Singular and Degenerate Cauchy Problems (Mathematics in Science and Engineering 127)

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Singular and Degenerate Cauchy Problems

ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION

This is Volume 127 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

Singular and Degenerate Cauchy Problems R.

w. Ccxrroll

Department of Mathematics University of Illinois at Urbana-Champaign

R. E. Showalter Department of Mathematics University of Texas at Austin

ACADEmIC PRESS

New York

San Francisco

London

1976

A Subsidiary of Harcourt Brace Jovanovich, Publishers

BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY B E REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, W I T HOU T PIIRMISSION IN WRITING FROM T HE PUBLISHER.

COPYRIGHT 0 1976.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York. New York 10003

Unifed Kingdom Edifion published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London N W l

Library of Congress Cataloging in Publication Data Carroll, Robert Wayne, Date Singular and degenerate Cauchy problems. (Mathematics in science and engineering ; ) Bibliography: p. Includes index. 1. Cauchy problem. I. Showalter, Ralph E., joint author. 11. Title. Ill. Series. QA 3 74.C 38 5 15’.353 76-46563 ISBN 0-12-161450-6

PRINTED IN TH E UNITED STATES O F AMERICA

Contents

vi i

PREFACE

1

Singular Partial Differential Equations

of EPD Type 1 .l. 1.2. 1.3. 1.4.

Examples Mean values in R" The Fourier method Connection formulas and properties of solutions for EPD equations in D ' a n d S' 1.5. Spectral techniques and energy methods in Hilbert space 1.6. EPD equations in general spaces 1.7. Transmutation

1 5

a 29 43 66

80

2

Canonical Sequences of Singular Cauchy Problems

a9

2.1. The rank one situation 2.2. Resolvants 2.3. Examples with mzo, = 0 2.4. Expressions for general resolvants 2.5. The Euclidean case plus generalizations

96 101 130 136

V

CONTENTS

3

3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.

Degenerate Equations with Operator Coefficients

Introduction The Cauchy problem by spectral techniques Strongly regular equations Weakly regular equations Linear degenerate equations Semilinear degenerate equations Doubly nonlinear equations

143 147 166 179 187 206 229

Selected Topics 4.1. 4.2. 4.3. 4.4. 4.5.

Introduction Huygens' principle The generalized radiation problem of Weinstein Improperly posed problems Miscellaneous problems

237 237 245 251 264

REFERENCES

215

INDEX

329

vi

Prefoce

In this monograph we shall deal primarily with the Cauchy problem for singular or degenerate equations of the form (u' = u, = au/at) (1.1)

A(t)u,, + BWu, + C ( t ) u= g

where 4-1 is a function of t , taking values in a separated locally convex space E, while A ( t ) , B ( t ) , and C ( t ) are families of linear or nonlinear differential type operators acting in €, some of which become zero or infinite a t t = 0. Appropriate initial data u ( 0 ) and u,(O) will be prescribed a t t = O , andg is a suitable €-valued function. Similarly some equations of the form

(1.2)

(A(-)u),+ , (B(.)u), + C(.)U = g

will be considered. Problems of the type (1.1) for example will be called singular if at least one of the operator coefficients tends to infinity in some sense as t + 0. Such problems will be called degenerate if some operator coefficient tends to zero as t + 0 in such a way as to change the type of the problem. Similar considerations apply to (1.2). We shall treat only a subclass of the singular and degenerate problems indicated and will concentrate on well-posed problems. If (1.1) or (1.2) can be exhibited in a form where the highest order derivative appears with coefficient one, then the problems treated will be of parabolic or hyperbolic type; if the highest order derivative cannot be so isolated, the equation will be said to be of Sobolev type. We have included a discussion of not necessarily singular or degenerate Sobolev equations for completeness, since it is not available elsewhere. The distinction between singular and degenerate can occasionally be somewhat artificial when some of the operators (or factors thereof) are invertible or when a suitable change of variable can be introduced (see, e.g., Example 1.3); however, we shall not usually resort to such artifices. In practice E frequently will be a space of functions or distributions in aregionclCclR"andO=ztcb<~. The study of singular and degenerate Cauchy problems is partially motivated by problems in physics, geometry, applied mathematics, etc., many examples of which vii

PREFACE

are given in the text, and there is an extensive literature. We have tried to give a fairly complete bibliography and apologize for any omissions. The material is aimed basically a t analysts, and standard theorems from functional analysis will be assumed (cf. Bourbaki [ l ; 21, Carroll [141, Horvath [ l l , Kothe [ l l , Schaeffer [ l l , Treves [ l ] 1; in particular Schwartz distributions will be used with the standard notations (cf. Schwartz [l I 1. There is also some material involving the abstract theory of Lie groups which is assumed but clearly indicated, with references, in Chapter 2; and there are enough concrete examples worked out in the text to illustrate the matter completely for the reader unfamiliar with Lie theory. The chapters are essentially independent; Chapters 1 and 3 begin with introductory material outlining content, motivation, and objectives, whereas Chapters 2 and 4 are organized somewhat differently. Some remarks on notation should perhaps be made here. If, in a given chapter, there appears a reference to formula ( x . y ) it means ( x . y ) of that chapter. I f a reference (x.y. z ) appears, it means formula (y. z ) of Chapter x. Theorems, Lemmas, Examples, Remarks, etc. will be labeled consecutively; thus one might have in order: Lemma x.y, Theorem x.y + 1 , Remark x.y +2, Lemma x.y + 3, etc. in a given section. The first author (R.W.C.) would like to acknowledge a professional debt to A. Weinstein, who initiated systematic work on Euler-Poisson-Darboux (EPD) equations (cf. Weinstein [ l ; 21 1 and whose further contributions to this theory have been essential to its development; his encouragement motivated this author's early work in the area. He was also instrumental in the undertaking of the present monograph. Thus this book is dedicated to Alexander Weinstein. The authors would like t o acknowledge the aid of the National Science Foundation from time to time in supporting some of the research presented here.

viii

Chapter 1 Singular P a r t i a l D i f f e r e n t i a l Equations o f EPD Type 1.1

Examples.

L e t us f i r s t g i v e some t y p i c a l examples o f

s i n g u l a r and r e l a t e d degenerate problems; t h e i r s o l u t i o n s , i n v a r i o u s spaces v i a d i v e r s e techniques, w i l l appear i n t h e course o f t h e book, u s u a l l y as s p e c i a l cases o f more general r e s u l t s . F u r t h e r examples o f degenerate problems w i l l appear i n Chapter 3. The s i n g u l a r Cauchy problem f o r EPD equations.

Example 1.1 L e t t 2 0 and x

Rn; we consider t h e problem

2m+lUm +t t

(1.1)

m u .

(1.2)

Urn(X,O)

tt

E

=

where t h e parameter m

f(x); E

m

U!(X,O)

=

0

C i s a r b i t r a r y f o r t h e moment,

notes t h e Laplace a p e r a t o r i n Rn

, and

Ax de-

c l a s s i c a l l y f and um were

numerical f u n c t i o n s ( n o t e t h a t t h e l e f t hand s i d e o f (1.1)

de-

notes t h e r a d i a l form o f a Laplace o p e r a t o r i n R n i f 2m + 1 = n

-

1).

These equations, f o r i n t e g r a l m, p r o v i d e a model f o r

some very i n t e r e s t i n g s c a l e s o f canonical s i n g u l a r Cauchy problems w i t h t h e i r n a t u r a l o r i g i n s i n L i e t h e o r y (see Chapter 2). They a l s o a r i s e i n physics as i s i n d i c a t e d i n Example 1.3 and o f course f o r m = -1/2 we have t h e wave equation; t h e i r connection w i t h mean values i s i n d i c a t e d i n S e c t i o n 2 and i n Chapter 2. v a r i o u s forms one can t r a c e t h e i r o r i g i n s back t o E u l e r [l], Poisson 111, and Darboux [l]. I n t h i s chapter we w i l l present

1

In

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

f i v e general techniques f o r t h e s o l u t i o n o f these problems and o f s i m i l a r a b s t r a c t s i n g u l a r problems; more general s i n g u l a r i t i e s i n t h e ut term and some n o n l i n e a r terms w i l l be treated.

The f i r s t

i s based on a F o u r i e r method i n d i s t r i b u t i o n spaces developed by one o f t h e authors ( c f . C a r r o l l [2; 3; 4; 5; 6; 8; 10; 11; 141). The second i n v o l v e s a s p e c t r a l technique i n a H i l b e r t space r e l a t e d t o t h e F o u r i e r method ( c f . C a r r o l l , l o c . c i t . )

while the

t h i r d i s based on t r a n s m u t a t i o n methods i n v a r i o u s spaces ( c f . C a r r o l l [24],

C a r r o l l -Donaldson [20]

, Delsarte

Lions [2; 31 Donaldson [l;3; 51, Hersh [2],

[l], D e l s a r t e -

Lions [l;2; 3; 4;

51, Thyssen [l;21). The f o u r t h i n v o l v e s t h e i d e a o f r e l a t e d d i f f e r e n t i a l equations ( c f . Bragg [2; 3; 4; 5; 91, Bragg-Dettman [l; 6; 7; 81, C a r r o l l [24],

Carroll-Donaldson [20],

Donaldson [l;3; 51, Donaldson-Hersh [6],

Dettman [l],

Hersh [l;2; 31).

There

are c e r t a i n r e l a t i o n s between t h e method o f t r a n s m u t a t i o n and t h a t of r e l a t e d equations and t h i s i s brought o u t i n C a r r o l l [24], Carroll-Donaldson [20], 251, Donaldson [4;

and Hersh [2] ( c f . a l s o C a r r o l l [18;

71, and Hersh [l;21).

19;

The f i f t h technique

i n v o l v e s "energy" methods f o l l o w i n g Lions [5] and y i e l d s some r e s u l t s o f Lions i n d i c a t e d i n C a r r o l l [8]. Remark 1.2

There are many a d d i t i o n a l papers devoted t o prob-

lems o f t h e form (1.1)

-

(1.2)

and we w i l l l i s t some o f them here;

f u r t h e r references can be found i n t h e b i b l i o g r a p h i e s t o these articles.

Some o f t h i s m a t e r i a l w i l l appear i n t h i s o r o t h e r

chapters.

I n g i v i n g such r e f e r e n c e s we have t r i e d t o a v o i d 2

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

r e p e t i t i o n of the c i t a t i o n s above and t h e r e will be considerable relevant material and further references i n the work on the Cauchy problem f o r a b s t r a c t Tricomi type problems t o be t r e a t e d l a t e r i n Chapter 3 (cf. Example 1.3), i n Remark 2.4 on mean values, and i n Chapter 4.

T h u s , l e t us c i t e here Agmon [2], Babenko [ l ] ,

Baouendi-Goulaouic [ l ] , Baranovskij [ l ; 2; 3; 4; 5; 6; 71, Barantsev [ l ] , Berezanskij [l], Berezin [l], Bers [ l ;

21,

Bitsadze

[ l ; 21, Blumkina [ l ] , Bresters [ l ] , Bureau [ l ; 21, Carroll [7; 9; 13; 18; 19; 21; 221, Carroll-Silver [15; 16; 171, Carroll-Wang

[I2],

31,

C h i [ I ; 21, C i b r a r i o [ l ; 2;

Cinquini-Cibrario [1;2],

Conti [1 ; 2; 31, Copson [ l ] , Copson-Erdelyi [Z],

Courant-Hi1 bert

[ l ] , Delache-Leray [ l ] , Diaz [5; 61, Diaz-Kiwan [7], Diaz-Ludford [3; 41, Diaz-Martin [9],

Diaz-Weinberger [2], Diaz-Young [ l ;

81,

Ferrari-Tricomi [ l ] , Filipov [l], Fox [ l ] , Frank'l [ l ] , Friedlander-Heins [ l ] , B. Friedman [ l ] , Fusaro [2;

3; 4; 5; 61, Gara-

bedian [ l ] , Germain [ l ] , Germain-Bader [2; 31, G i l b e r t [ l ] , Gordeev [ l ] , Gcnther [2; 31, Haack-Hellwig [1], Hariullina [ l ; 2; 31, Hairullina-Nikolenko [ l ] , Hellwig [ l ; 2 ; 31, Kapilevi?!

[l; 2;

3; 4; 5; 61, Karapetyan [l], Karimov-Baikuziev [ l ; 23, Karmanov [l], Karol [ l ;

21,

Kononenko [ l ] , Krasnov [ l ; 2; 31, Krivenko

[ l ; 23, Ladygenskaya [ l ] , Lavrentiev-Bitsadze [l], Lieberstein [ l ; 21, Makarov [ l ; 21, Makusina [ l ] , Martin [l], Mikhlin [ l ] , Miles-Williams [ l ] , Miles-Young [Z;

31, Morawetz [ l ; 21, Nersesyan

[l], Nevostruev [ l ] , Nosov [ l ] , Olevskij [2], Ossicini [ l ; Ovsyanikov [l], Payne [1], Payne-Sather [Z],

3

21,

P r o t t e r [2; 3; 4 ; 5;

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

61, Protter-Weinberger [1], [4],

Rosenbloom [l; 2; 31, Sather-Sather

S i l v e r [l], Smirnov [l;2; 3; 41, Solomon [l;21, Sun [l],

Suschowk [l], Tersenov [l;2; 3; 41, Tong [l],T r a v i s [1],

Tri-

Volkodarov [l],Walker [l;2; 31, Walter [l],Wang [l;

comi [1],

21, Weinstein [l;2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 13; 14; 15; 16; 17; 18; 19; 203, W i l l i a m s [l], Young [l;2; 3; 4; 5; 6; 7; 8; V

9; 10; 111, Z i t o m i r s k i j [l]. Example 1.3 equation. (1.3)

The degenerate Cauchy problem f o r t h e Tricomi

The o r i g i n a l Tricomi e q u a t i o n i s o f t h e form Utt

-

tuxx

=

0

and we w i l l be p r i m a r i l y concerned w i t h t h e Cauchy problem f o r a b s t r a c t v e r s i o n s of t h e more general e q u a t i o n utt + a(x,t)ut

(1.4)

+ b(x,t)u,

+ c(x,t)u

-

K(x,t)uxx

=

0

i n t h e h y p e r b o l i c r e g i o n t > 0 where i t i s assumed t h a t t K ( x , t )

0 for t

p

0 w h i l e K(x,O)

= 0; s u i t a b l e i n i t i a l data u(x,O)

>

and

u (x,O) w i l l be p r e s c r i b e d on t h e p a r a b o l i c l i n e t = 0. This ext ample w i l l be t r e a t e d as a degenerate problem i n Chapter 3 b u t i s d i s p l a y e d here because o f t h e connection o f (1.3) EPD equation o f i n d e x 2m

+ 1

t o a (singular)

= 1 / 3 under t h e change o f v a r i a b l e

T = 2/3 t3l2f o r t > 0 which y i e l d s

(1.5)

I

UTT +

3-r

UT

= uxx

S i m i l a r l y i n t h e r e g i o n t < 0 t h e change o f v a r i a b l e ~ = 2 / 3 ( - t )3/2

4

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1 i n t o uTT + 3* u

transforms (1.3)

+ uxx

~

= 0 which i s a s i n g u l a r

e l l i p t i c e q u a t i o n o f t h e type s t u d i e d by Weinstein and o t h e r s i n t h e c o n t e x t o f g e n e r a l i z e d a x i a l l y symnetric p o t e n t i a l t h e o r y (GASPT); c f . Remark 1.2 f o r some r e f e r e n c e s t o t h i s ,

The r e f e r -

ences above i n Remark 1.2 a l s o deal w i t h mixed problems where s u i t a b l e data i s p r e s c r i b e d on v a r i o u s types o f curves i n t h e e l l i p t i c ( t < 0) and h y p e r b o l i c ( t > 0) regions; such problems occur f o r example i n t r a n s o n i c gas dynamics.

We w i l l n o t deal w i t h t h e

e l l i p t i c r e g i o n i n general b u t some o t h e r types o f s i n g u l a r hyperb o l i c problems f o r EPD equations w i l l be discussed b r i e f l y i n Chapter 4. 1.2 (1.1)

-

Mean values i n Rn (1.2)

for x

E

tribution T

E

D

= DX I

We r e f e r now t o t h e EPD problem

Rn and 2m Let
case i n t h e theory. I

.

+ 1

= n

- 1 which i s a p i v o t a l

denote t h e a c t i o n i n Rn o f a d i s -

on a " t e s t f u n c t i o n " I$ E

D

E Dx; t h e same

b r a c k e t n o t a t i o n w i l l a l s o be used f o r p a i r i n g s E etc.

I

-

E, S

I

-

S,

One d e f i n e s a s u r f a c e ( o r s p h e r i c a l ) mean value o p e r a t o r

p x ( t ) ( t f i x e d f o r t h e moment) by t h e r u l e

L

0 E E,

n

1 xi,

wn = 2rn"/r(n/2) i s t h e surface area 1 o f t h e n-dimensional un t b a l l , and du = tn- dRn represents t h e n

where

=

surface area measure on t h e b a l l o f r a d i u s I x

= t.

The surface

mean value measure 1-1 ( t ) averages 41 over a sphere o f r a d i u s t cenX

tered a t the origin.

S i m i l a r l y one d e f i n e s a s o l i d mean value

5

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

operator A x ( t )

E

E

which averages @

E

I

( t s t i l l f i x e d ) by t h e r u l e

E over t h e e n t i r e b a l l 1x1 5 t.

Now we observe

t h a t =

and, a l l o w i n g t t o vary, t h e Y f o l l o w i n g c a l c u l a t i o n s are e a s i l y checked (v denotes t h e e x t e r i o r norma 1 )

-

1

(%)don

=

r

1

A@dx

n t = -
Note i n (2.3)

that i f

1x1 5 t then t h e E-E

D-D

I

I

E

A@> =

D with

(2.5)

= 1 i n a neighborhood o f

p a i r i n g < A x ( t ) , A p E = ,

AAx(t)

E

E

I

p x ( t ) and t

-f

=
in a

PE

I

i s computed i n D ).

By Lemma 2.3 below, (2.3) -+

)I

@>

p a i r i n g so t h a t D = ,

(i.e.,

t

t
and (2.4)

A x ( t ) belong t o C"(Ei)

;A f l x ( t )

xd p x ( t ) = t

6

imply t h a t the functions for t > 0 with

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

Theorem 2.1 px(t) (i.e.,

sd!-t

Let T

E

D i be a r b i t r a r y .

Then u:(t)

=

T s a t i s f i e s t h e EPD e q u a t i o n (1.1) w i t h 2m + 1 = n - 1 n 1) and t h e i n i t i a l c o n d i t i o n s um(o) = T w i t h m =

*

-

X

P ( o ) = 0. x Proof:

We r e c a l l t h a t (S,

T) +

*

S

T:

E

I

D

x

I -+

D

I

i s s e p a r a t e l y c o n t i n u o u s so e q u a t i o n ( 1 .l)r e s u l t s i m m e d i a t e l y by

-

d i f f e r e n t i a t i n g (2.5) and u s i n g (2.5) px(t)

-+

6 ( D i r a c measure a t x = 0 ) when t

dt d (px(t) *

T) = ; t AAx(t)

t (F) ( A x ( t ) * AT)

-+

0 as t

*

T = (:)[A6

-+

0.

Evidently

(2.6).

*

0 and we n o t e t h a t

-+

Ax(t)

*

TI = QED

For i n f o r m a t i o n a b o u t d i s t r i b u t i o n s and t h e d f f e r e n t ia t ion o f v e c t o r v a l u e d f u n c t i o n s l e t us r e f e r t o C a r r o l l [141, A. Friedman [Z],

v

Garsoux [l],G e l f a n d - S i l o v [5;

6;

71

H o r v l t h [l],

Nachbin [l],Schwartz [l; 21, Treves [l].

A function f:R

D e f i n i t i o n 2.2

E, E a general l o c a l l y

-+

I

I

convex space, has a s c a l a r d e r i v a t i v e f s ( t )E (E )

d

-df t( * ) ,

I

I

I

e > = for a l l e

b r a i c dual o f F).

I

E

E

I

(F

*

topology o f E then f

if

denotes t h e a l g e -

On t h e o t h e r hand i f d f = f I ( t ) I

*

E

E i n the

i s c a l l e d t h e s t r o n g d e r i v a t i v e o f f.

The f o l l o w i n g lemma i s a s p e c i a l case o f r e s u l t s i n C a r r o l l

[2; 5; 141 and Schwartz [2].

7

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

If f:R

L e m a 2.3

+

E

I

I

has a s c a l a r d e r i v a t i v e f S then f i s

strongly differentiable with f Proof: l i m i t s as t

I

I

= fs.

F i x to and s e t g ( t ) = -t

-

.

f ( t ) f ( to>

-

to

I n discussing

to i t s u f f i c e s t o c o n s i d e r sequences tn +. to s i n c e

any to has a countable fundamental system o f neighborhoods. hypothesis g ( t n )

I

+.

By

Using t h e f a c t t h a t E i s a

f s ( t o weakly. )

Monte1 space and a v e r s i o n o f t h e Banach-Steinhaus theorem ( c f . C a r r o l l [14])

i t follows t h a t g(tn)

I

+.

f;(to) s t r o n g l y i n E ' so

I

t h a t f s ( t o )= f ( t o ) . Remark 2.4

QED

A g r e a t deal o f i n f o r m a t i o n i s a v a i l a b l e r e l a t i n g

" s p h e r i c a l " mean values and d i f f e r e n t i a l equations, b o t h i n Rn and on c e r t a i n Riemannian manifolds.

I n a d d i t i o n t o references

i n c l u d e d i n t h e I n t r o d u c t i o n , Example 1.1,

Remark 1.2,

and work

on GASPT, r e l a t i v e t o such mean values, l e t us mention Asgeirsson

[l],C a r r o l l [l],Fusaro [l], Ghermanesco [l;21, G i n t h e r [l], Helgason [l;2; 3; 4; 5; 6; 7; 8; 9; 10; 111, John [l],O l e v s k i j

[l],P o r i t s k y [l], Walter [2], t h e i r bibliographies.

Weinstein [12],

Willmore [l]and

T h i s w i l l be discussed f u r t h e r i n a L i e

group c o n t e x t i n Chapter 2.

For some r e f e r e n c e s t o complex func-

t i o n t h e o r y and v a r i o u s mean values and measures r e l a t e d t o d i f f e r e n t i a l equations see Zalcman [l;21 and t h e b i b l i o g r a p h i e s there. 1.3

The F o u r i e r method.

We w i l l work w i t h a more general

e q u a t i o n than (1.1) which i n c l u d e s (1.1)

8

as a s p e c i a l case.

Thus

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

where the aa a r e constant (and r e a l ) and the sum i n (3.1) i s over

a f i n i t e set of mu1 t i - i n d i c e s a transform F:T

A

-+

T = FT:S

Schwartz [ l ] ) and f o r $ A

$ =

1

fm

-m

I -+

E

S

I

= (a1

. . ., a n ) .

The Fourier

i s a topological isomorphism ( c f .

S f o r example we use the form F$

$(XI exp - i<x,y>dx where <x,y>

=

A

n

1 xiyi '

=

Hence in par-

t i c u l a r F(Dk$) = ykF$ = y k $. We r e c a l l a l s o t h a t AxT = A 6 w i t h F6 = 1 and will w r i t e A(y) = F(Ax6.) = 1 aaya = 1 aayl8l a n I t w i l l be assumed t h a t A(y) s a t i s f i e s yn

*

T

...

.

Condition 3.1 IYI >

A(y)

=

1 aaya

i s real w i t h A(y) 2 a > 0 f o r

R o w

In p a r t i c u l a r Condition 3.1 holds when Ax = -A = -1 i 2D 2 -X k 2 1 0; so t h a t F(-A6) = 1 yk = lyI2. I t can a l s o hold f o r c e r t a i n h y p o e l l i p t i c operators A since then IA(y)l

-+

a s IyI

(cf.

-+

Hb'rmander [ l ] ) o r even nonhypoelliptic operators (e.g., A(y) = 2 2 a + y2 + + y n ) . One m i g h t a l s o envision a p a r a l l e l theory

...

when Re A(y) 2 a > 0 f o r IyI > Ro b u t we will not i n v e s t i g a t e t h i s here.

Let us f i r s t consider the problem f o r

on 0 5 t 5 b <

(b arbitrary)

9

~ " ( 0 )

E

C 2 ( S x' )

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where m

2

-1/2 i s r e a l (complex m w i t h Re m > -1/2 can a l s o be

t r e a t e d b y t h e methods o f t h i s s e c t i o n ) .

Taking F o u r i e r trans-

forms i n x one o b t a i n s (3.4)

ittm + Et i y

(3.5)

w

"m

T;

(0) =

+

im = 0

A(y)

"m

W&O)

0

=

-

and when T = 6 we denote t h e s o l u t i o n o f (3.2)

so t h a t $ ( t )

s a t i s f i e s (3.4)

(3.3)

h

-

by R m ( t )

The e x p r e s s i o n s

(3.5) w i t h T = 1.

R m ( t ) and i m ( tw)i l l be c a l l e d r e s o l v a n t s and one n o t e s t h a t i f

Rm(t)

O;,

E

0;

*

then Rm(t)

3.4).

@ ( t )E OM), w i t h s u i t a b l e d i f f e r e n t i a b i l i t y i n

(i.e.,

T = w m ( t ) s a t i s f i e s (3.2)

-

(3.3)

We r e c a l l h e r e t h a t OM i s t h e space o f

COD

(see Theorem f u n c t i o n s f on

Rn such t h a t Daf i s bounded by a p o l y n o m i a l (depending perhaps on a ) w h i l e f s -+ 0 i n OM whenever gD b

a 6

any a and any g

E

S.

0;

= F-'OM

C SI

+

and i s g i v e n t h e t o p o l o g y

t r a n s p o r t e d by t h e F o u r i e r t r a n s f o r m ( i .e., FT6

-+

mapping S I

Oc

x

S

I -+

-+

S

I

S

I

-+

0 i n Oc

c-f

I t i s known t h a t Oc i s t h e space o f d i s t r i b u t i o n s I

under c o n v o l u t i o n and t h e map (S,T)

+

S

*

T:

i s hypocontinuous.

Now t h e s o l u t i o n o f (3.4) ("R"(t)

T6

I

0 i n OM). I

0 u n i f o r m l y i n IRn f o r

-

A

(3.5) when T = 1 i s g i v e n by

= F(.,t))

10

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

where z = t

( c f . B. Friedman [ l ] and Rosenbloom [ l ] ) .

i s i n f a c t a function of z2 so t h a t i t i s

observes t h a t $ ( y , t )

m;evidently

not necessary t o define the square root

i s an e n t i r e function of the y k f o r yk by Hartog's theorem Lemma 3.2

One

-

E

R^m(-,t)

E (and hence of y

n

E (I:

cf. Hzrmander [Z]).

The funct on t

+

"m R (-,t):[o,b]

-+

OM i s cont nuous

(m 2 -1/2). i s continuous and one Y knows t h a t on bounded s e t s in OM the topology of O,., coincides with

Clearly t

Proof:

+

$(-,t):[o,b]

+.

E

t h a t induced by E (see Schwartz [2] and r e c a l l t h a t a set B C OM

is bounded i f Daf i s bounded by the same polynomial , depending perhaps on a, f o r a l l f E B ) . Consequently one need only show t h a t each DaR"m ( y , t ) i s bounded by a polynomial i n y , independent of t f o r t

E

[o,b].

W e choose here a straightforward method o f

doing t h i s and refer t o subsequent material f o r another, much more general , technique ( c f . Carroll [5; 81).

T h u s , one knows

t h a t f o r m > -1/2 (cf. Watson [ l ] )

(the case m = -1/2 i s t r i v i a l since z '"5 Writing la1 = a, +

. . . + an and

have from (3.7)

11

-1 / 2 ( z )

=

(2/.rr)'"

c, = 2r(m+l)/JiT r(m

+ -!-)2

cos z). we

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Now f o r IyI > R

0

a

(3.9)

Am

-R ayk

and since A(y) from (3.8) (3.10)

-

(y,t)

we w r i t e

1

a “m 5 7 R (y,t)

= (tL/2m)

ayk

2 a w i t h aA(y)/ayk

= Bk(y) a polynomial we have

(3.9)

I aim(y,t) I ayk

xbcm -lBk(Y)I 4 6

= clBk(Y)I

5 1

2 2 c Bk(Y)

+

A s i m i l a r argument obviously a p p l i e s t o majorize any D RIn( y , t ) f o r a

IyI > Ro by a polynomial i n y n o t depending on t. the power s e r i e s i n (3.6) may be used, since ( y , t ) continuous on the compact s e t ( l y l bound IDaim(y,t,)l

5 Ma.

5 Ro)

x

[o,b],

In Consequently lDaR (y,t)

For I y I 5 -+

Ro

Daim(y,t) i s

t o obtain a

I5

nomial = polynomial i n y, f o r a l l y, independent o f t

HIa E

+ poly[o,b].

QE D Lenma 3.3 t

E

[o,b]

(m

The f u n c t i o n t

-+

^Rm(-,t) belongs t o Cm(O,,)

2 -1/2) and

12

for

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Proof:

-

The formulas (3.11)

(3.12) a r e e a s i l y seen t o hold

i n E by u s i n g well-known recursion formulas f o r Bessel functions Y (cf. Watson [ l ] , Carroll [5]). Further we know by Schwartz [3;

? ( * , t ) E Co(OM) f'l C1(E) w i t h t +. iT(*,t)E Co(OEl) % 1 R ( - , t )E C (Old). Using (3.11) and Lemma 3.2 however we

41 t h a t i f t

then t

+.

see t h a t t

+.

+.

"m

Rt(*,t)

Co(OM) and in p a r t i c u l a r (3.11)

E

-

(3.12)

By i t e r a t i o n of this argument i t i s seen t h a t t

hold i n OM.

+.

"Rm(*,t) belongs t o Cm(On). Let T

Theorem 3.4 given by (3.6).

(3.2)

-

A

Then R m ( t )

W e know t

+.

Ah

A

the map (S,T)

+.

ST:OM x S

Using (3.11), t

E

= F

-1 R"m ( * , t )with i'"(y,t)

0; and w m ( t ) = R m ( t )

*

T satisfies

+

i'(*,t)?:[o,b]

+

S I i s continuous since

I

I

i s hypocontinuous ( c f . Schwartz +. S 2m t lAm Rt(-,t)?:[o,b] +. S I i s a l s o con-

From (3.4) i n E we see (using (3.11) again) t h a t t

tinuous. A

RFt(-,t)

S ' and R " ( t )

E

(3.3).

Proof:

[l]).

QED

=

-

A ( * ) [ W ' ( - , t )

which shows e x p l i c i t l y t h a t t Lemma 3.3) and t h a t t

+.

+

+.

A

R m ( = , t ) ] belongs t o Co(OM) 2 im(-,t) E C (OM) on [o,b] ( c f .

$(*,t)? s a t i s f i e s (3.4) in S I .

quently (3.2) holds i n S I and wT(o)

=

Conse-

0 since by (3.11) $(y,o)

=

QED

0.

To deal w i t h uniqueness we make f i r s t a few formal calculat i o n s following Carroll [8; 111; t h i s technique i s somewhat d i f -

ferent than t h a t used i n Carroll [5] b u t by c o n t r a s t i t can be generalized.

Thus f o r o <

T

5 t 5 b < 13

A

m

l e t Rm(y,t,T) and

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

^sm(y,t,-r)

be two l i n e a r l y independent s o l u t i o n s o f (3.4) w i t h

"m R (Y,T,T)

=

"m

"m

1, Rt(y,~,.r) = 0, S ( ~ , T , T ) = 0, and $(y,-r,-~) = 1.

Such (unique) numerical s o l u t i o n s e x i s t s i n c e t h e problem i s nons i n g u l a r on [ ~ , b ] f o r t > 0.

SO

that

^cm

Let

i s a fundamental m a t r i x f o r t h e f i r s t o r d e r system

corresponding t o (3.4)

+

(3.14)

ip =

w r i t t e n i n t h e form A

0; M(y,t) =

As i n Schwartz [3] i t i s e a s i l y shown t h a t

from which f o l l o w s i n p a r t i c u l a r

Given t h a t

^Rm

and

irnhave

s u i t a b l e p r o p e r t i e s ( s t a t e d and v e r i -

f i e d below) i t f o l l o w s t h a t i f

"m with w (T) = mulas (3.18)

A

T and ;"(T) t

-

(3.20)

i s a s o l u t i o n o f (3.4)

= 0 then u s i n g (3.16)

below make sense.

14

-

(3.17)

i n S' the f o r -

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

- ^m R (y,t,-c)?

t im(t)

-

= i m ( t ) "m R (y,t,T)?

From (3.20) i t then f o l l o w s f o r m a l l y t h a t any s o l u t i o n (3.4) i n S

I

"m with w (T) =

A

T

and

^m

w ~ ( T )=

0

(T

2 of

> 0) must be o f t h e

form i m ( t ) = i m ( y y t y T ) ? . We w i l l now proceed t o j u s t i f y t h i s and t o a t t e n d t o t h e case

T

= 0 as w e l l .

W e remark a l s o t h a t i f one

considers

w i t h i n i t i a l conditions given a t t =

A

T

as a b o v e ( f ( - ) c C o ( S I ) ) then

t h e procedure o f (3.20) f o r m a l l y y i e l d s t h e unique s o l u t i o n as (3.22)

$(t)

= i m ( y y t y T ) ?+ r ? ( y , t , < ) ? ( f ) d < T

We w i l l develop a procedure now which can a l s o be used i n o t h e r s i t u a t i o n s (see S e c t i o n 1.5).

It i s h e l p f u l t o have an ex-

p l i c i t example o f t h e technique however and hence we i n t r o d u c e i t here; i n p a r t i c u l a r t h i s w i l l make t h e considerable d e t a i l s more e a s i l y v i s i b l e and s h o r t e n t h e development l a t e r . 1/2 l e t P ( t ) = exp ( -

r y t

Thus f o r m 2

-

d5) = (t/b)2m+1 so t h a t P ( T ) / P ( t ) =

(-c/t)2m+1 5 1 f o r 0 S T 5 t 5 b and P ( t )

-+

0 as t

-+

0.

L e t us

c o n s i d e r equation (3.4) f o r km(y,t,T) w i t h i n i t i a l data a t

15

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

t = T > 0 as i n (3.13) and then, m u l t i p l y i n g by P ( t ) , we o b t a i n

(3.23)

& (Pi!)

+ PA^Rm = 0

This leads t o t h e i n t e g r a l equation (T 5 5 5 u 5 t), e q u i v a l e n t t o (3.4), w i t h i n i t i a l c o n d i t i o n s a t t = T as i n d i c a t e d ,

and e v i d e n t l y T = 0 i s p e r m i t t e d i n (3.24).

A s o l u t i o n t o (3.24)

i s given f o r m a l l y by the s e r i e s (3.25)

03

“m R (y,t,-c)

=

1

k=O

(-l)kJk

1

( c f . H i l l e [ l ] ) where Jk denotes t h e kth i t e r a t e o f the o p e r a t i o n J and Jo i s t h e i d e n t i t y . (3.26) we have R

Writing

imYp = 1 =

- ^RmYP-’

f

k=O

= (-1)’Jp-1

imYo = 1 and (-l)kJk and i f y

1 E

K C Cny w i t h K

compact, then I A ( y ) I C 5 cK and i n p a r t i c u l a r

5 cKF(t,-c) 5 cKF(t,o) = c K F ( t ) where F ( t , T) i s the double i n t e g r a l i n (3.27) and F(tl) 5 F ( t ) 5 F(b) f o r 0 S t, 5 t i b ( n o t e t h a t F ( t ) = have, using (3.27)

,

16

t2

Continuing we

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPDTYPE

1.

-

so t h a t

1:

h

Rmy'

-

-1

F(a)

0

,..

2 2 cKF ( t ) / 2 ! s i n c e i f F ( t ) =

Ic 5

($)2m+1dSdo t h e n

F

= F F' w i t h F(o) = F ( o ) = 0.

By

i t e r a t i o n one o b t a i n s

Hence t h e s e r i e s (3.25) [O 5

T

5 t S bl

x

converges a b s o l u t e l y and u n i f o r m l y on

K and s i n c e t h e terms Jp

1 are continuous h

and a n a l y t i c i n y t h e same i s t r u e o f Rm(y,t,-r). It ^m i s c l e a r t h a t R g i v e n by (3.25) s a t i s f i e s (3.24) and we can

i n (y,t,T)

state Lemma 3.5

"m w i t h R (Y,T,T)

The s e r i e s (3.25)

"m

= 1 and Rt(y.~,-r)

The maps (y,t,-c)

+

"m R (y,t,T)

r e p r e s e n t s a s o l u t i o n o f (3.24) = 0 (0

and (y,t,-c)

5

T

-t

5 t 5 b and m > -1/2). "m Rt(y,t,T)

are contin-

"m "m uous n u m e r i c a l f u n c t i o n s w h i l e R ( * , t , ~ ) and Rt(*,t,.r)

are anal-

ytic. Proof: for

ty

E v e r y t h i n g has been p r o v e d f o r

im and t h e

f o l l o w i m m e d i a t e l y upon d i f f e r e n t i a t i n g (3.24)

We examine now ( c f .

(3.24))

17

statements i n t.

QED

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

For

> 0 t h e r e i s o b v i o u s l y no problem i n p r o v i n g c o n t i n u i t y and

T

a n a l y t i c t y as i n Lemma 3.5 so we consider

T

-+

0.

Let

(3.31) Clearly

i s n o t continuous as 2m+l "m t h e r e i s no hope t h a t (y,t,i-) * e t ( y . t , . r ) t,T)

-f

H(t,.r)

i:(y,t,o)

from (3.30)

i s continuous on [o,b]

( y o y o ) equal t o - A ( y 2m o )+. 1m 2m+l "m

7 Rt (y,t,-c)

(3.31), x

(y,t)

<-

T i t

= H(t) =

-+

K w i t h l i m i t as ( y , t )

-+

-+

These p r o p e r t i e s

5 b.

a f t e r d i f f e r e n t i a t i n g (3.24)

i s s a t i s f i e d f o r 0 <-

i n t and equation (3.4)

w i l l be continuous

F u r t h e r from (3.30) y

i s analytic f o r 0

"in are t r a n s p o r t e d t o Rtt(y,t,T)

-

so

(0,O)

-+

We w r i t e H(t,o)

on [O s T 6 t 5 b] x K ( c f . Lemma 3.5).

zoand then c l e a r l y ,

(t,T)

T

<- t

5 b.

twice Summar-

i z i n g we have Lemna 3.6 and (y,t) * and y

*

For [O 5 T 5 t 5 b] t h e maps (y,t)

2m+l "m 7Rt(y,t,T)

"+'

$(y,t,r) t

are continuous w h i l e y

are analytic.

i m ( y y t y T )s a t i s f i e s (3.4)

-f

+

"m Rtt(yytyT)

"m Rtt(y,t,T)

The numerical f u n c t i o n

w i t h i m ( y , ~ , ~ )= 1 and "m R&Y,T,T)

im, assuming t h a t r e p l a c e d by c ) by

Next we w i l l d e r i v e some bounds f o r real.

I f we m u l t i p l y (3.23)

*m

P-'(S)Rg(y.S.r) (3.32)

(with t

and i n t e g r a t e b y p a r t s i t f o l l o w s t h a t

l i ~ ( y y t y T ) 1 +2 2(2m+l)

15t 1 ~ i ~ ( y , ~ , T2 d~ )l T

+ A(y) lkm(y,t,T)

=

18

A(Y)

= 0.

y is

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

d since dt

"m"m

= 2R Rt and

d dt

lq12

"m"m

= 2RtRtt

( c f . a l s o Section

"m

I n p a r t i c u l a r i f IyI > Ro we can say t h a t I R t ( y y t y T ) / 2 5

1.4).

A(y) and I^Rm(y,tyT)12 5 1. can be bounded on [O 5 ( c f . Lemma 3.5)

"m "m Since I R ( y y t y . r ) l and I R t ( Y y t y T ) l

5 t 5 b]

T

i t follows t h a t

"+'

?$yytyT) t

0

[im(yytyT)12

"m I R t ( y y t y T ) 1 2 5 c2 + A(y) f o r 0 5

t h e term

t o 5 IyI 5 R } b y c o n t i n u i t y

x

T

5 c1 and

5 t 5 b and a l l y.

we r e f e r t o (3.30)

For I y I > R o y I A ( y ) l = A(y) and f o r IyI

-

;R o y

(3.31)

To bound

t o obtain

I A ( y ) I i s bounded;

t h i s leads t o

"m

Lemma 3.7

"m

I R t ( u y t y T ) 1 2 5 c2

0 2 T

<

^m

s a t i s f i e s I R ( y y t y r ) 1 25 cl,

The f u n c t i o n R +

2m+l "m A(y), and F I R t ( y y t y - c ) I 5 c4 + c3 A(y) f o r

t i band all y

E

Rn.

"m L e t us examine now some s i m i l a r p r o p e r t i e s o f S

d i f f e r e n t i a t e (3.24)

in T

= -J

T

T

5 t

<-

I f we =

1)

+ A(y)U(tp)

where f o r m 2 - l / 2 , U ( t y T ) = -[I 2m for 0 5

.

*m t h e r e r e s u l t s ( s i n c e R (y,~,.)

T 2m 3 - (t)

b (note t h a t U ( t y T )

From (3.16) we have then

19

-t

i s continuous i n ( t y T )

-T l o g ( T / t )

as m

+

0).

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

A s o l u t i o n o f (3.35) i s g i v e n

(except perhaps where A(y) = 0).

03

?= 2

(3.36)

k=O

and t h e terms J k

(-l)kJk

U

U i n (3.36)

IU(t,-c)l 2 (3.26) - (3.29))

analytic i n y with analysis (cf.

a r e continuous i n (y,t,-c) c.

I t f o l l o w s by o u r p r e v i o u s

t h a t t h e s e r i e s i n (3.36)

verges a b s o l u t e l y and u n i f o r m l y on [0
D e f i n i n g now

and

.? by

'c

5 t 5 b] x

conC Cn

K (K

t h i s s e r i e s (whether o r n o t

A(y) = 0) t h e r e r e s u l t s

m( y , t , T )

For m 2 -1/2 th e expression S

Lemma 3.8 by (3.36)

s a t i s f i e s (3.35)

and 3.4);

defined

i t i s continuous i n (y,t,-c)

"m

and a n a l y t i c i n y f o r [0 5 T 5 t 6 b] a l o n g w i t h St (y,t,-c)

for

0 < T 5 t 5 b.

and, f o r T >

0, (3.17)

"m "m One has R = A(y)S (i.e., (3.16)) T "m "m holds w h i l e S (Y,T,T) = 0 w i t h St(y,-c,-c)

= 1; finally

P(y,t,o)

=

Proof:

o

f o r m > -1/2. One notes f i r s t t h a t t h e s e r i e s o b t a i n e d by term"m i n T i s t h e same as A(y)S

wise d i f f e r e n t i a t i o n o f (3.25) where m:

i s give n by (3.36).

Since Jk

(y,t,T)

U = 0 a t t = T for k

2

1 w i t h U(T,T) = 0 i t f o l l o w s t h a t &?(y,-c,~) = 0. On t h e o t h e r "m hand S (y,t,o) = 0 f o r m > -1/2 s i n c e U(t,o) = 0 w h i l e (3.17) holds f o r T > 0 s i n d e

im clearly

m "m ( s i n c e S (y,-c,-c) = 0) ST = UT UT(t,-c)

2m+l

= ,-U(t,-c)

-

-

s a t i s f i e s (3.35)

d J d-c

?=

U

T

1 and hence f o r T > 0

20

-

and hence J

irn But T*

1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE

( fr om (3.35) and t h e d e f i n i t i o n o f J ) , from which f o l l o w s "m St(y,~,~) = 1 f o r

$.

erties o f

> 0 and t h e c o n t i n u i t y and a n a l y t i c i t y prop-

T

To check (3.4)

one can use (3.17)

t o obtain f o r

T > O

which leads t o (3.4)

upon d i f f e r e n t i a t i o n i n t.

"m

Now ST i s determined by (3.17) and as "m

l i m ST may n o t e x i s t .

T

-+

QED 0 a priori

However l e t us l o o k f i r s t a t m

(3.40)

+s^m(y,t,r)

1 1 where 7 U ( ~ , T ) = $1 t 5 b.

=

-

1 (-l)kJk

k=I)

(c)2m3 2 T

Hence, as i n (3.26)

-

U(.,T)

*.

1

f o r m > 0 and 0 5

(3.29),

There i s l i t t l e hope t h a t (y,t,r)

5 5 5

t h e s e r i e s i n (3.40)

verges u n i f o r m l y and a b s o l u t e l y f o r 0 5 Cn.

T

T

+

5 t 5 b and y

-r1 "mS (y,t,T)

E

con-

K C

w i l l be

I 5 t 5 b ] s i n c e e.g., U(t,-r) i s n o t so T 1 continuous ( n o r i s J -r U(-,T), etc.). We can however l e t T

continuous on [0 S

0 i n (3.40)

T

f o r m > 0 t o obtain f o r t > 0

21

-f

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Consequently from (3.17)

for

Lemma 3.9 "m ST(y,t,o)

while y

+

For m > 0 and m = -1/2 th e 1 "m = -R (y,t,o) i s continuous f o r 2m "m ST(y,t,o) i s a n a l y t i c . F u rth e r ,

l y continuous f o r m

IBq(y,- t , -c)dr

Jci,

sion

2

% -1/2 w i t h S (y,t,B)

f u n c t i o n (y,t)

+

0 5 t 5 b and y "m S (y,t,?)

-

"m

K

E

i s absolute-

S (y,t,a)

=

and f o r 0 5 T 5 t S b and m 2 -1/2 t h e expres-

"m

Ts,(y,t,T)

i s continuous i n (y9t,-c) and a n a l y t i c i n y.

Proof: "m ST(y,t,o)

The c o n t i n u i t y and a n a l y t i c i t y i n d i c a t e d f o r "m and TS,(Y,t,T) f o l l o w immediately from (3.42) and

^m

(3.17). For 0 < ci, 5 B 5 t 5 b t h e formula S (y,t,B) S (y,t,a) = jb: (y ,t,r)d r "m i s obvious and by c o n t i n u i t y one can take l i m i t s as a

+

0.

I n t h e cases -1/2 < m

w i l l have an i n t e g r a b l e s i n g u l a r i t y a t f o r -1/2 < m < 0 and

-

T

5 0 we n o t e t h a t

1 U(t,-c) = 0 since T

QED

log-c f o r m = 0. A

I n o r d e r t o o b t a i n some bounds f o r Sm when

YE

m u l t i p l y (3.17) ( w i t h T re p l a c e d b y . 5 ) by im(y,t,E) d "m 2 -1s I = 2s"m"mS5 one o b ta i n s upon i n t e g r a t i o n d5

22

-

Rn we and s i n c e

1. SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

By Gronwall's lemma ( c f . Section 5 and Carroll [14]) there reA

s u l t s ISm(y,t,-c)12 5 c5 f o r 0 5

5 t <, b and y

T

E

Rn.

Conse-

A

quently by (3.17) and Lemma 3.7 (rS:(y,t,~)( 5 c6 on the same domain. c7

+

Finally from (3.38) we have

liyl 5

1 + bck"IA(y)l

5

c8A(y)* Lemna 3.10

For y

E

Rn and 0

S T i

t i b one has

A

ISm(y,t,r)12 5 c5 and lTiy(y,t,T)l 5 c7 + c8A(Y)*

The function t

Lemma 3.11

t

S b while ( t , T )

[0 5 T 5 t 5 b]

-+

+

Rm

(-,t,T)

J

SSF(=,t,S)

E

2

C (E ) for 0 5 Y and (t,-c)+ "m Rt(-,t,.r) :

E,, a r e continuous.

A

+

"m R

A

+

t

Co(Ey), "m

-+

-,t,T)

E

Similarly 5 ^m S (*,t,S)

E

Am S ( * , t , <E)

2 C (Ey),

C 1( E ) f o r 0

Co(E ), and 5 + R ( - , t , S ) E Y Y 1 s m ( * , t , S ) E C (Ey) and ( t L ) + q ( - , t , S )

E

-+

A

E

S

5

(t,S) r;

Co(E Y ) for

We w i l l i n d i c a t e the proof f o r some sample cases

and the rest follows i n a s i m i l a r manner. 23

To show (t,.c)+

+

t i b

0 < C 0 5 5 5 t 5 b. Proof:

T

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

"m Rt(-,t,.r) continuous w i t h values i n EY 2 while t ^R"(*,t,-r) E C (E ) f o r i n s t a n c e l e t y E KC Wn and Y enclose K i n t h e i n t e r i o r o f a compact p o l y d i s c K C Cn. If a = a a1 and l e t (al, an) w r i t e DO" = (a/ayl) (a/ayn) Y r = aK be t h e d i s t i n g u i s h e d boundary o f K. We have a Cauchy

$(.,t,.-)

and (t,-r)

-+

..

-+

. ....

... . .,

i n t e g r a l formula ( c f . Gunning-Rossi [l]) which leads t o

- Pk,tO,TO)l

P(<,t,T)

Irak!

=o"J, '

nd
ak+l

(Lk-yk)

By u n i f o r m c o n t i n u i t y i n (c,t,-r) (t,T)

-+

(to,To)

"m then R

(<,t,T)

-f

..

on K x [0 5

T

5 t 5 b] i f

"m R (c,to,-cCo)

u n i f o r m l y on

r

and

a"m a% consequently D R (y,t,-c) D R (y,to,To) u n i f o r m l y f o r y E K. I Y Y f o l l o w s t h a t (t,.r) im(*,t,T) E Co(E ). A s i m i l a r argument car Y "m o b v i o u s l y be a p p l i e d t o Rt etc. For d i f f e r e n t i a b i l i t y i n t o f -+

-+

^Rm(*,t,-r)

In

[R (*,t,T)

E

m/ A t = E f o r example one wants t o show t h a t AR Y "m - ?(-,to,T)]/At -+ Rt(-,to,-r) i n EY as A t = t to

-

-+

T h i s means t h a t we must prove t h a t D a ( A i m / A t ) Daim(y,to,r) Y t Y uniformly f o r y E K T h i s may be w r i t t e n (assuming t > tc Rn -+

.

f o r convenience)

B y t by t h e c o n t i n u i t y o f

5

^R"(*,~,T) t 5

Y E K, and hence t

-+

"m R (.,t r )

E E

24

for 1

we can make

IE-t0l

C (Ey).

s(E),

u n i f o r m l y fot

A s i m i l a r argument

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

"m may be a p p l i e d t o A R t / A t

For 0 5

L e m a 3.12 pointwise o r i n E

Y

of argument Q(*,t,T)

E

C

2

In Rtt,

T

5 t 5 b t h e formula (3.22)

etc.

QED

A

A

with f

E

Co(E ) where t

A

(Ev) w h i l e i n i t i a l c o n d i t i o n s

'

@(*,T,T)

J

A

-+

A

= T(*)

E

A

E and Q t ( = , ~ , ~ =)

0 are stipulated. Y s a r i l y unique b y (3.20).

Proof:

holds

i s r e p l a c e d by any numerical f u n c t i o n Q

if

s a t i s f y i n g (3.21)

(y,t,T)

A

-

P o i n t w i s e f o r (y,t,.r)

Such a f u n c t i o n $I i s neces-

f i x e d everything i s t r i v i a l

"m "m since o u r lemmas concerning p r o p e r t i e s o f R and S make a l l

-

c a l c u l a t i o n s i n (3.18) can l e t 3.8,

T +

(3.20)

legitimate f o r

T

> 0 and one

0 i n t h e r e s u l t i n g formulas ( c f . Lemnas 3.5,

1 "m

3.9 and n o t e t h a t terms - S (y,t,c)@(y,€,,T)

5

h

s i n c e @ s a t i s f i e s (3.21)

with

a r e continuous

l A

To work

@(Y,~,T) continuous).

5

3.6,

A

4

i n E we r e c a l l t h a t (S,T) + ST : x E E i s separately Y Y Y continuous w i t h E a Frechet space and hence t h i s map i s conY tinuous (see Bourbaki [2] and f o r v e c t o r valued i n t e g r a t i o n see Bourbaki [3; 41, C a r r o l l [14]). (3.18)

-

(3.20)

Lemma 3.13

Hence a l l of t h e o p e r a t i o n s i n

make sense i n E Y'

and (3.22) For 0 5

T

+

TS;(*,t,r)

C 1 (OM) w h i l e , f o r 0 <

T

S t 5 b,

Proof:

OED

5 t I. b we have (t,T)

"m Co(OM), ( t , T ) Rt(*,t,T) Em(*,t,.r) E Co(OM), T +

-+

E

Co(OM),

t E

T +

"m

+

R

E

(*,t,T)

Co(OM), Am S (*,t,.c)

"m R (-,t,-c) E

+

and E

T

2 C (OM),

-+

+

.1 R11( - , t , T )

C 1(OM).

Again we w i l l prove t h i s f o r some sample cases t o

25

E

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

i n d i c a t e a procedure which can then be a p p l i e d i n general.

t h e topology o f OM coincides

r e c a l l t h a t on bounded s e t s i n 0,

w i t h t h a t induced by E and t h a t i f t with t

[2;

-+

~ ) ~ ( - , t E) Co(OH) then t

We

+

+.

$(&,t)

+(*,t)

E

E

Co(OM)fl C1(E)

C 1 (OM) ( c f . Schwartz

I n order t o study Da^Rm f o r example Y i n (3.4) and d i f f e r e n t i a t e i n y t o o b t a i n

3; 41 and Lemma 3.3). by

we replace

im

where t h e P ( y ) a r e polynomials and Y

1.1

=

1 "k'

The i n i t i a l

c o n d i t i o n s a r e D"im(y,r,~) = D a ? ( y , ~ , ~ ) = 0 and by Lemma 3.12 Y Y t t h e unique s o l u t i o n o f (3.47) i s given by (3.22) i n the form

where the DYkm may be regarded as known f u n c t i o n s (by an i n Y d u c t i o n procedure). Now, r e f e r r i n g t o Lemmas 3.7 and 3.10, we know Is^m(y,t,S)12 IyI 5

1.1

-

5 c5 and by i n d u c t i o n t h e Dykm(y,E,r)

for Y 1 w i l l be bounded by polynomials i n y independent

(E,T).

Consequently a l l Da?(y,t,r) w i l l be bounded by p o l y Y nomials i n y depending o n l y on a so t h a t i m ( - , t , T ) i s bounded i n

of

OM f o r 0 (t,.c)

+.

5 T S t 5 b and hence from Lemma 3.11 we can say t h a t

"m R (-,t,.r)

E

Co(OM).

F u r t h e r i f one d i f f e r e n t i a t e s (3.48)

i n t there results

26

1.

for

T

SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE

5 c7 + c8A(y) f o r 0 5

> 0, w i t h I?$y,t,&)I

T 5

t

s by

from which f o l l o w s t h e d e s i r e d polynomial bound f o r D"@ s i n c e Y t one can l e t T + 0 i n t h e r e s u l t i n g i n e q u a l i t y f o r Y t F i n a l l y we have from (3.49) f o r T > 0

"m s i n c e St(y,t,t)

= 1 w h i l e Stt

"m But Stt

-

0.

"m

= -A(y)@

i s continuous i n (y,t,.c) f o r

T

>

2m+l "m -S and we need o n l y check t t

I

Again t h e r e s u l t i n g e s t i m a t e on ID"^Rm will Y tt Hence D"^Rm can be bounded by polynomials i n Y tt (t,-c) f o r 0 5 T 5 t -< b and consequently t +

h o l d as -c

+

0.

y independent o f

"m R (*,t,.c)

The r e s t f o l l o w s i n a s i m i l a r manner.

E

2 C (OM). QED

We can now prove uniqueness (and w e l l posedness) f o r t h e s o l u t i o n wm(t) = Rm(t)

*

T o f (3.2)

-

(3.3) g i v e n by Theorem 3.4.

As w i l l be seen l a t e r t h e r e a r e q u i c k e r ways t o prove uniqueness when one i s working i n a H i l b e r t o r Banach space b u t i n o r d e r t o study t h e growth o f s o l u t i o n s f o r example i t i s a b s o l u t e l y necessary t o work i n " b i g " spaces such as S

I

I

o r D and f o r t h i s

a more e l a b o r a t e uniqueness technique i s r e q u i r e d , such as we have developed.

I t i s s u f f i c i e n t now t o deal w i t h t h e s o l u t i o n

27

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

= 8(y,t,r)T

p(t,T)

form (3.5)

OM

x S

I

-+

o f (3.4) (3.20)

A

o f (3.4) w i t h i n i t i a l c o n d i t i o n s o f t h e

given a t t = S

I

We r e c a l l t h a t t h e map (.S,T)

T.

i s hypocontinuous and then, g i v e n any s o l u t i o n

C),

( w i t h t replaced by

m u l t i p l y by ?(y,t,<)

t h a t (3.22)

holds i n S

I

.

2

-1/2 and t h a t A(y) = F(Ax6) s a t i s -

2m+l m wm + -wt + Axwm = f tt t

with

E

Co(Sl) and i n i t i a l data wm(,)

( 0 5 T 5 t 5 b) i s given by

wm(t,r)

=

D e f i n i t i o n 3.15 (3.53)

with f(-)

E

Rm(*,t,T)

*

T +

1

t

= T

in S

E

S I and w ' l ] ( ~ )= 0

*

Sm(-,t,g)

T

f ( g ) dg

F o l l o w i n g Schwartz [3] we w i l l say t h a t

C o ( S 1 ) , wm(.)

= T

E

S',

m and w,(T)

u n i f o r m l y w e l l posed i f t h e s o l u t i o n (3.53) I

(3.21)

Hence we have proved

(3.52)

(3.53)

$ of

Then t h e unique s o l u t i o n o f t h e equation

f i e s C o n d i t i o n 3.1.

f(2)

as i n

i s n e c e s s a r i l y o f t h e form $(y,t,T)T.

Assume m

Theorem 3.14

"m

M

A

t o conclude t h a t

The same uniqueness argument works f o r any s o l u t i o n SO

ST :

-+

= 0

is

depends c o n t i n u o u s l y

on (t,-c,T,f). Theorem 3.16

wm(-r) = T

E

S',

The problem (3.52)

and

W;(T)

with f ( * )

E

Co(Sl),

= 0 i s u n i f o r m l y w e l l posed i n S I

w i t h s o l u t i o n (3.53). Proof:

T h i s f o l l o w s by hypocontinuity.

28

QED

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Remark 3.17 T

The formulas o f Lemma 3.3 can be composed with

I

A

E

S and a f t e r an inverse Fourier transformation we have

t

(3.54)

wmt ( t ) =

-

(3.55)

wp)

+w 2m m-1 ( t ) - w m ( t ) l

=

These recursion formulas a r e generalizations of formulas f i r s t developed by Weinstein [ l ; 3; 4; 6 ; 8; 9; 10; 11; 16; 181 when Ax = -A

They were systematically exploited by Carroll [2; 51

X’

i n a general existence-uniqueness theory and i t turns out t h a t

they have a group t h e o r e t i c significance (see Chapter 2 ) . 1.4

Connection formulas and p r o p e r t i e s of solutions f o r EPD I

equations i n D and S Ax = -Ax w i t h A(y)

given by (3.6).

=

I

.

Let us return now t o the case when

lyI2 =

1y i

and look a t the resolvant i m ( y , t )

We denote by s u p p S the support o f a d i s t r i b u -

tion S. Lemma 4.1

“m

The resolvant R ( y , t ) = 2mr(m+l)z-mJm(z), z =

t l y l , of (3.6) is an entire function of y type so t h a t R m ( * , t )

E

E:

and s u p p R m ( - , t )

fixed compact set lykl 5 6 f o r 0 2 t

Proof: y

E

any is

The o r d e r

p

S

E

Cn of exponential

i s contained i n a

b.

of an entire function g(y) =

given by (cf. F u k s [ l ] )

29

1 aaya,

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where la/ =

1 ak and

i f p(g) = 1 w i t h

where

llyllC = 1 l y k l C then

5

Now l o o k i n g a t @(y,t)

y.

g i s s a i d t o be o f exponential type =

1 aZkzzk as

a f u n c t i o n o f one

v a r i a b l e z = t ( y l we have by S t i r l i n g ’ s formula ( c f . Titchmarsh [l])

t h a t p = 1.

“m This means t h a t I R ( y , t ) l

im as a f u n c t i o n

> 0 where B i s the type o f 2 1/2 lYlC = Yk) IC (1 lYk1E)1/2

any

5 6 exp(B+E)lzlC f o r

E

I Bt 5

5

ZlYklC =

Since

o f z.

llyllc

One

has then

Bb = B

The conclusion o f t h e Lemma now f o l l o w s from t h e Paley-WienerSchwartz theorem Theorem 4.2

see Schwartz [l3) I f Ax = - A x ,

then wm(t) = Rm(-,t)

*

2

QED

-1/2

E

E I

DI.

-

(3.3)

E

in

D

I

D’

I

D on (t,T).

By Lemma 4.1 supp Rm(-,t) i s contained i n a f i x e d

compact s e t f o r 0 2 t 5 b so t h a t Rm(.,t)

T

0 5 t 5 b, and T

T i s a s o l u t i o n o f (3.2)

depending continuously i n Proof:

m

.

*

T makes sense f o r

Furthermore t h e s e t {Rm(*,t); 0 i t 6 b} i s bounded i n

(cf. Ehrenpreis [3],

[l]t h a t t h e map (S,T)

Schwartz [ l ] )

*

and one knows by Schwartz

D’ i s hypocontinuous.

T : El

x

DI

To check t h e d i f f e r e n t i a b i l i t y o f t

-+

Rm(.,t)

-+

S

30

-f

i n E l we r e c a l l

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

t h a t on bounded s e t s i n E

I

t h e topology c o i n c i d e s w i t h t h a t i n -

I

duced by Oc ( c f . Schwartz [l], p. 274, where t h e F o u r i e r t r a n s form i s shown t o e s t a b l i s h a t o p o l o g i c a l isomorphism between E

I

M and n o t e t h a t on s e t s B C Exp OM o f bounded exponen-

and Exp 0

t i a l t y p e t h e topology i s t h a t induced by OM).

The r e s u l t f o l -

QED

lows immediately.

For uniqueness i n Theorem 4.2 we can go t o a d i r e c t express i o n f o r "R"(y,t,r)

d e r i v e d i n C a r r o l l [2;

51 and then f o l l o w

Thus one can w r i t e f o r m

t h e procedure i n v o l v i n g (3.20).

2 -1/2

n o t an i n t e g e r

where t = tm,z1 =

TM, and

y,

=

r(m+l)r(-m)/Z,

while f o r

m 2 -1/2 an i n t e g e r we have

where Nm denotes t h e Neumann f u n c t i o n ( c f . Courant-Hi1 b e r t [Z], Watson [l]).Some a n a l y s i s o f t h e Bessel and Neumann f u n c t i o n s , which we o m i t here (see C a r r o l l [2; L e m a 4.3

Wnen A(y) = I y I

2

31

51 f o r d e t a i l s ) , leads t o

^m t h e r e s o l v a n t s R (y,t,T)

are

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

e n t i r e f u n c t i o n s of bounded e x p o n e n t i a l t y p e f o r 0 5 T 5 t 5 b w i t h {Rm(*,t,=)}

a bounded s e t i n E l .

i s bounded f o r 0 5

6 5

Lemmas 3.5,

we can say t h a t given A(y) = IyIL,

C E l i s bounded f o r 0 5 6 5 t 5 b and c e r t a i n l y 3.6,

3.7,

3.8,

3.9,

3.10,

3.11,

and 3.12 remain v a l i d .

Lemma 3.13 holds w i t h OM r e p l a c e d b y Exp OM = FE

-

we t a k e i n v e r s e F o u r i e r transforms i n (3.18) i n g c o n v o l u t i o n formulas w i l l be v a l i d f o r T Theorem 4.4

E

E

S i m i l a r l y t h e problem (3.52)

D

I

, given

I

c OM

(3.20) D

.

I

*

The s o l u t i o n wm(t) = R m ( - , t )

(3.3) when Ax = -Ax and T

and W:(T)

El

t 5 b.

Using f o r example (3.39) {S"(*,t,S)l

S i m i l a r l y {R:(*,t,S)}C

T

the resultHence we have E

by Theorem 4.2,

w i t h f ( * )E Co(D1),

= 0 i s u n i f o r m l y w e l l posed i n D

I

and if

w"(T)

D I o f (3.2)i s unique. = T

E

D',

w i t h s o l u t i o n given

b y (3.53). Going back now t o t h e r e s o l v a n t e x p l o i t t h e Sonine i n t e g r a

In R (y,t)

i n OM o r Exp OM we

Watson formula ( c f . Rosenbloom [l],

[l]) t o o b t a i n f o r m > p Z -1/2

We n o t e i n passing t h a t f o r m-p i n t e g r a l (4.6) consequence o f t h e r e c u r s i o n r e l a t i o n (3.12)

i s also a d i r e c t (cf. C a r r o l l - S i l v e r

[15; 16; 171, S i l v e r [l])and t h i s w i l l be u t i l i z e d i n Chapter 2 where an e x p l i c i t d e r i v a t i o n i s given; analogous formulas i n 32

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

o t h e r group s i t u a t i o n s w i l l a l s o be derived.

We r e c a l l n e x t

the formula ( c f . Watson [l])

2 0 an i n t e g e r .

for p

provided m f -1, -2,

T h i s leads t o t h e formula

. . ., -p.

We observe t h a t (4.8)

g i v e s re-

s o l v a n t s f o r va ues o f m < -1/2 and i t should be noted here t h a t

zrnJ-,(z

jm(z) =

-2m+l s a t i s f i e s jm + (T)jm.+

I

jm= 0 so t h a t ( c f .

A

R-m(y,t) = 2-mr(l-m)zmJ-m(z), f o r z = t m and any

(3.6))

nonintegral m

2

A

0, s a t i s f i e s t h e equation R i F + h

( -2m+l 7)Rtm A

A

A(y)"Rm = 0 w i t h i n i t i a l data R-m(y,o)

= 1 and R-m(y,o)

t

=

+ 0.

Now

from (4.8) w i t h n o n i n t e g r a l m < -1/2 we can choose an i n t e g e r p

so t h a t m + p 2 -1/2 and express I Rn i n terms o f d e r i v a t i v e s o f A

RmP,

whose p r o p e r t i e s a r e known from S e c t i o n 1.3.

I n particular

holds i n E and a l s o i n OM o r Exp OM(cf. Lemma 3.13 and t h e Y remarks before Theorem 4.4) so we can s t a t e , r e f e r r i n g t o (3.11), (4.8)

Theorem 4.5 o f t h e form (3.6) "m o f the R f o r m

For n o n i n t e g r a l m < -1/2 r e s o l v a n t s Am R (y,t) obey (4.8)

2 -1/2.

and thus i n h e r i t t h e p r o p e r t i e s

Also (4.6)

holds i n OM o r Exp Or,,.

Taking i n v e r s e F o u r i e r transforms i n (4.6) composing t h e r e s u l t w i t h T

E

S

I

or T

obtain

33

I

E.

and (4.8)

and

D (when Ax = -Ax) we

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(3.3)

in S

I

2

For m > p

Theorem 4.6

-1/2 t h e (.unique) s o l u t i o n o f (3.2)-

I

(or D ) f o r index m i s g i v e n by

-

where wp s a t i s f i e s (3.2)

(3.3)

i n S I ( o r 0') f o r index p.

-

S i m i l a r l y i f wm+P i s t h e (unique) s o l u t i o n o f (3.2) S

I

in

D ) f o r index m + p 2 -1/2 ( p an i n t e g e r ) then

(or

-

i s a s o l u t i o n o f (3.2)

m

(3.3)

I

+

-1,

. . ., -p.

Remark 4.7

(3.3)

in S

I

(or

I

D ) f o r index m provided

Formulas o f t h e form (4.9)

discovered by Weinstein ( l o c . c i t . ) , u s i n g d i f f e r e n t methods.

I n p a r t i c u l a r Weinstein observed t h a t

w i t h i n d e x m.

(3.54)

, leads

(4.9)

involves p = n

t o (4.10)

of descent (see e.g.,

-+

w"(t)

=

Weinstein's v e r s i o n o f

1 and i s o b t a i n e d by a g e n e r a l i z e d method

Weinstein [3]).

We n o t e a l s o here t h a t

t h e r e i s no uniqueness theorem f o r s o l u t i o n s o f (3.2) when m < -1/2 s i n c e ifm = - s (s>1/2) then t s a t i s f i e s (3.2)

-f

- (3.3)

um(t) = t

2s s w (t)

f o r i n d e x m w i t h um(o) = uF(o) = 0; here ws i s

t h e s o l u t i o n o f (3.2) Remark 4.8

t-2"CI-m(t)

This fact, plus a version o f

f o r example.

-

(4.10) were f i r s t

i n a classical setting,

i f w - ~s a t i s f i e s (3.2) w i t h i n d e x -m then t s a t i s f i e s (3.2)

-

-

(3.3)

f o r index s.

The e x c e p t i o n a l cases m = -1, -2,

34

. . . have

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

been t r e a t e d by Blum [l;23, Diaz-Weinberger [Z],

[lo],

Diaz-Martin

B. Friedman [l], M a r t i n [l],and Weinstein (.loc. c i t . )

we r e f e r t o these papers f o r d e t a i l s .

and

Here f o r m = -p w i t h p 2

1 an i n t e g e r we may t a k e

h

( c f . B. Friedman [ l ] ) . (3.5)

T h i s r e s o l v a n t R-p s a t i s f i e s (3.4)

f o r i n d e x m = -p w i t h T = 1 and one may w r i t e m

(4.12)

zpNp(z) =

1

k=O

ckzZk + zZp l o g z

^R'p

Consequently t h e 2pth d e r i v a t i v e o f mic s i n g u l a r i t y a t t = 0. (4.13)

-

h

w-P(t)

= R-P(-,t)

m.."

1

k=O

ckz

2k

i n t w i l l have a l o g a r i t h -

However, w r i t i n g f o r m a l l y

*

m

T

=

1

k=O

cktZk(AxS

T! we see t h a t t h e l o g a r i t h m i c term varnishes i f A

*

T)

= 0, i n which

event w - ~w i l l depend smoothly on t. We n o t e t h a t i f one takes p = 1 i n (4.8),

Remark 4.9

m r e p l a c e d by m

-

with

1, t h e r e s u l t i s e x a c t l y (3.12).

We w i l l now discuss some growth and c o n v e x i t y theorems f o r s o l u t i o n s o f (3.2)

-

(3.3)

when Ax = -Ax.

f i r s t discovered by Weinstein [9;

Such r e s u l t s were

10; 151 i n a c l a s s i c a l s e t t i n g

and subsequently g e n e r a l i z e d and improved i n c e r t a i n d i r e c t i o n s by C a r r o l l [l;2; 51 i n a d i s t r i b u t i o n framework; we w i l l f o l l o w

35

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(Other k i n d s o f growth and c o n v e x i t y

t h e l a t t e r approach.

theorems f o r s i n g u l a r Cauchy problems were developed by C a r r o l l [18; 19; 251 and w i l l be t r e a t e d l a t e r . )

R e f e r r i n g t o (4.6),

t h e b a s i c f a c t one r e q u i r e s f o r these theorems i s t h a t Rp(*,n)

2 0 f o r some p

2 -1/2 (i.e.,

Rp(-,n)

measure), i n which case a l l Rm(-,t)

should be a p o s i t i v e

2 0 f o r m 2 p.

For general

Ax n o t t o o much i s known i n t h i s d i r e c t i o n b u t when Ax f o l l o w s from S e c t i o n 2 t h a t Rn/'-'(=,n)

=

ux(n) 2 0.

Another

case o f i n t e r e s t i n v o l v e s t h e metaharmonic o p e r a t o r Ax = -A m n where R ( - , n ) L 0 f o r m > 7 r = (1~:)"~

-

Indeed f o r m >

1.

X

-

2 a

1 and

5 t one has (see C a r r o l l [5] and c f . O s s i c i n i [ l ] )

- r ( m + l ) ( t 2 -r2 ) m-n/2 ITn/2

while

n 7 -

it

= -Ax

p(*,t)= 0

for

Irl

t2m

m

1

k=O 2

2k [a2$-r2) Ik2 0 k! r(m-$+k+l)

2 t; i n p a r t i c u l a r then R m ( * , t )

E

El.

Now, general theorems o f growth and c o n v e x i t y i n v o l v i n g ask sumptions o f t h e form (-Ax) T

;1

0 were developed i n C a r r o l l [2;

51 u s i n g t h e n o t i o n s o f value and s e c t i o n o f a d i s t r i b u t i o n i n troduced by L o j a s i e w i c z [l].T h i s can be s i m p l i f i e d somewhat i n d e a l i n g w i t h Ax = -A

X

s i n c e AT

2

0 i m p l i e s t h a t T i s an almost

subharmonic f u n c t i o n . (see Schwartz [l]) and hence T a u t o m a t i c a l l y has a "value" almost everywhere (a.e.).

We r e c a l l b r i e f l y some

n o t a t i o n s and r e s u t s o f Lojas ewicz [l].

36

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE

A d i s t r i b u t i o n T i s s a i d t o have t h e value

D e f i n i t i o n 4.10 c at x

'$r+]>

i f l i m Tx +xx = c as X c:

for $

E

-+

+

0

D with

(i.e.,

=

I

0

Q ( x ) d x = 1 ) ; when T i s a

p o s i t i v e measure t h e n o t i o n o f value c o i n c i d e s w i t h t h a t o f denOne may f i x x

i n a distribution T i f l i m Tx = x,t 0 + o f T. St E D as X -+ 0 and S t i s s a i d t o be t h e s e c t i o n T xo ,t Thus one must have -+ <S,$> when $I E Dx, S,taAn $ E Dt, and $ ( x ) d x = 1. sity.

I

0

1

I

I t i s proved i n L o j a s i e w i c z [l]t h a t i f

Remark 4.11

1X , t

i n x we may f i x x = xo and t h e s e c t i o n

i s a measure then a.e.

I

F u r t h e r i f xo may be f i x e d f o r T l x o ,t t' x,t then xo may be f i x e d f o r ( a / a t ) T and (a/at)Tx,t I x ,t = X ,t i s a measure i n D

0

a/atTx ,t' 0

For t h e f o l l o w i n g theorems see C a r r o l l [5]. Theorem 4.12 o f (3.2)

-

(3.3)

f y i n g AT

2

0.

L e t w:(t)

= Rm(*,t)

f o r Ax = -Ax w i t h m 2

Then a.e.

*

T

$-

i n x the section t

decreasing f u n c t i o n of t w i t h wm ( t ) > Tx ' xO

n

Proof:

Using (3.11) (2.5)

Let m >

n

2-

1 and s e t p -

E

D 1 be t h e s o l u t i o n X

1 and T -+

w!

A

E

D' s a t i s -

( t ) i s a non0

0

-2

1 i n (4.6) t o o b t a i n

(transformed) and composing w i t h T we have ( c f .

where Rn"(*,t)

= Ax(t))

37

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

i n two v a r i a b l e s Now we may consider w m ( t ) as a d i s t r i b u t i o n wm x,t I 1 I I 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 Et(Dx) (since Dt(Dx) = D x,t C Di(D;) - cf. Schwartz [5]) and here one may extend u x ( t ) I

.

I

and A x ( t ) as even f u n c t i o n s o f t f o r t < 0. I n Dx one has t wm(t) - T = wT(r)dr which, by (4.16) w i t h AT 2 0, i s a p o s i t i v e

I

measure f o r t

wm

-

0

L

0.

."

Writing T = T

It i n D

x

Y

-

I

X

,t

i t follows that

T w i l l be a p o s i t i v e measure i n D f o r t L 0 and hence x,t m we can f i x x = x a.e. i n x f o r w and T together (cf. Remark 0 x,t 4.11 and r e c a l l t h a t T i s an almost subharmonic f u n c t i o n ) . For X,t

m such xo, (a/at)wx ,t = ( ~ y ) by ~ Remark ~ , ~ 4.11,

and, by (4.6),

0

m > 0 f o r t 2 0. Hence t + w; ,t = w ( t ) w i l l be a t X0,t xO 0 nondecreasing f u n c t i o n f o r t 2 0 o f bounded v a r i a t i o n (cf. (W")

We conclude t h a t w: Schwartz [l]). s p e l l t h i s o u t l e t (p

E

Dx, (p

z

ro

,t Z Txo f o r t 2 0 and t o

0, J i ( x ) d x = 1 and s e t $ , ( E )

(cf. D e f i n i t i o n 4.10).

Then from (4.16)

=

the contin-

( < ) > i s nondecreasing f o r C,t' A t 2 0. Consequently f o r t > 0 f i x e d <wm $ (C) 2 C,t' A + The r e s u l t f o r m = and l e t t i n g A + 0 we o b t a i n wm 2 Tx

uous f u n c t i o n t

-

+

<wm(t),$A> = <wm

X0,t

1 i s now immediate from (2.5)

0

.

and t h e remarks above.

L e t us now combine (4.8) w i t h Lemma 3.3 when A,

= -Ax

QED

in

order t o o b t a i n a formula f o r Rm(-,t) (m 2 -1/2) i n t e r n s of

38

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

Rm+P(*,t) and Rm+p+l(-,t),

where t h e i n t e g e r p i s chosen so t h a t

2 - 1. Some s i m p l e b u t t e d i o u s c a l c u l a t i o n s y i e l d (see C a r r o l l [2; 51 f o r d e t a i l s )

m + p 2

-

(3.2)

(3.3)

-

F o r -1/2 5 m < n/2

Lemma 4.13

T

(i.e.,

= 6

i n (3.3))

1 t h e r e s o l v a n t R"(*,t)

for

given by F-l applied t o

(4.8) can be w r i t t e n Rm(-,t) = (

(4.17)

+ (

q-1

q- 1

1

k=O

1

k=O

ak(Fl)t2k(A6)k)

Bk(n)t2k(A6)k)

*

*

Rm+P(-,t)

Rm+P+l(*,t)

where p i s an i n t e g e r chosen so t h a t m + p

[q]([u]

n 2 7 - 1,

q = 1 +

denotes t h e l a r g e s t i n t e g e r i n u ) , ak(m) 2 0,

Bk(m) 2 0, and

~1

0

(m) = 1.

n F o r - 1 / 2 5 m < 7 - 1 l e t t h e i n t e g e r p be n chosen so t h a t m + p 2 7 - 1. L e t q = 1 + and assume Theorem 4.14

T

E

D ' s a t i s f i e s (A)

-

s o l u t i o n o f (3.2) section t

+

T2

0 f o r 1 F k 5 q.

(3.3)

f o r Ax = - A ( t 2 0 ) t h e n a.e.

i n x the

rn

0

for t

->

0.

xO

Proof:

We c o n v o l u t e (4.17) w i t h

f e r e n t i a t e , u s i n g (3.11). p+p+2

T t o g e t wm(t) and d i f -

m+p+l (*,t), Since Rm+P(*,t), R

and

( - , t ) a r e p o s i t i v e measures we have a g a i n t h a t w T ( t )

Hence, as i n Theorem 4.12,

i n Dx f o r t 2 0. a.e.

I f wm(t) denotes t h e

wx ( t ) i s a nondecreasing f u n c t i o n o f t w i t h

wm ( t ) 2 T xO

k

[q]

m

for w

x,t

E

D

I

x,t

and t

+

w

X0't

39

= w

0

we may f i x x = xo

m ( t ) w i l l be a xo

->

SINGULAR AND DEGENERATE CAUCHY PROBEMS

nondecreasing f u n c t i o n o f bounded v a r i a t i o n (t

*

wm(t) = Rm+P(.,t)

+ l ( * , t ) where l ( . , t )

T

(we r e c a l l here t h a t ao(m) = 1). Rm+'(=,t)

*

w!+P(t)

L Tx

0

_> 0 i n D i f o r t L 0

From Theorem 4.12 we know t h a t

T = wm+P(t) s a t i s f i e s wm+P(t) 2 T

AT 2 0 (which holds since q 0

Further

0).

xO

L 2).

Hence, a.e.

i n x when

a.e. xO

i n x, w

m xO

.

(t) L QED

C o r o l l a r y 4.15

For -1/2 5 m <

n

7 - 1 choose p as i n

k Theorem 4.14 and assume ( A ) T 5 0 f o r 1 5 k 5 q

-

1.

i n x the section t

+

wm ( t ) s a t i s -

as i n Theorem 4.14 then a.e. f i e s wm ( t ) 2 T xO

for t

>-

Given w"(t)

xO

0.

xO

Proof:

We note f i r s t t h a t p L 1 so t h a t q

-

1 =

Then f r o m (4.17) one w r i t e s again wm(t) = Rm+P(-,t)

*

I

where l ( * , t ) 2 0 i n Dx f o r t 2 0 under our hypotheses.

L e t us w r i t e now Lm =

a /at

+2n a/at; t +

1.

T + l(*,t)

As i n a.e.

the p r o o f of Theorem 4.14 i t f o l l o w s t h a t wm ( t ) L T xO

[q] L i n x.

xO

then as noted

by Weinstein, f o r m f 0, L i = (2m)2t-2(2m+1) a2/aa(t-2m)y w h i l e

2 a 2/a 2 ( l o g t ) .

f o r m = 0, L: = t-

A

X

D

I

= -A we have Lkwm =

, for

m > n/2

-

R e f e r r i n g now t o (3.2) w i t h

awm and using (4.15), composed w i t h T

1 i t follows t h a t

40

E

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

since Lt = C2LCt.

Thus i f AT

t i v e measure f o r t t 0. m <

2

-

2

0 then Lh

wm(t) i s a p o s i 1 S i m i l a r l y from f4.17) w i t h - 1/2 5

--

1 we convolute w i t h T

+

q-1

1

k=1

Bk(m)t

2k

(R

E

D:

m+p+l

t o obtain

*

(-,t)

A~+’T)

k Hence ifA T 2 0 f o r 1 5 k 5 q then L i w m ( t ) i s a p o s i t i v e meal

sure i n Dx f o r t t 0. Theorem 4.16 (3.3)

f o r Ax = -A and l e t T

while, f o r

-

-

L e t wm(t) denote t h e s o l u t i o n o f (3.2)

1/2 5 m <

-

n

E

D

I

n

s a t i s f y AT 2 0 f o r m 2

-

1

1, A kT 2 0 f o r 1 i k 5 q = 1 +

[q]where t h e i n t e g e r p i s chosen so t h a t m + p t

n

-

1.

Then f o r t > 0, a.e. i n x, the s e c t i o n t wm ( t ) i s a convex xO n 2-n n function of t f o r -1/2 <, m < 7 f o r m ,t 2 - I, o f t-2m -+

f 0), and o f l o g t f o r m

(m

-

1

= 0.

By (4.18) - (4.19) we know t h a t f o r t > 0, I Z m (a /a-c )w ( t ) L 0 i n Dx under the hypotheses given, where T = n n t2+ f o r m 2 2 I, -c = t-2m f o r -1/2 I m < 7 1 (m 01,

Proof:

2

-

and T = l o g t f o r m = 0.

-

t

Now by Theorems 4.12 and 4.14 our

hypotheses i n s u r e t h a t x = xo may be f i x e d a.e.

i n x for

m w ( t 2 0) and by D e f i n i t i o n 4.10 w i t h Remark 4.11 i t f o l l o w s x,t t h a t a2wm /a? Z 0 i n D i f o r t > 0. Hence by Schwartz [l] xo ,t QED t -+ w l ( t ) i s a convex f u n c t i o n of t. 0

41

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Remark 4.17

O f p a r t i c u l a r i n t e r e s t i n t h e EPD frameword o f

course i s t h e wave e q u a t i o n when m = -1/2 and we see t h a t i n t h e hypotheses o f C o r o l l a r y 4.15 -1/2 + p

n

2 2

-

1 means p 2

-.n-12

Note here t h a t i n dimension n = 1, -1/2 5 m < -1/2 i s impossible so t h a t we are i n t h e s i t u a t i o n o f Theorem 4.12.

[q]so t h a t ,

n-1 p = 2;

1 =

g i v e n n even, p = n/2, while, f o r n odd ( n t 3),

thus f o r n even q

[!$!These I.

-

Now q

-

1 =

[-1n4+2

-

w h i l e f o r n odd q

1 =

r e s u l t s on t h e number o f p o s i t i v e Laplacians o f T

s u f f i c i e n t f o r a minimum ( o r maximum) p r i n c i p l e improve W e i n s t e i n ' s estimates and a r e comparable w i t h r e s u l t s o f D. Sather [l;2 ; 31. n even and

N

=

Sather i n [3] f o r example sets N =

n-3 for 2

n odd w i t h a =

fornodd; w h i l e a = [%'I n even w h i l e b =

1 [nI 4

=

f o r n odd.

2 T ( x ) f o r t 2 0.

for

[?If o r

N+l s i m i l a r l y he s e t s b = [T]

f o r 1 5 k 2 a and Akw;'/2(o) w-l/'(x,t)

[N+2 I 2

9

=

n even

[,In

for

Then he r e q u i r e s t h a t A kT t 0

2 0 for 0 5 k

5 b i n order t h a t

Thus our r e s u l t s seem a t l e a s t

e q u i v a l e n t and c e r t a i n l y more concise than those o f Sather.

For

f u r t h e r i n f o r m a t i o n on maximum and minimum p r i n c i p l e s f o r Cauchy t y p e problems we r e f e r t o Protter-Weinberger [l]and t h e b i b l i o graphy there; c f . a l s o Agmon-Nirenberg-Protter P r o t t e r [7],

[l],Bers [l],

Weinberger [l]. For growth and c o n v e x i t y p r o p e r t i e s

o f s o l u t i o n s o f p a r a b o l i c equations w i t h subharmonic i n i t i a l data see Pucci-Weinstein [l]. Remark 4.18

The e x i s t e n c e o f p o s i t i v e elementary s o l u t i o n s

( o r r e s o l v a n t s ) f o r second o r d e r h y p e r b o l i c equations i s i n 42

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

general somewhat n o n t r i v i a l and we mention i n t h i s regard D u f f

[l;21, i n a d d i t i o n t o C a r r o l l [2],

where t h e problem i s d i s -

cussed. Spectral techniques and energy methods i n H i l b e r t

1.5

F r s t we w 11 present some t y p i c a l theorems o f Lions f o r

space.

EPD t y p e equations, u s i n g "energy" methods, which were d i s cussed i n C a r r o l l [8].

Lions proved these r e s u l t s i n 1958 i n

l e c t u r e s a t t h e U n i v e r s i t y o f Maryland (unpublished).

There-

a f t e r f o l l o w some theorems i n H i l b e r t space based on a s p e c t r a l technique developed by C a r r o l l [4; 6; 8; 10; 111 (suggested by Lions as a v a r i a t i o n on C a r r o l l ' s F o u r i e r technique).

The two

approaches (energy and s p e c t r a l ) a r e n o t d i r e c t l y connected and l e a d t o weak and s t r o n g s o l u t i o n s r e s p e c t i v e l y . Thus, r e g a r d i n g energy methods ( c f . C a r r o l l [14] and Lions [5] f o r a more complete e x p o s i t i o n ) , l e t (u,v)

V

x

V

+.

B

a(t,u,v)

-+

:

be a f a m i l y o f continuous s e s q u i l i n e a r forms, V c

H,

where V and H a r e H i l b e r t spaces w i t h V dense i n H having a f i n e r topology. a(t,u,v)

D(A(t)) H).

One can d e f i n e a l i n e a r o p e r a t o r A ( t ) such t h a t

= (A(t)u,v),,

if V

+.

Assume t

a(t,u,v)

+.

a(t,u,v)

V (i.e.,

for u

E

D ( A ( t ) ) C V and

:V

-+

B i s continuous i n t h e topology o f

E.

1 C [o,b]

w i t h a(t,u,v)

V E

= a(t,v,u)

u

and

consider Problem 5.1

t

+.

A(t)u(t)

Find u ( - )

E

Co(H) on [o,b],

C2(H) on [o,b], and

43

u(t)

E

E

D(A(t)),

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where k t

-+

E

a,

q(-)

0, and f

E

D(A(t)) f o r t

Co(o,b]

E

i s r e a l valued w i t h q ( t )

Co(H) on [o,b].

03

as

Note here t h a t i f u(o) = T

E

then v i t ) = u ( t ) - T s a t i s f i e s (5.1)

[o,b]

E

-+

w i t h f replaced b y f ( * ) - A ( - ) T (assuming t h e l a t t e r i s continuous on [o,b]

- which

w i t h values i n H

holds i f A ( t ) = A).

T h i s can be transformed i n t o a weak problem (which i s more

d = t r a c t a b l e ) as f o l l o w s (= Problem 5.2 2 L (H) on [o,b],

Find u

).

L2(V) on [o,b],

+kq(u',h)"

0

for a l l h

E

u ( o ) = 0, u l f i

E

such t h a t g i v e n f/GE: L2(H)

jbla(t,u,h)

(5.2)

E

'

L2(V) w i t h hJii

Theorem 5.3

Let t

-+

E

-

I

1

( u ,h )H

L2(H), hl/J;i

a(t,u,v)

E

E

-

(f,h)Hldt = 0

L2(H), and h(b) = 0.

C 1 [o,b],

a(t,u,v)

= a(t,v,u)

Co(o,bl, q > 0, q ( t ) -+ as t + 0, Re k > 0, and a(t,u,u) 2 2 c l l l u l l v f o r a > 0. Then t h e r e e x i s t s a s o l u t i o n o f Problem 5.2. q

E

03

Proof:

Set h = e-Yt+l

I

f o r y r e a l ( t o be determined) and

b

(5.3)

Ey(u,+)

Let

=

0

1

{a(t,u,e-Yt+l)

+

k q ( t ) ( u l ,e-Yt+l)H

-

( u ,(e Y t + l ) l ) H l d t

F be t h e space o f f u n c t i o n s u

E

L2(V), u l G

2 u(o) = 0, w h i l e G i s t h e space o f 4 E L ( V ) , I I 1 2 2 9 4 E L (H), 9 / 4 E L (H), $(o) = 0, and

44

+' E + (b)

E

L2 (H),

2 L (V),

I

= 0.

Clearly

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

G CF and we p u t on G t h e induced topology o f F d e f i n e d by

,f)(

IIull;

+ q l l u ' l ! i ) d t = IIuIIF 2 so t h a t G i s a p r e h i l b e r t space.

0

Then i f u

E

F s a t i s f i e s E (u,$) =

Y

/

b

( f , e Yt$')Hdt f o r a l l $

G i t f o l l o w s t h a t u i s a s o l u t i o n o f Problem 5 . 2 .

i s a s e s q u i l i n e a r form on F x G w i t h u tinuous f o r 4

G fixed.

E

-

+ 2a(t,$,e-Yt$') (5.4)

b

(5.5)

yt$)

t

4

con-

= at(t,$,e-Yt$)

+Y

at(t,$,$)e-Ytdt

I

b

[-e-yt

0

=

y

i ( $ ,(e-Yt$')l)Hdt

0

=

: F.+

E (u,$)

= a(b,$(b),$(b))e-yb

0

b

-2Re

Now Ey(u,$)

so t h a t ,

ya(t,$,e-Yt$)

I

-

-f

Further, a(t,$,e

2Re I:a(t,$,e-Yt$')dt

E

0

&11$11: b

11$'(0)11

Hence, f o r y > c/a,

+ 2ye-ytII$'lli]dt

+

IEy($,$)I

0

e-ytII

2 811$11;

$

'

2

I] H

dt

'

( n o t e t h e second term

i n 2Re E ( $ & ) ) and by t h e Lions p r o j e c t i o n theorem ( c f .

Y

C a r r o l l [14], u

E

Lions [5],

Mazumdar [l;2 ; 3; 41) t h e r e e x i s t s

F s a t i s f y i n g E (u,$) =

Y

f(f,e-yt$')Hdt

(f,e-Yt$')Hdt,

-f

i s a continuous s e m i l i n e a r form on G.

0

Remark 5.4

since $

QED

For an i n t e r e s t i n g g e n e r a l i z a t i o n of t h e Lions

45

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

p r o j e c t i o n theorem and a p p l i c a t i o n s see Mazumdar [l;2; 3; 41. We not e t h a t i n Problem 5.2 q c o u l d become i n f i n i t e a t o t h e r p o i n t s on [o,b]

p ro v i d e d t h e "weight" f u n c t i o n s h a r e s u i t a b l y t h e problem can be solved d i r e c t l y as

chosen; i f q > 0 b u t q i n Lions [5].

1 Theorem 5.5 L e t t + a(t,u,v) E C [o,b], a(t,u,v) = 1 a(t,v,u), q E C (o,b],q > 0, q ( t ) + - as t + 0, Re k > 0, and 2 a(t,u,u) 2 a l l u l l v f o r a > 0. Then a s o l u t i o n o f Problem 5.2 i s unique. Proof:

L e t h be g i v e n by h ( t ) = 0 f o r t

= u on [o,s);

E +

0, l i m a(E,h(E),h(E))

Taking r e a l p a r t s i n (5.2),

+ I

where a(t,h,h)

I

s with h

I

-

kqh

then some c a l c u l a t i o n shows t h a t h i s admissable

i n Problem 5.2 and, as

= 2Re a(t,h,h

(2aq Re k

0

and s i n c e q ( t ) t h a t 2aq Re k u = 0 i n [o,so].

-+

-

(4

as t

-

z

2 0.

= 0 I

) + at(t,h,h).

Consequently

2 c)llhllVdt 5 0

+

0 one may choose so small enough so

which w i l l i m p l y t h a t h = 0 and

c > 0 i n [o,so]

I n t h i s step o n l y q

extend t h e r e s u l t u = 0 t o [so,b] methods ( c f . Li o n s [5])

= 8

w i t h f = 0, one o b t a i n s w i t h t h i s h,

j o at(t,h.h)dt

S

(5.7)

2

where q

E

Co(o.so]

i s needed; t o

one can r e s o r t t o standard E

46

C'[s,,bl

i s used.

QED

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Remark 5.6

There a r e a l s o versions of Theorems 5.3 and 5.5 2

2

due t o Lions when a ( t , v , v ) + Al1vllH 2 cill-vllv (cf. Carroll [81) b u t we will omit the d e t a i l s here.

We note t h a t no r e s t r i c t i o n s

have been placed on the growth of q ( t ) as t * 0; however the solution given by Theorem 5.3 i s a weak solution and not necess a r i l y a solution of (5.1).

We consider now the operator

for 0 5 as

T

-+

T

5 t 5 b <

0, B

E

my

Co[o,b],

where

ci E

and y

E

Co(o,b] w i t h

:I

Re a ( < ) d<

-+to

Let A be a s e l f a d j o i n t

Co[o,b].

(densely defined) operator in a separable Hilbert space H w i t h ( A u , u ) ~ + 6 1 I u l l i 2 0 and consider f o r Q

Problem 5.7

w(t) (5.9)

E

D ( A ) , and t

Find w -+

E

Aw(t)

E

Co[o,b]

2 C ( H ) on [ ~ , b ] ( 0 5 E

T

-5

t

b <

S

m),

C o ( H ) on [ ~ , b ] , such t h a t

( L + Q ( t ) A ) w = 0;

W(T) =

T

H;

E

Wt(~) =

0

I t i s t o be noted t h a t one may always assume A 1 0 since a

-

.v

change of variables A = A + 6 and y = y

-

SQ can be made.

Now

from the von Neumann spectral decomposition theorem ( c f . Dixmier [ l ] , Carroll [14]) i t follows t h a t t h e r e i s a measure v , a vmeasurable family of Hilbert sapces X an isometric isomorphism 0 : H

-+

ff =

I" -+

H ( X ) ( c f . Chapter 3 ) , and

H(A)dv(A), such t h a t A

i s transformed i n t o the diagonalizable operator of multiplication

47

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

-

IIBT(A)l[H(A)) and T

[

<

A2118Tll$A)dv

E

D ( A ) - i f and o n l y i f i n a d d i t i o n

where BT i s t h e image of T

E

H i n H ; we

w r i t e here BT f o r t h e f a m i l y (BT)(A) determined up t o a s e t o f v-measure zero.

For v a r i o u s o t h e r a p p l i c a t i o n s t o d i f f e r e n t i a l

equations see e.g., Lions [6],

Brauer [l],G i r d i n g [l],

B e r e z a n s k i j [2],

Maurin [l;2; 31.

Now by use o f t h e map 8 (5.9)

is

transformed i n t o t h e e q u i v a l e n t problem (L + AQ)Bw = 0;

(5.10) with t

-+

Bw(t)

E

ew(-c) = BT

C2(H) and t

BAw = BAB-'Bw = ABw).

-+

ABw(t)

E

E

H;

= 0

Bwt(-r)

Co(H) on [o,b]

(note t h a t

T h i s e s s e n t i a l l y reduces Problem 5.7 t o a

numerical problem, as i n t h e case o f t h e F o u r i e r transform, and we w i l l c o n s t r u c t s u i t a b l e " r e s o l v a n t s " Z(X,t,-c) ^m analogous t o t h e R (y,t,-c)

%l

and S (y,t,-c)

and Y(A,t,-c)

respectively o f Section

3 so t h a t f o r example ( L + AQ)Z = 0;

(5.11)

Z(A,T,T)

= 1;

= 0

Zt(A,-c,-r)

i n which event Bw = Z(A,t,-r)@T w i l l be a s o l u t i o n o f (5.10). Many o f t h e techniques o f Section 3 w i l l be employed and t h i s

w i l l enable us t o shorten somewhat t h e e x p o s i t i o n here.

exP and

L e t us assume Re a ?: 0 f o r 0 < t b (a(<)+ B(S))dS) (cf. (3.23)). t

(-1

48

<_

s and c o n s i d e r P ( t ) =

Then P ( t )

-+

0 as t

-+

0

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

55 u

for 0 5

( I n t h e terminology o f F e l l e r [2] t = 0 i s

5 b.

an entrance boundary.)

+

(PZt)t

(5.11) (5.13)

As i n (3.23)

-

= 1

Z(A,t,T)

/llP(5)/P(o)ldgdo

F(t,-r)

5 F(t,o)

of (5.13)

-

rt

1

1, rn 1, P (5)[Y (51 1'5

+AQ

( c f . (3.27))

-

y~

Setting F(t,r)

we have F(t,-c) <

h

=

(51I Z ( A ,5 1

Z.

J

F ( t ) 5 F ( t ) S F(b) f o r t 5 t 5 b.

The s o l u t i o n .

m

=

=

with

03

h

i s g i v e n again f o r m a l l y b y Z(A,t,T)

( c f . (3.25))

we o b t a i n from

P(y+AQ)Z = 0 and

which we w r i t e again i n t h e form Z = 1

:1

(3.24)

.

1 (-l)kJk 1

k =n

and one may proceed as i n S e c t i o n 113-to prove m

Lemma 5.8

Given Re

CL

2 0 on (o,s]

converges a b s o l u t e l y and u n i f o r m l y on (0 i

$ compact ( A of

r),

T

i s continuous i n (A,t,.c)

Proof: t i a t i n g (5.13)

and a n a l y t i c i n A.

The statements f o r Zt(A,t,-r) i n t ( c f . Lemma 3.5).

Sim

f o l l o w upon d i f f e r e n -

We n o t e a l s o t h a t i f

QED

S e c t i o n 3. NOW, as i n (3.30), a(t)Zt(A,t,o)

1

and represents a continuous f u n c t i o n

a n a l y t i c i n A , which s a t i s f i e s (5. 3).

(A,t,T),

Zt(A,t,-c)

E

1 (-l)kJk k=O t i b l x

the series

as t

-+

0.

we must examine t h e behavior o f Thus ( c f . (3.31))

49

we s e t

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

a ( t ) jt(P(<)/P(t))df T

now t h a t a

[o,sl, and

C (o,b]

E

H(t,T)

=

1

'

2 = a /a

E

0 then c l e a r l y ( c f . (5.13)

-+

L e t us assume

la1 2 kRe a on

Co[o,sl,

5 N f o r 0 5 5 5 t 5 s.

a(t)/a(S)I

H(o) e x i s t s as t

0

with

w i t h H ( t ) = H(t,o).

If l i m H(t) =

upon d i f f e r e n t i a -

t i o n ) l i m a t ) Z (X,t,o) = -H(o)[y(o) + AQ(o)]. Now c o n s i d e r tt dn) d5 = H(t,T) w i t h H ( t ) = H(t,o). Since JT

(-jga(n)

A

A

A

-

H ( t ) l can be made

a r b i t r a r i l y small f o r t s u f f i c i e n t l y small.

Indeed i f B(t,S)

B

Co[o,bl

E

(-1 5 rt

exp

A

i t i s e a s i l y seen t h a t I H ( t )

B(q) dn) ther;,

by c o n t i n u i t y , g i v e n

- 1I 5

such t h a t l B ( t , 5 )

E

E

=

there e x i s t s 6 ( ~ )

f o r t 5 ti(&). Hence f o r t 5

min(s,b(&)) one has

1

-15Rea(n)dn t

t

lG(t)-H(t)l 5

(5.14)

la(t)l

-

B(t,S)l

e

dS

-lSRea(n t )dn

t

5 EkN

11

0

dS = skN

Rea(5)e 0

A

To f i n d now H(o)

A

=

l i m H ( t ) as t

-+

A

of

H(t,T)

by p a r t s t o o b t a i n

Letting T

-+

0 we have

50

0 we i n t e g r a t e t h e d e f i n i t i o n

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPDTYPE

1.

so (5.16)

t

i

t But H(.t) = f(t,S)dS t 0 can be w r i t t e n as h ( t ) 5 1 = f(t,<)(l-@(S))dS. Evi-

where f(t,S)

h

= a ( t ) exp ( - / c r ( o ) d n ) .

10

A

d e n t l y h ( * ) i s continuous as t

-+

0 and choosing t smal

enough 4) 1

n

Hence ( 1 - @ ( o ) ) H ( t ) h

l i m H(t)

1 as t

-+

0 and i f @ ( o ) f 1 t h i s means

H ( t ) = H(o) = l / ( l - @ ( o ) ) as t

= l i m

follows t h a t t

-+

-+

w.

a(t)Zt(A,t,o)

l i m a(t)Zt(A,t,o)

=

0 from which

-+

i s continuous i n t as t

-+

0 with

I t may then a l s o be shown

f o r T >_ 0

i s continuous i n (A,t)

e a s i l y t h a t a(t)Zt(A,t,r)

f i x e d and a n a l y t i c i n A f o r ( t , r )

f i x e d (0 5 T 5 t 5 b); s i m i by (5.11).

l a r p r o p e r t i e s a r e then t r a n s p o r t e d t o Ztt(A,t,-c) Consequently Lemma 5.9

L e t ct

la1 2 kRea on [o,s], Then Z(A,-,T)

E

E

' 2 with 6 = a /a

E

and s a t i s f i e s (5.11)

and Ztt(A,t,r)

0

C [o,s],

<5

and l a ( t ) / a ( < ) l 5 N f o r 0 5

2 C [o,bl

w h i l e ct(t)Zt(X,t,r)

1 C (o,b]

for 0 5

t 5 s.

5

2 t 5 b

a r e continuous i n (A,t)

and a n a l y t i c i n A. Proof:

I t remains t o prove t h a t @ ( o )

t h e contrary. I R e @ ( t ) - 11

Then Re @

5

E

-+

1 and I m @

f o r t 5 B(E).

Now

51

-+

1

1 and we suppose

0 and we can make

-

1

-

-

-i

t

T

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

$(S)dS ( z < t ) w i t h lim r t

Hence l / a ( t ) =

= 0 as T

@(S)dSand f o r t

5

+

0 since Rea(-c) +

rnin(s,d(E)) and

E

m.

< 1 we

-10

can write

which v i o l a t e s the f a c t t h a t Rea(t) > 0 f o r t small.

QED

We can now derive some bounds f o r the resolvant Z a s i n Section 3 ( c f . (3.32)

-

(3.33) and Lemna 3 . 7 ) .

Here one mul-

t i p l i e s (PZ5)5 + P(y+hQ)Z = 0 by P - ' ( g ) T 5 ( h , 5 , ~ ) and i n t e g r a t e s from

T

t o t t o obtain

+

t

(y+hQ)ZF5dS = 0

T

Assume now t h a t Q

E

1 C [o,b] w i t h Q real and 0 < q 5 Q ( t ) .

Adding (5.19) t o i t s complex conjugate and noting t h a t d d 2 = - I Z [ we have 2Re(Z 7 ) = - I Z 1' with 2Re

5 55

(5.20)

d5

Zz5

5

IZt(h,t,e)12

d5

+ 2

-

I t i s convenient now t o write a + B = a + a where a1 = a on 1 2 . " (o,s], s < s , a1 = 0 on [s,b], and a1 = a [ ( ~ - t ) / ( s - s ) ]on [s,s]. Y

Then a1

E

Co(o,b] w i t h :[all 5 klReal on (o,s] and l a l ( t ) / a l ( 5 ) l 5

52

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

Then from (5.20) we obtain

N~ f o r 0 5 5 5 t 5 b.

I d5 + IZ 1') 5 c1

=

2 s u p IRea21, c 2

=

sup Iyl, and c3 =

SUP

IQ'I

and set

(on Co,bl).

I t may be assumed t h a t A 2 l / q s i n c e t h i s can always be assured by adding i f necessary a constant multiple of Q t o y (e.g., set A' = A

A'

+ 1 / q aed yl

2 l/q).

= y

-

($)a

so t h a t y l + X ' Q

=

y + AQ with

Then from (5.21) we have

+ (c2+c It +

where c

4

=

c4JTI

c1 + c2.

Now l e t us s t a t e a version of Gronwall's

l e n a which w i l l be useful ( s e e Carroll [14] o r Sansone-Conti [ l ] f o r proof).

Lemma 5.10 continuous.

Let u

E

Co, R

E

L 1 w i t h R L 0, and

Assume u ( t ) 5 + ( t )+

53

t

t

R(o)u(q)dn.

+ absolutely

Then

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Applying Lemma 5.10 t o (5.22) w i t h R = c4, t = + X Q ( t ) l z l 2 , and $ ( t ) = XQ(.r) + (IC3+c2)j:l;12dS

IZtI2

T~

u(t) = it fol-

lows t h a t (5.24)

IZt12

+

X Q ( t ) l Z I 2 <_ hQ(.r) exp c4(t--r)

where c5 = exp c4b.

I n particular

1 5 1 and u s i n g Lemma 5.10 again t h e r e l / q we have Aq r e s u l t s from (5.25)

2

Since X

1ZI2 < -

(5.26)

c5 where c6 = -(c

c5Q(-c) exp q +c ).

q 3 2

lZtlL 5

(5.27)

Lemma 5.11

-

(5.26)

c8X + c 9

Assume Q

IQI

for 0 5

E

1 C [o,b] $ =

N on [o,s].

(5.27)

Proof: sup

5

7

P u t t i n g t h i s i n (5.24) we o b t a i n

w h i l e la1 5 kRea on [o,s], la(t)/a(S)l

c6b = c

w i t h Q r e a l and O < q 5 Q ( t )

a '/a2 E Co[o,s],

and

Then Z g i v e n by Lemma 5.8 s a t i s f i e s T

5 t 5 b w h i l e l a ( t ) Z t l <_ clOA

There remains o n l y t h e l a s t statement.

on [o,b],

c13

= sup la21 on [o,b]

54

+ cll.

L e t c12 =

(recall that a + 6 =

1.

al

+

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

a 2 ) , and c14 = sup exp

Then d i f f e r e n t i a t e (5.13)

(-

:I

Re a2(q)drl) f o r 0 5 5 5 t 5 b.

i n t and f o r t

5 s we have ( c f .

(5.12))

5 c 15A + where (5.26)

has been used.

Remark 5.12 assumption $I

C16

E

QED

Under c e r t a i n m i l d a d d i t i o n a l hypotheses t h e

Co[o,s]

i s s u f f i c i e n t t o insure t h a t

l a ( t ) / a ( T ) l 5 N f o r 0 5 T 5 t 5 s (c f.

Lemmas 5.9 and 5.11).

I n p a r t i c u l a r t h i s holds i f $(o) f 0 o r i f

,.

$ / / @ IE

Co[o,s]

for

some s 5 s (see C a r r o l l [8] f o r d e t a i l s ) . We c o n s t r u c t an "associate" r e s o l v a n t Y(A,t,T)

"m ponding t o S (y,t,-c)

-

t a i n ( c f . (3.16)

i n S e c ti o n 3, by t h e same technique t o ob(3.17))

Then given Rea 2 0 on (o,s], t i a t e (5.13) where U

in

T

, corres-

(5.12)

t o o b t a i n Y(X,t,-c)

i s give n as i n (3.34)

holds and one can d i f f e r e n = U(t,.r)

by U(t,.r)

55

=

-

(J

Y)(A,t,-r)

(P(T)/P(a))do =

SINGULAR A N D DEGENERATE CAUCHY PROBLEMS

tI(u,T)du

and J i s d e f i n e d by (5.13).

'T

The formal s o l u t i o n i s

00

1

U (cf.

(-l)kJk k=O i s continuous i n ( t , T )

again Y =

(3.36))

we need o n l y c o n s i d e r t h e c r i t i c a l p o i n t

Thus d e f i n i n g e v i d e n t l y U(t,o)

T = 0.

and t o show t h a t U(t,T)

= 0 for t

2

0 (note t h a t

o u r assumptions p r e c l u d e a case where a ( t ) = 0 as when m = -1/2 i n S e c t i o n 3) one has l I ( u , ~ ) l 5 M f o r 0 5 T 5 u 5 b (cf. (5.12))

:1

and we w r i t e U(t,-c) Now l e t (t,T) as

T

-+

-+

,..

=

where I(u,T)

-+

Hence t h e terms Jk

0.

-+

u <

T.

0 f o r any u

9

0

M and we may invoke t h e Lebesque

bounded convergence theorem ( c f . Bourbaki [3]) U(t,T)

= 0 for

w

f o r to -> 0; then I(o,T)

(to,o)

1 I(o,T) 1 5

0 with

,.

I(u,.r)do

t o conclude t h a t

U w i l l be continuous i n (A,t,T)

and a n a l y t i c i n A w i t h lU(t,-c)l

5 Mb.

By p r e v i o u s a n a l y s i s (cf.

Lemmas 3.8 and 5.8) we have Given

Lemma 5.13

Recl

L 0 on (o,s]

the series Y =

00

1

(-l)kJk k=O t 5 b l x r,

U converges a b s o l u t e l y and u n i f o r m l y on (0 5 T 5

r cF

f u n c t i o n Y(A,t,-c) U

-

J

(5.30)

compact ( A

E

o f (A,t,-c),

r),

a n a l y t i c i n A , which s a t i s f i e s Y = One has (5.29)

Y (J g i v e n as i n (5.13)). Moreover Y(A,T,T)

holds.

T > 0, Yt(A,T,T)

Proof: t h a t Ut =

and represents a continuous

= 0, Y(A,t,o)

and f o r T > 0

= 0,

and, f o r

= 1.

T h i s goes as i n t h e p r o o f o f Lemma 3.8

P(T)/P(t),

UT

rt

= -1

( n o t e here

+ (a+B)U, and from (5.301,

56

cf.

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

Now YT i s d e f i n e d b y (5.30) t o i n f i n i t y as T

and does n o t n e c e s s a r i l y tend

o r a,(-c)Y(A,t,T),

a(T)Y(A,t,T),

We need o n l y examine

0 ( c f . (3.42)).

-+

as

T

As i n (3.40)

0.

-+

m1

al(T)Y(h,t,T)

=

I

where B 2 ( o , ~ ) =

1 (-l)kJk

C X ~ ( T ) U ( * , T so ) we l o o k f i r s t a t

k=O

aB2( 0 ,

and $1 = a;/a:

a0

small arguments (e.g.,

Lemma 5.11).

al(q)dn)

f i r s t t h e term A(t;-c) =

can compare here w i t h S e c t i o n 3 where a ( t ) = = (T/t)2m

-+

Thus assume A(t,-c)

0 as -+

o

t 5 s ) ; here B 2 ( o , ~ ) = exp

i s Cm i n ( a , ~ w ) i t h I B 2 ( o , ~ ) I 5 c14 (cf.

A(t,.r)

(-i

= $ for sufficiently

w

T

-+

0 as T

as

+

aZ(rl)do) We examine 0 and one

2m 7 and +

f o r m > 0 (Bz(t,r) -+

T

T

1 i n Section 3).

0 i n which event, from (5.31)

I

t h e r e r e s u l t s ( s i n c e B2/Bz = -az(.)) (5.32)

a,(.)

t

Bz(o,~)C1 +

az(4

+

(+:a1

$(o)le

(

0

do

~ +

1

T

as

T

-+

0, independently o f t.

But a,(a)/a,(o)

+

@ ( o ) (and B2(a,.r) 1 1) we have

(5.33)

l i m al(T)U(t,T) T-+O

=

0 as u

-+

0 and

+ $(a)

t a k i n g t small enough so t h a t 1

-+

1

rn 57

SINGULAR AND DEGENERATECAUCHY PROBLEMS

provided $(o) $ -1.

This checks w i t h Section 1 . 3 ( f o r m > 0) where 1/(1+$(0)) = o r 2m + 1 U ( t , T ) -t 2m T h u s when 2m T A ( t , - c ) * 0 a s T * 0 w i t h $ ( o ) $ -1 we have by (5.30) as T * 0

-

-. +

YT(X,t,o) = - ( $ ( o ) / ( l + $ ( o ) ) Z ( X , t , o ) ( c f . (3.42) and the d e f i n i t i o n o f Z i n Lemma 5.8). In general we cannot expect of course t h a t

A(t,T)

-t

0 as

* 0 o r t h a t $(o) $ -1 ( c f . Section 1.3). However, as w i t h ?(y,t,.r), ye can show t h a t YT(A,t,r) E L 1 in T ( w i t h Y ( A , t , . ) T

absolutely continuous) and we follow again Carroll [8]. m

consider al(T)Y(X,t,-r)

=

T

5

f o r some upper l i m i t

1

1(-l)kal(~)Jk

k=O

U(-,T)

Thus

and define

2

..,

(5.34)

where J n

Kn =

t

0

U(*,T)

0 by d e f i n i t i o n i f

a l ( ~ J)n =

U(0,T)d-c ..,

(Jn

T >

=

U)(t,-c)

(note here t h a t ( J n

U)(t,.c)

t). W e s e t again 1B2(u,~)1 5 c14 and

lal(~)l5 k l R e a l ( T ) so t h a t

..,

( f o r Fubini-Tonelli theorems see McShane [ l ] ) .

Similarly, s e t -

t i n g [ X I C 5 R and ( c 2 + Rc12) = c , we have ( c f . (5.13) f o r J )

58

=

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

. "

5 c,4k,cbF(t) 5 c14klcbF(b) Upon i t e r a t i o n we obtain ( c f . Section 3)

and t h u s

m

1 (-1)"'

m

1 (-l)n

jt

a l ( ~ ) ( J n U ) ( i , T ) d T converges n=O n=O 0 uniformly and absolutely. An elementary argument ( c f . Carroll =

[8]) then shows t h a t al(*)Y(A,t,*)E L1 w i t h

I

. "

(5.38)

t

m

..,

al(T)Y(A,tyT)dT

0

1 (-l)"Kn

=

n=O

. "

Lemna 5.14 -Z(A,t,T)

I f [ a l l 5 klReal on [o,s] 1 + (a+@)(T)Y(A,t,T) E L for 0 5

i s absolutely continuous w i t h Y ( h , t , - r )

+ = a'/,*

E

Co[o,s] and +(o)

4

=

then

I

T 5 T

T

-+

YT(X,t,.r) =

t <- b and Y ( X , t , * )

Yg(X,t,<)d<.

0

-1 with h ( t , T ) =

continuous w i t h YT(A,t,o) = - ( + ( o ) / ( l + + ( o ) ) ) Z ( A , t , o ) .

59

If

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Proof:

The a b s o l u t e c o n t i n u i t y f o l l o w s as i n Lemma 3.9

and we r e c a l l t h a t Y(A,t,o)

= 0 by Lerlima 5.13.

Bounds f o r Y can be o b t a i n e d as f o r Z. (5.30),

with

T

replaced by

4 , by V(X,t,<),

p l e x conjugate, and i n t e g r a t e i n

4 from

QED We simply m u l t i p l y

add t h i s t o i t s comT

t o t, u s i n g t h e r e -

d l a t i o n - l Y 1 2 = 2Re(Y V), t o o b t a i n ( r e c a l l t h a t d4 5

c1

+ B

= a1 + a 2 )

(5.39)

'T

R e c a l l i n g t h a t c1 = 2 sup

IY12

(5.40)

Rw2I we have, assuming Lemma 5.11, YI2d4

+

+ (cl

Under t h e hypotheses o f Lemma 5.11 5 c for 0 r; t b and X > 0.

Lemma 5.15

IYl

(X,t,T)

Proof:

1ZI2d4

T

5 c7b

2

1

t

5 (c

5 T

we have

r;

One a p p l i e s G r o n w a l l ' s lemma (Lemma 5.10) t o (5.40).

QED Now l e t us go t o t h e s o l u t i o n o f t h e Cauchy problem (5.9) (5.10),

r e c a l l i n g t h a t X can always be made l a r g e r than zero.

60

-

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

L e t Ls(E,F)

be t h e space o f c o n t i n u o u s l i n e a r maps E

t h e t o p o l o g y o f s i m p l e convergence, i.e., Ave * 0 i n F f o r each e

E

r e c a l l t h a t OH = ff =

ff(A)dv(A).

\

0

E.

Av

We w r i t e L,(E)

-+

F with

0 i n Ls(E,F) i f

-+

f o r LS(E,E) and

Now Z(-,t,T)

and Y(.,t,-r)

b e l o n g t o Ls(@D(A),ff), LS(ff), and, LS(eD(A)) s i n c e t h e y a r e

c2

bounded b y Lemmas 5.11 and 5.15 (on eD(A) we p u t t h e norm 2 " 2 I I ~ T l l H ( X ) d v Ywhere I l e T I I q X ) delle T II = IleTll;(,)dv . I

j

notes

0

IleT(x)II

Ifnow eT

-

A(eW)(X) = [Z(A,t,T) t

-+

Z(*,t,-c)BT

hence t h a t t

: [o,b] -f

Z ( A , t o y T ) ] ( ~ T ) ( X ) = (AZ)(BT)(X). -+

Z(-,t,T)

8D(A) i s "strongly"

: [o,b]

+

uous, one must show t h a t as t + t IlhA(ew)

Ilff

+

0.

e D ( A ) c o n s i d e r f o r example

E

continuous, and

Ls(eD(A)) i s "simply" continIIA(OW)llff

-+

0 and

But, f o r example, IIA(ew)llff

+

0 means

0

2 1 I l A ( e ~ ) ( X ) l l ~ -+ ( ~ 0) i n L ( v ) and s i n c e A(Bw)(X) 2 everywhere (cf. Lemma 5.8) w i t h IlA(Ow)(X) llff(X) 1 L (v) (cf.

(5.26)

i n Lemma 5.11)

IIA(ew)llH

-+

H(X)

E

one can a p p l y t h e Lebesgue

5.11,

t o conclude

to ( n o t e t h a t lleTl12

L 1 ( v ) ; t h u s one has shown t h a t t

CO(Ls(eD(A))) on [o,b]. Lemmas 5.8,

E

i s finite v H(X) A s i m i l a r argument a p p l i e s t o IlXA(ew) Ilff

0 as t

a l m o s t everywhere). 2 s i n c e X lleT/12

*

0 v almost 2 5 2c711eT11ff(h)

+

dominated convergence theorem ( c f . Bourbaki [3]) that

TO Show

5.13,

+

Z(*,t,T)

Analogous reasoning, u s i n g (5.29) w i t h and 5.15,

l e a d s t o ( c f . a l s o Remark

5.22) Lemma 5.16 0 L

T

2 t 5 b, t

Under t h e hypotheses o f Lemna 5.11 we have, f o r -+

Z(-,t,T)

E

CO(Ls(eD(A)) o r Co(Ls(ff)) (and

61

E

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

CO(Ls(eD(A),H))

trivially);

T

Y(*,t,-c)

-+

o r Co(LS(OD(A))); while t

Co(Ls(ff))

and

a(t)Z,(*,t,T),

T

+

ZT(*,t,.r)

E

-+

and

T

-+

Z(.,t,T)

t

Zt(*,t,T),

E

-+

CO(Ls(eD(A),H)).

We w i s h t o show now t h a t i n f a c t a / a t ( e w ) = a / a t ( z e T ) =

(1,

Z 8T i n H.

tt

For fixed ( A , T )

[Zq(X,q,r)

-

-

one has evidently A Z / A t

Zt(A,toT)

=

Zq(X,to,~)]dq)/At. Hence i n H ( h ) we have, v Y

alm8st everywhere, i n an obvious n o t a t i o n , Ah = ( A ( e w ) / A t ) t Z t ( h , t o , T ) e T ( h ) = [( l / A t ) l AZndnleT(h) and consequently

-

Now IleTIIH(,l = I / e T ( h ) l l H ( h l i s f i n i t e v almost everywhere and for (X,T) Irl-tol

fixed we can make (by Lemma 5.8) lAZ,l

5 6 ( ~ ) . Hence, f o r Y

so t h a t IIAhll

+

"

5 It-tol 5 6 ( ~ ) ,I l A A l 1 2 5

0 v almost everywhere a s t

we know t h a t llAh112 5 [ ( 2 / A t ) ( I 2 (c17X

+

a/at(ew)

t

-+

for

E E

2

2 ~~eT(A)~~H(h)

to. Also by (5.27)

(Cgx + ~ ~ ) ~ / ~ d ~ ] ~ I l e5T l l i ( ~ ) t ." 1 c i a ) I l e T ( h ) IIH(h)* Therafore l l A A I l 2 0 i n L ( v ) and = ZteT i n H. Similarly one checks a 2/ a t 2 ( O w ) , u s i n g -+

estimates f o r a ( t ) Z , ( h , t , T )

and X Z ( X , t , - c )

indicated above ( c f .

a l s o Lemma 5.9), t o e s t a b l i s h Theorem 5.17 Under t h e hypotheses o f Lemma 5.11, t -+ 2 Z ( = , t , - r ) E C ( L S ( e D ( A ) ) , H ) and there e x i s t s a solution of the Cauchy problem 5.9 (Problem 5.7) given by w ( t )

e-'[Z(=,t,~)eT(*)]

for T

E

D ( A ) ( 0 <-

62

T 5

=

t 5 b).

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

One notes a l s o t h a t by (5.30), ences thereto, T for

and Lemna 5.16 w i t h r e f e r -

(l/a(T))YT(*,t,T) cC0(LS(ff)) o r CO(Ls(OD(A)) 1 1 small w i t h T -+ Yf,-,t,~) E C (Ls(ff)) o r C (L,(OD(A)) f o r

T

T > 0.

-+

Now we note t h a t i f F i s b a r r e l e d then any separately

continuous b i l i n e a r map E

x F

-+

G i s hypocontinuous r e l a t i v e t o

bounded s e t s i n E (see Bourbaki [2]). b a r r e l e d so t h e map u : (A,h) uous on bounded sets

But a H i l b e r t space i s

Ah : Ls(F,G)

-+

Bc Ls(F,G).

Since 0 5

G i s contin-

x

F

T

5 5 S t 5 b is

-+

compact and t h e continuous image o f a compact s e t i s bounded we may s t a t e f o r example t h a t 5 Y(-,t,c)ew$). Yg(*,t,S)0w5(S),

5

+

(46)

and 5

-+

+

-+

Y(*,t,t)ew(<),

5

B(S))Y(-,~,S)@W~,5

Y(*,t,S)Bw

55

.

-

(3.20)

and see C a r r o l l

[a]

for

{Yewc5 + Y 0w I d 5 = 0

(5.42)

5 5

where 0w s a t i s f i e s (5.9).

(cf.

i s given by

The f o l l o w i n g formulas are then e a s i l y v e r i f i e d

using Lemna 5.14 ( c f . (3.18) de t a i 1s )

+

(5) are continuous (note

t h a t Y5eW5 = ( l / a ( < ) ) Y 5 a 0w5 and e w ( 5 ) = ew(5,T) Theorem 5.17).

-+

(5.30)).

Hence m u l t i p l y i n g (5.10)

by Y we have

Hence 0w(t) i s n e c e s s a r i l y o f t h e form Z(X,t,T)BT

and we have proved

63

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorem 5.18

The s o l u t i o n o f Problem 5.7 given by Theorem

5.17 i s unique.

As b e f o r e ( c f . (3.22) and Theorem 3.14) we have ( c f . C a r r o l l [8] f o r d e t a i l s and cf. a l s o Remark 5.22). Theorem 5.19

The unique s o l u t i o n o f (L+Q(t)A)w = f

Co(D(A)), W ( T ) = T

E

D(A),

E

wt(-c) = 0 i s given (under t h e hypo-

As i n D e f i n i t i o n 3.15 we w i l l say t h a t t h e problem (L+Q(t)A)w = f,

W(T)

= T

E

D(A), w ~ ( T =) 0, and f

E

0

C (D(A))

i s u n i f o r m l y w e l l posed i f w depends c o n t i n u o u s l y i n H on By h y p o c o n t i n u i t y and Lemmas 5.8 and 5.13 we can

(t,T,T,f). state

Theorem 5.20 W(T)

= T

D(A),

E

The problem (L+Q(t)A)w = f and w,(T)

=

2

T

5 t 5 b).

We can g e n e r a l i z e some o f t h e r e s u l t s o f Sec-

a2/at2 +

t i o n 1.3 t o t h e case o f L = place o f 3

Co(D(A)),

0 i s u n i f o r m l y w e l l posed under

t h e hypotheses o f Lemma 5.11 ( 0 5 Remark 5.21

E

(a(t)+B(t))

a/at + y(t) i n

2m+l / a t 2 + (7) a / a t ; t h e d e t a i l s are i n C a r r o l l [8].

Thus one considers f o r 0 5

T

5 t 5 b e.g.,

Lw + Q ( t ) A x

*

w

=

I

S , and w ~ ( T =) 0 w i t h C o n d i t i o n 3.1 on A(y) f u l -

W(T) = T

E

filled.

I t i s i n t h i s problem t h a t t h e a n a l y t i c i t y o f Z, Y,

etc. i n A i s u t i l i z e d ( c f . Lemmas 5.8,

64

5.9,

and 5.13).

0,

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

One proves t h a t t h e unique s o l u t i o n w

*

problem i s g i v e n by w ( t ) = Z(t,-c)

2 ' C ( S ) o f t h i s general

E

T where Z(t,-r)

=

F-'Z(A(y),t,-r)

and t h e problem i s u n i f o r m l y w e l l posed. Remark 5.22

I n C a r r o l l [ l o ; 111 some s e m i l i n e a r versions

o f t h e s i n g u l a r Cauchy problem were t r e a t e d by s p e c t r a l methods. L e t L be as above and c o n s t r u c t , under t h e hypotheses o f Lemma 5.11,

r e s o l v a n t s Z(A,t,-r)

Then consider

as above.

and Y(A,t,-c)

in H (L+Q(t)A)w+ f(t,w)

(5.45)

(0 5

T

5 t 5 b).

w,(T)

= 0

and

= e-'Z(-,t,T)e

Y(t,T)

=

belong t o Ls(H))

w ( t ) = Z(t,r)T

(5.46)

D(A);

E

F o l l o w i n g Theorem 5.19 t h i s leads t o t h e i n t e -

g r a l e q u a t i o n (where Z(t,T) e-'Y(-,t,r)O

w(T.) = T

= 0;

-

i:

Y(t,S)f(S,w(S))dS

Tw

=

Now p a r t s o f Lemma 5.16 can be somewhat improved by u s i n g e s t i mates a l r e a d y e s t a b l i s h e d p l u s some s t r o n g e r v a r i a t i o n s ; f o r example one can show t h a t ( t , t ) t

-f

Z(t,.r)

E

C 1(Ls(D(A 112 ),H)

can produce an e s t i m a t e T

+

Y(t,<)

( c f . (5.27)

I Y I 2 (A,t,-c)

Co(Ls(H,D(A 112 ) ) ) and

E

f o r Z and f o r Y one

h

5 c/A f o r A > 0 and 0 5

5 t 5 b, s t r o n g e r than t h a t o f L e m a 5.15

3).

c f . a l s o Chapter

One l o o k s f o r f i x e d p o i n t s of T u s i n g some m o d i f i c a t i o n s o f

a technique o f FoiaT-Gussi-Poenaru [l;21. assume [/;(S,V)

-

5

For example i f we

-, +.

f(S,v)

-f(<,U)l[

= A112f(SyA-1v)

E

Co(H) f o r v

5 k l ( < ) w l ( ~ ~ v - u ~where ~) kl 65

E

E

L

1

Co(H) w i t h and w1 i s an

SINGULAR AND DEGENERATE CAUCHY PROBLEMS E

Osgood f u n c t i o n (i.e.,

dp/wl(p)

-

Ilf(C,V)ll 5 k 2 ( E , ) ~ 2 ( j l v l l )where k2

P

f u n c t i o n ( i .e.,

e x i s t s a unique s o l u t i o n w

E

E

L 1 and w2 i s a Wintner

posed.

E

E

> 0) , then t h e r e

on [o,b].

with w

Co(D(A)) w h i l e (5.45)

Fur-

C2(H), w

E

E

i s uniformly well

The improved estimate on Y a l s o leads t o a s t r o n g e r ver-

s i o n o f Theorem 5.19 where o n l y f w

E

E

Co(D(A)) o f (5.46)

thermore, t h i s s o l u t i o n s a t i s f i e s (5.45) C1(D(A1l2)), and w

> 0) w h i l e

00

for

dp/w2(p) =

JE

for

=

0

2 C (H), w

E

C’(D(A’/2)),

a n o n l i n e a r term t o f(t,w)

and w

E

Co(D(A’/2))

E

Co(D(A)).

i s required while One can a l s o add

i n (5.45) which generates a compact

operator i n H and o b t a i n another existence r e s u l t f o r (5.45).

EPD equations i n general spaces.

1.6

We go now t o a

technique o f Hersh [l;21 i n Banach spaces which was generalized by C a r r o l l [18; 19; 24; 251 t o more general l o c a l l y convex spaces ( c f . a1so Bragg

[lo] , Carrol l-Dona1 dson

[203 , Dona1dson 11 ; 51 ,

Thus l e t E be a complete

Donaldson-Hersh 161, Hersh [ 3 ] ) .

separated l o c a l l y convex space and A a closed densely defined l i n e a r o p e r a t o r i n E which generates a l o c a l l y equicontinuous group T ( * ) i n E ( c f . remarks a f t e r D e f i n i t i o n 6.3). that T(t) T(t)e

E

E

This means

L(E) f o r t E R , T ( t ) T ( s ) = T ( t + s ) , T(o) = I , t

Co(E) f o r e

E

+

E, and given any (continuous) seminorm

p on E t h e r e e x i s t s a (continuous) seminorm q such t h a t p ( T ( t ) e ) 5 q(e) f o r

t

+

0 for e

It\ 5 E

t o<

00

(any to); a l s o l i m ( T ( t )

D(A) (dense).

-

I ) e / t = Ae as

We r e f e r here t o Komura [ l l and

Dembart [ll f o r l o c a l l y equicontinuous semigroups or groups and 66

SINGULAR PARTIALOIFFERENTIALEQUATIONSOF EPD TYPE

1.

remark t h a t i t i s a b s o l u t e l y necessary t o consider such groups i n " l a r g e " spaces E i n o r d e r t o deal f o r example w i t h growth propert i e s o f t h e s o l u t i o n s of c e r t a i n d i f f e r e n t i a l equations (see a l s o f o r example Babalola [l] , Komatsu [l]a Lions [71 a Miyadera 111 , O u c i i [ll, Schwartz [6]

a

Waelbroeck [l] , Yosida [ll, e t c . f o r

o t h e r "general" semigroups

-

f o r s t r o n g l y continuous semigroups

i n Banach spaces see H i l l e - P h i l l i p s [21). L e t us consider now t h e problem

wTt +

(6.1)

f o r wcC'(E).

2m+l

m = A2wm. wt , w"(0) = eED(A2);

wY(0) = 0

Following Hersh [l]one replaces A by a/ax and

looks a t

which i s a r e s o l v a n t equation as i n Section 3 whose s o l u t i o n

^Rm

= FRm i s given by (3.6) w i t h z = ty.

We r e c a l l t h e Sonine

i n t e g r a l formula (4.6) and p i c k p = -1/2, 2p + 1 = 0 = n Theorem 4.12).

-

which corresponds t o n 1, as t h e p i v o t a l index (note p = 7 - 1 as i n

C l e a r l y t h e unique s o l u t i o n o f (6.2) w i t h m =

-1/2 i s R - ' / 2 ( * , t ) (4.6),

u x ( t ) = 1/2[&(x+t) + ~ S ( x - t ) ] so t h a t , from

f o r m > -1/2

(6.3)

where c,

=

; I' ,e

Rm(=,t) =

1 ( 1 - A m - 7 vx(St)d6

2 r (m+l)

r(1/2)r(m o 2m-= cm] (1-6 ) 2 6(x-ct)d< -1

=

(note t h a t b(x?Ct) could be used i n the 67

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

l a s t expression i n (6.3) because o f symmetry i n v o l v e d i n vx - c f . (6.6)). (6.4)

Now one w r i t e s f o r m a l l y wm(t) =
T(=)e>

as t h e s o l u t i o n of (6.1) where f o r t f i x e d <, > denotes a p a i r i n g between t h e d i s t r i b u t i o n Rm(*,t) E

Co(E).

Note here t h a t f o r e

E

E

EI( o f order zero and T ( - ) e

2 D(A ), T ( - ) e

T ( x ) e = AT(x)e = T(x)Ae, and o f course t Section 1.3.

-+

E

2 C (E) w i t h

Rm(*,t)

E

d

Cm(Ei) from

I f we w r i t e o u t (6.4) i n terms o f (6.3) t h e r e

results (6.5)

1 2 ) m-- 2

wm(t) = C,,,/;~(~-S

< s(x-St),

T(x)e > dS

which coincides w i t h the formula e s t a b l i s h e d by Donaldson [l] using a d i f f e r e n t technique. wrn(t) = 2cm

/

2 m - -1 (1-S ) 2 cosh(A
where cosh A S t = 1 / 2 [ T ( S t ) cosh (ASt)e.

We can a l s o use t h e formula

+ T(-St)l w i t h < u x ( S t ) , T(x)e

> =

To v e r i f y t h a t (6.4) represents a s o l u t i o n o f (6.1)

one can o f course work d i r e c t l y w i t h the v e c t o r i n t e g r a l s (6.5)

or (6.6) b u t we f o l l o w Hersh [l]and C a r r o l l [18; 19; 24; 251 i n t r e a t i n g (6.4) as a c e r t a i n d i s t r i b u t i o n p a i r i n g . R:(*,t),

= AxRm(*,t) a r e a l l o f order less than o r

and R:,(-,t)

equal t o two i n E

Thus Rm(*,t),

I

w i t h supports contained i n a f i x e d compact

68

StNGULAR PARTtAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

set

K

xo +

=

Bl

Ix;

1x1 5

xol

for 0 5 t 2 b <

If S

f o r any f3 > 0.

E

m

and we l e t

A

K

1x1 5

= {x;

I

i s o f o r d e r l e s s than o’r equal Z A 1 t o two w i t h supp S C K we can t h i n k o f S E C (K) ( c f . Schwartz E

ill). Recall f u r t h e r t h a t on *K,C 2 ( E ) = C Z gA) E (see e.g., E

111) where t h e

E

Treves

2 topology on C @ E i s t h e topology o f u n i f o r m

convergence on p r o d u c t s o f equicontinuous s e t s i n ( C 2 ) ’ x E

I

.

It

i s easy t o show (see C a r r o l l [18] f o r d e t a i l s ) a

Lemma 6.1 tinuous f o r

The composition S + <S,p-: C 2 ( i ) ’

$E

C2(E) f i x e d .

(defined by (A x I )

1 $i

-+

E i s con-

The map A @ 1 : C2(E) = C O G E E

8ei =

1

8’ei)

i s continuous.

L e t us i n d i c a t e f i r s t t h e formal c a l c u l a t i o n s necessary t o show t h a t (6.4) i s a s o l u t i o n o f (6.1), t h e same t i m e some r e c u r s i o n r e l a t i o n s .

while establishing a t Their v a l i d i t y i s

e s s e n t i a l l y obvious upon s u i t a b l e i n t e r p r e t a t i o n ( c f . below). Thus, from (6.4) and (3.11) transformed,

w F ( t ) =
-2(m+lJ t

T(-)e> =

T(ii3-T

T(-)A2e>

1 (1-5 2 ) m+- 2

T(Ct)A*edE

S i m i l a r l y from (6.4) and (3.12) transformed

69

,t ) , T (

) e>

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

-

, T( * ) e > ]

=

2mt

-

[wm-l ( t )

Then d i f f e r e n t i a t i n g (6.7) and using (6.7)

-

wm( t ) l (6.8) we

The c a l c u l a t i o n s under t h e b r a c k e t <, > can be

o b t a i n (6.1).

j u s t i f i e d using Lemna 6.1 and i n o r d e r t o t r a n s p o r t A around i n (6.7) one can t h i n k of <, > as a d i s t r i b u t i o n p a i r i n g over the h

i n t e r i o r o f K (cf. Carroll [l8]). 2 and E as above ( e E D(A ) ) Theorem 6.2

The e q u i v a l e n t formulas (6.4) - (6.6) g i v e a

s o l u t i o n t o (6.1) f o r m 2 d i c a t e d by (6.7)

This proves, given A, T ( t ) ,

-

-

1/2 and t h e irecursion r e l a t i o n s i n -

(6.8) a r e v a l i d .

The q u e s t i o n o f uniqueness f o r (6.1) had seemed t o be r a t h e r more complicated than t h a t o f e x i s t e n c e b u t a new theorem o f C a r r o l l [26] (Theorem 6.5) gives a s a t i s f a c t o r y r e sponse.

There a r e several types o f theorems a v a i l a b l e ( c f .

C a r r o l l [18; 24; 25; 261, Carroll-Donaldson [201, Donaldson 111, Donaldson-Goldstein [2], Hersh [ l ] ) .

The f i r s t one we discuss

( f o r completeness) i s based on a v a r i a t i o n o f t h e c l a s s i c a l " a d j o i n t " method and simp1 i f i e s somewhat t h e p r e s e n t a t i o n i n C a r r o l l [18]; i t i s n o t as s t r o n g however, as Theorem 6.5. D e f i n i t i o n 6.3 adapted i f E

I

The space E (as above) w i l l be c a l l e d A-

*

i s complete and D(A ) i s dense.

When E i s A-adapted (A being t h e generator o f a l o c a l l y

70

SINGULAR PARTIAL DIFFERKNTIAL EQUATIONSOF EPD TYPE

1.

equicontinuous group i n E) i t f o l l o w s ( c f . Komura [ll)t h a t

*

generates a l o c a l l y equicontinuous group T ( t ) i n E

I

.

A*

The r e -

quirement o f completeness i s a l u x u r y which we p e r m i t ourselves for simplicity.

Indeed, f o r t h e d i s c u s s i o n o f l o c a l l y equi-

continuous semigroups s e q u e n t i a l completeness i s s u f f i c i e n t ( i n view o f t h e Riemann t y p e i n t e g r a l s employed f o r example).

Thus,

i n p a r t i c u l a r , i f E i s r e f l e x i v e ( n o t n e c e s s a r i l y complete) hence s e q u e n t i a l l y t h e n i t i s quasicomplete ( c f . S c h a e f f e r [l]), complete, as i s t h e r e f l e x i v e space E

I

*

, and

A

*

w i l l generate a

*

I

l o c a l l y equicontinuous group T ( t ) i n E w i t h D(A ) dense ( c f . Komura [ll).We remark a l s o t h a t i f E i s b o r n o l o g i c a l then E

I

i s i n f a c t complete ( c f . Schaeffer [l]). We use completeness

2 2 ^ b a s i c a l l y o n l y i n a s s e r t i n g t h a t C ( E ) = C BEE b u t a v e r s i o n o f t h i s i s p r o b a b l y t r u e f o r E s e q u e n t i a l l y complete; t h e i n crease i n d e t a i l and e x p l a n a t i o n throughout our d i s c u s s i o n does n o t seem t o j u s t i f y t h e refinement however. We c o n s i d e r now t h e r e s o l v a n t s Rm(-,t,.r) S e c t i o n 3 when A(y) = y S e c t i o n 4). Sm(*,t,T)

2

w i t h Ax = -A

and 2

X

= - 3 /ax

2

Sm(*,t,T)

of

(cf. also

We r e c a l l ( c f . Lemma 4.3) t h a t R m ( - , t , r )

and

belong t o E i , w i t h s u i t a b l e o r d e r s ( e x e r c i s e ) , and

Lemna 3.13 h o l d s w i t h OM replaced by Exp OM = F E

I

.

Thus l e t

w m ( t ) s a t i s f y (6.1) w i t h wm(0) = wF(0) = 0 and consider ( f o r an A-adapted E) (6.9)

um(t,<) = <Sm(*,t,<),

T*(*)el>

71

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where el

E

"2 D ( ( A ) ) (dense); wm i s any-

necessarily a r i s i n g from ( 6 . 4 ) . (6.10)

uF(t,<)

<(-Rm(-,t,<)

=

-vrn(tL)

+

+

*

I

(*)e>

2m+l m S ( - , t , < ) )T, ( * ) e > 5 *

*

T ( - ) e l > . Further, from (3.16),

*

v m ( t , < ) = - =

5

-

< and

has been replaced by

wm i s any-

i n i t i a l data; then i n t e g r a t e i n

(6.12) (6.13)

f:1

<

from 0 t o t ( c f . (3.18)

0

FE

2m W

= +

um(tY<)>Ed< =

( < I S

-1

(51, um(t,E)>EdE =

-1

t

1

t

i n (6.1), where t

solution of (6.1 ) w i t h zero

We note f i r s t t h a t

<w

um(t,<)

< S m ( - , t , < ) , AT*(-)e'> = -(A*)2

Now take (E, E l ) brackets <, >E with u " ( t , < )

(3.20)).

I

2m+l m 7 u (t,<)

where v m ( t , < ) = < R m ( - , t , < ) , (6.11)

Then from (3.17)

= < S ~ ( - , t , 5 ) .T

=

solution of ( 6 . 1 ) , not

t

I

<wF(<)Y

0

t

<Wm(<),

0

-

uT(tY<)>Ed5 (A

* 2 m U

(tY<)'Ed<

m

< w m ( < ) , vF(t,S)>EdS = - < w ( t ) ,

0

0

w;(<)Y

vm(tY<)>Ed<

E 0 f o r any el E Consequently, u s i n g (6.10), < w m ( t ) , " 2 D ( ( A ) ) (dense) - which implies t h a t w m ( t ) E 0 - s i n c e we w i l l

have from ( 6 . 1 ) , (6.12), and (6.13)

72

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

Theorem 6.4

I f E i s A-adapted then t h e s o l u t i o n t o (6.1),

g i v e n by Theorem 6.2,

i s unique.

This uniqueness theorem has been i n c l u d e d p r i m a r i l y t o i l l u s t r a t e t h e a d j o i n t method.

I f can be supplanted by t h e

f o l l o w i n g r e c e n t r e s u l t of C a r r o l l [26] which was motivated by (and improves) a r e s u l t i n C a r r o l l [18] based p a r t i a l l y on a technique o f F a t t o r i n i [l]. Theorem 6.5

For Rem >

-

1/2 l e t wm be any s o l u t i o n of

6.1)

w i t h zero i n i t i a l data and l e t A generate a l o c a l l y equicont nuous group T ( x ) i n E as above. Proof:

Then wm(t)

5

0.

We n o t e f i r s t t h a t o u r existence-uniqueness c a l -

c u l a t i o n s i n Sections 3-4 a r e v a l i d f o r s u i t a b l e complex m (Rem 2

-

1/2), as i n d i c a t e d p a r t i a l l y i n Section 5, b u t we w i l l

not s p e l l out the details.

Thus o u r demonstration i s s t r i c t l y

j u s t i f i e d only f o r real m 2

-

Sm(*, t, s ) w i t h Ax = 6.4.

-

1/2.

We use R m ( * , t ,

s ) and

2 2 Ax = - a /ax as i n t h e p r o o f of Theorem

Define, w i t h b r a c k e t s as i n (6.4),

(6.15)

R(t,s)

=
T ( - ) wm(s)>

(6.16)

S(t,s)

= <Sm(-,t,s),

T ( * ) w:(s)>

73

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

-

Then e v i d e n t l y ( c f . (3.16)

(3.17) and t h e c a l c u l a t i o n s l e a d i n g

t o Theorem 6.4) (6.17)

RS(t,s) =

-

<Sm(*,t,s),

+
Ss(t,s)

(6.19)

(R+S),(t,s)

Consequently $ ( t , o )

m

R (*,t,t)

T(*)W:(S)>

= <-Rm(-,t,s)

+ <Sm(*,t,s),

+ 2m+l

m

S (-,t,s),

T(*)w;(s)>

T(*)w;,(s)> =

=

T ( * ) A2wm(s)>

$,(t,s)

$(t,t)

= 6 we have $(t,o)

=

0

and r e c a l l i n g t h a t S"(-,t,t) = 0 = $(t,t)

=

0 with

= wm(t).

We sketch here i n passing some r e l a t i o n s o f (4.6)

(IED

and (4.10)

t o t h e Riemann-Liouville (R-L) i n t e g r a l ( c f . C a r r o l l [18; 191, Donaldson [l], and Rosenbloom [l]); questions o f uniqueness f o r i n E can be t r e a t e d i n t h i s c o n t e x t b u t Theorem 6.5 i n -

(6.1)

cludes e v e r y t h i n g known.

The r e l a t i o n o f

EPD equations t o R-L

i n t e g r a l s was o f course known by Diaz, Weinberger, Weinstein, etc. many years ago and some f a c t s a r e o f obvious general i n t e r We r e c a l l f i r s t ( c f . Riesz [l]) t h a t t h e R-L i n t e g r a l o f

est. f

E

Co o r " s u i t a b l e " f

(6.20)

(Iaf)(t) =

E

L1 i s d e f i n e d f o r Re a > 0 b y

t m1 io ( t - s ) a - ' f ( s ) d s

74

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

One can o b v i o u s l y deal w i t h f

E

Co(E) f o r example i n the same

manner and we w i l l assume f i s E valued i n what follows. Re a > -p (p

for f

E

2

For

1 an i n t e g e r ) t h e " c o n t i n u a t i o n " o f I' i s given by

Cp(E) o r f

CP-'(E) w i t h f ( ' ) ( = )

E

E

L 1 (E) " s u i t a b l y " .

= IatBa except p o s s i b l y when B i s a negative One knows t h a t IaIB

i n t e g e r B = - p a where IqPf= f(')a f('-')(o)

i n which case f ( o ) =

i s r e q u i r e d i n order t h a t

= 0

...=

i n general,

=

i s d e f i n e d t o be t h e i d e n t i t y . E v i d e n t l y ( d / d t ) ( I " f ) = and Io I"-1 f f o r any a. We record now some e a s i l y e s t a b l i s h e d f a c t s . Lemna 6.6

If f

E

Cn+'(E) w i t h -n < Re a 5 -n

+ 1 and

a

n o n i n t e g r a l then (6.22)

(I"f)'(t)

= (Ia-'f)(t)

I f f ( o ) = 0 o r a = -n w i t h f (6.23)

(Iaf)l =

-*-+

Cn+'(E)

(Iafl)(t) then

= Ia-lf

Iafl

(when Re a > 0 (6.22) w i t h -n < Re a 5 -n

E

=

-

(6.23) h o l d f o r f

+ 1 or

a = -n ( o r f

E

E

1 C (E)).

If f

E

Cn(E)

Co(E) when Re a > 0)

then (6.24)

t(I"f)(t)

Proof:

= (I"(sf))(t) + a(Iat'f)(t)

Routine computation y i e l d s (6.22)

(6.24) f o l l o w s by an i n d u c t i o n argument.

75

-

(6.23) and

QED

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Now, r e f e r r i n g t o t h e Sonine i n t e g r a l formula (4.6) ( c f . a l s o (6.3) and (6.6)), (6.25)

Rm(*,-r1/')

we can w r i t e , a f t e r a change o f v a r i a b l e s ,

I

m- 1

f

=

(T-S)

's

_1_ _1_ 'R

2(-,s1/2)ds

0

as before.

where cm =* . (6.26)

p(t) =

w i t h wm(t1/')

m m

L e t then

1/2

= ;

i t f o l l o w s from (6.25) t h a t

f o r Re m > -1/2 1 (6.27)

Wm(t) = (I?W-')(t)

S e t t i n g i m ( t ) = wm(t'/')

zt'/'

1

i t i s easy t o show (note t h a t

=

( a/ a t ) *

Lemma 6.7

Given wm, w",

and Wm as above, where wm s a t i s f i e s

t h e d i f f e r e n t i a l equation i n (6.1), (6.28)

4tqt

+ 4(m+l)wy

= A2im

(6.29)

4tWyt

-

= A2Wm

Remark 6.8 [18;

a/at'/*

4(m-l)W:

one has f o r Wm

E

C2(E)

A previous uniqueness argument ( c f . C a r r o l l

191) f o r n o t n e c e s s a r i l y A-adapted E proceeded from w" any

s o l u t i o n - o f (6.1) w i t h zero i n i t i a l data t o Wm determined by 1 (6.26) and d e f i n e d W - l " = I-m-'i W", t o g e t h e r w i t h t h e corresponding w-'".

Then, u s i n g Lemma 6.6,

given s u i t a b l e ( b u t ex-

cessive) hypotheses of d i f f e r e n t i a b i l i t y and " r e g u l a r i t y " on Wm,

76

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

1.

W-l"

was shown t o s a t i s f y (6.29) w i t h index -1/2,

and a unique-

ness theorem f o r t h e corresponding w - l l 2 was e s t a b l i s h e d (weaker This l e d t o uniqueness f o r wm under s u i t a b l e

than Theorem 6.5). hypotheses.

One f e a t u r e o f t h e technique i n v o l v e d embedding Wm

i n a "Weinstein complex'' determined by ( c f . (4.10)

where p i s an i n t e g e r ( 0 5 p S c1

a r e chosen so t h a t -1/2 = m

1 7 < Re

m < k +

R = k + 2). hold.

3 7, k

(4.10)

fl+a-p(o)

= -1,

a ) and Re

+

a

0, 1, 2,

-

a > 0 or a = 0

( a and

R which means t h a t i f k +

. . ., then

3 a = k +7

propagates i n i t i a l values and (6.7)

and WT+a-P(o) (m+a-p 2

-

and t o study t h i s one notes from (6.28)

T1 )

-

- m with - (6.8)

are c r i t i c a l i n (6.30)

(6.29) t h a t Wm cor-

responds t o some w - ~( f o r any index m) which we w r i t e i n the form (6.31 )

Wn ( t ) = i - n ( t ' l 2 ) / r (n+l )

f o r some i - n ( * ) s a t i s f y i n g the d i f f e r e n t i a l equation i n (6.1) w i t h index -n.

The connection i s somewhat i n t e r e s t i n g and i t i s

s u f f i c i e n t t o i n d i c a t e t h i s f o r t h e c r u c i a l index 1/2 (note w1/2 = w-1/2 and i n (6.26) W-'"(O) t i n s u r e W-'/2(o)

= 0).

= 0 does n o t a u t o m a t i c a l l y

Thus d i f f e r e n t i a t i n g (6.31) w i t h index 1/2

and n o t i n g t h e obvious d i f f e r e n c e between f t ( t ' / ' ) (d/dt)f(t'/') (6.32)

and f ( t 1 / 2 ) t =

we o b t a i n ( c f . (6.26))

Ft 1 -1/2w1/2(t1/2)

+

t 1/2w1/2(t1/2)

77

= ;-l/2(t1/2)t

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

whereas from (6.8) w i t h index 1/2 we have ( s i n c e

(6.33)

2t

=

1/2( p ) ,

-

Consequently from (6.32) (6.34)

a / a t 112

we have

= 2 t1 / 2 w Y 2 ( t1/2)

w-1’2(t1’2)

and thus given

(6.33)

W-’/~(O)

= 0 there results

0

Therefore i-1’2(t1’2)t0 as t -+

means b y (6.31)

-t

0 s i n c e w;’/’(<)

= W-l/‘(t)

t h a t W;/‘(t)

-t

-+

0 as t

-f

0 which

0, as desired.

In any event we have e x i s t e n c e and uniqueness determined by Theorems 6.2 and 6.5 and some new growth and c o n v e x i t y theorems w i l l f o l l o w , w i t h r e a l i s t i c examples, o f a t y p e f i r s t developed by C a r r o l l [18;

19; 251.

Thus, r e f e r r i n g t o (6.6)

-

and t a k i n g now m r e a l so t h a t cm > 0 f o r m > -1/2 we

(6.7),

have under t h e hypotheses o f Theorem 6.2 Theorem 6.9

Let

E be a space o f f u n c t i o n s o r equivalence

classes o f f u n c t i o n s and assume e 0 ( m >_ -1/2).

Proof: (6.36)

2 D(A2) w i t h cosh (At)A e >_

Then w m ( * ) i s a nondecreasing f u n c t i o n o f t. From (6.6)

m

E

wt(t)

=

-

(6.7) we have f o r m

tA’ 2(m+17 w“+’(t)

1

m-+ 1

m+l 78

t

-1/2

2 cosh(A6t)A e dS 2 0

QED

SINGULAR PARTIAL DIFFERENTIAL EQUATIONSOF EPD TYPE

1.

Next we r e c a l l t h a t ( f r o m S e c t i o n 4) L i =

2m

a/at

+

t

a 2/ a t 2

= (2m) t -2(2m+1) a2/a2(t-2m) f o r m

t

0 w h i l e Lot =

t-2 a2/a2(10g t ) . The d i f f e r e n t i a l equation i n (6.1) can be

w r i t t e n as L k P = A2wm so we have from (6.6) 1 1 ni-(6.37)

Lmw t m ( t ) = 2cm

1

(1-5

w i t h m > -1/2 r e a l

cosh(A5t)A2e d5

0

Theorem 6.10

Under t h e hypotheses o f Theorem 6.9,

a convex f u n c t i o n o f t-2m for m Proof:

+ 0 and o f l o g t f o r m

~"(0)i s = 0.

The r e s u l t f o r m = -1/2 i s obvious s i n c e w - ' l 2 ( t )

cosh(At)e and t h e r e s t f o l l o w s from (6.37).

=

QED

L e t E = Co(IR) w i t h t h e topology o f u n i f o r m

Example 6.11

The o p e r a t o r A = d/dx generates a

convergence on compact sets.

l o c a l l y equicontinuous group ( T ( t ) f ) ( x ) = f ( x + t ) which i s n o t

I f f L 0 then e v i d e n t l y T ( t ) f 2 0 f o r t

equicontinuous.

t o f u l f i l l t h e hypotheses o f Theorems 6.9 f i n d e = f such t h a t A 2 f = f

I I

we need o n l y

and we do indeed want t o

+

E t o p e r m i t such growth.

L e t E = C"( R3)

Example 6.12

R so

2 0; such f u n c t i o n s are abundant,

b u t o f course t h e y increase as t work i n " l a r g e " enough spaces

- 6.10

E

x

C"( R3) where C"( R3) has

the Schwartz topology and r e c a l l t h e mean value operators p,(t) and A x ( t ) d e f i n e d i n S e c t i o n 2 which we extend as even f u n c t i o n s f o r t negative. (6.38)

A

=

Define 0 (A

1

) ; A 2 =

79

A (0

0 A1

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where A i s the three-dimensional Laplacian.

Then A generates a

l o c a l l y equicontinuous group T ( t ) i n E a c t i n g by convolution (i.e., T ( t ) ?

=

T(t)

* 1) determined

by

One notes again t h a t T ( t ) i s not equicontinuous.

From (6.39) i t

follows t h a t cash A t

(6.40)

so t h a t cosh A t

=

72

u,(t) + gt 2 A 2A x ( t ) 0 provided

?2

0 and A2? 2 0.

Given

7

=

fl ( f ) t h i s means t h a t f i 2 0 w i t h A f i 2 0. T h u s i n order t o apply 2 Theorems 6.9 - 6.10 we want ? such t h a t A 2' f 2 0 and A4? 2 0 (i.e., 2 A f i 2 0 and A f i 2 0) w h i c h i s possible e.g., when f i ( x ) = i exp l y j x j . Again the necessity of using "large" spaces E i s apparent.

W e note a l s o from (6.39) t h a t T ( t ) and T ( - t ) behave

q u i t e d i f f e r e n t l y r e l a t i v e t o the preservation of p o s i t i v i t y . Analogues of these growth and convexity theorems f o r other "canonical" singular Cauchy problems appear i n Carroll [18; 19; 251 ( c f . a l s o Chapter 2 ) . 1.7

Transmutation.

The idea of transmutation of operators

goes back t o Delsarte and Lions ( c f . Delsarte [ l ] , DelsarteLions [2; 31, Lions [ l ; 2 ; 3; 4;

51) with subsequent contributions 80

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

by Carroll-Donaldson [ZO],

Hersh 131, Thyssen [l;23, e t c .

The

s u b j e c t has many y e t unexplored r a m i f i c a t i o n s and i s connected w i t h t h e q u e s t i o n o f r e l a t e d d i f f e r e n t i a l equations ( c f . BraggDettman, l o c . c i t . ,

Carroll-Donaldsoc [20],

Hersh [3]); we expect

t o examine t h i s more e x t e n s i v e l y i n C a r r o l l [24].

This section

t h e r e f o r e w i 11 be p a r t i a1 l y h e u r i s t i c and t h e word " s u i tab1 e"

w i l l be used o c c a s i o n a l l y when convenient w i t h p r e c i s e domains of d e f i n i t i o n u n s p e c i f i e d a t times. One f o r u m l a t i o n goes as f o l l o w s .

L e t Dx = a/ax and Dt =

a / a t and c o n s i d e r polynomial d i f f e r e n t i a l operators P Q = Q(D)

(D = D, o r Dt).

= P(D) and

One says t h a t an o p e r a t o r B transmutes

P i n t o Q i f ( f o r m a l l y ) QB = BP.

B w i l l u s u a l l y be an i n t e g r a l

o p e r a t o r w i t h a f u n c t i o n o r d i s t r i b u t i o n k e r n e l and i n f a c t one o f t e n assumes t h i s a p r i o r i , a l t h o u g h i t i s perhaps t o o r e s t r i c t i v e ( c f . Remark 7.3).

One p i c k s a space o f f u n c t i o n s f ( t h e

choice i s v e r y i m p o r t a n t ) and considers t h e problem

(7.2)

@(x,o) = f ( x ) ;

where 1 5 k 5 m

-

k Dt@(x,o) = 0

1 w i t h m = o r d e r Q.

p u t @ ( o , t ) = ( B f ) ( t ) and w r i t i n g $ ( x , t ) sul t s

81

x,t

2

0

I n o r d e r t o d e f i n e B we = P(Dx)@(x,t) t h e r e r e -

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

One may w i s h t o extend f and P (D )f a p p r o p r i a t e l y f o r x < 0 when d e s i r a b l e o r necessary. (7.5)

There r e s u l t s f o r m a l l y

uJ(o,t)

and e v i d e n t l y i t have const ant c o e f f i c i e n t s .

-

t h a t (7.1)

(7.2)

I t i s however necessary t o suppose

have a unique s o l u t i o n i n o r d e r t o w e l l d e f i n e

-

B; t h e same uniqueness c r i t e r i o n a p p l i e s t o (7.3)

(7.4)

but

t h i s w i l l o f t e n be t r i v i a l i f f i s chosen i n t h e r i g h t space. Theorem 7.1 (7.4))

Suppose t h e problem (7.1)

has a unique s o l u t i o n

+

-

(7.2)

(resp. (7.3)

(resp. $) w i t h f i n a s u i t a b l e

space so t h a t (7.5) makes sense and d e f i n e g e n e r i c a l l y ( B f ) ( t ) = +(o,t).

Then B transmutes P(D) i n t o Q(D). 2 L e t P(D) = D

L e t us i n d i c a t e some simple a p p l i c a t i o n s .

-

and Q(D) = D so t h a t (7.1)

(7.2)

become

We prolong f as an even f u n c t i o n and take p a r t i a l F o u r i e r t r a n s A 2^ s ( F f = f ) t o o b t a i n +t = - s @ w i t h i ( s , o ) = f ( s ) . A

forms x

+

A

The s o l u t i o n i s + (s ,t)

2 f ( s ) exp ( - s t ) and i f A

=

A

82

A

-

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

(7.7)

2

i t i s known t h a t FxR(x,t) Hence ( R ( * , t )

*

= exp ( - s t ) ( c f . Titchmarsh [Z]).

f ( - ) ) ( x ) = @(x,t) and r e c a l l i n g t h a t f i s even

there r e s u l t s

(7.8)

@ ( o y t )= ( B f ) ( t ) =

( c f . Carroll-Donaldson [ZO]).

(7.5) w i l l h o l d provided f

I

c

f(S)K(S,t)dS

An easy c a l c u l a t i o n shows t h a t

(0) =

0 and o f course f, f

I

, and

f

I 1

must have s u i t a b l e growth a t i n f i n i t y .

Now t h e r e i s no d i f f i c u l t y i n extending t h e transmutation i d e a t o v e c t o r f u n c t i o n s f w i t h values i n a complete separated l o c a l l y convex space E.

For example suppose t h a t A i s a s u i t a b l e

(closed, densely d e f i n e d ) o p e r a t o r i n E w i t h g(A) a reasonable o p e r a t o r f u n c t i o n and l e t P(D)w = g(A)w where w takes values i n E ( a c t u a l l y i n D(g(A)) s i n c e A may n o t be everywhere defined).

I f B transmutes P i n t o Q then f o r m a l l y Q(D)Bw = BP(D)w = Bg(A)w = g(A)Bw p r o v i d e d t h a t w s a t i s f i e s t h e c o n d i t i o n s necessary f o r t h e t r a n s m u t a t i o n and t h a t Bw

E

D(g(A)).

I n t h i s event u = Bw w i l l

s a t i s f y Q(D)u = g(A)u and t h e r e a r i s e s t h e general q u e s t i o n o f when t h i s s i t u a t i o n can p r e v a i l .

I f B i s an i n t e g r a l o p e r a t o r o f

Riemann type w i t h a f u n c t i o n k e r n e l then g(A) can be passed under t h e i n t e g r a l s i g n and t h i s p a r t o f t h e q u e s t i o n i s t r i v i a l up t o t h e p o i n t where i n i t i a l o r boundary values a r i s e .

As an ex-

ample l e t us consider t h e case ( c f . Carroll-Donaldson [ZO])

83

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(7.9)

2 2 P(D)w = D w = A w;

(7.10)

2 Q(D)u = Du = A u;

w(o) = uo;

wt(o) = 0

u ( 0 ) = uo

IfA2 = B t h e s e a r e o f c o u r s e wave and h e a t e q u a t i o n s and t h e

c o n n e c t i n g f o r m u l a i s w e l l known, b u t we w i l l d e r i v e i t v i a t r a n s mutation.

F i r s t f o r m a l l y we use (7.8)

obtain u = Bw =

(7.11)

t r a n s m u t i n g D2 i n t o D t o

c

w(<)K(S,t)dS I

and we n o t e t h a t wt(o) = 0, c o r r e s p o n d i n g t o f ( 0 ) = 0, as r e q u i r e d i n (7.8). sumed.

S u i t a b l e g r o w t h o f w, wt, and wtt w i l l be as2 2 F u r t h e r A Bw = aA w f o r t > 0 w h i l e f o r t = 0 one must

require u

0

E

D ( A ~ ) . Consequently

Theorem 7.2 g i v e n by ( 7 . U )

wt,

and wtt

Given w a s o l u t i o n o f (7.9) s a t i s f i e s (7.10),

p r o v i d e d uo

i t follows t h a t u E

2 D(A ) w h i l e w,

grow s u i t a b l y a t i n f i n i t y .

Such r e s u l t s a r e n o t new o f course, a t l e a s t i n more c a s s i c a l forms, and we r e f e r t o Bragg-Dettman, Donaldson [ZO], Remark 7.3

Hersh [3],

loc. cit.,

Lions, l o c . c i t . ,

One can r e l a x (7.2)

Carrol

-

and r e f e r e n c e s t h e r e .

and s t u d y n o t w e l l posed

Cauchy problems ( o r o t h e r problems) w h i c h l e a d t o a unique s o l u t i o n @ ( c f . C a r r o l l - D o n a l d s o n [ZO]);

f o r example growth c o n d i -

t i o n s on @ and f c o u l d be imposed (see h e r e C a r r o l l [24]).

Lions

i n h i s e x t e n s i v e i n v e s t i g a t i o n s , chooses spaces o f f u n c t i o n s f 84

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE

where the transmutation i s an isomorphism in the sense t h a t

there i s an inverse transmutation B - ' sending Q ( D ) i n t o P ( D ) ( i . e . PB-

era

=

B-lQ) b u t t h i s seems t o be an unnecessary luxury in gen-

, although obviously of g r e a t i n t e r e s t ( c f . Section 4 . 3 ) .

The choice of spaces f o r f i s i n any event of paramount importance f o r the transmutation method t o work.

I f B i s an i n t e g r a l opera-

t o r w i t h a function o r d i s t r i b u t i o n kernel one can begin with this a s a s t i p u l a t i o n and then discover the d i f f e r e n t i a l problem w h i c h the kernel must s a t i s f y ( n o t necessarily a simple Cauchy

problem), and we r e f e r t o Carroll [24; 251, Carroll-Donaldson [20],

Hersh [3], Lions, loc. c i t . , e t c . f o r f u r t h e r information

( c f . a l s o (7.25)).

We w i l l conclude this section w i t h a version of Lions trans2 2 mutation method sending P(D) = D i n t o Lm = D + t D (cf. a l s o Section 4.3). T h u s we look f o r a function $ ( x , t ) s a t i s f y ing

(7.12)

@xx = 4 t t

+

?m + 1 t t ;

and again we extend f t o be even w i t h f

I

(0)=

0.

This i s of

course t h e one dimensional EPD equation whose solution is given by ( 6 . 3 ) a s ( c f . Theorem 4.2)

(7.14)

$(o,t) = (Bf)(t)

=

2cmt-2m

1

t

0

85

( t -T

1 ) m-Ff(-r)d.r

SINGULAR AND DEGENERATE CAUCHY PROBEMS

which agrees w i t h Lions [l;2; 3; 51. (7.15)

1

t u = Bw = 2c t-2m ( t m

Consequently

1 ) m-Pw(-c)d-c

-T

0

i s t h e v e c t o r v e r s i o n o f (7.14) and we have Theorem 7.4 g i ven by (7.15) Proof:

I f w s a t i s f i e s (7.9) w i t h uo

E

D(A2) then u

s a t i s f i e s Lmu = A 2u w i t h u ( o ) = uo and u t ( o ) = 0. I

E v i d e n t l y (Bw)(o) = w(o) and (Bw) ( 0 ) = 0 ( c f . L i o n s

[l]where an e x p l i c i t v e r i f i c a t i o n o f (7.5)

i s also given

-

the

I

c o n d i t i o n w (0) corresponding t o f ( 0 ) = 0 i s c r u c i a l here as i n t Again A2 can be passed under t h e i n t e g r a l (7.8) and (7.11)). s i g n i n (7.15) as b e fo re i n (7.11). Remark 7.5

QED

Changing v a r i a b l e s i n (7.15) we o b t a i n (6.6)

when W (T ) i s w r i t t e n as W(T ) = cosh

Awe.

A c t u a l l y i n t h i s case t h e r e i s a l s o an i n v e r s e t r a n s m u t a t i o n o p erat or B - l b u t we w i l l n o t go i n t o d e t a i l s here (see Lions, l o c . cit.. f o r an exhaustive study o f t h e m a t t e r ) . I n Lions [3] 2 i t i s a l s o shown how t o transmute P(D) = D i n t o Q(D) = Mm = 2 D + D + p ( t ) D + q ( t ) and v i c e versa, and we w i l l sketch t some o f t h e c a l c u l a t i o n s here (p and q a r e assumed t o be conti n uous) ( c f . a l s o Hersh [S]).

Thus l e t $(x,t)

be t h e s o l u t i o n

of

(7.16) (7.17)

$(x,o)

= f(x);

$ t ( ~ , ~ =) 0

86

(x,t 2 0 )

1.

SINGULAR PARTIAL DIFFERENTIAL EQUATIONS OF EPD TYPE I

w'here f i s extended as an even f u n c t i o n w i t h f ( 0 ) = 0.

Then

again t he t rans mu ta ti o n o p e ra to r B i s determined by ( B f ) ( t ) = and s a t i s f i e s f o r m a l l y t h e tra n s m u t a t i o n r e l a t i o n MmB =

r$(o,t)

BD2, w h i l e t o c o n s t r u c t B L i o n s uses t h e f o l l o w i n g ingenious

"trick."

He sets

(7.18)

B(cos x s ) ( t ) = 0 (s ,t)

and from t h e t r a n s mu ta ti o n r e l a t i o n p l u s t h e f a c t t h a t ( B f ) ( o ) = f ( o ) we o b t a i n (7.19) w i t h e(s,o)

2 MmO(s,t) + s 0 ( s , t ) = 1 , w h i l e et(s,o)

s o l u t i o n s e(s, t )

= 0 = 0 a u t o m a t i c a l l y (.for unique

see Section 5).

Assuming now (we o m i t v e r i f i -

c a t i o n ) t h a t B i s d e fi n e d by a ( d i s t r i b u t i o n ) k e r n e l b ( t , x ) w i t h support i n t h e r e g i o n

1x1

5 t, which i s l e g i t i m a t e here b u t i s

i n p a r t i n c i d e n t a l t o t h e tra n s mu ta ti o n concept, one obtains, s i n c e f ( x ) = cos xs i s even, and b ( t , * ) w i l l be even ( v e r i f i c a t i o n f o l l o w s from (7.25) f o r example) (7.20)

0(s,t)

= 2

t

b(t,x)

cos xs dx = 2

b(t,x)

cos xs dx

0

from which f o l l o w s , by F o u r i e r i n v e r s i o n i n a d i s t r i b u t i o n sense, (7.21)

b(t,x)

Theorem 7.6

=

1

O(s,t)

cos xs ds

0

The tra n s mu ta ti o n o f D

2

i n t o Mm i s determined

by an operat or B w i t h k e rn e l d e fi n e d by (7.21),

87

where €Ii s known.

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

F u r t h e r p r o p e r t i e s a r e developed by Lions [3;

51 and a gen-

e r a l t ransmut ati o n t h e o r y w i l l be developed i n C a r r o l l [24]. Hersh's technique ( c f . Hersh 131) i s based on s p e c i a l f u n c t i o n s and k e r n e l s and leads f o r m a l l y t o t h e same r e s u l t as t h a t o f Theorem 7.6. w(t,A)

L e t us sketch t h i s f o r completeness.

Thus l e t

be t h e s o l u t i o n o f 2 Mmw + A w = 0;

(7.22)

w(o,A)

= 1;

wt(oyA) = 0

w h i l e u s a t i s f i e s ( c f . (7.10)) 2 2 D u + A u = 0;

(7.23) THen u(t,A)

=

u(0,A)

= 1;

ut(o,A)

=

0

cos A t and Hersh proposes a formula m

(7.24)

w(t,A)

b(t,x)u(x,A)dx

=

b(t,x)

=

cos Ax dx

m ,

Since (7.22) hol d s we have, p ro v i d e d b ( t , - ) w i t h reasonable smoothness (7.25)

,

Mmb(t,x) = bxx(t,x)

Th i s shows symmetry o f b (t,x ) (7.26)

has compact support

w(t,A)

=

2

[

b (t,x )

i n x and (7.24) becomes ( c f . (7.20)) cos Ax dx

Again (7.25) can be solved, as can (7.19),

by t h e methods o f

L

D e f i n i n g now B by ( B f ) ( t ) = b(t,x) f ( x ) dx we 2 have f o r m a l l y from (7.25) MmB = BD The i d e a then i s t o r e p l a c e Section 1.5.

.

A 2 by s u i t a b l e o p e ra to r f u n c t i o n s -g(A).

88

Chapter 2 Canonical Sequences of Singular Cauchy Problems 2.1

The rank one situation.

In t h i s chapter we will

develop the group theoretic version of the EPD equations studied i n Chapter 1.

This leads t o many new classes of equations and

results parallel t o those of Chapter 1 as well as t o some i n teresting new situations.

Moreover i t exhibits the main results

of Chapter 1 i n t h e i r natural group theoretic context.

The

ideas of spherical symmetry, radial mean val'ues and Laplacians, etc. inherent i n EPD theory have natural counterparts i n terms of geodesic coordinates and one can obtain recursion relations,

Sonine formulas, etc. group theoretically.

The results are

based on Carroll [21; 221 in the semisimple case and were anticipated i n p a r t by e a r l i e r work of Carroll [18; 191, CarrollSilver [l5; 16; 17) and Silver [ l ] f o r some semisimple and Euclidean cases.

The group theory i s "routine" a t the present

time and r e l i e s heavily on Helgason's work (cf. Helgason [ l ; 2 ; 3; 4 ; 5; 6; 7; 8; 9; 101) b u t one must of course refer t o basic

material of Harish-Chandra (cf. Warner [ l ;

21

for a sumnary) as

well as lecture notes by Varadarajan, Ranga Rao, e t c . ) ; other specific references t o Bargmann [ l ] , Bargmann-Wigner [2], Bhanu-Munti [l], Carroll [27; 281, Coifman-Weiss [l], EhrenpreisMautner [ l ] , Flensted-Jensen [1], Furstenberg [ l ] , Gangolli [ l ] , Gelbart [ l ] , Gelfand e t a l , [ l ; 2 ; 3; 41, Godement [ l ] , Hermann 89

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

[ l ] , Jacquet-Langlands [l], Jehle-Parke [l], Kamber-Tondeur [l], Karpelevic" [ l ] , Knapp-Stein [l], Kostant [l], Kunze-Stein [ l ; 23, Lyubarskij [ l ] , Maurin [l], McKerrell [l], Miller [ l ; 23, Naimark [ l ] , Pukansky [ l ] , RGhl [l], Sally [ l ; 23, S i m s [ l ] , Smoke [ l ] , Stein

113,

Takahashi [l], Talman [ l ] , Tinkham [ l ] ,

Vilenkin [ l ] , Wallach [ l ] , Wigner [ l ] , etc., as well as the fundamental work of E. Cartan and H. Weyl, are t o be taken f o r granted, even i f n o t mentioned explicitly.

The basic Lie theory

i s developed somewhat concisely i n t h i s section; for a rather more leisurely treatment we refer t o Bourbaki [5], Carroll [23], Hausner-Schwartz [l], Helgason [ l ; 2; 31, Hochschild [ l ] , Jacobson [1], Loos [ l ] , Serre [ l ;

21,

V

Tondeur [ l ] , Zelobenko

Ell, etc. W e will s t a r t o u t w i t h the f u l l machinery f o r the rank one semisimple case, following Carroll [21; 221, and l a t e r will give extremely detailed examples f o r special cases.

T h i s avoids

some repetition and presents a "clean" theory immediately; the reader unfamiliar w i t h Lie theory m i g h t look a t the examples f i r s t where many d e t a i l s and definitions are covered.

We delib-

erately omit the treatment of invariant differential operators acting in sheaves o r i n sections of vector bundles even t h o u g h t h i s i s one of the more important subjects i n modern work (some references are mentioned above).

The preliminary material will

be expository and specific theorems will n o t be proved here. The Eucl idean group cases have been covered in Carroll -Si 1 ver [15; 16; 171 and especially Silver [ l ] s o that we will only give

90

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

a few remarks l a t e r about t h i s a t t h e end o f t h e chapter; t h e b a s i c r e s u l t s a r e i n any event i n c l u d e d i n Chapter 1.

Thus

l e t G be a r e a l connected noncompact semisimple L i e group w i t h f i n i t e c e n t e r and K a maximal compact subgroup so t h a t V = G/K -

.

-

d

L e t g = k + p be a

i s a symmetric space o f noncompact type.

Cartan decomposition, a c p a maximal a b e l i a n subspace, and we

-

-

w i l l suppose u n t i l f u r t h e r n o t i c e t h a t dim a = rank V = 1. One s e t s A = exp a, K = exp k, and

IsA for

N

-

= exp n where n =

A > 0 where t h e gA a r e t h e standard r o o t spaces corres-

ponding t o p o s i t i v e r o o t s a and p o s s i b l y 2a i n t h e rank one case.

1

One s e t s p = - c m A f o r 2 x element Ho

E

X > 0 where mA

= dim gA and we p i c k an

a such t h a t a(Ho) = 1. Thus p =

1 (H ma

+ m2a)a and

we can i d e n t i f y a Weyl chamber as a connected component a + C I

a c a w i t h (o,m) to

u

E

*

a

i n w r i t i n g a ( t H o ) = t where

by u ( t H o ) = U t .

u

E

R corresponds

The Iwasawa decomposition o f G i s G =

KAN which we w r i t e i n t h e form g = k ( g ) exp H(g) n ( g ) where t h e n o t a t i o n at = exp t H o i s used.

I

L e t M (resp. M ) denote t h e

c e n t r a l i z e r (resp. n o r m a l i z e r ) o f A i n K so t h a t t h e Weyl group I

i s W = M /M and t h e maximal boundary o f V i s B = K/M ( t h u s M = {k

E

H

K; AdkH = H f o r

E

a) and M

I

=

{k

E

K; Adka c a )

-

see

t h e examples f o r s p e c i f i c d e t a i l s ) . There are n a t u r a l p o l a r c o o r d i n a t e s i n a dense submanifold o f V a r i s i n g from t h e decomposition G = (kM,a)

-+

G+K (A+

=

exp a+) p r o v i d e d by t h e diffeomorphism

kaK : B x A+-+ V (one c o u l d a l s o work w i t h t h e decom-

p o s i t i o n G = KAK).

Thus t h e p o l a r coordinates o f ~ ( g )=

91

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

ak2)

V a r e (klM,a)

E

g i v e n v = gK -1

E

where

:G

IT

V and b = kM

+ .

V i s t h e n a t u r a l map.

B one w r i t e s A(v,b)

E

k ) and t h e F o u r i e r t r a n s f o r m o f f

=

2 L ( V ) i s d e f i n e d by

E

Y

F f = f where ( u s i n g Warner's n o t a t i o n ) Y

f(u,b)

(1.1) for

u

E

a

*

and b

= E

r 1, f(v) B.

exp (iu+p)A(v,b)dv

--

A l l measures a r e s u i t a b l y normalized i n

t h i s treatment. T h i s s e t s up an i s o m e t r i c isomorphism f f 2 * between L ( V ) and t h e space L2(a: x B) ( c f . here below f o r a+) w i t h i n v e r s i o n formula

Here t h e 1 / 2 comes from t h e o r d e r o f t h e Weyl group (which i s two) and c ( u ) i s t h e standard Harish-Chandra f u n c t i o n .

For a

general expression o f C(M) we can w r i t e ( c f . a l s o (3.14)) c ( u ) = I(ip)/I(p)

where ( c f . G i n d i k i n - K a r p e l e v i c [l],Helgason

[3; 81, Warner [l;21)

Here B(x,y)

= I'(x)r(y)/I'(x+y)

i s t h e Beta f u n c t i o n and one

d e f i n e s (v,a) i n t h i s c o n t e x t i n terms o f t h e K i l l i n g form B(-,*)

as B(H ,Ha)

A(h) = B ( j , H )

V

where f o r X

for H

E

l i n e a r maps o f a i n t o

E

*

aC, HA

E

*

aC i s determined by

a ( n o t e here t h a t aC i s t h e space of R w h i l e aC i s t h e c o m p l e x i f i c a t i o n o f a

which i s f o r m a l l y t h e s e t o f a l l sums A + i u f o r

92

A,u

E

a).

Then

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

*

a+ i s t h e preimage o f a+ under t h e map A chamber i n a later.

* . *

HA and i s a Weyl

S p e c i f i c examples o f c ( p ) w i l l be w r i t t e n down

Now a / W

- a+* (.recall W

= (1,s)

o r e q u i v a l e n t l y sat = a-t = a t ’ )

where, f o r $

+.

where sH = -H f o r

H

E

a

and one can w r i t e

L2 (B),

E

The q u a s i r e g u l a r r e p r e s e n t a t i o n L o f G on L2 (V), d e f i n e d by L ( g ) f ( v ) = f(g-’v)

(1.5)

L =

decomposes i n t h e form

r Ja*,W

‘plC(w)

dp

where L a c t s i n H by t h e same r u l e as L. L i s i n f a c t i r P Fc 1-1 r e d u c i b l e and u n i t a r y and i s e q u i v a l e n t t o t h e s o - c a l l e d c l a s s one p r i n c i p a l s e r i e s r e p r e s e n t a t i o n induced from t h e p a r a b o l i c subgroup MAN by means o f t h e c h a r a c t e r man

-f

a”

= exp i p l o g a.

We r e c a l l here a l s o t h e d e f i n i t i o n o f t h e mean value o f a f u n c t i o n @ over t h e o r b i t o f g r ( h ) = gu under t h e i s o t r o p y = gKg-’ subgroup Iv

Thus w r i t i n g MhC$ = Mu$ we have

a t v = .rr(g).

( r e c a l l i n g t h a t r ( h ) = u)

93

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

( c f . Helgason [l]) when D

D(G/K).

E

The symbol D(G/K) denotes

t h e l e f t i n v a r i a n t d i f f e r e n t i a l o p e r a t o r s on G/K which a r e d e f i n e d as follows.

If $ : G

G i s a diffeomorphism and f i s

-+

a f u n c t i o n on G one sets f $ ( g ) = ( f

while i f D i s a -1 l i n e a r d i f f e r e n t i a l o p e r a t o r we w r i t e DQ : f + (Of$ )$. Thus

G o r on V = G/K.

$-')(g)

W r i t i n g akg = gk f o r g

G and k T

space of d i f f e r e n t i a l o p e r a t o r s on G such t h a t D D i s denoted by D(G/K);

E

K, t h e

= D and D"k =

t h i s i s i d e n t i c a l w i t h t h e space o f

l i n e a r d i f f e r e n t i a l o p e r a t o r s on G/K ( c a l l e d l e f t i n v a r i a n t ) such t h a t

T

D

= D (the condition

proof obvious).

Dak

D i s automatic on G/K

=

-

Now t h e zonal s p h e r i c a l f u n c t i o n s on G a r e

d e f i n e d by t h e formula

(u

E

*

a )

. ..,

and we can w r i t e by i n v a r i a n t i n t e g r a t i o n 41 (9) = $ (gK) (ac-

u

..,

u

..,

t u a l l y t h e $ (9) a r e K b i i n v a r i a n t i n t h e sense t h a t $ ( 9 ) =

u

* -

..,

$ ( k g k ) ) and K i s u n i m d u l a r

u

-

u

c f . Helgason [l], Maurin [l],

-

Nachbin [2], Wallach [l],o r Weil [l] and thus

.,K CI$

f(k-')dk

-u

f(ikk)dk = ..,

f(k)dk. I t i s known f u r t h e r t h a t $,(g) = K ( g - l ) ( c f . Harish-Chandra [2] o r Warner [l;21) w h i l e t h e

E

=

C"(G/K),

@u X ($ ) u ..,u of

1

I,

for D

and a r e c h a r a c t e r i z e d b y t h e i r eigenvalues AD = E

D(G/K), w i t h $u(n(e)) = 1, p l u s t h e b i i n v a r i a n c e

(I~.We now demonstrate a lemma which has some i n t e r e s t i n g

consequences.

94

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

h The F o u r i e r t r a n s f o r m o f 1.1 = MU

Lemma 1.1

.rr(h)) i s g i v e n b y FMh = (M‘f)(v)

-

A

1-I

(h) and i f h = kak w i t h a

Proof:

A+ then

o f u = .rr(h).

h The a c t i o n o f M o r MU as a d i s t r i b u t i o n i n E ’ ( V )

determined by ( c f . (1.6)) b e i n g t h e i d e n t i t y i n G).

<M h ,$>

=

h (14 $ ) ( w ) where w

Thus i n (1.6)

A

is

r ( e ) (e

=

take Q = exp (ill+ p )

A

where b = kM.

Since A(ekhK,kM)

t h e u n i m o d u l a r i t y o f K ( c f . (1.8) (1 -9)

E

(u =

( M a f ) ( v ) so t h a t (Muf)(v) depends o n l y on t h e r a d i a l

=

component a i n t h e p o l a r decomposition (kM,a)

A(w,b)

E’(V)

E

-

(Mh$)(w) =

I

= -H((kh)-’;)

one has by

and comments t h e r e a f t e r )

exp [-(ip+p)H(h -1 k -1,- k)dk

K

j

=

exp [-(ip+p)H(h -1 kk)dk =

K

,. = $

-1-1

(h-’)

=

-

0

1-I

j

e-(i’+p)H(h-lk)dk

K h (h) = FM

h One notes here t h a t M = MU works on f u n c t i o n s i n V such as exp (i’+p)A(v,b) placed by 14‘

= $

h

= M

h and <M ,I$> i s p r e c i s e l y (1.1) w i t h f r e -

and $ = exp (iu+p)A(w,b);

no i n t e g r a t i o n over

V i s i n v o l v e d s i n c e v = w i n $, o n l y a d i s t r i b u t i o n e v a l u a t i o n A

i s o f concern here.

-

F i n a l l y we n o t e t h a t w i t h h = kak as above

t h e r e f o l l o w s from (1.6) (1.10)

(Mhf)(v) =

J K f(gkiaK)dk

=

i,

f(gkaK)dk = (Maf)(v)

T h i s l e a d s t o a symmetric space v e r s i o n o f theorem o f Zalcman [l],g e n e r a l i z i n g an o l d formula o f P i z z e t t i ; here G/K

95

QED

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

i s of a r b i t r a r y rank and we refer t o Helgason [l ; 23 o r Warner [ l ; 23 f o r general information on h i g h e r rank s i t u a t i o n s . Theorem 1.2

Suppose the o p e r a t o r s

(Pizzetti-Zalcman).

A k w i t h eigenvalues X k g e n e r a t e D(G/K) as an a l g e b r a ; then l o -

cal l y (1.11)

(MUf)(V)

=

bqu,

Ak)f)(V)

00

1 Pn(u,X k ( 4 v ) ) i s expressed i n terms o f i t s n=l eigenvalues a s $ (u,Xk).

where $,(u)

= 1 +

P

Zalcman [ l ] , i n an Rn c o n t e x t , has g e n e r a l i z e d t h i s kind o f theorem considerably Proof:

.

h

One n o t e s from Helgason [ l ] t h a t (M @,)(v)

-

i,

=

= z X ( g ) $ X ( h ) and the l o c a l expres$X(gk.rr(h))dk = ;Jgkh)dk K O3 sion (Muf)(v) = ([1 + 1 Pn(u,Ak)]f)(v) holds. Consequently n=l we o b t a i n $ X ( h ) = applying t h i s l o c a l e x p r e s s i o n t o

-

m

1 P , ( U , A ~ ( $ ~ ) =) $,(u) and (1.11) follows. One should n=l remark a l s o t h a t the polynomials P a r e without cons t a n t terms. 1 +

n

2.2

Resolvants.

The o b j e c t s o f interest n a g e n e r a l i z e d

EPD theory a r e the r a d i a l components o f a b a s i s f o r the H

v spaces

of (1.4), m u l t i p l i e d by a s u i t a b l e weight funct on, the r e s u l t

o f w h i c h we w i l l denote by @ ( t , u ) ( c f . C a r r o l l [21; 221, C a r r o l l S i l v e r [15; 16; 171, S i l v e r [ l ] ) and we mention t h a t m can denote

96

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

a m u l t i i n d e x here.

F i r s t we remark t h a t V i s endowed w i t h t h e

Riemannian s t r u c t u r e induced by t h e K i l l i n g form B ( * , * )

(but

t h e Riemannian s t r u c t u r e does n o t p l a y an i m p o r t a n t r o l e here, e.g.,

$(*,*)

c o u l d serve

-

see Example 3.2) and f o r rank

V

=

1, D(G/K) i s g e n e r a t e d by a s i n g l e Laplacian A, determined by

-

t h e standard Casimir o p e r a t o r C i n t h e enveloping algebra o f g ( c f . Remark 3.15 about C).

We l o o k a t t h e r a d i a l component AR o f

passing t h i s from t h e coord n a t e t i n at E A t o r ( A ) i n an a obvious manner, and s e t t i n g Mt = M w i t h Ro(t,p) = FMt = $,,(at) A,

-

A

one o b t a i n s an eigenvalue e q u a t i o n ( c f . Helgason [5]) (2.1)

[of. +

(ma+mZa)cotht Dt + mZatanht D

141

t u

The s o l u t i o n o f (2.1), a n a l y t i c a t t = 0, i s e.g., A

(2.2)

^Ro(tyu) = (I (exp t H o ) = F(G,B,y,

u

"0

"0

where e v i d e n t l y R ( 0 , ~ =) 1 and Rt(o,p)

B

=

-

2

sh t )

0 ( 6 = (ma+2m2a+ 2 i p ) / 4 ,

= ( m +2mZa-2iu)/4,

and y =(ma+m2a+1)/2). The general EPD a "0 s i t u a t i o n i n v o l v e s embedding R i n a "canonical" sequence o f "m l l r e s o l v a n t s " R (t,p), f o r m > 0 a p o s i t i v e i n t e g e r o r a m u l t i index, such t h a t t h e r e s o l v a n t i n i t i a l c o n d i t i o n s (2.3)

"m R ( 0 , ~ =) 1;

are s a t i s f i e d .

"m Rt(oYp) = 0

There w i l l a l s o be "canonical" r e c u r s i o n r e l a t i o n s

97

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

“m between t h e R of i n d i c e s d i f f e r i n g by k1 o r k 2 which w i l l a r i s e group t h e o r e t i c a l l y f r o m c o n s i d e r i n g a f u l l s e t o f b a s i s elements i n the H

spaces.

1-I

2 Thus we must f i r s t determine a b a s i s f o r L ( B ) and t h i s i s w e l l known (thanks a r e due here t o R. Ranga Rao f o r some h e l p f u l information).

We l e t { I T ~ , V ~ Iw i t h dim VT = dT be a complete s e t

o f i n e q u i v a l e n t i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s o f K and l e t

M VT C V T be t h e s e t o f elements f i x e d by M.

One knows by a r e -

s u l t o f Kostant [l]t h a t dim V y = 1 o r 0 i n t h e rank one case ( c f . a l s o Helgason [5]) and f o r t h e s e t

T E

we l e t w i be a b a s i s v e c t o r f o r VTM w i t h

T where dim

wY(1

2

normal b a s i s f o r VT under a s c a l a r p r o d u c t < example t h e c o l l e c t i o n o f f u n c t i o n s

(T E

T)kM

= 1

i 5 dT) an o r t h o >T.

-+

!V

Then f o r

<wi,aT(k)w;>T

is

2 known t o be a b a s i s f o r L ( B ) and we d e f i n e ( c f . (1.4) w i t h v = atK and b = kM) n -

(2.4)

$;,(atK)

= El 9T(BT:-ip:a -1 )w T

t

where BT

E

j

HomM(VT9$) i s determined by t h e r u l e Bw‘:

= 61,s

(Kronecker symbol) so t h a t B T r T ( k )-1 wT. = <W:,IT (k)w;>T = J J T <~~(k)-’w;,w;>~ (note t h a t $(k)dk = $(km)dm)db and K B M

J

I (I

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

H(g-’km)

= H(g-’k)

w i t h nT(km) = nT(k)nT(m)).

above

The

are E i s e n s t e i n i n t e g r a l s as d e f i n e d i n Wallach [l]and t h e A .

( a K) a r e t h e r a d i a l components o f b a s i s elements i n H t FC ( c f . below). I t i s p o s s i b l e t o o b t a i n an e x p l i c i t e v a l u a t i o n o f M,T

these f u n c t i o n s , u s i n g r e s u l t s o f Helgason [5;

111 as f o l l o w s .

One d e f i n e s ( c f . Helgason 151)

Then i t i s proved i n Helgason [5;

f o r x = gK.

111 ( t o whom we

g r a t e f u l l y acknowledge some conversation .on t h e m a t t e r ) t h a t

where Y

,T

( x ) i s d e f i n e d by

j

e (-iA-p)H(g-’k) < wl,T a T ( k ) w i >T dk K Thus r e c a l l i n g t h e p o l a r decomposition (kM,a) + kaK we have yA,T(x) =

(2.7)

Theorem 2.1

;’

(2.8)

MrT

h

General b a s i s elements i n

(iatK) =

1

K

e (iu-p)H(at k

dT- l l 2 fT,J -’.(katK)

I n p a r t i c u l a r one has s i n c e ti1

H me FC

-1^-1

A

=

A

G’

k ) <wi,aT(k)w1>., T

dk

A T = < w i , ~ ~ ( k ) w Y-FC,T(atK) ~>~

;’

-’

( a K) = ( a K) = Y (atK) t 1-I.T t ,T ( t h i s “ c o l l a p s e ” was observed i n VaT

= <W?,IT ( l ) w T > ,j J T 1 T s p e c i a l cases by S i l v e r [l]). A

Now l e t KO = {nT,VTl f o r T ET (i.e.,

dim VTM = 1 ) and f o l l o w -

i n g Helgason [5] we use a p a r a m e t r i z a t i o n due t o Johnson and

99

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Wallach (see Helgason [5] f o r references and c f . a l s o Kostant h

[l]).I f mZcl = 0, KO = {(p,q)> w i t h p ..

1, KO = {(p,q)> w i t h (p,q)

E

E

Z+ and q = 0; i f mZa =

Z+ x Z where p

f

q

E

i f mZa =

22,;

h

3 o r 7, KO = {(p,q)>

w i t h (p,q)

E

Z+ x Z+ where p

&

q

E

One

22.,

s e t s R = (ip+p)(HO) and then ( c f . Helgason [5; 111) Theorem 2.2

The r a d i a l components o f b a s i s elements i n

H

IJ

are given by

= cI.1'

STt h P t

ch-'tF(-,--p

R+

where t h = tanh, ch = cosh, and

+

R+p-q+l -m2a

+

ma+m

+1

,th2t)

F i s t h e standard hypergeometric

function. Note here t h a t o u r R i s t h e n e g a t i v e o f t h e R i n Helgason [5;

111 where R = ( i A - p ) ( H ) = ( - i p - p ) ( H

i n (1.1))

0

0

) ( c f . here o u r n o t a t i o n

and r e c a l l t h a t a(Ho) = 1 w i t h at = exp tHo.

now sets dZa = -4q(q+mZcl-1) and da = -p(p+m +mZa-l) i t i s shown i n Helgason [5] t h a t Y ( t ) = Y

the d i f f e r e n t i a l equation

100

-1-I ,T

I f one

+ q(q+mZcl-l)

(atK) a l s o s a t i s f i e s

2.

(2.11)

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

ytt

+ (ma+in2a)coth t Yt + m2a tanh t i t

+

[d sh-2t + d2ash-22t + p(Ho) 2 + v(Ho) 2 19' = 0 a

where sh = s i n h , and we have used t h e i d e n t i t y c o t h 2 t = h c o t h t + tanh t). R e c a l l i n g t h a t p ."(-in21 a+m 2a ) one sees t h a t the t r i v i a l representation 1 g i v e s r i s e t o (2.1),

T, corresponding t o p = q = 0,

E

so t h a t (since c

R+I-M,

.w

1) we should have +I 2a , a 2a , t h t). This =

m +m

(atK) = +(at) = ch-'t F(L/2, ,1 i s borne o u t b y t h e Kummer r e l a t i o n F(a,b,c,z)

y-lJ

w i t h z = -sh 2 t, a = 6

c,z/z-1)

( l - ~ ) -= ~ch-'t

and c

-

b = y

B

=

(l-z)-aF(a,c-b,

b , = 8 , and c = y, so t h a t

= R/2,

-

=

(ma+2+2iv)/4 = (R+1-m2a)/2;

. "

hence Y

-v,1

2.3 h

(atK) = +(a ) as i n d i c a t e d . t = 0.

Examples w i t h

Now t o c o n s t r u c t r e s o l v a n t s

A

( a K) one m u l t i p l i e s t h e t by a s u i t a b l e f a c t o r i n o r d e r t o o b t a i n t h e r e s o l v a n t i n i -

Rm(t,v) = RPsq(t,p) from t h e Y

y-v ,T

-!J,T

. "

t i a l c o n d i t i o n s (2.3)

and t o produce I$ (a,)

v

when p = q = 0.

These requirements are n o t alone s u f f i c i e n t t o produce t h e "cano n i c a l " r e s o l v a n t s s i n c e one needs t o i n c o r p o r a t e c e r t a i n group t h e o r e t i c r e c u r s i o n r e l a t i o n s i n t o t h e t h e o r y which serve t o " s p l i t " t h e second o r d e r s i n g u l a r d i f f e r e n t i a l equation f o r t h e A

Rp,q

a r i s i n g from (2.11)

i n t o a composition o f two f i r s t o r d e r

equations ( c f . here I n f e l d - H u l l [ l ] ) .

Even then we remark t h a t

t h e r e s o l v a n t s w i l l n o t be unique since one can always m u l t i p l y ^Rm(t,p) by a f u n c t i o n $, I+

0

( t ) E 1.

E

C2 such t h a t $,(o)

=

1, $ i ( o ) = 0,

T h i s w i l l simply g i v e a d i f f e r e n t second o r d e r

101

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

s i n g u l a r d i f f e r e n t i a l equation f o r t h e new r e s o l v a n t and d i f -

-

ferent s p l i t t i n g recursion r e l a t i o n s b u t the resolvant i n i t i a l c o n d i t i o n s w i l l remain v a l i d and t h e r e d u c t i o n t o 0 (a ) f o r m = U

0 i s unaltered.

t

Thus we w i l l choose the s i m p l e s t form o f r e s o l -

vant f u l f i l l i n g t h e s t i p u l a t i o n s imposed w h i l e r e f e r r i n g t o these r e s o l v a n t s and equations as canonical. Example 3.1 -p

2

, d2,

Take t h e case where ma = 1 , mZa = 0, da =

= 0, and p = 1/2.

T h i s corresponds t o G = SL(2,IR)

with

K = S O ( 2 ) and t h e r e s o l v a n t s a r e given i n C a r r o l l [18; 19; 223,

C a r r o l l - S i l v e r [15; 16; 171, S i l v e r [l]as

where 5 = c h t and P i m denotes t h e standard associated Legendre f u n c t i o n o f t h e f i r s t kind.

Now we r e c a l l a formula ( c f . Snow

111, P. 18)

2 S e t t i n g z = t h t i n (3.2) and n o t i n g t h a t 1 have fi = sech t and 1 as above.

-

-

2 2 t h t = sech t we

n / ( l + Z ) = 5-1/<+1 w i t h 5 = c h t

Now one observes t h a t

102

(t,U)

i n (3.1)

i s symmetric

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

t h i s i s e x h i b i t e d f o r example i n the form of

i.n !J (and t ) and

t h e r e s o l v a n t (3.1) when one uses t h e formula ( c f . Robin [l]) 1 1 2-mshmt m+-+ip m+--ip 2 2 (3.3) P-m (cht) = F( , ,m+l,-sh t ) ip-

'2

z

= F(b,a,c,z))

Hence (3.1) can be w r i t t e n ( r e c a l l t h a t F(a,b,c,z) (3.4)

1 1 "m m+T+ip m+q-ip R (t,u) =F( , , m + 1, -sh 2 t ) Now, r e t u r n i n g t o (3.2) and (2.9)

T

-

-

(2.10), we w r i t e f o r and R = i u

( p , ~ ) w i t h ct = i d 2 , B = 1/2(p + 1/2),

= c

R

-!J,T

t h P t ch- t (

1 t sech t ) -

1/2

2(ct+B)

1

+2 + p,

= c-!J ,Ts h P t ( y ) - ' - P F ( i p

+

ill +

1

7, p+l,

5- 1

5+1)

R e w r i t i n g t h e f i r s t equation i n (3.1) w i t h p replaced by -!J i n t h e r i g h t hand s i d e we o b t a i n (3.6)

i m ( t , p ) = ($!-)

-R-m F (1z + ip,z+ 1 ip+m,m+l,-) 5-1

5+1

Then i d e n t i f y i n g p w i t h m we can say from (3.5) h

(3.7)

RP(t,p) =

sh-Pt Y

-!J,T

103

( a K) t

-

(3.6)

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

For completeness we check t h e d i f f e r e n t i a l e q u a t i o n s a t i s f i e d by t h e Rp o f (3.7),

g i v e n t h a t (2.11) holds.

An elementary c a l c u l a -

t i o n y i e l d s then

which agrees w i t h p r e v i o u s c a l c u l a t i o n s ( c f . C a r r o l l [21; C a r r o l l - S i l v e r [15;

223,

16; 171, S i l v e r [l]).The canonical r e c u r s i o n A

r e l a t i o n s a s s o c i a t e d t o t h e Rp o f (3.1),

(3.4),

(3.6),

o r (3.7)

can o f course be r e a d o f f from known formulas f o r t h e associated Legendre f u n c t i o n s f o r example b u t t h e y can a l s o be o b t a i n e d group t h e o r e t i c a l l y ( c f . Example 3.5 o f b a s i s elements i n t h e

-

Theorem 3.8)

H spaces. P

by u s i n g a f u l l s e t

For now we s i m p l y l i s t them

i n t h e form

i!

+ 2p c o t h t

^Rp

= 2p csch t

iP-'

E v i d e n t l y t h e composition o f these two r e l a t i o n s y i e l d s (3.8) and t h i s i s t h e sense i n which we speak o f " s p l i t t i n g " t h e r e s o l v a n t equation (3.8). Example 3.2 Example 3.1.

We g i v e now e x p l i c i t m a t r i x d e t a i l s t o c l a r i f y

(The E u c l i d i a n case can a l s o be t r e a t e d group

t h e o r e t i c a l l y as i n C a r r o l l - S i l v e r [15;

16; 171 and e s p e c i a l l y

S i l v e r [1] and a s i m p l e example i s worked o u t a t t h e end o f t h i s chapter; t h e r e s u l t s o f course agree w i t h those o f Chapter 1.)

104

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

-

Thus l e t G = SL(2,lR)

-

and K = SO(2) w i t h L i e algebras g = sR(2,R)

We r e c a l l t h a t G i s connected and semisimple,

and k = so(2).

-

c o n s i s t i n g o f r e a l 2 x 2 matrices o f determinant one, w h i l e the

- -

matrices i n t h e compact subgroup K a r e orthogonal; g consists o f r e a l 2 x 2 m a t r i c e s o f t r a c e zero and k c g i s composed o f skew We w r i t e V = G/K and s e t

symmetric matrices.

X =

(3.10)

1 7

-

O

1 1 ., Y = 1z

1

0

so t h a t k = lR Z and we w r i t e p = { J R X +.lRY) f o r the subspace

-

--

( o f v e c t o r spaces) w i t h [k,k] Cartan i n v o l u t i o n 8 :

-

5 +n

b r a automorphism o f g.

- -

One has a Cartan decomposition g = k + p

o f g spanned by X and Y.

-t

-

- -

c k, [p,p] c k , [k,p] c p and the

6-

Q

-

(6 E

k,

n

E

p ) i s a L i e alge-

Recall here t h a t i f P = exp p then G =

PK i s t h e standard p o l a r decomposition o f g

E

G i n t o a product The K i l l i n g

o f a p o s i t i v e d e f i n i t e and an orthogonal m a t r i x . form B(5,n) = t r a c e ad5

adn

(=4 t r a c e

"

CQ)

-

i s negative d e f i n i t e

on k and p o s i t i v e d e f i n i t e on p ( w i t h B(k,p) = 0).

One checks

e a s i l y t h a t X and Y form an orthonormal basis i n p f o r t h e s c a l a r product

1 ((6,~))= p(5,n); we

repeat t h a t t h e 1/2 f a c t o r

i s o f no p a r t i c u l a r s i g n i f i c a n c e here i n c o n s t r u c t i n g resolvants, etc. and i s used mainly t o be c o n s i s t e n t w i t h some previous work and w i t h t h e e x p o s i t i o n i n Helgason [l]. Now s e t t i n g Xa = X

+

Z, X-a

= X

- Z, and Ha

standard ( o r "canonical") t r i p l e ( c f . Serre [ l ] )

105

= 2Y we have a

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

= RX,

where t h e r o o t subspaces g,

(5

i z e d by t h e r u l e gA = w h i l e t h e map a : t Y r o o t ) i n t h e dual a

*

and g

.-E

-a

RX

=

are character-,

g; ad H< = X(H)S f o r a l l H

a = RY),

E

t determines then an element ( c a l l e d a

-+

( n o t e t h a t ,(Ha)

-a i s a l s o a Cartan subalgebra of g.

= 2 and

R

N

a

*

=

Ra).

Here a i s a maximal a b e l i a n subspace o f p and i n t h i s case a = .--

{I,,g,

> 01,

.--

N = exp n, and A

g = k a n

(3.13)

e t t

-

< IT)

= exp B Z exp t Y exp

[b

=

The Iwasawa decomposi-

= exp a.

e

t i o n o f G can be w r i t t e n ( 0 2

-.

Set now n = g,

exa

which we express as b e f o r e i n t h e form g = k(g) exp H(g)n(g).

[-:

Next we s e t M = {+ 1

I4 = M {k

E

u {+

:]I

f o r t h e c e n t r a l i z e r o f A i n K and

w i l l be t h e n o r m a l i z e r o f A i n K ( t h u s M =

K; AdkH = H f o r H

- . - -

I

E

a and M = { k

E

K; Adk a c a) ).

Again

I

W = M /I4 has o r d e r two and B = K/M i s e s s e n t i a l l y K w i t h t h e angle v a r i a t i o n c u t i n h a l f . b = kM

E

B A(v,b)

= -H(g-lk)

We r e c a l l t h a t f o r v = gK and here p = 1/2

106

l m A A (A>O)

E

V and equals

2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

1

Thus p ( t Y ) = t / 2 and f o r 1-1

p.

R, 1-1

E

"* a

E

i s determined again

-*

by t h e r u l e u ( t Y ) = tu(Y) = 1-1t (such l~ o f course exhaust a ct

- 1-1

(1.1)

= 1).

The F o u r i e r t r a n s f o r m F f o f f

E

and

L2 ( V ) i s given by

and a+ i s d e f i n e d as i n S e c t i o n 1, as i s t h e s c a l a r prod-

u c t (X,v) = B(HA,HV).

When 1-1

E

-* a

= 2 i t follows t h a t H

s i n c e B(Y,Y)

i s c h a r a c t e r i z e d as above then = p1

*

Y

and o b v i o u s l y a+ can 1-1 "*" thus be i d e n t i f i e d w i t h ( 0 , ~ ) . E v i d e n t l y f o r 1-1, v E a one has

Example 3.3

A geometrical r e a l i z a t i o n o f t h e present s i t u a -

t i o n can be described i n terms o f G a c t i n g on t h e upper h a l f a b p l a n e I m z > 0 by t h e r u l e g z = (az+b)/(cz+d) f o r g = ( c d) G.

E

Then K i s t h e i s o t r o p y subgroup o f G a t t h e p o i n t z = i

(i.e.,

K

i = i ) and t h e upper h a l f plane can be i d e n t i f i e d

w i t h G/K under t h e map gK d e s i c p o l a r coordinates

-+

g

(T,e)

i ( c f . Helgason [ l ] ) .

I n geo-

a t i t h e Riemannian s t r u c t u r e i s

2 2 2 2 described by ds =d-c + s i n h .cde where we use t h e m e t r i c tensor determined by gv(<,q)

for

= $(C,q)

5,

q

E

p.

I f w = r ( e ) (which

corresponds t o i under o u r i d e n t i f i c a t i o n s i n c e r ( e ) = K) then, denoting l e f t t r a n s l a t i o n i n G by

t h e Riemannian s t r u c t u r e i s g' determined b y g V ( ( T )*<,(T~)*T-I) = gw(S,q) ( f o r t h e n o t a t i o n ( T )* 9 9 see e.g. Kobayashi-Nomizu [ l ; 21). The geodesics on V through

w =.rr(e) a r e o f t h e form y : s

t h e square .rr(exp

T'

-+

T

(exp sC)w and i f

5

= aX

+ BY

E

o f t h e geodesic d i s t a n c e from w t o (exp <)w =

5 ) i s equal t o

a2

+ B2 ( s i n c e X and Y form an orthonormal

b a s i s f o r p under t h e s c a l a r product ((<,q)) = 1/2 B(5,n)) 107

and

p

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

the polar angle i s determined by tan 8 = B/a.

Many calculations

can easily be made using standard formulas and one can describe the action g

i for example i n general (relevant material for

the computations appears i n Gangolli [l], Helgason [ l ; 2 ; 31, and

21).

Kobayashi-Nomizu [ l ;

We will spare the reader the results

of o u r computations since they are essentially routine.

Let us

however remark that the volume element dv i n (1.1) can be written i n this example as dv = shT d.r d8 ( u p t o a normalization factor). F i r s t we refer back t o the formula for c(v)

Example 3.4

given a f t e r ( 1 . 2 ) and mention t h a t i n general (cf. Helgason [2; 31, Warner [Z])

where

n=

Y

On = g-ahere,

c(-ip) = 1.

for 5

exp

n, and

dii i s normalized by

Here 8 i s the Cartan involution 8 : 5 + rl

Y

E

K=

k and rl

E

+

E

- rl

p and one observes t h a t i t is an involutive

Y

automorphism of g such that

-.-.

(c,rl)

-+

-B(€,,Orl)

i s s t r i c t l y positive

definite on g

x

Since

1 / 2 , ( i p , a ) = i p , and ( p , a ) = 1/4 there results

(cl,a) =

g.

In terms of Beta functions we obtain

(cf. Bhanu Murti [l])

108

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

-

Thus we can r e c o v e r f ( v ) f r o m f ( u , b ) Example 3.5

u s i n g (3.16)

and (1.2).

We now want t o show how t h e r e s o l v a n t s (3.1)

can be o b t a i n e d by a cumbersome and " c l a s s i c a l " method i n o r d e r t o f u r t h e r c o n f i r m (.3.7) as a l e g i t i m a t e r e s o l v a n t ( t h e equival e n c e o f (3.1)

and (3.7)

has a l r e a d y been e s t a b l i s h e d ) .

The

t e c h n i q u e has a c e r t a i n i n t e r e s t and moreover we w i l l o b t a i n a

H

w h i l e showing how t h e r e c u r s i o n r e l a t i o n s (3.9) 1-1 can be o b t a i n e d group t h e o r e t i c a l l y . Some o f t h e more t e d i o u s f u l l basis f o r

c a l c u l a t i o n s w i l l be o m i t t e d .

We remark t h a t Theorem 2.1

does

n o t seem c o n v e n i e n t h e r e t o o b t a i n a f u l l s e t o f b a s i s elements

in

Hu s i n c e t h e u n i t a r y i r r e d u c i b l e r e p r e s e n t a t i o n s o f K

(see V i l e n k i n [l]).Thus

a r e one dimensional o f t h e form ein4 h

h

= SO(2)

h

h

f i r s t we can remark t h a t atk0 = k a n where a = as w i t h (3.17)

es = ( c h t t sh t cos 0 )

(cf..Helgason

[l]and Warner [l;23).

Thus pH(atke) = p(sY) =

s/2 and f o r ~ ( s V ) = us one has (iu-p)H(atkg) (3.18)

e

(i1-1-

2I) s

= e

=

(ch t

-+ sh

t

COS

0)

ill-

I

Now, g i v e n b a s i s v e c t o r s $m(b) = exp im(0-n) f o r example i n

LL(B) (recall t h a t B t i o n o f (3.13), in

-

K w i t h t h e a n g l e v a r i a t i o n , i n o u r nota-

c u t f r o m ( 0 , 4n) t o (0, IT)), we g e t b a s i s v e c t o r s

H by means o f (1.4) ( t h e f a c t o r o f exp ( - i m ) i s i n s e r t e d f o r

u

convenience i n c a l c u l a t i o n ) . o b j e c t s " when v = atK i n (1.4)

I n p a r t i c u l a r we o b t a i n " r a d i a l and s i n c e A(gK,kOM) = -H(g-'kO)

109

we

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

where s ' i s determined as i n

consider H(ailkg) = s'Y i n (1.4). (3.17) b u t w i t h t replaced by -t. b a s i s vactors i n H

(3.19)

9

u =

u

w i l l be I

( i u - p ) s Yeimede

e

-imr

=

Thus the r a d i a l p a r t o f the

21T

1

TT

(ch t

-lT

-

I iv- 7 e irned0 sh t cos 0 )

'

iv- 1 - m' .o (ch t ) (see V i l e n k i n [ l ] ) .

%l

I n order now t o produce r e s o l v a n t s R ( t a u )

as i n Section 2.2 from Pm

a0

-

1/2(ch t ) o f (3.19)

consider t h e growth o f these f u n c t i o n s as t

+ .

one must f i r s t

0 and i n t r o d u c e

s u i t a b l e weight f a c t o r s i n o r d e r t o s a t i s f y t h e r e s o l v a n t i n i t i a l c o n d i t i o n s (2.3)

(note t h a t

a / a t = ( ~ ~ - 1 ) 3/22). l/~

L e t us r e c a l l

i n t h i s d i r e c t i o n t h a t ( c f . V i l e n k i n [l]and Robin [l]) (3.20)

where P!

P:am(~) =

r ( k t m t l ) r ('R-mtl r2(R+i)

)

R

pm,o(z)

i s the standard associated Legendre f u n c t i o n o f the

f i r s t kind.

Then we can e x h i b i t t h e r e s o l v a n t s i n t h e form

110

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

z + l R-m

z-1 F( -R ,m-R ,m+l -,)z+l

= (T)

where z = c h t, R = -1/2 + i p , rn

2 0 i s an i n t e g e r , and F denotes

a standard hypergeometric f u n c t i o n ( c f . C a r r o l l - S i l v e r [15; 171 and C a r r o l l [18;

191).

16;

R e c a l l i n g t h a t r(R+m+l)Pim(z) =

r ( a - m + l ) P ~ ( z ) f o r m 2 0 an i n t e g e r , we remark i n passing t h a t , for m

E

C n o t a n e g a t i v e i n t e g e r , r e s o l v a n t s can be determined

b y t h e formula

(which reduces t o (3.21)

when m i s an i n t e g e r ) and t h i s should be *

taken as t h e generic expression f o r Rm i n t h i s case ( c f . (3.1)). Evidently the

^Rm

o f (3.21)

s a t i s f y (2.3).

We r e c a l l t h e n e x t w e l l known formulas

(3.23)

,

dP! mz ( ~ ~ - 1 )" ~ + /2 dz (22-1)

P!

=

(a+m) (R-m+l )P!-'

( c f . V i l e n k i n [l]and Robin [l])and these, w i t h (3.22), (3.9),

where p i s r e p l a c e d by m ( n o t e t h a t R(R+1) =

lead t o

- ( 1B + u 2 ) ) ;

consequently (3.8) w i l l f o l l o w as before. The most i n t e r e s t i n g f a c t however about (3.9)

i s t h a t these

r e c u r s i o n r e l a t i o n s have a group t h e o r e t i c s i g n i f i c a n c e (cf. 111

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

-

C a r r o l l - S i l v e r [15; 16; 171) and t h e v a r i a b l e t i n (3.8)

(3.9)

may be i d e n t i f i e d w i t h t h e geodesic d i s t a n c e T o f Example 3.3. For purposes o f c a l c u l a t i o n i t i s convenient t o c o n s i d e r X+ = -X + i Y ;

(3.25)

X

and t o n o t e t h a t (H, X+,

-

-

= -X

iY;

-

H = -2iZ

X-) form a "canonical" t r i p l e r e l a t i v e -.

t o t h e r o o t space decomposition gt = kt

Y

+ t X + + CX - , kt

= (CH, i n

t h e sense t h a t (3.26)

= 2X+;

[H,X+]

[H,X-]

= -2X

[X+,X-]

;

(one notes t h e d i f f e r e n c e here between (3.12) -.

9Q; = Q;Ha

+ t$ +

(H,X+,X-)

tX,

= H

c o m p l e x i f i e d as

and t h e decomposition above r e l a t i v e t o

based on (3.26)

-

such decompositions occur r e l a t i v e

t o any Cartan subalgebra such as (CH o r ICY as i n d i c a t e d i n Serre

[l]f o r example).

It w i l l be u s e f u l t o e x p l o i t t h e isomorphism

Q between S L ( 2 , R ) and t h e group SU(1,l)

o f unimodular q u a s i u n i 1 i t a r y m a t r i c e s g i v e n by Q(g) = g = m-lg m where m = (i l). Thus A

g =

(y E)

+

6

=

a b (6:)

w i t h a = 1/2[a+6+i(B-y)]

and b =

A

1 / 2 [ ~ + y + i ( a - 6 ) ] w h i l e d e t g = d e t g = 1 ( c f . V i l e n k i n [l]). There i s a n a t u r a l p a r a m e t r i z a t i o n o f SU(1,l) a l i z e d E u l e r angles

h

(I$,T,$)

so t h a t any g

w r i t t e n i n t h e form

112

E

i n terms o f gener-

SU(1,l)

can be

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

where 0 I cp < ZIT, 0 5

<

T

h

pressed as g = (cp,~,$).

I: $ < ZIT;

and -2n

m,

t h i s w i l l be ex-

I n p a r t i c u l a r , s e t t i n g wl(s)

=

exp sX,

wz(s) = exp sY, and w3(s) = exp sZ, we have Qwl(s) = (o,s,o), = (IT/Z, s, -v/Z),

Qw,(s) that

to,.,$)-’

and Qw ( s ) =(s,o,o) 3

= (o,o,s).

We n o t e

and ( ~ ~ , o , o ) ( o , T , o )(o,o,$)

= (IT-$,T,-IT-@) A

A

(+,T,$) w h i l e ifg1 = (0, T ~ , O ) and g2 =

(cp2,~2,~)

= A

h

t h e n gigz =

with

(cp,T,$)

t a n cp = ch

(3.28)

= chTl

ch T

’

tan

T~

sh -rz s i n cpz sh -r2 cos cp + sh T 1 ch 2

+ sh

ch T~

T~

sh

T~

Tz ’

cos cpZ ;

sh T, s i n cpz = sh -r1 ch T~ cos

cpZ + ch

T~

sh

‘c2

A

Now g o i n g t o t h e n o t a t i o n o f Example 3.3 w i t h p = exp (ax = a2 +

T‘

$,

and t a n

e

= @ / aan

+

BY),

easy c a l c u l a t i o n shows t h a t

A

Q ( p ) = (O,-r,-O). A

Consequently t h e geodesic p o l a r c o o r d i n a t e s o f A

h

~ ( p )= pK can be r e a d o f f d i r e c t l y f r o m t h e E u l e r angles o f Q(p). Now t h e r e p r e s e n t a t i o n L

!J

H

o f G on

u

a r e p r e s e n t a t i o n , which we a g a i n c a l l L

W

u’

u

E

( c f . (1.5))

induces

o f i o n dense subspaces

H o f d i f f e r e n t i a b l e f u n c t i o n s by t h e r u l e

u

R e c a l l h e r e t h a t L,,(h)f(n(g)) a p p l y (3.29) f(n(gi(s)))

= f(h-lv(g))

=

f(v(h-lg))

and t o

f o r 5 = X, Y , o r Z one wants t h e n t o d i f f e r e n t i a t e w i t h r e s p e c t t o s where gi(s) 113

= wi(-s)g

=

wil(s)g

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(and v

=

W e work i n the geodesic polar coordinates

T(g)).

( ~ ( s ) , e(s)) o f .rr(g(s))

A

=

A

T ( p ( s ) k ( s ) ) = ~ ( p ( s ) )and consider

therefore Q ( g ( s ) ) = Q ( ; ( s ) ) Q ( k ( s ) ) = (e(s), ~ ( s ) , -e(s))

(o,o,ds)) =

(W, ~ ( s ) ,+(s)-e(s))

=

( e ( s ), T ( s ) ,$ts)).

In par-

t i c u l a r Q ( g l ( s ) ) = Q(exp(-sX))Q(g)=(o,-s,o)(8,T,o)(o,o,u-8) where Q ( g ) = Q ( p ) Q ( k ) = (O,T,-e)(o,o,u).

Then one uses (3.28)

t o compute (o,-s,o)(O,T,o) w i t h

T =

T~ = T,

and 0,

e

=

S imi 1arly Q( g2 ( s ) )

$ = e(s),

~ ( s ) , T~ = -s,

(evidently $ ( s ) i s of no i n t e r e s t here). ( ~ / ,-s 2 ,- ~ / 2 )(8, T ,-8) ( o ,o,

=

U)

= ( ~ / ,o 2 ,o)

(o,-s,o)(~-T/~,T,o)(o,o,u-~) and (3.28) can be applied t o the middle two terms.

Finally Q ( g 3 ( s ) ) = (-s,o,o)(B,T,o)(o,o,u-8)

e write H+ = which lends i t s e l f immediately t o calculation. W 1 Lv(X+), H- = L ( X ), and H3 = 4 ( H ) and using the relation u 2 v I d/ds f(T(s),e(s))ls=O = f T T ( 0 ) + f e 0 ( 0 ) a routine calcula-

tion yields (cf. Carroll-Silver [16]) In geodesic polar coordinates (.r,8) on V

Proposition 3.6 one has (3.30)

H+ = e-ie [-i coth

T

a/ae

+

a/aT]

(3.31)

H- = e i e

T

a/ae

+

a1a-c-j

(3.32)

H3

=

i

From genera

[i coth

a/ ae known facts about irreducib e unitary principal

series representations o f G , o r SU(1,1), and t h e i r complexifications (see e.g.

Miller [ l ;

21 o r

Vilenkin [l])

114

-

other

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

representations contribute nothing new i n t h i s context

-

it is

natural now t o look f o r "canonical" dense d i f f e r e n t i a b l e basis vectors in H of the form

u

(3.33)

H+f:

satisfying

fk(T,e)

= (m-R)f:+

H3fi = mfi

where R = -1/2 + i p ( u > 0 ) and m = 0 , k l , k2,

....

( I t seems

excessive here t o go i n t o the general representation theory of SL(2, R ), o r SU(1,l); the references c i t e d here -and previously a t the b e g i n n i n g of t h e chapter

-

a r e adequate and accessible.)

I t i s then easy t o v e r i f y , using (3.30)

-

(3.32), t h a t the f o l -

lowing proposition holds. Proposition 3.7

Canonical b a s i s vectors i n H can be taken

u

i n the form

(3.34)

f i ( T , e ) = ( - l ) m exp(-im8) P

where R = -1/2

t

R 0 ,m

(ch

ip(u > 0) and m = 0, 21,

T)

k2,

....

In view of (3.20) and the d e f i n i t i o n (3.21) ( o r (3.22)) of

8 we

can write f o r m 2 0, z = ch

Using (3.30)

-

T,

and R = -1/2 + iu

(3.31) one can then e a s i l y prove ( c f . Carroll-

S i l v e r [16])

115

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorem 3.8

The " c a n o n i c a l " r e l a t i o n s (3.33)

a r e e q u i v a l e n t t o t h e r e c u r s i o n r e l a t i o n s (3.9)

and H-

f o r H,

"m for R

.

I n par-

t i c u l a r t h e v a r i a b l e t i n Example 3.1 can be i d e n t i f i e d w i t h t h e geodesic d i s t a n c e Proof:

T,

from w = K t o k a K. UJt

The r e c u r s i o n r e l a t i o n s (3.9)

low immediately from (3.30)

-

(3.31)

(and hence (3.8))

and (3.34)

suggests o f course t h e i d e n t i f i c a t i o n o f

T

-

(3.35).

A

n

This

and t b u t one can g i v e

We w r i t e k , a k 2 = pk

a computational p r o o f also.

fol-

n

so t h a t kla =

3

pk4 and f o r p = exp (aX+BY) we o b t a i n from Example 3.3 t h e d i s -

tance r 2 = a2 + B

2

n

between w = T ( e ) = K and pK t o g e t h e r w i t h

angle measurements cos

e

=

a/T

and s i n 8 =

B/T.

Set now kl =

a = aty and k4 = kg, u s i n g t h e n o t a t i o n o f (3.13); kJ, r e s u l t a number o f equations connecting t h e v a r i a b l e s

there

( T y t , $ y $ y a y f 3 ) y t h e s o l u t i o n o f which i n v o l v e s t h e i d e n t i f i c a t i o n o f t w i t h T.

Thus t h e r a d i a l v a r i a b l e t i n t h e n a t u r a l p o l a r

expression ( k M y a ) f o r u = T(h) = kUJatK i s i n f a c t t h e geoJ , t d e s i c r a d i u s T o f T ( h ) = pK. The a c t u a l c a l c u l a t i o n s a r e someA

what t e d i o u s b u t completely r o u t i n e so we w i l l o m i t them. We n o t e now t h a t t h e second e q u a t i o n i n (3.9) i n t h e form (shZmt @ ) ' = 2mshzm-lt.

QED

can be w r i t t e n

8-l which y i e l d s ,

i n view

of (2.3), (3.36)

shZmt i m ( t y u )= 2m

1:

sh2m-1

One r e w r i t e s t h e i n t e g r a n d i n (3.36),

116

rl

In-1 R (rlyil)drl

i n terms o f (3.36)m-1

and

2.

CANONICAL SEQUENCES OF SINGUALR CAUCHY PROBLEMS

i n t e g r a t e s o u t t h e rl v a r i a b l e ; i t e r a t i n g t h i s procedure we o b t a i n I f in 2 k 2 1 a r e i n t e g e r s t h e "Sonine" formula

Theorem 3.9 (3.37)

shZmt km(t,p) =

r

m-k+l

r

i s a d i r e c t consequence o f (3.9). have

jt(cht-chrl)k-'

I n p a r t i c u l a r f o r m = k we

t

(3.38)

shZmt @(t,p)

=

2%

0

(cht-chn)m-lsho

s p h e r i c a l f u n c t i o n s (1.8)

and u s i n g t h e n o t a t i o n o f (3.17)

we see t h a t ( c f . (3.19))

,.,

(3.39)

+v(at)

where P

=

&

I

ip-

27T

( c h t + s h t cose)

ip-

0

'

-

1

dB = P o

,(cht)

1v-7

1 -

i s t h e standard Legendre f u n c t i o n . ,.. a l s o by Lemma 1.1 t h a t FMat = FMt = ( a ), which

'p-7l ( c h t )

We r e c a l l

io(q,u)do

R e f e r r i n g back t o t h e formula f o r t h e zonal

Remark 3.10

(3.18)

I

k

= Po

SO

'(cht)

+U

t

equals i o ( t , p ) . Using (3.38)

we can d e f i n e Rm(t,-)

integrating i n E'(V)

=

F

-l*m

R (t,v)

E I ( V ) and

E

( c f . C a r r o l l 1141) we have f o r m

2

1 an

integer, (3.40)

shZmt Rm(t,-) = Zmm

(cht-cho)m-lshn

We r e c a l l t h a t i n t h e p r e s e n t s i t u a t i o n D(G/K)

M drl rl

i s generated by

t h e Laplace-Beltrami o p e r a t o r A which a r i s e s from t h e Casimir

117

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

.

o p e r a t o r C i n t h e enveloping a l g e b r a E(g) (here A corresponds t o 2C due t o o u r choice o f Riemiannian s t r u c t u r e determined by 2 2 2 and C = 1/2(X +Y -Z )

1/2B(-,*)

-

see Remark 3.15 below on Casi-

-3

m i r operators).

I n E(g ) we have then a l s o 2C = ($1)'

B

+ X-X,)

$X+X-

+

and i n geodesic p o l a r coordinates ( c f . Proposi-

t i o n 3.6) (3.41)

A = H:

+ $H+H-

+ H-H,)

a2

=

+ coth

T

a/a-c

+ csch2T a2/ a 0 2 We n o t e here t h a t an a l t e r n a t e expression f o r A i s given by A = H+H

.

+ H'3 -

-

H3 (cf.

Theorem 3.11

C a r r o l l - S i l v e r [15;

16; 171 and S i l v e r

Ill).

For v f i x e d (Muf)(v) depends o n l y on t h e geo-

d e s i c r a d i u s T = t o f u = n ( h ) = k a K and can be i d e n t i f i e d

9 t

w i t h t h e geodesic mean value M(v,t,f). Proof:

T h i s can be proved b y t e d i o u s c a l c u l a t i o n based on

t h e coordinates i n t r o d u c e d i n p r e v i o u s examples b u t i t i s simply a consequence o f Lemma 1.1 and Theorem 3.8. A

Indeed, s e t t i n g

AA

A

h

g = k ak w i t h h = k a k we have v = n ( g ) = k aK and u = n ( h ) = 1 2 1 t 2 1 klatK w i t h (3.42)

a ( M u f ) ( v ) = (M t f ) ( v ) = ( M t f ) ( v )

A

A

A

A

We n o t e t h a t t h e d i s t a n c e from v = klaK t o k aka K i s t h e same as 1 t

118

2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS A

t h e d i s t a n c e from w = n ( e ) = K t o katK = (exp E ) w = pw, which as

n d i c a t e d i n Theorem 3.8, ~n

V '

Hence as k

d e s c r i b e a geodesic " c i r c l e " around

v a r es t h e p o i n t s klak.rr(at) n

w i l l s i m p l y be T = t.

h

k aw o f r a d i u s T = t; (3.42) i s , upon s u i t a b l e normal z a t i o n

1 a l r e a d y provided, t h e d e f i n i t i o n o f M(v,t,f). Remark 3.12

QED

The v a l i d i t y o f t h e Darboux equation ( .7) f o r and D = A i n spaces o f constant

geodesic mean values M(v,t,f)

n e g a t i v e curvature, harmonic spaces, etc. has been discussed, u s i n g p u r e l y geometrical arguments by G h t h e r [l],O l e v s k i j [l], and Willmore [l]( c f . a l s o Fusaro [l],G'u'nther [2;

31, Weinstein

and Remark 5.6).

[12],

We can now define, f o r m c o n v o l u t i o n ) o f Rm(t,.) means o f (3.40).

E

2 1 an i n t e g e r , a composition ( o r

E'(V) w i t h a f u n c t i o n f ( - ) on V by

Thus f o r v = n ( g ) we w r i t e

(3.43) where <

,>

denotes a d i s t r i b u t i o n p a i r i n g as i n Lemma

( c f . He gason [2] f o r a s i m i l a r n o t a t i o n ) . (Rm(t,=

1.1

Then, s e t t i n g

# f ( * ) ) ( v ) = um(t,v), we have from (3.40) a Sonine

formula (3.44)

sh2mt urn(t,v)

Since F

= R(R+l)FT w i h

Ehrenpre s-Mautner [1])

R

=

-1/2 + i u when T

I

E

t h e r e f o l l o w s from (3.8),

119

E (V) (cf. (3.9),

(3.43)

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

and t h e d e f i n i t i o n o f u" = u"(t,v)

F o r m 2 1 an i n t e g e r and f

Theorem 3.13

= um(t,v)

t i o n um(t,v,f)

= um(t,v,f)

=

(Rm(t,-)

E

2 C ( V ) t h e func-

# f(*))(v) satisfies

qq)sh [A-m(m+l)]~"+~

(3.45)

u;

(3.46)

uy + 2m c o th t um = 2m csch t urn-'

(.3.47)

utt + (2m+l) c o th t uy + m(m+l)um = Aum

(3.48)

um(o,v) = f ( v ) ;

=

m

where (3.45),

and (3.48)

=

0

h o l d f o r m 2 0.

The n o t a t i o n here d i r e c t s t h a t (AMt # f ) ( v ) =

Proof: Av(Mtf)(v)

(3.47),

uT(o,v)

= (MtAf)(v)

(see e.g.

(1.7)

and r e c a l l a l s o t h a t A =

6d + d6 i n t h e n o t a t i o n o f de Rham [1] i s f o r m a l l y symmetric) w h i l e (3.48) example.

i s immediate fro m t h e d e f i n i t i o n s and (3.45)

for

One can a l s o prove Theorem 3.13 d i r e c t l y from t h e de-

f i n i t i o n (3.44)

and t h e Darboux e q u a ti o n (1.7) w i t h o u t u s i n g t h e

F o u r i e r t r a n s f o r m a t i o n ( c f . C a r r o l l - S i l v e r [15; (3.46)

f o l l o w s immediately from (3.44)

(3.45)

and (3.47)

16; 173).

by d i f f e r e n t i a t i o n w h i l e

r e s u l t then by i n d u c t i o n u s i n g (1.7)

r a d i a l Laplacian DU = A = m = 0 t oget her w i t h (3.46)

a 2/ a t 2 +

Indeed

coth t

and t h e

a / a t ( c f . (3.41)) f o r

(see C a r r o l l - S i l v e r [16] f o r d e t a i l s ) . QED

D e f i n i t i o n 3.14

The sequence o f s i n g u l a r Cauchy problems

120

2.

(3.47)

-

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

sequence. (3.8)

f o r m 2 0 an i n t e g e r w i l l be c a l l e d a canonical

(3.48)

The canonical r e s o l v a n t sequence corresponding t o

under i n v e r s e F o u r i e r t r a n s f o r m a t i o n i s ( c f . a l s o (2.3))

+ ( 2 1 ~ 1 c) o t h

(3.49)

RTt

(3.50)

Rm(o,*) = 6 ;

t RY + rn(m+l)Rm = ARm m

= 0

Rt(0,*)

where 6 i s a D i r a c measure a t w = .rr(e). t i o n s (3.9)

S i m i l a r l y from t h e equa-

we have

2(m+l) sh [A-m(m+l)]Rm+’

(3.51)

Rt =

(3.52)

R Y + 2m c o t h t R” = 2m csch t Rm-’ Remark 3.15

We r e c a l l a t t h i s p o i n t a few f a c t s about .-.

Casimir o p e r a t o r s f o r a semisimple L i e algebra g ( c f . Warner [l]).

. . ., Xn i s any b a s i s f o r gE l e t gij .-.

I f X1,

= B(Xi,X.)

where J be t h e elements o f

B(=,*) denotes t h e K i l l i n g form and l e t gij the matrix (g’j)

i n v e r s e t o (g..). 1J

The Casimir o p e r a t o r i s then

d e f i n e d as

T h i s o p e r a t o r C i s independent o f t h e b a s i s {Xi} .-.

c e n t e r Z o f t h e enveloping algebra E(gt).

Another f o r m u l a t i o n

. “ -

can be based on a Cartan decomposition g = k

Cartan subalgebra j o f g

.-.

.

such j always e x i s t .

(i.e.,

. - . . - . - . . - .

j = j f) k +

L e t a b a s i s Hi

121

and l i e s i n t h e

+ p and a e s t a b l e

jfl p

= j- + jp);

k ( 1 <, i 5 m) f o r j-. and a k

SINGULAR AND DEGENERATE CAUCHY PROBLEMS = -6ij for b a s i s H.(m+l<jlR) f o r j be chosen so t h a t B(Hi,H.) J P J 1 5 i, j 5 m and B(Hi,H.) = fiij f o r m + 1 5 i,j 5 R. Pick r o o t J ,. vectors Xa i n g so t h a t B(Xa,X,a) = 1 (where [Xa,X-,] = Ha).

t

Then

When G has f i n i t e c e n t e r w i t h K a maximal compact subgroup then the operator i n D ( G / K )

determined by C i s the Laplace-Beltrami

operator f o r the Riemannian s t r u c t u r e induced by t h e K i l l i n g form B(*,-)

(see Helgason [l]and cf.

Example 3.16

(3.41)).

We consider now the case where G = SOO(3,1) =

SH(4) i s the connected component o f the i d e n t i t y i n t h e Lorentz group L = SO(3,l)

and K = SO(3) x SO(1) Z SO(3).

The r e s u l t i n g

Lobazevskij space V = G/K has dimension 3 and rank 1.

I n R4 G

c o n s i s t s o f so c a l l e d proper Lorentz transformations which do n o t

2 2 reverse the time d i r e c t i o n . Thus s e t [x,x] = -r (x,x) = -xo + 2 2 2 + x2 + x3 and t h i n k o f xo as time. Then G corresponds t o 4 x 4 x1 matrices o f determinant one l e a v i n g [x,x] t h e s i g n o f xo.

I n particular i f

2 p o s i t i v e l i g h t cone r (x,x)

i n v a r i a n t and p r e s e r v i n g

n4 denotes

the i n t e r i o r o f the

> 0, xo > 0, then G : R4

+

R4.

The

LobaEevskij space V can be i d e n t i f i e d w i t h t h e p o i n t s o f t h e 2 "pseudosphere" r (x,x) = 1, xo > 0 (which l i e s w i t h i n R4). coordinates

122

If

2.

(3.55)

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

x1 = r sh

e3

s i n €I2 s i n

-

e 1’

x3 = r sh €I3 cos €I

x2 = r sh €I3 s i n €I2 cos €I1; with 0

5

€I1 <

ZIT, 0 5 O 2 <

IT, 0

s c r i b e d i n R4 then coordinates

5

ei(i

x0 = r ch €I3

e3

<

m,

and 0 5 r <

= 1,2,3)

m

are pre-

can be used on V and

t h e i n v a r i a n t measure dv on V i s g i v e n by dv = sh

(i= 1,2,3).

2;

2

e3

s i n O2lIdei

We w i l l now f o l l o w Takahashi [l]i n n o t a t i o n because

o f h i s more “canonical” f o r m u l a t i o n b u t r e f e r a l s o t o V i l e n k i n

[l]f o r many i n t e r e s t i n g geometrical i n s i g h t s . ,.. Thus as a b a s i s o f t h e L i e algebra g o f G we t a k e

Io (3.56)

Y1 =

Y3 =

‘1 3

1

0

O 0

0’ 0 Y

Y2 =

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0)

0

0

0

1’

0

0

0’

0

0

0

0

0

1

0

-1

0

0

3

‘12

=

0

0

0

0

1

0

0

0,

0

0

0,

0

0

0

0’

0. 0

0’

0

0

0

1

0

0

0

0

0

1

0

-1

0,

0

0

0

0

0

-1

0

0,

Y

‘23

Y

3

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Note t h a t i f E

denotes the matlrix w i t h 1 i n the (p,q) Pq (pth row and qth column) w i t h zeros elsewhere then Y P E ( p = 1,2,3) and X E (p
-

position

+

EoP Then

Y

R Y 3 we

~

Y

.

.

,

+ p and K

have a Cartan decomposition g = k

Y

= exp k.

Take now

a = R Y 1 as a maximal a b e l i a n subspace o f p and s e t X

P

+ X

= Yp

1P

(p = 2,3) so t h a t

(3.57)

x2

=

’0

0

1

0’

0

0

1

0

1 - 1

0

0

0

0,

,o

0

;

x3 =

0

0

0

1

0

0

0

0

1

-1

0

0,

-

Then w r i t i n g n = RX, + R X an Iwasawa decomposition o f g i s 3 ,.L given b y g = k + a t n and w r i t i n g A = exp a w i t h N = exp n one

- -

has G =

KAN.

We w i l l use the n o t a t i o n at = exp t Y 1 so t h a t

-

sh t

0

0

sh t

ch t

0

0

0

0

1

0

( 0

0

0

1

‘ch

(3.58)

at =

One can take Y1 = Ho i n the general rank one p i c t u r e w i t h a(Ho) = 1 since [Yl,X2]

= X2 and [Y, ,X3]

= X3.

Thus a, which i s the

1 o n l y p o s i t i v e r o o t , has m u l t i p l i c i t y two and p = -in a = a 2 a p(tY1) = t.

We r e c a l l t h a t M = (kek; AdkH = H f o r H

t h a t Mi = (kEk; Adka C a).

E

with

a1 and

Since exp AdkH = k exp Hk-’ we see

124

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

t h a t AdkH = H f o r H a for H

E

a i m p l i e s k exp Hk-l = exp H w h i l e AdkH

a i m p l i e s k exp Hk”

E

mediate t h a t M = {exp

m =

(3.59)

0

= exp Hi f o r H I

E

a.

E

It i s im-

c o n s i s t s o f matrices

1

0

A

m

;

E

SO(2)

A

whereas M

I

exp I T X ~ ~ ) where , exp ITX

= M U {exp vXl2,

( p = 2,3) 1P Thus W = M /M has the p r o p e r t i e s i n d i c a t e d beI

sends at t o a,t.

f o r e i n t h e general rank one s i t u a t i o n and B = K/M can be i d e n t i -

2

f i e d w i t h the two sphere S c R

3

.

This i d e n t i f i c a t i o n can be

s p e l l e d o u t i n a t l e a s t two ways and we w i l l adopt the one leadi n g t o the s i m p l e s t i n t e g r a l expressions l a t e r .

Thus, f o l l o w i n g

Takahashi [l], we w r i t e u = exp 0X12 and v = exp I$X23 f o r 0 5 0 I$ 0 5 IT and -IT < I$ 5 TI ( i n p a r t i c u l a r v E M). Then any k E K can

0

be w r i t t e n i n t h e form k = m u v f o r m E M so t h a t i f m = v 0 0 JI then kll = cos 0 , k12 = s i n 8 c o s b k13 = s i n 0 s i n I$, kZ1 = -sin 8

COSI),

(kll,k21,k31)

and k31 = s i n 8 s i n +

:k

+

One can pass the map k

R3 t o q u o t i e n t s t o o b t a i n a map K/M

+

+

S

x =

2

v u M and x + ( J I , 0 ) ; here -IT < JI < TI and 0 < 0 < IT $ 0 2 on the domain o f uniqueness. The i n v a r i a n t measure on S corres-

where k

+

l ponding t o t h i s i d e n t i f i c a t i o n i s given by ds = ( 4 1 ~ ) - sined0dJI and the f u n c t i o n s

125

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

with c = [(2n+l)(n-1n)!/(n+m)!]"~, -n 5 m 5 n, and n = 0,1,2, m ,n 2 2 form a complete orthanormal system i n L ( S ) ( c f . Talman

. . .,

[l]and V i l e n k i n [ l ] ) . We can now g i v e e x p l i c i t formulas f o r a canonical s e t o f radial objects i n

H, ( c f . (1.4)).

Thus l e t v = atK and b = kM

w i t h k = v+ue and consider ( c f . (3.19)) IT

A

(3.61)

@;"(atK)

=

-IT

I

(iP-l)H(a;'k)

IT 0

e

'n ,m

( e , $ ) s i n 8 dOd$

It f o l l o w s from general formulas i n Takahashi [l]t h a t

-

exp H(a-'v u ) = ch t t $ 0

I

1

2

IT

(ch t

-

sh t cos e)ip-lPn(cos

f 0 and P:(cos

reduces t o R (t,w)

=

u

io(t,p)

."

0 ) = Pn(cos 0 ) .

FMt = @ (a,)

= 1 and Po(cos 0 ) = 1 whereas,

calling that @

e ) s i n e de

o

"0

t h a t (3.62)

(3.63)

IT

= 0 for m

eimd$

-IT

0 3 0

,

h~

= 2 '0 n

c

becomes

"m 0', n (atK) = @ 'n (atK) = Z n ( t ) 1-I

(3.62)

since

sh t cos 8 and t h e r e f o r e (3.61)

1-I

when n = 0 since

s e t t i n g k = v, u v

+ @ @

and r e -

i s an even f u n c t i o n o f t, t h e expression f o r

can be w r i t t e n i n t h e form ( c f . Takahashi [l])

"0

We n o t e

R (t,p) =

I

(iu-l)H(ai'k)

e K

126

dk

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

=

1 JrO( c h t - s h t

cose)iv-l sinede = Z o ( t )

v

-

2ipsh t [(cht + s h t ) ”

=

sinpt/vsht

-

(cht

-

~ht)~’]

To evaluate (3.62) i n general one has recourse t o various formulas for special functions and we are grateful here t o R. G. Langebartel for indicating a general formula for such integrals i n terms of Meijer G functions (cf. Luke [l]).

The details are some-

what complicated so we simply s t a t e the r e s u l t here.

We note that f o r n

=

Thus, since

0 t h i s formula yields (cf. Magnus-

Oberhettinger-Soni [l])

(3.65)

=q

Z;(t)

sh-’t

=

1

1

l(ch t ) i v-7L

P-’

(L) sh-’t[(ch 21 v 1 21U

= (-)

t + sh t ) i v

sh-’t(e i v t

i n accordance w i t h (3.63).

-

-

(ch t + sh t)-iv]

e-ivt)

Further l e t us remark that (cf.

Robin [l])

127

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(3.66)

P

--1

1/2 t F l +( i vv 7 l-ill 2, 3 -sh 2 t )

l(ch t)=&sh 2

ill7

and hence from (3.65)

one has Z E ( t ) =

F(Y),Y, $

-sh 2 t)

as r e q u i r e d by (2.2). "n To o b t a i n re s o l v a n ts R (t,p) m u l t i p l y t he Z n ( t ) o f (3.64) ll Since one has f o r example

(3.67) (cf.

P

--

:(ch

t) =

ip-7

n+21

'sh(t3)

2

r(n +

$

F (n +r l,+Fi p, n ++l - -i p, - s 3h 2

-

i t i s n a t u r a l t o ta k e ( c f . (3.21) n+l+iu n+l-iu

h

Rn(t,u)

We now use (3.23) Some r o u t i n e in. Theorem 3.17

3

= F ( T F n + p h

-

(3.24)

2

t o obtain recursion relations f o r the

c a l c u l a t i o n y i e l d s then The re s o l v a n ts

in for the

case o f t h r e e dimen-

iu-1)

^R!

(3.22)

t)

s i o n a l Lobazevskij space a re d e fi n e d by (3.68)

(3.69)

2t )

Robin [l]), i n o r d e r t o s a t i s f y t h e r e s o l v a n t i n i t i a l condi-

t i o n s (2.3), (3.68)

by an a p p ro p r i a t e w e i g h t f a c t o r .

-n 1

-n- 1

. . . we now

f o r n = 1, 2,

+ (2n+l) c o t h t

in = (2 n + l ) 128

csch t

and s a t i s f y (v

=

2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

h

Rt:

(3.71)

+(2n+2) c o th t

s:

+ n(n+2)^Rn = v(v+2)^Rn

These equations a re i n agreement w i t h S i l v e r [l]where they are obt ained i n a d i f f e r e n t manner and t h e r e s o l v a n t s a r e exof B ,Y V i l e n k i n [l]. One can now proceed as b e f o r e t o determine r e s o l -

pressed somewhat d i f f e r e n t l y i n terms o f t h e f u n c t i o n s Pa

vants Rn and canonical sequences o f s i n g u l a r Cauchy problems. S i m i l a r l y one can expand t h e m a t r i x th e o r y t o deal w i t h h i g h e r dimensional Lobazevskij spaces ( c f . S i l v e r [l])b u t we w i l l n o t s p e l l t h i s o u t here ( c f . Se c ti o n 4). i n g t h e r e c u r s i o n r e l a t i o n s (3.69)

-

The q u e s t i o n o f deduc-

(3.70)

from group t h e o r e t i c

i n f o r m a t i o n about t h e i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s o f G i n t h e f-l

u

w i l l be d e f e r r e d f o r t h e moment.

Remark 3.18 2.1

-

Going back t o th e general format o f Sections

2.2, Example 3.16 corresponds t o t h e case ma = 2 , mZa = 0,

p = 1, da = -p(p+l),

dpa = 0, and n = p.

Then u s i n g t h e Kumner

r e l a t i o n i n d i c a t e d a t t h e end o f S e c ti o n 2.2 w i t h z = t h L t now,

- (p,o)

2 we have z/z-1 = -sh t and f o r T (3.72

Y

-u,T

(a K) t = c

-uYT

t h P t ch-iu-lt

129

F(+,

ip+l+

iu+2+p

3 2 ,p+Z,th t )

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Hence we can write from (3.68) (3.73)

A

R P ( t , u ) = c-l

-uYT

exactly as i n (3.7).

shePt Y

-u ,'c (atK)

P u t t i n g this i n (2.11) yields (3.71) and

the recurrence relations follow as before from (3.23)

2.4

Expressions f o r general resolvants.

m and m2a = 0 so t h a t d2a = 0, while

T

Theorem 4.1

p =

- (p,o).

m/2, R = i v +

T, da

-

(3.24).

Let f i r s t ma = = -p( p+m-1)

, and

Resolvants f o r the case ma = m and m2a = 0 are

given by A

(4.1)

R P ( t , u ) = c-l

-uYT

= ch-P-R

s h w P t Y -u

,f(atK)

R+p+l +m+l 2 ,p 7 t h t )

t F(+,R+

These s a t i s f y the resolvant i n i t i a l conditions as well as the differential equation and sDlittinq recursion relations below. (4.2)

A

A

(4.4)

A

RtPt + (2p+m) coth t R! + [p(p+m) + p

A

2

+ ($m 2 ]R" p - - 0

A

R! + (2p+m-1) coth t Rp = (2p+m-l)csch t RP-'

130

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2.

Proof: and (3.73)

Equation (4.1)

f o l l o w s from d e f i n i t i o n s ( c f .

f o r m o t i v a t i o n ) , (2.9)

(3.7)

t h e Kumner r e l a t i o n F(a,b,c,z)=

(l-z)-aF(a,c-b,c,z/z-l),

and an extended v e r s i o n o f (3.67);

r e c u r s i o n formulas (4.3)

-

( c f. Robin [l]f o r d e t a i l s ) . o r (4.3)

-

(4.4)

the

f o l l o w from t h e known r e l a t i o n s

F i n a l l y (4.2)

r e s u l t s from (2.11)

(4.4).

QE D

I n t h e s i t u a t i o n now when mZa = 1 t h e s i t u a t i o n becomes somewhat d i f f e r e n t .

We r e c a l l t h a t

T

-+

(p,q) w i t h (p,q)

E

Z+

x

Z and

2 We take ma = m so t h a t da = -p(p+m) + q and dpa = q E 22., 2 m For t h e r e s o l v a n t s we use (3.7) again ( c f . -4q w i t h p = 2 + 1.

p

2

(3.73)

a l s o ) w i t h (2.9)

t o obtain

= ch-P-2XF(x+y,

x +

y,

y, t h 2 t )

where x = -1( i p + 2 + 1 ) = R and y = p + 2 + 1. An elementary 2 2 2 computation now y i e l d s th e re s o l v a n t e q u a t i o n from (2.11) i n t h e form (4.8)

*R;iq

+ [(Pp+m+l)coth t + tht]c!7q

+ [p(p+m+2) + p 2 + (!)+1)2 131

+ q 2 ~ e c h 2 t ] " R 9 q= 0

SINGULAR AND DEGENERATE CAUCHY PROBEMS

Theorem 4.2 given’by (4.7)

For the case

= 1 with

%

= in resolvants a r e

s a t i s f y i n g the resolvant i n i t i a l conditions

a

(2.3) and (4.8).

There a r e various spl i t t i n g recursion r e l a t i o n s

which we l i s t below.

(4.9)

p

-

^Rp’q

2x1 t h t

sh2t t h t

(4.10)

A

RPtYq = 2(y-l)coth t sech2t

ip-2sq

2(x+-l)(y-x

+ [ 2 ( 1 - 6 ) ~ 0 t ht + 1

-y-1)

Y-2

(4.11)

(4.12)

A

Rp’q

t

= -q t h

t

SPsq -Z(y-l)coth tiPsq

+ Z(y-l)csch t ^Rp-lSq-’ (4.13)

(4.14)

A

Rp’q

t

= qth

t

iPsq -Z(y-l)coth

+ E(y-l)csch t

t

iPsq

ip-l’q+l 132

6p+zq

- qlth

t]

2.

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

2 coth t iPsq + -=x+y) 9+ 1

h

RPtSq = q t h t

(4.15)

h

R!’q

(4.16)

= -q t h t

(y-x Proof: mula

d

iPsq + 2 coth t ( X + Y ) q-1

-

The r e c u r s i o n r e l a t i o n s are derived using the f o r -

F(a,b,c,z)

= (ab/c)F+ = (ab/c)F(a+l ,b+l ,c+1 ,z)

and var-

i o u s c o n t i g u i t y r e l a t i o n s f o r hypergeometric f u n c t i o n s ( c f . Magnus-Oberhettinger-Soni

[l].Thus t o o b t a i n (4.9)

-

(4.10) one

uses the e a s i l y derived formulas (4.17)

( l - z ) F + = F + w ,cz c+l F(a+l,b+l,c+2,z)

(4.18)

abz(1-z)F+ = c(c-l)F(a-1 ,b-1 , c - ~ , z )

+ For (4.11)

-

c

[

J

-

( z

-

(c-bz-l)]F

(4.12) one uses t h e formulas

(4.19)

b(1-z)F+ = CF + (b-c)F(a+l ,b,c+l ,z)

(4.20)

abz(1-z)F+ = c ( c - l ) F ( a - 1 ,b,c-1,z)-c(c-bz-l)F

whereas f o r (4.13) (4.21)

-

(4.14)

we u t i l i z e

abz(1-z)F+ = c(c-l)F(a,b-1,c-1

133

,z)

-

c(c-aa-1)F

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(4.22)

-

a(1-z)F+ = CF

and f i n a l l y f o r (4.15)

-

(c-a)F(a,b+l ,c+1 ,z) (4.16) one has

(4.23)

bZ(l-z)F+ = c ( z (b-c))F + ~ - ~ +- ~

(4.24)

abz(1-z)F+

=

F(a+l ,b-1 ,c,z)

-bco F(aa-b-1

Let us a l s o remark b r i e f l y about the s p l i t t i n g phenomenon.

Thus

f o r example (4.19) y i e l d s (4.25)

F

I

=

c(1 a -z)-[cF

+ (b-c)F(a+

whereas (4.20), a f t e r an index change (4.26)

I

[cF

F (a+l,b,c+l ,z) = z ( l

gives

-

(c-bz)F(a+l ,b,c+l ,z)]

Now d i f f e r e n t i a t e (4.25), insert (4.26), and then use (4.25)

Multiplying by z(1-z) one

again t o eliminate F ( a + l , b , c + l , z ) .

obtains then the hypergeometric equation z(1-z)F [c

-

(a+b+l)z]F

I

-

abF

=

I 1

+

I t i s easy t o show t h a t i f t h e

0.

hypergeometric equation s p l i t s i n t h i s manner then so does (4.8) under the composition of (4.11)

-

(4.12), f o r example.

QED

For completeness we will write down some formulas f o r t h e = 3 o r 7 b u t will omit t h e recursion r e l a t i o n s s i n c e 2a the pattern i s exactly as above. T h u s f o r m2a = 3 we s e t ma = m

cases m

and r e c a l l t h a t (p,q)

E

Z+

x

Z+ w i t h p

134

?

q

E

22.,

One has

2. CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

p =

-m2 + 3 and R

q(q+2).

(2.9)

=

iu +

We use (3.7)

p w i t h d2a = -4q(q+2) and da = -p(p+m+2) + h

and (3.73)

again t o d e f i n e R p y q and from

this yields

1 where x = R/2 = $ i u

+%+ 3)

and y = p +

e q uat ion a r i s i n g from (2.11) *

R!tq

(4.28)

m -i+

2.

The d i f f e r e n t i a l

i s then

+ [(2p+m+3)coth t + 3 t h t]i!yq = [p(p+m+6)

+ p 2 + =(!)+3)

For mZa = 7 one has p = !+

2 + q(q+2)sech2t]RPyq = 0 A

7 and s e t t i n g ma = m i t f o l l o w s

2

t h a t da = -p(p+m+6) + q ( q t 6 ) w i t h dZa = -4q(q+6). and (2.9)

from (3.7)

again R+

h

(4.29)

+

RPYq(t,u) = ch-P-a t F ( 9 ,

R;tq

,p+%+4,

2 th t )

th e d i f f e r e n t i a l e q u a ti o n f o r R P y q i s

h

+ [(Ep+m+7)coth t + 7th 2

+ [p(p+m+l4) + p +

Theorem 4.3 by (4.27)

Rtp-q-6

*

and from (2.11) (4.30)

I n t h i s case

t]6!’q

m 2 (9+ 7) + q ( q + 6 ) s e ~ h ~ t ]= ~0~ ’ ~

For m2a = 3 (resp. 7) t h e r e s o l v a n t s a r e given

(resp. (4.29))

and s a t i s f y (4.28)

135

(resp. 4.30)).

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

These r e s u l t s completely s o l v e t h e rank 1 case and t h e conn e c t i o n o f t h e F o u r i e r t h e o r y t o t h e associated s i n g u l a r p a r t i a l d i f f e r e n t i a l equations has been i n d i c a t e d a l r e a d y ( c f . Theorem 3.13 and Chapter 1). 2.5

The Euclidean case p l u s g e n e r a l i z a t i o n s .

here C a r r o l l - S i l v e r [15;

We f o l l o w

16; 171 and w i l l i n d i c a t e r e s u l t s o n l y

f o r one simple Euclidean case; a complete e x p o s i t i o n appears i n S i l v e r [l]. Thus l e t G = R2xySO(2) and K = SO(2) where (;,a)

(;,8)

=

(&y(ct)?,ct+B)

i s a semidirect product w i t h

( c f . Helgason [l], M i l l e r [l;21, and V i l e n k i n [l]) ground information).

f o r back-

Thus G = M(2) i s t h e group o f o r i e n t a t i o n

and distance preserving motions o f R2 and i s a s p l i t extension o f R2 ( w i t h a d d i t i v e s t r u c t u r e ) by K so t h a t R 2 and K are subgroups o f G w i t h K compact and

R2 normal ( c f . Rotman [l]).

Elements g = (;,a) can be represented i n t h e form 'cos ct (5.2)

g =

s i n ct

, o

- s i n ct cos

ct

0

and m u l t i p l i c a t i o n i s f a i t h f u l .

."

xl' x2

1, As generators o f t h e L i e alge-

b r a g o f G we t a k e

136

2.

1 l;i 1; ; b;; j

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

0

(5.3)

al =

0

0

0

0

0

a2

0

a3 =

0

0

-1

0'

1

0

0

0

0

0,

with multiplication table

."

Thus g i s s o l v a b l e ( b u t n o t n i l p o t e n t , n o r semisimple).

Now s e t

V = G/K and s i n c e (;,a)(;,@) = (z,a+B) t h e r e i s an obvious g l o b a l a n a l y t i c diffeomorphism V f(g-%)

g-';

R

2

.

One d e f i n e s as b e f o r e L ( g ) f ( x ) =

where g = (;,a) and g - l = (y(-a)(-;),-a).

= f(y(-a)(;-;))

One t h i n k s here o f

-+

;;as

= g - l r ( h ) = IT(g-'h)

r ( h ) = IT(;,@) = = r(y(-a)(-;)

so t h a t

IT((;,O)(O,@))

+ y(-a);,@-a)

= y(-a)(;-G). . "

If f i s d i f f e r e n t i a b l e then L induces a r e p r e s e n t a t i o n o f g as i n we have f i r s t , f o r $ = ( x1 ) = x2 +. ( f o r s i m p l i c i t y o f n o t a t i o n ) , exp(-tal)x =

W r i t i n g Ai

(3.29). -+

r ( x , @ ) = (xl,x2) (xl-t,x2),

= L(ai)

e x p ( - t a 2 ) z = (xl,x2-t),

Consequently we o b t a i n A1 = -a/axl, x2 a/axl-xl

a/ax2.

(3.12)

and (3.26))

(5.5)

[H,H+]

A2 = -a/ax2,

W r i t i n g H+ = A1 + i A 2 , H

2iA3 i t f o l l o w s from (5.4)

= 2H+;

+.

and exp (-ta3);

= y(-t)x.

and A3 =

- = A1 -

iA2, and H =

t h a t ( n o t e t h e c o n t r a s t here w i t h

[H,H-]

= -2H ;

[H+,H-]

= 0

The Riemannian s t r u c t u r e on V i s described i n (geodesic) p o l a r c o o r d i n a t e s by ds2 = d t 2 + t 2d8 2 ( c f . Helgason [ l ] ) geometry i s " t r i v i a l . "

and t h e

There f o l l o w s immediately ( c f . V i l e n k i n

137

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Lemma 5.1 (5.6)

I n p o l a r coordinates ( t , e ) it3

H+ = -e

[a/at +

H- = -e (5.7)

A = H H

+-

[a/at

-

i

H = -2i

a/ae;

a/ael

1 a/at + a 2/ a t 2 + t

=

As b e f o r e (cf.

-i0

Lt a/ael;

on V

1 t

a 2/ae 2

P r o p o s i t i o n 3.7) we work w i t h (dense) d i f f e r -

e n t i a b l e b a s i s " v e c t o r s " f i i n H i l b e r t spaces tl a u n i t a r y i r r e d u c i b l e representation L

u

u

where G provides

and these are c h a r a c t e r -

i z e d by t h e c o n d i t i o n s ( c f . M i l l e r [l;23, V i l e n k i n [I]) =

iuf;+l;

H-f;

= ipfi-l;

= mfi

H3f;

where H3 = 1/2 H and 1-1 i s r e a l (see C a r r o l l - S i l v e r [15;

16; 171

Here we can w r i t e L =

and S i l v e r [ l ] f o r f u r t h e r d e t a i l s ) .

(1.5) ) , b u t emphasize t h a t t h e semi simp1 e t h e o r y does n o t apply. Theroem 5.2

There r e s u l t s (cf.

Vilenkin [l])

Canonical b a s i s v e c t o r s i n H

u

i n t h e form (5.9)

can be w r i t t e n

f i = (-i)m exp (ime)Jm(pt)

Proof:

Take f i = eimOwL(t)

requirement i n (5.8).

which w i l l assure t h e t h i r d

Then t h e w i must s a t i s f y

138

2.

(cf. (5.7

CANONICAL SEQUENCES OF SINGULAR CAUCHY PROBLEMS

-

The solutions J,(ut)

(5.8)).

are chosen for f i n i t e -

ness cond tions and the factor ( - i ) mmakes valid the recursion relations indicated i n (5.8) (cf. Vilenkin [l]); the resolvants and s a t i s f y age given by "R(1-I,t)= (i)m2mr(m+l)(pt)-mwi(t) 2 (1.3.11) - (1.3.12) with A(y) replaced by u (resp. y by 1-11 similarly (1.3.4) Remark 5.3

-

(1.3.5) hold f o r

^Rm

with

?=

1.

QED

I t i s easy t o show that the mean value as de-

fined by (1.6) coincides i n t h i s case with the mean value u X ( t ) (cf. formula (1.2.1)

-

and see Carroll-Silver [16] for d e t a i l s ) .

Thus the results of Chapter 1 may be carried over t o t h i s group theoretic situation , which i s equivalent. W e conclude t h i s chapter with some "generalizations" of the

growth and convexity theorems (1.4.12) and (1.4.16) in the special case V = SL(2,R) /SO(2) Silver [15; 16; 173).

(cf. Theorem 3.13 and Carroll-

F i r s t i t i s clear that i f f _> 0 then,

referring t o (3.43), ( M t # f ) ( v ) teger (here we assume f

E

C'(V)

=

(Mtf)(v) 2 0 for m 2 0 an in-

f o r convenience l a t e r ) .

Hence

u m ( t , v , f ) = u m ( t , v ) 2 0 when f 2 0 by (3.44) f o r m > 0 an integer. Now write Am = A a M t, Theorem 5.4

- m(m+l) so Let A,f

t h a t by

3.45) we have

recall Mt

=

L 0, m 2 0 an integer; then u m ( t , v ) i s

monotone nondecreasing i n t f o r t 2 0. 139

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

AM f = M ( A f ) by (1.7)

Proof:

rl

rl

and Lemma 1.1.

QED

Now l e t $ be any f u n c t i o n such t h a t d$/dt = csch2m+1t so t h a t d/d$ = sh2m+1t d / d t ( c f . Weinstein [12]).

Then (3.47) can

be w r i t t e n

a 2/a$ 2 um ( t , v , f )

(5.11)

= ~

h um(t,v,A,f) ~

~

~

t

Consequently t h e r e f o l lows

L

Iff,A,,

Theorem 5.5

0 then um(t,v,f)

i s a convex f u n c t i o n

o f $. Working i n a harmonic space Hm(cf. Ruse-Walker-

Remark 5.6 Willmore [ l ] ) ,

Fusaro [l]proves (M = M(v,t,f)

3.11) (5.12)

Mtt

+

(w

+ log' g(t)'l2)Mt

where g = d e t (9. .), gij 1J

as i n Theorem

= AM

denoting t h e m e t r i c tensor, and g depends

on the geodesic distance t alone (A denotes t h e Laplace-Beltrami

I f A f 2 0, M w i l l be nondecreasing i n t and a convex

operator).

f u n c t i o n of $ where $ ' ( t ) = t'-m/g(t)'/2.

Weinstein 1123 works

2 i n spaces o f constant negative c u r v a t u r e -a (which a r e harmonic) and proves s i m i l a r theorems f o r (5.13)

Mtt

+

a(m-1) c o t h (at)Mt = A M

We r e f e r t o Helgason [4] f o r such "Darboux" equations i n a group context; t h e extension t o "canonical sequences" i s due t o C a r r o l l

140

2.

CANONICAL SEQUENCES OF SINGUALR CAUCHY PROBLEMS

[21; 221, Carroll-Silver [15; 16; 171, and Silver

141

113.

This page intentionally left blank

Chapter 3 Degenerate Equations w i t h Operator Coefficients 3.1

Introduction.

In the preceding chapters we were con-

cerned w i t h Cauchy problems f o r evolution equations with operator c o e f f i c i e n t s t h a t were permitted t o become i n f i n i t e o r singular. We turn now t o the dual s i t u a t i o n i n which the operator c o e f f i -

cients a r e permitted t o degenerate i n some sense.

The examples

t o follow w i l l i l l u s t r a t e some typical p a r t i a l d i f f e r e n t i a l equat i o n s t h a t occur i n the i n i t i a l boundary. value problems t o which our a b s t r a c t r e s u l t s will apply. Example 1.1

Various diffusion and f l u i d flow models lead t o

the p a r t i a l d i f f e r e n t i a l equation

with nonnegative real c o e f f i c i e n t s .

This equation can be e l l i p -

t i c (ml = m2 = 0 ) , parabolic (ml > 0, m2 = 0 ) , o r of pseudoparab o l i c o r Sobolev type (m2 > 0 ) , and the type may change w i t h position = x and time

=

t.

The equation (1.1) w i t h m2 > 0 was

proposed i n 1926 by Milne [l].

Similar equations have been pro-

posed f o r numerous o t h e r a p p l i c a t i o n s by Barenblat-ZheltovKochina [ l ]

, Benjamin

Nachman-Ting [ l ] ,

[ l ] , Benjamin-Bona-Mahoney [2],

Buckmaster-

Chen-Gurtin [ l ] , Coleman-No11 [1], Huilgol

[l], Lighthill [ l ] , Peregrine [ l ; 21, Taylor [ l ] , and Ting [2].

T h i s equation appears below i n Examples 3.5, 4.5, 4.6, 4.7, 5.10, 143

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

5.13,

5.14,

5.15,

Example 1.2

6.21 and 7.6.

A 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 analogous t o

(1.1) b u t c o n t a i n i n g a second o r d e r t i m e d e r i v a t i v e was i n t r o duced i n 1885 b y Poincarg.

S i m i l a r equations o f t h e form

= f(x,t)

w i t h nonnegative c o e f f i c i e n t s have been proposed; here An i s t h e Laplacian i n x

E

Rn and An,1

A p p l i c a t i o n s o f (1.2)

denotes t h e f i r s t n-1 terms o f An.

a r e g i v e n by Boussinesq [l], L i g h t h i l l [l],

Love [l],and Sobolev [l]. Equations o f t h e form (1.2) s i d e r e d below as Examples 3.9, Remark 1.3

4.8 and 6.22

(cf.,

a r e con-

Theorem 3.8).

The name SoboZev equation has been used exten-

s i v e l y t o designate p a r t i a l d i f f e r e n t i a l equations o r , more gene r a l l y , e v o l u t i o n equations w i t h a n o n t r i v i a l o p e r a t o r a c t i n g on the highest order time derivative.

This operator i s usually--not

always--el l i p t i c i n t h e space v a r i a b l e . Example 1.4

C e r t a i n a p p l i c a t i o n s l e a d t o problems w i t h

standard e v o l u t i o n o r s t a t i o n a r y equations i n a r e g i o n G o f Rn b u t w i t h a c o n s t r a i n t i n t h e form o f a p a r t i a l d i f f e r e n t i a l equat i o n on a lower dimensional submanifold (e.g., G.

Thus one may seek a s o l u t i o n o f

144

t h e boundary) o f

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

DtU(xyt)

- AnU(X,t)

= f(x,t),

x

E

G,

(1.3) Dtu(syt) +

au/aN

= d i v ( a ( x ) grad u(s,t)),

where S i s a n-1 dimensional submanifold,

s

E

S

N denotes a normal,

and t h e divergence and g r a d i e n t a r e given i n l o c a l coordinates Such a problem was g i v e n by Cannon-Meyer [l]t o describe

on S.

a d i f f u s i o n process which was " s i n g u l a r " on S ( c f . Example 6.21). Example 1.5

Problems s i m i l a r t o t h e above b u t o f second

o r d e r i n time occur i n t h e form -Anu(x,t)

= f(x,t),

x

E

G,

s

E

aG

(1.4) 2 D t u ( s y t ) + au/aN = 0,

t o d e s c r i b e g r a v i t y waves ( c f . Whitham [l]).One can view t h i s as a degenerate problem i n which t h e c o e f f i c i e n t o f utt i s t h e boundary t r a c e operator.

I t can be handled as t h e preceding

( c f . Showalter [5]) o r d i r e c t l y as by Friedman-Shinbrot [S] o r Lions

[lo],

Ch. 1.11.

Example 1.6

Degenerate p a r a b o l i c equations a r i s e i n various

forms o t h e r than (1.1).

Problems from mathematical genetics

lead t o

where t h e l e a d i n g o p e r a t o r degenerates a t t = 0 and t h e second

145

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

degenerates on t h e boundary o f t h e i n t e r v a l [ O , l ] Rosenkratz-Singer [2],

Friedman-Schuss [l],Kimura-Ohta [l],

Levikson-Schuss [l], and Schuss [l;23 f o r (1.5) problems). Also, (1.5)

(cf. Brezis-

and r e l a t e d

i s considered below i n Example 5.15.

De-

generacies can occur i n p a r a b o l i c problems as t h e r e s u l t o f nonlinearities.

This i s the s i t u a t i o n f o r the equation

of flow through porous media ( c f . Aronson [l]and Example 6.23). Example 1.7

Wave equations occur w i t h degeneracies, t y p i c a l -

l y e i t h e r i n t h e form o f (1.2)

t h e form o f (1.1.4).

w i t h cl(x,t)

? 0 and c2 z 0 o r i n

The l a t t e r s i t u a t i o n w i l l be discussed i n

S e ct ion 3.2 below. Each o f t h e preceding examples i s a r e a l i z a t i o n i n an app r o p r i a t e f u n c t i o n space o f one o f t h e a b s t r a c t Sobolev equations (1.7)

~d ( M ( u ) ) + L ( u ) = f

f o r c e r t a i n choices o f t h e operators.

These two equations w i l l

be considered i n v a ri o u s forms i n t h i s chapter.

I n S e c t i o n 3.2

we use s p e c t r a l and energy methods t o discuss (1.8)

when C i s

t h e i d e n t i t y and B and A a re ( p o s s i b l y degenerate) o p e r a t o r p o l y nomials i n v o l v i n g a c l o s e d densely d e f i n e d s e l f a d j o i n t operator. This covers t h e s i t u a t i o n o f (1.1.4). 146

The e q u a t i o n (1.7)

is

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

c a l l e d s t r o n g l y regular when M i s i n v e r t i b l e and M-lL i s c o n t i n uous on some space.

S i m i l a r l y , (1.8)

C - l B and C - l A a r e continuous. i e d i n Section 3.3;

i s s t r o n g l y r e g u l a r when

The s t r o n g l y r e g u l a r case i s stud-

t h e o p e r a t o r s a r e p e r m i t t e d t o be time-

dependent and n o n l i n e a r .

We c a l l t h e equations weakly regular

when t h e i r l e a d i n g o p e r a t o r s are o n l y i n v e r t i b l e .

We s h a l l g i v e

w e l l -posedness r e s u l t s i n Section 3.4 f o r 1i n e a r weakly r e g u l a r equations by means o f t h e c l a s s i c a l generation t h e o r y f o r l i n e a r semigroups i n Hi1 b e r t space.

S e c t i o n 3.5 t r e a t s degenerate

l i n e a r time-dependent equations by t h e energy methods o f Lions [5].

A p p l i c a t i o n s i n c l u d e Examples 1.1,

1.4,

1.6,

second o r d e r

e v o l u t i o n equations o f p a r a b o l i c t y p e ( c f . Theorem 5.9), many o t h e r s ( c f . Showalter

[Z]).

and

Related n o n l i n e a r problems a r e

discussed i n Sections 3.6 and 3.7 by methods o f monotone n o n l i n e a r operators.

These techniques a l s o a l l o w t h e treatment o f r e l a t e d

v a r i a t i o n a l i n e q u a l i t i e s ( c f . Theorems 3.12 and 6.6). t i o n contains a l i s t o f references t o r e l a t e d work.

Each secThe p a r t i -

t i o n o f these references i s inexact, so one should check a l l sect i o n s f o r completeness. 3.2

The Cauchy problem by s p e c t r a l techniques.

t o Remarks 1.1.3

and 1.1.4

Referring

we w i l l deal f i r s t w i t h an a b s t r a c t

v e r s i o n o f t h e degenerate Cauchy problem i n t h e h y p e r b o l i c r e g i o n w i t h data g i v e n on t h e p a r a b o l i c l i n e , f o l l o w i n g Carroll-Wang [12]

C a r r o l l [7; 9; ( c f . a l s o f o r example Berezin [l],Bers [l],

341, Conti [3],

Krasnov [l;2; 31, Lacomblex [l], F r a n k ' l [l],

147

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Walker [l;2; 31, Wang [l;21 as w e l l as o t h e r p r e -

P r o t t e r [2],

v i b u s c i t a t i o n s i n Remark 1.1.4). I I

(2.1)

+

U

Thus c o n s i d e r

+ Aq(1)u

AaS(t)ul + A'R(t)u

( I

denotes d / d t )

= f

where A i s a c l o s e d densely d e f i n e d s e l f a d j o i n t o p e r a t o r i n a

t

separable H i l b e r t space H w i t h (Ah,h)

1=

A-'

(B(t)

E E

L(H), q(1) = a ( t ) + B ( t ) l where a ( t ) vanishes as t

L(H), S ( t )

(see below). seek u(*)

cllh112, c > 0, a,@ 2 0,

E

E

L(H), and R ( t )

E

0

-+

L(H)) w h i l e f i s " s u i t a b l e "

I t i s assumed t h a t a l l o p e r a t o r s comnute and we 2

I

w i t h u(o) = u ( 0 ) = 0 ( o t h e r

C ( H ) s a t i s f y i n g (2.1)

-

i n i t i a l c o n d i t i o n s can a l s o be t r e a t e d b u t we o m i t t h i s here cf. however (2.32)).

We w i l l use f i r s t t h e technique o f Banach

algebras and s p e c t r a l methods developed i n C a r r o l l E 4 ; 6; 8; 10; 11; 14; 29; 30; 311, Carroll-Wang [12],

and C a r r o l l - N e u w i r t h [32;

A r e n s - C a l d e r h [l], 331; c f . a l s o references t h e r e t o Arens [l], Dixmier [l],Foias [3],

V

Gelfand-Raikov-Silov

and Waelbroeck [2]. R i c k a r t [l],

[8],

Lions [5; 61,

We o b t a i n r e s u l t s " s i m i l a r " t o

those o f Krasnov [l;2; 31 and P r o t t e r [2] b u t i n a more general o p e r a t o r t h e o r e t i c a l frameword; moreover i n o u r development a(

0 )

need n o t be monotone and some new f e a t u r e s a r i s e (see Remarks 2.10

- 2.12

and c f . a l s o Theorems 2.16 and 2.17).

L e t us assume f o r i l u s t r a t i v e purposes t h a t S ( t ) = S s ( t ) , R(t) = R r ( t ) ,

and B ( t ) = B b ( t ) where B, R, S , and

1 commute

in

L(H) and a r e normal; hen e by a r e s u l t o f Fuglede [l](l,B,R,S,

* * *

B ,R ,S ) a r e a commuting f a m i l y and we denote by A t h e u n i f o r m l y

148

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

closed

*

algebra generated by t h i s f a m i l y and t h e i d e n t i t y I.

Note t h a t f o r any h

E

H, ABlh = AlBh = Bh a u t o m a t i c a l l y f o r ex-

ample, and t h e c o m n u t a t i v i t y o f say A w i t h B means t h a t f o r h

E

D ( A ) , Bh

D(A) w i t h ABh = BAh.

E

r, and s belong t o Co[o,T] Remark 2.1

and a

E

It w i l l be assumed t h a t b,

1

C [o,T]

where T <

m.

A few remarks about f i n i t e l y generated commuta-

t i v e Banach algebras A w i t h i d e n t i t y w i l l perhaps be u s e f u l i n what f o l l o w s (cf.

A

+

$

C a r r o l l [14] and references t h e r e ) .

be a continuous homomorphism so t h a t k e r

maximal i d e a l i n A w i t h A/ker ker

+Z 6

+

Let

+

:

i s a (closed)

and one i d e n t i f i e s I$ w i t h

+. The s e t QA ( c a l l e d t h e c a r r i e r space)

o f maximal i d e a l s

can then be i d e n t i f i e d w i t h a weakly closed subset o f t h e u n i t ball i n A

I

A

and we w r i t e +(a) = a(+) for a

i f A i s a commutative al,

. . ., an,

*

al,

A.

A

Since I(+) = 1,

a l g e b r a i n L(H) w i t h generators ao,

. . ., an,*

*

E

and I (ao s e l f a d j o i n t ) , one can

show t h a t QA i s homeomorphic t o t h e (compact) j o i n t spectrum A

UA A*

a,(+)

=

A

ak(O)

and a,(+)

continuous on QA

. . ., "*a n ( + ) ) > C $2nt1

A

= {a(+)>= {(ao(+),

al(+),

i s real.

- uA and A w i l l

where

A

The f u n c t i o n s ak w i l l then be be i s o m e t r i c a l l y isomorphic t o

C(QA) where C(QA) has t h e u n i f o r m norm.

I n o u r s i t u a t i o n above

7 and we a s s o c i a t e t h e complex v a r i a b l e s ( z

we have u A C$

z),

A

. . .,S

A

t o (I(+),B(+),

c1 = max(IIBII,IIRII,IISII) responds t o X C(U,)

-+

-+

m

A*

for k

2

1.

One notes t h a t zo

and we w i l l c a l l t h e map

A t h e Gelfand map.

0'

' *'

( + ) ) w i t h h = l / z o 2 c and l z k l 5

r

:

+

-+

0 cor-

a : C(QA) =

A f u r t h e r n o t i o n o f use here ( c f . also

149

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

i s t h a t o f decomposing H by an i s o m e t r i c isomor-

Section 1.5) phism 0 : H

-+

H =

I@ H(z)dv which d i a g o n a l i z e s A and r may be uA

extended, as an i s o m e t r i c isomorphism, f o r example t o a l l bounded B a i r e f u n c t i o n s B(uA) w i t h values i n t h e von Neumann algebra A (here A ’ l C L(H

i s t h e bicommutant o f A and i s t h e c l o s u r e o f

A i n say Ls(H))

we w i l l use uA f o r convenience i n measure

t h e o r e t i c arguments i n s t e a d o f QA.

I I

The so c a l l e d b a s i c measure

v a r i s e s n a t u r a l l y ( c f . Dixmier [l]o r Maurin [l;31) and we w i l l n o t g i v e d e t a i l s here. A v-measurable f a m i l y o f H i l b e r t spaces H(z) on uA c o n s i s t s o f a c o l l e c t i o n o f f u n c t i o n s z E

h(z)

-+

H(z) such t h a t t h e r e i s a v e c t o r subspace F C I I H ( z ) w i t h z

1) h ( z ) Ilff(z)measurable all h

E

F implies g

E

(h

E

F), ( h ( * ) , g ( - ) )

H(z)

-+

measurable f o r

F, and t h e r e i s a fundamental sequence hn

E

F such t h a t t h e c l o s e d subspace o f H(z) generated by hn(z) i s H(z).

2 Now under t h e p r e s e n t circumstances ff = L,,(uA,ff(z))

H i l b e r t space w i t h norm

[I

-

I l h ( ~ ) 1 ) ~ d v ]=~ l’l ~ h l l H and a Lebesque

0-

Diagonalizable

dominated convergence t h e o r h w i l l be v a l i d . operators G

E

is a

..

L(H) are d e f i n e d i n t h e obvious manner ( c f . Section

1.5) w i t h G = 0-lG0

E

- G(z)

L(H) where G

E

L(H(z)), a l l argu-

ments being c a r r i e d o u t i n Ls(H) f o r example. Now i n connection w i t h (2.1)

we c o n s i d e r t h e equation (under

our assumptions S ( t ) = s ( t ) S , etc.),

o b t a i n e d by a p p l y i n g 0 t o

(2.1) (2.2)

A l

u

I

n

ni

+ haz3s(t)u + h b z 2 r ( t ) u + h[a(t)+zozlb(t)]u

150

n

= 0

3.

where

^u

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

- z3'

= eu and f o r example

We w r i t e z = ( z

1' ' [l]( c f . a l s o

z6) and by Coddington-Levinson [l]o r Dieudonn; Chapter 1 ) t h e r e e x i s t unique s o l u t i o n s Z(t,-r,z,A) o f (2.2)

such t h a t Z(T,T,Z,X)

= 1, Z (T,T,z,A)

0, and Yt(TyTyzyX) = 1 f o r 0 5 c 5 X <

S t 5

T

<

00,

l z k l 5 cl,

and

The f u n c t i o n s Z and Y w i l l be continuous i n (t,T,z,h)

a.

i n t h i s r e g i o n and we need o n l y check t h e i r behavior as A *

a.

To t h i s end we c o n s t r u c t a "Green's" m a t r i x as i n Sections 1.3 and 1.5 ( c f . (1.3.13))

i n t h e form

which w i l l s a t i s f y t h e e q u a t i o n ~a A ( t y T y z Y A )+ A 1 ~ 2 H A ( t y Z , h ) G A ( t , ~ ~ Z y X ) = 0

(2.4)

(2.5)

HA(

Y

Y

A) =

1

As i n Chapter 1 i t f o l l o w s t h a t ( c f .

a

(2.6)

ea(t,T,Z,A)

1

0

a(t)+zozl b ( t ) + A B - l z p r ( t )

a-1 X 'z3s(-l t )

(1.3.15))

-X1/ZGA(ty~,z,X)H (T,z,X) A

= 0

A

A

A

*

with $

S e t t i n g u1 = u and u2 = X-'/2c' from

zt

+ ~ ' / 2 H A ( t , z , ~ ) i i=

-i t h e

) we o b t a i n f o r m a l l y

s o l u t i o n of (2.1)

151

' )

and Y(t,-c,z,A)

= 0, Y(T,T,z,X)

t

T

-

i n t h e form

=

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

f = ef.

I n o r d e r t o check t h e behavior o f Y we r e p l a c e t by (2.2)

f o r Y and m u l t i p l y by

v5 (<,-c,z,X);

5 in

upon t a k i n g r e a l p a r t s

and assuming now t h a t a ( t ) i s r e a l valued one o b t a i n s

Now note t h a t IrhBYy I

5

i n t e g r a t i o n from

T

5

w i l l apply

X2'

-

2 28

X

lYI2

+

I Y5 I 2 )

so t h a t , upon

t o t,

Assume now t h a t Re(z3s(<)) 1 so t h a t

(Irl

2 0 i f a > 0 w i t h 28 <, 1 and t a k e X >-

5 X ( r e c a l l X >, c and i f c < 1 a f u r t h e r argument see below).

If

= 0 then t h e A"

corporated i n t o t h e r i g h t hand s i d e o f (2.9) obvious manner.

term can be i n -

o r (2.10)

i n an

We have then ( s i n c e l z k l 5 c1 f o r k 2 1 )

152

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

t

( Y t I 2 + Aa(.t)IYI2 5 1 + 2cl

(2.10)

1YS12dS

T

It

Aa(S)IY12dS t o t h e r g h t hand s i d e o f

u s i n g G r o n w a l l ' s lemma (Lemma 1.5.10)

E(t,S)

=

IYtI2

and

one obtains, s e t t ng

exp 2cl(t-S),

(2.12)

2.10)

+ Aa(t)lYI2 5 E(t,r) +

I

t

AP(S)E(t,S)lY12dt

T

The f o l l o w i n g lemma now g i v e s a somewhat sharper e s t i m a t e on

IYI2

than can be o b t a i n e d b y a d i r e c t a p p l i c a t i o n o f t h e Gronwall

lemma; t h e p r o o f however i s a simple v a r i a t i o n . Lemma 2.2 for 0 <

m

and A

Aa(t)lYI2 5

E(t,T)

T 5

(2.13)

Given (2.12) w i t h P

t 5

T<

'1 exp

2 0 and a > 0

(1: #

i t follows t h a t

da)

We f i r s t o m i t t h e term I Y t I 2 on t h e l e f t hand s i d e t and s e t X(t,T) = AP(S)E(t,S)lY12dg so t h a t X+(t,r) =

Proof: of (2.12)

AP(t)lY12 +

I

t

[

'9 AP(S)Et(t,S)IYI dS

r

m u l t i p l y i n g (i.12)

(with

-

lYtlZ

L

= A P ( t ) l Y 1 2 + 2clX(t,T).

d e l e t e d ) by P ( t ) and u s i n g t h e

l a s t r e l a t i o n f o r Xt we o b t a i n (2.14)

.a(Xt-2clX)

Then

5 PE + PX 153

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

T h u s defining F ( t , T ) = exp ( from (2.14), (FX)t

2cl]dS) f o r

2 (P/a)EF.

0 we have

B u t E(t,T)F(t,T) = exp(-/:[:]dt)

and hence (FX)t 5 -(e~p(-l:[:]dS))~

since F(T,T)X(T,T) = 0.

T >

from which follows

T h i s may be written

w h i c h proves the lemna

QED

We observe from (2.11) t h a t P/a so t h a t exp

(2.17)

(I

t

T

[:Id<)

= #exp

=

c1 a ' / a + (,)[lrl

( j t c l [ & +a

2 +xlb12] 1

&Id<) Aa

T

I f c < 1 we can c a r r y through the e s t i m a t e s w i t h lrlz replaced by

I r I 2A28-1

when A < 1 ( r e c a l l 28 5 1 ) and from Lemma 2.2 we ob-

t a i n , u s i n g (2.17), Lema 2 . 3

Given (2.12) w i t h P 2 0 and a(-c) > 0 f o r

T

> 0

we have f o r 28 5 1

."

where c1 = c 1 max (1, c"-l)

exp

Define now +(t,.)

" -

d ..

A

T

-kf a

=

dg and a s

and c2 = exp 2clT.

-

exp c T

-F

'

1' T

a

and $(t,T,A) =

0 these f u n c t i o n s may o f course

154

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

become i n f i n i t e . $(T,T,c)

N o t i n g t h a t $(t,T)

5 $(T,T) and $(t,T,A)

5

we can s t a t e

C o r o l l a r y 2.4

Let

= $-’(T,T)

$(T)

and

= $-’(T,T,c);

$(T)

then

where Y = Y(t,.r,z,A) a(.)

> 0 for

T

i s the unique s o l u t i o n o f (2.2) w i t h

P 2 0,

2 0 satisfying

> 0, 2 8 5 1, A 2 c, and Re(z3s(<))

= 0 w i t h Y t ( ~ , ~ , z Y A ) = 1.

Y(T,T,z,A)

We s e t now W(t,T,z,A)

=

Q(T)

Y(t,-c,’z,A)

and observe t h a t t h e e s t i m a t e (2.19)

(Q(T)

W i s continuous i n (t,T,z,A)

I

then ( c f . Remark 2.1) continuous since

(t,.r)

-+

+

W(t,T)

where Q = ($$a) 1/2

holds f o r 0 as

T

= 0-lWB

-+

T =

0 also while

If h

0). :. h

+

E

H then

A1/‘Wh

2 1/2 I l A W0hlI dv w i l l converge a p p r o p r i a t e l y by

Lebesque dominatedaA convergence f o r example ( i n f a c t A’/’W can be demonstrated, so r(A1/2W) needed

-

is

E

A

I I

B(aA)

E

b u t t h i s w i l l n o t be

n o t e here t h a t A l l 2 and W a l s o c o n u t e on uA). One can

now prove Theorem 2.5

Under t h e assumptions o f Coro l a r y 2.4 (and

Lemma 2.3) we have, w i t h y = max (l/Z,cx)

155

and 0 5

T

5 t 5 T <

m,

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Proof:

We need o n l y check t h e bounds s i n c e t h e r e s t w i l l From

f o l l o w from t h e Lebesque dominated convergence theorem.

-

(2.12) and (2.16)

IYtI2 5

(2.21) SO

that

[o,T]); T

+

lWtl

2

=

(2.17)

#

IQYtI2

a c t u a l l y (t,.r)

one has

E(t,Y)$(t,T)$(t,T,x)

5 c3 where c 3 -f

Wt(t,.r)

E

0 and t h i s i s u s e f u l l a t e r .

= c2 max a ( t ) (max on

Co(Ls(H)) s i n c e

Q(T)

-t

0 as

For t h e l a s t e s t i m a t e we go

back t o (2.2) t o o b t a i n t h e i n e q u a l i t y ( r e c a l l t h a t 2f3 5 1)

5 c4xa + c5x 1/2 follows.

We again observe t h a t (t,.r)

-t

Wtt(t,T)

E

QED Now we consider (2.7)

and w i l l show t h a t i t r e p r e s e n t s a

s o l u t i o n o f t h e Cauchy problem (2.1) when f

E

Co(D(Ay)).

Thus

s e t t i n g h(c) = f ( < ) / Q ( c ) (2.7) may be w r i t t e n i n t h e form t W(t,<)h(c)d< and we can c o n s i d e r t h i s as a Riemann t y p e u(t) = 0

( v e c t o r valued) i n t e g r a l ( i n o r d e r t o c a r r y t h e c l o s e d o p e r a t o r s

Aa, A',

and A under t h e i n t e g r a l s i g n s ) .

The f o l l o w i n g computa-

t i o n s are then j u s t i f i e d by Theorem 2.5 and remarks i n i t s proof.

F i r s t u ( o ) = 0 and

156

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Theorem 2.6

Assume a

a ( o ) = 0, b y r, s 0, P

E

E

Co[o,T],

1

C [o,T],

a ( t ) > 0 r e a l f o r t > 0,

28 5 1, y = max(a,l/2),

2 0, and f / QE Co(D(Ay)). and u

C'(D(Ay)),

E

2 C (H), u

(2.25)

E

C0(D(Ay+'I2)).

E

F o r uniqueness we w i l l g i v e s e v e r a l r e s u l t s . (2.6)

F i r s t from

one o b t a i n s ( c f . 1.3.17) Y

T

= -Z+A"Z~S(T)Y

Duplicating our esti-

and some bounds f o r Z must be e s t a b l i s h e d . mates l e a d i n g t o (2.9) we have (2.26)

t

IZtI2 +

2Re(Xaz3s(S

T

and under t h e same assumptions and procedures as b e f o r e i t f o l lows t h a t ( c f . (2.27)

2

Then t h e r e e x i s t s a s o l u t i o n o f

g i v e n b y (2.7) w i t h u ( o ) = u t ( o ) = 0, u

(2.1)

Re(z3s(t))

(2.9)

-

(2.12))

l

JT

157

1

t

A a ( t ) l Z 1 2 + lZtL2 5 Aa(-r) + 2c

T

1Z512dE

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Then by a v e r s i o n o f Lemmas 2.2 and 2 . 3 one o b t a i n s

Consequently we have proved Lemna 2.7

Under t h e hypotheses o f C o r o l l a r y 2.4,

@ ( T ) v J ( T ) l Z l 25 c2. We s e t q = (@vJ)”‘

with

V(t,T)

=

e-1q(T)Z(t,T,z,A)8

by Lebesque dominated convergence again V(t,T) V(t,T)

E

A

I I

1/2 w i t h

lY,l

Next we

again t h a t

so t h a t I Q ( T ) Z ~5 ~c7A1l2 w h i l e from (2.25)

5

L(H) (and

can be shown b u t again t h i s i s n o t needed).

observe from (2.6)

a

E

so t h a t

.“

5 c6 f o r a 2 1/2.

lYTl

5 c&

a-7

for

The case a 5 1/2 i s essen-

t i a l l y t r i v i a l and w i l l n o t be discussed f u r t h e r .

Thus u s i n g

Lebesque dominated convergence again we can s t a t e t h a t f o r T > 0, a- 1 1 T +. Y(t,.r) = e”Ye E C (Ls(D(A z),H)) while T + Z(t,r) = e-lZ0

E

C 1 (Ls(D(A 1/2 ),H)).

Therefore i f u i s a s o l u t i o n o f (2.1)

w i t h data p r e s c r i b e d a t T > 0 we operate on (2.1) w i t h Y(t,C) (changing t t o 5 i n (2.1)) and i n t e g r a t e t o o b t a i n f o r u y t -.1 Co(D(A ’I), u E C’(D(Ay)), and u E C 2 ( H )

158

E

DEGENERATE EQUATIONSWITH OPERATOR COEFFICIENTS

3.

JT

JT

Our hypotheses and lemmas i n s u r e t h a t e v e r y t h i n g makes sense and u s i n g (2.25) w i t h (2.30)

p l u s a n o t h e r i n t e g r a t i o n by p a r t s t h e r e

r e s u l t s f r o m (2.31) u ( t ) = Z(t,-clu(r) + Y ( t y T ) u t ( T ) +

(2.32)

i:

Y(t,S)f(S)dS

As i n S e c t i o n 1.5 f o r example we a r e making use i n (2.31) and

o f t h e h y p o c o n t i n u i t y o f s e p a r a t e l y c o n t i n u o u s maps E

(2.32)

F

+.

F i s barreled.

G when

Now as T

+.

x

0, b y h y p o c o n t i n u i t y again

we have ( n o t e t h a t t h i s i s s t a t e d b a d l y i n Carroll-Wang [12]) Theorem 2.8 t i n u o u s u/q (2.1)

Assume t h e hypotheses of Theorem 2.6 w i t h conI

+

0 and u /Q

-f

0 as

T

-f

0.

Then t h e s o l u t i o n o f

g i v e n b y Theorem 2.6 i s unique.

I n t h e e v e n t t h a t q ( t ) > 0 f o r 0 5 t 2 T a somewhat s t r o n g e r uniqueness theorem f o r (2.1)

can be p r o v e d ( c f . Carroll-Wang

I

1121).

Assume u ( o ) = u ( 0 ) = f = 0, w i t h u s a t i s f y i n g t h e con-

c l u s i o n s o f Theorem 2.6,

and t h e n we d e f i n e t h e o p e r a t o r i n

L(H) by (2.33)

1

t

L(t,.r)

O p e r a t i n g on (2.1)

= 0 - l exp(-Aaz3

w i t h L(t,S),

T

s(S)d<)e

where t has been changed t o 5 i n

159

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(2.1) we o b t a i n (2.34)

u ’ (t) =

-1

t

0

L(t,E)CABRr(E)+Aa(E)+Bb(S)lu(.C)dE

Since IlAull and IIABuII a r e bounded i n Theorem 2.6 we have by a we1 1 known i n e q u a l i t y

( r e c a l l t h a t q > 0 means Z(t,T)u(T)

0

-+

lr21 a dC <

0 a u t o m a t i c a l l y i n (2.32)

bounded and u

+

w

and as

T

0 +

hd.!a

dc <

NOW

a).

0 s i n c e qZ i s

w h i l e t h e term Y(t3?)ut(T) can be w r i t t e n

0

’0

But Q = q a 1 j 2 so q > 0 i m p l i e s a(T)’I2Y(t,T)

where 6 < 1/2.

Ls(H,D(A 1/2 ) ) by C o r o l l a r y 2.4 and u’(.)/(radE)6

-+

0 as

T

-+

0.

E

heorem 2.5 w h i l e by (2.35)

Hence we have

0

Theorem 2.9

L e t u be a s o l u t on o f (2.1)

satisfying the

c o n d i t i o n s of Theorem 2.6 w i t h q ( t ) > O f o r O < t < T < m and (ra(E)dE)6/a(r)’/2

bounded ( 6 < 1/2).

160

Then u i s unique.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

3.

Remark 2.10

We emphasize here t h a t a ( - ) i s n o t r e q u i r e d t o

be monotone, i n c o n t r a s t t o P r o t t e r [2] o r Krasnov [l],since by P(t) 2 0 for a l l

(2.111,

x

merely i m p l i e s a

I

2

-cllrl

2

(examples

o f nonmonotone a are given i n Carroll-Wang 1123 and Wang [l;23). The c o n d i t i o n o f P r o t t e r [2]

( c f . a l s o Berezin 111, Bers [ l ] ) ,

phrased here i n t h e form t r ( t ) / a ( t ) ’ / ’

-+

0 as t

-+

0 (with a(-)

monotone), f o r s o l u t i o n s o f a numerical version o f (2.1)

t o be

w e l l posed i n a l o c a l uniform m e t r i c , has i t s analogue here i n t h e c o n d i t i o n s on + ( t ) .

Thus f o r example i f A

tm,A B R ( t ) u = r ( t ) u x = tnux, and A”S(t)

A

-.-32/ax 2 , a ( t )

= s(x,t)

- b ( t ) then i t f o l l o w s t h a t t rT( t ) / a ( t ) ” ‘

1 B(t)

0 if n > m (?n-m+l

-

:1

1 whereas j;lr,‘/a

t2n-ni+l

-.21

d< =

) so t h a t + ( t )

=

with = t

=

n + l ~ -+

2n-mdS = (1/2n-m+l)

k exp (c,t 2n -m+l /Zn-m+l) > 0

2 c o n d i t i o n o f Krasnov [l]involvesmonotone -- 1 a ( t ) = O(tm) and r ( t ) = O ( t 2 0 but y ( t ) ) where y ( t ) * 0 as t

when n >

Tm -

The L

-+

Krasnov i s d e a l i n g w i t h weak solutions.

Now our existence and

uniqueness c o n d i t i o n s are phrased i n terms o f u/q, u-/Q, and f/Q where q2 = +$ and Q

2

= $$a which permits a r a t h e r p r e c i s e com-

parison between t h e behavior o f f, u, and u

I

as t

+

The 9

0.

term seems somewhat curious however since f o r example i f h

Ibl

2 k

<

m

then

:1

I b l /a de

2

( i 2 / - m + l ) ( T -m+l-t-m+’)

tmso t h a t $ ( t ) = 0 (exp k t-mtl/-m+l) 1

+

for a(t) =

0. f o r m > 1 which i m -

poses growth c o n d i t i o n s on b ( t ) i n order t h a t $ ( t ) > 0. seems t o i n d i c a t e t h a t t h e r o l e o f the

i)

This

term should be i n v e s t i -

gated f u r t h e r and some comnents on t h i s a r e made i n Remark 2.12.

161

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Walker, i n as y e t unpublished work, i n v e s t i g a t e s b y s p e c t r a l methods ( u s i n g Riesz o p e r a t o r s Rk d e f i n e d by a/axk = i ( - A ) 1/ 2

Rk)

t h e q u e s t i o n o f d i r e c t i o n a l o s c i l l a t i o n s and w e l l posedness ( c f . a l s o Walker 11; 2; 31) I n Wang [2] i t i s p o i n t e d o u t t h a t i f a ( - )

Remark 2.11

i s monotone near t = 0 ( l o c a l l y monotone) then (2.35)

can be

*

r e p l a c e d by I l u l ( t ) l I 5 c 1 0 t 1 / Z a ( t ) 1 / 2 f o r small t so t h a t i n (2.36)

a1/2(T)Y(t,.r)-l/2 with (TI a T h i s produces a somewhat s t r o n g e r

one c o u l d w r i t e Y(t,T)ul(T)

ul(s)/a1/2(T)

+.

0 as

T

+.

0.

=

v e r s i o n o f Theorem 2.9 when a ( - ) i s l o c a l l y monotone. a r e given i n Wang [2] t o show t h a t A ( t ) =

Examples

a ( S ) d S / a ( t ) may n o t

even be bounded f o r nonmonotone a ( t ) = O ( t m ) . Remark 2.12 (2.9)

F i r s t we assume z1 and b ( t ) a r e r e a l so t h a t

becomes, a f t e r e l i m i n a t i o n o f t h e A" term as before, and

adding a term clX

I

Now assume I b ( t ) / b ( t ) l

5 c 3 on [o,T]

162

h

and s e t c = max(cl,c3);

3. DEGENERATE EQUATIONSWITH OPERATOR COEFFICIENTS

then, i f zlb(S)

?

0 one can w r i t e (for 2f3

A

A

S e t t i n g E(t,<) (cf.

5 1 and A 2 1 )

= exp c ( t - C ) we o b t a i n from Gronwall's lemma again

(2.12))

(2.41)

t A

E ( t ) 5 k(t,T)

A

+ A / P(C)E(t,S)jY12dC T

*

Assuming P 2 0 Lemma 2.2 a p p l i e s t o (2.41) term l Y t l 2 + z l b ( t ) l Y / 2 i n B ( t ) ) t o y i e l d

(upon o m i t t i n g the

and as i n Lemna 2.3 one o b t a i n s ( c f . (2.17)) Lemna 2.13 A

P

I

Assume I b /bl

2 0, a 2 0, and zlb

2 0.

A

5 c3 w i t h c

= max(cl,c3) A

while Y

Then f o r c4 = exp cT and c1 =

max( 1 ,c 28-1 )

C o r o l l a r y 2.14

D e f i n i n g $(t,T)

the hypotheses o f Lemma 2.13 imply

163

and $(T) as i n C o r o l l a r y 2.4

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Thus i t i s p o s s i b l e t o e l i m i n a t e t h e JI t e r m i n Q and q ( c f . Theorems 2.5,

2.6,

i f one assumes zlb

and 2.8,

Remarks 2.10 and 2.11,

1 0 w i t h I b l / b l <, c3 i n a d d i t i o n t o a ' 2 T h i s means l o c a l l y (i.e.,

(cf . Remark 2.10).

-cllrl'

and Lemma 2.7)

t h a t i f z1 2 0 then b L 0 with e i t h e r b b(T) exp c3(t--r) o r b

I

5

0 with b(t)

F

I

near t =

T)

1 0 and 0 5 b ( t ) 5

b(-r) exp ( - c 3 ( t - ~ ) ) ; I

s u i t a b l e o s c i l l a t i o n s i n t h e s i g n o f b a r e o f course allowed. On t he o t h e r hand i f z1 5 0 then b 5 0 w i t h

= -b so e i t h e r

o r b' i 0 w i t h b ( t ) 5

b' L 0 w i t h 0 L b ( t ) 2 b(T) exp (-c 3 (t--r) ) b(-r) exp c3(t--r).

Ib l

We r e f e r t o C a r r o l l [34] f o r f u r t h e r remarks.

Remark 2.15

I n C a r r o l l [7; 91 some weak degenerate problems The n o t a t i o n i s

a r e solved usin g Lions type energy methods.

chosen here t o conform t o these a r t i c l e s r a t h e r than t o e a r l i e r p a r t s o f t h i s s e c t i o n and most t e c h n i c a l d e t a i l s w i l l be omitted. Thus l e t Q > 0 be a numerical f u n c t i o n i n Co(o,T], i n c r e a s i n g as t

-+

t

-+

0.

m;

with

)I

0 (a p r i o r i Q need n o t approach i n f i n i t y b u t 1 L e t q > 0 belong t o C (o,T]

i t usually w i l l ) .

T <

with q(t)

-+

0 as

L e t V c H be H i l b e r t spaces ( H separab e f o r s i m p l i c i t y ) ,

V dense i n

H w i t h a f i n e r topology, and a(t,-,

continuous s e s q u i l i n e a r forms on V and a(t,u,u)

2 ,> a l l u l l v .

f i x e d and l e t t

-+

B(t)

Hermitian operators.

Assume t E

-+

x

) a family o f

V w i t h a(t,v,u)

a(t,u,v)

C'(Ls(H)) on [o,T]

E

1

C [o,T]

=

'm

f o r (u,v)

be a f a m i l y o f

L e t w > 0 be a numerical f u n c t i o n t o be

determined such t h a t w ( t )

-+

00

as t

164

-+

0.

L e t Fs be t h e H i l b e r t

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

s 5 T t o be determined, such t h a t

space o f f u n c t i o n s u on [o,s], U ( O ) = 0, Jlu

1

2 2 = L (H), and wu E L (V) w i t h norm I 2 FT $u I H ) d t ; a l l d e r i v a t i v e s are taken i n D (H) ( c f . .

E

C a r r o l l [14] o r Schwartz [5]).

h s a t i s f y i n g h ( s ) = 0, h/$

..

LL(V).

We d e f i n e

..

(2.45)

ES(u,h) =

..

I

S

0

Ls(h) =

E

L e t HS be the space o f functions

L2( H), h ' / $

E

L2( H), and qh/w

-

+ (B(t)u',h)

{qa(t,u,h)

I

E

I

( u ,h ) ) d t ;

fS

-(f,h)dt

JO

where f i s given w i t h Jlf

E

L2 (H) (here (

,

notes the s c a l a r product i n H (resp. V)). to find u

E

..

Fs such t h a t Es(u,h)

since q > 0 on say [s/2,T]

..

find u

E

, 1))

de-

The f i r s t problem i s

= Ls(h) f o r a l l h

E

Hs.

Then

one can apply standard techniques f o r

nondegenerate problems ( c f . Lions [5])

..

(resp. ( (

..

t o proceed stepwi se and

FT s a t i s f y i n g ET(u,h) = LT(h) f o r a l l h

s e r i e s o f t e c h n i c a l lemmas such a u

E

E

HT.

After a

FT can be found using the

Lions p r o j e c t i o n theorem (see C a r r o l l [9] f o r d e t a i l s ) .

Another

s e r i e s o f t e c h n i c a l lemnas w i l l y i e l d uniqueness and one can state Assume t h e c o n d i t i o n s above w i t h q ( t ) 2 t (( d V $ 2 ( < ) ) ' - E ( 0 < E < 1 ) w h i l e Q = (q'VJ2/q)I d W 2 ( E ; ) 0 o . . . e x i s t s . Then t h e r e e x i s t s a unique s o l u t i o n u E FT of ET(u,h) = t

..

Theorem 2.16

L (h) for a l l h T

:$

E

HT, based on a f u n c t i o n w

165

4

L2 (w

E

Co(o,T]).

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

We remark t h a t w measures t h e r a t e o f how r a p i d l y u ( t ) as t

+

+

0

0 b u t i t s p r e c i s e d e s c r i p t i o n i s somewhat complicated

( c f . below).

Another theorem based on energy methods can be

o b t ained as f oll o w s .

rt

k(t) = on (o,s]

JO

@hdS f o r h

E

We d e f i n e K- t o be t h e space o f f u n c t i o n s 5 1 E C [o,s] where @ > 0 Hs w i t h s u i t a b l e

while @(+ t )0 as t

+

For s u i t a b l e choice o f t h e

0.

numerical f u n c t i o n 6 > 0 w i t h b ( t )

+ m

as t

+

0 we p u t a p r e h i l -

(We not e t h a t i f v = @/q then o u r w above can be taken t o be I 2-E1 w2 = v / v ( 0 < E, < l).)F o r s u i t a b l e $,6as above (e.g.,

2 (5) and'b 2 = - "c ( l / v )

j:d
@ =

Theorem 2.17

E

), Ks C Fs and one can s t a t e A

Under t h e hypotheses o f Theorem 2.16 t h e r e

e x i s t s a unique s o l u t i o n u all h

I

"

n

E

KT s a t i s f y i n g ET(u,h) 1

.,

= LT(h) f o r

HT such t h a t I l u l l A 5 c ( / l $ f l i d t ) 1 / 2 KT 0

3.3

S t r o n g l y r e g u l a r equations.

We consider f i r s t t h e

n o n l i n e a r equation I

(3.1)

Fl(t)u ( t ) = f ( t , u ( t ) )

i n a separable and r e f l e x i v e Banach space V. Ia :[o,a] I

L(V,V

).

For each t

we a re g i v e n a continuous l i n e a r o p e r a t o r M ( t )

E

E

Denote by Bb(uo) t h e c l o s e d b a l l i n V centered a t uo

w i t h r a d i u s b > 0, and suppose we are g i v e n a f u n c t i o n f : I ,x I

.

A s o l u t i o n o f (3.1)

i s a f u n c t i o n u : 1,

y ) a b s o l u t e l y continuous, d i f f e r e n t i a b e a.e.

166

-f

V which on I,,

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

3.

has i t s range i n B ( u ) , and s a t i s f i e s (3.1) b o

a.e.

on I,.

f i c i e n t c o n d i t i o n s f o r t h e Cauchy Problem f o r (3.1)

Suf-

t o be w e l l -

posed a r e g i v e n i n t h e f o l l o w i n g . Theorem 3.1 k ( - ) : 1,

+

Assume t h e r e i s a p a i r o f measurable f u n c t i o n s 1 ( o , m ) and Q ( - ) : Ia [ l , m ) w i t h Q ( * ) / k ( - ) E L ( I a , R ) -f

such t h a t

Then any two s o l u t i o n s u 1 ( * ) , u 2 ( - ) o f (3.1) w i l l s a t i s f y t h e estimate

I n p a r t i c u l a r , t h e Cauchy problem o f (3.1) w i t h u ( o ) = uo has

a t most one s o l u t i o n . Proof: M(t) : V

+

Since k ( t ) > 0, t h e c o e r c i v e e s t i m a t e (3.2) i m p l i e s V I i s a b i j e c t i o n and

11

M(t)-’

Thus, we o b t a i n from (3.1) llu;(t)-u;(t)

11

f(t,ul(t))-f(t,u2(t))

t

E

11

V

I

IIL(,,lYv)

/Iv

S k(t1-l.

5 k(t1-l

5 k ( t ) - l Q ( t ) ~ ~ U l ( t ) - u 2 ( t ) ~ ~,, ,a.e.

I,. Since u l ( * )

-

u

2

(

0

)

i s absolutely continuous w i t h a 167

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

I

summable d e r i v a t i v e , we have Ilul(t)-u2(t)llV 5 t l ~ / U ~ ( O ) - U ~+ ( O I() ul(s)-u;(S)IIVds. ~~~ Therefore t h e bounded function Z ( t ) Z(o)

+

0

= Ilul(t)-u2(t)llv

Itk(s)-’g(s)Z(s)dsy

t

s a t i s f i e s the i n e q u a l i t y Z ( t ) 5 E

Ia. The e s t i m a t e (3.4)

follows

0

i n m e d i a t e l y by t h e Gronwall i n e q u a l i t y .

I n a d d i t i o n t o t h e hypotheses o f Theorem 3.1

Theorem 3.2 assume t h a t t and t h a t t

+

+.

QED

<M(t)x,y>

i s measurable f o r each p a i r x, y

i s measurable f o r x

Bb(u0) and y

E

F i n a l l y , l e t bo > 0 and c > 0 s a t i s f y Ilf(t,uo)II

V’

rC

t

E

Icy and

k ( t ) - ’ Q ( t ) d t 5 b/(bo+b).

E

V

E

V.

,

5 Q ( t ) b o y a.e.

Then t h e r e e x i s t s a

0

which s a t i s f i e s u ( o ) = uo. (unique) s o l u t i o n u ( * ) of (3.1) on Ic Proof:

We f i r s t show t h a t f o r every f u n c t i o n u : Ic +

Bb(uo) which i s s t r o n g l y (= weakly) measurable i n V, t h e f u n c t i o n t

+

Ll(t)-l

0

f(t,u(t))

: Ic V i s measurable.

we have < I $ , M ( t ) - l f ( t , u ( t ) ) >

For each I$

-+

=
cated a d j o i n t i s measurable when M ( t ) - ’

-1

*

) cp>.

E

VI

The i n d i -

i s , and so i t s u f f i c e s

t o show t h e m e a s u r a b i l i t y o f each f a c t o r .

To show f ( t , u ( t ) )

i s measurable we consider f i r s t t h e case where u ( * ) i s countablywhere { G : j 2 1) i s a valued; thus u . ( t ) = x . f o r t E G J J j’ j measurable p a r t i t i o n o f I,. L e t t i n g @ . denote t h e c h a r a c t e r i s t i c J function o f G we o b t a i n f ( t , u ( t ) ) = l { f ( t , x . ) I $ . ( t ) : j f 1) jy J J on I and t h i s i s c l e a r l y measurable. The case o f general C’

u ( * ) f o l l o w s from t h e s t r o n g c o n t i n u i t y o f x

+

P(t,x),

s i n c e any

measurable u i s t h e s t r o n g l i r r i t o f countably-valued rlieasurable

168

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

functions.

To show t h e second f a c t o r i s measurable, l e t m 2 1

and c o n s i d e r t h e r e s t r i c t i o n o f M : 1,

+

L(V,V’)

t o t h e s e t Jm o f

Since M i s measurable on t h i s s e t ,

t h o s e t €IC w i t h k ( t ) 2 l/m.

t h e r e i s a sequence o f c o u n t a b l y - v a l u e d f u n c t i o n s Mk : Jm M(J,)

C L(V,V’)

such t h a t

Lz

-+

Mk(t) = M(t) strongly f o r t V

I

Jm.

E

we have

on o f 14-l

to

desired re-

The p r o o f o f Theorem 3.2 now f o l l o w s s t a n d a r d arguments. L e t X be t h e s e t o f u

E

C(Ic,V)

w i t h range i n Bb(uo).

the function t * M ( t ) - l f ( t , u ( t ) ) and i t s a t i s f i e s t h e e s t i m a t e

For u

E

i s measurable Ic + V ( f r o m above)

IIM(t)-’

0

2 k(t1-l

f(t,u(t))lIv

-

Q ( t ) (bo+b) on I,. Hence, we can d e f i n e F : X + X by [ F u ] ( t ) t M(s)-lf(s,u(s))ds, t E I,, and i t s a t i s f i e s t h e e s t i m a t e uo + 0

(3.5)

11 [ F u l ( t ) - [ F v l ( t ) l l v

X

I

=

t

5

0

k ( s ) - l Q ( ~ ) l ~ U ( S ) - V ( S ) I ~ ~ds,

u,v

E

x,

t

E

B u t any map F o f a c l o s e d and bounded subset X o f Lm(Ic,V)

I,. into

i t s e l f which s a t i s f i e s (3.5)

i s known t o have a unique f i x e d -

p o i n t , u ( c f . C a r r o l l [14]).

This fixed-point i s c l e a r l y the

OED

d e s i r e d s o l u t i on.

A s o l u t i o n e x i s t s on t h e e n t i r e i n t e r v a l 1, i n c e r t a i n 169

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Two such s i t u a t i o n s a r e g i v e n below.

situations.

Theorem 3.3 w i t h Bb(uo) = V.

Assume a l l t h e hypotheses o f Theorem 3.1 h o l d Also, suppose t

+

<M(t)x,y>

a r e measurable Ia + R f o r e v e r y p a i r x,y bo > 0 be g i v e n and s a t i s f y I l f ( t , u o ) l l V I

functions u I,.

If u

E

Set g ( t ) = bo exp{ E

C(Ia,V)

rt

5 Q ( t ) b o a.e.

f o r which ]lu(t)-udl,

E

V,

on I,.

and l e t X be t h o s e

5 g(t)

X, t h e e s t i m a t e I l ~ l ( s ) - l f ( s , u ( s ) ) I I V

k(s)-’Q(s)(llu(s)-udlv

on I, w i t h u ( o ) = uo.

k(s)-’Q(s)dsI

JO

+

V, and l e t uo

E

Then t h e r e e x i s t s a unique s o l u t i o n o f (3.1) Proof:

and t

-

bo f o r t

E

5

+ bo) shows t h a t Fu d e f i n e d as above belongs

t o C(Ia,V) and f u r t h e r m o r e s a t i s f i e s 11 [ F u ] ( t ) - u d l v 5 t k ( s ) - l Q ( s ) g ( s ) d s = g ( t ) - b o , so Fu E X. Thus F has a unique 0

fixed-point u

E

X which i s t h e s o l u t i o n o f t h e Cauchy problem

f o r (3.1).

QED

Remark 3.4

The p r e c e d i n g r e s u l t s p e r m i t t h e l e a d i n g opera-

That i s , t o r s t o degenerate i n a v e r y weak sense a t any to E I a’ 1 (3.2) must be m a i n t a i n e d w i t h k ( * ) - ’ E L ( I a , R ) On t h e o t h e r

.

hand we have p l a c e d no upper bounds on t h e f a m i l y { M ( t ) t h e y may be s i n g u l a r ( c f .

I n t r o d u c t i o n ) on a s u i t a b l e

: t

E

Ial;

s e t of

p o i n t s i n Ia. The L i p s c h i t z c o n d i t i o n (3.3) f(t,uo)

t o g e t h e r w i t h t h e e s t i m a t e on

i n Theorem 3.3 impose a g r o w t h r a t e on f ( t , u )

l i n e a r i n u.

which i s

Such hypotheses a r e f r e q u e n t l y a p p r o p r i a t e , espec-

i a l l y i n l i n e a r problems such as Example 1.1 which o c c u r

170

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

3.

However they a r e not appropriate f o r

frequently in practice.

nonlinear wave propagation models, and these applications c a l l f o r solutions global in time. The t h i r d order nonlinear equation

Example 3.5 (3.6)

Ut

-

+ ux + uux = 0

Uxxt

provides a model f o r the propagation of long waves of sma 1 amplitude.

The interest here i s i n solutions of (3.6) f o r a l l t 2 0.

Similarly, models with other forms of n.on1inearity i n add tion t o dispersion and dissipation occur i n the form (3.7)

u

t

-

u

xxt

l u l q + buPux = cu

+ a sgn(u)

xx'

W e r e f e r t o Benjamin [ l , 2 3 , Dona [2,3, 4, 51, and Showalter [ l o , 19, 211 f o r discussion of such models and r e l a t e d systems. I n i t i a l boundary value problems f o r (3.6) and (3.7) can be resolved in the following a b s t r a c t forni. Theorem 3.6

Let a > 0 and assume given f o r each t

[O,a] an operator M ( t ) Banach space.

E

L(V,V

I

+

1,

Assume there i s a k > 0 such t h a t

E

V the function

E

I,,

<M(t)x,y> i s absolutely continuous w i t h

(3.9)

d dt <M(t)x,x>

=

), where V i s a separable reflexive

each M(t) i s symmetric, and f o r each p a i r x,y

t

E

5 m(t)llxll

2

,

171

a.e. t

x

E

V,

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

f o r some m ( - )

E

1 L (Ia,R).

f o r each p a i r x,y

E

L e t f: 1, x V

V the function t

-+

-+

v~

Ilf(t,x)-f(t,y))I

f o r x,y

B and a.e.

E

t

I

be g i v e n such t h a t

and f o r each bounded s e t B i n V t h e r e i s a that

V

a(-)

E

i s measurable, 1 L ( I a y R ) such

5 Q ( t ) Ilx-Alv, and I l f ( t , x ) I I I,.

E

5 O(t),

F i n a l l y , assume t h e r e e x i s t K ( - )

and L ( * )

E

1 L (Ia) with L ( t )

2 0 a.e. such t h a t < f ( t , x ) , x > 5

K(t)/Ixlf

+

L(t)llXll, x

Then f o r each uo

unique s o l u t i o n u : 1, Proof:

E

V.

+

V there e x i s t s a

E

V o f ( 3 1 ) w i t h u ( o ) = uo.

Uniqueness and l o c a

from Theorem 3.1 and Theorem 3.2,

e x i s t e n c e f o l l o w immediately respectively.

The e x i s t e n c e

o f a ( g l o b a l ) s o l u t i o n on 1, can be o b t a i n e d by standard c o n t i n u a t i o n arguments i f we can e s t a b l i s h an a p r i o r i bound on a s o l u tion.

But i n t h e present s i t u a t i o n t h e a b s o l u t e l y continuous

function a ( t )

I

5

Z C2L(t)/k’/2]

< M ( t ) u ( t ) , u ( t ) > s a t i s f i e s ( c f . Lemma 5.1) a ( t ) =

+ <Ml(t)u(t),u(t)> 5 [(ZK(t)+m(t))/k] a ( t ) +

o(t)”*;

hence we o b t a i n

’0

with H ( t )

=

’0

(K+(t) + (1/2)m(t))/k.

The estimates (3.10) and

(3.8) p r o v i d e t h e d e s i r e d bound on a l l s o l u t i o n s o f (3.1) w i t h given i n i t i a l data uo. Remark 3.7

on 1, OED

Equations s i m i l a r i n form t o (3.G) have been

used t o “ r e g u l a r i z e ” h i g h e r o r d e r equations as a f i r s t s t e p i n s o l v i n g them ( c f . Bona [4],

Showalter [17]).

172

The advantage over

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

standard (e.g.

, parabolic)

r e g u l a r i z a t i o n s i s t h a t the order o f

t h e problem i s n o t increased by t h e r e g u l a r i z a t i o n . case o f t h i s i s t h e "Yosida

An i m p o r t a n t

approximation" AE o f a maximal ac-

c r e t i v e ( l i n e a r ) A i n Banach space g i v e n by AE = ( I + E A ) - ~ A , E > I

Thereby, one approximates t h e e q u a t i o n u ( t ) + A u ( t ) = 0 by

0.

one w i t h A r e p l a c e d by t h e bbunded A E y or, e q u i v a l e n t l y , solves I

t h e equation (I+EA)uE(t) + AuE(t) = 0 and then l o o k s f o r u(t) =

1i m

E4+

T h i s technique was used f o r a

u E ( t ) i n some sense.

n o n l i n e a r time-dependent e q u a t i o n by Kato [2] and t o study t h e nonwell-posed backward Cauchy problem i n Showalter [ l G ,

201;

a l s o see B r e z i s [3]. The preceding r e s u l t s immediately y i e l d corresponding r e s u l t s f o r second o r d e r e v o l u t i o n equations o f t h e form (3.11)

M(t)u

I 1

( t ) = F(t,u(t),ul(t)) L e t t h e f a m i l y o f operators ( M ( t ) > and t h e

Theorem 3.8

L e t uoyul

f u n c t i o n s k ( - ) and Q(*) be g i v e n as i n Theorem 3.2. V and t h e f u n c t i o n F : 1, x Bb(uo)

t V.

1)

+

measurable f o r x1

Suppose we have IIF(t,x)-F(t,y)ll F(t,uoyul)l~v,

f o r x = (xl,x2)

x

VI

Bb(ul)

+

V

I

be given w i t h

Bb(ul). y E 2 2 5 Q(t)[llxl-YIIlv + IIx2-YdIv1~

E

Bb(u0), x2

E

E

5 c l ( t ) , and

a n d y = (y1,y2) i n B ( u ) x Bb(ul) b o

and a.e.

t

E

Then t h e r e e x i s t s c > 0 and a unique u : Ic + V which i s Ia. 173

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

continuously d i f f e r e n t i a b l e w i t h u ( t ) u

I

I

E

Bb(uo). u ( t )

E

Bb(ul),

,

i s ( s t r o n g l y ) a b s o l u t e l y continuous and d i f f e r e n t i a b l e a.e.

(3.11)

I

holds a.e.

on Ic and u ( o ) = u o y u ( 0 ) = ul.

The preceding f o l l o w s from Theorem 3.2 a p p l i e d t o t h e equation M(t)U'(t) = f(t,U(t))

in V z

[M(t)xlyM(t)x2]

5

and f ( t , x )

V x V w i t h M(t)x

[M(t)x2,F(t,xl

!

The e s t i m a t e

,x,)].

l i m i t s t h e growth r a t e o f t h e l e a d i n g o p e r a t o r ( c f . Re-

(3.12)

T h i s hypothesis can be d e l e t e d when V i s a H i l b e r t

mark 3.4).

space; we need o n l y r e p l a c e M ( t ) by t h e Riesz map V

-+

V ' i n the

f i r s t f a c t o r o f each o f M ( t ) and f. I n i t i a l boundary value problems f o r p a r t i a l d i f f e r e n t i a l equations a r i s e i n t h e form (3.11)

i n various applications.

Tww

c l a s s i c a l cases a r e g i v e n below; see L i g h t h i l l [l]o r Whitham

[l]f o r a d d i t i o n a l examples o f t h i s type. The equation o f S. Sobolev [l]

Example 3.9 (3.13)

+ uzz = 0

A3utt

describes t h e f l u i d motion i n a r o t a t i n g vessel.

I t i s on ac-

count o f t h i s equation t h a t t h e term "Sobolev equation" i s used i n the Russian l i t e r a t u r e t o r e f e r t o any equation w i t h s p a t i a l d e r i v a t i v e s on t h e h i g h e s t o r d e r t i m e d e r i v a t i v e (cf., Example 1 .2). Example 3.10 (3.14)

u

tt

-

The equation

aA3utt

-

A3u = f ( x , t )

174

however,

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

3.

was i n t r o d u c e d by A.E.H. t i o n s i n a beam.

Love [l]t o d e s c r i b e t r a n s v e r s e v i b r a -

The second t e r m i n (3.14)

represents r a d i a l

i n e r t i a i n t h e model. F i n a l l y , we d e s c r i b e how c e r t a i n d o u b l y - n o n l i n e a r e v o l u t i o n equations (3.15)

M(t,u’(t))

=

f(t,u(t)),

0 5 t < _ T ,

i n H i l b e r t space can be s o l v e d d i r e c t l y b y s t a n d a r d r e s u l t s on monotone n o n l i n e a r o p e r a t o r s .

The t e c h n i q u e a p p l i e s as w e l l t o

t h e corresponding v a r i a t i o n a l i n e q u a l i t y I

<M(t,u ( t ) )

(3.16)

-

f(t,u(t)),

v

-

I

u (t).

2 0,

v

E

K(t),

w i t h a p r e s c r bed fami y o f c l o s e d convex subsets K ( t ) c V, so we c o n s i d e r t h i s more general s i t u a t i o n . L e t V be r e a l H i l b e r t space and f o r each a 2 0 we l e t Va be t h e H i l b e r t space o f square summable f u n c t i o n s from [O,T] i n t o T V w i t h t h e norm I l v l l a = I l v ( t ) l l i e - 2 a t dt)”2. F o r each t E

(1

[O,T]

0

we a r e g i v e n a p a i r o f ( p o s s i b l y n o n l i n e a r ) f u n c t i o n s f ( t , * ) ,

M(t,*) from V i n t o V

I

.

The f o l l o w i n g elementary r e s u l t i s funda-

mental f o r t h i s technique. Lemma 3.11 uous M : Va

Assume [Mv](t)

E M(t,v(t))

I

+

d e f i n e s a hemicontin-

V a and t h a t t h e r e i s a k > 0 w i t h

175

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(3.17) L e t uo Va

+

V

2 2 kllx-yllv,

<M(t,x)-M(t,y),x-y> E I

a

=

V and assume [ f ( v ) ] ( t )

f(t,uo+

a.e.

t

E

[O,T],

x,y

E

V.

v(s)ds) defines f :

and t h a t t h e r e i s a Q > 0 w i t h

(3.18)

<- Q (1 X-Y 11 v >

11 f ( t , x ) - f ( t > Y ) l l

Then t h e o p e r a t o r T

=M-

f : Va

a.e.

t

E

[O,T],

x,y

E

V.

I

-+

Va i s hemicontinuous and

s t r o n g l y monotone f o r "a" s u f f i c i e n t l y large. Proof:

For u,v

E

Va we have

The c o n d i t i o n o f u n i f o r m s t r o n g m o n o t o n i c i t y on M ( t , - ) <Mu-Mv,u-v>

2 c IIu-vII,

2

, u,v

E

Va,

gives

2

hence we obtai'n

E c - Q ( T / ~ ~ ) ~ / ~ I ~u,v ~ u E- vV a' ~ ~ ~The , d e s i r e d r e s u l t holds f o r a >

QED

TQ~/ZC~.

176

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Theorem 3.12

L e t t h e hypotheses o f Lemma 3.11 h o l d and as-

sume i n a d d i t i o n t h a t we a r e g i v e n a f a m i l y o f nonempty c l o s e d convex subsets K ( t ) o f V f o r w h i c h t h e c o r r e s p o n d i n g p r o j e c t i o n s P(t) : V

+

K ( t ) a r e a measurable f a m i l y i n L(V) ( c f . C a r r o l l

Then t h e r e e x i s t s a unique a b s o l u t e l y c o n t i n u o u s u [14]). I 2 L (0,T;V) w i t h u E Vo, u ( o ) = uo, and s a t i s f y i n g (3.16). Proof: Va * V {v

E

I

a

W i t h a chosen as i n Lemma 3.11

E

K ( t ) , a.e.

and nonempty i n Va,

t

E

[O,T]).

(3.19)

Define K 5

Then K i s closed, convex

so i t f o l l o w s f r o m Browder [5]

t h a t t h e r e e x i s t s a unique w

E

2 0,

Vo =

operator T :

i s hemicontinuous and s t r o n g l y monotone.

Va : v ( t )

[14])

, the

E

(cf. Carroll

K such t h a t v

E

K.

F i n a l l y , t h e measurable f a m i l y o f p r o j e c t i o n s ( P ( t ) > i s used t o show t h a t (3.19)

i s e q u i v a l e n t t o (3.16) w i t h u ( t ) = uo

rt

J o w ( s ) d s , 0 5 t 5 T.

i.

QED

The p r e c e d i n g r e s u l t i s f a r from b e i n g b e s t p o s s i b l e , b u t s e r v e s t o i l l u s t r a t e t h e a p p l i c a b i l i t y o f t h e t h e o r y o f monotone nonlinear operators t o t h e situation. t h e doubly-nonl i n e a r e q u a t i o n

and more genera l y , t h e

neq ua it y

177

A s i m i l a r r e s u l t holds f o r

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(3.21)

-

<M(t,u(t)) u(t)

E

'0

K(t),

-

rt

1, f(S,u(S))dS,

v-u(t)> a.e.

2 0, v

M(t,-)

K(t),

t E COYTI,

under e s s e n t i a l l y t h e same hypotheses as Theorem 3.12. be d e s i r a b l e t o have such r e s u l t s on e i t h e r (3.16)

E

I t would

o r (3.21) w i t h

p e r m i t t e d t o be degenerate, o r a t l e a s t n o t u n i f o r m l y

bounded ( c f . Remark 3.4).

We s h a l l r e t u r n t o s i m i l a r problems

i n Banach space w i t h a s i n g l e l i n e a r o p e r a t o r ( p o s s i b l y degenera t e ) i n S e c t i o n 3.6 below. ng Theorem 3.12)

A l s o , a v a r i a t i o n on Theorem 3.8

g i v e s e x i s t e n c e r e s u l t s f o r second o r d e r

u t i o n e q u a t i o n s and i n e q u a l i t i e s . Remark 3.13

Theorems 3.1,

3.2 and 3.3 a r e f r o m S h o w a l t e r

and Theorem 3.4 i s f r o m S h o w a l t e r [18,

191.

Theorem 3.8 was

u n p u b l i s h e d and Theorem 3.12 i s due t o Kluge and Bruckner [l]. F o r a d d i t i o n a l m a t e r i a l on s p e c i f i c 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 and on general e v o l u t i o n e q u a t i o n s o f t h e t y p e c o n s i d e r e d i n t h i s s e c t i o n we c i t e t h e above r e f e r e n c e s and Amos [l],Bardos-BrezisB r e z i s [Z],

Bhatnager [l],Bochner-von Nuemann [l],Bona [l],

B r i l l [l],Brown [l],C a l v e r t [l],Coleman-Duffin-Plizel

[23,

C o l t o n [l,2, 31, Davis [l,23, Derguzov [l],Dunninger-Levine 111, E s k i n [l],Ewing [l,2, 31, W. H. F o r d [l,21, Fox [31, Gajewski-Zacharius [1,

2, 31, Galpern [l,2, 3, 4, 51, Grabmuller

[l],tiorgan-Wheeler [l],I l i n [l],Kostyuzenko-Eskin [l],Lagnese

[l,2, 3, 4, 5, 6, 7, 8, 91, Lebedev [l],L e v i n e [3, 5, 6, 7, 8,

91,

Lezhnev [l],L i o n s [5,

9,

lo], 178

Maslennikova [l,2, 3, 4, 5,

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

61, tkdeiros-Menzala [l],M i k h l i n [Z],

Fliranda [l],Neves [l],

Prokopenko [l],Rao [l,2, 4, 51, Rundell [l,2, 31, Selezneva

[l],Showalter [l,7, 8, 9, 11, 12, 13, 141, S i g i l l i t o [l,2, 31, Stecher-Rundell [l],T i n g [l,31, Ton [l],Wahlbin [l],Zalenyak One c o u l d a l s o i n c l u d e c e r t a i n r e f e r e n c e s f r o m

[l,2, 3, 41. Remarks 4.10,

5.18,

6.24 and 7.10 t o which we r e f e r f o r a d d i t i o n -

a l material. 3.4

Weakly r e g u l a r e q u a t i o n s

We r e s t r i c t o u r a t t e n t i o n

t o a c l a s s o f l i n e a r e v o l u t i o n e q u a t i o n s o f t h e form Mul(t) t L u ( t ) = f ( t ) ,

(4.1)

and o b t a i n we1 1-posedness r e s u l t s when as s t r o n g as L.

t > 0,

M i s not (necessarily)

A l t h o u g h t h e semigroup techniques employed h e r e

w i l l be used f o r n o n l i n e a r degenerate problems i n S e c t i o n 3.6 below, we i n t r o d u c e them i n t h e s i m p l e r s i t u a t i o n o f (4.1) where we s h a l l be a b l e t o d i s t i n g u i s h t h e a n a l y t i c s i t u a t i o n from t h e s t r o n g 1y-continuous case. L e t Vm be a complex H i l b e r t space w i t h s c a l a r - p r o d u c t (-,*), and d e f i n e t h e isomorphism ( o f R i e s z ) f r o m V , o n t o i t s a n t i d u a l V,

I

x,y

of c o n j u g a t e - l i n e a r continuous f u n c t i o n a l s by <Mx,y> E

I

-f

(x,y),,

Suppose D i s a subspace o f Vm and t h e l i n e a r map

. ,V

I f u E V and f o m c o n s i d e r t h e problem o f f i n d i n g a u ( * )

L :D

=

Vm i s given.

E

C((o,m),Vm)

E

C([o,m),

I

a r e g i v e n , we 1 Vm) f ) C ((o,m),

Vm) which s a t i s f i e s (4.1) and u ( o ) = uo.

To o b t a i n an elementary uniqueness r e s u l t , l e t u ( * ) be a 179

SINGULAR AND DEGENERATE CAUCHY PROBEMS

s o l u t i o n o f (4.1)

with f

-2Re,

t > 0.

( u ( t ) , u(t))A’2 tained.

0 and n o t e t h a t D,(u(t),u(t)),,,

=

Hence, if L i s monotone then I l u ( t ) I l m =

i s nonincreasing and a uniqueness r e s u l t i s ob-

Recall t h a t L monotone means Re

Z

0, x

E

D.

This

computation suggests t h a t Vm i s an a p p r o p r i a t e space i n which t o The equation i n V

seek well-posedness r e s u l t s .

-1 u l ( t ) + M oLu(t) = M-lf(t),

(4.2)

lil

t > O

i s e q u i v a l e n t t o (4.1) and suggests c o n s i d e r a t i o n o f t h e o p e r a t o r -1 A = M OL w i t h domain D ( A ) = D. We see from (4.3)

(Ax.Y),

,

=

x

E

D, y

E

Vm

t h a t L i s monotone i f and o n l y i f A i s a c c r e t i v e i n t h e t l i l b e r t space Vm.

I n t h i s case, -A generates a c o n t r a c t i o n semigroup

on Vm i f I + A i s s u r j e c t i v e ; then t h e Cauchy problem i s w e l l posed f o r (4.2). C a r r o l l [14],

These o b s e r v a t i o n s prove t h e f o l l o w i n g ( c f .

H i l l e [2],

Theorem 4.1 H i l b e r t space Vm,

o r Kato [l]).

L e t M : Vn, L : D

-t

V,

I

* Vm be t h e Riesz map o f t h e complex

I

be g i v e n monotone and l i n e a r w i t h I

domain i n V m y and assume M + L : D each f

E

C([O,m),V,)

I

and uo

E

+

Vm i s s u r j e c t i v e .

Then f o r

D t h e r e i s a unique s o l u t i o n u ( - )

o f (4.1) w i t h u(o) = uo. The Cauchy problem f o r (4.1)

i s solved above by a s t r o n g l y

continuous semigroup o f c o n t r a c t i o n s .

T h i s semigroup i s a n a l y t i c

when t h e o p e r a t o r A i s s e c t o r i a l (Kato [1]) 180

and we describe t h i s

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

s i t u a t i o n i n the following. Theorem 4.2

Suppose M i s t h e Riesz map o f t h e H i l b e r t space

Vm, V i s a H i l b e r t space dense and continuous i n Vm, and L : V V ' i s continuous, l i n e a r and coercive: f o r some c > 0. and uo

E

V,

Re L cllxIl;,

x

Then f o r e v e r y Hb'lder continuous f: O[).,

-+

-+

V,

E

V,

I

w i t h u(o) =

t h e r e i s aunique, s o l u t i o n u ( = ) o f (4.1)

uO*

Each o f t h e two preceding r e s u l t s i m p l i e s well-posedness of t h e Cauchy problem f o r a l i n e a r second-order e v o l u t i o n equation (4.4)

Cu

I 1

I

( t ) + Bu ( t ) + A u ( t ) = f ( t ) ,

The i d h a i s t o w r i t e (4.4)

Theorem 4.3

as a f i r s t - o r d e r system i n t h e form

L e t Va and Vc be complex H i l b e r t spaces w i t h

Va dense and continuous i n Vc,

Vc

and denote by A : Va

I

-f

' 0.

t

I

Va and C :

-+

L e t B be l i n e a r and monotone

Vc t h e r e s p e c t i v e Riesz maps. I

from D(B) C Va i n t o V a and assume A + B + C : D(B) -+ V i i s sur1 j e c t i v e . Then f o r each f E C ( C O , ~ ) , V L ) and p a i r uo E Va, u1 D(B) w i t h Auo + Bul c([o,m),va)

E

Vc t h e r e i s a unique s o l u t i o n u ( * )

n c 1((oym),va) n

cl([o,m),vc)

n c2((o,m),vc)

E

of (4.4)

I

w i t h u ( o ) ) = u o and u ( 0 ) = ul. Proof:

E

I

Define Vm = Va x Vc;

then we have V,

181

I

I

I

= Va x Vc and

SINGULAR AND DEGENERATE CAUCHY PROBLEMS I

t h e Riesz map M : Vm [xl,x2]

E

I

I

[xl,x2]

I

+

Vm by L([xl ,x2])

I n o r d e r t o a p p l y Theorem 4.1 we need o n l y t o v e r i f y t h a t

L i s monotone.

E

Va and C : Vc

fly x2

E E

I

Va and f2 E Vc, D(B),

then s e t x1 = x2 + A - ’ f l

D w i t h (M+L)[xl ,x2] = [fl ,f2].

I

-+

then we can

QE D

L e t t h e H i l b e r t spaces and operators A: Va

Theorem 4.4 I

-

so B monotone i m p l i e s

I

F i n a l l y , i f fl

t o o b t a i n [x, ,x2]

But an easy computation

[x, ,x2]> = Re

s o l v e (A+B+C)x2 = f2

L e t B : Va

Vc be g i v e n as i n Theorem 4.3.

be continuous and l i n e a r w i t h B + AC : Va

E

=

I

shows Re
uo

Vc),

i s c l e a r l y e q u i v a l e n t t o Mw ( t ) + L w ( t ) = [O,f(t)],

L i s monotone and M + L i s s u r j e c t i v e .

A > 0.

I

E

With M and L so defined, t h e

I).

E

= [Axl,Cx2],

V a x D(B) : Axl + Bx2

E

and d e f i n e L : D

+ Bx2],

system (4.5) t > 0.

Vm i s g i v e n by M[xl,x2]

L e t D E {[x,,x,]

Vm.

r e c a l l i n g Vc C Va, [-Ax2,Axl

+

u1

E

Vc,

n c2 ((O,m),Vc)

Va

Va c o e r c i v e f o r some I

+

t h e r e i s a unique s o l u t i o n u ( = )

c 1 ((0,m),Va) f l c 1 “O,m),Vc)

I

+

I

+

Then f o r e v e r y H6lder continuous f : [O,m)

v a’

+

of (4.4)

E

Vc and p a i r

C([O,m),Va)n w i t h u(0) =

I

uo and u ( 0 ) = ul. Proof:

F o l l o w i n g t h e p r o o f o f Theorem 4.3,

can a p p l y Theorem 4.2 w i t h V i s V - e l l i p t i c f o r some X > 0.

=

we f i n d t h a t we

Va x Va i f we v e r i f y t h a t AM + L But AB

p r e c i s e l y what i s needed.

+ C being V a - e l l i p t i c i s QED

B r i e f l y we i n d i c a t e some examples o f p a r t i a l d i f f e r e n t i a l equations f o r which corresponding i n i t i a l - b o u n d a r y value problems 182

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

a r e s o l v e d by t h e preceding r e s u l t s .

The examples a r e f a r from

b e s t p o s s i b l e i n any sense and a r e d i s p l a y e d here o n l y t o suggest t h e types o f equations t o which t h e r e s u l t s apply. Let m(*)

Example 4.5

Lm(O,l) w i t h m(x) > 0 f o r a.e.

E

and d e f i n e Vm t o be t h e completion o f C;(O,l)

(0,l)

s c a l a r - p r o d u c t (u,v) s e t

=

rl

I

m

m(x)u(x)mdx.

E

0

u.(x)vl(x)dx,

u,v

V.

E

x

E

with the

L e t V = HA(0,l) and

Then Theorem 4.2 shows

JO

t h a t t h e i n i t i a l -boundary va 1ue problem m(x)Dtu(x,t) u(0,t)

=

-

u(1,t)

Dxu(x,t) z =

=

t > 0,

f(x,t),

0,

U(X,O) = uo(x),

O < X < l ,

i s well-posed f o r a p p r o p r i a t e f(-,.) and measurable uo w i t h m1/2uo

E

2 L (0,l).

The i n i t i a l c o n d i t i o n above means

lim"m(x) l u ( x , t ) - u o ( x )

ta+

Jo

I 2 dx

= 0 and t h e equation holds a t each

I

t > 0 i n t h e space Vm o f measurable f u n c t i o n s v on ( 0 , l ) w i t h

m-1'2v

E

2 L (0,l).

I 2 Note t h a t Vm C L ( 0 , l ) C V m and t h e equation

may be e l l i p t i c on a n u l l s e t (where m(x) = 0). 1 L e t Vm = Ho(O,l) w i t h t h e scalar-product

-5--J o l u ( x ) W + m u l ( x ) v l ( x ) ) d x where Exam l e 4.6

(u,~),,,

=

m > 0.

(More general

3 Set Lu = Dxu on D E c o e f f i c i e n t s c o u l d be added as above.) 2 1 {u E H (0,l)fl Ho(O,l) : c u ' ( 0 ) = ~ ' ( 1 ) )where c i s given w i t h IcI

5

1.

Then 2 Re = J uI ( o ) l

-

lu1(1)I2

2 0,u

E

D, s o

L i s monotone; we can show M + L i s s u r j e c t i v e , so Theorem 4.1 shows t h e problem

183

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

+ D,u(x,t) 3

(Dt-mDxDt)u(x,t) 2 u(0,t)

= u(1,t)

u(x,o)

= U,(X)

= 0,

c

0 < x < 1,

= f(x,t),

ux(O,t)

=

i s well-posed f o r a p p r o p r i a t e f ( * , * ) and u

0'

t

ux(l,t),

2

0,

T h i s equation i s a

1 i n e a r r e g u l a r i z e d Korteweg-deVries equation ( c f . Bona [4]).

2 Take Vm as above and l e t V = Ho(O,l)

Example 4.7 D:

: V

-+

1;

V ' d e f i n e d by =

2 Dxu(x)Dxv(x)dx, u,v

with E

L

=

V.

Then Theorem 4.2 g i v e s existence-uniqueness r e s u l t s f o r t h e metap a r a b o l i c (Brown [l]) problem (Dt

- mDxDt)u(x,t) 2 = u(1,t)

when uo

E

H:(0,1)

and t

+ Dxu(x,t) 4

= ux(O,t)

+

=

=

f(x,t),

ux(l,t)

f ( * , t ) : [O,m)

+

O < X < l , t > 0,

= 0,

H-'(O,l)

i s H6lder

con t i nuous. Example 4.8

2 L (0,l)

1 L e t Va = Ho(O,l)

added as i n Example 4.5.

+ bul, u

E

rl

J",

dxy v c = uydx; p o s i t i v e c o e f f i c i e n t s c o u l d be

w i t h =

by Bu = r u

11,

w i t h =

Let r

Va.

monotone whenever Re(r)

E

$, b

E

R and d e f i n e B: Va

Since Re<Bu,u> = Re(r)

2

0.

The o p e r a t o r A + B

,B

Ilully'

+ C : Va

+

C

Vc

is I

+

Va i s

continuous, l i n e a r and coercive, hence s u r j e c t i v e , so Theorem 4.3 shows t h e problem

184

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

3.

2 x,t)-Dxu( x,t)

t)+rDtu( x,t)+bDxDtu( y 0 , t )

(4.9)

= u(1,t) =

U(X,O)

= 0,

damped wave equation.

u1

E

0 < x < 1, 1 Ho(O,l)

and t

-+

T h i s i s a c l a s s i c a l problem f o r t h e

f(*,t) E C ([0,~),L~(0,1)).

E

= u 1( x ) ,

1 2 (O,l), Ho"H

E

1

s e t B = EA,

t > 0,

u0(x), Dtu(x,o)

i s w e l l posed w i t h uo

,

= f ( x ,t)

I f i n s t e a d o f t h e above choice f o r B we

> 0, then from Theorem 4.4

f o l l o w s well-posedness

f o r t h e " p a r a b o l i c " problem rDtU(X,t)-EDXDtU( 2 2 u(0,t)

X,t)-Dxu( 2 x,t)

= u(1,t)

= f (x,t)

,

O < X < l ,

= 0,

u(x,o) = u0(x),

t > 0,

Dtu(x,o)

=

u,(x),

o f c l a s s i c a l v i s c o e l a s t i c i t y ( c f . Albertoni-Cercignani t h e data i s chosen w i t h u

0

f(*,t)

:

[O,m) -+

L'(0,l)

E

1 H o ( O Y l ) , u1

E

2 L (O,l),

[l]).Here

and t

-t

H6lder continuous.

We r e t u r n t o consider t h e p a r a b o l i c s i t u a t i o n o f Theorem 4.2 and d e s c r i b e a b s t r a c t r e g u l a r i t y r e s u l t s on t h e s o l u t i o n . a d d i t i o n t o t h e hypotheses o f Theorem 4.2,

assume given a H i l b e r t

space H which i s i d e n t i f i e d w i t h i t s dual and i n which V, and c o n t i n u o u s l y embedded. I

I

H = H C VmCV

I

.

: Mu

E

HI.

i s dense

Thus we have t h e i n c l u s i o n s V c V m C

1 L e t M be t h e (unbounded) o p e r a t o r on H ob-

t a i n e d as t h e r e s t r i c t i o n o f M : Vm {u E V,,

In

-t

I 1 Vm t o t h e domain D(M ) =

Similarly, the r e s t r i c t i o n o f L : V

1 t h e domain D ( L ) = { u

E

V : Lu

E

185

-t

V

I

to

H I w i l l be denoted by L1. Each

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

o f D(M 1) and D(L 1 ) i s a H i l b e r t space w i t h t h e induced graph-norm and t h e y a r e dense and continuous i n Vm and V, r e s p e c t i v e l y .

We

s h a l l describe a s u f f i c i e n t condition f o r the s o l u t i o n u ( * ) o f w i t h f ( * ) = 0 t o belong t o C([O,m),Vm)/)Cm((O,m),D(L

(4.1)

1) ) .

The arguments depend on t h e t h e o r y o f i n t e r p o l a t i o n spaces and f r a c t i o n a l powers o f o p e r a t o r s ( c f . C a r r o l l [14] f o r references). S p e c i f i c a l l y , t h e o p e r a t o r L1 i s r e g u l a r l y a c c r e t i v e and c o r r e s ponding f r a c t i o n a l powers Le, 0

5

L

0

1, can be defined.

Their

corresponding domains a r e r e l a t e d t o i n t e r p o l a t i o n spaces between 1 0 5 8 5 1, D(L ) and H by t h e i d e n t i t i e s D(Llme) = [D(L),H;O], and corresponding norms a r e e q u i v a l e n t , Theorem 4.9

I n a d d i t i o n t o t h e hypotheses o f Theorem 4.2,

we assume D(L 1-e ) C D(M 1 ) f o r some 8 , 0 < 8 5 1. uo

E

V,

cm( (0,m)

t h e s o l u t i o n u ( * ) o f (4.1)

with f(*)

, ~1)d

Remark 4.10

= 0 belongs

Theorem 4.1 i s new i n t h e form given.

c l o s e l y r e l a t e d Theorem 4.2

i s from Showalter [6];

[7, 111 f o r an e a r l i e r s p e c i a l case.

Theorem

to

The

c f . Showalter

4.3 i s t h e l i n e a r

case o f a r e s u l t i n Showalter [5] and Theorem 4.4 Theorem 4.9

Then f o r each

i s new.

i s presented i n Showalter [6, p t . 111 w i t h a l a r g e

c l a s s o f a p p l i c a t i o n s t o i n i t i a l -boundary value problems,

The

a d d i t i o n a l hypothesis i n Theorem 4.9 makes L s t r i c t l y s t r o n g e r than M; t h e r e s u l t i s f a l s e i n general when

e

= 0.

M i k h l i n [2], work we r e f e r t o C o i r i e r [l],K r e i n [l],

[l]and Showalter [17]. 186

For r e a t e d Phil ips

3.

3.5

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

L i n e a r degenerate equations.

We s h a l l consider appro-

p r i a t e Cauchy problems f o r t h e equation

whose c o e f f i c i e n t s { M ( t ) )

and { L ( t ) ) a r e bounded and measurable

f a m i l i e s o f l i n e a r operators between H i l b e r t spaces.

We show

( e s s e n t i a l l y ) t h a t t h e problem i s well-posed when t h e { M ( t ) ) are nonnegative, (L(t)+XM(t)) are c o e r c i v e f o r some X > 0, and t h e o p e r a t o r s depend smoothly on t.

Some elementary a p p l i c a t i o n s

t o i n i t i a l - b o u n d a r y value problems are presented i n o r d e r t o i n d i c a t e t h e l a r g e c l a s s o f problems t o which t h e r e s u l t s can be a p p l i e d (cf.

Examples 1.1,

case of (5.1)

1.4 and 1.6).

Consideration o f t h e

on a product space leads t o a p a r a b o l i c system

which c o n t a i n s t h e second o r d e r equation ( c f . Example 1.2) (5.2)

2 %C(t)u(t)) dt

+ $B(t)u(t))

+ A(t)u(t)

= f(t).

C e r t a i n h i g h e r o r d e r equations and systems can be handled s i m i l a r ly.

I n o r d e r t o describe t h e a b s t r a c t Cauchy problem, l e t V be a separable complex H i l b e r t space w i t h norm I l v l I ; V dual, and t h e a n t i d u a l i t y i s denoted by .

I

i s the anti-

W i s a complex

H i l b e r t space c o n t a i n i n g V and t h e i n j e c t i o n i s assumed continuous w i t h norm 5 1.

L(V,W)

denotes t h e space o f continuous l i n e a r

o p e r a t o r s from V i n t o W and t h a t f o r each t

E

T i s t h e u n i t i n t e r v a l [0,1].

Assume

T we are g i v e n a continuous s e s q u i l i n e a r form

187

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

on V and t h a t f o r each p a i r x,y

a(t;*,*)

i s bounded and measurable.

E

E

2

2

L (T,V) t h e f u n c t i o n t

g r a b l e ( c f . C a r r o l l [14], f o r each t

E

measurable.

p. 168).

E

E

V

E

Let {V(t) : t

2 L (T,V)

W and f

+

a(t;u(t),v(t))

i s inte-

S i m i l a r l y , we assume g i v e n

W t h e map t E

E

+.

m(t;x,y)

on W such

i s bounded and

T I be a f a m i l y o f c l o s e d subspaces

2 o f V and denote by L ( T , V ( t ) )

uo

K R l l x l l - llyll f o r a l l x,y

T a continuous s e s q u i l i n e a r form m(t;*,*)

t h a t f o r each p a i r x,y

E

R(t;x,y)

-+

Standard m e a s u r a b i l i t y arguments then show t h a t f o r

T.

any p a i r u,v

those (I

V t h e map t

By u n i f o r m boundedness t h e r e i s a

number KR > 0 such t h a t IR(t;x,y)l and t

E

t h e H i l b e r t space c o n s i s t i n g o f

f o r which (I(t)

E

L 2 (T,V l ) be given.

V ( t ) a.e.

A

on T.

Finally, l e t

s o l u t i o n o f t h e Cauchy prob-

lem (determined by t h e preceding data) i s a u

E

L 2 (T,V(t)) such

that

=

for all v

E

lo

1 < f ( t ) , v ( t ) > d t .+ m(O;uo,v(o)

L2(T,V(t))/)

A family {a(t;-,*)

1

H (T,W) w i t h v ( 1 ) :t

E

T I o f continuous s e s q u i l i n e a r forms

on V i s c a l l e d r e g u l a r i f f o r each p a i r x,y a(t;x,y)

= 0.

E

V the function t

i s a b s o l u t e l y continuous and t h e r e i s an M ( - )

such t h a t f o r x,y

E

1 L (T)

E

V we have a.e.

188

t

E

T.

-t

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

The f o l l o w i n g " c h a i n r u l e " w i l l be used below. Lemma 5.1

Let {a(t;*,-)

Then f o r each p a i r u,v

: t

1 H (T,V)

E

E

T} be a r e g u l a r f a m i l y on V.

the function t

a(t;u(t),v(t))

-+

i s a b s o l u t e l y continuous and i t s d e r i v a t i v e i s g i v e n by I

a(t;u(t),v

( t ) ) , a.e.

Proof: Fix x

V.

E

=

E

L(V,V)

by (a(t)x,y),,

Dt(a(t)x,yn)v,

t

E

T.)

a.e:t

E

T.

a(t;x,y),

absolute c o n t i n u i t y o f t I t & ( s ) x ds, t

E

T.

+

(The e s t i m a t e

*

V

The weak

Thus a ( - ) x i s s t r o n g l y a b s o l u t e l y continuous

1 H (T,V).

( u ( t ) ,a ( t ) v I v

E

a ( t ) x t h e n shows a ( t ) x = a ( o ) x +

and s t r o n g l y d i f f e r e n t i a b l e a.e. E

E

F o r each n 2 1

shows i t i s i n L'(T,V).

'0

Let u

x,y

The map & ( * ) x i s weakly, hence s t r o n g l y , mea-

s u r a b l e and t h e e s t i m a t e (5.4)

on T.

F o r each v

E

V we have ( a ( t ) u ( t ) , v ) v

=

a b s o l u t e l y continuous, s i n c e t h e above d i s c u s -

s i o n a p p l i e s as w e l l t o t h e a d j o i n t a " ( t ) . i s weakly a b s o l u t e l y continuous.

Hence, t

-+

a(t)u(t)

The s t r o n g d i f f e r e n t i a b i l i t y o f

a ( - )f r o m above i m p l i e s D t [ a ( t ) u ( t ) ] E

=

shows ( & ( t ) x y y ) v i s d e f i n e d and continuous a t e v e r y y

and a.e.

t

+

T.

V and l e t { yn : n Z 1) be dense i n V.

define (&(t)x,yn)v (5.4)

t

Define a ( t )

E

+ a(t;ul(t),v(t))

= al(t;u(t),v(t))

uta(t;u(t),v(t))

=

i ( t ) u ( t ) + a ( t ) u l ( t ) , a.e.

T, hence t h e i n d i c a t e d s t r o n g d e r i v a t i v e i s i n L 1 (T,V).

From

t h i s i t f o l l o w s t h a t a ( * ) u ( - ) i s s t r o n g l y a b s o l u t e l y continuous and t h e d e s i r e d r e s u l t now f o l l o w s e a s i l y .

189

QED

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Our f i r s t r e s u l t g i v e s e x i s t e n c e o f a s o l u t i o n w i t h an a p r i o r i estimate. Theorem 5.2

L e t t h e H i l b e r t spaces V ( t ) C V C W , sesqui-

l i n e a r forms m(t;*,-) uo

W and f

E

: t

{m(t;-,=) m(t;x,y) x

E

W.

E

2

L (T,V

' ) be given as above.

Assume t h a t

T I i s a r e g u l a r f a m i l y o f H e r m i t i a n forms on W:

E

'my x,y W,

=

on W and V, r e s p e c t i v e l y ,

and k ( t ; - , - )

E

a.e.

t

E

T, and m(o;x,x)

2 0 for

Assume t h a t f o r some r e a l X and c > 0

(5.5)

Rek(t;x,x)

2

+ hm(t;x,x)

+ m-(t;x,x)

2

x

CllXIl"'

E

V(t),

a.e.

t

E

T.

Then t h e r e e x i s t s a s o l u t i o n u o f t h e Cauchy problem, and i t s a t i s f i e s IIuII

5 const.(IIfII

2

I

L (T,V) L (T,V where t h e constant depends on A and c. Proof:

1

+

m(o;uo,uo))

1/2

,

Note f i r s t t h a t by a standard change-of-variable

argument we may r e p l a c e k ( t ; - , - )

by !L(t;-,*)

+ Xrn(t;*,=).

There-

f o r e we may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t A = 0 i n (5.5):

+ m'(t;x,x)

2 ~ 1 1 x 1 1 ~x.

ale. t E T. Now 1 2 I l u ( t ) / 1 2 d t and l e t define H = L2(T,V(t)) w i t h t h e norm IIuIIH = 0 2 F E {$ E H : $ I E L2(T,W), $ ( 1 ) = 0) w i t h t h e norm 11011, =

2Rek(t;x,x)

190

E

V(t),

3. DEGENERATE EQUATIONS WITH OEPRATOR COEFFICIENTS

sesquilinear, L : F

-+

I$ i s c o n j u g a t e - l i n e a r , and we have t h e

continuous. Finally, f o r

0

i n F we have f r o m Lemna 5.1 and 5.5 w i t h A =

These e s t i m a t e s show t h a t t h e p r o j e c t i o n theorem o f L i o n s [5, p. Thus, t h e r e i s a u

371 a p p l i e s here. L(@) f o r a l l @

E

E

H such t h a t E(u,@) =

F and i t s a t i s f i e s l l u l l H 5 ( 2 / n i i n ( ~ . l ) ) I I L ~ ~ ~ ~ .

But t h i s i s p r e c i s e l y t h e desired r e s u l t . Remark 5.3 m(t;x,x)

QED

The hypotheses o f Theorem 5.2 n o t o n l y p e r m i t

t o v a n i s h b u t a l s o t o a c t u a l l y t a k e on n e g a t i v e values.

T h i s w i l l be i l l u s t r a t e d i n t h e examples. Remark 5.4 r e p l a c e d b y 2X

(5.7)

An easy e s t i m a t e shows t h a t (5.5)

+ a) i f

ReR(t;x,x)

we assume t h a t f o r

some c > 0 and a 2 0

2 2 cllxlIv,

+ Xm(t;x,x)

T h i s p a i r of c o n d i t i o n s i s t h u s s t r o n g e r t h a n (5.5) frequently i n applications. i m p l y m(t;x,x)

2 0 for all t

Note t h a t (5.6) E

holds ( w i t h X

T.

191

b u t occurs

and m(o;x,x)

2 0

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

We p r e s e n t an elementary uniqueness r e s u l t w i t h a r a t h e r severe r e s t r i c t i o n o f m o n o t o n i c i t y on t h e subspaces { V ( t ) > .

This

h y p o t h e s i s i s c l e a r l y "ad hoc" and i s a l i m i t a t i o n o f t h e t e c h n i que which i s n o t i n d i c a t i v e o f t h e b e s t p o s s i b l e r e s u l t s .

In

p a r t i c u l a r , much b e t t e r uniqueness and r e g u l a r i t y r e s u l t s f o r t h e very

s p e c i a l case o f m(t;x,x)

C a r r o l l [14],

= (x,x),

C a r r o l l - C o o p e r [35],

have been o b t a i n e d ( c f .

C a r r o l l - S t a t e [36],

t h e n o v e l t y here i s i n t h e forms m ( t ; - , - )

L i o n s [5]);

determining t h e leading

( p o s s i b l y degenerate) o p e r a t o r s ( M ( t ) ) . L e t t h e H i l b e r t spaces V ( t ) C V C W,

Theorem 5.5

l i n e a r forms !L(t;*,-) uo

E

W and f

E

2

L (T,V

2 0, x

Rem(t;x,x)

'

) be g i v e n as above.

V ( t ) , a.e.

E

on V and W,

and m ( t ; * , * )

t

E

sesqui-

respectively,

Assume t h a t

T, t h a t {!L(t;=,*)

: t

E

T) i s

a r e g u l a r f a m i l y o f H e r m i t i a n forms on V ( t ) : x,y and f o r some r e a l A and c > 0, R(t;x,x) ~ 1 1 x 1 1 ~x.

V ( t ) , a.e.

t

E

o f subspaces { V ( t ) : t

E

TI

(5.9)

E

T

E

t >

T,

t,T

T.

E

V(t),

+ ARem(t;x,x)

a.e.

t

E

2

F i n a l l y , assume t h a t t h e f a m i l y

i s decreasing: imply V ( t ) C V ( - r ) .

Then t h e r e i s a t most one s o l u t i o n o f t h e Cauchy problem. Proof: uO

L e t u ( - ) be a s o l u t i o n o f t h e Cauchy problem w i t h

= 0 and f ( * ) = 0.

By l i n e a r i t y i t s u f f i c e s t o show t h a t

192

T

3.

u ( - ) = 0.

[o,s]

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Let s

E

(0,l)

and v ( t ) = 0 f o r t

shows v ( t )

E

2 L (T,W)

V ( t ) f o r each t

and v ( 1 ) = 0.

,flR(t;v'(t),v(t))dt (5.8)

rs

E

T.

(O,l),

E

S i n c e u(.)

-

1:

E

['u(r)dr

9

L (T,V)

for t

E

and (5.9)

t

Also, v ( t ) = - u ( t ) f o r t so we have v '

E

E

L2(TyV(t))C

i s a s o l u t i o n we have by (5.3)

rn(t;u(t),u(t))dt

= 0.

Lemma 5.1 and

g i v e us t h e i d e n t i t i e s

j 1 2 Rem( t ;u ( t ) ,u ( t ) ) d t =

I

Then v

[s,l].

E

and v ' ( t ) = 0 f o r s

(0,s)

-

and d e f i n e v ( t ) =

1:

Dt%( t ;v ( t ) ,v ( t ) )

+ R'(t;v(t),v(t))}dt

{ZRem(t;u(t),u(t))

- R ' ( t ;v ( t ) ,v ( t ) ) 1dt ,

+ Q(o;v(o),v(o))

0.

=

'0

L ~

As b e f o r e , we may assume %(t;x,x) D e f i n e W(t) = for t

E

cllW(s)

(0,s)

11'

1:

u(.r)d-r; t h e n W(s) = - v ( o )

1 1 x ~ E ~V ( t )~ , t~ E . T. and W(t) - W(s) = v ( t )

and we o b t a i n t h e e s t i m a t e

5 %(o;W(s),W(s))

,f M ( t ) { l l w ( t ) l 1 2

+

:j

ZRem(t;u(t),u(t))dt

S

5 2

+

I/W(s)/121dt.

0

Choose s

0

> 0 so t h a t 2

have IIW(s) 11'

-

5 (2/(c

1

S

2

M ( t ) d t < c.

0rS

Then f o r s 8 [o,so]

O M ( t ) d t ) > j S 1 4 ( t ) l l W ( t ) 1I2dt.

t h e Gronwall i n e q u a l i t y we conclude t h a t W(s) = 0 f o r s hence u ( s ) = 0 f o r s

E

[o.so].

From

0

JO

Since

E

[oyso].

M(*) i s i n t e g r a b l e on T we

c o u l d use t h e a b s o l u t e c o n t i n u i t y o f t h e i n t e g r a l

193

we

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

I

T+ So

choose so > 0 i n t h e above so t h a t T 2

0 with

T

M ( t ) d t < c/2 f o r every

T

+ so 5 1.

Then we a p p l y t h e above argument a f i n i t e

T.

number o f times t o o b t a i n u ( - ) = 0 on

QED

A fundamental problem w i t h degenerate e v o l u t i o n equations i s t o determine t h e p r e c i s e sense i n which t h e i n i t i a l c o n d i t i o n i s

A c o n s i d e r a t i o n o f t h e two cases o f m(t,x,y)

attained.

Lzt

=0

and m(t;x,y) of

:( X , Y ) ~

shows (see below) t h a t we may have t h e case

u(t) = u

0

i n an a p p r o p r i a t e space or, r e s p e c t i v e l y , t h a t These remarks

lim+ u ( t ) does n o t n e c e s s a r i l y e x i s t i n any sense. t-4 i l l u s t r a t e t h e importance o f t h e f o l l o w i n g . Theorem 5.6

L e t u ( * ) be a s o l u t i o n o f t h e Cauchy problem

and assume t h e subspaces { V ( t ) t h e r e i s a to E (0,1] Then f o r e v e r y

T E

: t

T I a r e i n i t i a l l y decreasing:

E

such t h a t (5.9)

(o,to)

and v

holds f o r

V(T) we have t

E

t,T +

E

[o,to].

m(t;u(t),v)

1 i s i n H ( 0 , ~ )(hence i s a b s o l u t e l y continuous w i t h d i s t r i b u t i o n ) m(o;u(o) d e r i v a t i v e i n L2 ( 0 , ~ ) and

:

(){V(T)

T

Proof: t

E

CT,~].

-

uo,v)

> 0) i s dense i n W, then m(o;u(o)

Define v ( t ) = Then v ( * )

E

for t

2 L (T,V(t))f)

E

= 0.

-

R(*;u(-),v)

t

Dtm(*;u(*),v)

194

uo,u(o)

-

uo) = 0.

(o,T) and v ( t ) = 0 f o r

H1(T,W)

and v ( 1 ) = 0, so

That i s , we have t h e i d e n t i t y (5.10)

Thus, i f

=

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

i n t h e space o f d i s t r i b u t i o n s on ( 0 , ~ f)o r e v e r y v

f i r s t and l a s t terms o f (5.10)

E

V(T).

The

2

) so, then, i s a r e i n L ( 0 , ~ and

t h e second and we t h u s have p o i n t w i s e v a l u e s a.e.

i n (5.10)

and

hence t h e e q u a t i o n

=

for g

E

j l < f ( t ) ,v>g( t ) d t

L2 ( 0 , ~ ) and v

E

V(r).

1 Suppose now t h a t $ E H ( 0 , ~ i)s g i v e n w i t h v ( * ) as above so as t o o b t a i n f r o m (5.3) /)(t;u(t),v)$'(t)dt v E V(T).

=

S i n c e (5.11)

these r { D t m ( t ; u ( t )

:1

:1

$(T)

0.

=

Define

-

R(t;u(t),v)$(t)dt

< f ( t ) , v > $ ( t ) d t t m(o;uo,v)$(o)

where

h o l d s w i t h g ( t ) = $ ( t ) , we o b t a i n from

,v)$(t)

t m(t;u(t),v)$'(t)Idt

= -m(o;uo,v)$(o).

0

The i n t e g r a n d i s t h e d e r i v a t i v e o f t h e f u n c t i o n t m(t;u(t),v)$(t)

-+

1 i n H ( 0 , ~ and ) $(r) = 0, so t h e d e s i r e d r e s u l t

QED

f o l l ows.

Another s i t u a t i o n which a r i s e s f r e q u e n t l y i n a p p l i c a t i o n s and t o which t h e above t e c h n i q u e i s a p p l i c a b l e i s t h e f o l l o w i n g . Theorem 5.7

L e t u ( * ) be a s o l u t i o n o f t h e Cauchy problem

and assume t h e r e i s a c l o s e d subspace Vo i n V w i t h V o C f ) { V ( t ) t

E

TI.

L(V,Vi)

Vo,

t

D e f i n e t h e two f a m i l i e s o f l i n e a r o p e r a t o r s { L ( t ) ) C and { M ( t ) I C L ( W , V i )

E

:

by

T; <M(t)x,y> = m(t;x,y),

x

195

E

W, y

= R(t;x,y), E

Vo,

t

E

x T.

E

V, y

E

Then i n

SINGULAR AND DEGENERATE CAUCHY PROBLEMS I

t h e space of Vo-valued d i s t r i b u t i o n s on T we have (5.1) function t

-f

M(t)u(t) : T

and t h e

I

-t

Vo i s continuous.

The p r o o f o f Theorem 5.7 f o l l o w s from (5.10) t h e remark t h a t each term i n (5.1),

with v

as w e l l as l . I ( - ) u ( - ) ,

E

Vo and

is in

Our l a s t r e s u l t concerns v a r i a t i o n a l boundary c o n d i t i o n s . Suppose we have t h e s i t u a t i o n of Theorem 5.7 and a l s o t h a t (5.9)

L e t H be a H i l b e r t space i n which

holds.

imbedded and Vo i s dense.

We i d e n t i f y H

E

H, v

Then from (5.11) rT

E

Vo.

and (5.1)

Dtm(t;u(t) , v ) $ ( t ) d t =

JO

each @

E

w i t h H by t h e Riesz

H 4 Vo and t h e i d e n t i t y = 2 Assume f E L (T,H), T > 0 and v E V ( T ) . we obtair:

:1

R(t;u(t),v)@(t)dt +

( L ( t ) u ( t ) + Dt(M(t)u(t)) ,V)H$(t)dt for

This g i v e s us t h e f o l l o w i n g .

C;(O,T).

Theorem 5.8 (5.9)

Assume t h e s i t u a t i o n o f Theorem 5,7 and t h a t

holds.

L e t H be g i v e n as above and f E 2 each T > 0 and v E V(T) we have i n L ( 0 , ~ ) (5.12)

V i s continuously

I

theorem and hence o b t a i n V o + (h,v)H f o r h

I

R(t;u(t),v)

L 2 (T,H).

Then f o r

+ Dtm(t;u(t),v)

We b r i e f l y i n d i c a t e how t h e preceding theorems can g i v e c o r responding r e s u l t s f o r second o r d e r e v o l u t i o n equations which a r e "parabol i c " .

These r e s u l t s are t i m e dependent analogues o f Theo-

rem 4.4.

196

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

L e t Va and Vc be separable complex H i l b e r t

Theorem 5.9

spaces w i t h Va dense and c o n t i n u o u s l y imbedded i n Vc. {a(t;=,-)

:t

E

TI, {b(t;-,-)

: t

E

{ A ( t ) l and {B(t)IcL(Va,V:)

and { C ( t ) > CL(Vc.Vi)

5.7.

: t

Assume t h a t { a ( t ; * , - )

t 0 for x

E

I

Xa(t;x,x)

+

(5.14)

2Reb(t;x,x)

f(-),g(.)

E

TI

V a y and V c y

E

as i n Theorem

T I and ( c ( t ; * , * )

a (t;x,x)

T I are

E

Assume t h a t f o r some X and c > 0

Va.

2

2 cllxllvay

+ Xc(t;x,x)

Va and a.e. 2 l L (T,Va)

: t

2 0 and

r e g u l a r f a m i l i e s o f H e r m i t i a n forms w i t h a(o;x,x)

E

E

Define corresponding fami J i e s o f 1 i n e a r operators

respectively.

f o r each x

:t

T I and { c ( t ; * , - )

be f a m i l i e s of continuous s e s q u i l i n e a r forms on Va,

c(o;x,x)

Let

t

T.

E

I

t

+ c (t;x,x) Then f o r uo

E

cIIxII 2 , 'a

Va,

vo

E

V c y and

given, t h e r e e x i s t s a p a i r u ( * ) , v ( * )

E

2

L (T,Va) which s a t i s f i e s t h e system

I

i n t h e sense o f Va-valued d i s t r i b u t i o n s on T and s a t i s f i e s t h e i n i t i a l c o n d i t i o n s A(o)u(o) = A(o)uo, C(o)v(o) = C(o)vo. Proof:

T h i s f o l l o w s d i r e c t l y from Theorem 5.2 and Theorem

5.7 w i t h m(t;X,?) and !L(t;Y,a

v

V(t)

f

a(t;xl

= a(t;x,,y2)

v0

5

va

x

,yl)

-

+ c(t;x2,y2)

a(t;x2,yl)

va.

for

X,y E

+ b(t;x2,y2)

W E Va x Vc

for

~,TE QED

197

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

There are two n a t u r a l r e d u c t i o n s o f t h e f i r s t o r d e r system (5.15)

t o a second o r d e r equation, and we s h a l l d e s c r i b e these.

F i r s t , we s e t g(

0 )

=

0 i n Theorem, 5.9 and n o t e than t h a t b o t h

C ( - ) v ( * ) and D t ( C ( * ) v ( * ) ) + B(*)v (.)

1

'

belong t o H (T,Va).

Thus

v ( = ) i s a s o l u t i o n o f t h e second o r d e r e q u a t i o n

and t h e i n i t i a l c o n d i t i o n s (Cv)(o) = C(o)vo, (Dt(Cv) + B v ) ( o ) = -A(o)uo.

Second, we s e t f ( - )

A(t)

i s independent of t, i.e., and Hermit ian A 1 H (T,Va) (5.17)

I

E

=0

L(Va,Va).

5

i n Theorem 5.9 and assume A ( t )

A f o r some n e c e s s a r i l y c o e r c i v e

From (5.15)

i t follows that u(-)

E

and i s a s o l u t i o n o f t h e second o r d e r e q u a t i o n Dt(C(t)u'(t))

+ B(t)ul(t)

t Au(t) = g ( t ) I

and t h e i n i t i a l c o n d i t i o n s u ( o ) = uo, ( C ( * ) u ) ( o ) = C(o)vo. We s h a l l pre s e n t some elementary a p p l i c a t i o n s o f t h e precedi n g r e s u l t s t o boundary value problems f o r p a r t i a l d i f f e r e n t i a l equations i n t he form (5.2).

These examples w i l l i l l u s t r a t e

some l i m i t a t i o n s on th e types o f problems t o which t h e theorems apply. Example 5.10 For o u r f i r s t example we choose Vo = V ( t ) = 1 1 V = Ho(T), W = { @ E H (T) : @ (1 ) = 03 and H = L2(T). Recall t h e est imat e

198

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Let a : T

T be a b s o l u t e l y c o n t i n u o u s and d e f i n e t h e H e r m i t i a n

+.

forms m(t;@,$)

= ~ ~ ( t ) @ ( x ) ~ @,$ d x Ey

W,

t

E

T.

Then m(*;@,$)

I

i s a b s o l u t e l y c o n t i n u o u s and s a t i s f i e s m (t;@,$) = ' 1 I @ ( a ( t ) ) 1 2 a ' ( t ) , @ E W, a.e. t E T. S i n c e a E L (T), t h e e s t i m a t e (5.18) F o r @,$

shows t h a t t h i s i s a r e g u l a r f a m i l y o f forms. 1 1 V, we s e t !L(t;@,$) = @ ( x ) $ ' ( x ) d x , t E T, From t h e

E

1

0

we1 1 -known i n e q u a l i t y

w i t h A = 0.

i t f o l l o w s t h a t t h e p r e c e d i n g f a m i l y s a t i s f i e s (5.7)

W and F

F i n a l l y , l e t uo

E

F(-,t),

Then t h e hypotheses o f Theorem 5.5

t

T.

E

ness) a r e s a t i s f i e d .

E

L 2 ( T x T) be g i v e n and s e t f ( t ) = (on unique-

Suppose a d d i t i o n a l l y t h e r e i s a number a,

0 < a < 2, such t h a t I

(5.20)

(t) 2 a

-

2,

a.e. I

Then (5.18) @

E

i t f o l l o w s t h a t (5.5)

hence, t h e hypotheses o f Theorem 5.2 Let u(*,t)

T.

I 2 1) 2 , L (TI h o l d s w i t h A = 0,

(on e x i s t e n c e ) a r e s a t i s f i e d .

= u ( t ) be t h e s o l u t i o n o f t h e Cauchy problem.

i s t h e unique g e n e r a l i z e d s o l u t i o n o f t h e e l l i p t i c -

p a r a b o l i c boundary v a l u e problem, ut a ( t ) ; -uXx = F ( x , t ) , u(x,o)

E

2 (2-0)11@ shows t h a t 2!L(t;@,@) + m (t;@,@)

V, so by (5.19)

Then u ( x , t )

t

E

T, so u ( * , t )

0 < x <

uxx = F ( x , t ) ,

a ( t ) < x < 1; u(O,t)

= uo(x), 0 < x < a ( o ) .

ti2(T) f o r each t

-

= u(l,t),

t

E

T;

The s o l u t i o n u ( t ) belongs t o and u x ( - , t )

199

a r e continuous

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

across t h e c u r v e x = a ( t ) , t Remark 5.11

E

T. p e r m i t s a ( - ) t o decrease,

The e s t i m a t e (5.20)

h o l d s i f and

b u t n o t v e r y r a p i d l y , whereas t h e e s t i m a t e (5.6) I

o n l y i f a ( t ) t 0 a.e.

on T.

The c u r v e x = a ( t ) i s noncharac-

on T; t h e example a ( t ) = 1/2 + [ ( t

t e r i s t i c a.e.

-

1 / 2 ) / 4 ] 1/ 3

shows i t may a c t u a l l y be c h a r a c t e r i s t i c a t c e r t a i n p o i n t s . Remark 5.12

The c o n c l u s i o n s o f Lemma 5.1

i s a b s o l u t e l y c o n t i n u o u s f o r each p a i r x,y

o n l y t h a t a(*;x,y) V.

h o l d ifwe assume

Hence we need n o t assume an e s t i m a t e l i k e (5.4)

{m(t;.,-)>

i n o r d e r t o o b t a i n Theorem 5.2.

a s s e r t i o n as f o l l o w s :

E

on t h e f a m i l y

One can p r o v e t h i s

( 1 ) use t h e c l o s e d graph and u n i f o r m

boundedness theorems t o o b t a i n an e s t i m a t e

'0

( 2 ) approximate u

I

2 i n L (T,V)

Lebesgue theorem w i t h (5.21) case o f c o n s t a n t v

E

by s i m p l e f u n c t i o n s and use t h e t o obtain the r e s u l t f o r the special

V; approximate v

I

by s t e p f u n c t i o n s and use

t h e r e s u l t s o f ( 1 ) and ( 2 ) t o o b t a i n t h e g e n e r a l r e s u l t .

The de-

t a i l s a r e s t a n d a r d b u t i n v o l v e some l e n g t h y computations. Example 5.13

Our second example i s s i m i l a r b u t a l l o w s t h e

e q u a t i o n t o be o f Sobolev t y p e i n p o r t i o n s o f T

x

T.

Choose t h e

spaces and t h e forms a(t;-,*)

E

W, d e f i n e

m(t;Qy+3)

E

T, where

as b e f o r e . F o r Q,$ I r@(t) I = r ( t ) Q ( x ) m d x + Jo 4 (x)+ (x)dx, t 0

200

ci

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

and B map T i n t o i t s e l f , b o t h a r e a b s o l u t e l y continuous, and B i s nondecreasing.

For $I,$

W , m(-;$I,$)

E

uous s o Remark 5.12 a p p l i e s here. sup{l$I(s)12 : 0 5 s 5 B ( t ) )

i s absolutely contin-

Using t h e e s t i m a t e

2 ~ ~ ( t ) l $ I ' ( x ) 1 2 d x ,$I E V,

show t h a t t h e e s t i m a t e (5.5)

holds i f a

I

one can

has an e s s e n t i a l lower

( 0 )

bound ( p o s s i b l y n e g a t i v e ) and i f t h e r e i s a numberu, 0 < u < 2, I

-

such t h a t a ( t ) 2 u W and F

2 where a ( t ) > B ( t ) .

Thus, f o r each uo

E

L2(T x T) i t f o l l o w s from Theorems 5.2 and 5.5 t h a t

E

t h e r e e x i s t s a unique g e n e r a l i z e d s o l u t i o n o f t h e e l l i p t i c parabolic-Sobolev boundary value problem, ut 0 < x < a ( t ) , x < B(t);

-

-Uxxt

= 0, t E T;

= u(1,t)

B(o)).

uxxt

-

uxx = F,

uxx = F, a ( t ) < x < B ( t ) ;

uXx = F, B ( t ) < x < a ( t ) ; -uXx = F, a ( t ) < u(0,t)

-

u(x,O)

X,

-

B ( t ) < x < 1;

= uo(x). 0 < x < max{a(o),

a ( t ) E 0) show t h a t i f B i s p e r m i t t e d

Examples (e.g.,

t o decrease, then a s o l u t i o n e x i s t s o n l y i f t h e i n i t i a l data uo s a t i s f i e s a c o m p a t i b i l i t y condition.

T h i s i s t o be expected

s i n c e t h e l i n e s "x=constant" a r e c h a r a c t e r i s t i c f o r t h e t h i r d o r d e r Sobolev equation. Example 5.14

We s h a l l seek c o n d i t i o n s on t h e f u n c t i o n

m(x,t) which a r e s u f f i c i e n t t o apply o u r a b s t r a c t r e s u l t s above t o t h e boundary v a l u e problem a/at(m(x,t)u(x,t)) (x,t) a.e.

E

x

E

T

x

T.

T; u(O,t)

uxx = F(x,t),

= 0; ~ ( x , o ) ( u ( x , o ) - u ~ ( x ) ) = 0,

1 We choose t h e spaces Vo = V ( t ) = V = Ho(T) and t h e

forms k ( t ; = , - ) valued m ( - , * )

= u(1,t)

-

2 as above and l e t H = W = L (T). E

Lm(T x

Assume t h e r e a l

T) i s such t h a t m ( x , - ) i s a b s o l u t e l y 201

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

continuous f o r a.e. t

E

T, where

M(-)

x

E

1 L (T).

E

H e r m i t i a n forms m(t;$,$) gular.

L e t uo

E

H, F

Then i t f o l l o w s t h a t t h e f a m i l y o f

1:

: m(x,t)$(x)mdx,

E

5 M ( t ) f o r a.e.

T and s a t i s f i e s Imt(x,t)I

2 L (T

x

$,$

E

W, i s re-

T), and s e t f ( t ) = F ( * , t ) .

Theorems 5.6 and 5.7 show t h a t t h e boundary value problem above i s t h e r e a l i z a t i o n o f our a b s t r a c t Cauchy problem. I n o r d e r t o o b t a i n uniqueness o f a g e n e r a l i z e d s o l u t i o n from Theorem 5.5, n o t e t h a t t h e forms !L(t;*,*)

are c o e r c i v e over V,

so Theorem 5.5 i s a p p l i c a b l e i f and o n l y i f a d d i t i o n a l l y we ass ume

(5.22)

m(x,t)

L

0

x,t

E

T,

2 0 for a l l

since t h i s i s equivalent t o m(t;=,-)

$

E

W, t

E

T.

To o b t a i n e x i s t e n c e from Theorem 5.2 i t s u f f i c e s t o assume m(x,o)

a.e.

2 0,

x

E

T, and

T

x

T I > -2~r

(5.23) ess i n f { m t ( x , t )

: (x,t)

E

f o r then we o b t a i n t h e e s t i m a t e m'(t;4,$)

41

E

V, f o r some number

0,

2

0 < a < ZIT

2

.

t a i n (5.5) w i t h X = 0 and c = a / 2 ~

.

2

2

(a

-

,

2~r')Il 0

Then from (5.19)

I$l 2, we ob-

Thus, (5.22) g i v e s unique-

ness and (5.23) i m p l i e s e x i s t e n c e o f a g e n e r a l i z e d s o l u t i o n o f t h e problem. C l e a r l y n e i t h e r o f t h e c o n d i t i o n s (5.22), (5.23) i m p l i e s t h e other.

I n p a r t i c u l a r (5.23) p e r m i t s m(x,t)

202

t o be negative;

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

then t h e e q u a t i o n i s backward p a r a b o l i c .

This i s t h e case f o r

t h e equation ~ ( ~ ‘ / 4 ) ( 1 - 2 t ) ~ u ( x , t ) )= uxx

(5.24)

= - ( 3 2~/2)(1-2t)‘;

f o r which mt(x,t)

and e x i s t e n c e f o l l o w s . 1/2,

0 i t terval.

But m(=,.)

hence, (5.23)

i s satisfied

i s nonnegative o n l y when

so uniqueness i s claimed o n l y on t h i s s m a l l e r i n -

To see t h a t uniqueness does n o t h o l d beyond t = 1/2, = ( ~ 2 t ) e - ~~ p { l - ( 1 - 2 t ) - ~ s} i n ( T X ) i s a solu-

n o t e t h a t u(x,t) t i o n o f (5.24);

a second s o l u t i o n i s o b t a i n e d by changing t h e t o zero f o r a l l t > 1/2.

values o f u ( x , t )

The l e a d i n g c o e f f i c i e n t i n t h e e q u a t i o n (5.25)

$$(T

2/ N ) ( 1 - t 2 ) ’ / 2 u ( x , t ) )

s a t i s f i e s (5.22),

= uxx

( N > 0) 2 > -27~

hence uniqueness f o l l o w s , b u t mt(x,t)

o n l y on t h e i n t e r v a l [0, 1-(4N)-‘).

Hence, t h e e x i s t e n c e o f a

s o l u t i o n f o l l o w s on t h e i n t e r v a l [O, 1-(4N)-2-u]

f o r any u > 0;

choosing N l a r g e makes t h e i n t e r v a l o f e x i s t e n c e as c l o s e as However t h e f u n c t i o n u(x,t)

d e s i r e d t o [O,l]. exp(2N(1-t)’/2)

sin

w i t h uo(x) = s i n

(TX)

(TX)

-1/2

= (l-t)

i s t h e unique s o l u t i o n o f t h e problem

on any i n t e r v a l [0, 1-u], u > 0, and t h i s

f u n c t i o n does n o t belong t o L

2

(T x T).

Hence, t h e r e i s no s o l u -

t i o n on t h e e n t i r e i n t e r v a l . Example 5.15

L e t G be a smooth bounded domain i n Rn and

r ( x ) be a smooth p o s i t i v e f u n c t i o n on G such t h a t f o r those x 20 3

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

near the boundary a G , r ( x ) equals t h e distance from x t o aG. Parabolic equations of the form

t u t + r(x)L(t)u

(5.26)

=

F(x,t),

X E G ,

t E T

a r i s e i n mathematical genetics ( c f . Kimura and Ohta [ l ] , Levikson They a r e t y p i c a l l y degenerate i n t ( a t t

and Schuss [ l ] ) .

and degenerate i n x ( a t x e l l i p t i c operator.

E

=

0)

3G); L ( t ) i s a strongly uniformly

W e i n d i c a t e how such problems can be inSuppose we a r e given r e a l -

cluded i n the preceding theorems.

valued absolutely continuous R. . ( t ) ( l < i, j S n ) and R . ( t ) ( O < j i n ) 1J J n 2 2 with R o ( t ) 2 0 and 1 Rij(t)ziyj 2 c ( l z l I + + I Z n / 1, i ,j=l c > 0, z E qn. Take Vo = V = H 1o ( G ) and define a regular family

...

of coercive forms by R ( t ; @ , $ )

n

1 R.(t)Dj$ J

$)dx, @,I) E V.

j=o

n

1 R i j ( t ) D i @ D . $J + G i ,j=1 The corresponding e l l i p t i c operators =

{

a r e given by

-

L(t) =

(5.27)

n

1

Rij(t)DjDi i ,j=l

+

n

1 R.(t)D j'

j=o

J

t

Let m(x,t) be given a s i n Example 5.14, set W = H 1o ( G ) , m(t;@,$)

=

1

G

ity (5.28)

( m ( x , t ) / r ( x ) 2 ) @ ( x ) m d x , $,$

II@/rll 2 5 C l l @ l l W ,

W.

T.

and define

From the inequal-

$

L

i t follows t h a t m ( t ; * , * )

E

E

E

W,

i s a regular family of Hermitian forms

T ) and define f ( t ) = F ( m , t ) / r ( * ) f o r t E T. From (5.28) i t follows t h a t f E L 2 (T,V ' ). Finally, assume uo E

on W.

Let F

E

L2(G

x

204

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

From Theorems 5.6 and 5.7 we see t h a t o u r a b s t r a c t

Hi(G).

Cauchy problem i s e q u i v a l e n t t o t h e i n i t i a l - b o u n d a r y v a l u e prob1em La t( w rl (x x , t ) )

+ r(x)L(t)u(x,t)

m(x,o)(u(x,o)-uo(x))

= F(x,t),

0,

=

x

E

G,t

s

E

aG,

x

E

G.

E

T

(The f i r s t e q u a t i o n has been m u l t i p l i e d by r ( x ) and t h e l a s t by r(x)'

E x i s t e n c e and uniqueness f o l l o w f r o m t h e a d d i t i o n a l

> 0.)

2 0, mt(x,t)

assumptions m(x,t)

2 0, a.e.

x

E

G, t

E

T.

These

a r e n o t b e s t p o s s i b l e f o r e x i s t e n c e ; t h e y can be weakened as i n Example 5.14. Remark 5.16

The e q u a t i o n (5.29)

permits t h e leading operat-

o r t o be s i n g u l a r and t h e second t o be degenerate i n t h e s p a t i a l variable,

w h i l e t h e l e a d i n g o p e r a t o r may be degenerate i n time.

Remark 5.17

A m i n o r t e c h n i c a l d i f f i c u l t y a r i s e s from t h e i s g i v e n w i t h t h e t i m e d e r i v a t i v e on u, n o t

f o r m i n which (5.26) on ( t u ) .

We f i r s t n o t e t h a t t u t = ( t u ) ,

r e s u l t i n g second o p e r a t o r L ( t )

-

C~TU

T

=

"(TU),

-

u b u t see t h a t t h e

u / r m i g h t n o t be coercive.

However, i n t h e change of v a r i a b l e given by tut =

-

T =

t" t h e l e a d i n g t e r m i s

a u and t h e second o p e r a t o r r e s u l t -

i n g as above i s g i v e n by L ( T ~ / " ) - au/r. t h a t t h i s l a s t operator i s coercive f o r

205

From (5.28) CI

i t follows

> 0 s u f f i c i e n t l y small.

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Remark 5.18 Showalter [2].

Theorems 5.2,

5.6,

5.7 and 5.8 a r e from

Theorem 5.9 which c o n t a i n s (5.16)

See Showalter [Z]

new.

5.5,

and (5.17)

is

f o r examples i n h i g h e r dimension w i t h

e l l i p t i c - p a r a b o l i c i n t e r f a c e o r boundary c o n d i t i o n s and o t h e r c o n s t r a i n t s on t h e t i m e - d i f f e r e n t i a l along a submanifold o r p o r t i o n o f t h e boundary.

For a d d i t i o n a l m a t e r i a l on s p e c i f i c par-

t i a l d i f f e r e n t i a l equations o r e v o l u t i o n equations o f t h e t y p e considered i n t h i s s e c t i o n we r e f e r t o Baiocchi [l], Browder [4], Cannon-Hi 11 [Z],

F i chera [2]

, Ford [l] , Ford-Wai d

[2]

, Friedman-

Schuss [l],Gagneux [l],Glusko-Krein [l],Kohn-Nirenberg [l], L i o n s [5, 91, O l e i n i k [2],

Schuss [l],V i c i k [110 A l s o see

references g i v e n i n preceding s e c t i o n s f o r r e l a t e d l i n e a r problems and i n t h e f o l l o w i n g s e c t i o n s f o r n o n l i n e a r problems. 3.6

S e m i l i n e a r degenerate equations.

We s h a l l present some

methods f o r o b t a i n i n g existence-uniqueness r e s u l t s f o r t h e a b s t r a c t Cauchy problem f o r t h e equation

where M i s continuous, s e l f - a d j o i n t and monotone.

The m s t we

s h a l l assume on t h e n o n l i n e a r N i s t h a t i t i s monotone, hemicontinuous, bounded and coercive.

S p e c i f i c a l l y ; 1 e t . V be a r e f l e x -

i v e r e a l Banach space w i t h dual V

V to V (6.2)

I

.

I

and l e t N be a f u n c t i o n from

Then N i s s a i d t o be monotone i f 2 0,

206

u,v

E

v,

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

hemicontinuous i f f o r each t r i p l e u,v,w

E

V the function s

-+

R t o R , bounded i f t h e image o f

i s continuous from

each bounded s e t i n V i s a bounded s e t i n V - , and c o e r c i v e i f f o r each v

E

V l i m (/llull)

(6.3)

=

II u ll-

.

+a0

L e t H be a H i l b e r t space i n which V i s dense and continuously I d e n t i f y H and i t s dual and thereby o b t a i n t h e i n c l u -

imbedded.

sions V C H C V

I

.

We suppose

M i s a continuous, s e l f - a d j o i n t and

monotone o p e r a t o r o f H i n t o H; thus t h e square r o o t M112 o f M i s defined.

H = L2 (o,T;H),

LP(o,T;V), u

E

D(Lo) = { u

closure o f L

0

u

E

,> 2, l / p + l / q

Let p

V

V : du/dt

E

in V

x

D(L) then M1/2u

V'.

=

00,

and d e f i n e V =

2 = L (o,T;VI).

E

H I and denote by L : D(L)

L e t Lou = dMu/dt f o r E

V

I

the

A standard computation shows t h a t i f

C(o,T;H),

E

1, 0 < T <

'

t h e space o f continuous H-valued

f u n c t i o n s on [o,T]. Lemma 6.1

For u

E

V and f

E

V

, the

f o l l o w i n g are equiva-

lent: (a)

for all v

D(L), M1/2u(o) = 0, and L u = f,

u

E

E

V w i t h dv/dt

Proof:

E

H and v(T)

= 0.

C l e a r l y (a) i m p l i e s ( b ) :

t h a t t h e i d e n t i t y i n ( b ) holds f o r u

207

we need o n l y observe E

D(Lo), than pass t o t h e

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

general case o f u

D(L) by c o n t i n u i t y .

E

rt

Conversely, i f ( b ) i s

t r u e then we d e f i n e u n ( t ) = n e n ( s - t ) u ( s)ds, f n ( t ) = t JO n j en(s-t)f(s)ds. Then un E D(L0), Loun = fn, and we have 0

(un,fn)

(u,f)

-+

in

v

x

v

I

.

QED

The b a s i c i d e a here i s t o c o n s i d e r t h e l i n e a r o p e r a t o r A :

D(A)

-+

V ’ d e f i n e d by

AU =

Lu and D ( A ) = { u

D(L) : M1I2u(o) = 01.

E

That i s , A = ( d / d t ) M w i t h zero i n i t i a l c o n d i t i o n . closed, densely-defined and l i n e a r .

Lemna 6.1 shows A i s t h e

a d j o i n t of t h e o p e r a t o r -Md/dt w i t h domain {v

A

, be

tone.

E

V : dv/dt

E

H,

That i s , A i s t h e a d j o i n t o f a monotone o p e r a t o r , so

v(T) = 0).

*

Clearly A i s

ng t h e c l o s u r e o f a monotone operator, i s n e c e s s a r i l y monoSince t h e above i m p l i e s t h a t A i s maximal monotone by a of B r e z i s [I]we o b t a i n t h e f o l l o w i n g from Browder [l]:

theore

for N : V

I

V

-+

monotone, hemicontinuous, bounded and coercive,

the operator A + N i s surjective.

The preceding remarks g i v e t h e

following result. L e t t h e spaces V c ti C V

Theorem 6.2 and monotone M

E

-+

LP(o,T;V)

-1

for a l l v E

E

0

self-adjoint

Lq(0,T;V1)

there

and v(T) = 0.

Also

E

such t h a t

/

T rT = Jo 0

Lp(oYT;V) w i t h d v / d t

C(o,T;H)

For each f

V 1 be given.

rT <Mdv/dt,u> +

(6.4)

M112u

E

, the

L ( H ) and t h e monotone, hemicontinuous, bounded

and c o e r c i v e 14 : V exists a u

I

E

and M1l2u(o) = 0.

208

2 L (o,T;ti)

The s o l u t i o n i s unique i f N

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

i s s t r i c t l y monotone, i.e., Remark 6.3

i f

> 0, x,y

The c o n d i t i o n expressed by (6.4)

E

V, x

+ y.

i s equivalent

t o t h e Cauchy problem

(6.5)

$u(t)

+ N(u(t))

(M’j2u)

(0) = 0

= f(t)

i n which t h e e q u a t i o n s h o l d i n Lq(0,T;V’)

Remark 6.4 i n i t i a l data u

0

and i n H, r e s p e c t i v e l y .

The e x t e n s i o n o f Theorem 6.2 t o t h e case o f E

H ( n o t n e c e s s a r i l y homogeneous) i s g i v e n i n

B r e z i s [l]. Then a t e r m ( M ’ / 2 ~ o , M 1 / 2 ~ ( ~ ) ) Hi s added t o (6.4) and t h e i n i t i a l c o n d i t i o n i n (6.5)

becomes M’12u(o)

=

M 1/2

uo.

A l s o , t h e i n i t i a l c o n d i t i o n i n Theorem 6.2

can be r e p l a c e d by

t h e p e r i o d i c c o n d i t i o n M1/2u(o) = M1/’u(T)

by an obvious modi-

f i c a t i o n o f t h e o p e r a t o r A. We c o n s i d e r b r i e f l y t h e Cauchy problem t h a t a r i s e s when t h e s o l u t i o n i s c o n s t r a i n e d t o a s p e c i f i e d convex set. t o a variational inequality.

This leads

F o r such problems, t h e “ n a t u r a l ”

hypotheses on t h e n o n l i n e a r o p e r a t o r N : V

+

V

I

i s t h a t i t be

i f l i m u = u i n V weakly and l i m sup 5 0, t h e n f o r e v e r y v E V j

pseudomonotone:

(6.6)

Remark 6.5

5 l i m i n f . J J

The pseudomonotone o p e r a t o r s i n c l u d e t h e semi-

monotone o p e r a t o r s of Browder [6],

209

the operators o f calculus-

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

o f - v a r i a t i o n s i n t r o d u c e d by Leray and L i o n s [l],and t h e demimonotone o p e r a t o r s o f B r e z i s [4].

The pseudomonotone o p e r a t o r s

a r e p r e c i s e l y t h o s e which a r e most n a t u r a l f o r e l l i p t i c v a r i a tional inequalities. To i n d i c a t e t h e r e l a t i o n s h i p between Theorem 6.2

and r e s u l t s

t o f o l l o w , we s h a l l p r o v e t h a t e v e r y hemicontinuous and monotone operator N : V

V

I

So, suppose t h e n e t I u . 1 J i n V s a t i s f i e s l i m u . = u ( w e a k l y ) and l i m sup 5 0. J J J Since N i s monotone i t f o l l o w s t h a t 2 . J J J The r i g h t s i d e converges t o z e r o so we have l i m = 0. J J L e t v E V. F o r 0 < t < 1 and w = ( 1 - t ) u + t v we have by mono+

i s pseudomonotone.

t o n i c i t y < N ( u . ) - N(w),u.-w> 2 0 and t h i s shows t + J J J Taking t h e l i m i n f + t 2 . J The and d i v i d i n g b y t g i v e s l i m i n f 2 . J J r e s u l t f o l l o w s by u s i n g h e m i c o n t i n u i t y t o l e t t + 0 above.

A v a r i a t i o n a l i n e q u a l i t y problem c o r r e s p o n d i n g t o (6.5) find u

a g i v e n convex subset K o f V i s t h e f o l l o w i n g :

E

and

K with

M1/*u(o) = 0 f o r which

MU)

(6.7)

+ N(u)

-

f,v-u>dt

for all v

E

K.

A technical d i f f i c u l t y w i t h t h i s formulation i s

t h a t t h e f u n c t i o n a l e q u a t i o n (6.7) E

V

I

2 0

0

might n o t imply t h a t (d/dt)Mu

, so we s h a l l weaken t h e problem a p p r o p r i a t e l y .

s o l u t i o n o f (6.7)

and i f v

E

K w i t h (d/dt)Mv

21 0

E

V

I

If u i s a

t h e n we o b t a i n

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

IT d

I’

IT d < d M V ) + N U - f ,V-U>dt = <-( MU)+Nu-f ,v-U>dt + dt :t Jo dt 0 2 (l/E)<M(v-u),v-u>l;. The l a s t term i s nonnegative i f we r e -

1,

q u i r e Mv(o) = Mu(o) = 0.

Under hypotheses t o be given below, we

s h a l l c o n s i d e r t h e (weak) v a r i a t i o n a l i n e q u a l i t y (6.8)

u

K,

E

I

T <&(Mv)+Nu-f,v-u>dt

2 0

0

v

for all

K such t h a t (d/dt)Mv E V

E

L e t t h e spaces V C H

C

V

I

and V

I

and Mv(o) = 0.

C

ff C V

I

be given as above.

L e t K ( t ) be a c l o s e d convex subset o f V f o r each t sume 0 {v

E

K ( t ) and K(s) c K ( t ) f o r 0 <, s

V : v(t)

E

E

<- t <, T.

E

[o,T];

as-

Define K =

K ( t ) a.e.l.

The semigroup o f r i g h t s h i f t s g i v e n by O
s

u

s < t 2 T

u(t-S),

i s s t r o n g l y continuous on each o f V , U and V

, it

I

E V

i s a contrac-

t i o n semigroup on 14, and i t s a t i s f i e s (6.10)

G(s)KC K

by t h e assumption t h a t t h e convex s e t s a r e increasing. -L t h e generator o f (G(s) : s 2 D(L,V)

( r e s p e c t i v e l y , D(L,ff),

Denote by

I

01 on V ( H , V ) and i t s domain by

D(L,V’)).

Thus, we have L = d / d t

w i t h n u l l i n i t i a l c o n d i t i o n i n each o f V , H, and

V I .

From (6.10)

and t h e r e p r e s e n t a t i o n f o r t h e r e s o l v e n t o f -L by t h e formula cw

(I+eL)-’

E-’exp(-t/E)

=

G ( t ) d t we o b t a i n

0

21 1

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(I t EL)-~KcK,

(6.11) Suppose M

E

I

M

L(V,V

> 0,

E

E I

L(V,V ) i s nonnegative and s e l f - a d j o i n t and d e f i n e

) by [Mv](t)

defined by A = L

o

= Mv(t).

Then we consider t h e o p e r a t o r

M, D(A) = { v

E

v

: Mv

I

D(L,V )>.

E

The p l a n

i s t o a p p l y C o r o l l a r y 39 o f B r e z i s [4] t o solve t h e a b s t r a c t variational inequality (6.12)

u

E

for a l l v

E

D(A)

K,

n K.

2 0

The problems (6.8)

and (6.12)

are certain-

l y e q u i v a l e n t w i t h o u r choice o f data and we are l e d t o t h e f o l -

lowing r e s u l t . L e t t h e spaces V

Theorem 6.6 and monotone

c

H C V ' and t h e s e l f - a d j o i n t

I

M

L(V,V ) be g i v e n as above.

E

L e t K be t h e closed

convex s e t i n V o b t a i n e d as above from an i n c r e a s i n g f a m i l y { K ( t ) : 0 S t 5 T I o f c l o s e d convex subsets o f V, and assume 0 K.

Let N : V

f o r each f

E

-+

V

I

be pseudomonotone, bounded and coercive.

Lq(o,T;V')

t h e r e e x i s t s a s o l u t i o n o f (6.8).

E

Then The

s o l u t i o n i s unique i f N i s s t r i c t l y monotone. Proof:

The r e s u l t f o l l o w s i m n e d i a t e l y from B r e z i s [4] a f t e r

we v e r i f y t h a t t h e l i n e a r A i s nonnegative ( = monotone) and comp a t i b l e w i t h K. have j:$Vv,v>dt

I

Note f i r s t t h a t f o r those v E V w i t h Mv E V we = ( l / B ) < M v ( s ) , ~ ( s ) >T( ~ . Thus f o r v E D ( A ) i t

f o l 1ows

21 2

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

>_ 0.

j I < A v , v > d t = (1/2)<Mv(T),v(T)>

(6.13)

T h a t A i s c o m p a t i b l e w i t h K means t h a t f o r each u

E

K there

e x i s t s Cunl C K f l D ( A ) such t h a t l i m ( u n ) = u i n V and

5 0.

l i m sup

un = ( I + n

-1

L)

-1

Thus, l e t u

We have l i m ( un ) =

u.

E

K and d e f i n e f o r n 2 1 L(,

un ( t ) = n

and (6.11)

shows un E K . The r e p r e s e n t a t i o n above g i v e s t Mun(t) = n j en(s-t)Mu(s)ds, hence, LMu, = MLun. F i n a l l y , s i n c e 0

M i s monotone we have -n-l<MLun,Lu n> 5 0, so Remark 6.7

= -
-1

Lun> =

A i s c o m p a t i b l e - w i t h K.

QE D

I f we change t h e domain o f L a p p r o p r i a t e l y , we

can r e p l a c e t h e i n i t i a l c o n d i t i o n i m p l i c i t i n (6.8)

by a c o r r e s -

ponding p e r i o d i c c o n d i t i o n . We r e t u r n now t o t h e Cauchy problem f o r t h e e v o l u t i o n equat i o n (6.1) where M i s g i v e n a s i n Theorem 6.6 b u t w i t h weaker r e s t r i c t i o n s on t h e n o n l i n e a r o p e r a t o r N. assume N : V

+

V

I

i s o f t y p e M:

l i m (Nu.) = f i n V

I

J

S p e c i f i c a l l y , we s h a l l

i f l i m ( u . ) = u i n V weakly,

J weakly, and l i m sup j j

<_ ,

then

Nu = f. Remark 6.8

The o p e r a t o r s o f t y p e M i n c l u d e weakly c o n t i n -

uous o p e r a t o r s (i.e.,

c o n t i n u o u s f r o m V weakly i n t o V ’ weakly)

and t h e pseudomonotone o p e r a t o r s , hence, t h e o p e r a t o r s o f Remark 6.5. We show t h a t pseudomonotone o p e r a t o r s a r e o f t y p e M.

21 3

Let

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

{ u . ) be a n e t ( o r sequence i f V i s separable) as i n t h e d e f i n i t i o n J above o f "type M" and assume N i s pseudomonotone. Then l i m sup

5 0, so f o r every v

5 l i m i n f

J

J

E

V we o b t a i n from (6.6)

But v

5 .

E

V i s arbitrary

so Nu = f f o l l o w s from t h i s i n e q u a l i t y . Much o f t h e n o t a t i o n o f Theorem 6.6 w i l l be used t o prove the following result. Theorem 6.9 self-adjoint V

-f

V

I

M

be type

Lq(o,T;VI)

E

L e t t h e spaces V L(V,V

I

C

H C V

I

and t h e monotone

) be g i v e n as i n Theorem 6.6.

M, bounded and coercive.

there e x i s t s a solution u

E

Let N :

Then f o r each f LP(o,T;V)

E

o f t h e Cauchy

problem dMU) + N(u) = f (6.14) (Mu)(o) = 0.

I f N i s s t r i c t l y monotone ( c f . Theorem 6.2) t h e s o l u t i o n o f (6.14)

i s unique. Proof:

We g i v e a f i n i t e - d i f f e r e n c e approximation t o (6.14)

o f i m p l i c i t type.

Thus, (6.14) i s approximated by a correspond-

i n g f a m i l y o f " s t a t i o n a r y " problems ( c f . Chapter 11.7 o f L i o n s

[lo]).

Note t h a t t h e Cauchy problem (6.14)

equation i n V (6.15)

i s equivalent t o the

I

A(u) + N(u) = f,

A=LoAl 21 4

3.

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

where -L i s t h e g e n e r a t o r o f t h e s t r o n g l y - c o n t i n u o u s semigroup (6.9)

in V

I

and M

I

E

L ( V , V ) i s o b t a i n e d ( p o i n t w i s e ) f r o m M as

before. To approximate (6.15) tion i n V

S i n c e m(x,y) the estimate E

+ NuS = f.

[(I-G(s))/s]Mu,

(6.16)

v

we l e t s > 0 and c o n s i d e r t h e equa-

l

=

:1

<Mx,y> i s a s e m i - s c a l a r - p r o d u c t on V we o b t a i n dt

=

:1

m(v(t-s) , v ( t ) d t 5

V , which shows t h a t t h e o p e r a t o r [I-G(s)]M

6.13).

Thus, t h e o p e r a t o r [ ( I - G ( s ) ) / s ]

bounded and c o e r c i v e .

<Mv,v>dt,

i s monotone ( c f .

M + N i s o f t y p e M,

T h i s i m p l i e s by B r e z i s [4] o r L i o n s

t h a t i t i s s u r j e c t i v e , hence, (6.16) s > 0.

o

:1

[lo]

has a s o l u t i o n us f o r each

From (6.16) we o b t a i n 5 5 I l f l l * l l u , I I ,

so t h e c o e r c i v i t y o f N shows t h a t {us : s > 0) i s bounded i n V . Since N and M a r e bounded and V i s r e f l e x i v e t h e r e i s a subset ( w h i c h we denote a l s o by {us : s > 0)) f o r which l i m ( u s ) = u, l i m N(us) = g and l i m M(us) = Mu weakly i n t h e a p p r o p r i a t e spaces.

Let L (6.16)

*

*

denote t h e a d j o i n t o f t h e o p e r a t o r L i n V

t o any v

E

*

I

.

D(L , V ) and t a k e t h e l i m i t t o o b t a i n

<Mu,L v> + = ,

v

E

*

D(L ,U).

T h i s shows Mu

E

and t h a t (6.17)

We a p p l y

LMu + g = f

Since N i s o f t y p e M, i t s u f f i c e s f o r e x i s t e n c e t o show

21 5

I

D(L,V )

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

l i m sup <_ 0.

[(I-G(s))/slMweobtain <[(I-G(s))/s]Mu,us-u>.

But from (6.16)

and t h e m o n o t o n i c i t y o f

-

5 I

Since A h E D(L,V

) the l e f t factor o f the

l a s t term converges s t r o n g l y and t h i s g i v e s t h e d e s i r e d r e s u l t . Uniqueness f o l l o w s e a s i l y from t h e equivalence o f (6.14) (6.15);

we need o n l y n o t e t h a t

Remark 6.10

and

A i s monotone.

QED

The e x t e n s i o n t o nonhomogeneous i n i t i a l data

i s g i v e n i n Bardos-Brezis [l]w i t h r e l e v a n t r e g u l a r i t y r e s u l t s on t h e s o l u t i o n .

The problem w i t h p e r i o d i c c o n d i t i o n s Mu(o) =

Mu(T) i s s o l v e d s i m i l a r l y . Our n e x t c l a s s o f r e s u l t s w i l l be o b t a i n e d from t h e generat i o n t h e o r y o f n o n l i n e a r semigroups and i t s extensions t o e v o l u t i o n equations w i t h m u l t i v a l u e d operators.

The l i n e a r semigroup

t h e o r y i s inadequate f o r degenerate problems, b u t i t s p r e s e n t a t i o n above i n Section 4 p r o v i d e d an elementary m o t i v a t i o n f o r t h e techniques and c o n s t r u c t i o n s t o f o l l o w .

Specifically, i t indi-

cates t h e " o p t i m a l " hypotheses from which one m i g h t expect t h e Cauchy problem t o be well-posed and i t suggests t h e c o r r e c t space i n which t o l o o k f o r such r e s u l t s .

Also, we s h a l l n o t present

here a n o n l i n e a r v e r s i o n o f t h e a n a l y t i c s i t u a t i o n o f Theorem 4.2 and Theorem 4.4. L e t E be a v e c t o r space w i t h a l g e b r a i c dual E

E

-+

E

*

be nonnegative (monotone) and s e l f - a d j o i n t .

kernel o f M and q : E

+

*

and l e t M : L e t K be t h e

E / K t h e corresponding q u o t i e n t map.

symetric bilinear function m : E

21 6

x

E

+

R given by m(x,y)

The =

DEGENERATE EQUATIONS WITH OEPRATOR COEFFICIENTS

3.

<Mx,y> determines a s c a l a r p r o d u c t mo on E / K by

L e t W be t h e H i l k r t space completion o f E / K w i t h moy and denote by mo t h e (extended) s c a l a r product on Let E

I

W.

be t h e H i l b e r t space o b t a i n e d as t h e s t r o n g dual o f

t h e space E w i t h t h e seminorm induced by m.

Note t h a t since E / K I

i s dense i n W we may i d e n t i f y t h e dual spaces (E/K) q

*

: W

I

+

E

I

be t h e continuous dual o f q : E

dense range, q

*

i s necessarily injective.

vanishes on K, hence, g = f o q f o r some f q

*

preserving.

F i n a l l y , i f Mo : W

-f

determined by <Mox,y> = mo(x.y), (6.19)

W

W.

E

(E/K)

I

.

Let

Since q has

Also, each g

i s s u r j e c t i v e , and i t f o l l o w s from (6.18) I

-+

= W

I

.

that q

E

E

I

T h i s shows

*

i s norm-

i s t h e Riesz isomorphism

then we o b t a i n t h e i d e n t i t y

*

M = q Moq

r e l a t i n g t h e f u n c t i o n s considered above. Assume we a r e g i v e n f o r each t

[o,T] a nonempty s e t

E

D ( t ) C E and a ( n o t n e c e s s a r i l y l i n e a r ) f u n c t i o n N ( t ) : D ( t )

-+

E

I

Then f o r each such t we de’fine a m u l t i v a l u e d f u n c t i o n o r r e l a t i o n

N o ( t ) on q [ D ( t ) ] (6.20)

x

W

I

as t h e composition

No(t) = (q*)-l

0

N(t)

0

q -1

.

F i n a l l y , we d e f i n e t h e composite r e l a t i o n W x W w i t h domain q [ D ( t ) ]

f o r each t

21 7

E

[o,T].

( t ) = Mi’

o

No(t

on

Note t h a t No t )

.

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

and A ( t ) are f u n c t i o n s i f and o n l y i f x,y i m p l y N ( t ) x = N(t)y.

E

D ( t ) and Mx = My

The f o l l o w i n g diagram i l l u s t r a t e s t h e var-

ious relationships. N(t) E > D ( t ) --------tE

I

M

=

We s h a l l c o n s i d e r t h e s e m i l i n e a r e v o l u t i o n equation d x(MU(t)) + N(t,u(t))

(6.21)

A s o l u t i o n o f (6.21) I

O<_t
= 0,

i s a f u n c t i o n u : [o,T]

Mu : [o,T]

+

E

a.e.,

E

D ( t ) f o r a l l t, and (6.21)

u(t)

+

E such t h a t

i s a b s o l u t e l y continuous, hence, d i f f e r e n t i a b l e i s s a t i s f i e d a.e.

on

The Cauchy problem i s t o f i n d a s o l u t i o n o f (G.21)

[o,T].

which (Mu)(o) i s s p e c i f i e d i n E

I

for

.

We f o l l o w t h e idea from S e c t i o n 4 o f reducing (6.21) “standard” e v o l u t i o n e q u a t i o n w i t h

M

= identity.

and we d e f i n e v = q

0

u, then (6.19)

t i o n o f (6.21)

to a

I f u i s a solu-

and (6.20)

imply *

(6.22)

(q

*

Since q for t

E

MeV)

and M

0

[o,T],

I

= -N(t,U(t))

E

-

q

*

No(t,V(t)).

a r e l i n e a r isometries, we see t h a t v ( t ) v : [o,T]

+

E

q(D(t))

W i s a b s o l u t e l y continuous, and

218

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS I

(6.23)

v (t)

i s s a t i s f i e d a.e.

E

- A(t,v(t)) on [o,T].

We c a l l such a v ( - ) a s o l u t i o n o f

Conversely, suppose v i s a s o l u t i o n o f (6.23).

(6.23).

each t a t which (6.23) f o r some u ( t )

E

holds, (6.20)

D ( t ) and (6.22)

Then f o r

shows t h a t q ( u ( t ) ) = v ( t )

holds,

*

By choosing u ( t )

E

D(t)

w i t h q ( u ( t ) ) = v ( t ) , hence, Mu(t) = q Mov(t), f o r a l l remaining t

E

[o,T],

we o b t a i n a s o l u t i o n u o f (6.21).

T h i s proves t h e

f o l l owing. Lemma 6.11. t

E

[o,T]

there i s a u ( t )

(6.21). v = q

I f v i s a s o l u t i o n o f (6.23), E

then f o r each

D ( t ) such t h a t u i s a s o l u t i o n o f

Conversely, f o r each s o l u t i o n u o f (6.21), 0

the function

u i s a s o l u t i o n o f (6.23).

Corol1ar.y 6.12

L e t uo

E

D(o).

There e x i s t s a s o l u t i o n v o f

w i t h v ( o ) = q(uo) i f and o n l y i f t h e r e e x i s t s a s o l u t i o n

(6.23)

w i t h (Mu)(o) = Mu.,

u of (6.21)

There i s a t most one s o l u t i o n v

of (6.23) w i t h v ( o ) = q(uo) i f and o n l y i f f o r every p a i r o f s o l u t i o n s ul, follows N(t,u,(t))

u2 o f (6.21)

t h a t Mul(t)

w i t h (Mul)(o)

= Mu2(t) f o r a l l t

= N(t,u2(t))

a.e.

= (Mu2)(o) = Muo i t E

[o,T],

hence,

on [o,Tl.

Results on t h e Cauchy problem f o r (6.21) from corresponding r e s u l t s f o r (6.23).

can now be obtained

Turning f i r s t t o t h e

q u e s t i o n o f uniqueness, we f i n d t h a t a s u f f i c i e n t c o n d i t i o n f o r t h e mo-seminorm on t h e d i f f e r e n c e o f two s o l u t i o n s t o be noninc r e a s i n g i s t h a t each A ( t ) be a c c r e t i v e : 219

i f [x,,w,]

and [x,w,]

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

belong t o A ( t ) , t h e n mo(w1-w2.x,-x2)

2 0.

above c o n s t r u c t i o n f o r which (6.21)

The success o f t h e

and (6.23)

correspond t o one

another i s r e f l e c t e d i n t h e following. The r e l a t i o n A ( t ) i s a c c r e t i v e i f and o n l y i f

Lemma 6.13

t h e f u n c t i o n N ( t ) i s monotone. Proof:

L e t w1

E

A(xl) and w2

w i t h x j = q ( u j ) and N ( t , u .J) (6.24)

m0(W1

=

q

*

E

A(x2).

M ow j y j

= 1,2.

E

D( t )

Then we have

-w2 ,xl -x2) = 1

so A ( t ) i s a c c r e t i v e i f N ( t ) i s monotone. D ( t ) t h e r e i s a unique p a i r w1,w2 1,2.

Choose u1 ,u2

Then [q(uj),wj]

E

E

Conversely, i f u1,u2

*

= q Mowj, j J Hence N ( t ) i s

W w i t h N(t,u.)

A ( t ) and (6.24)

holds.

monotone i f A ( t ) i s a c c r e t i v e . Remark 6.14

E

=

QED

N ( t ) i s s t r i c t l y monotone i f and o n l y i f i t i s

i n j e c t i v e and A ( t ) i s a s t r i c t l y a c c r e t i v e f u n c t i o n . Theorem 6.15 monotone f o r each t

L e t N ( t ) be monotone and M + N ( t ) s t r i c t l y E

[o,T].

Then f o r each uo

a t most one s o l u t i o n u ( - ) o f (6.21) Proof:

E

D(o) t h e r e i s

w i t h (Mu)(o) = Mu.,

Since A ( t ) i s a c c r e t i v e f o r each t by Lemma 6.13,

uniqueness h o l d s f o r (6.23).

The r e s u l t t h e n f o l l o w s f r o m QED

C o r o l l a r y 6.12. We c o n s i d e r now t h e e x i s t e n c e o f s o l u t i o n s .

220

Specifically,

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

we s h a l l r e c a l l t h e e x i s t e n c e r e s u l t s o f Crandall and Pazy [l] as t h e y a p p l y t o t h e r a t h e r s p e c i a l H i l b e r t space s i t u a t i o n o f (6.23).

(The monograph o f H. B r e z i s [3] p r o v i d e s an e x c e l l e n t

i n t r o d u c t i o n t o t h i s t o p i c w i t h an e x t e n s i v e b i b l i o g r a p h y ; c f . a l s o t h e r e c e n t monograph o f Browder [7].!

The equation (6.23)

has a (unique) s o l u t i o n v w i t h v ( o ) s p e c i f i e d i n q(D(o)) under t h e f o l l o w i n g hypotheses: Each A ( t ) i s a c c r e t i v e and I + A A ( t ) i s s u r j e c t i v e f o r a l l A > 0. It follows that J A ( t )

of

(I+AA(t))-’

i s a f u n c t i o n d e f i n e d on a l l

w. i s independent o f t;

The domain o f A ( t ) , q[D(t)], we denote i t by q[D]. There i s a monotone g : [o,m)

[ o , ~ ) such t h a t

+

11 JA(t,x)-JA(s,x) 5 A [ t - s [ g ( I[xllw) .(l+ i n f rllyllw : y E A ( s , x ) I ) ,

t,s 2 0, x

E

W,

O
L e t M be nonnegative and symmetric from t h e

*

l i n e a r space E t o i t s dual E

.

I

Denote by E t h e dual o f t h e

l i n e a r t o p o l o g i c a l space E w i t h t h e seminorm <MX,X>~’~; E l i s a

IlfllEl =

H i l b e r t space w i t h norm given by x

E

E, <Mx,x> 5 1).

For each t

(possibly nonlinear) N ( t ) : D ( t )

sup {II

[o,T] assume we a r e given a

E

+

E

I

w i t h domain D ( t ) C E.

Assume f u r t h e r t h a t f o r each t, N ( t ) i s monotone and D(t)

+

E

I

i s surjective; M[D(t)]

:

M + AN(t) :

i s independent o f t; and f o r

221

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

some monotone i n c r e a s i n g f u n c t i o n g : [o,m)

Then f o r each uo

E

+

[o,m) we have

D(o) t h e r e e x i s t s a s o l u t i o n u o f (6.21)

with

(Mu)(o) = Muo. From C o r o l l a r y 6.12 we see t h a t i t s u f f i c e s t o show

Proof:

has a s o l u t i o n v w i t h v ( o ) = q(uo), and t h i s w i l l f o l l o w

(6.23)

i f we v e r i f y t h e hypotheses on CA(t)) above.

Also, M [ D ( t ) ]

each A ( t ) i s a c c r e t i v e by Lemma 6.13. o f t i m p l i e s t h e same f o r q[D(t)].

*

F i r s t note t h a t independent

To determine t h e range o f

*

*

I + A A ( t ) we have ( q Mo)(I+AA(t)) = q Mo + A N o ( t ) = q Mo + AN(t)q-’ I

=

*

(M+AN(t))q-l on q[D(t)].

The range o f t h e above i s

E and q Mo i s a b i j e c t i o n , so I + A A ( t ) i s n e c e s s a r i l y onto The e s t i m a t e on J A ( t , x ) f o l l o w s from (6.25)

and we r e f e r t o

Showalter [5] f o r t h e r a t h e r l e n g t h y computation. Remark 6.17

W.

QED

I n c o n t r a s t t o preceding r e s u l t s o f t h i s sec-

t i o n , N need n o t be c o e r c i v e o r even d e f i n e d everywhere on

V.

Thus i t may be a p p l i e d t o examples where N corresponds t o a n o t n e c e s s a r i l y 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 ( c f . Theorem 4.1 and Example 4.6). Second o r d e r e v o l u t i o n equations w i t h ( p o s s i b l y degenerate) o p e r a t o r c o e f f i c i e n t s on time d e r i v a t i v e s can be r e s o l v e d by t h e preceding r e s u l t s .

These c o n t a i n as a s p e c i a l case a f i r s t 222

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

o r d e r s e m i l i n e a r e q u a t i o n which i s n o n l i n e a r i n t h e time d e r i v a tive. Theorem

6.18

L e t A and C be symmetric continuous l i n e a r I

o p e r a t o r s f r o m a r e f l e x i v e Banach space V i n t o i t s dual V ; asl I

Denote by Vc t h e ( H i l b e r t

sume C i s monotone and A i s coercive.

Let B : V

space) dual o f V w i t h t h e seminorm '".

-+

V I be

a ( p o s s i b l y n o n l i n e a r ) monotone and hemicontinuous function. I

Then f o r each p a i r u1,u2 E V w i t h Aul + B(u2) E Vc t h e r e e x i s t s I 1 a unique v E L (o,T;V) such t h a t Cv : [o,T] -+ Vc i s a b s o l u t e l y d continuous, &Cv)

+ B(v) : [o,T]

-+

V

I

i s (a.e.

equal t o ) an ab-

s o l u t e l y continuous f u n c t i o n , Cv(o) = Cu,,[$Cv)+B(v)](o)

= Aul,

and

Proof: [Ax,Cy], Ax + B(y)

On t h e product space E E V x V consider M[x,y] I

hence, E l = V ' x Vc. I

E

Vc) and N :

D

+

Define D = ([x,yl E

I

by N[x,y]

E

=

V x D(B) :

= [-Ay,Ax+B(y)].

Then

a p p l y Theorem 6.16 t o o b t a i n e x i s t e n c e o f a s o l u t i o n u = [w,v] 2 of (6.21). Note t h a t C + AB t h A i s monotone, hemicontinuous I

and coercive, hence onto V ' ,

and t h i s shows M

( c f . p r o o f o f Theorem 4.3).

The second component v o f t h i s solu-

t i o n u w i t h Mu = M[u,,u2]

=

N[x2,y21

AN i s s u r j e c t i v e

i s t h e s o l u t i o n o f (6.26).

f o l l o w s from C o r o l l a r y 6.12 s i n c e M[xl,yl] N[xl,~ll

t

i m p l y [xl,yll

= [x2,y21.

223

= M[x2,y2]

Uniqueness and

QED

SINGULAR AND DEGENERATE CAUCHY PROBEMS

Remark 6.19

The f i r s t component o f t h e s o l u t i o n u o f (6.21)

i n t h e preceding p r o o f i s t h e unique a b s o l u t e l y continuous

w : [o,T]

-f

dw V w i t h B ( a ) : [o,T]

s a t i s f y i n g w(o) = ul,

(6.28)

+

Remark 6.20

I

V c a b s o l u t e l y continuous and

-f

I

Bw ( 0 ) = Bu2,

dw B ( a ) + Aw(t) = 0,

a.e.

E

[o,Tl.

The s p e c i a l case t h a t r e s u l t s from choosing C =

0 i s o f i n t e r e s t and should be compared w i t h (6.21). c a l l y , (6.26)

t

and (6.27)

Specifi-

become f i r s t o r d e r equations i n which

t h e time d e r i v a t i v e a c t s on t h e n o n l i n e a r term. v e r t i b i l i t y o f A was never used i n s o l v i n g (6.26). f i c i e n t f o r e x i s t e n c e o f a s o l u t i o n o f (6.26)

Also, t h e i n It i s suf-

t h a t B + AA be

c o e r c i v e f o r each A > 0; however, uniqueness may then be l o s t take A = 0).

(e.g.,

We b r i e f l y sketch some a p p l i c a t i o n s o f t h e r e s u l t s o f t h i s s e c t i o n t o i n i t i a l - b o u n d a r y value problems.

L e t G be a bounded

domain i n Rn w i t h smooth boundary aG, Wp(G) be t h e Sobolev space of u

E

Lp(G) w i t h each D.u = au/ax.

D u

= u.

g

Lq, q = p / ( p - l ) ,

J

J

E

Lp(G), 1 5 j 5 n, and

L e t f u n c t i o n s N.(x,y) be given, measurable i n x E G J Suppose f o r some C > 0, c > 0, and and continuous i n y E Rn+’ 0

E

.

p

L 2, we have

224

DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

3.

n +

for x

n}, l e t V N :

V + V

(6.32)

n+l

, and

0 5 k 5 n.

L e t Du = {D.u : 0 j J be a subspace o f Wp(G) c o n t a i n i n g Ci(G), and d e f i n e

G, y,z

E

g ( x ) 2 CIYIP,

I

R

E

by

=

1

1

j=o

N.(x,Du(x))D.v(x)dx, G J J

The r e s t r i c t i o n o f Nu t o C:(G)

-

-

E

v.

i s t h e d i s t r i b u t i o n on G g i v e n by

n

- 1 D.N.(.,Du)

NU =

(6.33)

u,v

+ No(*,D$).

j=1 J J

The d i v e r g e n c e theorem g i v e s t h e ( f o r m a l ) Green' s f o r m u l a

lam

v ( s ) d s , v E V, where t h e conormal d e r i v a t i v e n au on aG i s g i v e n by - = 1 N.(*.Du)nj and (nly nn) i s t h e a' j=1 J I u n i t outward normal. The o p e r a t o r N : V -t V i s bounded, hemicon-

=

. . .,

t i n u o u s , monotone and c o e r c i v e ; c f . Browder [3, 61, C a r r o l l [14] o r Lions

[lo]

f o r d e t a i l s and c o n s t r u c t i o n o f more general op-

e r at o r s . Example 6.21 define

M :V

(6.34) L e t uo

Let m(-)

-t

Lm(G), mo(x) 2 0, a.e.

x

E

G, and

E

v.

V by

<Mu,v> = E

E

I

m(x)u(x)v(x)ds, l G

V w i t h N(uo)

=

ml"h

f o r some h

225

u,v E

2 L (G).

Then each of

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorems 6.2,

6.6,

and 6.16 g i v e s e x i s t e n c e o f t h e s o l u t i o n t o

the e l l i p t i c - p a r a b o l i c equation (6.35)

-

+ N(u(x,t))

D,(m(x)u(x,t))

= 0

w i t h i n i t i a l c o n d i t i o n m(x)(u(x,o)-uo(x)) boundary c o n d i t i o n depending on V.

= 0, x

I f V = W:(G),

E

G, and a the D i r i c h l e t

boundary c o n d i t i o n i s obtained, w h i l e t h e ( n o n l i n e a r ) Neumann c o n d i t i o n au/aN = 0 r e s u l t s when

V

= Wp(G).

c o n d i t i o n i s o b t a i n e d by adding t o (6.32)

The t h i r d boundary

a boundary i n t e g r a l .

The assumption t h a t m ( * ) i s bounded may be relaxed; c f . BardosB r e z i s [l]o r Showalter [5].

r

J aGu ( s ) v ( s ) d s Wp(G)

t o (6.34)

and i f

I f we add t h e boundary i n t e g r a l

V i s t h e space o f f u n c t i o n s i n

which are c o n s t a n t on aG, we o b t a i n t h e boundary c o n d i t i o n

o f f o u r t h t y p e ( c f . A d l e r [l])

This problem i s s o l v e d by Theorems 6.9 and 6.16. p r o p r i a t e terms t o (6.32)

and (6.34)

By adding ap-

i n v o l v i n g i n t e g r a l s over

p o r t i o n s o f t h e boundary ( o r over a m a n i f o l d S o f dimension n-1 in

c) one o b t a i n s

s o l u t i o n s o f e q u a t i o n (6.35)

s u b j e c t t o degen-

erate parabolic constraints D,(m(s)u(s,t))

-

div(a(s)grad u(s,t))

- -

s

E

s,

where t h e i n d i c a t e d divergence and g r a d i e n t are i n l o c a l

226

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

c o o r d i n a t e s on S and t h e c o e f f i c i e n t s a r e nonnegative ( c f . Showalter [2,

51). 2 Choose V = Wo(G)

Example 6.22 =

n

r

j=1

G

and d e f i n e

Dju(x)D.v(x)dx, J

u,v

E

v,

where each c . E Lm(G) i s nonnegative, 0 5 j 5 n. L e t B = N be J given by (6.32) and assume (6.29), (6.30) w i t h 1 < p 5 2. Then Theorem 6.18 g i v e s a unique s o l u t i o n o f ' t ' ? e equation (6.36)

n

- j =1l Dj(cjDjv)]

Dt{Dt[cov

."

+ N(v))

n

- j =11D?v = J

0,

w i t h D i r i c h l e t boundary c o n d i t i o n and a p p r o p r i a t e i n i t i a l condi. "

tions.

Note t h a t

N i s g i v e n by (6.33).

i n c l u d e t h e wave equation (4.9), (4.10),

Special cases o f (6.35)

t h e v i s c o e l a s t i c i t y equation

p a r a b o l i c equations (4.6),

and Sobolev equations w i t h

f i r s t o r second o r d e r t i m e d e r i v a t i v e s .

A s p e c i a l case o f (6.36)

Example 6.23 erable interest.

has been o f eonsid-

j S n, and NO(x,s) = m ( x ) I s I P - l sgn ( s ) where m

n e g a t i v e and 1 < p (6.37)

N

S p e c i f i c a l l y , we choose C 5 0,

I 2.

E

= 0 for 1

j

Lm(G) i s non-

This gives t h e equation

Dt(m(x)Iv(x,t)IP-l

sgn v ( x , t ) )

-

Av(x,t)

=

0.

The change o f v a r i a b l e u = I v l P - ' sgn(v) p u t s t h i s i n t h e form

227

5

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(6.38)

D,(m(x)u(x,t))

-2=

with q

(2-p)/(.p-l)

-

n

(4-1)

2 0.

1 D.()u)q-2Dju(x,t)) j=1 J

= 0

This e q u a t i o n a r i s e s i n c e r t a i n

d i f f u s i o n processes; i t i s "doubly-degenerate"

since t h e leading

c o e f f i c i e n t may vanish and t h e power o f u may vanish independently.

Note t h a t t h e second o p e r a t o r i s n o t monotone so t h e equa-

t i o n i s n o t i n t h e form o f (6.21). Remark 6.24

[lo]

Theorem 6.2

i s from B r e z i s [l]. See Lagnese

f o r a v a r i a t i o n on Theorem 6.6 and corresponding p e r t u r b a -

t i o n r e s u l t s on such problems.

Theorem 6.9 i s contained i n t h e

r e s u l t s o f Bardos-Brezis [1] along w i t h o t h e r types o f degenerate e v o l u t i o n equations n o t covered here ( c f . Lagnese [5] f o r c o r responding p e r t u r b a t i o n r e s u l t s ) . a r e from Showalter [5]. Lions

[lo]

Theorems 6.15,

We r e f e r t o Aronson

6.16 and 6.18

[l],Dubinsky [l],

and R a v i a r t [l]f o r a d d i t i o n a l r e s u l t s on (6.38)

and, s p e c i f i c a l l y , t o Strauss [l]f o r a d i r e c t i n t e g r a t i o n o f s e m i l i n e a r equations o f t h e form ( c f . (6.37)) (6.39)

Dt(N(u)) + A ( t ) u ( t ) = f ( t )

w i t h n o n l i n e a r N and time-dependent l i n e a r { A ( t ) ) i n Banach spaces.

Cf.

B r e z i s [l],Kamenomotskaya [l],and Ladyienskaya-

Solonnikov-Uralceva [2] f o r a r e d u c t i o n o f c e r t a i n ( S t e f a n ) freeboundary value problems t o t h e form (6.39), Lions-Strauss [8] and Strauss [2, (6.28).

hence, (6.36),

and

31 f o r a d d i t i o n a l r e s u l t s on

Crandall [3] and Konishi [l]have s t u d i e d s p e c i a l cases

228

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

o f (6.39)

by d i f f e r e n t methods.

3.7

Doubly n o n l i n e a r equations.

We b r i e f l y consider some

f i r s t o r d e r e v o l u t i o n equations o f t h e form (7.1)

$(u(t))

t N(u(t)) = 0

where ( p o s s i b l y ) b o t h operators are nonlinear.

Some s o r t o f de-

generacy i s d e s i r a b l e (and necessary f o r a p p l i c a t i o n s ) so we s p e c i f i c a l l y do n o t make assumptions on M o f s t r o n g monotonicity as was done i n Theorem 3.12.

Moreover, we can i n c l u d e c e r t a i n

r e l a t e d v a r i a t i o n a l i n e q u a l i t i e s (e.g.,

by l e t t i n g one o f

t h e operators be a s u b d i f f e r e n t i a l ) so we p e r m i t t h e operators t o be m u l t i v a l u e d f u n c t i o n s o r r e l a t i o n s as considered i n Section 3.6. L e t B be a r e a l r e f l e x i v e Banach space, l e t D(M) and D(N) be subsets o f B and suppose M : D(M) m u l t i v a l u e d functions.

-+

That i s , M

B and N : D(N)

c D(M)

x

-+

B are

B and N C D(N) x B.

(When M and PI a r e f u n c t i o n s , we i d e n t i f y them w i t h t h e i r respect i v e graphs.)

We c a l l t h e p a i r o f f u n c t i o n s u,v a s o l u t i o n t o

t h e d i f f e r e n t i a l " i n c l usion"

if u(t)

E

D(M)fl

D(N), v : [o,m)

-+

hence, s t r o n g l y - d i f f e r e n t i a b l e a.e.,

N are f u n c t i o n s , then (7.1) whereas i f o n l y

B i s L i p s c h i t z continuous, and (7.2)

holds.

i s c l e a r l y s a t i s f i e d a.e.

I f M and on [o,m),

M i s a f u n c t i o n we r e p l a c e t h e e q u a l i t y symbol i n 229

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(7.1)

M[D(M) f l D ( N ) ] denote t h e

L e t D( A)

by an i n c l u s i o n .

i n d i c a t e d image i n B and d e f i n e a r e l a t i o n A : D (A ) composition A = N some z

E

D(M)

fl

t h e p a i r (u,v)

0

M- 1

.

T h a t i s , (x,y)

D ( N ) we have (z,x)

E

E

o f (7.3),

a u(t)

E

U ( A ) and (7.3)

holds.

E

N.

Thus, i f

then i t f o l l o w s t h a t

so we c a l l a L i p s c h i t z f u n c t i o n v : [ o , ~ ) if v ( t )

B by t h e

A i f and o n l y i f f o r

E

M and (z,y)

i s a s o l u t i o n o f (7.2),

-f

-f

B a s o l u t i o n o f (7.3)

Conversely, i f v i s a s o l u t i o n

t h e d e f i n i t i o n o f A shows t h e r e e x i s t s f o r each t L 0 D(M) fl D ( N ) f o r which t h e p a i r u,v

i s a solution of

(7.2). L e t M and N be m u l t i v a l u e d o p e r a t o r s on t h e

Theorem 7.1

r e a l r e f l e x i v e Banach space B; assume M + N i s o n t o B and t h a t

II (x1-x2)+s(Y1-Y*)11

2

II

x p 2

II

f o r s > 0,

Y

(7.4) and (z.,Y.) J J Then f o r each uo

E

E N , (z.,x.) J J

D ( M ) n D(N) and vo

s o l u t i o n p a i r u,v o f (7.2) Proof:

E

From (7.4)

E

j

=

1,2.

M(uo), t h e r e e x i s t s a

w i t h v ( o ) = vo.

By t h e p r e c e d i n g remarks

has a s o l u t i o n .

M,

t s u f f i c e s t o show (7.3)

i t f o l l o w s t h a t t h e sane e s t i m a t e

h o l d s f o r s > 0 whenever ( x . , y . ) E A; t h i s i s p r e c i s e l y t h e J J statement t h a t A i s a c c r e t i v e ( c f . C r a n d a l l - L i g g e t t [2]).

230

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Furthermore, i t i s easy t o check t h a t (x,y) i f f o r some z

E

D(M) n D ( N ) we have (z,y-x)

E

I + A i f and o n l y

E

N and (z,x)

E

M,

M + N.

so i t f ollows t h a t t h e range o f I + A equals t h e range o f

Thus I + A i s h y p e ra c c re ti v e and i t f o l l o w s from r e s u l t s o f C r a n d a l l - L i g g e t t [2] t h a t (7.3) vo

E

has a unique s o l u t i o n f o r each

D(A).

QED

Remark 7.2 then i t f ollows

11

If u ,v and u2,v2 a re s o l u t i o n s o f (7.2), 1 1

v l (t)-v 2 (t)

11

5 Ilv,(o)-v,(o)

11.

Ifvl(o)

=

v2(o) and e i t h e r M - l o r N - l i s a fu n c ti o n , then t h e s o l u t i o n p a i r u,v i s unique. Remark 7.3

I n a H i l b e r t space H w i t h i n n e r product (*,*),,

th e c o n d i t i o n (7.4)

i s equivalent t o whenever ( z .,x .) J J

(x1-x2Y Y1-YZ)H

E

M

(7.5) and ( z

y ) j yj

E

N

f o r j = 1,2.

For l i n e a r f u n c t i o n s t h i s i s t h e r i g h t angle c o n d i t i o n ( c f . Grabmuller [l],Lagnese [2, 4,

81,

and Showalter [6, 14, 15, 16,

201). The d i f f i c u l t y i n a p p l y i n g Theorem 7.1 t o i n i t i a l - b o u n d a r y value problems a r i s e s from t h e assumption (7.4) which r e l a t e s t h e operat ors M and N t o each other.

A desirable alternative

which we s h a l l d e s c ri b e i s t o p l a c e hypotheses on each o f t h e o perat ors independently.

The r e s u l t s o f S e c t i o n 3.6 were

231

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

o b t a i n e d when t h e l e a d i n g ( l i n e a r ) o p e r a t o r was symnetric.

The

n o n l i n e a r analogue o f t h i s i s t h a t t h i s o p e r a t o r be a d i f f e r e n t i a l o r , more g e n e r a l l y , a s u b d i f f e r e n t i a l . subdifferential a t x

E

Recall t h a t t h e

E o f an e x t e n d e d - r e a l - v a l u e d l o w e r semi-

c o n t i n u o u s convex f u n c t i o n $ on t h e l o c a l l y convex space E i s t h e s e t a$(x) o f those u

(Cf. B r e z i s [3],

E

E

I

verifying

Ekeland-Ternan [l],L i o n s

w i l l be g i v e n below.

Some examples

I n g e n e r a l , t h e s u b g r a d i e n t a$ i s a m u l t i -

valued operator from (a subset o f ) E i n t o Theorem 7.4

[lo].) E

I

.

L e t V1 and V2 be r e a l separable r e f l e x i v e

Banach spaces w i t h V1 C V2, i n c l u s i o n i s compact.

Let

V1 i s dense i n V2, (I

1

and $z be convex c o n t i n u o u s (ex-

tended) r e a l - v a l u e d f u n c t i o n s on V1 and V2, sume t h e i r s u b d i f f e r e n t i a l s

N

and assume t h e

=

a$l and M E

respectively.

As-

a$z a r e bounded ( c f .

and t h a t N i s " c o e r c i v e " i n t h e sense t h a t

S e c t i o n 3.6)

: l u l v -+ -1 > o f o r some p, 1 < p < m. v1 1 Suppose we a r e g i v e n uo E V1, vo E M(uo), and f E Lm(0,T;V;) l i m i n f I$~(U)/IUIP

with df/dt

E

a p a i r o f functions u L~(O,T;V;)

a.e.

where l / p + l / q = 1.

Lq(0,T;V;) E

Lm(o,T;V1),

satisfying

on [o,T]

and v ( o ) = vo. 232

v

E

Then t h e r e e x i s t s

Lm(0,T;V;)

w i t h dv/dt

E

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Remark 7.5

The compactness assumption above l i m i t s a p p l i c a -

t i o n o f Theorem 7.4 t o p a r a b o l i c problems; c f . examples below. We i l l u s t r a t e some a p p l i c a t i o n s o f t h e p r e c e d i n g r e s u l t s t h r o u g h some examples o f i n i t i a l - b o u n d a r y v a l u e problems.

p

W1’p(G),

T : V

-f

(6.30)

V

I

2

2 , and d e f i n e t h e n o n l i n e a r e l l i p t i c o p e r a t o r

by (6.32),

and (6.31).

2

mj(x)

V

I

, and

the closure o f Ci(G) i n

L e t V = W;”(G),

Example 7.6

0, a.e.

x

E

d e f i n e D(M)

m2T(u); D(N)

{u

2

E

where t h e f u n c t i o n s N.(x,y) s a t i s f y (6.29), J F o r j = 1,2, l e t m . ( * ) E Lm(G) be g i v e n w i t h J G, and l e t k > 0. S e t H = L 2 ( G ) so V C H

c

2

{u

E

V : mlu + m2T(u)

V : (k/ml(*))Tu

E

E

H I , M(u)

mlu +

H I , N(u) :( k / m l ( * ) ) T ( u ) .

M and N a r e ( s i n g l e - v a l u e d ) n o n l i n e a r o p e r a t o r s i n t h e

Then

H i l b e r t space H.

To a p p l y Theorem 7.1,

i n i t s e q u i v a l e n t form (7.5).

( M U - M V ~ N U - N V =) ~ k + dx 2 0, so (7.4)

holds.

I

F o r u,v

we f i r s t v e r i f y (7.4) E

D(M) f l D(N) we have

( km2(x)/ml ( x ) ) ( T ( u ) - T ( v ) G To show M + N i s o n t o H, l e t w E H and

consider the equation

The c o e f f i c i e n t s a l l belong t o Lm(G), so t h e o p e r a t o r on t h e l e f t s i d e o f (7.7) to V

I

, hence,

Since m,(*)

u

E

i s monotone, hemicontinuous and c o e r c i v e from V s u r j e c t i v e , so t h e r e i s a s o l u t i o n u

E

V o f (7.7).

and m 2 ( - ) a r e bounded, i t f o l l o w s from (7.7)

D ( M ) n D(N) and t h a t M(u) + N(u) = w.

that

Thus, Theorem 7.1

g i v e s e x i s t e n c e o f a weak s o l u t i o n o f an a p p r o p r i a t e i n i t i a l 233

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

boundary value problem c o n t a i n i n g t h e equation

where T[u]

i s g i v e n by (6.33).

Flote t h a t t h e c o e f f i c i e n t s m1

and m2 may vanish over c e r t a i n subsets o f G. We a p p l y Theorem 7.4 w i t h V1 = WAYp(G) and V2 = n La(G), ct 5 P, Q1(u) = l / p 1 IDju(X)IPdx, @,(u) = G j=l lu(x)j"dx, and a p p ro p ri a te f : [o,T] -+ W-lgq(G). The c o r l/ct IG responding s u b d i f f e r e n t i a l s a re g i v e n by 34, ( u ) = Example 7.7

we o b t a i n exist e n c e o f a s o l u t i o n o f an i n i t i a l - b o u n d a r y value problem f o r t h e equation

Example 7.8 define @,(u)

= l/a

@1and f be given as above b u t -c u (x)l"dx, u E V2, where u ( x ) = 0 when

L e t V1,V2,

iGl

+

u ( x) < 0 and u+(x) = u ( x ) when u ( x ) >, 0.

Theorem 7.4 g i v e s

e x i st ence f o r an i n i t i a l - b o u n d a r y value problem f o r

Example 7.9

2 max {1/2,1ulH/21.

L e t V2 be a H i l b e r t space and s e t @,(u) The s u b d i f f e r e n t i a l i s g i v e n by

lulH

<

' 9

I u I = 1,

lulH ' 234

=

3. DEGENERATE EQUATIONS WITH OPERATOR COEFFICIENTS

Ifv1 and N are g i v e n as i n Theorem 7.4,

t h e r e i s then a s o l u -

t i o n p a i r u,v which s a t i s f i e s t h e a b s t r a c t " p a r a b o l i c " equation I

-u ( t ) + f ( t ) t i o n f(t)

E

E

N ( u ( t ) ) where l u ( t ) l > 1 and t h e " e l l i p t i c " equa-

N ( u ( t ) ) where l u ( t ) l < l.

Remark 7.10

Theorem 7.1 i s an e x te n s i o n o f t h e correspond-

i n g r e s u l t o f Showalter [4] when M and N a r e f u n c t i o n s . 7.4 and Examples 7.7,

Theorem

7.8 and 7.9 a re from Grange-Mignot [l].

See Barbu [l]f o r r e l a t e d a b s t r a c t r e s u l t s and examples where M and N are s u b d i f f e r e n t i a l s which a re assumed r e l a t e d by an assumption s i m i l a r t o (7.4)

b u t d i s t i n c t from it.

(7.9) was s t u d i e d d i r e c t l y i n R a v i a r t [2]; L i o ns

[lo].

The equation

c f . Chapter IV.1.3

of

A doubly-nonl i n e a r parabolic-hyperbol i c system i s

discussed by Volpert-Hudjaev [l],and p a r a b o l i c systems a r e considered by Cannon-Ford-Lair [3].

These l a s t two references

i n d i c a t e how c e r t a i n doubly-nonl i n e a r equations a r i s e i n a p p l i c a t i o n s ; f o r a p p l i c a t i o n s o f equations o f t h e type (7.8), Gajewski-Zachari us [2].

235

cf.

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Chapter 4 Selected Topics 4.1

Introduction.

I n t h i s chapter we w i l l discuss

b r i e f l y some f u r t h e r questions a r i s i n g i n t h e study o f s i n g u l a r o r degenerate equations.

Some o f t h e p r e s e n t a t i o n w i l l be i n a

more " c l a s s i c a l " s p i r i t and we w i l l f o l l o w t h e o r i g i n a l papers i n d e s c r i b i n g problems and r e s u l t s .

Proofs w i l l o f t e n o n l y be

sketched o r even o m i t t e d e n t i r e l y , m a i n l y f o r reasons o f space and time, b u t s u i t a b l e references w i l l be p r o v i d e d ( w i t h o u t a t tempting t o be exhaustive i n t h i s r e s p e c t ) ; f u r t h e r r e l e v a n t i n formation can o f t e n be o b t a i n e d from t h e b i b l i o g r a p h i e s i n these references.

The m a t e r i a l considered by no means covers a l l pos-

s i b l e questions and has been s e l e c t e d b a s i c a l l y t o be represent a t i v e o f t h e h i s t o r i c a l development and t o r e f l e c t some c u r r e n t research trends i n t h e area.

Thus we w i l l deal f o r example w i t h

Huygens' p r i n c i p l e , r a d i a t i o n problems, i n i t i a l - b o u n d a r y value problems, nonwellposed problems, e t c . , among o t h e r questions; t h e s e c t i o n t i t l e w i l l i n d i c a t e t h e problem under c o n s i d e r a t i o n . 4.2 and (1.4.10)

Huygens' p r i n c i p l e . with A

X

R e f e r r i n g back t o (1.4.8)

= -A l e t us suppose m t p =;

-

1 with

Ta

f u n c t i o n so t h a t wm+P(t) i s g i v e n i n terms o f a s u r f a c e mean value u x ( t ) want m 2 n

-

-

*

T o f T(<) over t h e sphere [ x - < l = t (here we a l s o

1/2 t o i n s u r e t h e uniqueness o f wm).

Thus i f

(2m+l) = 2p+l i s an odd p o s i t i v e i n t e g e r t h e value wm(t) =

wm(x,t) depends o n l y on t h e values o f T(5) on t h e s u r f a c e

237

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

I x-s

=

141.

To picture t h i s one can draw the retrograde "light" cone

t, as was noticed by Diaz-Weinberger [2] and Weinstein

from ( x , t )

E

Rn

x

R+ t o the singular hyperplane t

a t the intersection. where m =

-

= 0 and look

We note here t h a t f o r the wave equation

1/2,n = 2p + 1 is required above which reproduces a

well known fact.

This phenomenon furnishes an example of what i s

known as the minor premise i n Huygens' principle.

In general

Huygens' principle can be enunciated i n various versions involving other features as well ( c f . Baker-Copson [ l l , Courant-Hi1 bert

111, Hadamard [ l ] , Lax-Phillips [ l ] ) b u t we shall simply say here t h a t a linear second order partial differential operator

L(Dx , D t ) ( k = 1 , ...,n ) i s of Huygens' type i n a region R C k Rn x R i f the value u ( x , t ) o f the solution of Lu = 0 a t a point

(x,t)

E

R depends only upon the Cauchy data on the intersection

of the space l i k e i n i t i a l manifold with the surface of the characteristic conoid w i t h vertex ( x , t ) .

T h i s formulation and

others play an important role i n wave propagation for example and the matter has been studied extensively. ( I n addi t i o n t o the above references we c i t e here only some more o r l e s s recent work by Bureau [l], Douglis [ l ] , Fox [ l ] , Giinther [ l ; 2 ; 3; 4; 51, Helgason [4], Lagnese [ l l ; 1 2 ; 131, Lagnese-Stellmacher [141, Solomon [ l ;

21,

and Stellmacher [ l ] . )

I n dealing w i t h Huygens'

principle f o r singular problems we will follow Solomon [ l ;

21

only make a few remarks about other work on singular problems later.

Thus l e t us consider the operator 238

and

4.

(2.1

1

SELECTED TOPICS

E"' = a2/at2 +

where Rem >

-

1/2 o r m =

a/at + c(t)

-

1/2 and c ( * ) i s a complex valued

f u n c t i o n belonging t o L ' ( o , b ) (O
(2.2)

We deal here w i t h the

D'

m Emum = Au ; um(0) = T

E

D';

uF(0) = 0

where A i s the Laplace o p e r a t o r i n Rn and since c

(

0

E

)

L 1, um

i s r e q u i r e d o n l y t o s a t i s f y t h e d i f f e r e n t i a l equation almost everywhere (a.e.).

Solomon [l;23 employs t h e F o u r i e r technique

developed i n Chapter 1 and looks f o r resolvants Z;(t) (2.2) w i t h T = 6.

S e t t i n g A2 =

f u n c t i o n i m ( y , t ) = FxZ:(t) (2.3)

Em^Zm + A2^Zm = 0;

A s o l u t i o n ?(y.t)

"

1 yk2

satisfying

= l y I 2 one then looks f o r a

1 satisfying "m

Z (y,o) = 1;

iy(y,o) = 0

o f (2.3) i s obtained by c o n v e r t i n g (2.3) i n t o

an i n t e g r a l equation as i n Chapter 1 ( c f . (1.3.24)) and using a Neumann s e r i e s such as (1.3.25)

( c f . a l s o Section 1.5).

Using

techniques s i m i l a r t o those o f Chapter 1 t h e s o l u t i o n i s obtained i n the f o l l o w i n g form (see Solomon [l],pp. 226-231). Theorem 2.1 y

E

The unique s o l u t i o n o f (2.3) f o r t

E

[o,b]

and

Rn can be w r i t t e n as

(2.4) where

"m Z (y,t) = im(y,t) + i m ( y , t )

im i s given

by (1.3.6)

w i t h z = t l y l and 239

im i s a "smooth"

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

perturbation s a t i s f y i n g the estimate

f o r IyI > A0 and o S t 5 b (ko and ho a r e independent o f ( y , t ) ) . The corresponding unique r e s o l v a n t Z!(t)

E l has support con-

E

*

t a i n e d i n t h e b a l l 1x1 5 t and u m ( t ) = Z!(t) s o l u t i o n o f (2.2) i n D Remark 2.2

I

.

The e s t i m a t e (2.5) i s u s e f u l i n determining t h e i n p a r t i c u l a r f o r Rem + 1/2 >

o r d e r o f t h e d i s t r i b u t i o n Z!(t); n

-

T i s t h e unique

1 i t f o l l o w s t h a t Z!(t)

w i l l be continuous i n x f o r 0 < t i b.

We w i l l say now t h a t (2.2) i s o f Huygens' t y p e i f and o n l y i f t h e support o f Z!(t)

i s concentrated on t h e sphere 1x1 = t.

Note here t h a t t h i s d e f i n i t i o n depends on t h e choice o f i n i t i a l data when m =

-

1 / 2 and n = 1 f o r example s i n c e t h e one dimen-

s i o n a l wave o p e r a t o r i s n o t a Huygens' o p e r a t o r i n t h e sense p r e v i o u s l y d e l i m i t e d f o r a r b i t r a r y Cauchy data u(x,o) ut(x,o).

0, 1, 2,

Solomon uses t h e d i s t r i b u t i o n s 6 ( ' ) ( \ x l

..., d e f i n e d

2

and

2 -t 1, v = V

f o r t > 0 and n > 1 by ( c f . Gelfand-Silov

[51)

Here 1x1 = p , 8 denotes t h e n

-

-

1 angular v a r i a b l e s (el,

...,

i n p o l a r s p h e r i c a l coordinates w i t h + ( p , B ) = + ( x ) , and Rn i s t h e s u r f a c e o f t h e u n i t sphere i n Rn.

240

For n = 1 one w r i t e s

4.

dV)(1 x

(2.7)

SELECTED TOPICS V

=

L[& wax)

[6(x+t) + 6(x-t)]

21tl

c l e a r l y the d i s t r butions

&'((XI 2- t2 )

a r e c o n c e n t r a t e d on t h e

sphere 1x1 = t.

Then t h e main r e s u l t i n Solomon [ll i s g i v e n by

Theorem 2.3

A necessary and s u f f i c i e n t c o n d i t i o n f o r ( 2 . 2 )

-

t o be o f Huygens ' t y p e i s t h a t n

(2m+l) = 2N + 1 w i t h N L 0 an

i n t e g e r w h i l e t h e r e e x i s t s a r e s o l v a n t Z:(t)

where c,

= (-1 )Nn-n'2r(m+l),

a.

=

o f the form

1 , and f o r v = 1 ,

. . ., N

(if

N > 0 ) one has E-m(aN) = 0 w i t h 4(ta\: + vav) = E-m(av-l)

(2.9)

Remark 2.4

0, 1, 2, (2.10)

V

We r e c a l l from G e l f a n d - S i l o v [51 t h a t f o r v

... FS(')(

where c(v,n)

2v-n+221/2(n-2)-vJ 1 (2) -(2 n-2) -v and z = t l y l . Then examples t o

1x12-t2) = ;(v,n)lyl

= 7(-1/2)v(2n)n/2 1

i l l u s t r a t e Theorem 2.3 can be g i v e n as f o l l o w s . t h a t Z:(t), a

V

=

Suppose f i r s t

expressed b y (2.8), has p r e c i s e l y one t e r m ( i . e . ,

= 0 f o r v _> 1 ) .

Then by ( 2 . 9 ) E-m(ao) = c ( t ) = 0 s i n c e a.

1 and we a r e i n t h e EPD s i t u a t i o n .

i n (2.8) w i t h o f n e c e s s i t y n i n t e g e r as i n d i c a t e d .

-

=

L e t N be t h e i n t e g e r i n v o l v e d

(2m+l) = 2N + 1 an odd p o s i t i v e

To check t h e f o r m o f Z:(t)

g i v e n by ( 2 . 8 )

i n t h i s case we use (2.10) t o o b t a i n Zm(y,t) = cmt-ZmaoFG(N)

241

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

2 2 1 (1x1 - t ) and s i n c e $n-2)

- N

Am

= m t h i s y i e l d s Z (y,t)

"m 2mr(m+l)z-mJm(z) = R (y,t) as r e q u i r e d by (1.3.6).

E-m(ao) = c ( t ) o r c ( t ) = ( t a , ) '

+ al

= 0 we have ta;

and E-m(al)

+;

=

+ 1 2m)

= a;'

Combining t h e s e two e q u a t i o n s one has al

c ( t ) al = 0.

2mal

N e x t suppose

av = 0 f o r v _> 2 ) .

t h a t ( 2 . 8 ) has no more t h a n two terms ( i . e . , From (2.9) and t h e r e q u i r e m e n t E-m(al)

=

I '

t

+

9

+t2 2 al=

ct 0

for

cto

any c o n s t a n t .

' " w /w t h i s becomes w 2 (log w)' with c ( t ) = 2 (log Now i n terms o f w we have a l ( t ) = ables al(t)

=

2

Making a change o f v a r i ct 2m+l ' o 7 -2 w = 0.

w)" and s u i t a b l e w can be found as f o l l o w s .

If

= 0 we g e t

cto

w ( t ) = cr + f3t2m+2 where a and B a r e a r b i t r a r y c o n s t a n t s b u t may depend on m.

I n o r d e r t h a t t h e r e s u l t i n g a,

be such t h a t (2.8)

s a t i s f i e s t h e r e s o l v a n t i n i t i a l c o n d i t i o n s we must have

-

while f o r m =

B = 0.

1 / 2 i t i s necessary t h a t

If

cto

c1

j o

f 0 the

s o l u t i o n s w o f t h e d i f f e r e n t i a l e q u a t i o n f o r w w i l l be o f t h e k f o r m w = exp ( 4 2 ) F ( - ;T; - k;

T)

where

T

= tal/',

k = 2m+l

, and

F (y; 6; t ) i s any s o l u t i o n o f Kunnner's c o n f l u e n t hypergeometric equation tF"

+

(6

-

t ) F' + y F = 0.

Upon e x a m i n a t i o n o f t h e be-

h a v i o r o f such F(y;6; t ) as t + o i t f o l l o w s t h a t f o r t h e r e s u l t i n g a, t o s a t i s f y t h e a p p r o p r i a t e i n i t i a l c o n d i t i o n s F must be k k t a k e n i n t h e f o r m F = alF1 ( - 2; - k; T ) + f3 F2 ( - 2; - k; T) where lF1y

F2 i s any s o l u t i o n o f t h e Kumner e q u a t i o n independent o f a f

0,

and when m =

-

1/2 ( i . e .

k =

s e r v a t i o n s can be f o u n d i n Solomon [1;21.

242

0)

6 =

0.

F u r t h e r ob-

SELECTEDTOPICS

4.

We w i l l deal here e x p l i c i t l y o n l y w i t h some work

Remark 2.5

o f Fox [l]on t h e s o l u t i o n and Huygens' p r i n c i p l e f o r s i n g u l a r Cauchy problems o f t h e t y p e

w i t h data um(x,o) = T ( x ) and u;(x,o)

= 0 g i v e n on t h e s i n g u l a r

Except f o r some r e s u l t s o f Gunther ( l o c . c i t . )

hyperplane t = 0.

most o f t h e o t h e r work on Huygens' p r i n c i p l e mentioned e a r l i e r deals w i t h n o n s i n g u l a r Cauchy problems so we w i l l n o t discuss i t here; Gunther's work on t h e o t h e r hand i s expressed i n a more geometric language which r e q u i r e s some background n o t assumed o r developed i n t h i s book and hence t h e d e t a i l s w i l l be omitted. Thus l e t

r

=

r

(x,t;

C)

= t

2

- n1 ( x k -

Ck)

k= 1 L o r e n t z i a n d i s t a n c e between a p o i n t ( x , t )

(5,o) i n t h e s i n g u l a r hyperplane t =

0.

2 E

be t h e square o f t h e

Rn x R

and a p o i n t

We denote by (2.11)-,,

t h e e q u a t i o n (2.11) w i t h index -m and observe t h a t i f u - ~ satisf i e s (2.11)-,, 1.4.7).

( 5k/xk)

then um =

t-2m

-m u s a t i s f i e s (2.11) ( c f . Remark

Fox composes t h e t r a n s f o r m a t i o n s zk =

r

( k = 1,

. . ., n )

1

/4xkCk and yk =

t o produce new v a r i a b l e s (z,r,y)

p l a c e o f (x,t,C)

( s u i t a b l e regions a r e o f course d e l i n e a t e d ) . n m- s o l u t i o n o f (2.11)-,, i s sought i n t h e form u-" = r 2 V(z) where V(z) i s t o be determined.

in

A

A f t e r some computation one shows

t h a t such umma r e s o l u t i o n s i f V ( z ) s a t i s f i e s a c e r t a i n system o f n o r d i n a r y d i f f e r e n t i a l equations and such a V(z) can be found i n t h e form V = FB where FB i s t h e L a u r i c e l l a f u n c t i o n given f o r

243

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

l z k l < 1 by ( c f . Appell (2.12)

-

Kamp6 de F 6 r i e t 111)

where ak =

1

+

1

n

-

FB = FB (a, 1-a, m

+ 1, z )

and (8,s) = r(R+q)/I'(R).

(1-4Ak)'/2

The s e r i e s

d e f i n i n g FB converges u n i f o r m l y i n ( a , m , z ) on closed bounded regions where l z k l < 1 and m t h a t s i n c e ak and 1

-

-

n

. . ..

+ 1 # 0, -1, -2,

a k a r e complex conjugates f o r

Xk r e a l ,

t h e products ( a k ,p k ) ( l - a k y p k ) a r e r e a l , and thus FB

m and t h e Ak a r e r e a l .

I f Rem >

n 2 -1

and l x k l >

One notes

It1

s r e a l when > 0 it

makes sense then t o c o n s i d e r as a p o s s i b l e s o l u t i o n o

the sin-

g u l a r Cauchy problem f o r (2.11) t h e f u n c t i o n . (2.13)

m uX(x,t,T)

= k,

where km = r(mc1)/vn/2r(m-"+l) 2 i s a s o l u t i o n o f (2.11).

n

and v!(x,t,C)

=

Itl-2mrm-T FB

We assume here t h a t T ( - ) i s s u i t a b l y

d i f f e r e n t i a b l e and denote by S ( x , t )

the region o f the singular

hyperplane t = 0 c u t o u t by t h e c h a r a c t e r i s t i c conoid w i t h vertex (x,t)

( t h u s S(x,t)

i s an n-dimensional sphere o f r a d i u s

It1 and c e n t e r x i n t h i s hyperplane determined b y r ( x , t , S ) 2 0 ) . As described, w i t h l x k l >

It1

2 0, S ( x , t )

does n o t i n t e r s e c t any

o f t h e hyperplanes xk = 0 and l i e s i n an o c t a n t o f t h e hyperplane t = 0.

Note here a l s o t h a t t h e f a c t o r r(m-;+l)-l

n p l a y e d o f f a g a i n s t (m - ? +

S 1, Ipk)-'

i n k,

can be

i n FB t o remove any d i f 1 n t 1 = -p. Now f o r Rem > + 1 one checks f i c u l t i e s when m244

5

4.

e a s i l y t h a t Lyu:

SELECTEDTOPICS

T(<)L!)r(x,t,<)d<

km

=

= 0 when T

(

0

)

is

continuous, s i n c e a l l boundary i n t e g r a l s which a r i s e w i l l vanish. U s i n g a n a l y t i c c o n t i n u a t i o n and a f t e r s e v e r a l pages o f i n t e r e s t i n g a n a l y s i s Fox shows t h a t ( f o r m # -1

¶

-2,

.. .)

i f T(-)

E

n where p i s t h e s m a l l e s t i n t e g e r such t h a t Rem 2 2 - p - l \

c{2+p)

m

t h e n u A Y d e f i n e d b y (2.13),

s a t i s f i e s t h e s i n g u l a r Cauchy prob-

lem f o r (2.11) ({2+p) = 0 i f 2 + p i 0). s o l u t i o n uy(x,t,T)

i s p r o v e d f o r 2m

+ 1

Uniqueness o f t h e 2 0 and i t i s shown t h a t

Huygens' p r i n c i p l e h o l d s when, f o r some nonnegative i n t e g e r p,

- p - 1, xk = a k ( l - a k ) satisfy m = n where ak E ~ l y . . . y p t l ~ and , 1 ak i n + p. F o r m # -1, -2, 1 t h e s o l u t i o n ,u: a l t h o u g h n o t unique f o r Rem < - 1/2, i s t h e t h e parameters m and

xk

...

unique a n a l y t i c c o n t i n u a t i o n of t h e unique s o l u t i o n s uy w i t h

2m + 1

2 0 and t h e c r i t e r i a above f o r Huygens' p r i n c i p l e r e -

main v a l i d ( i n f a c t necessary and s u f f i c i e n t ) . 4.3

The g e n e r a l i z e d r a d i a t i o n problem o f Weinstein.

One

can r e a d e x t e n s i v e l y on t h e s u b j e c t o f r a d i a t i o n f o r t h e wave e q u a t i o n and we mention f o r example C o u r a n t - H i l b e r t [ll LaxP h i l l i p s [l] Somnerfeld [l] and r e f e r e n c e s t h e r e . ¶

¶

The usual

f o r m u l a t i o n ( c f . W e i n s t e i n (61) p r e s c r i b e s a f u n c t i o n f ( * ) w i t h f ( t ) = 0 f o r t < 0 and asks f o r a f u n c t i o n u s a t i s f y i n g utt = Au i n R P x I R w i t h u(x,o) = ut(xYo) = 0 and such t h a t

(3.1) where r

dun =

lim r4

2

= 1x1'

=

n

1 xi¶2 1

-

wnf(t)

w n = 2 ~ ~ / ~ / r ( n / 2 and ) , theintegral i n

245

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

(3.1) i s o v e r a sphere of r a d i u s r i n l R n .

When n = 3 f o r

example one has t h e well-known s o l u t i o n u ( x , t )

= f(t-r)/r

and

one i s l e d t o l o o k f o r s o l u t i o n s w h i c h depend o n l y on t and r. T h i s e n t a i l s a r a d i a l f o r m o f A and y i e l d s a s i n g u l a r problem i n two v a r i a b l es u

(3.2)

tt

=

'rr

+ -n

- 1 r 'r

w i t h u = 0 f o r r 2 t and f o r r = 0, u s h o u l d have a s i n g u l a r i t y o f t h e f o r m r-(n-2).

Now r e p l a c e n

-

1 by 2m

+

1 and c o n s i d e r a

g e n e r a l i z e d r a d i a t i o n problem (3.3)

tt

=

m 'rr

+

2m + 1 r r

where we r e c a l l t h a t i f urn s a t i s f i e s ( 3 . 3 ) w i t h i n d e x m t h e n uqm = r2mum s a t i s f i e s (3.3) w i t h i n d e x -m ( c f . Remark 1.4.7). I f 2m

+ 1

=

n

-

1 t h e n 2m = n - 2 so t h a t a s o l u t i o n u - ~f i n i t e

near r = 0 corresponds t o a s o l u t i o n u" w t h a s i n g u l a r i t y o f t h e d e s i r e d t y p e as r

.+

0.

Thus, t h i n k i n

o f s u i t a b l y negative

values o f Rem (see Theorem 3.2 f o r t h e p r e c i s e range Rem < 0), one l o o k s f o r a s o l u t i o n u m ( t , r ) o f (3.3) s a t i s f y i n g (3.4)

urn(t,o) = f ( t ) ; u " ( t , t )

where f ( t ) = 0 f o r t < 0.

= 0

T h i s t y p e o f problem has been s t u d i e d

i n p a r t i c u l a r by D i a z and Young [l;81 , L i e b e r s t e i n [l;21, W e i n s t e i n [ 6 ] , and Young [6; 121. L i o n s [ 4 ] , Suschowk [l], We w i l l f o l l o w L i o n s [4] h e r e i n t r e a t i n g t h e g e n e r a l i z e d

246

4.

SELECTEDTOPICS

r a d i a t i o n problem because o f t h e g e n e r a l i t y o f h i s approach and t h e use o f t r a n s m u t a t i o n operators, which serves t o augment F i r s t , as a sketch o f t h e procedure, we consider

S e c t i o n 1.7.

-

f o r Rem >

1/2

m

(3.5)

v (t,r) =

~n r z(z2-r2)m-1/2 2 m+1/2

m

um(t,z)dz

Then, i f u" s a t i s f i e s (3.3) w i t h u m ( t , r ) = 0 f o r t 5 r, i t m f o l l o w s t h a t vTt = vrr w i t h v m ( t , r ) = 0 f o r t 5 r.

v m ( t , r ) = F m ( t - r ) where Fm(s) = 0 f o r s i 0.

Consequently

Now ( c f . below)

(3.5) can be i n v e r t e d i n t h e form

For Rem <

-

1 / 2 (3.6) i s v a l i d i n t h e usual sense w h i l e b o t h

(3.5) and (3.6) can be extended by a n a l y t i c c o n t i n u a t i o n t o every m

E

C.

P u t t i n g F m ( t - z ) i n (3.6) and i n t e g r a t i n g by p a r t s

one o b t a i n s

For Rem

L

0 such u m ( t , r ) do n o t converge when r

and hence urn, i s i d e n t i c a l l y zero. hand, as r (3.8)

+

+

0 unless Fm,

For Rem < 0, on t h e o t h e r

0

um ( t , r )

+.r(-m+1/2) 1 k-2m-1(Fm)l(t-z)dz

so t h a t , r e c a l l i n g t h e d e f i n i t i o n o f t h e R S e c t i o n 1.6 ( c f . (1.6.20)

f o r example),

247

-

L integral i n

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

m u (t,o)

(3.3)

i s required.

I-2m(Fm)' = f ( t )

=

-2m-1 m S i n c e I-2m(Fm)' = I ( F ) we have Fm determined

u n i q u e l y by Fm = ( r ( - m + l / 2 ) / r ( - 2 m ) )

12m+1(f) and p u t t i n g ( 3 . 9 )

i n (3.7) one o b t a i n s m

um ( t , r )

(3.10)

1

=

-m-1/2(12mf)(t-z)dz

which i s e q u i v a l e n t t o W e i n s t e i n ' s s o l u t i o n ( c f . W e i n s t e i n [61) and demonstrates uniqueness a'l so. I n L i o n s 141 t h e f o r m a l c a l c u l a t i o n s above a r e rephrased and j u s t i f i e d i n terms o f t r a n s m u t a t i o n o p e r a t o r s as f o l l o w s . (We r e c a l l f r o m S e c t i o n 1.7 t h a t B transmutes P i n t o Q i f QB = BP.)

Now L e t R = (0, m) and D:(R)

=

D+(R)' be t h e space o f

d i s t r i b u t i o n s on R w i t h s u p p o r t l i m i t e d t o t h e r i g h t ( c f . Schwartz [l]and n o t e t h a t D+(R) s i g n i f i e s Cm f u n c t i o n s w i t h n o t n e c e s s a r i l y compact s u p p o r t b u t equal t o z e r o n e a r t h e origin).

F o r simp1 i c i t y we w i l l occasiona l y use a f u n c t i o n

n o t a t i o n f o r d i s t r i b u t i o n s i n what f o l l o w s i n w r i t i n g f o r example < T , p = .

T(<)

Then f o r

T

s e t MT(S) =

E

which means t h a t <MT,p = < T ( x ) , 2x@

x2) > f o r @

i t i s easy t o see t h a t M i s an isomorphism $ ( f i x )

( o r DL(Rx)

+.

+.

E

D(RE);

D'(R5)

$ ( Q ) ) w i t h i n v e r s e M-lS(x) = S ( x 2 ) ( i . e . ,

- 5

<M-~s,$)> = <s(E,), 1

5-1/2

for

$

E

qnx)).

c a l c u l a t i o n y i e l d s M f T = ( 4 5 I? + 2D ) MT f o r

5

Dx = d/dx, e t c . ) .

5

T

A simple E

D'(Rx) ( h e r e

Now r e f e r r i n g t o t h e s t a n d a r d spaces D 248

I

,

SELECTEDTOPICS

4.

D i , and D: o v e r (l/r(p))

R as i n Schwartz [ll, we s e t f o r p

Pf(xP-’)lx>b

Schwartz [ l ] ) .

Then Y

E

P Y

E, Y

E

P D i ( t h e n o t a t i o n P f i s explained i n

$.a

= o f o r x < 0, Y-R =

=

f o r R a non-

I

: Q: -+ D+ i s a n a l y t i c and Y P P * y q = I S e t now Z ( x ) = Y ( - x ) so t h a t Z E D 9 Z = 0 f o r x > 0, yP+q P P P P Z-, = (-I)‘#& f o r R a n o n n e g a t i v e i n t e g e r , etc.; i n p a r t i c u l a r

negative integer, p

-+

.

DZ

= -Zp-l and XZ = - P Z ~ + ~ .F i n a l l y one n o t e s t h a t T S*T i s P P I a c o n t i n u o u s l i n e a r map Di(n) D-(fi) when S E D- i s z e r o f o r -+

I

-+

1 S(x-y)T(y)dy) m

x > 0 ( f o r m a l l y (S*T) ( x ) =

*

Zp

JX

L (D:(a),

E

DL(S2)) t h e map T

Now l e t Lm = DE t @-$

,L

that

D-(n) I

:

-f

Dx (L,

i n t o Lm on D

(a)

For m

(i.e.,

DH ,

2

H,

= M-

-+

1 O

E

Q: L i o n s con-

* z-ln-1/2

0

M o f D2

= LmHm) such t h a t Hm:

I

I

D-(O)

= L i i n Chapter 1 ) and n o t e

D (S2) i s continuous.

s t r u c t s a transmutation operator I

Zp*T.

I

-+

and we denote by

D ( Q ) i s an isomorphism and m

-+

H, i s an e n t i r e a n a l y t i c

The i n v e r s e H, = Hi’ = 2 M, which n e c e s s a r i l y s a t i s f i e s D H, = HmL, on

f u n c t i o n w i t h v a l u e s i n L(Dl(S2), D l ( 0 ) ) .

* zm+1/2

v-’ D- ( a ) , e n j o y s 0

0

s i m i l a r p r o p e r t i e s and i n f u n c t i o n n o t a t i o n one

can w r i t e (3.11)

ff,T(x)

(3.12)

HmT(x) =

=

mr ~x ( y ~ - x ~ ) - ~ - ~ / ~ T (Rem y ) d
F o r o t h e r values o f m continuation.

2

E

m

I:(yZ-x2)m-1/2r(r)dy;

Rem >

-

1/2

$ one extends t h e s e formulas by a n a l y t i c

I t i s important t o note t h a t i f T

249

E

Dl(n) i s

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

L i o n s phrases

zero f o r x > aT t h e n HmT = HmT = 0 f o r x > aT. t h e g e n e r a l i z e d r a d i a t i o n problem as f o l l o w s . Let P

Problem 3.1

c R x R be t h e open h a l f p l a n e ( t , r )

where r > 0 and Q

cP

Rem < 0 f i n d urn

D l ( P ) s a t i s f y i n g (3.3) w i t h supp u m c Q and

um(-,r)

+

E

be t h e s e t Q = { ( t , r )

f i n D I as r

+

0 where f

E

E

P; t 2 r ) .

:D i s g i v e n w i t h f

For

= 0 for

t < 0.

The problem i s r e s t r i c t e d t o Rem < 0 s i n c e i t i s shown n o t t o be m e a n i n g f u l f o r o t h e r m Theorem 3.2

4

E

There r e s u l t s

( c f . L i o n s [4]).

Problem 3.1 has a unique s o l u t i o n urn depending I

continuously i n D (P) 0 n . f

D,

E

I

( f as d e s c r i b e d ) .

F o r m a l l y t h e p r o o f goes as f o l l o w s .

One w r i t e s vm =

( 1 @Hm)um so t h a t vm i s g i v e n by (3.5) f o r Rem > - 1/2 say ( c f . (3.12)). v"

E

m t o (3.3) one has vtt = vrr

A p p l y i n g 1 @ H,

D ' ( P ) has i t s s u p p o r t i n Q.

as b e f o r e w i t h Fm

E

and

Consequently v m ( t , r ) = F m ( t - r )

D I and Fm(s) = 0 f o r s < 0.

Since H,

i s an

m isomorphism w i t h H i ' = Hm we have u = ( 1 C9 Hm)vm and f o r Rem <

-

1 / 2 say we o b t a i n f o r m a l l y by (3.11) ( c f . ( 3 . 6 ) )

where < S L Y @ > = ( Z / r ( - m - l / Z ) ) Rem <

-

1/2, and @

E

D-.

i:

~(z~-r~)-~-~/~@ f o(r z r) d> z0,

The f u n c t i o n m

-+

S i can i n f a c t be

extended by a n a l y t i c c o n t i n u a t i o n t o be e n t i r e w i t h v a l u e s i n

250

4.

SELECTED TOPICS

D i and, f o r Rem < 0, a s r + O , i n .D,

I

Sk

Hence f r o m (3.13) as r

+

+

(Zr(-2m-l)/r(-m-l/Z))

Y-2m-l

0

and t h i s l i m i t must equal f ( c f . ( 3 . 9 ) ) .

Consequently t h e ex-

p l i c i t s o l u t i o n o f Problem 3.1 f o r Rem < 0 i s (3.15)

um(t,r)

w h i c h can be r e w r t t e n i n a f o r m e q u i v a l e n t t o (3.10)

(cf

Lions [41). 4.4

I m p r o p e r l y posed problems.

I m p r o p e r l y posed problems

such as t h e D i r i c h l e t problem f o r t h e wave e q u a t i o n or t h e Cauchy problem f o r t h e Laplace e q u a t i o n have been s t u d i e d f o r some t i m e and have r e a l i s t i c a p p l i c a t i o n s i n p h y s i c s .

For a

n o n e x h a u s t i v e l i s t o f r e f e r e n c e s see e. g., A b d u l - L a t i f - D i a z [lI, Agmon-Nirenberg [31, B o u r g i n [ll, B o u r g i n - D u f f i n [21, B r e z i s - G o l d s t e i n [ 5 ] , Dunninger-Zachmanoglou [21, Fox-Pucci [21 , John [23, L a v r e n t i e v [ Z ] , [121, Ogawa [1;23,

L e v i n e [l;5; 10; 111, Levine-Murray

Pucci [21 , S i g i l l i t o 141

by Payne [31 ( c f . a l s o

-

and a r e c e n t survey

Symposium on nonwellposed problems and

l o g a r i t h m i c c o n v e x i t y , S p r i n g e r , L e c t u r e notes i n mathematics, Vol. 316, 1973).

I n p a r t i c u l a r t h e r e has been c o n s i d e r a b l e

i n v e s t i g a t i o n o f i m p r o p e r l y posed s i n g u l a r problems by DiazYoung

[lo]

Dunninger-Levine [lI, Dunninger-Weinacht

[31

T r a v i s [ll, Young [4; 5; 7; 8; 113, e t c . and we w i l l d i s c u s s 251

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

some o f t h i s l a t t e r work. The most general a b s t r a c t r e s u l t s seem t o appear i n Dunninger-Levine [l]and we w i l l sketch t h e i r technique f i r s t . L e t E be a r e a l Banach space and A a closed densely d e f i n e d l i n e a r o p e r a t o r i n E (as i s well-known t h e use o f spaces

R instead o f

i n v o l v e s no l o s s o f g e n e r a l i t y ) .

E over

We s h a l l denote

by cs ( A ) t h e p o i n t spectrum o f A. P D e f i n i t i o n 4.1

The E valued f u n c t i o n urn(-) i s s a i d here t o

be a s t r o n g s o l u t i o n o f

on (o,b) i f urn(-) i s norm continuous on (o,b), weakly, and f o r any e

I

E

E

urn(*)

E

C2(E)

I

m I 2m+1 rn I m ' + - + = 0

(4.2)

t

m on ( o , b ) , where
I d2 e > = - am, e l > , etc., and w r i t i n g dt2 p ( t ) = t2m+1 11 A u m ( t ) l l f o r m > - 1/2 w i t h p ( t ) = t [I A u m ( t ) l l 1 f o r m < - 1/2, one has p ( * ) E L (o,b).

The n o n s i n g u l a r case m = 1/2 i s encompassed i n forthcoming work o f Dunninger-Levine and w i l l n o t be discussed here. now p = I m l and l e t (An)

{An)

denote t h e p o s i t i v e r o o t s o f

b) = 0 f o r m 2 0 o r m <

Jp(%

Set

-

1/2; i f - 1/2 < m < 0 l e t

denote t h e p o s i t i v e r o o t s o f J

results

252

-P

(6b )

= 0.

Then t h e r e

4.

Theorem 4.2

SELECTEDTOPICS

L e t um be a s t r o n g s o l u t i o n o f (4.1) as i n I f m > - 1 / 2 assume

D e f i n i t i o n 4.1 w i t h um(b) = 0.

11 tlI

t2m+1(

u y l l + t l I u m I I ) - + 0 as t

that

u y l l + Ilum 11

t h e sequence {An}

-+

-+

0 as t

s a t i s f i e s An

0 while f o r m < +

0.

4

S I

-

1/2 assume

Then urn :0 i f and o n l y i f

( A ) f o r a l l n. P The p r o o f can be sketched as f o l l o w s . Consider t h e eigen-

v a l u e p r o b l em

(4.3)

$ ( b ) = 0; t 1 I 2 $ ( t ) bounded

and t a k e f o r m 2 0 o r m <

-

/ 2 t h e so u t i o n s

(4.5)

$n,p ( t ) = Jp(%t,

while f o r

-

(4.6)

$n,-p

=o

1/2 < m < 0 choose t h e s o l u t i o n s (t) = J

-P

(.Ant); J

-P

Assume f i r s t t h a t , f o r some n, An

( q b ) = 0

E CJ

P

(A) w i t h Avn = hnvn and

set tmmJp(Ant)vn; (4.7)

urn(t) =

t-mJ-p(Jj;nt)vn;

m 2 0 or m <

-

1/2

-1/2 < m < 0

Such um a r e s t r o n g s o l u t i o n s o f ( 4 . 1 ) i n t h e sense o f D e f i n i t i o n

0 i s c o r r e c t , as s p e c i f i e d i n t h e

4.1 and t h e i r b e h a v i o r as t

-+

hypotheses o f Theorem 4.2.

Hence An

E

u ( A ) f o r some n i m p l i e s

P t h e e x i s t e n c e o f n o n t r i v i a l um s a t i s f y i n g t h e hypotheses o f

253

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

Theorem 4.2 which means t h a t urn An

4 uP (A)

f

0 i n Theorem 4.2

f o r a l l n.

On t h e o t h e r hand suppose An

4 uP( A )

by J,n t h e a p p r o p r i a t e f u n c t i o n s $I,

t i o s n o f Theorem 4.2 and f o r some 6 (4.8)

V p ) =

The u:(6)

a r e we1

b 6

tW1J,,(

p+l

1 -while 2

11

E

-f

s a t i s f y i n g t h e condi-

(o,b)

set

0, tm1J,,(t)

O(t) f o r m <

=

1 L (o,b)

norm topology o f E as ,6

-

0(t2"+')

Hence t

and vf: = l i m $(ti)

E

D(A) and furthermore y:

-f

exists i n the

= lim

A v:(6)

For e

I

I

E

=

+

'n

b 6

-f

0

= yf: by

E fixed a routine calculation

y i e l d s now (4.9)

so t h a t

e x i s t s as 6

under o u r hypotheses on p ( t ) i n D e f i n i t i o n 4.1 w i t h Av: t h e closedness o f A.

for

Since A i s closed i t can be

0.

-f

=

passed under t h e (Riemann t y p e ) i n t e g r a l s i g n i n (4.8) vf:(6)

(4.6).

d e f i n e d by v i r t u e o f t h e hypotheses on u"

t"+'Qn(t)

qJn(t)llum(t)

E

-

from (4.5)

t ) u m (t ) d t

and one notes eas l y t h a t , as t

-

f o r a l l n and denote

or

Suppose urn i s a s t r o n g s o l u t i o n o f (4.1)

m >

implies

tmtlqJn(t)
~~tm''Qn(t)dt 254

,e'>dt

SELECTEDTOPICS

4.

m

1

and i n view o f t h e hypotheses one o b t a i n s as 6 -+ 0 = n I m m 1 m a l i m h = h . T h e r e f o r e <(A-hn)vn,e > = 0 f o r n n n m any e l E E l which i m p l i e s AvY = Anvn which b y o u r assumptions means v:

= 0.

B u t one knows t h a t t h e Bessel f u n c t i o n s qn form

a complete o r t h o g o n a l s e t on (o,b) t i o n t ( c f . Titchmarsh [3])

f o r any e l

E

m 1 and hence f r o m =

= 0 f o r a l l n i t f o l l o w s t h a t

Ibtm’$,,(t)
r e l a t i v e t o t h e w e i g h t func-

E l f r o m which r e s u l t s u m ( t )

=

0.

O t h e r boundary c o n d i t i o n s o f t h e t y p e uF(b) t a u m ( b ) = 0 a r e a l s o c o n s i d e r e d i n Dunninger-Levine [l]and uniqueness f o r weaker t y p e s o l u t i o n s i s a l s o t r e a t e d .

One n o t e s t h a t v e r y

l i t t l e i s r e q u i r e d o f t h e o p e r a t o r A i n t h e above r e s u l t .

The

t e c h n i q u e does n o t i m m e d i a t e l y e x t e n d t o more general l o c a l l y convex spaces E because o f t h e norm c o n d i t i o n s . We go now t o some more c o n c r e t e problems which a r e r e l a t e d t o Theorem 4.2

i n an o b v i o u s way.

F i r s t we s k e t c h some r e s u l t s

o f Young [4] on uniqueness o f s o l u t i o n s f o r t h e D i r i c h l e t problem r e l a t i v e t o t h e s i n g u l a r h y p e r b o l i c o p e r a t o r d e f i n e d by 2m+l uL, = u + +-A 1 1 ( a i ju ~ . ) t ~cu where c and t h e a i j tt t t 1 1 1 .i a r e s u i t a b l y smooth f u n c t i o n s o f x xn) and m i s (xly

-

Let R c

real. for x

E

R

. . .,

be bounded and open, Q =fi x(o,b),

c(x) 2 0

R, and assume t h e m a t r i x ( a i j) i s symmetric and p o s i t i v e

d e f i n i t e ; s u f f i c i e n t smoothness o f t h e boundary taken f o r granted.

255

r

o f R w i l l be

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

L e t Lmum = 0 i n Q w i t h um = 0 on rx[o,b)

Theorem 4.3 m u (x,o)

= 0.

Then um

=0

1

-

Suppose m >

and assume urn

E

and

2

C ( Q ) 0 C1(q).

q.

in

The p r o o f uses t h e f a c t t h a t any s o l u t i o n um o f Lmum = 0 belonging t o C

2

f i e s uF(x,o) =

f o r t > 0 and t o C

o

+ -2 1

f o r any m

1

for t 2 0 necessarily satis-

(cf.

[I], W a l t e r [I]).

FOX

NOW

integrate the i d e n t i t y

1

2u L u = ( u 2 t + t m

(4.10)

a i ju

i,j

o v e r t h e r e g i o n Qs = Qx(o,s),

X

Wt2

ux + c u ' ) ~ + 2 2m+l i

j

s 5 b y and use t h e d i v e r g e n c e

theorem t o o b t a i n , when Lmu = 0,

I

(4.11)

aQs

[(up.l.aiJux.ux ~ Y J 1

+ 2(2m+l)I where v = (v

,

. . .,

QS

vn,vt)

c o n d i t i o n s p l u s u"'(x,o) t (aiJ)

i,j

p1 t d2 x d t = 0 i s t h e e x t e r i o r normal on

boundary Qs. P u t t i n g o u r urn i n (4.11)

1 a i j u mxi

1 a i juxi utv J. l d a

+cu2 )vt-2 j

=

x j + c ( um ) 2 ]l:t,sdx

=

and u s i n g t h e boundary

I,

i s p o s i t i v e d e f i n i t e , c 2 0, and

J

[(u:)~ + R l( ut) d x d t = 0. S i n c e 1 m > - - both o f these

0 we have t h e e q u a t i o n

+ 2(2m1)

aQs

2

i n t e g r a l s a r e z e r o and by c o n t i n u i t y t h e f i r s t t e r m t h e n vani s h e s f o r 0 5 s 5 b. Theorem 4.4

C'(0)

Hence urn s 0 i n

If m 5

- 71 t h e n

0 since

um(x,o) = 0.

any s o l u t i o n urn

E

2

C ( Q ) fl

m o f Lmum = 0, w i t h u = 0 on t h e boundary o f Q, vanishes

256

SELECTED TOPICS

4.

f 0, where t h e

i d e n t i c a l l y i f and o n l y i f JLm(%b)

nonzero e i g e n v a l u e s o f t h e problem Av =

-

r.

$2 w i t h v = 0 on

z . ( a ij vxi iY J

An a r e t h e

xJ

t

cv i n

There i s an obvious r e l a t i o n between Theorems 4.4 and 4.2 where A corresponds t o t h e o p e r a t o r determined by Aw = cw

2

- 1 (aijwx.)x

i n E = L (L?)w i t h w = 0 on

i,j i j Theorem 4.4 goes as f o l l o w s .

r.

The p r o o f o f

F i r s t suppose t h e r e i s a nonzero

e i g e n v a l u e An such t h a t J-m(Q)

= 0 w i t h c o r r e s p o n d i n g eigen-

Then um(x,t) = t - m J - m ( ~ t ) v n ( x ) s a t ' i s f i e s

f u n c t i o n vn(x).

Lmum = 0 and t h e s p e c i f i e d boundary c o n d i t i o n s .

Conversely i f

0 f o r any An one i n t e g r a t e s t h e i d e n t i t y wLmu

J-,,(%b)

-

-

- 1 [ l a i j(

U ~ ~ W - U WOver ~ ~ t)h e] ~cy~ j i 1 b e n c l o s e d by Q between t h e p l a n e s t = s and t = b ( 0 < l i n d e r Qs

uMmw = ( u t w - u w t + - -2m+l Uw)

s < b).

Here Mm i s t h e f o r m a l a d j o i n t o f L,

(2mtl)(w/t)t

-

l(aijw

jQ:( =

By t h e divergence theorem one

'j

obtains (4.12)

+ cw.

)

g i v e n by Mmw = wtt

wLmu-uMmw) d x d t

b[(utw-uwt aQS

-+ 2mt1 w)vt-la

t"

ij

( u w-uwx )v . I d a 'i

i J

Now l e t Lmum = 0 w i t h urn = 0 on t h e boundary o f Q and s e t wm(x,t)

= tmt1J-,(%t)vn(x)

where An i s a nonzero e i g e n v a l u e

of t h e problem i n d i c a t e d i n t h e statement o f Theorem 4.4.

It

i s e a s i l y checked t h a t Mmwm = 0 so t h a t t h e l e f t hand s i d e i n (4.12)

vanishes and s i n c e wm = 0 on

257

r

we have

-

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

m l m 0 wt and p a r e bounded w i t h um(x,o) = 0 (and Now as s m - 21 ) ; t h e e v a l u a t i o n i n ( 4 . 1 3 ) a t t = s tends ut(x,o) = 0 f o r m +

t o zero and we o b t a i n

s i n c e um(x,b) = 0.

Consequently f o r n = 1, 2 ,

...

uy(x,b)vn(x)dx = 0 and by t h e completeness o f t h e vn(cf. R H e l l w i g [3]) i t f o l l o w s t h a t uF(x,b) = 0. Now one may i n t e g r a t e m b (4.10) over Qs and since u (x,b)

=

=

m ut(x,b)

=

0 there r e s u l t s

1 m 2 - ( u ) dxdt 2(2m+1)jQb t t S

Since m 5

-

1 i t f o l l o w s t h a t f o r any s, 0 5 s 5 b, 2 +c(um)']/

theorem i s immediate.

t=s

dx = 0 and t h e conclusion o f t h e

Young a l s o t r e a t s t h e Neumann problem by s i m i l a r t e c h n i ques.

T r a v i s [l]uses a d i f f e r e n t method w i t h more general

boundary c o n d i t i o n s and breaks up t h e range o f m i n t o p a r t s corresponding t h e t h e Weyl l i m i t c i r c l e and l i m i t p o i n t s i t u a t i o n s ( c f . Coddington-Levinson [ l ] ) . s i n g u l a r e i gen va 1ue problem

258

Thus one considers t h e

4.

SELECTEDTOPICS

,Itzm+’ I$( t ) I ‘dt

<

$(b) = 0

m;

I t can be shown t h a t f o r m 5 -1 o r m

L

1 t h e problem (4.16)

has

a p u r e p o i n t spectrum and t h e e i g e n f u n c t i o n s f o r m a complete 2m+l o r t h o g o n a l s e t on (o,b) r e l a t i v e t o t h e w e i g h t f u n c t i o n t

.

Furthermore i f f has f i n i t e tZm+’ norm on (o,b) (i.e.¶ b $m+l I f ( t ) l 2 d t < m ) t h e n t h e Fredholm a l t e r n a t i v e ( c f . H e l l w i g

[

’0

2m+l $’ [3]) h o l d s f o r t h e nonhomogeneous e q u a t i o n ( t = tZm+’f. I n t h e l i m i t

Atzm+’$

c i r c l e case

-

I

+

1 < m < 1 there i s

always a p u r e p o i n t spectrum w i t h a complete o r t h o g o n a l s e t o f e i g e n f u n c t i o n s $ n ( r e l a t i v e t o t h e w e i g h t f u n c t i o n tZm+’) f o r the problem (4.17)

(t

2m+l$l)’

I

where W($,J,,t)

= $J,

+

t2rn+lA$ = 0;

-

$ J, and J, = + ( t y A o ) i s a p o s s i b l y complex

$ ( b ) = 0;

I

v a l u e d s o l u t i o n o f t h e d i f f e r e n t a1 e q u a t i o n ho s a t i s f y i n g t

2m+ 1 W ( $ ¶ T y t ) + 0 as t

-f

n (4.17)

for X =

0 (such J, e x i s t ) .

The

Fredholm a1 t e r n a t i v e a l s o h o l d s a g a i n f o r t h e corresponding nonhomogeneous problem. We t a k e now a i j ( * )

E

C

s u f f i c i e n t l y smooth boundary

1

i n a bounded open s e t R

r

where aiJ

u1El2 w h i l e c ( = ) i s c o n t i n u o u s i n R. 259

= a j i and Fa

cRn ij

with

Cisj 2

L e t a/av denote t h e

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

transverse d e r i v a t i v e a/av

r

i s t h e e x t e r i o r normal t o

Let m

Theorem 4.5 E

cos (v,x.) J

and l e t Lmu = utt +

a/ax. where v 2m+l

1

-pt -

) + cu as before. x .1 'j

l(aiju

u"

laij(x)

=

C2(cx

(o,b)),

um(x,b) = 0 f o r x

4

(-1,l)

E

r

+ a ( x ) u m = 0 on

aum/av

r bt2m+l

R while

3x

and Lmum = 0 i n x

(o,b),

m 2 Iu (x,t)l d t <

a.

(o,b) w i t h and Then

JO

(9)

f 0 where p = I m l and {An) i s P t h e s e t o f p o s i t i v e eigenvalues o f - l ( a i j v x ) x + cv = hv i n u" 5 0 i f and o n l y i f J

R w i t h av/av

+ crv

= 0

on

i

r.

j

The p r o o f i s s i m i l a r i n p a r t t o t h a t o f Theorem 4.4.

Thus,

i f t h e r e e x i s t s a p o s i t i v e eigenvalue An o f t h e e l l i p t i c problem i n d i c a t e d i n t h e statement o f Theorem 4.5 such t h a t J 0 then s e t um(x,t) = tmmJ (%t)vn(x)

P f u n c t i o n corresponding t o An.

(q) =

P where vn i s t h e eigen-

E v i d e n t l y u" s a t i s f i e s Lmum = 0

and t h e r e q u i r e d boundary c o n d i t i o n s .

On t h e o t h e r hand suppose

L e t u" be a s o l u t i o n o f J ( 5 b ) 0 f o r a l l An as described. P Lmum = 0 s a t i s f y i n g t h e c o n d i t i o n s s t i p u l a t e d i n Theorem 4.5. L e t +,(t) (4.16)

and vn(x) be t h e normalized e i g e n f u n c t i o n s f o r t h e

and t h e e l l i p t i c eigenvalue problem i n t h e statement o f

Theorem 4.5 r e s p e c t i v e l y .

Then t h e s e t C+,(t)v,(x)l

is a

complete orthonormal s e t f o r t h e r e a l H i l b e r t space H = 2 L(R ,

x

(o,b))

= (h;

:1 1,

t2m+1h2(x,t)dxdt <

H we can w r i t e um(x,t) = mula f o r aij.

m)

and s i n c e urn E

la..+.(t)vi(x) w i t h t h e standard f o r 1J J um(x,t)vi(x)dx so t h a t Now d e f i n e h i ( t ) =

R

260

4.

(t2m+’h;(t))l

+

SELECTED TOPICS

Ait2m+1hi(t)

= 0, hi(b)

e i g e n v a l ue c o r r e s p o n d i n g t o vi. formula ] I t 2 m ’ h f ( t ) d t Ai

=

= 0, where Ai

By t h e P a r s e v a l - P l a n c h e r e l

r l a 1. .JI 2 <

m

and t h e r e f o r e h i ( t )

i s n o t an e i g e n v a l u e f o r t h e problem (4.16),

Jp(qb)

$ 0.

i s the

0 if

if

i.e.,

Consequently um(x,t) z 0 s i n c e t h e vi(x)

a r e com-

2

p l e t e i n L (R) and um i s c o n t i n u o u s i n R x (o,b). Remark 4.6

We remark f i r s t ( c f . T r a v i s [ l ] ) t h a t f o r m 5 -1

t h e problem t r e a t e d i n Theorem 4.5 i s e q u i v a l e n t t o Lmum = 0,

r‘

3um/3v + oum = 0 on Remark 4.7

x (o,b)

and um(x,b)

= um(x,o)

= 0.

L e t us n o t e i n p a s s i n g t h a t , i n general, i f L

i s a s i n g u l a r second o r d e r o r d i n a r y d i f f e r e n t i a l o p e r a t o r i n t such t h a t t h e n u m e r i c a l i n i t i a l v a l u e problem Lu + Anu = 0, u ( o ) = 1, u t ( o ) = 0, has a s o l u t i o n u n ( t ) , when A,

i s an e i g e n -

v a l u e o f t h e ( c l o s e d densely d e f i n e d ) o p e r a t o r A i n a H i l b e r t space E, w h i l e t h e c o r r e s p o n d i n g s e t o f e i g e n v e c t o r s {en) o f A forms a complete orthonormal s e t i n E, then f o r m a l l y t h e E valued function w ( t ) = lan(Lun++,un)en

1 anun(t)en

s a t i s f i e s Lw + Aw =

= 0 w i t h w(o) = l a n e n and wt(o)

t h e an t o be t h e F o u r i e r c o e f f i c i e n t s o f w(o) = e

Lw + Aw

a formal s o l u t i o n o f

Choosing

= 0. E

D(A) one has

= 0 w i t h w(o) = e and wt(o)

=

0.

Uniqueness q u e s t i o n s c o u l d t h e n be handled i n terms o f t h e L

2

formal a d j o i n t L

I

o f L and t h e o p e r a t o r A

t h a t Ae = A(lanen) = lanAen = CanAnen f o r e ( e y en ) we have (Ae,ek)

=

*

E

.

Indeed g i v e n

D(A) w i t h an =

~anAn(en,ek) = akAk = Ak(e,ek)

2 61

so

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

ek

*

*

A ek

D(A ) w i t h

E

=

-

Hence A

Akek.

*

has a complete o r t h o -

normal s e t o f e i g e n v e c t o r s ek w i t h eigenvalues

Tk.

Assume then

I

t h a t t h e problem L v + Tnv = 0, v(b) = 0, v ( b ) = 1 has a solut t i o n v n ( t ) on [o,b]

f o r any b.

Here, g i v e n Lu = u

I

Bu w i t h u ( o ) = u ( 0 ) = 0 and v a C

i

I

I

0 and v ( b ) = 0 we d e f i n e L v = v b t i that u v d t = -u(b) + O

I 1

I

+ au +

2 function s a t i s f y i n g v(b) =

I I

-

(w)

I

+ Bv and observe

n

Now w r i t e w = l b n v n ( t ) e n where e = w ( b ) = lbnen i s an a r b i t r a r y

*

D(A ) ) .

element o f some dense s e t (e.g.,

Then i f (L+A)w = 0

I

w i t h w(o) = w ( 0 ) = 0 we consider f o r m a l l y t h e e q u a t i o n (4.18)

0 =

b

A

((L+A)w,w)dt

= lbn

0

+

I

b

(Lw,vnen)dt

0

1

b

l (w,L v n e n ) d t l + l b

0

I t f o l l o w s t h a t w(b) = 0 f o r any b and hence w

z 0.

The c a l c u -

l a t i o n s have been formal b u t they a p p l y t o r e a l i s t i c s i t u a t i o n s . Theorem 4.8

m Assume m E (-l,l), Lmu = 0, urn

aum/av + a(x)um = 0 on

r

x

l i m t2m+1W(um,~,t) = 0 as t

(o,b),um(x,b) +

E

C2(sZx(o,b)),

= 0, and

0 where $ i s a p o s s i b l y complex

262

SELECTEDTOPICS

4.

valued s o l u t i o n o f t h e d i f f e r e n t i a l equation i n (4.17) A

0

= A s a t i s f y i n g l i m t2m+1W(q,~,t) = 0 as t

+

0.

f o r some

Then u" :0

i f and o n l y i f Rn f Ak where Rn and Ak a r e t h e eigenvalues o f (4.17)

and t h e e l l i p t i c problem i n t h e statement o f Theorem 4.5

respect iv e l y

.

The p r o o f here i s s i m i l a r t o t h a t o f Theorem 4.5.

Finally

T r a v i s [l]s t a t e s some r e s u l t s r e l a t e d t p Theorems 4.3 and 4.4 and some comparison i s i n s t r u c t i v e (see Remark 4.10). Theorem 4.9

Let u

3um/3v + a ( x ) u m = 0 on for x

E

r

x

(o,b),

w h i l e um(x,o) = um(x,b) = 0

R when m < 0 o r Ium(x,o)l <

when m 2 0.

s a t i s f y Lmum = 0 and

C2(Ex(o,b))

E

m

and um(x,b) = 0 f o r x

E

R

(q)

f 0 where t h e P An a r e t h e eigenvalues o f t h e e l l i p t i c problem i n the statement Then urn E 0 i f and o n l y i f J

of Theorem 4.5 and p = Im Remark 4.10

Theorem 4.9

-

i s independent o f Theorem 4.3 f o r example even when

0.

m I n Theorem 4.3 one assumes ut i s continuous i n

whereas Theorem 4.9 assumes o n l y c o n t i n u i t y o f u:

[o,b))

1/2 < m <

Ex

-

-

But uT(x,t) = O ( t -2m-1) as t

x

[o,b]

on E x (o,b).

Consider f o r example um(x,t) = t'mJ( t ) s i n x f o r P m 2m+l m m m which s a t i s f i e s utt + T~ uxx = 0 w i t h u (o,t)

0 and um (x,o) = 0.

r

(rephrased f o r urn = 0 on

+

1/2 < m < 0 = u"(r,t)

=

0 so t h e hy-

potheses o f Theorem 4.3 do n o t h o l d whereas Theorem 4.9 ( r e phrased) w i l l apply.

On t h e o t h e r hand Theorem 4.3 does n o t

require conditions a t t = b f o r m >

-1. S i m i l a r l y T r a v i s [l] 2

s t a t e s a r e s u l t f o r a Neumann problem (ut(x,o) 263

= ut(x.b)

= 0)

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

r e l a t e d t o a theorem o f Young [4]. 4.5

Miscellaneous problems.

Various Cauchy problems w i t h

boundary data f o r EPD equations have been s t u d i e d and we w i l l mention a few r e s u l t s .

2m urn Solve Lmum = utt + ___ t t

Problem 5.1

m >

-

1/2,

o

F i r s t c o n s i d e r ( c f . Fusaro [41)

+

< x < r , t > 0, um(x,o) = f ( x ) ( f

0, and um(o,t) = u m ( r , t ) = 0.

E

-

um = 0, where xx 2 c 1, u! (x,o) =

One assumes a l s o f ( o ) = f(n) = 0.

Then by s e p a r a t i o n o f v a r i a b l e s Fusaro [41 o b t a i n s a s o l u t i o n i n t h e form (5.1)

um(x,t) =

where bn =

]:[

I

IT

r(m+l)

[F] J m ( n t ) s i n n x -m

m

1

n=l

bn

Using t h e formula (1.3.7)

f ( x ) sinnx dx.

0

a change o f v a r i a b l e s $ = r/2

-

with

0 and p u t t i n g t h i s i n (5.1) one

obtains

cos

2m

$ d $

where 8 denotes t h e Beta f u n c t i o n .

There r e s u l t s ( c f . Fusaro

[41) Theorem 5.2

A unique s o l u t i o n o f Problem 5.1 i s g i v e n by

(5.2). The p r o o f o f Theorem 5.2 i s s t r a i g h t f o r w a r d and w i l l be omitted.

I f one s e t s 5 = s i n $ and M(x,a,f)

f ( x - o ) ] we can w r i t e (5.2) i n t h e form

264

=

21 [ f ( x + a ) +

4.

SELECTEDTOPICS

+ ~]-1~1(l-~)m-"2M(x,Et,f)dg

um(x,t) = 2 ~ [ + , m

(5.3)

0

which c o i n c i d e s w i t h (1.6.3).

We remark here t h a t f i s extended

from ( O , I T ) as an odd p e r i o d i c f u n c t i o n o f p e r i o d ZIT. Another t y p e o f such "mixed" problems i s solved by Young 191 ; here mixed problem means mixed i n i t i a l - b o u n d a r y value probThus con-

lem and does n o t mean t h a t t h e problem changes type. sider m + ___ 2m+1 Solve Lmum = utt t

Problem 5.3 f(x,y,t)

m > f

E

-

i n t h e q u a r t e r space x > 0, t > 0,

m 1/2, u (x,y,o)

C2, g

E

C

2

A t p o i n t s (x,y,t)

03

xx < y <

= 0, and u"(o,y,t)

= u:(x,y,o)

, and g(y,o)

-

t

= gt(y,o)

-

=

m,

YY where

= g(y,t) with

= 0.

where x 2 t > o t h e presence o f g i s

n o t f e l t and Young takes t h e s o l u t i o n i n t h e form given by DiazLudford [31

where D i s t h e domain bounded by T = Oandthe r e t r o g r a d e chara c t e r i s t i c cone R = v e r t e x a t (x,y,t)

(t-T)'

while

^R

-

-

(x-c)'

= (tt-r)'

-

= 0 (t-oO)

(y-0)'

(x-c)'

-

(y-n) 2

.

with I n the

r e g i o n where 0 < x < t formula (5.4) no l o n g e r g i v e s t h e s o l u t i o n s i n c e i t does n o t i n v o l v e g and Young uses a method o f M. Riesz as adapted by Young [l;21 and Davis [l;21 t o a d j o i n a d d i t i o n a l terms t o t h e um o f (5.4).

We w i l l n o t go i n t o t h e d e r i v a t i o n o f

t h e f i n a l formula b u t simply s t a t e t h e r e s u l t ; f o r s u i t a b l e m 265

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

t h e t e c h n i q u e can be extended t o more space v a r i a b l e s .

The

s o l u t i o n i s g i v e n by t h e f o r m u l a

22m+l

--_ 'lr where R,

-

= (t-.)'

I

a ax

T

(x+<) 2

I\

v a l u e o f R ( r e s p . R) f o r

D'. i s t h e domain where

<

5

-

(y-rl)',

i* =

= 0,

> 0,

Ro ( r e s p .

T

(ttT)2

-

Go)

denotes t h e 2 (x+E)2 - (y-rl) y

> 0, bounded by t h e r e t r o g r a d e

*

D C D), and

cone w i t h v e r t e x a t ( - x , y , t ) ( t h u s

T i s the region

i n t e r c e p t e d on t h e p l a n e 5 = 0 by t h e cone R = 0

*

that for x 2 t > 0 D g r a l s o v e r D and

Theorem 5.4

*

and T a r e empty w h i l e as x

D cancel each o t h e r s i n c e D

*

(T +

Note

> 0).

0 the inteh

+

D and (R,R)

+

The s o l u t i o n o f Problem 5.3 i s g i v e n by (5.5).

I t i s i n t e r e s t i n g t o n o t e what happens when a l l o f t h e d a t a

i n an EPD t y p e problem a r e a n a l y t i c .

Thus f o r example Fusaro

[3] c o n s i d e r s f o r t > 0, 1 5 k 5 n, and x = (xl,

(5.6)

. . .,

u"tt + 2m+l t um t = F(x,t,um,Dku",u~,Dku~,DkDeum); m u (x,o)

=

f(x);

m

Ut(X,O)

266

= 0

Xn)

SELECTEDTOPICS

4.

where Dk = a/axk w h i l e F and f a r e a n a l y t i c i n a l l arguments. Theorem 5.5

-1, -3/2,

for m

The problem (5.6)

....

-2,

C o r o l l a r y 5.6

has a unique a n a l y t i c s o l u t i o n

. . .)

I f F(x,t,u,

=

G(x,u,Dku,D

k Dau ) +

+ h ( t ) u t w i t h g even i n t and h odd i n t, t h e n t h e problem

g(x,t)

has f o r a l l m f -1,

(5.6)

-3/2,

...a

-2,

unique a n a l y t i c s o l u -

. . . there exists

I f m = -3/2,-5/2,

t i o n and i t i s even i n t.

a unique s o l u t i o n i n t h e c l a s s o f a n a l y t i c f u n c t i o n s which a r e even i n t. Fusaro [3] uses a Cauchy-Kowalewski t e c h n i q u e t o p r o v e these r e s u l t s and we r e f e r t o h i s paper f o r d e t a i l s .

One s h o u l d n o t e

a l s o t h e f o l l o w i n g r e s u l t s o f L i o n s [l],o b t a i n e d by transmutat i o n , which i l l u s t r a t e f u r t h e r t h e r o l e p l a y e d b y even and Cm f u n c t i o n s o f t i n t h e uniqueness q u e s t i o n f o r EPD equations. L e t E,

be t h e space o f Cm f u n c t i o n s f on [o,m)

f(2n+1)(o) = 0 (thus f (-0,m))

and g i v e E,

E

E,

such t h a t

admits an even Cm e x t e n s i o n t o

t h e topology induced by

E on

(-,a).

Let H

be a H i l b e r t space and A a s e l f a d j o i n t o p e r a t o r i n H such t h a t f o r a l l h c D(A) (Ah,h) Lm=D

*'+'

+-

t

+ allh112 Z 0 f o r some r e a l a. S e t t i n g

D and p u t t i n g t h e graph t o p o l o g y on D(A) L i o n s

proves Theorem 5.7

For any m

e x i s t s a unique s o l u t i o n um

E E

f, m =f -1, -2, E,

267

h

BT D(A)

-3,

. . ., t h e r e

o f Lmum + hum = 0 f o r

SINGULAR AND DEGENERATECAUCHY PROBLEMS

t 2 0 w i t h um(o) = h

Here

T

and uT(o) = 0.

D(A")

denotes t h e p r o j e c t i v e l i m i t t o p o l o g y o f Grothendieck

( c f . Schwartz [5]) i s nuclear.

E

which c o i n c i d e s w i t h t h e

I f m f -1,

Coo f u n c t i o n s on [o,m)

-3/2,

w i t h t h e standard E-type topology then any

s o l u t i o n urn E E + B ~ D(A) o f E

topology s i n c e E,

. . . and E+ i s t h e space o f

-2,

A

h

E

L~U"'

+ hum = o s a t i s f y i n g um(o)

=

A

D(A")

and uT(o) = 0 i s a u t o m a t i c a l l y i n E * B T D ( A ) so t h e *

corresponding problem i n E+ BT D ( A ) has a unique s o l u t i o n . t h e e x c e p t i o n a l values m = -1, -2, Theorem 5.8

For

. . . Lions [l]proves

-3,

Let m = -(p+l) with p

2

0 an i n t e g e r so t h a t h

2m + 1

There e x i s t s a unique s o l u t i o n urn E E + q D ( A ) -(2p+l). m m 2p+l m + Au = 0, um(o) = uY(0) = = Ut u ( 0 ) = 0,

of u L,

...

DZP+*um(o) = h

E

D(Am), and DZP+3um(o) = 0.

I n connection w i t h t h e uniqueness q u e s t i o n we mention a l s o t h e f o l l o w i n g r e c e n t r e s u l t o f Donaldson-Goldstein [Z].

Let H

be a H i l b e r t space and S a s e l f a d j o i n t o p e r a t o r i n H w i t h no n o n t r i v i a l n u l l space.

L e t Lm = D2

+ 2m+l D again and d e f i n e urn t

t o be a t y p e A s o l u t i o n o f m LmU + S2um = 0;

(5.7)

if urn

E

2

um(o) = h

E

D(S2);

m ut(o) = 0

and a t y p e B s o l u t i o n i f urn E C1 ([o,m),H)n

C ([o,m),H)

C2( (oYrn),HI. Theorem 5.9 solution f o r m

The problem (5.7)

2 -1. 268

can have a t most one type' A

4.

SELECTED TOPICS

F u r t h e r , by v i r t u e o f (1.4.10) ( c f . Remark 1.4.7),

and t h e formula um =

t-2mu-m

f o r m < -1/2 n o t a n e g a t i v e i n t e g e r t h e r e i s

no uniqueness f o r t y p e B s o l u t i o n s .

We n o t e here t h a t f o r -1 <

m < -1/2 we have 1 < -2m < 2 and um = t -2mu-m = O(t-2m) -+ 0 as t

+

0 w i t h u" = O(t-2m-1 t

)

-+

0 b u t uYt = O(t-2m-2)

-+

m

so such a

m 2 1 u i s n o t o f t y p e A. F o r m = -1 we l o o k a t u - l = t u w i t h -1 -1 O However from Remark u ( 0 ) = u'; ( 0 ) = 0 and utt E C ([o,m) ,H). 1.4.8

we see t h a t t h e s o l u t i o n u-' a r i s i n g from t h e r e s o l v a n t

R - l (which c o u l d be s p e c t r a l i z e d as i n S e c t i o n 1.5) does n o t have a continuous second d e r i v a t i v e i n t unless t h e l o g a r i t h m i c term 2 vanishes, which corresponds t o S h = 0 i n t h e p r e s e n t s i t u a t i o n .

does n o t seem t o have a type A s o l u t i o n f o r

Thus i n general (5.7)

2

m = -1 and i f i t does t h e s o l u t i o n a r i s e s when S h = 0.

i f L-lu-l

then U;;(O) -S2um1

[3]),

-+

+ S2u-l = 0 w i t h u - l ( o ) = h and u;l(o)

Indeed

i s well defined

+ 0 so L lu-l = u-l - 1 u-l = tt t t 2 -1 2 0 which means t h a t S u ( 0 ) = S h = 0 ( c f . Weinstein

= l i m l/t

u,'(t)

as t

which i s precluded unless h = 0.

2 -- 0 then u - ' ( t )

S h

=

h and u - ' ( t )

0 with

Note t h a t i f h

= h

+ t 2h w i l l b o t h be solu-

t i o n s o f (5.7). We go now t o some work o f Bragg [2; 3; 4; 51 on index s h i f t i n g r e l a t i o n s f o r s i n g u l a r problems.

T h i s i s connected t o t h e

i d e a o f " r e l a t e d " d i f f e r e n t i a l equations developed by Bragg and Dettman, l o c . c i t . , Section 1.7).

and described b r i e f l y i n Chapter 1 ( c f .

Consider f i r s t f o r example ( a 2 0,

269

u 2

1)

SINGULAR AND DEGENERATE CAUCHY PROBEMS

T h i s i s a r a d i a l l y symmetric form of the EPD equation when

=

n f o r example since the r a d i a l Laplacian i n R n has the form 2 +n-1

r Dr* As i n i t i a l data we take

AR = Dr

(5.9)

U (ay’)(ryo) =

u(ay’)(ryo) t = 0

T(r);

From the recursion r e l a t i o n s of Chapter 1 ( c f . (1.3.11),

(1.3.54),

(1.3.55),

(1.4.6),

(1.4.10),

(1.3.12),

and Remark 1.4.7) we

see t h a t there a r e c e r t a i n “automatic” shifts i n the a index. For example the Sonine formula (1.4.6) w i t h a l al + a 2 (5.10)

=

=

2 p + 1 and

2m + 1 y i e l d s f o r al 2 0, a2 > 0

u

(a1+a2 ,!J)

a2-2

( r , t ) = c1,2t -al -a2 It(t2-,,2) 21; 0

‘ U

(a1 YU)

( r Y r l ) drl

al+l a2 Bragg [5] obtains t h i s r e s u l t u s i n g = 2 / f 3 ( ~, 7). 1 2 Laplace transforms and a r e l a t e d r a d i a l heat equation

where c

(5.11)

v: = vyr +

$!-v;:

v’(r,o)

=

T(r)

The connection formula f o r a 2 1 i s given under s u i t a b l e hypotheses by (cf. Bragg [5], Bragg-Dettman [6])

(here L;’

2

i s the inverse Laplace transform w i t h kernel est ) arid

(5.10) follows d i r e c t l y by a convolution argument.

setting I v ( z ) = exp, ( - 1F v ~ i ) J v ( i z ) and Ku(r,S,t) 270

Similarly, =

4.

SELECTEDTOPICS

2 2 exP C - ( r +E ) / 4 t l I u / 2 - 1 ( r E / 2 t ) , one can show P t h a t v’(r,t) = r,E,t)T(E)d< s a t i s f i e s (5.11), and u s i n g oKJ (5.12) t h e r e r e s u l t s from known p r o p e r t i e s o f Iv (a)-lrl-’/2<’/2

1

Let a 2 0 w i t h u(~’’)

Theorem 5.10 o f (5.8)

-

(5.9).

s a t i s f i e s (5.8)

a continuous s o l u t i o n

Then

= (5.9)

w i t h i n d e x 1-1 + 2.

On t h e o t h e r hand a

d i r e c t s u b s t i t u t i o n shows t h a t i f u (ay4-’)(r,t)

-u

w i t h index 4

r2-’u(ay4-’)(r,t)

and u (ay4-’)(r,0) s a t i s f i e s (5.8)

=

-

s a t i s f i e s (5.8)

r’-2T(r) (5.9)

then u ( a y ’ ) ( r , t )

=

except possibly a t r =

0. Using t h e i n d e x s h i f t i n g r e l a t i o n s i n d i c a t e d on a and p

-

Bragg [5] proves a uniqueness theorem f o r (5.8)

L 1 and c o n s t r u c t s fundamental s o l u t i o n s .

a 2 0 and 1-1

o f r e l a t e d d i f f e r e n t i a l problems such as (5.8) ( o r (5.8)

-

(5.9)

mulas l i k e (5.12) 3;

(5.9) when

-

(5.9)

The i d e a and (5.11)

w i t h i n d e x changes), h a v i n g c o n n n e c t i o n f o r ( o r (5.10)),

has been e x p l o i t e d by Bragg [2;

41 t o produce a v a r i e t y o f i n d e x s h i f t i n g r e l a t i o n s i n v o l v i n g

a t t i m e s t h e g e n e r a l i z e d hypergeornetri c f u n c t i o n s

where (a),

=

a(a+l)

. . . (a+n-1),

II:(yj),,

=

1 i f p = 0, and

none o f t h e 6 . a r e nonposi t i v e i n t e g e r s ( c f . A p p e l l -Kampe/ de J 271

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

F g r i e t [l]).We w i l l mention a few r e s u l t s i n t h i s d i r e c t i o n w i t h o u t g i v i n g p r o o f s and here we r e c a l l t h a t i f 0 = zDZ then

w

F s a t i s f i e s the d i f f e r e n t i a l equation P q IOnl(O+B.-l)-zn~(O+a.)}w q = 0. Thus l e t P(x,D) = J A: ‘n Dn where Dk = a/axk and [XI = l X i raX(x)D1 =

...

5 m.

Let

= t D l l q ( t D + B . - l ) and 02(pya,t,Dt) = t I I y ( t D +a ). t 1 t J t j Then given P(x,D) and Q(x,D) o f t h e type i n d i c a t e d w i t h o r d e r s

el(qYB,t,Dt)

R1 and

R2 r e s p e c t i v e l y and r = max (p+R1 ,q+R2) w i t h p 5 q one can

prove f o r example (cf.

Bragg [4])

Let u

Theorem 5.11

E

Cr i n ( x , t )

f o r t > 0 w i t h u and a l l

i t s d e r i v a t i v e s through o r d e r r bounded and l i m u ( x , t )

t

-

P(x,D)0,(p-1,a,t,Dt)u = 0. -1 -1 rrn -0 Then f o r a > 0 t h e f u n c t i o n v(x,t) = r ( a ) e u P u(x,tu)du P p 10 P82(p,art,Dt)v = 0 and v(x,o) = T(x). s a t i s f i e s Qel(q,B,t,Dt)v +

0.

= T ( x ) as

Suppose Q(x,D)01(q,f3,t,Dt)u

-

S i m i l a r l y , under s u i t a b l e hypotheses, i f [Q0 (4-1 ,B,t,Dt) 1-Bq q P02(p,a,t,Dt)]u = 0 and w(x,t) = t r(Bq)Ui1[S u(~,l/s)]~+~

-A

then one has [Qel(q,B,tyDt)

a , and

formulas i n (p,q),

-

P02(p,a,t,Dt)]w

= 0.

Index s h i f t

are a l s o proved and when a p p l i e d t o

EPD equations f o r example w i l l y i e l d some o f t h e standard r e c u r sion relations.

5

=

We n o t e t h a t under t h e change o f v a r i a b l e s

1 t 2 t h e EPD e q u a t i o n utt + a ut = Pu becomes [
a+ 1 -1) 2

-

5

gP]u = 0.

s h i f t s i n a and

5

+

Using t h e R-L i n t e g r a l , continuous index

B f o r s i m i l a r a b s t r a c t problems are t r e a t e d i n

Bragg [3] where a more s y s t e m a t i c t h e o r y i s developed; nonhomogeneous problems a r e s t u d i e d i n Bragg [2].

272

4.

SELECTEDTOPICS

We conclude w i t h a few remarks about n o n l i n e a r EPD equations. I n K e l l e r [l]i t i s shown t h a t f o r m > 0 and c e r t a i n n o n l i n e a r f t h e Cauchy problem L u = utt + m

(5.15)

u(x,o)

"+'

u t t

2 c Au = f ( u ) ;

uo(x);

=

R2x

does n o t have g l o b a l s o l u t i o n s i n

E

R

c R n with u

= 0

on

s u f f i c i e n t l y smooth boundary t i o n and f o r each r e a l @ Set (@,I)) =

(0,m)

for arbitrary ini-

-

Levine [2] s t u d i e s t h i s problem f o r

t i a l data uo. and x

= 0

Ut(X;O)

E

r r.

1 < m 5 0

[o,T) where R i s bounded w i t h

x

L e t f be a r e a l valued C

C2(E) define G(@) =

1

R

such t h a t f o r a l l @

E

func-

(~(x)f(r)dz)dx.

1R@ ( x ) $ ( x ) d x and assume t h e r e i s a constant r

1

O

ct > 0

C2(E)

I t can be shown t h a t i f , f o r some ct > 0 and some monotone i n -

) , (5.1G) c r e a s i n g f u n c t i o n h, f ( z ) = l ~ 1 ~ ~ + ' h ( zthen

i s valid.

Assuming t h e e x i s t e n c e o f a l o c a l s o l u t i o n f o r each uo v a n i s h i n g on

r

-

-+

Let

G(uo) > as t

-+

- 21 < m 5

0 and assume (5.16)

holds.

If

R 1 i s a c l a s s i c a l s o l u t i o n o f Lmu = f ( u ) w i t h

u(x,o) = u ( x ) , ut(x,o) 0

C2(E)

Levine proves

Theorem 5.12

u : R x [o,T)

E

1 2 7 ~ R I V u o ldx,

= 0, and u = 0 on

then n e c e s s a r i l y T <

T-.

273

r

x

[o,T), w h i l e

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Index Accretive, 180, 219, 220, 222 A adapted, 70 Adjoint method, 70 Almost subharmonic function, 36, 38 Analytic d a t a , 266 Analytic semigroup, 180 Antidual, 179, 187 A p r i o r i estimate, 190 Associate resol van t 55 Backward Cauchy pro lem, 173 Backward parabolic, 203 Baire function, 150 Basic measure, 150 Basis elements, 99, 109, 115, 126, 138 Biinvariance, 94 Boundary, 91 , 106, 25 Boundary condition of t h i r d type , 226 Boundary condition of fourth type, 226 Bounded, 207 Calculus of v a r i a t i o n s o p e r a t o r , 210 Canonical recursion r e l a t i o n s , 97, 104, 129 Canonical sequence, 97, 101, 121, 129 Canonical t r i p l e , 105, 112 C a r r i e r space, 149 Cartan decomposition, 91, 105, 121, 124 Cartan involution, 105, 108 Cartan subalgebra, 106, 112, 121 Casimir o p e r a t o r , 97, 117, 121 Cauchy i n t e g r a l formula, 24 Cauchy Kowalewski theorem, 267 Center, 121 C e n t r a l i z e r , 91, 106, 124 Character, 93 C h a r a c t e r i s t i c , 200 C h a r a c t e r i s t i c conoid, 238, 244, 265 Coercive, 181, 202, 207 Compatible o p e r a t o r , 212, 213 Connection formulas, 29 Contiguity r e l a t i o n s , 133 Composition, 119 Darboux equation, 93, 119, 120, 140 Degenerate problem, i i i Demi mono tone , 21 0 Diagonalizable operator, 47, 150 329

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

D i f f e r e n t i a l i n c l u s i o n , 219, 229 D i f f u s i o n , 143, 145 D i r e c t i o n a l o s c i l l a t i o n s , 162 D i r i c h l e t c o n d i t i o n , 226 D i r i c h l e t problem, 255 D i s t i n g u i s h e d boundary, 24 Doubly degenerate, 228 Doubly n o n l i n e a r e q u a t i o n s , 177, 229 Eigenvalue e q u a t i o n , 97, 101, 253, 257, 259, 261 E i s e n s t e i n i n t e g r a l , 99 E l l i p t i c p a r a b o l i c problem, 199, 226 E l l i p t i c p a r a b o l i c Sobolev problem, 201 E l l i p t i c region, 5 Energy methods, 43, 164 Entrance boundary, 49 E n v e l o p i n g a l g e b r a , 97, 118, 121 E u c l i d e a n case, 136 E u l e r angles, 112 E u l e r Poisson Darboux (EPD) e q u a t i o n , 1, 4, 9, 67 E x c e p t i o n a l cases, 34 E x p o n e n t i a l t y p e , 29 F i n i t e p a r t ( P f ) , 249 Flow, 143, 146 F o u r i e r t r a n s f o r m , 9, 92 F r a c t i o n a l powers, 186 Fredholm a1 t e r n a t i v e , 259 Fundamental m a t r i x , 14 Gas dynamics, 5 GASPT, 8 G e l f a n d map, 149 G e n e r a l i z e d h y p e r g e o m e t r i c f u n c t i o n , 271 G e n e r a l i z e d r a d i a t i o n problem o f W e i n s t e i n , 245 Genetics, 145, 204 Geodesic, 107, 116, 19, 137, 140 G r a v i t y waves, 145 Green's m a t r i x , 151 Gronwall ' s 1emma, 53 Growth and c o n v e x i t y theorems, 35, 42, 78, 139 H a r i s h Chandra f u n c t on, 92, 108 Harmonic space, 119, 140 H a r t o g ' s theorem, 11 Hemicontinuous. 175. 177, 207, 223 Huygen s ' o p e r a t o r , 240 Huygens' p r i n c i p l e , 237 Huygens' t y p e o p e r a t o r , 238 H y p e r a c c r e t i v e , 231 Hyperbol ic r e g i on, 4 Hypergeometric e q u a t i o n , 134, 242 Hypocontinuous, 10, 13, 28, 30, 63, 159 Hypoelliptic, 9 330

INDEX

Improperly posed problems, 251 Index s h i f t i n g , 269 Induced represe n ta ti o n , 113, 137 I n t e g r a t i n g f a c t o r , 16, 49 I n t e r p o l a t i o n spaces, 186 I n v a r i a n t d i f f e r e n t i a l operators, 90, 94, 122 I n v a r i a n t in t e g r a t ion, 94 I s o t r o p y group, 93, 107 Iwasawa decomposition, 91 , 106, 124 J o i n t spectrum, 149 K i l l i n g form, 92, 105, 122 Kortweg deVries equation , 184 Kummer r e l a t i o n , 101, 129, 131 Laplace operator, 1, 97, 117, 120, 121, 138, 270 Laplace Belt ram i o p e ra to r, 117, 122, 140 Laplace transform, 270 L a u r i c e l l a f u n c t i o n , 243 L i g h t cone, 122, 238 L i m i t c i r c l e , 258 L i m i t p o i n t , 258 L i o n s ' p r o j e c t i o n theorem, 45, 165, 191 L i p s c h i t z c o n d i t i o n , 167, 170 Lobacevs k i j space, 122 , 128 Local l y equi continuous semi group, 66, 79 Long waves, 171 Lorentz distance, 243 L o rent z group, 122 Maximal a b e l i a n subspace, 91 , 106, 124 Maximal compact subgroup, 91, 122 Maximal i d e a l , 149 Maximal monotone , 208 Maximum p r i n c i p l e , 42 Mean value, 5, 8, 93, 97, 118, 119, 139, 237 Metaharmonic, 36 Metaparabol ic , 184 Measurable f a m i l y o f H i l b e r t spaces, 47, 150 Measurable f a m i l y o f p r o j e c t i o n s , 177 M e t r i c t ensor, 107, 140 Mixed problems, 5 Monotone, 177, 180, 206, 220 M u l t i v a l u e d f u n c t i o n , 217 M u l t i v a l u e d ope ra to r, 216, 230 Negative curvatu re , 119, 140 Neumann c o n d i t i o n , 226 Neumann problem, 258, 263 Neumann series, 16, 20, 49, 56, 239 Nonlinear EPD equations, 65, 273 Nonlinear semigroup, 216 Normalizer, 91, 106, 124 Operator o f type M y 213, 215 331

SINGULAR AND DEGENERATE CAUCHY PROBLEMS

O r b i t , 93 Osgood f u n c t i o n , 66 Paley Wiener Schwartz theorem, 30 Parabolic l i n e , 4 , 147 Parabolic subgroup, 93 Parseval Plancherel formula, 261 P e r i o d i c c o n d i t i o n , 209, 216 P i z z e t t i Zalcman formula, 96 Point spectrum 252 Polar c o o r d i n a t e s , 91, 95, 99, 113, 116, 137, 240 Pol a r deoomposi t i on , 105 P o s i t i v e measure, 36 P r i n c i p a l series, 93, 114 Protter c o n d i t i o n , 161 Pseudomonotone, 209, 212 Pseudo parabol i c , 143 Pseudosphere, 122 Q u a s i r e g u l a r r e p r e s e n t a t i o n , 93 Radial component, 95, 96, 97, 99, 100, 110, 246 Radial i n e r t i a , 175 Radiation, 245 Rank, 91 Recursion formulas, 12, 29, 33, 34, 70, 120, 129, 130, 132, 272 Regular family of s e s q u i l i n e a r forms, 188, 204 Regul a r i z e , 172 Regularly a c c r e t i v e , 186 Related d i f f e r e n t i a l equations, 2 , 81, 269 R e l a t i o n , 217 Resolvant, 10, 14, 29, 31, 48, 55, 67, 71, 96, 101, 111, 115, 128, 130, 132, 135, 151, 239, 241 Resolvant i n i t i a l c o n d i t i o n s , 10, 1 7 , 97, 110, 128, 132, 139 Riernann L i o u v i l l e i n t e g r a l , 74, 272 Riemannian structure, 97, 107, 118, 137 Riesz map, 174, 179, 180, 196, 217 Riesz o p e r a t o r s , 162 Right a n g l e c o n d i t i o n , 231 Root s p a c e s , 91, 106, 122, 124 Rotating vessel , 174 Scalar derivative , 7 Section o f a d i s t r i b u t i o n , 37 S e c t o r i a l o p e r a t o r , 180 Semi d i rect product, 136 Semi 1i n e a r degenerate e q u a t i o n s , 206 , 228 Semi 1i near s i n g u l a r problems, 65 Semimonotone, 209 Semisirnple Lie group, 91 S e p a r a t e l y continuous, 7 , 63, 159 Sequential completeness, 71 Simple convergence, 61 Simply continuous , 61 332

INDEX

Singular e l l i p t i c equation, 5 S i n g u l a r problem, iii Sobolev e q u a t i o n , i v , 143, 144, 174, 200, 227 Sobolev space, 224 Sonine i n t e g r a l f o r m u l a , 32, 34, 76, 117, 119, 270 S p e c t r a l decomposition, 47, 150 S p e c t r a l techniques, 47, 147, 162 S p l i t t i n g r e l a t i o n s , 101, 104, 130, 132 S t a t i o n a r y problem, 214 S t r i c t l y monotone, 209, 214 Strong d e r i v a t i v e , 7 S t r o n g l y c o n t i n u o u s , 61, 168 S t r o n g l y monotone, 176, 229 S t r o n g l y r e g u l a r , 147, 166 S u b d i f f e r e n t i a l , 232, 234, 235 Subgradient, 232 Symmetric space, 91 Test function, 5 Topology E, 69 Topology T , 268 Transmutation, 2, 80, 248 Transverse d e r i v a t i v e , 260 Tricomi equation, 4 Type A s o l u t i o n , 268 Type B s o l u t i o n , 268 U n i f o r m l y w e l l posed, 28 Unimodular, 94 Unimodular q u a s i u n i t a r y m a t r i c e s , 112 Value o f a d i s t r i b u t i o n , 37 V a r i a b l e domains, 175, 177, 192, 194 V a r i a t i o n a l boundary c o n d i t i o n s , 196 V a r i a t i o n a l i n e q u a l i t y , 175, 178, 209, 211 V e l l i p t f c , 182 V i b r a t i o n s , 175 V i s c o e l a s t i c i t y , 185, 227 Von Neumann a l g e b r a , 150 Wave equation, 1 , 42, 146 Weak problem, 44, 47, 161, 164 Weakly c o n t i n u o u s o p e r a t o r , 213 Weakly r e g u l a r , 147, 179 W e i n s t e i n complex, 77 Weyl chamber, 91 Weyl group, 92, 106, 125 W i n t e r f u n c t i o n , 66 Yosida a p p r o x i m a t i o n , 173 Zonal s p h e r i c a l f u n c t i o n , 94 A 0 C D

6 7 8 9

F 6 H 1 1

1 2 3 4 5

E O

333

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