, and
J
A
r a p i d d e c r e a s e o f f u i n W by a p p l y i n g P r o p o s i t i o n
DEFINITION
1.7:
+(x)
(u
We put
P
Let
r
E P(x)
1. WFu
J
(1.7.2)
I
thepe e x i s t s bourhood W o f SUP
5Ew
(I+
&',(x)
C
, so)
ci E
co
C"(X) 0
i n ?i
EX x
.
]'l
to u
(uj) converges in f l ( X ) t o
For a12 (xo
'S
such t h a t
hence t h e
1.4.
be a closed cone i n X x (R" \ 0)
sequence (u.) converges i n
(1.7.1)
j
.
We say t h a t a
E ~ ; ( x )
if
u
(R" \ 0) not belonging t o r,
w i t h a(xo) # 0 , and a conic neigh\ 0 such t h a t , for a l l N ,
n IS(F 1 G(5)- C Y U ~ ( ~ ) I + O
i f
j + + m
506
(CHAP. 8 )
MICROLOCALISATION
By a r g u i n g as i n t h e p r o o f o f C o r o l l a r y 1 . 6 , it can b e shown t h a t condition ( 1 . 7 . 2 ) i s equivalent t o
(1.7.3)
For any
E
such t h a t
(supp cy x W )
(1 +
SUP
5EW
c:(X)
and f o r any c l o s e d cone W i n B n \ 0
nr
, then,
=
- Cj(s)l-+
1 ~ 1 11 ~
?U(SI
o
i f
for all N, j++-.
By a r g u i n g a s i n t h e p r o o f o f P r o p o s i t i o n 1 . 4 , it can be
f
shown t h a t for
(1.7.4)
E
:
C:(X)
+(XI i n t o
from
fu
u> -I
The mapping
PROPOSITION 1.8:
.
+(x)
Let u
i s s e q u e n t i a l l y continuous
8i(X); then there e x i s t s a
E
Furthermore, it sequence u E C E ( X) such t h a t u + u i n o 8 ' , ( X ) , j j i s possibZe t o choose the u I s w i t h support i n an a r b i t r a w j neighbourhood of supp u
.
PROOF: p
u
j
E
C;(R")
n
u
F i r s t we assume t h a t
pj
* u.
Let f
E
in X
and
W
IRn\ 0 such t h a t (V x
fu
.
Let and
C r ( X ] and l e t W b e a c l o s e d cone i n Bn\ 0
r
6.
Let
V
b e a neighbowhood o f
a c l o s e d c o n i c a l neighbourhood o f
nr 6 .
u n i t y i n t h e neighbowhood of l a r g e , we have :
&l(X)
pj(x) = j" p ( j x )
w i t h i n t e g r a l 1; we p u t
such t h a t [supp f x W ) fi supp f
E Pr(X) n
supp
Let cy f
.
* U.) = f ( p j *
E
C:(X)
W
in
be e q u a l t o
For j s u f f i c i e n t l y (CYU))
4 ~ l ( -l ^p(T/j))+T
,
and t h u s
A
0
(SEC. 1)
Since
1
507
SINGULAR SPECTRUM
-
- i[v/j)
D ,
and remains bounded, when j
+
+
m,
we proceed e x a c t l y as i n t h e proof o f P r o p o s i t i o n 1 . 4 t o establish that
and t h u s t h a t u
j
+
u, i n
16i(X).
When
u
no l o n g e r has compact
s u p p o r t , we consider a l o c a l l y f i n i t e covering o f X by r e l a t i v e l y compact open s e t s i n X and
%,
Bk
*j =
also
~ i c, 8, E
w i t h C ak = 1
Cz(X,)
= 1 i n t h e neighbourhood of supp ak.
E C:(X,)
then
3 , and
%,j .13 Wku
such t h a t
@k%j
is in
c"(x)
Suppose
j
i n b k ( X ) when
and t e n d s towards
u
in
+
+
m;
pr(x);
l e t K . be an e x h a u s t i v e sequence of compact s u b s e t s o f X and l e t 3
Xj
E
that
it t h e n f o l l o w s immediately * j' t e n d s towards u i n & , (; X ) , F i n a l l y , t h e
be e q u a l t o u n i t y i n K
C"(X) o
'pj =
xjqj
support p r o p e r t y f o r t h e t r u n c a t i o n , s t a r t i n g from
DEFINITION 1.9:
Q ' s is o b t a i n e d by means of a new
j
(1.7.4).
Let r be a closed cone i n X x (R"\ 0)
and l e t F be a closed subset of X. P+X1
= Eu E +(XI
We s h a l l usually assume t h a t
We put
1 supp
u E Fl
h j X rc F
0
( t h i s i s not an
actual r e s t r i c t i o n i n view o f Proposition 1.2), and i n B;,,(x) we s h a l l consider the notion o f sequential convergence induced by
&$(XI.
508
2.
(CHAP. 8 )
MICROLOCALISATION
THE FUNDAMENTAL THEOREMS
We s t a r t by s t u d y i n g t h e s i n g u l a r spectrum o f an i n t e g r a l d i s t r i b u t i o n a s s o c i a t e d w i t h a phase and an amplitude ( s e e Chapter 111, S e c t i o n 7 ) .
Let
THEOREM 2 . 1 :
E
A
be the d i s t r i b u t i o n defined
a(X)
by
,u >
where (2.1.11
Q
WFA
1
=
ei~(Y~e)a(x,B)u(x)dx
i s a phase and
a
c
1e#
Ux,c:(x,e))
(.,el
dB
(u
an amplitude.
E
0
E
Then
,
cone SUPP a
, cp;(x,e)
PROOF:
Let F denote t h e r i g h t - h a n d s i d e o f ( 2 . 1 . 1 ) .
have (xo
, 5,)
E
X x
(R” \ 0) which
C:(X))
=
03
,
Suppose we
It can
does n o t b e l o n g t o F .
e a s i l y b e shown t h a t t h e r e e x i s t s a neighbourhood V of xo i n X, a c l o s e d conic neighbourhood W of 5, i n IRn\ 0 and a conic neighbourhood such t h a t
?
of
c
E
= E(x,e)
Cp~(X,O)
1W
cone supp a
1 0 # o , Cp$x,e)
=
01
u
when x
E
V and ( x , 9 )
E
C.
(Note t h a t
t h i s proves t h a t F i s c l o s e d ) .
By t r u n c a t i o n , we can assume t h a t a ( x , 9 ) = 0 f o r 1 9 1 L e t x(x,9) be a Cm f u n c t i o n i n degree 0 i n 0 , such t h a t supp
WF A
x
c
F.
= WF A,
x
X x
S
1.
(RN \ 0) , homogeneous of
= 1 i n t h e neighbourhood of C , and
From P r o p o s i t i o n 7.7 o f Chapter 111, we have
, where
A
1
i s d e f i n e d by t h e phase
q,
and. t h e
(SEC. 2 )
For a
amplitude Xa.
C z ( V ) , we have
E
.
b(x,e) = w(x)X(x,e)a(x,B)
with
-tf0
cp$X,8)
5 09
FUNDAMENTAL THEOREMS
for 6
E
By c o n s t r u c t i o n , we have
W and ( x , e )
cone supp b ;
E
we t h e n
deduce, by c o n t i n u i t y and homogeneity w i t h r e s p e c t t o (6',5), t h a t t h e r e e x i s t s a c o n s t a n t c > 0 such t h a t
Iqi(x,@]
- gl
5
.(I91
+ 151)
for
5
E
W and ( x , e )
E
cone supp b .
Consider t h e d i f f e r e n t i a l o p e r a t o r
It can e a s i l y be shown, d e n o t i n g by
m
t h e degree of t h e amplit-
s o t h a t we have
5 Ew
for (x0
9
5,)
, I$\
2
1
and
m
- k/2 < - N
, and
t h i s shows t h a t
& W F +a
REMARK 2 . 2 :
By t r u n c a t i o n , it can e a s i l y b e deduced from
Tneorem 2 . 1 t h a t , w i t h t h e same assumptions, we have more precisely
(CHAP. 8 )
MICROLOCALISATION
510
WFA
c
18f
E(x, c ~ : ( X , Q I I
0
I
(x,eI E ess where
9 , the
01 ,
.
-m
is S
a
REMARK 2 . 3 : fixed
=
denotes t h e s m a l l e s t c l o s e d cone i n X x IRn
e s s supp a
o u t s i d e o f which
, (p;(x,e)
a
SUPP
The p r o o f of Theorem 2 . 1 a l s o proves t h a t , f o r mapping a * A i s s e q u e n t i a l l y c o n t i n u o u s from
&i(X)
Sm(X x IRn ) i n t o
where
1040
A = h9;(x,e)l
-
+p,el
03
-
Let us now c o n s i d e r t h e t e n s o r product of two d i s t r i b u t i o n s .
Let X be an open s e t i n IRn , and Y an open
THEOREM 2 . 4 :
s e t i n IRp , and suppose WF(u Q v ]
(2.4.1)
(WFU
AX)
u E
PROOF:
(x
x WFV-)
u
( S U P P ~ Ux W F V )
x YI x
Suppose we have
(xo
, 5,)
#
0 and
a(x ) 0
5 , i n B n \ 0;
WFu
;
& rapidly suppose B
a ( I , q )
have
R"+" (x,
E
(5
(x
with
9
u
(WFUx
x
R"]
,
R'I
,
not belonging
Y
5, f 0
F i r s t we assume t h a t
we can t h e n c o n s i d e r a
vj
SUW,
x (Y x
c, , Y , ,
E
C:(X)
with
d e c r e a s i n g i n a c o n i c neighbourhood W of E
C:(Y)
~$(C)
i s such t h a t B(yo) # 0 .
07)
f i n d a c o n i c neighbourhood C o f ( C o y 1
Then
= supp u x {O]
suppou
t o t h e right-hand s i d e o f ( 2 . 4 . 1 ) . and
.
E @(Y)
c
wherc we have p u t f o r example and i d e n t i f i e d
,v
.
Since
no) in
IRn+'
5, #
We
0 , we can
\ 0 such t h a t
.
,n )
E
C implies
5
E
W and
151 +
5
c
IC\ , where
c
is a
(SEC. 2 )
FUNDAMENTAL THEOREMS
positive constant.
Thus
with r e s p e c t t o ( 5 ,
n)
yo
qo
$1 0 w i t h
(y
have 5 , = 0 and thesis; that
(yo
hence u
(X,
I
Yo
@
REMARK 2 . 5 :
Y x ($ \ 0)
k WFV
, %)
.
We argue s i m i l a r l y when
F i n a l l y , when, f o r example, w e
, then
E WFv
kW
Suppose
.
i s r a p i d l y decreasing
xo
k
U
by hypo-
r 1' r 2
a r e c l o s e d cones i n X x
Let Fly F2 be c l o s e d s u b s e t s of X , Y .
(x')
n t i a l l y continuous from
EXAMPLE 2 . 6 :
,
u
.
Q vl
above proof a l s o shows t h a t t h e mapping (u,.)
x = ( x ' , xn)
supp
v i s zero i n t h e neighbourhooh o f ( x o , y o ) s o
5, *
#
.
WF(u O v )
\)
0'
A
au(6) @ v( 7 )
i n C , and t h i s shows t h a t
, 5, , \) k
(x0
A
511
x'
WF($(x') 9 Y ( x n ) )
x
.-t
92' F2Cy)
(Rn \
0),
The
u Q v i s seque-
into
From Example 3 . 2 of Chapter I , p u t t i n g
=
, we
( X ~ . . . , X ~ ~ )
have
c
b = o , g f 0 ' 3
u
Cx'=O
,
xn>09
r'fo9a-0,
( i t i s shown i n E x e r c i s e ( 5 . 1 ) t h a t t h i s i n c l u s i o n i s a c t u a l l y an equality).
We s h a l l now show t h a t , under c e r t a i n assumptions, it i s p o s s i b l e t o d e f i n e i n a n a t u r a l manner t h e i n n e r product of two d i s t r i b utions.
H-'(IRn
The argument employed i s analogous t o t h a t which allows
) t o be i d e n t i f i e d w i t h t h e d u a l o f H S ( I R n ) .
For
512
(CHAP. 8)
MICROLOCALJSATION
The mapping of
THEOREM 2 . 7 :
< u1
which associates (ul,,
, u2
> =
into C ,
C:(X)
x C(:X)
u,(x)
u2(x) dx
with
extends uniquely i n t o a mapping ( a l s o denoted by
U2),
< . , . > ) o f
n supp rl , r2 i n
u2
supp u1
cones
i n t o CC which, f o r a l l closed
compact}
X x
(R" \ 0 ) such t h a t rl
V
n r2
#
=
, for
any compact subset K of X, i s sequentialzy continuous w i t h respect
t o (ul
I
PROOF:
1
u2
+1(XI x P (XI r2
Uniqueness:
a) V
n
WF u1
E
,
WF u2 = $t
when
Let
u1
supp u,
n
SUPP u1
, u2
E P(X)
SUPP u2
n
SUPP u2
c K.
be such t h a t
compact
.
We can
f i n d a neighbourhood V1 of supp u1 i n X and a neighbourhood V2 of supp u2 i n X such t h a t V
-
1
V
n
2
i s r e l a t i v e l y compact i n X .
From
P r o p o s i t i o n 1 . 8 , we can f i n d two sequences ( uj , k ) k c N such t h a t U
J, k
and
rj
u
in
5
&' (X) w i t h
supp rj C We must t h e r e f o r e have
= WFu . ) . J
< u,
, u2
>= l i m k
b)
ment of t h e theorem.
rJo =
[E
1
(xo
CD
Let
For x0
, C) E rj]
( f o r j = 1,2,
Ulk(x)U&(x)dx
d +
Existence:
vJ
EK
, r2
and K be a s i n t h e s t a t e -
we put
and we c o n s i d e r an open c o n i c neigh-
?"
bourhooa
jo
such t h a t
where y
H
i n X such t h a t V x y
0
jy
rl0 n r20 =
N
r
of
jo neighbourhood V o f x
r
51.3
FUNDAMENTAL THEOREMS
(SEC. 2 )
denotes t h e complement o f
j
along with a
j
*
r
jo
does n o t i n t e r s e c t
in IRn\
Suppose
0.
we have a f i n i t e c o v e r i n g o f t h e compact s e t K by neighbourhoods V = V
o f t h i s t y p e , and t h a t we have f u n c t i o n s a
k
E
k
C z ( V k ) such
t h a t C a,- = 1 i n t h e neighbourhood o f K , a l o n g w i t h f u n c t i o n s R
8,
km
C (V,)
E
For ful
I
Bk
such t h a t
Ed
u2)
rl
(X) x
= 1 i n t h e neighbourhood o f supp ax.
d
(X)
with
r2
o b s e r v i n g t h a t t h e i n t e g r a l s a r e a b s o l u t e l y convergent :
i n view
A
A
1 . 6 , ak u,
i s r a p i d l y d e c r e a s i n g i n yl, 8 k u 2 i s A A r a p i d l y d e c r e a s i n g i n y2, and t h u s %u1(C) p,u2(-t) is rapidly
of C o r o l l a r y
u
y1
decreasing i n
V
.
R" \ 0
=
y2
Using ( 1 . 7 . 3 ) ~ it f o l l o w s
immediately t h a t t h e i n t e g r a l s depend ( s e q u e n t i a l l y ) c o n t i n u o u s l y on
(u,
I
u2) E
rl
(X) x
9
(X)
and f i n a l l y , t h a t t h e r i g h t -
r2 u,(x)
hand s i d e of (2.7.1) c o i n c i d e s w i t h
CONSTRUCTION 2.8: if
E c
E' c
E' = {(x of (X x
Let X , Y be open s u b s e t s o f IRn
, we
(XxY) x Rw
(X x 1
R")
R")
x (Y x Rp)
I:
Y
X
9
I
(Y x
fl 1 RP )
u2(x) dx
when
, IRp
;
define
by
[x v
5
9
Y
9
-M E El
-
A s u b s e t E'
d e f i n e s a r e l a t i o n of Y x Rp w i t h X x En :
514
(CHAP. 8)
MICROLOCALISATION
-
We c o n s i d e r a continuous l i n e a r o p e r a t o r
A : Co(Y)
&*(X)
,
KA E
If
(X x Y)
denotes t h e kernel
o f A, we have by d e f i n i t i o n
(2.8.1
I
for
u E
41, u > C;(X)
v E C;(Y)
I
a
We now seek t o e x t e n d t h e d e f i n i t i o n of
&"(Y)
E
2.7.
p ( Y ) and u
WF(uQv) Theorem
E
E
t o certain
C"(X), we r e c a l l t h a t 0
and s u p p ( u ~ v ) =
supp0u x WF v
C
Av
by u s i n g t h e e q u a l i t y ( 2 8.1) and Theorem
distributions v For v
>
U Q V
2.7 t h u s a l l o w s t h e i n n e r product
supp u x supp v
< KA
uQv
>
t o be
d e f i n e d when
WF'KA
tl
supp KA
( S U P P ~ Ux W F V)
n (supp
=
6
These c o n d i t i o n s a r e r e a l i s e d f o r a l l u
(2.8.2)
(2.8.3)
compact
u x supp v) E
.
Cz(X) i f and o n l y i f
W P K A , WF v c X x (Rn \ 0) The p r o j e c t i o n
n
X '
supp KA
n (X
x supp v)
4
X
i s proper.
Let us assume t h a t t h e s e c o n d i t i o n s a r e r e a l i s e d ;
if u
j
is a
.
sequence il
j
B
51 5
FUNDAMENTAL THEOREMS
(SEC. 2 )
C"(X), Remark 2.5 shows t h a t
which t e n d s t o 0 i n
&',(X
v tends t o 0 in
0
C
x Y), where
I
, uj @ v
and Theorem 2 . 7 then shows t h a t
< KA
Tnus formula ( 2 . 8 . 1 ) d e f i n e s
as an element o f
Suppose we have I',
Av
c l o s e d s u b s e t o f Y such t h a t such t h a t t h e p r o j e c t i o n
r
WF*KA
n
supp KA
(X x
-
.
#(X)
and F, a
, and
X x (Rn\ 0)
F)
X
i s proper. +
Av i n
and t h i s proves t h e f i r s t two p a r t s o f :
THEOREM 2 . 9 :
Suppose we have an operator A : c"(Y)-P 0
(XI.
i ) We define an extension ( i t s e l f denoted by A) of A into
an operator from {v
E
1 WFK' ,
&"Y)
SUPP K~
n (x
putting
ii)
Let
r
WF v
x SUPP
V)
,u >
C
X x (R" \ 0)
=X
x proper) i n t o
-
u
e K,,
be a cZosed cone i n
8
&'(XI,
> for ~ Z uZ
Q V
by E
WF*K~ supp K~
r cx n
x ( R ~I 0)
(X x F )
=X
x
poper
.
Then t h e operator A i s s e q u e n t i a l l y continuous from $,F(y)
into
P (XI.
In s i t u a t i o n ( i ) ,we have
cz(x).
Y x (Rp \ 0) and l e t F be
a closed subset of Y such t h a t
iii)
,
tends t o 0.
8; yF(Y), t h e n Avk
We s e e s i m i l a r l y t h a t i f vk -+ v i n
&'(X),
>
Y x (Rp \ 0)
a c l o s e d cone i n
x WF v
(X x {O])
516
(CHAP. 8)
MICROLOCALISATION
(2.9.1 )
(WF'K~
WF(AV)
u
WF V ]
(WF'K~
SUPP~V)
where the right-hand s i d e is a conic closed subset of X x (Rn\ 0). We s h a l l a c t u a l l y e s t a b l i s h a more g e n e r a l r e s u l t t h a n
Let
THEOREM 2.10: (X x Y ] x (RnW
\
0)
x
a closed subset of
(iii):
be a closed cone i n
C
,r a
closed cone in Y x (# \ 0)
x Y, and F a closed subset of Y.
,
s
Suppose
that
=x
The projection S f'l(X x F)
(2.10.2) -OjX
VQ put A = (C closed cone i n
,
U (c'
(F x
, and
0)
lo])) .
the mapping
+,$YI
into
%(XI
*
PROOF:
F i r s t , we prove t h a t A i s c l o s e d i n
suppose
(xj
, F,)
E
Y
cj'v
Yj
and, since x
j
X x (Rn \ 0) ;
is a sequence t e n d i n g t o ( x , C ) i n
A
X x ( IRn \ 0); we c o n s i d e r a sequence (XJ
(yj
, qj) E r such t h a t , 'Y,] E s , Y J E F
9
Tlj] E Z'
+
x i n X , ( 2 . 1 0 . 2 ) shows t h a t y
j
;
we have
(xj
j
remains i n a
by e x t r a c t i n g a subsequence, we can t h e r e -
compact s u b s e t of Y ; f o r e assume t h a t y
Then A i s a
i s sequentially continuous from
AV
x y) x
,
p r ~ j y rc F
)'I (Rn
X x
( K ~ v) +S(X
,
c c s
x Y
proper
X
-+
y i n Y.
If t h e
1' ) . 1 J
a r e not bounded, we
517
FUNDAMENTAL THEOREMS
(SEC. 2 )
can assume, a f t e r e x t r a c t i n g a d i f f e r e n t subsequence , t h a t
ITj:
+ m,
and t h e r e f o r e t h a t
r
E
we t h e n o b t a i n (y, v)
and
(x, 0, y,
i m p o s s i b l e i n view o f ( 2 . 1 0 . 1 ) .
Iqjl
The
which i s
7) E 2
a r e t h e r e f o r e bounded,
and f u r t h e r m o r e , a f t e r e x t r a c t i n g a subsequence, we can suppose
If
7f
0 we have ( y , T )
, 5 , y j 9J. * ( X E r , and t h e r e f o r e (x, 6)
If
7=
0 we have (y, 7)
E F x {O], and t h e r e f o r e
qj
that
(x, f )
+.
E C',
KA E.8!!s(X
, $)
(X,
that
(xo
'1 i n IR
(F
X
P
,
(xj
8
{O]) c A .
x Y ) and v E
o f xo i n X , and.
WF(AV)
Y
I
E Z'
grYtF(Y), t h e n Av E % ( X ) .
.
Let K
~2 = ~ o j yri;l
i s a compact s u b s e t o f F ;
9
TIE C . rCA .
We s h a l l now prove t h a t i f
E X x (R" \ 0) , n o t b e l o n g i n g t o A ;
, t:)
5
for y
K~ 0
E
1
Let
we s h a l l now show
be a compact neighbowhood
;
i n view of (2.10.21,K~
K2, we denote by
Co t h e f i b r e
of C a t (xo, y o ) and by To t h e f i b r e of T a t yo; by u s i n g an a r g p n e n t analogous t o t h a t employed t o prove t h a t A i s c l o s e d , it
i s e a s y t o show t h a t t h e r e e x i s t s a c o n i c open neighbowhood u
To of To i n IRp\ 0 , a conic open neighbowhood ? of Co i n 0
Bn+' \ 0 , a conic neighbowhood W o f > 0 , a neighbowhood
E
y
0
in Y , such t h a t
5'0 i n IRn \
0 , a number
V1 o f xo i n X and a neighbourhood V2 of
518
(CHAP. 8 )
MICROLOCALISATION
and a l s o , i n view o f (2.10.1),such t h a t
F~ n
(2.10.5)
I
Go
-T
tsI
I
5
-
~ITD
By s h r i n k i n g V1 and V2, we can a l s o impose
v1
=
K1
w
where y denotes t h e complement of
r0
i n IRp\
0 and u t h e comple-
L)
ment o f
c 0 i n I R ~ + P 0. \
By p a r t i t i o n o f u n i t y i n t h e neighbourhood o f t h e compact s e t K2,
we s e e t h a t when
< AV
cy
E
, c ~ ( 4e-iX*f'>
where
i s a f i n i t e sum of terms o f t h e form
I
< f K*
(2.10.81
E
@
Cr(V2)
t h e above t y p e .
has s u p p o r t s u f f i c i e n t l y n e a r xo,
Cz(X)
,
f
9
E
dx)e
C:(vl
-ix.
f'
x V2)
@(YIV
, V1
> and V
2
being of
I n accordance w i t h t h e proof of Theorem 2 . 7 , t h e
i n n e r product ( 2 . 1 0 . 8 ) i s e q u a l t o
/<(*, 71
(2.10.9)
t h ' - 61
6 (-TI1 86 dT
0
fi
We note t h a t , i n view o f (2.10.7), fKA i s r a p i d l y d e c r e a s i n g outside
?0 )
and t h a t , for
5' fixed,
25'-
n
6) pv(-rl)
is rapidly
decreasing with re s p e ct t o ( 5 , q ) o u t s i d e
[(r, T) 1
-71
€yo ,
s e1~1'j
i n view of (2.10.6), and
(SEC. 2 )
519
FUNDAMENTAL THEOREMS
t h e r e f o r e t h a t t h e i n t e g r a l i s a b s o l u t e l y convergent i n view o f
L e t W ' be a conic open neighbourhood o f 5 ' i n W n \ 0
(2.10.5).
0
such t h a t
w e s h a l l assume h e n c e f o r t h t h a t 5 '
c W;
E
W'.
Since w e have, f o r a l l N , an upper bound of t h e form
\as'-511
5
CN(l
+ '!I
I + Itl)4
when 5
4 W,
it i s easy t o
show t h a t
i s r a p i d l y decreasing i n 5 ' .
We s h a l l t h e r e f o r e r e p l a c e ( 2 . 1 0 . 9 )
by t h e i n t e g r a l
This lemma i m p l i e s t h a t (2.10.10) i s r a p i d l y d e c r e a s i n g i n 5 ' : i n o r d e r t o s e e t h i s , we argue a s i n t h e proof of P r o p o s i t i o n
1 . 4 , by performing s e p a r a t e i n t e g r a t i o n s f o r and f o r
2
15'
1/
.
2
5
Ir;'
1/
2
I n o r d e r t o prove Lemma ( 2 . 1 0 . 1 1 ) ,
A
we f i r s t n o t e t h a t f K A ( E , r l ) i s r a p i d l y d e c r e a s i n g i n ( 5 , q ) f o r f
EW and
171 2 e l S l
171 and
5 elf1
-71
-TI E A
ro, t h e n
i n view of (2.10.4); when
dye ,
( a n d t h e r e f o r e i n (5,r)) N
,
(0,q)
A
a/(-q)
i s ra pi dl y decreasing i n
i n view of (2.10.6)); when 5
ea
E
q
W and
from (2.10.31, and t h u s
f K A ( c , T ) i s r a p i d l y d e c r e a s i n g i n ( 5 , ~ i) n view o f ( 2 . 1 0 . 7 ) .
520
(CHAP. 8 )
MICROLOCALISATION
c o n t i n u i t y with respect t o (KA
E pA(X)
Av
We have t h u s f i n a l l y shown t h a t
, v)
;
the sequential
i s proved i n e x a c t l y t h e same
way.
REMARK 2 . 1 1 : If u
We now c o n s i d e r t h e s i t u a t i o n of Theorem 2 . 9 .
CI(X), we can c o n s t r u c t a neighbourhood ? o f F i n Y such
E
n
t h a t supp KA
(supp u x
7
we c o n s i d e r a sequence v k w i t h supp vk
,u >
c
?
E
i s compact. C”(Y) o
For
such t h a t yk
( s e e Proposition 1 . 8 ) .
v +
,
E q,F(Y) v in
aI(Y)
Then
.
lim k d + m REMARK 2 . 1 2 : For many o p e r a t o r s A , t h e s e t =
= WF’KA
WFxKA
(y x {O])
i s empty;
t h e n formula ( 2 . 9 . 1 )
simplifies t o
(2.12.11
WF(AV)
t WF’KA
WFV
0
A s a p a r t i c u l a r c a s e o f t h e two p r e c e d i n g theorems, we now s t u d y t h e i n v e r s e image of a d i s t r i b u t i o n by a C f : X
+
Y be a Cm mapping and l e t
*
t h e o p e r a t o r d e f i n e d by f v = v
0
111 and from Theorem 2 . 1 , we have
m
mapping;
-c”(x)
f* : c:(Y) f.
let be
From Example 8 . 5 i n Chapter
*
WF’K
C
f
, tf’(xI71 , f(x), 7) I x E X , 71 E Rp \ 0 3 . S i n c e supp K * = {(x , f ( x ) ) I x E X ] , t h e p r o j e c t i o n {(x
f supp K X -*
f imply :
i s p r o p e r , and Theorems 2.9 and 2.10 t h e r e f o r e
(SEC. 2 )
FUNDAMENT&
-.
Suppose f : X
THEdREM 2 . 1 3 ( I n v e r s e Image)
mapping and t h a t f* : C z ( Y )
* f v
image operator defined by Nf
=
1
TI)
m(x)
x E
521
THEOREMS
=
x
v
+
Y is a
Cm
C"(X)
is the inverse
f
We put
7 E R ~ 0\
tf*(x)q-
.
03
Then f * extends uniquely i n t o an operator ( i t s e l f denoted by f*) from
any closed cone f*
I
{v E p ( Y )
d]
WFV fl Nf
r in
Y x (Rp
\
into
which does n o t i n t e r s e c t N f ,
0)
+(Y)
is sequentially continuous from
is as above, then t A = {(x
,
cone i n X x (Rn &$Y)
x E X
f'(x)T)
\
0)
.
into J$X)
REMARK 2 . 1 4 :
for a l l v
E
Nf = @.
When
, and
I
f*
(fo()
'TI) E
v
.R'(x).
r}
If
r
i s a closed
I n p . a r t i c u l a r , we have
when f
into
is sequentiaZZy continuous from
The assumption
&'(Y)
such t h a t , f o r
>'(X)
f
WFv
n Nf
=
d
is satisfied
i s a submersion, s i n c e we t h e n have
i s a diffeomorphism, t h e s e q u e n t i a l
c o n t i n u i t y of t h e o p e r a t o r
f*
: p ( Y ) -a(X)
defined i n
Theorem 2.13 shows t h a t t h i s o p e r a t o r c o i n c i d e s w i t h t h e o p e r a t o r d e f i n e d i n S e c t i o n 7.1, Chapter I .
It can b e seen s i m i l a r l y t h a t i f
x
i s a diffeomorphism o f a n
open s e t Z i n IRn o n t o X , t h e n , under t h e assumptions of Theorem 2.13, w e have
522
EXAMPLE 2.15 (Convolution K e r n e l ) . f : Rn X Rn
For
(CHAP. 8 )
MICROLOCALISATION
E
v
n
-R
P(Rn)
Let
be t h e submersion d e f i n e d by f(xsy) = x-y
, we
E P(Rn
u = f*v
consider
abuse of n o t a t i o n we w r i t e
u(x,y) = v(x-y)
of t h e operator
*
tp
-V
cp
, and
x Rn)
.
by an
u i s the kernel
of c o n v o l u t i o n w i t h
,
v
and
we have
WF'u
C
5 ,
{[x,
y ,
5)
1x
I
Y ERn
s
(X-Y
9
5)
EWFv3.
( E x e r c i s e 5 . 2 shows t h a t t h i s i n c l u s i o n i s a c t u a l l y a n e q u a l i t y ) . I n t h e c a s e where
, Example
v = 6(x') @Y(X,-,]
2.6 t h e r e f o r e
shows t h a t
WF'U
C b
EXAMPLE 2.16:
f'(x) # 0 .
I
5
9
Let
x
f
, 51 1 5 f
E
,R)
C"[X
u
03
be such t h a t
If 6 is t h e Dirac measure a t 0 i n B, t h e n Theorem
2 . 1 3 a l l o w s us t o d e f i n e
f*6 = 6(f)
Approximating 6 by a sequence o f C
m
Ea(X)
, and
we have
f u n c t i o n s , and u s i n g t h e
s e q u e n t i a l c o n t i n u i t y o f t h e o p e r a t o r f * , it can be shown t h a t
6 ( f ) can be d e f i n e d i n terms o f o s c i l l a t o r y i n t e g r a l s as f o l l o w s :
(SEC. 3 )
MANIFOLDS
523
f(x) = x,,
This e b r e s s i o n shows t h a t i n t h e c a s e
,
6(f)
coincides with t h e d i s t r i b u t i o n 6(xn = 0 ) defined i n Section 7.2, Chapter I , and consequently ( 2 . 1 4 . 1 ) i m p l i e s t h a t 6( f) c o i n c i d e s i n t h e general case with t h e d i s t r i b u t i o n defined i n Section 7.2, Chapter I .
THE CASE OF MANIFOLDS
3.
LEMMA 3.1:
n
Let X , Y be open sets in R and l e t f : X
be a diffeomorphism.
PROOF:
For v
-+
Y
a'(Y) , We have
E
The i n c l u s i o n o f - t h e l e f t - h a n d s i d e of ( 3 . 1 . 1 ) i n t h e
right-hand s i d e f o l l o w s from Theorem 2.13, and t h e e q u a l i t y ( 3 . 1 . 1 )
i s t h e n deduced by r e p l a c i n g
f
by f - I .
This lemma shows t h a t ,
i n t h e following d e f i n i t i o n , t h e choice of t h e l o c a l c h a r t i s immaterial:
DEFINITION 3.2:
.
u E #(X)
We d e f i n e
Suppose X i s a COD manifold and suppose
* \
WFu c T X
0 by t h e c o n d i t i o n :
(x,g)
E WF u i f and only i f t h e r e e x i s t s a l o c a l c h a r t
x
3
t X
v
j
N
that
x(x,{]
induced by
Exercise
7 C Rn
o f X t o t h e neighbourhood of
E WF[~.-'*(U,~)] , where
x
such
*
i s t h e cha rt of T X
x.
5 . 4 g i v e s an i n t r i n s i c c h a r a c t e r i s a t i o n of t h e s i n g u l a r
(CHAP. 8)
MICROLOCALISATION
524
spectrum.
It i s an immediate r e s u l t t h a t WFu i s a c o n i c c l o s e d s e t i n
T* X \ 0 , and t h a t i t s p r o j e c t i o n o n t o X i s sing supp u.
If
r
i s a c l o s e d cone i n T* X \ 0, we a g a i n p u t
=
flr(X)
u
E
j chart
{u E
1 WFIJ
9 (x)
,J?;(X) converges i n
x
C
“IV
‘jlv
E jy
’
We s a y t h a t a sequence
B;(X)towards
of X ( d e n o t e d a s a b o v e ) ,
a;(?), where
.
u
E
8;(~) if, for
any
converges towards 5 i n
j 5 , r a r e obtained by t r a n s p o r t of
IV ‘
The g e n e r a l i s a t i o n of Theorem 2 . 4 ( t e n s o r p r o d u c t ) t o t h e c a s e i n which
X and Y a r e manifolds i s immediate.
By p a r t i t i o n of u n i t y
and l o c a l c h a r t s , we can a l s o g e n e r a l i s e P r o p o s i t i o n 1 . 8 ( r e l a t i n g t o t h e density of
C:(X))
and Theorem 2.7 ( r e l a t i n g t o
t h e c o u p l i n g o f two d i s t r i b u t i o n s ) to t h e c a s e i n which X i s a
For t h e g e n e r a l i s a t i o n of Theorem 2 . 7 , it i s n e c e s s a r y
manifold.
t o suppose t h a t X i s endowed w i t h a p o s i t i v e d e n s i t y
X x
(Rn \ O )
by T*X
\
u,
and t o
0.
Let X, Y be two manifolds endowed w i t h p o s i t i v e d e n s i t i e s
u,w.
If A : c~(Y)-B~(x)
i s a continuous l i n e a r o p e r a t o r ,
we r e c a l l ( s e e S e c t i o n 8, Chapter I ) t h a t t h e k e r n e l KA
E fl(X
x Y) o f A i s c h a r a c t e r i s e d
, for
cp
E
C:(X),
E $(y),
MANIFOLDS
by
C
K I ~ (tp
8 $)(P @
V)
>
=
525
tpp
8
.
>
I n view of ( 8 . 2 . 1 ) , Chapter I , t h e s i n g u l a r spectrum o f KA i s independent o f v .
C o n s t r u c t i o n 2 . 8 and Theorems 2 . 9 and 2.10
r e l a t i n g t o t h e e x t e n s i o n of A g e n e r a l i s e word f o r word i f we
(X x Y) x Rw
replace
Y x Rp by
* T Y.
,
by T*(X x Y)
X x
R"
by
*X
T
and
( i t < )of Theorem 2.9 and
[To prove p a r t
Theorem 2.10, c o n s i d e r p a r t i t i o n s o f u n i t y
% = I i n
X
,
i n Y by Cm f u n c t i o n s w i t h compact s u p p o r t which a r e
C Bj = 1
From t h e analogue of
l o c a l l y f i n i t e i n open c o o r d i n a t e p a t c h e s .
=
assumption ( 2 . 8 . 3 ) , we have A v locally finite;
AV =
thus
, the
A(Bjv)
5SA J
Bjv , t h e
sum b e i n g
sum b e i n g once
It i s t h e r e f o r e s u f f i c i e n t t o s t u d y t h e
again l o c a l l y f i n i t e .
s i n g u l a r spectrum o f a t e r m cliA B . v . J
Now i n t h i s c a s e we c a n
use l o c a l c h a r t s t o reduce t o t h e known s i t u a t i o n
X =
R"
,Y
= Rp
m
1.
If
f :
X
+
Y is a C
mapping, we t h e n
deduce t h e g e n e r a l i s a t i o n o f Theorem 2 . 1 3 r e l a t i n g t o t h e i n v e r s e image o p e r a t o r f * , by p u t t i n g exists x
E
y
X with
P
f(x)
Nf = [ ( y , r l )
, %,(x)tf
I
E T*Y \ 0
01
.
/there
We s h a l l now
d e s c r i b e t h e p a r t i c u l a r c a s e where X i s a sub-manifold of Y and f :
X
-+
Y i s t h e canonical i n j e c t i o n .
THEOREM 3 . 3 ( R e s t r i c t i o n t o a sub-manifold):
Suppose
Y
is
a manifold and X i s a submanifold of Y. Let NX
C(YIT1
E T*Y
\
1Y
E
I
n\Tvx
=
conormai! t o X, deprived o f i t s null seotion.
01 be the bundle Then th e operator y
526
MICROLOCALISATION
of r e s t r i c t i o n t o X(defined by yv = v
IX
(CHAP.
for v
uniqueZy i n t o an operator (aZso denoted by [v
E 9 (Y) 1
cone
r of r
] i n t o S'(X)such t h a t , f o r any cZosed
WFv fl NX =
T Y \ 0 n o t i n t e r s e c t i n g NX, y is sequentially Contin-
8;1(Y)
fi(x).
into
If
r i s as above, the p r o j e c t i o n
n
IX
from
n
uous from A of
y)
extends
C:(Y))
E
8)
n
i n t o T X i s a cZosed cone i n T X \ 0 , and
iaZZy continuous from &;(Y)
y
i s sequent-
I n particuzar, we
i n t o &;(X).
have
The i n v e r s e image theorem now a l l o w s u s , i n c e r t a i n c a s e s , t o d e f i n e t h e product o f two d i s t r i b u t i o n s .
rl
For
I
r2
c T"X
c;?) E
r2 3
we put
rl + r2
(b q
=
+
$1 1
E ~ EI r.
(X
I
-
Let X be a mmifoZd.
THEOREM 3.4 ( M u l t i p l i c a t i o n ) :
muZtipZication operator $(XI
x
c;(x)
3 cul
#
u21
(X
,
.
The
u1u2 E C0"CXl
extends wziqueZy i n t o an operator from
~ ( u , u21 E P ( X )
x P(X>
1
+W F U ~c
W F U ~
d'(X) which, for a21 cZosed cones
r, + r2
*
rl, r 2
T*X \
01
into
n
i n T X \ 0 satisfying
c T X \ 0 , i s sequentialZy continuous from
&p d (XI i n t o a(xl . r2 x
1 ;
rl, r 2
are as above, then A =
(r, + r2)
U
rl
U
r2 i s
a cZosed
(SEC. 3 )
527
MANIFOLDS
cone i n T*X \ 0, and the m u l t i p l i c a t i o n i s sequentially contin-
(x)
B'
uous from
rl
(x)
B'
x
into
3
( x ) . I n p a r t i c u l a r , we
r2
have
PROOF:
The uniqueness f o l l o w s from t h e d e n s i t y p r o p e r t y o f
CE(X).
Let f : X -X
f ( x ) = (x,x);
we have
1
Nf = C(X,X,61,f2) u1
, u2
E
x X be t h e mapping d e f i n e d by
(S&I
, we
C:(X)
*X
c T
*
9
c1 +
= 01
*
u1 Q u2 E
have
u p 2 = f (u, @ u2)
For
.
For
Pr(X)w i t h
{o) x r 2 ) - x (rl x x x 10)) Nf n r = 0 s i g n i f i e s t h a t
\ 0 ;
*
u1u2 = f (u, Q u 2 )
x (X x
and t h e c o n d i t i o n
r, + r2
0
c l e a r l y have
E Pr j(X) (j P 1,2) , we
r = (rl x r2)
4
from Remark 2 . 5 ,
i f such i s t h e c a s e , we p u t
, and
t h i s concludes t h e proof i n view o f
Remarks 2 . 5 and 2 . 1 2 .
Suppose X,Y a r e two manifolds endowed w i t h p o s i t i v e d e n s i t i e s p, v
.
m
Let f : X + Y be a C
i n v e r s e image o p e r a t o r
mapping.
f* : c:(Y)
-c"(x)
By t r a n s p o s i t i o n o f t h e
c 9 (XI
, we
o b t a i n t h e d i r e c t image o p e r a t o r (which depends on t h e d e n s i t i e s
528
(CHAP. 8)
MICROLOCALISATION
We have
supp Kf
=
*
{(f(x)
1x
X)
E
X]
Theorems 2 . 9 and 2 . 1 0 ( g e n e r a l i s e d t o manifolds ) t h u s imply :
Let X , Y be two Cm manifolds
THEOREM 3.5 ( D i r e c t i m a g e ) :
endowed with p o s i t i v e d e n s i t i e s u, v ; l e t f mapping m d
(x)
fw: C , " ( X ) - ~
defined b y ( 3 . 4 . 1 ) .
: X
f
closed cone that
r
-,
Y
from
proper) i n t o 8 " ( ~ For ) . any
i s proper m d P r o j X r
Y
'A = ~ [ y , . q )f T*Y \ (x, tfl(x)T)
4
)
i n T*X \ 0 and f o r m y closed subset F o f X such
: F
flF
Cw
Then f, extends ( i n accordance w i t h
1 lsuppu : supp u
E .Bte"x)
Y be a
the d i r e c t image operator
Theorem 2 . 9 ) i n t o an operator ( a l s o denoted by f,
c.
-+
E r
o
1
3 x E x with
c
F, then
y = f(x)
,
i s a cZosed cone i n T*Y \ 0 , and
U(Fx {O])]
f, i s sequentialZy continuous from
b' r ,F(x) i n t o ~ C Y ) . In
particular, w e have
REMARK 3.6:
I n t h e c a s e where
* )
-1
can be shown t h a t f, = ( f through
f
.
When
X = Y x
f
i s a diffeomorphism, it
when t h e measures 1~ and v correspond
,
)L
= v Q 8
,
where
a r e p o s i t i v e d e n s i t i e s on t h e manifolds Y , Z , and when
v, 0
(SEC. 3)
529
MANIFOLDS
p : Y x Z
+
Y i s t h e p r o j e c t i o n , we o b t a i n t h e f o l l o w i n g s p e c i a l
case o f Theorem 3.3:
COROLLARY 3.7 ( I n t e g r a t i o n along f i b r e s ) :
Suppose Y Let
two manifolds endowed with p o s i t i v e d e n s i t i e s v,B. p* : C,"(Y x 2)
(P,U)(Yl
-
i,
=
Z are
be the operator defined by
C:(Y)
c p
(u E
v
U(Y,ZI
.
x 211
Then p+ extends ( i n accordance w i t h Theorem 2 . 9 ) i n t o an operator
(also denoted by pn) from
-
: SUPPU
For any closed cone
r
*
(y
zr c F,
,z ,7 ,
IF
:F
then A = [ ( y
0) E
r]
B'(Y).
i n T (Y x Z ) \ 0 and f o r any cZosed subset
F of Y x Z such t h a t Rody
Y proper) i n t o
-
, q)
Y
i s proper, and
I 3z
E T*Y \ 0
i s a cZosed cone i n
* T Y ?\ 0
EZ
with
and p+ is
sequentially continuous from 8' (Y x Z ) i n t o .,?;(Y).
In
r,F
p a r t i c u l a r , we have
We s h a l l now i n v e s t i g a t e t h e composition o f o p e r a t o r s .
Suppose
X Z y X, X1 a r e manifolds endowed w i t h t h e p o s i t i v e d e n s i t i e s lily
u, u2.
A* :
Suppose we have o p e r a t o r s
cz(s)- B"X>
,
A, :
c;(xI
-
B"x,)
.
We
5 30
(CHAP. 8)
MICROLOCALISATION
s e e k t h e c o n d i t i o n s under which A1
0
A2 i s meaningful as an OD
i n t o ag(X,).
o p e r a t o r from C:(X2)
u
I=
For u2
E
C (X2), we have 0
, with
A2u2 E B ' ( X ]
WFu
,
c WF' K
( c f . (2.9.1))
suppou2
A2 and
Chapter I ) .
Theorem 2 . 9 t h e r e f o r e a l l o w s A u t o b e d e f i n e d when
1
I
YUF' KA
(WF' K 1 A2 The p r o j e c t i o n
n (xl
supp K~ 1
suppo~)
*
c T XI \
supp u211
x (supp K~
2
0
.
- x1
i s proper.
m
These c o n d i t i o n s a r e s a t i s f i e d f o r a l l u2
where we have p u t
E
Co(X2) i f and o n l y i f
(SEC. 3 )
5 31
MANIFOLDS
E T*X\ 0
{(x,g)
=
wF;( KA1
with
K A2
{fx,c)
5
with Under t h e above c o n d i t i o n s ,
there exists
, 0 , x , t) E WF'KA 11 E T*X\O I t h e r e e x i s t s % E X 2
(xl
(3.7.2) WF
1
(X I
A,(A2
I Y2)
9
9
9
0) E WF KA 2
E J1(X,)
i s w e l l defined
s a) sequende and Theorem 2.10 allows us t o see t h a t i f ( u ~ i ~ which t e n d s t o
0
in Cz(X2), t h e n A,(A2u2)
-
0 in 8'(X1);
we t h e r e f o r e have
THEOREM 3.8 (Composition): Suppose X1, X , X2 are manifolds, and
A1 :
,
cr(x) - p ( x , )
A2 : C,"(%)
---P(X)
are
operators such t h a t wFi
KA1
"
(cf 3.7.2)
wF
A2 =
.
We assume t h a t the projection
Then A1
0
A2 i s well defined ( i n accordance with Theorem 2.9),
an operator from C;(X,)
REMARK 3.9:
A1
0
i n t o &'(x,).
Theorem 2 . 1 0 a l s o a l l o w s t h e dependence of
A2 on A1, A2 t o be i n v e s t i g a t e d :
c l o s e d cone i n
*
T (x,
x X)
suppose t h a t C1 i s a
\ o , c2 i s a c l o s e d cone i n
as
532
(CHAP. 8)
MICROLOCALLSATION
T*(X x
5)\
F, i s a c l o s e d s u b s e t o f X 1 x X and t h a t F2
D,
i s a c l o s e d s u b s e t of X x x
F%XI
x
projection
c
1
5 c F2
PIPj,
n (XI x Diag qxn sx= b
Then f o r u2
t h a t i n (3.7.2)).
J' =lgF1
we assume t h a t
(F, x F2)
i s p r o p e r , and t h a t
(KAl,
,
c F1
s;
K )-A1 A2 (XI x X) x
(X x X) x
and t h a t t h e
$) -XI
( u s i n g n o t a t i o n analogous t o
m
E
C ( X 2 ) , t h e mapping 0
A2 u2 i s s e q u e n t i a l l y continuous from
9
(X x
%, F2
5)
into
#(XI]
We conclude t h i s s e c t i o n by i n v e s t i g a t i n g
1
WF(K *1
r e l a t i n g t o operator composition;
f :
x1
defined by
%
x X x f(xl
5 -XI X , 5 , XI x 5, XI
=
p : XI x X x
x
5
XI,
x
x
we then have
i . e . formally
A2
p1
we p u t
x X x X x
-XI
, x , x2)
positive densities
O
We adopt the conditions o f Theorem 3.8
THEOREM 3.10:
Let
x
x
(xl
5
be the mapping
, x , x 51,
be the projection.
and l e t The manifolds
are endowed r e s p e c t i v e l y with the
, p , + , p1 8 % , p,
8p 8%
.
533
PROOF:
F i r s t , we n o t e t h a t i f K1 and K2 a r e i n C
(Au2)(2) =
d e f i n e d by
for
u* E C p 2 1 , K(xq
I
5
%)
K1(X1? x 1 q x L X 2
E
i
=
*
I
m
0'
then A i s
x2Iu,("2h
*
Thus
KI(X1
I
4
$(x
I
%llL
8
and t h i s e s t a b l i s h e s ( 3 . 1 0 . 1 ) i n t h i s p a r t i c u l a r case. proceed t o t h e g e n e r a l c a s e ;
QD
l e t UJ E
ct(xj)
We now
(j = 1 s 2)
.
We have
It i s e a s y t o show t h a t neighbourhoods V1, V2 o f supp K1, supp K2 can be c o n s t r u c t e d such t h a t
13.10.3)
V,
1 7 [Supp
U,
X
[v2
o
SUpp
U,))
i s compact ( s e e
Remark 2 . 1 1 ) .
I n view o f P r o p o s i t i o n 1 . 8 , t h e r e e x i s t sequences
KkI
E C(X.:,
x XI
,
E CX(:
x
$)
such t h a t
534
MICROLOCALISATION
Kjk c
SUPP
'I
Vj
K
'
8)
i n B' where j = 1, 2 and j rj be t h e o p e r a t o r w i t h k e r n e l K j k , and l e t
WK
jk
= WF K
* let A jy jk I$ be t h e k e r n e l o f A lk
j
(CHAP.
A
0
2k'
From ( 3 . 1 0 . 2 ) , ( 3 . 1 0 . 3 ) and
Theorem 2 . 7 , we have
, ulpl
@ u2b
* (Klk
Kk = fp,
>
, up1 @u2b
lim < Kk
=
>
k - w
@$k)
which t h u s g i v e s ( 3 . 1 0 . 1 ) .
,
( 3 . 1 0 . 3 ) and Remark 2.11 imply
The i n c l u s i o n given for WF'K i s
o b t a i n e d from ( 3 . 1 0 . 1 ) by a p p l y i n g i n t u r n Theorems 2 . 4 and 2.13, and C o r o l l a r y 3.7.
4.
THE CASE OF PSEUDO-DIFFERENTIAL OPERATORS
Let X be a manifold and P a p s e u d o - d i f f e r e n t i a l o p e r a t o r i n X. Since
Projx
diagonal of X
xWF Kp I= x
sing supp
Kp i s c o n t a i n e d i n t h e
X , it i s p o s s i b l e , by u s i n g l o c a l c h a r t s and
t a k i n g account of ( 8 . 2 . 1 ) of Chapter I , t o reduce t h e problem t o n t h e c a s e where X i s an open s e t i n lR when we i n v e s t i g a t e WF K P
.
I n p a r t i c u l a r , Theorem 2 . 1 shows t h a t
WF'Kp
c Diag(T*X \ 0) x (T*X\O)
p r o j e c t i o n of WF'K
P
.
We denote by WF P t h e
*
onto one or o t h e r of t h e two f a c t o r s T X \ 0.
(SEC. 4)
535
PSEUDO-DIFFERFNTIAL OPERATORS
*
It can be seen t h a t WF P i s a c l o s e d cone i n T X \ 0 , and t h a t
=
WF'Kp
.
Diag (WFP x WFP)
The r e s u l t s o f t h e preceding
s e c t i o n t h e n imply
PROPOSITION 4 . 1 :
Let P ,
Q
be two pseudo-differential
operators i n a manifold X , and Zet u i f
E
I f P i s proper o r
.%'(X).
u has compact support, we have
(4.1.11
c WF P
#F(P u)
n
.
c WFu
WFu
I f P o r Q is proper, we have WF(PQ)
(4.1.21
c WF P
n
WFQ
.
We s h a l l now g e n e r a l i s e t h e p r o p e r t y o f e x i s t e n c e o f a para-
m e t r i x , encountered i n S e c t i o n 7.2, Chapter I V .
THEOREM
4.2
(Microlocal p a ra m e t ri x ):
foZd and suppose we have P cZosed cone rm(P) of P
r
L m ( X ) e Z Z i p t i c of degree
m
in a
i n T*X \ 0 ( t h i s means t h a t t h e ' p r i n c i p a z symboZ
is e l z i p t i c i n a conic neighbourhood of r l .
there e x i s t s a proper Q
PROOF:
E
Suppose X i s a mani-
E
L-m(X) such t h a t
C l e a r l y , we can suppose t h a t P is p r o p e r .
Let
Then
5.36
p
MICROLOCALISATION
(CHAP.
Sm(T*X \ 0 ) b e a p r i n c i p a l symbol o f P , and l e t
E
c o n i c neighbourhood o f
7.
r
8)
b e an open
i n T*X \ 0 such t h a t a m ( P ) i s e l l i p t i c
*
- . Let Qo L-m(X) be p r o p e r , w i t h p r i n c i p a l - 1 f s-1 (r) Lo(X) i s p r o p e r symbol q . We have PQo = I + Ro, where Ro in
E
We know t h a t t h e r e e x i s t s q
pq
SWm(T X \ 0) such t h a t
E
E
N
and w i t h p r i n c i p a l symbol o f degree 0 e q u a l t o z e r o i n T ; can t h e r e f o r e decompose R
n r
WF R1
proper,
- R2, w i t h R1
i n t o Ro = R1
0
,
P
E
F$
L-'(X)
E
Lo(X)
p r o p e r ( t h i s decomp-
n o s i t i o n i s t r i v i a l when X i s an open s u b s e t of IR
and i n t h e
g e n e r a l c a s e i s deduced from t h i s by p a r t i t i o n o f u n i t y ) .
B
E
Lo(X) be p r o p e r such t h a t
+ (I -
W = RIB
we have Since
B
-
+ (I +
R2)B = RIB
M
S i m i l a r l y , we c o n s t r u c t Q
WF(KP - I) n r
R)
P u t t i n g Q = QoB,
, with R
E
L-m(X).
d
=
L-m(X)
E
,
p r o p e r , such t h a t
by c a l c u l a t i n g
;
N
Q P Q
Q(PQ)
D
-
~ F ( Q a31
=
@P)Q, we s e e t h a t
n r
=
COROLLARY 4.3: u
$ .
Let
c WFR, does n o t i n t e r s e c t r , we o b t a i n (4.2.1).
WF(RIB)
N
k 2 O
we
E
B'(X).
PROOF:
Then
d , which
t h u s g i v e s (4.2.2).
Let X be a manifozd, Zet P
WF u c
Suppose we have
WF(P
(x,
U
u)
, f,)
E
Char
*
T X
\ 0
WF(Pu) U C h a r P. A s P i s e l l i p t i c a t ( x o y 5,) shows t h a t t h e r e e x i s t s Q
QP
I
I
+R ,
(x,
4.2 i m p l i e s t h a t WF u
E
, &) c
E
Lm(X) and
p,.
not belonging t o
, the
above theorem
Ldm(X) p r o p e r , such t h a t WFR
.
AS u = QPu - Ru, Theorem
WF(Pu) U WFR
and t h e r e f o r e t h a t
(SEC. 4 )
4
(xo, 6,)
537
PSEUDO-DIFFERENTIAL OPERATORS
We s h a l l now g i v e a c h a r a c t e r i s a t i o n o f t h e WF
WFu.
o f a d i s t r i b u t i o n on a manifold by means of p . d . 0 . ' ~ ;
this
c h a r a c t e r i s a t i o n c o u l d be t a k e n as t h e d e f i n i t i o n o f WF u ( s e e HORMANDER [ 5 1 ) .
4.4:
PROPOSITION
Let X be a manifold and l e t u
E
B'(X).
Xhen
(4.4.1)
PROOF:
The i n c l u s i o n o f t h e l e f t - h a n d s i d e o f
right-hand s i d e follows from C o r o l l a r y 4.3. (Xo
, to)
*X
ET
, not
\ 0
belonging t o WF u.
(4.4.1) in the
Conversely, suppose Let
r
be a conic
%+
neighbourhood o f (xo, t o ) i n T X \ 0 which does n o t i n t e r s e c t
WF u, and l e t P WF P c
r.
E
L o ( X ) be e l l i p t i c a t (xo, 5,)
such t h a t
From ( 4 . 1 , 1 ) , we have
W F ( P ~ Jc W F P
n
=
WFU
d ,
When X i s an open s u b s e t o f IRn
and t h u s P u
, we
E
c"(x).
can d e f i n e t h e e s s e n t i a l
s u p p o r t o f a p s e u d o - d i f f e r e n t i a l o p e r a t o r , p r e c i s e l y , by u s i n g
i t s complete symbol: THEOREM 4 . 5 :
Let
x
be an open s e t i n I R ~and l e t P be a
pseudo-differential operator i n X.
Then hT P i s the e s s e n t i a l
support of the complete symbol a ( P ) of P ( i . e . the smallest conic
538
PROOF:
We can w r i t e
a
where
i s a symbol r e p r e s e n t i n g o ( P ) .
W F ' K ~c ( ( x WF p
, f , x , 51 1 5 f supp cr(p).
ess
C
0 , (x
such t h a t (xo
V
, !&, ,
Xo
, 5,) ,- i)
WF P
From Remark 2 . 2 ,
, 61
E
ess SUPP
(xg
, i.e.
A
-
where
i n t h e neighbourhood o f xo.
a
,p
E
C(:V)
cr(Q)(x
P
TI)
(R"
0)
i n En\ 0 such m
E
C (X x X) has 0
ar e equal t o unity
We have
where o ( Q ) i s t h e complete symbol o f Q = a P 6 . have
x
we t a k e i n p a r t i c u l a r
, p(y)
a(x)
thus
There e x i s t s a neighbourhood
t h a t f K i s r a p i d l y d e c r e a s i n g in W x W when f P
f(x,y)
'3;
such t h a t
.
wF$
, 8)E X
of xo i n X and a conic neighbourhood W of 5,
s u p p o r t i n V x V;
B
In order t o e s t a b l i s h t h e inclusion
i n t h e o t h e r d i r e c t i o n , we c o n s i d e r
(xo
(CHAP. 8)
MICROLOCALISATION
-
eixr c r ( Q ) [ e ,
7$ 8C
We t h e r e f o r e
(SEC. 5 )
5 39
ADDITIONAL NOTES
so that
Iw-
n T
lu(Ql(E
I
71)
]u(Ql(x 9
5
, TI1
d6
+
fW -J
A
Iu(Q)(t
L e t W' b e a c o n i c neighbourhood of 5
-
W'
c W.
We have seen t h a t
d e c r e a s i n g f o r 5 , rl d e c r e a s i n g for rl
Iv(Q)(i
, q) 1
degree o f
IS
- 71
Q 2
c(
E
E
CJ
0
[a) ( 5
- 71
P
Tl)1 B;
rn
i n IR"\ 0 such t h a t
- 'll , Tl)
i s rapidly
thus i n W the integral i s rapidly
W;
W.
By combining t h e immediate upper bound
CN(l
+
5
16\)"(1
+ 1q11"
where
m
is the
i s a r b i t r a r y , with t h e i n e q u al i t y
and N
151 + 1711)
for 5
g!
W and rl E W '
( w i t h c > O), we
s e e t h a t i n En \ W t h e i n t e g r a l i s r a p i d l y d e c r e a s i n g f o r rl
E
W'.
It can be shown i n similar f a s h i o n t h a t t h e d e r i v a t i v e s of o(Q) a r e r a p i d l y decreasing i n W ' .
We t h u s have f i n a l l y
(xo, 5 , ) g! e s s supp a ( P ) s i n c e c(P)z o ( Q ) (mod S - 7 s u f f i c i e n t l y near x
5.
0
for
x
.
A D D I T I O N A L NOTES AND EXERCISES
5.1
CONCLUSION OF THE CALCULATION OF wF(6(xV) 8 Yfx,)) ( s e e Example 2 . 6 )
540
c)
(CHAP. 8)
MICROLOCALISATION
Conclude, t a k i n g account o f Example 2 . 6 , t h a t WFU
= ~ ~ 0 3 U ~ X ’ = xO n >, o ,
cx=o,
5.2
.
q = o 1
f’#O,
CONCLUSION OF THE CALCULATION OF WF’(v(x-y)) FOR E .8’(IRn) (see Example 2.15)
v
Suppose v
E .P(R”)
and
a)
Show t h a t
b)
Deduce from t h i s t h a t
<
v(X-y],
where we have p u t
c)
rp(x) $(x-y)
f(5,
.
> = < 1, cp > C
- 5)
= < 1, rp
T(x, y] = rp(x) $(x-y)
>
~(x-y]
v, $
.
>
fi
,
$V(5)
.
Conclude, t a k i n g account of Example 2.15, t h a t
5.3
we p u t
c
=
W!=qV(X-Yll
f
rp, $ E 4 R n )
(x,
I# Y, I1
I
x, Y
E R“,
(x-Y,
I 1 E WF
v
3
0
CONSTRUCTION OF A DISTRIBUTION WHOSE SINGULAR SPECTRUM I S A D I R E C T I O N I N T*(X) \ 0.
R”
I
E S(f-’]
g ( T ) = 1 for
R
x
R”-’
with f ( 0 ) T
,
x
(t9 y )
# 0, and l e t g
t 1, g ( . r ) =
o
for
T
, E
2 1/2.
5=
(7,
7)
.
C m ( I R ) be such t h a t
We p u t :
Let
(SEC. 5 )
ADDITIONAL NOTES
5 41
We now propose t o show t h a t :
Show t h a t h ( 5 ) i s r a p i d l y d e c r e a s i n g ( a s
a)
c o n i c neighbourhood o f any p o i n t q0
#
0 , o r such t h a t rl
0
= 0 and
( T
0
'c0
, no)
151
+
in a
m)
i n R" such t h a t
< 0.
Show t h a t for any multi-index a, t h e r e e x i s t s C
b)
+
2
c1
0 such
that
Deduce from t h i s t h a t f o r any m u l t i - i n d e x 8 , t h e d i s t r i b u t i o n
d + x D u is that u
c)
1 i n L (33")
E Cw[Rn \ 0)
when /a1 i s s u f f i c i e n t l y l a r g e , and t h u s with
Sup
Deduce from t h i s t h a t u
p a r t i c u l a r Proposition
5.4
1xa
x
E Rn
4
C"(IRn)
[x,,
.[)I
c +
03
.
and t h a t , u s i n g i n
1 . 4 , we have t h e e q u a l i t y (5.3.1).
INTRINSIC CHARACTERISATION OF THE SINGULAR SPECTRUM
Let X be a manifold o f dimension Let
D$
So] E
T*(X]
\
0
.
n
, and
let u
E
d?'(X).
We now propose t o show t h a t
t h e f o l l o w i n g c o n d i t i o n s ( i ) ,( t i ) , a r e e q u i v a l e n t :
542
(ii)For a l l p 2 1, f o r a l l .A
4 E
C"(X
x
@, R)
such t h a t
E
IRp, f o r any f u n c t i o n
, there
5o
lo] =
dx +(x,,
e x i s t s a neighbourhood V o f xo i n X and a neighbourhood A of A i n IRp such t h a t :
<
u(x),
.e[x*
a ( x ] e-"
t o T when
T +
+
for a l l
a,
CY
E
> is
8)
(CHAP.
MICROLOCALISATION
, the
n,(x)]
C:(V,
0
quantity
r a p i d l y decreasing with respect
uniformly f o r A
A.
E
For t h e p r o o f , it is c l e a r l y s u f f i c i e n t t o c o n s i d e r t h e c a s e X=IR".
a)
Show t h a t ( i i )i m p l i e s ( i )by t a k i n g
*b, 11 b)
-
p
P
n
,
*
as i n ( i i ) .
Assume t h a t ( i ) i s s a t i s f i e d , and c o n s i d e r
We know t h a t t h e r e e x i s t s a neighbourhood Vo o f x 0
of 5
0
such t h a t , f o r a
decreasing with respect t o
Consider $
E
,
.
x.1
bourhood W
lo = 5,
C:(V]
T
when
T +
+
and a neigh-
6?(~ 5)
C:(V):
E
0
m,
is rapidly
uniformly f o r 5 E Wo.
such t h a t $ = 1 i n supp a.
Show t h a t :
r
Show t h a t t h e r e e x i s t s a neighbourhood V c V bourhood W c Wo o f such t h a t :
0
o f xo, a neigh-
5 0 ' a neighbourhood A of A 0 and a number c
> 0
5)
(SEC.
543
ADDITIONAL NOTES
By i n t e g r a t i n g by p a r t s , u s i n g t h e d i f f e r e n t i a l o p e r a t o r
15
- $:(x,
< 5
- $:(x,
I],
axa >
,
deduce from t h i s t h a t
for any i n t e g e r N t h e r e e x i s t s C N 2 0 w i t h 1 1 ( ~ s
%
5, 111
5ERn\W
,
~
A E A
.
~+ 1(5 1 P1'
for
7
,
I
2
By decomposing t h e i n t e g r a l which appears i n t h e l e f t - h a n d s i d e o f ( 5 . 4 . 1 ) i n t o t h e sum o f an i n t e g r a l i n B n \ W and an i n t e g r a l i n W , conclude t h a t t h e c o n d i t i o n ( i i )i s s a t i s f i e d .
5.5
TRACE CALCULATIONS
Let X be a compact manifold endowed w i t h a d e n s i t y take f points.
m
E
C ( X ; X) and we denote by
We suppose t h a t each f i x e d p o i n t
t h a t it s a t i s f i e s
det(1
- f!(x)] & 0 .
o p e r a t o r o f i n v e r s e image under
a)
Fix(f)
Let P be a p . d . 0 .
the operator f
* 0
of degree
.
dx
We
t h e s e t of i t s f i x e d
x i s simple, i . e . We denote f
*
the
.
f
m
on X.
We a s s o c i a t e w i t h it
P , t h e k e r n e l of which we denote by
K
.
*
f P Show, by u s i n g Example 8.5 o f Chapter I11 and t h e theorems r e l a t i n g t o t h e s i n g u l a r spectrum, t h a t t h e d i s t r i b u t i o n
*
K
f P
admits a r e s t r i c t i o n t o t h e diagonal
A
5:
{ (x,
Furthermore, i f C denotes t h e conic c l o s e d set
x)IX
E x 'J
.
544
c
(x,
5;
E
u
P
*X
511 (x, 51
x,
\ 01, t h e n
E T
Lm[XJ --tr(f*
P9
8)
(CHAP.
MICROLOCALISATION
t h e mapping
I
m i s continuous when we endow U Lm(X) w i t h t h e n o t i o n of convergence (We have i d e n t i f i e d K
induced by ak(X x X ) .
f*pl
*with
a
d i s t r i b u t i o n on X , by t h e p r o j e c t i o n o f A o n t o X ) . I n t h e p a r t i c u l a r c a s e where P = I d e n t i t y , show t h a t we have
K
6X
=
det(1
f* I A
1
wpp(K
[Note t h a t
C
Fjx(f)
-
f9[xJl
and c a l c u l a t e t h e a c t i o n of
f"( A t h i s d i s t r i b u t i o n on a t e s t f u n c t i o n w i t h s u p p o r t i n a small c o o r d i n a t e p a t c h domain n e a r a f i x e d p o i n t of l o c a l e x p r e s s i o n ( 8 . 5 . 1 ) of Chapter I11
f
, by
using t h e
1.
More g e n e r a l l y , l e t E be a v e c t o r bundle on X and l e t 8 be a
b)
k
morphism of b u n d l e s : f E
P
E
Lm(X; E,
E)
+.
With any p . d . 0 .
E.
T p r e o f
we a s s o c i a t e t h e o p e r a t o r
m
from C (X ; E) which a c t s on a s e c t i o n
=
(Tp u)(x)
ex
.
Pu(f(x)]
u
* o~
v i a t h e formula
,
x
E X
.
The k e r n e l d i s t r i b u t i o n o f Tp i s a d i s t r i b u t i o n
E B9[X x X; E E*] ( E x e r c i s e 1 1 . 5 , Chapter I V ) . "rP as i n a ) t h a t it i s p o s s i b l e t o d e f i n e t h e r e s t r i c t i o n E ";
E
k
@
E*1
and t h a t t h e t r a c e mapping o f E
i n t o C allows us t o d e f h e t h e d i s t r i b u t i o n
bace(5
PI A
Show
I E Jqx)
,
@
E
(SEC. 5 )
545
ADDITIONAL NOTES
trace(Tp]
Show t h a t t h e mapping
P E U Lm(X ; E, E)
-
< trace(KT
defined by
4 A) , 1
dx >
E
c
i s continuous ( i n t h e same sense a s i n a ) ) and t h a t f o r P = I d e n t i t y we have:
5.6
THE LEFCHETZ-ATIYAH-BOTT FORMULA:
Let ( E k , Pk) be an
e l l i p t i c complex over a compact manifold X endowed w i t h a d e n s i t y dx
.
a)
Prove t h a t t h e i d e n t i t y endomorphism I = ( I k )i s t h e l i m i t of E
endomorphisms (Sk) of t h e complex ( i n t h e s e n s e of E x e r c i s e 1 1 . 5 , Chapter IV) w i t h Cm k e r n e l s , t h e l i m i t b e i n g understood i n t h e s e n s e of t h e convergence, as where
[Let
:IG
(%) be (I
=
E
0 , of t h e k e r n e l s S k ( x , y ) i n
E -+
Z
P
- ()
%
0 , towards I
m
Finally, define
k
5)
I
(x,
S) -E T * X \ O
i n t h e sense of E
k e r n e l s which converge, as
& X x X ;%
zl
@
$) .
Sk by t h e e q u a l i t y
Take a mapping f
E
Cm( X,
X) , o f which a l l t h e f i x e d p o i n t s
a r e simple (see E x e r c i s e 5 . 4 ) , and f o r any
1,
show t h a t t h e o p e r a t o r s
a g a i n c o n s t i t u t e a p a r a m e t r i x of t h e complex
when p k denotes o p e r a t o r s w i t h C
b)
Xp
a p a r a m e t r i x of t h e complex;
E
E -+
{(x, 5 ;
k
t a k e a morphism o f
546
(CHAP.
MICROLOCALISATION
bundles
f* Ek
k '
-5
.
8)
A s s o c i a t e w i t h t h e s e t h e endo-
morphism T of C m ( X ; E k ) d e f i n e d by Tk = Bk k
4 0
f
.
Assume t h a t
t h e (T )Is d e f i n e an endomorphism o f t h e complex.
Then, show
k
t h a t we have t h e e q u a l i t y :
where we have p u t
[Use E x e r c i s e 11.5 of Chapter I V t o prove t h e e q u a l i t y when Tk is r e p l a c e d by t h e o p e r a t o r T
S'
k' k
t h e n deduce
(5.5.1)
w i t h Cm k e r n e l , and
( 5 . 5 . 1 ) by proceeding t o t h e l i m i t when
E
+
0.
Use E x e r c i s e 5 . 4 t o prove ( 5 . 5 . 2 ) 1 .
c)
I n t h e c a s e o f t h e De Rham complex (Example 9.4, Chapter I V ) ,
take
Ak T* f
ek
, and
[-
t5.5.3.1
o b t a i n t h e c l a s s i c a l L e f c h e t z formula
Trace
(Hk
f) sign [ d e t ( 1
= x Gi*(P) where
Hk
4
f denotes t h e endomorphism induced by f
cohomology group $(X,
C)
E
C;
by p u t t i n g
the.identity:
u
on t h e k t h
.
[Let E be t h e vector-valued space of dimension let A
- Txf) 1
m
on
C
and
i n t r i a n g u l a r form, show t h a t we have
(SEC. 5 )
det(X1
ADDITIONAL NOTES
- u]
=
k
hmk (- 1)
547
Trace (A
k
U)
,
Use t h i s i d e n t i t y w i t h h = 1 t o make t h e right-hand s i d e of ( 5 . 5 . 2 ) e x p l i c i t ) 1.
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INDEX
Adjoint mixed problem, 464 Algebra of p.d.o.'s, 207 Amplitude, 192 Asymptotic expansion, 152 Boundary-value problem, 271 Calder6n projector, 290, 322 Calder6n-Vaillancourt theorem, 267 Cauchy's formula, 308 Cauchy problem, 337, 359, 378, 384, 389 Cauchy-Dirichlet problem, 439 Cauchy-Kovaleski theorem, 330 Cauchy-Neumann problem, 439 Cauchy-Riemann operator, 74 Characterisation of p.d.o.'s, 196 Characterisation of the singular spectrum, 541 Characteristic hypersurface, 331 Characteristic set, 151 Classical p.d.o., 218 Classical symbol, 148 Cohomology group, De Rham, 250 Commutator , 210 Compatibility conditions, 480 Complete symbol, 205 Complex, elliptic, 243 Composition, 531 Conic subset, 141 Convolution, 13 Convolution kernel, 522 Correct operator, 346 Cotlar-Knapp-Stein lemma, 266 Decomposition into plane waves, 185 Density, 51 De Rham cohomology group, 250 De Rham complex, 244 Differential operator, 5, 55 Dirac measure, 45 Direct image of a distribution, 528 Dirichlet ' s principle , 84 Dirichlet's problem, 83, 122, 272, 302, 305, 395, 397 Douglis-Nirenberg, ellipticity in the sense of, 226
-555-
556
INDEX
Elliptic, strongly, 124 Elliptic boundary-value problem, 295, 320 Elliptic boundary-value problems with interfaces, 320 Elliptic complex, 243 Elliptic differential operator, 40 Elliptic p.d.0. , 221 Elliptic regularity theorem, 235, 299 Elliptic symbol, 149 Ellipticity in the sense of Douglis-Nirenberg, 226 Energy inequality, 361, 379, 404, 442, 493 Evolution equations , 325 Evolution operator, 348 Fourier transformation, 15, 19 Fredholm operator, 239 Friedrich's lemma, 253 Fundamental solutions, 33, 74, 76, 77, 78 Ggrding's inequality, 231 Generator, infinitesimal, 385 Goursat-Beudon theorem, 332 Green's formula, 84, 304 Group, strongly continuous, 402 Heat operator, 76, 347, 395 Hilbert operator, 308 Hille-Yosida theorem, 391 Hodge's theorem, 247 Holmgren's theorem, 333 Homogeneous distributions, 72 HBrmander's inequality, 134 Hyperbolic Cauchy problems , 344, 359 Hyperbolic operator, 349 Hyperbolic operator, strictly, 350, 368 Hypersurface, spacelike, 372 Hypoellipticity, 38 Index of an operator, 241, 304, 306 Infinitesimal generator, 385 Integral operator associated with a phase, 167 Integration along fibres, 529 Intrinsic characterisation of p.d.0. Is, 196 Intrinsic characterisation of the singular spectrum, 541 Inverse image of a distribution, 45, 80, 521 Jump formula, 11 Kernel distribution, 25 Kernels theorem, 25, 59
INDEX
Lacunas (for waves), 184 Laplace operator, 74 Lax-Milgram theorem, 133 Lax-Mizohata theorem, 338 Lefchetz formula, 265 Lefchetz-Atiyah-Bott formula, 545 Left-elliptic operator, 223 Left parametrix, 224 Leibniz's formula , 6 Leibniz-Hzrmander formula, 6 Local regularity, 310 Local uniqueness, 486 Lopatinski condition, 301, 424, 435, 465, 483, 499 Lopatinski's determinant , 301, 437 Manifold, 62, 523 Manifold with boundary, 62 Maxwell's equations , 367, 498 Measure, Dimc, 45 Microlocal parametrix, 535 Mixed problem, 409, 478 Morse's lemma, 178 Multiplication of distributions, 526 Neumann problem, 272, 305, 347 Oblique derivative problem, 316 Operator of principal type, 135 Oscillatory integral, 141, 183 Paley-Wiener theorem, 17 Paley-Wiener-Schwartz theorem, 20, 71 Parabolic Cauchy problems , 378 Parabolic operator, 378, 382 Parametrix of a differential operator, 39 Parametrix of an elliptic complex, 246 Parametrix of a p.d.0. , 221 Parametrix, left , 224 Parametrix, microlocal, 535 Parametrix, right , 224 Parametrix, two-sided, 224 P . d . 0 . (pseudo-differential operator), 192 Peetre's inequality, 90 Petrovski, operator correct in the sense of, 346 Petrovski, operator parabolic in the sense of, 382, 384 Phase function, 162 Poincar6's inequality, 131 Poisson bracket , 210 Poisson operator, 287 Positive symmetric systems, 495
557
558
INDEX
Potential, surface, 280 Potential, volume, 286 Principal symbol, 6, 49, 189, 206, 216 Principal type, operator of, 135 Product of distributions, 526 Propagation speed, 375, 480, 493 Proper operator, 257 Pseudo-differential operator, 192, 534 Regular distributions, 12 Regular open sets, 62 Regularising operator, 28 Regularising sequence, 3, 91, 110 Regularity, elliptic, 235, 299 Regularity, global, 310 Regularity, local, 310 Regularity theorems 280, 286 Regularity theorem, converse of, 311 Reillich's theorem, 95 Resolvent, 390 Restriction of a distribution, 525 Right-elliptic operator, 223 Right parametrix, 224 SchrWinger ' s operator , 77, 407, 408 Schwartz's kernels theorem, 25 Sectional trace, 275 Seeley's extension, 64 Semigroup of operators, 384 Sheaf, 7 Singular integral, 193 Singular spectrum, 21 Singular support, 7 Sobolev's injection theorem, 89 Sobolev spaces, 126, 233 Solution, strong, weak, 255 Spacelike hypersurface, 372 Spectrum, singular, 21, 501 Stationary-phase theorem, 171, 188 Strictly hyperbolic operator, 354 Strong solution, 255 Support 0f.a distribution, 7 Support of a function, 2 Support , singular 7 Symbol, 140 Symbol, complete, 205 Symbol, principal, 6, 49, 206, 216 Symbol, uniform, 182, 256, 261 Symbolic calculus, 207, 216 Symmetric system, 495
INDEX
Symmetrisable operator, 360 Symmetriser, 448 Trace, at the boundary, 116 Trace, on a hypersurface, 115 Trace calculations, 543 Trace operator, 116, 287 Trace, sectional, 275 Trace theorems, 112 Transmission assumption, 287 Two-sided parametrix, 224 Uniform Lopatinski condition, 421 Uniform symbol, 182, 256, 261 Wave operator, 78, 187, 257, 371, 405 Weak solution, 255 Well-posed Cauchy problem, 337
559
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