Second Order Linear Differential Equations in Banach Spaces

SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS IN BANACH SPACES

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (99)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester

108

SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS IN BANACH SPACES

H. 0.FAlTORINI University of California at Los Angeles USA.

1985

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

OElsevier Science Publishers B.V., 1985 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87698 7

Publishers:

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Sole distributors for the U.S.A. and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC 52 VAN DERB ILT AVE NUE NEW YORK, N.Y. 10017 U.S.A.

Library of Congress Cataloging in Publication Data

Fattorini, H. 0. (Hector O.), 1938Second order l i n e a r d i f f e r e n t i a l equations i n W a c h spaces. (North-Holland mrthematics s t u d i e s ; 108) (Notas de m a t d t i c a ; 99) Bibliography: p. 1. Differential equations, P a r t i a l . 2. Differential equations, Linear. 3. Banach spaces. I. Title. 11. Series. 111. Series: lVDtas de m a t d t i c a (Amsterdam, Netherlands) ; 99. w . 1 8 6 no. 108 510 s ~515.3'541 84-28658

wav1

ISBn 0-444-87698-7

PRINTED IN THE NETHERLANDS

V

PREFACE

An initial value or initial-boundary value problem u t = Au, u

=

uo

for

(1)

t = 0

A

is a partial differential operator in the space variables x l ' . . . can be recast in the form of an ordinary differential initial value

where

x rn problem

~ ( 0 =) u0'

u'(t) = Au(t), where

A

(2)

is thought of as an operator in a function space E

and the

boundary conditions, if any, are included in the definition of the space E or of the domain of A . I f

E is suitably chosen, solutions of (2) will

exist for sufficiently many initial data on

uo

in the norm of

uo

and will depend continuously

E. This yay of looking at (1) was initiated by Hille

and Yosida in the forties and resulted in the creation and development of semigroup theory, now an integral part of most advanced treatments of parabolic and hyperbolic partial differential equations. A second order initial value or initial-boundary value problem

utt = Au, u = uo, u

=

u1

for

(3)

t = 0

can be reduced in the same way to an ordinary differential initial value problem ~"(t) = Au(t), where

A

~(0) = u0, u'(0)

=

u

(4)

1

is defined as in (2). This, however, can often be avoided reducing

(3) to a first order system following the "take the derivative as a new function" rule one learns in elementary theory of partial differential equations. That this trick always works, at least if one measures the derivative in a new norm, is in fact one of the results in Chapter I11 of this book. Moreover, the choice of this norm is usually natural and has physical meaning. However, reduction to first order is of no particular help in a problem as elementary as the growth of solutions of u"(t)

=

(A

+

cI)u(t)

PREFACE

vi

in terms of the growth of the solutions of

u"(t)

=

Au(t).

In other

problems, such as singular perturbation, direct consideration of second order equations leads to simpler and more inclusive theories. Finally, the formalism associated with ( 4 ) has proven useful in other fields, such as the control theory of hyperbolic equations. These and other reasons give motivation to the development of a theory of second order differential equations in Banach spaces. This work presents a few facts on that theory and some applications.

NO claim of completeness is made, either in the text or in the references; many important results have been left out and many important papers are not mentioned. Chapter I expounds semigroup theory; Chapter I1 presents cosine function theory, which stands in relation to the second order equation ( 4 ) as semigroup theory stands in relation to the first order equation (2). Chapter I11 deals with the reduction of ( 4 ) to a first order system mentioned above and other related topics. The next four chapters are on applications; in Chapter IV we treat the initial-boundary value problem (3) with

A a second order uniformly elliptic partial differential operator in a domain of m-dimensional Euclidean space, with either the Dirichlet boundary condition or a variational boundary condition. Chapter V treats the second order equation ( 4 ) in Hilbert spaces, where many special results are available; there are applications to equations with almost periodic and periodic solutions. Chapters VI and VII are on singular perturbation problems, with applications to diverse physical situations. Finally, in Chapter VIII we touch upon the theory of the "ctmplete" second order equation u"(t)

t Bu'(t)

+

Au(t)

=

0

(5)

without going too far into it; mostly, we search for the correct definition of correctly posed initial value problem for (5). Some shortcuts through the book are possible, and we do not bother to indicate them explicitly; for instance, Chapter 1 1 1 is only briefly needed in Chapters IV and V and not used at all in Chapters VI and VII. Some effort has been made to make this book as self-contained as possible; nothing isneededexcept the elementary theory of Banach and Hilbert spaces and some acquaintance with parabolic partial differential equations. The functional calculus for self adjoint operators is only used in Chapters IV and V and in exercises in other chapters. The exercises throughout the book cover parts of the theory not in the text or related facts of interest; references are included for the less

PREFACE

vii

immediate.

I am glad to acknowledge my thanks to many colleagues who read parts of the book and suggested improvements and to the Instituto Argentino de

MatemAtica, Consejo Nacional de Investigaciones Cientificas y TQcnicas, Argentina, for its hospitality during March I983 and August 1984, at which time the actual writing was concluded. Finally, and most important of a l l , the undertaking of this project would have been impossible without the understanding support of the National Science Foundation, which support extended during the entire time it t o o k to complete it. As always, my wife Natalia was encouraging, patient and understanding and to her go my very special thanks.

Buenos Aires, August 1984

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ix

CONTENTS

PREFACE. LIST OF SYMBOLS. CHAPTER I.

............................................. ..........................................

V

xiii

THE CAUCHY PROBLEM FOR FIRST ORDER EQUATIONS. SEMIGROUP THEORY

....... .................... ..........................

51.1

The Cauchy problem for first order equations.

1

51.2

The Cauchy problem in

6

(-m,m).

51.3 The Hille-Yosida theorem. 51.4 51.5 51.6

.................................. The inhomogeneous equation. ........................ Miscellaneous comments. ............................ Semigroup theory.

7 13

15 18

CHAPTER 11. THE CAUCHY PROBLEM FOR SECOND ORDER EQUATIONS. COSINE FUNCTION THEORY

311.1

The Cauchy problem for second order equations.

511.2 The generation theorem.

511.3 Cosine function theory.

........................... ...........................

....................... .............. ...........................

511.4

The inhomogeneous equation.

511.5

Estimations by hyperbolic functions.

511.6

Miscellaneous comments.

CHAPTER 111.

....

24 28 32 35

37 38

REDUCTION OF A SECOND ORDER EQUATION TO A

FIRST ORDER SYSTEM. PHASE SPACES. 5111.1 Phase spaces.

....................................

5111.4

........... Resolvents of fractional powers. ................. Translation of generators of cosine functions. ...

5111.5

The principal value square root reduction.

43

5111.2 Fractional powers of closed operators.

50

5111.3

56

5111.6 9111.7 5111.8

....... Second order equations in Lp spaces. ............. Analyticity properties of bb(t). ................. Other square root reductions. ....................

59 62 71

80 86

CONTENTS

X

5111.9

Miscellaneous comments.

95

. * . . . . . . . . . . . . . . . . . I . . . .

CHAPTER IV. APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS 5IV.l Wave equations: the Dirichlet boundary condition. sIV.2 5IV.3

5IV.4 5IV.5 sIV.6 sIV.7

SIV.8 sIV.9

CHAPTER V.

. The phase space. ................................ The Cauchy problem. ............................. Wave equations: other boundary conditions. ...... The phase space. ................................ The Cauchy problem. ............................. Higher order equations. ......................... Higher order equations (continuation). .......... Miscellaneous comments. .........................

100

104 109 112 116 117

118

120 124

UNIFORMLY BOUNDED GROUPS AND COSINE FUNCTIONS

IN HILBERT SPACE

...... ... Uniformly bounded groups in Hilbert space. ...... Almost periodic functions. ...................... Almost periodic groups in Hilbert space. ........ Banach integrals. ...............................

133

5V.6 Uniformly bounded cosine functions in Hilbert space.

145

sv. 1 g.2

5v.3 sv.4

5v.5 sv.7

The Hahn-Banach theorem: Banach limits.

Almost periodic cosine functions in Hilbert space...

.........................

SV.8 Miscellaneous comments.

126 128

138 142 153 158

CHAPTER VI. THE PARABOLIC SINGULAR PERTURBATION PROBLEM sVI.1

Vibrations of a membrane in a viscous medium.

sVI.2

Singular perturbation. Explicit solution of the perturbed equation.

5VI.3 sVI.4 sVI.5 5VI.6

............................

The homogeneous equation: convergence of u(t;E) Convergence of u'(t;E)

..

165 166

....

and higher derivatives...

...

The homogeneous equation. Rates of convergence.....

171 180 192

Singular integrals of liilbert space valued functions and applications to inhomogeneous first order equations.

.....................................

202

5VI.7 The inhomogeneous equation: convergence of u(t;E) and u'(t;E).

...................................

210

sVI.8 Correctors at the initial layer. Asymptotic series. 5VI.9 Elliptic differential operators.

218

sVI.10 Miscellaneous comments.

233

............... ........................

228

xi

CONTENTS CHAPTER VII. OTHER SINGULAR PERTURBATION PROBLEMS sVIT.1

A singular perturbation problem in quantum

mechanics.

.....................................

sVII.2 The Schrodinger singular perturbation problem..

sVII.3 sVTI.4 sVII.5 sVII.6 sVII.7 sVII.8

Assumptions on the initial value problem.

....

238 239

......

24 1

......

245

The homogeneous equation: convergence results

................ Elliptic differential operators. ............... The inhomogeneous equation. .................... Miscellaneous comments. ........................ Verification of the hypotheses.

250 258 262 265

CHAPTER VIIT.

THE COMPLETE SECOND ORDER EQUATION sVIII.1 The Cauchy problem. ........................... sVIII.2

Growth of solutions and existence of phase spaces.

270 271

sVIII.3 Exponential growth of solutions and existence of

................................. ................. sVIII.5 Miscellaneous comments. ....................... BIBLIOGRAPHY. ................................................. phase spaces.

§VIII.4 Construction of phase spaces.

276 289 298 303

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xiii

LIST OF SYMBOLS

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1

CHAPTER I

THE CAUCHY PROBLEM FOR FIRST ORDER EQUATIONS. SEMIGROUP THEORY

$1.1 The Cauchy problem for f i r s t order equations.

E

We denote by with domain

D(A)

be complex and

a general Banach space and by

E

in

D(A)

Unless otherwise s t a t e d

w i l l be dense i n

E.

We shall use i n t h e s e q u e l t h e

t o i n d i c a t e functions

t h e i r i n d i v i d u a l values a t with values i n t h e i i m i t as

-t

t

are

u(t), f ( t ) ,

0

t h e l i m i t i s one-sided a t [0,

-t

m).

t = 0.

t

will

E

u f t ) , t + f ( t ) , ...;

etc.

of t h e quotient of increments

e x i s t s and i s a continuous function of other than

t

i s continuously d i f f e r e n t i a b l e i n

E

h

a l i n e a r operator

E.

...

symbols u(^t),,);('f

A

and range i n

u(t)

A function

t 2 0

h-l(u(t

i n t h e norm of

i f and only if

+ E;

h)

-

u(t))

of course,

Similar d e f i n i t i o n s a r e used i n i n t e r v a l s

A (stron_g or genuine) s o l u t i o n of t h e a b s t r a c t d i f f e r -

e n t i a l equation ~ ' ( t =) Au(t) in

[0,

that

i s a continuously d i f f e r e n t i a b l e E-valued f u n c t i o n u ( t )

m)

u(t)

(1.1)

E

D(A)

and (1.1) i s s a t i s f i e d f o r

t >_ 0.

such

Solutions a r e

correspondingly defined i n other i n t e r v a l s . The Cauchy o r i n i t i a l value problem f o r (1.1)i n

t >_ 0

is that of

f i n d i n g s o l u t i o n s s a t i s f y i n g t h e i n i t i a l condition u ( 0 ) = uo

.

The Cauchy problem f o r (1.1)i s w e l l posed ( o r properly posed) i n

t? 0

i f and only i f t h e following two assumptions hold:

(a) f o r any

-fying

(Existence). uo

(1.2).

E

D

There e x i s t s a dense subspace

t h e r e e x i s t s a solution

u(t^)

of

D

(1.1)

of

E

such t h a t

t r_ 0 s a t i s -

2

FIRST ORDER EQUATIONS

(b)

There e x i s t s a nonnegative, f i n i t e

(Continuous dependence).

function

t

defined i n

C(t^)

10

such t h a t

Ilu(t)': 5 c(t)llu(o)ll f o r any s o l u t i o n of (1.1).

(1.3) i s assumed t o hold f o r

Note that o r not

u(0)

Also, s i n c e t h e equation must hold i n p a r t i c u l a r f o r

D.

E

s o l u t i o n of (l.l), whether

t = 0 we must have D

.

5 D(A)

(1.4)

F i n a l l y , t h e Cauchy problem f o r (1.1)i s uniformly w e l l posed (or

t 5 0

uniformly properly posed) i n

t5 0

C(f)

i s nondecreasing i n

C($)

i s bounded on compacts of

E(t)

=

sup {C(s); 0 <_ s

5

i f ( a ) and ( b ) hold and t h e function

(obviously, it i s enough t o assume t h a t

t 2 0

s i n c e we may r e p l a c e it by

t).

We n o t e t h a t condition ( b ) i n t h e w e l l posed c a s e i s equivalent t o pointwise continuous dependence of t h e s o l u t i o n s on t h e i r i n i t i a l values, s i n c e no r e l a t i o n between d i f f e r e n t values of

i s postulated.

C(z)

In

the uniformly w e l l posed case, t h e dependence i s continuous on bounded s u b i n t e r v a l s of 0

5 t <_

T <

Let

u

t >_ 0

due t o t h e f a c t t h a t

C(t)

-. D,

E

t 1 0 .

5

C(T)

in

Define S(t)u = u(t) ,

where

u(<)

i s t h e s o l u t i o n of (1.1)w i t h u ( 0 ) = u.

i s a bounded operator w i t h in

E,

(1.5)

S(t)

ilS(t)l:

5

C(t).

Since

can be extended t o a bounded operator i n

denote by t h e same symbol) without i n c r e a s e of norm. function

tion or

S($)

Obviously,

D(S(t)) = D E

S(t)

i s dense

(which we

We c a l l t h e operator

t h e propagator ( o r s o l u t i o n operator or fundamental soluh

evolution operator) of (1.1). If

u(t)

i s an a r b i t r a r y s o l u t i o n

of (1.1)t h e n U(t)

= S(t)u(O)

This i s nothing but t h e d e f i n i t i o n of

S(t)

(t

2

when

0)

. u(0)

(1.6) E

D.

To prove

3

FIRST ORDER EQUATIONS

(1.6) i n t h e g e n e r a l c a s e , l e t

[u

1

be a sequence of elements i n

un (^t) S(t)u(O) = l i m S ( t ) u n = l i m u n ( t ) = u ( t )

such t h a t Then

un

-+

t h e s o l u t i o n of (1.1)w i t h

u(O),

D

un(0) = un.

by ( b ) .

THEOREM 1.1. Let t h e Cauchy problem f o r (1.1)be w e l l posed i n

hen

t 5 0.

(i)

s(s +

S(0) = I,

(ii) There e x i s t s a c o n s t a n t

~

with

depending on

C6

(iii)

t

5

s(;)

0

5

Co

u 0

and l e t

u(e)

from (1.6). dense i n

[0,

we have

o

( s t r o n g l y continuous i n

To prove t h e second

u(0) = S ( t ) u

we deduce t h a t

Since

u ( 0 ) = u.

Fix

is a

u(2)

u(s) = S(s)S(t)u

Noting t h a t a l l o p e r a t o r s involved are bounded and

D

is

t h e second e q u a l i t y (1.7) f o l l o w s .

E,

uo

E

E,

{un)

a sequence i n

D

with

Since

t 1 0,

pointwise i n

of

(1.7)

i s uniformly w e l l posed).

u ( g ) = S(s^ + t ) u .

S(t>un

S(;)uo

.

6 z 0

be t h e s o l u t i o n of (1.1)with

We prove n e x t (iii). Let

I+,.

t >

(1.7) i s obvious.

and consider t h e f u n c t i o n

s o l u t i o n of (1.1)with

un +

t2 0

The f i r s t e q u a l i t y

D

E

( s , t >_ 0)

such t h a t

i s s t r o n g l y continuous i n

Proof.

t

sfs)s(t)

such that, f o r every

w

i f t h e Cauchy problem f o r

pick

=

( i f t h e Cauchy problem f o r (1.1) i s uniformly

6

w e l l posed t h e r e e x i s t s

t)

-+

(1.lo)

S(t)u()

S(^t)uo i s s t r o n g l y measurable t h e r e .

Accordingly,

i s almost s e p a r a b l y valued, t h a t i s , t h e r e e x i s t s a n u l l s u b s e t m)

such that

Xo = { S ( t ) u o ; t d, do)

i s s e p a r a b l e ( f o r t h i s and

o t h e r r e s u l t s on measurable v e c t o r valued f u n c t i o n s s e e for i n s t a n c e HILLE-WILLIE generated by

[1957:1,Ch. 1111. ru01 IJ Xo

It follows t h a t t h e c l o s e d subspace

is s e p a r a b l e and t h e r e e x i s t s a sequence

f t n ;n 2 l’j contained i n t h e complement of do such that t h e s e t (uol (J ( S ( t n ) u O ; n 2 11 i s fundamental i n Eo ( f i n i t e l i n e a r

Yo =

Eo

FIRST ORDER EQUATIONS

4

Yo a r e dense i n E 0 ). L e t now t k do ( n = 1,2, ...); then S ( t ) u o and S ( t ) S ( t n ) u O =

combinations of elements of

t + tn )! do

that

S ( t + tn)uO belong t o eo

Eo.

-

eo = do U (do

Define

t l ) U (do

-

such

t2) U

...;

i s a n u l l s e t and it follows from t h e preceding arguments that

c_

S(t)Eo

0 < Cr

We show next t h a t i f

(t p

Eo

-

there exists

5c

!lS(t)II

-

eo)

(1.11)

(a <_ t 5 B)

(t

(1.12)

*

If t h i s were not t h e case we could f i n d a sequence

and a numerical sequence

such t h a t

C = C

iiunll = 1 Fun> C_ E, IlS(t n)unll 5 n

[a,f31 such t h a t

in

( n = 1,2,...). Applying t h e argument leading t o (1.11)we can construct f o r each

c_

En

n

a null set

#

t

for

union of t h e

en;

E

m(t)

= IlS(t)II,

Em

em = el U e2 U

SUP

.. ..

m(t,)

> n and i f

#

t

S(t)En

S(t)

t

i n the

>0

denote by

Em;

since

IIS(:)vnil

with

t

Given

i s t h e supremum of t h e sequence

m(Z)

a dense sequence i n t h e unit sphere of Moreover,

S ( t ) E m E Em for

t h e norm of t h e r e s t r i c t i o n of

i s separable,

with

i s t h e closed subspace generated by t h e

i s separable and

complement of t h e n u l l s e t

En

and a separable subspace

hence i f Em

n’

en

to

El, {vn]

Em and is i t s e l f measurable.

em we have

m(s

+ t)

=

u E Em, hdl 5 11 5 SUP f lb(s>vli; v E Em, b l l < _ m ( t > l Accordingly, a c o n t r a d i c t i o n r e s u l t s from:

1/s(s)S(t)uli;

< - m(s)m(t).

LEW 1.2.

t

defined i n

Let

1 0

m(<)

and such that

m(s + t ) where

e

be a nonnegative f i n i t e measurable function

5

is a null set i n

interval

[a,p ] ,

o

( s > 0, t ,d e ) ,

m(s)m(t) (0,

c a< B <

m).

Then

m(Z)

(1.13)

i s bounded i n every

m.

+ t

)! e ,

(1.13) implies m(s)m(t) 5 m(a), thus e i t h e r m ( s ) 5or m ( t ) 2Hence if d is t h e s e t of a l l t i n [ 0 , a ] with m(t) >_ we have eU d U ( a - d) 2 [ O , a ] s o t h a t Id] + ] a - dl >_ a , where 1.1 indic a t e s Lebesgue measure. But Id1 = la - d l , hence Id1 >_ a/2. Assume now t h a t m($) is unbounded i n [ a , f31, so that t h e r e e x i s t s Proof.

Let

a sequence

[anl

a > 0.

If

t h e r e with

s

= a, t

m(an) +

m.

.

Applying t h e argument above we

5

FIRST ORDER EQUATIONS

deduce t h e existence of a measurable set

dn

[0, p ]

in

with

.Jm(an)

i n dn, which c o n t r a d i c t s t h e a/2 and m ( t ) 2 /dnl 1 an/2 f a c t t h a t m(t) i s everywhere f i n i t e . This completes t h e proof of

Lemma 1.2. End of proof of Theorem 1.1. Let 0

I h / <_ r , ( S ( t O + h)

in

5

0

< CY <

-

< to

-

0

to > 0,

E,

E

0 < r < to,

We obtain from ( 1 . 7 ) t h e e q u a l i t y

r.

-

S ( t O ) ) u = S ( t ) ( S ( t O+ h

t <_ B.

bounded i n

@

u t)

-

S(tO

-

valid a t least

t))u,

I n view of (1.12) t h e f u n c t i o n on t h e r i g h t hand s i d e i s

5 t 5 B;

it i s a l s o e a s i l y seen t o be s t r o n g l y measurable.

I n t e g r a t i n g we o b t a i n

-

(B

a ) l l S ( t O+ h ) u -S(tduII

5

rB

C

ilS(to + h

-

t)u

-

S(tO

-

t ) u l l d t , (1.14)

,J cy

where t h e r i g h t hand s i d e tends t o zero i n view of t h e mean c o n t i n u i t y of Bochner i n t e g r a b l e functions (HILLE-PHILLIPS [1957:1, p. 861). t h e n completed t h e proof of ( i i i ) in t h e g e n e r a l case.

We have

If t h e Cauchy

problem f o r (1.1) i s uniformly well posed t h e convergence i n ( 1 . 1 0 ) i s uniform on f i n i t e s u b i n t e r v a l s of

[O,

thus

m)

i s continuous i n

C(:)u

t 2 0. F i n a l l y , we show (ii). Observe f i r s t t h a t , by v i r t u e of (iii) f o r each

u

E

E and

S > 0

the function

S(i)u

i s continuous, hence bounded

in

6 5 t <_ 1; it follows from t h e uniform boundedness p r i n c i p l e (DUNFORD-SCHWARTZ [1958:1, p. 521) t h a t Ils(t)II 5 C6 in 6 5 t 5 1. Let

t 2 1/2

be a r b i t r a r y ,

we have

S ( t ) = ~ ( n / e ) s ( t- n/2> = ~ ( 1 / 2 ) " ~ ( t n/2)

t h e l a r g e s t i n t e g e r with

n/2

-

llS(t)ll <_ C exp (n log C ) in

n

5

C exp ( W t )

with

<_ t . Using (1.7)

s o that i f

c

=

c1/2 '

which i s (1.8)

w = 2 log C,

t 2_ 1/2; obviously, t h e i n e q u a l i t y can be extended t o t i 6

( p o s s i b l y with a d i f f e r e n t constant) o r t o is uniformly w e l l posed.

t 2 0 when t h e Cauchy problem

This completes t h e demonstration of Theorem 1.1.

We comment b r i e f l y on s e v e r a l obvious consequences. case, t h e f u n c t i o n that if

{un(;)],

C(t")

u(;)

in (1.3) can be replaced by a r e s o l u t i o n s of (1.1)and

exp ( - w t ) u n ( t ) + exp ( - u t ) u ( t )

uniformly i n

problem f o r (1.1)i s simply w e l l posed, uniformly i n

t 2 6

f o r each

6 > 0.

bargained f o r i n t h e d e f i n i t i o n .

I n t h e uniform

Co exp ( & )

un(0)

t 20.

exp ( - u t ) u ( t )

4

u(0)

so then

When t h e Cauchy +

exp ( - & ) u ( t )

This is considerably more t h a n we

6

FIRST ORDER EQUATIONS

The Cauchy problem i n

$1.2

my.

(-m,

We d e c l a r e t h e Cauchy problem f o r (1.1)w e l l posed i n

-a

< t <

m

If ( a ) and ( b ) i n t h e previous s e c t i o n hold with t h e following modifications:

t h e solutions i n ( a ) a r e solutions i n

t 10,

and

t.

for a l l

(1.3) holds i n If

(-m,

and

C($

uniformly w e l l posed i n

t.

be defined f o r a l l

(-m,

(-m,

m)

r a t h e r than i n

t 2 0

(or

t h e Cauchy problem f o r (1.1) i s

m))

Plainly, t h e propagator

m).

defined

C(t)

a r e both nondecreasing i n

C(-;)

j u s t bounded on compacts of

(-m,

with a f i n i t e function

m)

S(t)

can now

It i s remarkable t h a t t h e notions of w e l l posed

and uniformly w e l l posed problem coalesce i n t h e p r e s e n t s i t u a t i o n , a s t h e next r e s u l t shows. THEOREM 2.1. (-03,

m).

Then

Let t h e Cauchy problem f o r (1.1)be well posed i n (i)

(ii) There e x i s t constants

Co,

w

such that

( t h u s t h e Cauchy problem f o r (1.1)i s uniformly w e l l posed i n (iii)

s(;>

a ) ) .

(-m,

i s s t r o n g l y continuous f o r a l l t. t ? 0.

The proof of (2.1) is t h e same a s that f o r t h e case

To show

(iii)we note that t h e assumptions imply that t h e Cauchy problems f o r

both (1.1)and U ' ( t ) = -Au(t) a r e w e l l posed i n ' t

2

0,

(2.3)

u(t^)

s i n c e t h e correspondence

~ ( - 2 ) maps

-t

s o l u t i o n s of (1.1) i n t o s o l u t i o n s of (2.3) and v i c e versa. be t h e propagator of (2.3) i n

t 2 0.

s (t)= Applying Theorem l . l ( i i i ) t o h

we deduce t h a t

S(t)

S(t^)

kt

~-(t)

We v e r i f y e a s i l y t h a t

S(-t)

and t o

(t 2 0)

.

is s t r o n g l y continuous i n t < 0

t > 0

in

S ( - c ) = S-(t^)

and

t > 0.

However,

S(h)u = S ( t + h)S(-t)u,

follows.

In p a r t i c u l a r , t h e Cauchy problems f o r (1.1)and ( 2 . 3 ) a r e

thus s t r o n g c o n t i n u i t y a t

t

= 0

FIRST ORDER EQUATIONS

7

t L 0 which y i e l d s ( 2 . 2 ) by i n t e r v e n t i o n o f Obviously, t h i s i n e q u a l i t y implies t h a t t h e Cauchy

uniformly w e l l posed i n Theorem l . l ( i i ) .

problem i s w e l l posed i n

(-m,

thus completing t h e proof of

m),

Theorem 2.1.

$1.3

The Hille-Yosida theorem. Assume that t h e Cauchy problem f o r (1.1) i s w e l l posed i n

that

is closed.

A

number with

Re

X>

Let

w

R(X)u =

(1.9), X

be t h e constant i n

E

Define an operator i n

w.

[

e

-Xt

S(t)u d t

and

t 2 0

a complex

by

(u

E

.

E)

(3.1)

I

Since t h e norm of t h e integrand is bounded by

ColIu/I exp ( w

is a bounded operator i n

u

Assume now t h a t

E.

s o l u t i o n of (l.l), s o t h a t

Then

D.

E

-

Re X)t, R ( X ) h

S(t)u

is a

s l ( t ) u = AS(t)u. (l) Using a w e l l known r e s u l t

on i n t r o d u c t i o n of closed operators under t h e i n t e g r a l s i g n (HILLE-PHILLIPS

[1957:1,p. 831) we obtain A

LT

-

e-‘%(t)u

dt =

L~

.,

= e -XT S(T)u

for

T > 0.

R(X)u

E

ment of

Letting

D(A)

E,

and

T

-t

00

e-XtAS(t)u d t = JOT e-XtSt ( t ) u d t

-

u

+ X L T

e-XtS(t)u d t

and using closedness of

AR(X)u = XR(X)u

choose a sequence

- u.

{unj

-

-

A ) R ( X )= I

I n p a r t i c u l a r , (3.2) shows t h a t

w e l l one-to-one. Then

(XI

-

-u

(XI

-

X

E

p(A),

A)-l = R(X), The f a c t t h a t

p(A)

A):D(A)

t h e resolvent s e t of p(A)

u. Then

-t

i s onto.

E

u

A

E D(A)

It is a s with

Au = Xu.

/lu(t)II =

u = 0.

W e have then

and t h a t

contains t h e half plane

R(X;A) = Re

X>

w.

i s nonempty allows f o r t h e complete i d e n t i f i c a -

t i o n of t h e subspace Do of “admissible i n i t i a l d a t a ” c o n s i s t i n g of a l l uo such t h a t X€p(A),

S($)uo

and

R(X)E C_ D ( A )

(3.2)

i s a s o l u t i o n of (1.1)with

s o that

-t

.

(exp (Re X)t)(/ull which c o n t r a d i c t s (1.9) u n l e s s proved t h a t

un

so that

To s e e t h i s assume t h e r e e x i s t s

u ( t ) = ( e m (Xt))u

i s an a r b i t r a r y e l e -

such t h a t

D

R(X)un -+ R(X)u, AR(X)un= XR(X)un u n + XR(X)u (XI

u

Now, i f

in

we deduce that

A

is a s o l u t i o n of (1.1). This is done a s follows.

E

E

If

R(X;A)S(t)u is a s o l u t i o n of (l.l), h e n c e i t f o l l o w s f r o m ( 1 . 6 ) t h a t

a

FIRST ORDER EQUATIONS

(t > 0)

R(X;A)S(t)u = S(t)R(X;A)u

u

This e q u a l i t y extends t o a l l

E by denseness of

E

(3.3) f o r u

Write

(XI

=

-

v E D(A)

A)v,

D

(3.3)

and implies t h a t

.

( t 2 0)

S(t)D(A) C_ D(A)

.

(3.4) (XI

and then apply

-

A)

to

The r e s u l t is

both s i d e s ,

AS(t)u = S(t)Au

u

valid f o r

E

Jb

S(t u - u =

for

u

E

D.

Apply

(3.5 we obtain

Making use of

D(A).

R(X;A)

t

t S'(s)u

"

t o both s i d e s and u s e

R(X;A)(S(t)u

u

E;

E

(3.3):

the r e s u l t i s

t

-

U)

S(S)AR(X;A)U

=

Since both s i d e s a r e bounded operators of to all

S(s)Au ds

ds =

u

.

ds

t h e e q u a l i t y can be extended

using (3.3) again and applying

(XI

-

t o both s i d e s ,

A)

we obtain

s(t)u

-

u

=

rt

AS(s)u ds

(3.6)

JO

for a l l

u

E

D(A).

(1.1)f o r any

u

Obviously, t h i s implies t h a t E

A,

D(, = D(A) We can obtain estimates for R ( X ; A ) formula

= (-l)nn!R(X;A)nfl

R(X;A)(II)

S(;)u

i s a s o l u t i o n of

hence

.

(3.7)

and i t s powers using t h e well known and d i f f e r e n t i a t i n g

(3.1) under

t h e i n t e g r a l s i g n ( t h i s can be e a s i l y j u s t i f i e d on t h e b a s i s of t h e dominated convergence theorem).

results for

where ities

u

E

E

and

Re

X>

The formula

0.

Using (1.9) we obtain

Co and w a r e t h e constants i n (1.9). We show next t h a t inequal(3.9) a r e as well s u f f i c i e n t f o r uniform well posedness of t h e

9

FIRST ORDER EQUATIONS

Cauchy problem for (1.1).

3A

THEOREM 3 . 1 (Hille-Yosida).

f o r (1.1)i s 5

and

w

The Cauchy problem

t 2 0 with propagator S ( t ) u(A) i s contained i n t h e half-space

uniformly w e l l posed i n

s a t i s f y i n g ( 1 . 9 ) i f and only i f Re X

b e closed.

R(X;A)

(3.9).

s a t i s f i e s inequalities

The proof of t h e other h a l f below

We have a l r e a d y proved one h a l f .

i s perhags not t h e s h o r t e s t a v a i l a b l e but can be adapted with minor changes t o equations o t h e r than (1.1) ( s e e Chapter 11). We begin by constructing s o l u t i o n s of (1.1)given s u f f i c i e n t l y "smooth" i n i t i a l conditions.

u

E

D(A 3 ),

> w, 0.

w'

Let

Define ( 2 )

N

u(t;u) = u

+

tAu

t2 2 +A u 2 w'+im

+

(3.10)

eXt

Ju'-im

&i

3u ax - R(X;A)A

( t 5 0)

,3

.

A deformation of contour shows t h a t t h e i n t e g r a l i n (3.10) vanishes for

t

5 .O,

thus N

u(0,u) = u ~ ~ ~ ( t=; O u ()e q ~ (~w ' t ) )

Obviously,

as

t

.

(3.11)

+

m.

An e a s i l y j u s t i f i a b l e

d i f f e r e n t i a t i o n under t h e i n t e g r a l s i g n shows that continuous d e r i v a t i v e in

t > 0;

;(E;u)

on t h e o t h e r hand

A

has a can be introduced

under t h e i n t e g r a l s i g n with convergence of t h e r e s u l t i n g i n t e g r a l , s o that

<(t;u)

E

D(A)

"U(t;u)

and

-

AE(t;u) =

+

&1

-t2 A 3u 2 w'+im X t

w'-im

= o Accordingly,

u"(t;u)

-

(XI

,3

-

A)R(X;A)A3u dA

( t 2 0 ) .

is a s o l u t i o n of (1.1). A n argument s i m i l a r t o t h e

one leading t o t h e proof t h a t

R(X)

in

(3.1) coincides with R(X;A)

(or

a d i r e c t i n t e g r a t i o n of ( 3 . 1 0 ) ) shows t h a t

L

e

u ( t ; u ) d t = R(X;A)u

This e q u a l i t y w i l l be used t o obtain bounds on inversion formula below.

u(-)

.

(3.13) by use of t h e

Post

FIRST ORDER EQUATIONS

10

Let

LEMMA 3.2.

t >_ 0

E-valued continuous f u n c t i o n defined i n

f ( t ) = O ( e q (at))

such that

(Sf)(:)

f(?)

b e t h e Laplace transform of

as

t

+

f o r some

w

CY

and l e t

t,

(Sf)(X) =

e-Xtf(t)

dt

.

Then

t > 0.

uniformly on compacts of

The proof i n t h e s c a l a r c a s e can b e found i n WIDDER [1946:1, Chapter V I I ] .

The extension t o v e c t o r valued functions i s immediate.

End of proof of Theorem 3.1.

We apply t h e inversion formula (3.14)

in (3.10); i n view of (3.13) we obtain

t o t h e function

(3.15)

s o t h a t , u s i n g i n e q u a l i t i e s (3.9) w e o b t a i n IllY(t;u)ll

5 Collu/I lim

n

Define N

N

S(t)u = u(t;u) for

u

E

ment of

D ( A 3 ),

u(t)

s(%) we

operator such t h a t

(t

0)

t h e solution i n (3.10).

can extend each E ( t )

g($)

.

Arguing as i n t h e t r e a t -

t o a bounded everywhere defined

i s s t r o n g l y continuous i n

t 2 0

and

C l e a r l y , t h e proof of Theorem 3 . 1 w i l l be complete i f we show t h a t any solution

u(t)

of (1.1)admits t h e r e p r e s e n t a t i o n U(t) = iqt)U(O)

.

(3.18)

11

FIRST ORDER EQUATIONS

This is done as follows.

It r e s u l t s from t h e d e f i n i t i o n of for all

R(X;A)S(t)u = g ( t ) R ( X ; A ) u

u

thus f o r a l l

D(A 3 )

E

g(t) u

E

that

E.

Ace ord i n g l y ,

-S ( t ) A u = A i ( t ) u m(t^)

On t h e o t h e r hand,

(u

t 2 0)

D(A),

E

.

(3.19)

i s a continuously d i f f e r e n t i a b l e

= S(i)R(h;A)3

W(t)

o p e r a t o r valued f u n c t i o n t h a t s a t i s f i e s

Accordingly, i f

= Am(t).

i s a n a r b i t r a r y s o l u t i o n of (1.1)we have

u(^t)

d as m ( t

-

+ m ( t - s)Au(s)

(3.19), hence R(X;A)3u(t)

after

-

(XI

Applying

3.3.

= 0

(0

5 s 5 t)

(3.20)

= m(O)u(t) = h ( t ) u ( O ) = R(X;A)3S(t)u(0).

u(A)

(-m,

be c l o s e d .

A

uniformly w e l l posed i n i f and only i f

s)u(s)

t o b o t h s i d e s (3.18) r e s u l t s .

A)3

Corresponding t o t h e c a s e

THEORFM

-

s ) u ( s ) = -Am(t

(-m,

m)

we have

a)

The Cauchy problem f o r (1.1)&

w i t h propagator

i s contained i n t h e s t r i p

satisfying (2.2)

S(%) /Re

XI 5 w and R(X;A)

w,

n

satisfies

5

IIR(X;A)n(l Proof.

Co( /Re

XI -

( IRe XI

=

0,1,. . . )

.

(3.21)

We have a l r e a d y observed t h a t i f t h e Cauchy problem f o r (1.1)

i s w e l l posed i n

t h e n t h e Cauchy problem f o r ( 2 . 3 ) ( o f course

m)

(-m,

a l s o f o r (1.1))i s w e l l posed i n

t 2 0.

i n e q u a l i t i e s (3.21) f o l l o w from Theorem

Since

3.1.

R(X;-A)

= -R(-X;A),

Conversely, we deduce from

t h e same theorem t h a t i f (3.21) holds t h e n t h e Cauchy problem f o r (1.1) and (3.2) i s w e l l posed i n

t 2 0.

t 2 0 with

S

and

S- s a t i s f y i n g (1.9) i n

T h i s i s e a s i l y s e e n t o imply t h a t t h e Cauchy problem f o r (1.1)i s

w e l l posed i n

(-a,

m),

t h u s completing t h e proof of Theorem 2.1.

For t h e s a k e of b r e v i t y i n f u t u r e s t a t e m e n t s , w e i n t r o d u c e t h e A closed, densely defined operator A i s s a i d t o

following n o t a t i o n .

1

belong t o & + ( C 0 , w ) posed i n

t 2 0

i f t h e Cauchy problem f o r (1.1)i s uniformly w e l l

and t h e s o l u t i o n o p e r a t o r

S(c)

satisfies

(1.9).

We

12

F I S T ORDER EQUATIONS

1 1 &+(w) = U {&+(Co,w); C > 13 ( n o t e t h a t Co < 1 i s h p o s s i 01 1 The n o t a t i o n s f o r t h e case b l e ) and &+ = U {Q+(w); -m c w < a].

a l s o write

-m

c t <

a r e K1(C0,w)

m

(Co t h e constant i n ( 2 . 2 ) ) ,

d(u),

X1; w e

s h a l l not employ any s p e c i a l symbol f o r operators t h a t make t h e Cauchy problem only w e l l posed. The following r e s u l t i s an inmediate consequence of t h e Hille-Yosida theorem.

Let

THEOREM 3.4.

X Xo, t 2 0 and

h a l f plane

Re

tinuous i n

be an operator such t h a t

A

km

such that

u

E

(t 1

1

A E & + ( C ,w) 0

and

0)

(3.22)

*

E

S(t)

X>

(Re

e-XtS(t)u d t = R(X;A)u

Then -

contains t h e

an operator valued f u n c t i o n s t r o n g l y con-

S(t)

wt lls(t)lI <_ Coe

Assume t h a t , f o r each

p(A)

l o ).

(3.23)

i s t h e s o l u t i o n operator of (1.1).

We o b t a i n t h e i n e q u a l i t i e s (3.9) from (3.23) i n t h e same way 1 a s from (3.1); applying Theorem 3.1 it r e s u l t s that A E &+(C,,w). Let S(t) be t h e propagator of (1.1). Then (3.23) holds f o r both S and S Proof.

whence

S =

REMARK

s

follows from uniqueness of I a p l a c e transforms.

3.5.

I n e q u a l i t i e s (3.9) follow from t h e i r real counterparts

The proof i s an elementary e x e r c i s e i n Taylor s e r i e s . weaken, s a y Theorem 3.4 by p o s t u l a t i n g (3.23) only f o r

This allows us t o

X z w, although

t h e advantage does not seem t o b e very s i g n i f i c a n t . REM4RK 3.6.

Replacing powers of t h e r e s o l v e n t by d e r i v a t i v e s ,

i n e q u a l i t i e s (3.9) can be w r i t t e n t h u s : llR(k;A)(n)l\

5

COn!(Re

X

-

w)

-(n+l)

(Re

X>

W,

n

s

0,1,...)

.

(3.25)

Although t h i s form i s l e s s p r a c t i c a l , s i m i l a r i t y with generation theorems f o r other equations becomes more apparent ( s e e Section 11.2).

13

FIRST ORDER EQ,UATIONS

91.4

Semigroup t h e o r y . Given two Banach spaces

a l l l i n e a r bounded o p e r a t o r s from (uniform o p e r a t o r ) topology.

A

E-valued f u n c t i o n

F

and

E

we denote by

into

E

F

(E;E)

t 2 0

defined i n

t h e space of

endowed with i t s usual

We u s u a l l y a b b r e v i a t e

S(^t)

(E;F)

to

(E).

i s c a l l e d a semigroup

i f (1.7) h o l d s , that i s i f

s(s +

s(0) = I ,

(s, t 2 0)

t ) = S(S)S(t)

(4.1)

*

Equations ( 4 . 1 ) a r e o f t e n c a l l e d t h e e x p o n e n t i a l ( f u n c t i o n a l ) equations. We have s e e n i n t h e preceding s e c t i o n s t h a t t h e s o l u t i o n o p e r a t o r of a w e l l posed Cauchy problem i s a s t r o n g l y continuous semigroup.

As t h e

following r e s u l t shows, t h e converse is as w e l l t r u e . THEORFM 4.1.

t L 0.

S(-)

b e a semigroup s t r o n g l y continuous i n

Then t h e r e e x i s t s a unique c l o s e d , densely d e f i n e d o p e r a t o r

such t h a t

E

1

E+

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

S(t)

Proof.

A

We d e f i n e t h e i n f i n i t e s i m a l g e n e r a t o r

A

of

S(t)

by t h e

formula 1 AU = lim I;(S(h) kt0+

The domain of

A

a r b i t r a r y and

a > 0

u

c o n s i s t s of a l l

E

E

-

such t h a t

r a S(S)U

=

ds

a Jo

The second e q u a t i o n ( 4 . 1 ) implies t h a t

hence

ua

E

a+ h

-

I)ua =

D(A)

S ( S ) U ds

ua -+ u

as

(4.3) e x i s t s .

For u

-

.

+

(4.4)

h

S(s)u ds),

(with AU& =

But

(4.3)

define ua

g 1( S ( h )

.

I)u

a

( a f o r t i o r i , D(A))

-+

0,

1 g(s(a) - I)U)

.

t h u s t h e s e t of a l l elements of t h e form

is dense i n

E.

(4.5) ua

FIRST ORDER EQUATIONS

14 We prove next t h a t

t 2 0

for h, 0 ,

i s closed.

A

thus if

u

E

D(A)

u

If

then

u

after (4.5). Au -+ v. n

E

Let

E

as w e l l and

D(A)

.

(4.6)

E,

{u n

D(A)

be a sequence i n

un + u,

such t h a t

Then 1 j-(S(h)

1 1 ) u = l i m h(S(h) n+

-

h + O+

Taking limits a s

Let now

u

E

-

we see t h a t

shows t h e closedness of

ht

E we have

S(t)u

AS(t)u = S(t)Au Hence, for any

E

h

1 ) u = l i m (Au,) n n+m

u

E

D(A)

.

h

= v

and t h a t

Au = v,

which

A.

D(A).

Integrating

(4.6) i n

0

ds

S(s)u

S(s)Au ds =

5

s

5t

we o b t a i n

tAut = S ( t ) u

=

-

u

h

using

(4.6) i n t h e l a s t e q u a l i t y .

differentiable i n

t

>0

This shows t h a t

with

,

S ' ( t ) u = S(t)Au = AS(t)u

so that

t >_ 0.

S(^t)u i s a s o l u t i o n of (4.2) i n

This proves ( a ) i n t h e

d e f i n i t i o n of uniformly w e l l posed Cauchy problem. dence p r o p e r t y ( b ) i s checked a s f o l l o w s . for a l l

u,

llS(t)ull

t 2 0.

0 <_ s

5t

and

S(z)u

$3(t)il

i s continuous

- s)u(s) - s)Au(s) -

AS(t

u ( t ) = S(t)u(O) This completes t h e proof.

5

0

thus

) : ( u

i s an a r b i t r a r y

i s continuously d i f f e r e n t i a b l e

v(s) = S(t

v'(s) = S(t

t

must be a s w e l l bounded on

Using t h i s we can show that i f

s o l u t i o n of ( 4 . 2 ) t h e n in

Since

The continuous depen-

must be bounded on bounded s u b s e t s of

by t h e uniform boundedness theorem compacts of

i s continuously

S(t)u

.

-

s)u(s) = 0,

so that

FIRST ORDER EQUATIONS The c a s e tion

A

E

El;

< t <

-m

E

m.

t

is

&eJ

S(

-)

be a group s t r o n g l y continuous i n

Then t h e r e e x i s t s a unique c l o s e d , densely d e f i n e d o p e r a t o r

such t h a t

Q1

s,

conversely, we have

THEOREM 4.2.

A

(E)-valued func-

A

and s a t i s f y i n g ( 4 . 1 ) f o r a l l

t

The propagator of ( 4 . 2 ) i s a s t r o n g l y continuous group

c a l l e d a group. when

i s handled i n a similar way.

m)

(-w,

defined f o r a l l

S(t^)

15

i s t h e e v o l u t i o n o p e r a t o r of (4.2).

S(:)

We omit t h e p r o o f , which i m i t a t e s t h a t of Theorem 4.1. I n t h e f u t u r e , " s t r o n g l y continuous semigroup" means a semigroup s t r o n g l y continuous i n

t 5 0,

cates continuity i n

< t <

continuous i n

-w

t > 0, t

only w e l l posed i n

$1.5

while " s t r o n g l y continuous group" i n d i We s h a l l make no u s e of semigroups only

m.

which a r e n a t u r a l l y a s s o c i a t e d with Cauchy problems

2

0.

.

The inhomogeneous equation Let

f ( % ) b e a continuous

t

E-valued f u n c t i o n d e f i n e d i n

5

0.

S o l u t i o n s ( s t r o n g or genuine) of t h e inhomogeneous equation

a r e d e f i n e d i n t h e same way as s o l u t i o n s of (1.1). S i n c e t h e d i f f e r e n c e of two s o l u t i o n s of (5.1) i s a s o l u t i o n of (l.l),i f most one s o l u t i o n of

u(0) = uo Let S(%)

u ( : )

A

E

5:

there is at

(5.1) s a t i s f y i n g t h e i n i t i a l c o n d i t i o n

.

(5.2)

be an a r b i t r a r y s o l u t i o n of ( 5 . 1 ) s a t i s f y i n g ( 5 . 2 ) and l e t

be t h e propagator of (1.1). Given

t h a t the function

S(t

-

G)u(;)

t > 0

f i x e d we show e a s i l y

i s continuously d i f f e r e n t i a b l e i n

G)U'(~)

= 0 5 s 5 t with d e r i v a t i v e - S ' ( t - $)u(i) + S(t S ( t - hs)(u'(hs) - A u ( i ) ) = S ( t - z ) f ( i ) . I n t e g r a t i n g we o b t a i n

u ( t ) = S(t)uo

+ Jot

S(t

However, t h e converse i s n o t i n g e n e r a l t r u e : s o l u t i o n of (5.1) more t h a n c o n t i n u i t y of below a c l a s s i c a l r e s u l t .

f

-

s ) f ( s ) ds

.

t o make (5.3) a genuine

i s needed.

We prove

(5.3)

16

FIRST ORDER EQUATIONS

LEMMA 5.1.

Assume t h a t one of t h e following two c o n d i t i o n s

is

-s a t i s f i e d : (a)

f(t)

(b)

f(6)

D(A)

E

and

u

Suppose i n a d d i t i o n t h a t

2 (5.1) 2 Proof.

0

Af(t^) a r e continuous i n

f(;),

i s continuously d i f f e r e n t i a b l e i n 0

E

0

Then ( 5 . 3 )

D(A).

0

5 t 5 T,

or

5 t 5 T. i s a genuine s o l u t i o n

5 t 5 T. S(t^)uo i s a s o l u t i o n of (1.1)when

Since

u0

E

may d i s r e g a r d t h e f i r s t t e r m on t h e r i g h t hand s i d e of (5.3).

our claim it i s enough t o show that and u(t) = Assume ( a ) holds.

Lt

Lt

(Ads)

+

u(t>

A~(Z)

D(A),

E

( 0 <_ t

f ( s ) ) ds

D(A)

we

To prove

i s continuous

.

5 T)

(5.4)

Then

['( 1

n S

Au(s)ds =

.,

-

r)Af(r)dr

0

r"

=

\

S(s

(S(t

-

r)f(r)

-

f ( r ) ) dr

,

0

j'

which i s nothing i f n o t (5.4). On t h e o t h e r hand, suppose t h a t ( b ) i s v e r i f i e d . bY P u t s , u(t) =

-

t i

rt

=

j$ Lt-'

Then, i n t e g r a t i n g

S ( r ) f ( s ) d r ) ds

Lt (J"'

S(s)f(O)ds +

u o

S ( r ) f ' ( s ) d r ) ds

Y

Hence Au(t) = S ( t ) f ( O )

-

t f(0) +

-

(S(t

s)

-

I ) f ' ( s ) ds

i

= S(t)f(O)

-

f(t)

+

[t

S(t

-

s ) f ' ( s ) ds

.

I n t e g r a t i n g and r e v e r s i n g t h e o r d e r of i n t e g r a t i o n we o b t a i n (5.4). D e t a i l s a r e omitted.

.

17

FIRST ORDER EQUATIONS

Obviously, formula (5.3) makes sense w i t h much weaker conditions on f.

15 p c

Let

a.

The space

c o n s i s t s of a l l (equivalence)

Lp(O,T;E)

c l a s s e s o f ) s t r o n g l y measurable functions

f ( ^ t ) defined i n

0

5

t

5

T

such t h a t

endowed w i t h t h e norm

II*lIp;

c o n s i s t s of a l l (equivalence

Lm(O,T;E)

c l a s s e s o f ) e s s e n t i a l l y bounded s t r o n g l y measurable functions endowed w i t h t h e norm

llflIm

= e s s . SUP llf(t)ll

.

OitLT All t h e spaces

Lp(O,T;E)

are Banach spaces

[1957:1,

(HILLE-PHILLIPS

P. 891). If

f

element of

L'(0,T;E)

is, say, a function i n

not be d i f f e r e n t i a b l e anywhere or

t). W e s h a l l declare u ( t )

any

and

uo

u(t)

LEMMA 5.2.

Co,

w

Assume 0

5 t <_

T.

t h e constants i n Proof.

uo

E

E,

( ~ ( 2 ) may

may f a i l t o belong t o

D(A)

for

t o be a generalized s o l u t i o n of (5.1).

The following r e s u l t concerns c o n t i n u i t y p r o p e r t i e s of

continuous i n

i s an a r b i t r a r y

(5.3) i s not i n g e n e r a l a s o l u t i o n of ( 5 . 1 )

E,

f

E

L1(O,T;E).

Then

u( t^) u(;)

. 3

Moreover, we have

(1.9).

(5.5) i s r a t h e r obvious.

To prove t h e rest of Lemma 5.2 we

note that ib(t')

for

0 < t < t'.

-

u ( t ) / / <_

rt'lb(t'

-

s ) i i lif(s)ll ds

The f i r s t i n t e g r a l can be made small when

t'

-

t + 0

by c o n t i n u i t y of t h e i n d e f i n i t e i n t e g r a l of sunrmable f u n c t i o n s ; t h e second y i e l d s t o s t r o n g c o n t i n u i t y of

S(z)

and t h e uniform boundedness theorem.

18

$1.6

FIRST

OIlDER EQUATIONS

Miscellaneous comments. Theorem

3.1 was discovered independently by HILLE [1948:1] and

YOSlDA [1948 :11 i n t h e p a r t i c u l a r case where

= 1. The proof f o r t h e 0 g e n e r a c a s e was discovered, a l s o independently, by FELLER 11953:1],

MIYADERA [1952:1] and PHILLIPS [1953:1].

C

I n all t h e s e papers, Theorem

3.1 i s formulated i n t h e l a n g u a g e of s t r o n g l y continuous semigroups and t h e i r generators r a t h e r t h a n t h a t of a b s t r a c t d i f f e r e n t i a l equations and t h e i r propagators. HILLE [1952:11.

The proof of Theorem

3.1 presented here follows

E a r l i e r r e s u l t s on semigroups of operators were obtained

by NATHAN, FUKAMIYA and GELFAND ( f o r more information on t h e h i s t o r y of t h e subject see HILLE-PHILLIPS [1957:1] or t h e author [1983:1, p.

91 1.

Theorem 1.1 i s due independently t o MIYADERA [1951:1] and PHILLIPS [1951:1], who improved on l e s s general statements of DUNFORD and HILLE. I n i t s present form, t h e notion of uniformly well posed ( a b s t r a c t ) Cauchy problem i s due t o LAX ( s e e LAX-RICHTMYER [1956:1].

The notion of

a b s t r a c t Cauchy problem was introduced by HILLE [1952:2 1 ( s e e a l s o HILLE [1953:11,

[1953:21,

[1954:11,

[1954:21,

[1957:11, PHILLrPS [1954:11)

although h i s d e f i n i t i o n of a w e l l posed Cauchy problem i s somewhat d i f f e r e n t from t h a t employed here.

For a thorough discussion of t h e ab-

s t r a c t Cauchy problem and t h e c l a s s i c a l Cauchy problem of solving a p a r t i a l d i f f e r e n t i a l equation with i n i t i a l d a t a prescribed on a noncharacter;istic surface see t h e author [1983:1, p.

54 1.

I n p a r t i c u l a r , t h e present

formulation of t h e Cauchy problem o r i g i n a t e s i n t h e concept of properly posed Cauchy problem formulated by HADAMARD

EXERCISE 1. L e t

Snow t h a t

A

E

&,

A

[1923:1].

be a bounded operator i n a. Banach space

E.

t h a t is, t h a t t h e Cauchy problem f o r

u'(t) i s uniformly well posed i n

-m

C

(6.1)

= Au(t)

t <

w.

The group

S ( t ) generated by

A

(or, equivalently, t h e propagator of (6.1)) i s given by

t h e s e r i e s (6.2) converging uniformly on compact subsets of Prove t h a t

S(t)

can a l s o be expressed i n t h e form

-m C

t <

m.

FIRST ORDER EQUATIONS

[ eAtR(h;A) r

where

dh

19

,

(6.3)

i s a simple closed curve (or t h e union of a f i n i t e number of

t h e s e ) o r i e n t e d counterclockwise and e n c l o s i n g t h e spectrum

its (their) interior.

if

(1)

1

>

b)

+’

ti)

2

u(A)

in

Show t h a t IlS(t)ll

5

Ils(t)ll

5

, where

> -w-

w

= sup {Reh;

+

- = inf

li)

h

{Reh;

a(A)]

,

(6.6)

E rs(A)]

,

(6.7)

E

Produce a n example where ( 6 . 4 ) ( r e s p . ( 6 . 5 ) ) does n o t hold with =

(resp.

W

1-

-u-). Show t h a t

=

c a n be extended t o a f u n c t i o n

S(t)

+

S(5) holomorphic ( a s a (E)-valued f u n c t i o n ) f o r all i s continuous i n t h e norm of

S(t)

5;

i n particular

(E).

h

EXERCISE 2.

Let

t = 0

(continuity a t

of ( E )

b e a semigroup continuous i n t h e topology

S(t)

suffices).

Show t h a t t h e i n f i n i t e s i m a l

h

generator for

n

=

IlhR(A;A)

A

S(t)

of

1 and c o n t i n u i t y of

- 111 < 1

A

for

bounded i n v e r s e ) . and

i s a bounded o p e r a t o r ( H i n t : at

S(t)

t

=

0

use formula

t o show t h a t

hR( A;A)

sufficiently large so t h a t

Show t h a t

has a

(6.2)

admits t h e r e p r e s e n t a t i o n s

S ( : )

(3. 8)

(6.3). EXERCISE

3. Let

E = C

0

be t h e Eanach space o f all

(-m,m)

(complex-valued) continuous f u n c t i o n s

and such t h a t

U(X)

4

0

as

1x1

-

a,

u(x)

defined i n

-m

<x c

m

endowed with t h e norm

Define S ( t ) u ( x ) = u(x + t ) Snow t h a t

S ( : )

(-m


i s a s t r o n g l y continuous group i n

infinitesimal generator.

(6.9)

m)

E.

Identify i t s

20

FIRST ORDER EQUATIONS

EXERCISE

4.

Prove t h e r e s u l t of Exercise

inf'initesimal g e n e r a t o r ) i n t h e space

6.3 (and i d e n t i f y t h e 15 p <

LP(-m,m),

m.

Show t h a t

(6.9) i s not strongly continuous ( o r even s t r o n g l y measurable) i n Lm( -a,m) . EXERCISE

5.

Define S(t)u(x)

i n t h e space

C

0

=

u(x + t )

(t

2

(6.10)

0)

of all continuous f i n c t i o n s defined i n x

[O,m)

tending t o zero at i n f i n i t y , endowed with t h e

supremum norm.

2

0

Show t h a t

i s a s t r o n g l y continuous semigroup and i d e n t i e i t s i n f i n i t e s i m a l

S(t)

generator.

Do t h e same i n t h e spaces

EXERCISE

6.

LP(O,m)

(t

5

x <

m.

(0

5

x < t)

m)

(t

=

2

0).

(6.11)

i s a s t r o n g l y continuous semigroup i n t h e spaces

S(:)

(15 p C

m)

functions defined i n u(0) = 0

15 p <

Define

S(t)u(x)

Show t h a t

Lp(O,m),

and i n t h e space

x

(but not i n

2

0,

C

070

of all continuous

[O,m)

tending t o zero a t i n f i n i t y and such t h a t

Co[O,m)).

I d e n t i f y t h e i n f i n i t e s i m a l generator

i n each case.

EXERCISE Show t h a t

7.

A E Et

Let

A

be a normal operator i n a H i l b e r t space

i f and only i f

u

+

(defined by

H.

(6.6) i s f i n i t e .

Prove t h e formula (6.12)

for t h e semigroup

generated by

S(c)

of t h e i d e n t i t y associated with

A,

where

P(dh)

i s the resolution

Using t h e f u n c t i o n a l c a l c u l u s for

A.

normal operators show t h a t

Show t h a t i f

A

EXERCISE 8. Show t h a t

A

E

&

i s s e l f a d j o i n t t h e n each Let

A

S(t)

i s self adjoint.

be a normal operator i n a H i l b e r t space

i f and only i f

W+

and

w-

E.

(defined by ( 6 . 7 ) ) a r e

FIRST ORDER EQUATIONS

21

Prove t h a t t h e formula (6.12) holds i n

finite.

-m


and t h a t

(6.13) and

hold.

iA i s a s e l f a d j o i n t o p e r a t o r t h e n each

Show t h a t i f

EXERCISE

9.

Let

H, A

H i l b e r t space

i s normal.

S(t)

be a s t r o n g l y continuous semigroup i n a

S(.f)

i t s infinitesimal generator.

Show t h a t

(3.8) show t h a t EXERCISE 1 0 .

i s normal ( H i n t :

A

(HILLE [1938:1],

i s self a d j o i n t

Assume t h a t each u s i n g t h e f i r s t formula

i s normal).

R(A;A)

hypoteses of Exercise 9 , A

S(t)

.

i s unitary

SZ.-NAGY [1938:11).

assume t h a t each

(Hint:

R(A;A)

show t h a t

Under t h e

i s self adjoint.

S(t)

Then

i s s e l f adjoint f o r

h red). EXERCISE 11.

(STONE [1932:11)

9, assume t h a t each EXERCISE 1 2 . Banach space

S(t)

Let

E.

Under t h e hypoteses of Exercise

i s unitary.

iA

Then

is self adjoint.

be a strongly continuous semigroup i n a

S(;)

Define w = l i m

2t l o g

.

Ils(t)\\

(6.15)

t - m

Show t h a t t h e l i m i t e x i s t s ( t h e value

(a) hi

c

W.

If

(b)

hi1

>

hi

'11

=

-m

i s allowed)

and

then

(6.16) (6.16) may n o t hold f o r

although for

(I)'

<

fJ,

g e n e r a t o r of

(r)

(6.6).

H i l b e r t space and MERCISE 13. serninrouD

S(<)

=

(1)

itself. ( c )

(6.16) cannot hold

i s a bounded o p e r a t o r t h e n

A

defined by

W'

Using Exercise 6.1 show t h a t i f t h e i n f i n i t e s i m a l

(d)

A

Show t h a t

(6.17) h o l d s

a s well i f

E

is a

i s a normal o p e r a t o r .

Show, by means o f a n example, t h a t t h e r e e x i s t s a i n separable H i l b e r t space

H

with i n f i n i t e s i m a l

22

FIRST ORDER EQUATIONS

generator

A

such t h a t

>

(r)

S(i)

More g e n e r a l l y , we can c o n s t r u c t

lib+,

i n such a way t h a t

where

h >0

i s a r b i t r a r i l y preassigned.

See ZAECZYK

[1975:11;

e a r l i e r (although l e s s elementary) example can be found i n HILLEPHILLIPS

[1957:1, p . 6651 where

EXERCISE

14.

n(A)

3.:?

Prove Lemma

i s i n f a c t empty ( 1 )

so t h a t

u s e the f a c t that

(Hint:

with

(6.18) EXERCISE 15.

Prove t h a t t h e “ r e d ” i n e q u a l i t i e s (3.24) imply t h e i r

complex c o u n t e r p a r t s ( 3 . U ) .

(Hint:

if

Re h >

f~

express

R(A;A)

by

means of i t s Taylor s e r i e s

with

p

real.

EXERCISE

R(A;A)

=

Then l e t

p

16. Let

i n f i n i t e s i m a l generator, Show t h a t i f

u

E

D(A)

be a s t r o n g l y continuous semigroup,

S(t)

t h e niimber defined by

(11

(6.15),

its

A

w’ >w,

0.

then

(6.20) t h e l i m i t being uniform on compact s u b s e t s of

l i m i t is

EXERCISE 17.

Let

E

such t h a t

h(h ; A ) / / 5 u

E

D(A)

For

t = 0

the

be a ( n o t n e c e s s a r i l y densely defined) l i n e a r

A

operator i n a Banach space

Show t h a t i f

t > 0.

(6.3)).

(compare with

$u

C/h

R(X;A) exists for

( h > w)

.

X >

w

(6.21)

then

l i m

A-

+m

hR(h;A)u = u

.

(6.22)

and

FIRST ORDER EQUATIONS

23

FOOTNOTES TO CHAPTER I (1) We d e n o t e h e r e by

not t h e d e r i v a t i v e of

( E ) ) applied t o valued funct i on s .

u.

S'(;)u S(;)

t h e d e r i v a t i v e of t h e f u n c t i o n

S(t^)u

(which may f a i l t o e x i s t i n t h e norm of

The same o b s e r v a t i o n a p p l i e s t o o t h e r o p e r a t o r The c o r r e c t n o t a t i o n

(S(i)u)'

becomes cumbersome

later. ('2)

Roughly speaking,

R( i;A)u.

u(t)

i s t h e i n v e r s e Laplace t r a n s f o r m of

However, t h e c o r r e s p o n d i n g i n t e g r a l may n o t be c o n v e r g e n t

-1

-2

...

t h u s we u s e t h e w e l l known formula R(h;A)u = h u + A Au + + A- mAm -1 u + hmmR(h;A)Amu f o r m = 3, which a l l o w s d i f f e r e n t i a t i o n w i t h respect t o

t.

24

CXIIPTER I1

THL CAUCHY PROELEM FOR SECOND ORDER EQUATIONS COSINE FUIVCTION THEORY

The Cauchy problem f o r second o r d e r e q u a t i o n s .

$11.1

A t h e o r y f o r t h e equation

u"(t)

(1-11

Au(t)

=

t h a t p a r a l l e l s c l o s e l y t h a t of (1.1.1)can be developed without undue difficulty.

We prove t h e fundamental r e s u l t s i n t h i s chapter.

s o l u t i o n of (1.1)in t E-valued f u n c t i o n u(:)

30

A

i s a twice c o n t i n u o u s l y d i f f e r e n t i a b l e

such t h a t

u(t)

E

D(A)

and (1.1)h o l d s t h e r e ;

s o l u t i o n s i n d i f f e r e n t i n t e r v a l s a1.e defined accordingly.

The Cauchy

problem i s now t h a t of f i n d i n g s o l l t i o n s of (1.1)t h a t s a t i s f y t h e i n i t i a l conditions u(0)

=

u0

, u'(0)

=

u1

(1.2)

The Cauchy problem i s w e l l posed o r p r o p e r l y posed i n

2

t

i f and only

0

if (a)

uo,ul

E

D

There e x i s t s a dense subspace there e x i s t s a solution

D

of

u ( 7 ) of

E

such t h a t for any

(1.1)2

t

2

satisfying

0

(1.2). (b)

t

2

0

There e x i s t s a nonnegative, f i n i t e f u n c t i o n

C(<)

defined i n

such t h a t

llu(t)ll

5

C ( t ) ( / l u ( ~ ) l l+

IlU~(O>ll>

The d e f i n i t i o n s of well posed problem in posed problem i n

[0,m)

and

(-my=)

(-m,m)

(t

2

(1.3)

0)

and uniformly well

a r e obvious analogues of t h e f i r s t

o r d e r case and we omit t h e d e t a i l s . We d e f i n e now two p r o p a g a t o r s o r s o l u t i o n o p e r a t o r s

c(<)

and

8(<)

25

SECOND ORDER EQUATIONS

a s follows: C(t)U

for u

D,

E

u(t)

where

v(f)

(1.4)

u(t)

(1.1) with

i s t h e s o l u t i o n of

u ( 0 ) = u, u ' ( 0 ) = 0 ; 3 ( t ) solution

=

i s defined i n t h e same way i n r e l a t i o n t o t h e v ( 0 ) = 0, v ' ( 0 ) = u .

satisfying

@ ( t )and

Both

8(t)

a r e extended by c o n t i n u i t y t o bounded, everywhere defined operators E

satisfying

IlC(t)ll

1. C ( t ) ,

Il8(t)il

5

It follows from t h e

C(t).

definitions that S(t)u

u(:)

If

ds.

=J:@(S)U

i s an a r b i t r a r y s o l u t i o n of

(1.5)

(1.1) we have

The proof i s t h e same a s i n the f i r s t order case.

u(i)

to

=

C(2 + t ) u

and

v(i)

=

S(i + t ) u

Applying t h i s e q u a l i t y

(1.5)

(and u s i n g

i n the

second i n s t a n c e ) we i n s t a n t l y o b t a i n

+ S(s)@'(t)u,

(1.7)

= @ ( s ) d ( t ) u+ S ( s ) @ ( t ) u ,

(1.8)

@(s + t ) u = @(s)C(t)u

a(s for

u

D

E

+ t)u

(obviously,

(1.8) may be extended t o all u

E

E

by

continuity). The next r e s u l t shows (among o t h e r t h i n g s ) t h a t t h e two notions of p r o p e r l y posed problem coalesce i n t h e second order case.

THEOFEM 1.1. Let t h e Cauchy problem for t

2

&

0.

Then

( i ) The Cauchy problem for

so t h a t

(-m,m>,

@(;)

(1.1) be well posed i n

(1.1)i s u n i f o r d y well posed

i s s t r o n g l y continuous i n

@(;)

(-,m).(ii)

s a t i s f i e s t h e cosine f u n c t i o n a l equations C(0) hi?)

=

I, C(S

+

t) + C(~-t)=2@(s)@(t)

There e x i s t constants

C ,m 0

(-m

<

S,

t <

m)

.

(1.9)

such t h a t

(1.10)

26

SECOND CRDER EQUATIONS

-

u(c)

(1.3)

obvious t h a t

a solution i n

t

u(-G)

with

u'(0)

extended t o a s o l u t i o n i n

2


-m

0

=


-m

u ( t ) = -u(-t), t < 0)

( r e s p . by

i f it i s v e r i f i e d i n t

identity

u(0) = 0 )

u(t)

u(-t), t > 0

by

i s verified i n

(a)

thus

=

E

(1.7) ( f o r u

The proof of Given

E

t

for

D)

u

E

0


m

(1.11) and a d d t h e two

(1.9)

(1.11): t h e r e s u l t i s E

E

applied t o

by c o n t i n u i t y .

(iii) i m i t a t e s c l o s e l y t h a t of Theorem 1.1.1 ( i i i ) .

t h e r e e x i s t s a n u l l subset

E

-m

.

-t

and

and t h e e q u a l i t y can be extended t o d l u

D,

can be

It f o l l o w s from t h e above c o n s i d e r a t i o n s t h a t

0.

resulting e q u a l i t i e s using

u

On t h e o t h e r hand,

m.

(resp. with

@ ( - t=)@ ( t )8,( - t ) = - 8 ( - t ) Write

0.

maps s o l u t i o n s i n t o s o l u t i o n s , t h u s it i s

can be extended t o

1. 0

3

t

(1.1) i s w e l l posed i n

Proof: Assume t h e Cauchy proklem for The a s s i g n a t i o n

subspace

Eo

sequence

(tn] ,k d o

0

of

{u,}

such t h a t t h e s e t

such t h a t t h e

[O,m)

L r@(t)uo; t { do]

{u,)

generated by

d

i s s e p a r a b l e and a

U F@(tn)uo]

i s fundamental

-

i n Eo; accordingly, i f e0 = ( d o tl) u ( d o + tl) U ( d o - t2) U . . . , e 0 i s a n u l l s e t and it f o l l o w s from t h e second e q u a t i o n (1.9) t h a t

@W0 E Eo We show next t h a t

#

(t

eo)

(1.12)

*

Il@(t)lli s bounded on i n t e r v a l s

Assuming t h e o p p o s i t e we o b t a i n a bounded sequence

Tun]

5 E,

\l@(tn)unl/ 2 n.

j\unl/ = 1 with

[-@,PI, B <

3 n

{t

m.

and a sequence

Operating e x a c t l y a s i n t h e

proof of Theorem 1.1.1 we can t h e n c o n s t r u c t a s e p a r a b l e subspace

un

E containing all t h e @ ( t ) E , 5 Em f o r t e,; of

t

extended t o n e g a t i v e

and a n u l l s e t

since not

in

m(t),

Chapter I, t h e f u n c t i o n

C(-t)

=

t h i s r e l a t i o n can be

t h e norm of t h e r e s t r i c t i o n of

Em i s defined and measurable €or a l l

if

t f e m , m(s + t ) 5 E

eft)

A s i n t h e semigroup c a s e i n

-1,.

to

+ sup { l l @ ( s - t ) u I I ; u

Em

e m such t h a t

t;

moreover,

SUP

~ 2 l I @ ( s ) ~ ( t ) u ul l ;c E,,

E,

/Iu11 5 13

5

2m(s)m(t)

lluII +

m(tn)

@(t)

>n

and

5 1.3

m(s-t).

The contra-

d i c t i o n i s t h i s time obtained with t h e a i d of t h e following r e s u l t :

LEMMA 1 . 2 . defined i n

-00

Let m(;)
m

be a nonnegative f i n i t e measurable f u n c t i o n such t h a t

m(-t)

=

m(t)

27

SECOND ORDER EQUATIONS

where e [-@,@I,

i s a null set i n

B <

m.

a > 0, s + t = a, t f e .

Proof: Let =

+ m(a- 2 t )

2m(a- t ) m ( t )

must occur: (c)

a)

m(a- 2 t )

m ( t ) 2-/2.

t c [-a,a]

-

3 [-a,a],

m(a)

[-p, a 3 U [ a , @ ] with o r i g i n we note t h a t , m(s- t ) s

if

.(a)

we have

m(i)

near

5

2

2a.

m(f)

m.

2rn(s)m(t) + m(s

To show boundedness n e a r t h e

+ t ) for t

-e;

(1.13)

the

taking

t >

o

t t h e proof of Lemma 1 . 2 i s complete.

- @ ( t o= )2 @ ( t ) ( @ ( t 0+

CX C f3

(a-2d)

Arguing as i n

i s even we deduce from

End o f proof of Theorem 1.1. Let

Take

t h e s e t of a l l

u

e U d U (a-d)

u

E

E, t o and

A few manipulations with t h e cosine f u n c t i o n a l e q u a t i o n

@(to -t h )

+ m(s-t) =

i s bounded on every s e t of t h e form

0 < 0: < p <

since

2m(s)m(t)

- t ) 2 .im(a)/2 2 1 and d i s

4 / d / = I d ] + / a - d / + l a - 2dl

we deduce t h a t

inequdity

m(a

m(a)/2 ( b )

Accordingly,

thus

Lemma 1 . 1 . 2

2 2

Since

one o f t h e following t h r e e i n e q u d i t i e s

m(t) L G ) / 2

with

f i x e d and

i s bounded on i n t e r v a l s

m(c)

(-a,m).

and i n t e g r a t e i n

t h u s again c o n t i n u i t y of

h

01

@(<)u

h

arbitrary.

(1.9) show

that

- t ) - @(to - t ) )- ( @ (+t ho- 2 t ) - @ ( t- o2t)). 5 t 5 f3; we o b t a i n

for a l l

t

follows from i t s mean con-

tinuity. To show ( i i ) we apply once more t h e uniform boundedness p r i n c i p l e

Il@(t)l\ i s bounded i n 0 5 t f 1. Choose 0,W l/@(t)ll f COeWt i n 0 5 t 5 1 and 211@(l)l/e-W+ e-2w f 1.

and conclude t h a t such t h a t Assume t h a t

(1.10)

t h u s it h o l d s f o r

all

t

proof.

2

0;

since

holds f o r

0 ft

5

n

+

0 ft

5n

.

Then

1. By induction,

(1.10) i s v a l i d f o r

@ ( - t=) @(t),it i s v a l i d for a l l

t.

This ends t h e

28

SECOND ORDER EQUATIONS

The comments a f t e r Theorem 1.1.1 have an obvious c o u n t e r p a r t here: we omit t h e d e t a i l s .

bound on

Note a l s o thltt

y i e l d s an exponential

(1.10)

(1.5):

through

E;

(1.14) if

#

w

for

0;

511.2

=

we o b t a i n

0

The g e n e r a t i o n theorem. The following analogue of The'wem 1.3.1

Let

THEOREN 2.1.

A

uniformly w e l l posed i n

The Cauchy problem f o r

with propagator

(-m,m)

R(h2;A)

i f and only i f

be c l o s e d .

holds.

e x i s t s i n t h e h a l f space

Proof: The f i r s t t a s k i s t o show t h a t

(1.1)

@(t) satisfying Reh > and

2

(1.10)

i s nonempty ( i n t h e

p(A)

f i r s t o r d e r case, t h i s was a byproduct of t h e proof and d i d not r e q u i r e a s e p a r a t e argument ).

cp(t)

such t h a t

Choose a twice continuously d i f f e r e n t i a b l e f u n c t i o n

cp(t) = 1 i n

If

u

E

D =

then

@(c)u

A@(t)u, AQ(A)U

that

A

=

5

-h+

D(A)

t

2

2

and d e f i n e

=lo-

e A t cp(t)@(t)u dt

(2.2)

r"

e-htcp(t)@"(t)u d t =

-hu +

h2Q(A)u

- 0

h2Q( A)u

i s closed and

Q(A)E

t h e form

in

i s twice continuously d i f f e r e n t i a b l e and

(e-htcp"(t)

Since

1, w ( t ) = 0

t h u s we o b t a i n i n t e g r a t i n g by p a r t s twice t h a t

+s,' =

5t 5

?m

Q(A)u

@"(t)u

0

- 2he- j t c p ' ( t ) ) @ ( t ) u + M( A)LL

Q(A)

and extend

and

dt =

. M(h)

(2.3) a r e bounded we can e a s i l y show

(2.3) t o all

u

E

E;

we r e w r i t e it i n

29

SECOND OKDER EQUATIONS

A simple e s t i m a t i o n y i e l d s

A

for

2

6 > 0.

r e a l s o l a r g e t h a t t h e r i g h t hand s i d e of

A

Take now

- h-'M(h))-l

(2.5)

i s l e s s than

(2.4)

on t h e r i g h t by it we o b t a i n

1. Then

A

(I

-1 2 ( A I-A)P(A)

e x i s t s and multiplying

I ,

=

(2.6)

2

P( A) = Q( A ) ( I - h-'M(h))-', so t h a t A I - A i s onto. We can show t h a t P(h) i s 1 - 1 i f we assume i n a d d i t i o n t h a t h > W ; t h e n i f 2 ( h I - A ) u = 0 t h e function u ( t ) = cosh(ht) s o l v e s (1.1) b u t does

where

not obey

(1.3) w i t h C ( t )

We have t h e n shown t h a t

= Co exp(IJt), a c o n t r a d i c t i o n unless u = 0 . 2 Arguing R(A ;A) e x i s t s and e q u a l s h-lP(A).

now e x a c t l y as i n Theorem 1.1.1 we show t h a t @(t)D(A) 5 D ( A )

which a g a i n i m p l i e s t h a t

R(A;A)@(t) = @ ( t ) R ( A ; A )

and

A@(t)u = @(t)Au

Making use of

u (1.3.6)

for

u

E

E

D;

D

(2.7)

we o b t a i n

t h e e q u a l i t y i s extended t o

and i m p l i e s t h a t

u

E

D(A)

@ ( i ) u i s a s o l u t i o n of

i n t h e same way as (1.1) f o r any

i n o t h e r words,

D(A);

where

(2.7)

.

2

0 of every u

i s t h e space of a11 " a d m i s s i b l e i n i t i a l values" c o n s i s t i n g

D

2

D1

t i o n of t h e space

S(G)u

such t h a t

@ ( i ) u i s a s o l u t i o n of

such t h a t

D

(1.1). (The i d e n t i f i c a -

of a l l "admissible i n i t i a l d e r i v a t i v e s "

s a t i s f a c t o r y and w i l l be postponed u n t i l t h e next c h a p t e r ) . Let now

Reh >

W.

Define

/' 0

m

R(A)u for

u

E

E.

If

u

i s a s o l u t i o n i s c o n s i d e r a b l y more complex and l e s s

u

E

D(A)

and

=

T > 0

e-At@(t)u d t we have

30

SECOND ORDER EQUATIONS

.

+ A2 rTe-AtC(t)u d t JO

Since

@"(t)u

@(t)Au,

=

@'(T)u

=

rT

@ ( t ) A u dt

0

a n d we can t a k e l i m i t s i n

R(A)u c D ( A )

AR(A)u

with

u

extended t o all

(2.9)

F

E

=

so t h a t

h 2

T

2s

2 A R(1)u

-

m

,

t o conclude t h a t

- k; again,

5 D(A)

R(A)E

t h i s e q u a l i t y can be

and

.

-1 2 ( A I-A)R(A) = I

(2.10)

-

showing t h a t ( A I A ) i s onto. We have a l r e a d y proved b e f o r e t h a t 2 h I - A i s one-to-one, so t h a t A p(A) i f Reh > w and R(h)

=

D i f f e r e n t i a t i n g iinder t h e i n t e g r a l s i g n we deduce t h a t

M(A2;A).

m

(AR( A2;A))(")u

=

[

(-1)"

"

t

(1.10) i n

2

(2.11)

( 1 . 3 . 8 ) ) and. a l l the inequalities

( s e e t h e j u s t i f i c a t i o n of using

tne-At@(t)u d t

0

(2.1) follow

0.

Solutions o f

We prove now t h e o t h e r h a l f of Theorem 2.1.

(1.1)

a r e defined by t h e formula ( 2 )

-u ( t ; u )

=

t2

u + 7 Au +

t 4 A 2u + 1

2.

At

W'+im

R(A2;A)A3u

c ~ l u W ' - j m

(2.12)

(t for

u

E

D(A3)

s o l u t i o n of

and

0'

>

L

0)

- c .

The checking t h a t

w,O.

(1.1) i s done j u s t a s i n Theorem

t h e same way t h a t

dh

A5

llL(t;u)ll

= O(exp

-u ( ~ , u )

= U,

(w't))

-

u'(O,U)

u(t;u)

is actually a

1.3.1; a l s o , we prove i n

as = 0

t

-

m,

(2.13)

and

L m e - h t L ( t ; u ) d t = hR(h2 ;A)u

for

Re7 >

W'.

It follows from t h i s e q u a l i t y and from Lemma 1.3.2 t h a t

31

SECOND (\WF:Y FCUATIONS

(2.14) hence, using i n e q u a l i t i e s

(2.1),

l l h t ; ~ ) l l5 C o I I ~ I / l i m n-

p-y,1

-(n+l)

wt

wt

= cOllulle

ft 1 0 )

.

(2.15)

m

Condition ( a ) i n the d e f i n i t i o n of uniformly well posed problem f o r (1.1) i s v e r i f i e d a s follows:

if u

u

0’ 1

E

D(A3),

i s given by

(1.2)

u(t)

G(t;u )

=

0

-1

Lt-

@(t)U

(E)

u

E

D(A3)

- valued

We s h a l l show t h a t i f 0

we define

(2.16)

= i(t;u)

E,

ohtaining a

s t r o n g l y continuous function with

-S ( t ) u 2

(b)

a n d extend it by c o n t i n u i t y t o all of

A second operator valued function

t

.

u ( s ; u ~ )d S

To check t h e continuous dependence statement

for

a solution satisfying

u(;)

-8 ( t )

=Lt

i s defined by

.

“@s)u d s

i s an a r b i t r a r y s o l u t i o n of

(1.1) i n

we must have

u(t)

=

E(t)u(O) + Z(t)u’(O).

(2.18)

This i s done a s i n the f i r s t order case and we only sketch t h e d e t a i l s . The f i r s t step i s t o show t h a t and consequently with functions

h(t)

A;

c(t)

= “@t)R(h;A)3 and

=

“(t)R(A;AP

commute with

h ( t ) = i(t)R(A;A)3

continuously d i f f e r e n t i a b l e and s a t i s f y h’(t)

z(t)

and

R(X;A)

t h e second i s t o note t h a t t h e operator valued

h’(t)

= AZ(t)R(A;Af’=A h ( t ) .

=

a r e (twice)

h ( t ) , and the equality

Accordingly

32

SECOND OFTIER EQUATIONS

and

(2.18)

h ( 0 ) = R(A;A)3,

follows noting t h a t

h(0)

=

This com-

0.

p l e t e s t h e proof of Theorem 2.1.

REMARK 2.2. Theorem 2.1 shows i n p a r t i c u l a r t h a t we have t h e r e l a t i o n 2 2 t h e region t o t h e l e f t o(A) 5 ;Reb 5 w } = {A;Reh 5 b) - (Imh)2/4m23, 2 I n p a r t i c u l a r , if o f a p a r a b o l a p a s s i n g through t h e p o i n t s U2, 2 2iw

{w

.

a(A)

o=O,

i s contained i n t h e n e g a t i v e r e a l axis.

A n o t a t i o n s i m i l a r t o t h a t f o r t h e f i r s t o r d e r c a s e w i l l be u s e f u l here.

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

g2(Co,o) w

2

for

The following analogue of Theorem

&

THEOREM 2.3. t h e h a l f plane continuous i n

t

2

km

u

E

2 Q

(Co,w)

wt Cge

@(

=

for

< 0 by Theorem 2 . 1 ) .

1.3.4 holds:

(t

L

:1 1 )

hR(h2 ;A)u

2 R( A ; A )

exists i n

(2.20)

0).

(Reh

> ho).

(2.a)

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

The proof i m i t a t e s t h a t o f Theorem

REMARK 2.4.

8(u)

E

eht@(t)u d t

F

05

and such t h a t

0

Assume t h a t , f o r each

A

if


an o p e r a t o r valued f u n c t i o n s t r o n g l y

ll@(t>ll5

Then

W

-a

i s t h e union of a l l

8 ( W )

be a n o p e r a t o r such t h a t

A

hot @(:)

Reh

2 K (CO,w)

to

g2 i s t h e union of all t h e

and

2 Q ( 0 ) i s empty f o r

(note t h a t

0

The c l a s s

(1.10).

21

C0

belows

(1.1) i s (uniformly) w e l l posed i n

t h e Cauchy problem for and

A

A c l o s e d , d e n s e l y defined o p e r a t o r

(1.1).

1.3.4 and we omit it.

I n e q u a l i t i e s ( 2 . 1 ) follow from t h e i r r e a l c o u n t e r p a r t s

( s e e Remark 1.3.5).

$11.3 A

Cosine f u n c t l o n t h e o r y . (E)-valued f u n c t i o n

C(i)

defined i n

-m


C m

i s called a

c o s i n e f u n c t i o n o r cosine o p e r a t o r f u n c t i o n i f t h e c o s i n e f u n c t i o n a l equations

(1.9) hold, t h a t i s , if

33

SECOND ORDER EQUATIONS

+ t) +

C ( 0 ) = I, C(s

s,t

f o r all

in

The propagator

(-m,m).

(3.1)

C ( s - t ) = 2C(s)C(t)

of a well posed second

C(t)

o r d e r Cauchy problem i s a s t r o n g l y continuous c o s i n e f u n c t i o n ; we s h a l l show i n t h i s s e c t i o n t h a t , j u s t as i n t h e f i r s t o r d e r case, t h e converse

i s also true.

c(G)

THEOREM 3.1. &eJ -m

A


c(f)

(S2 such t h a t

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

1 -

Au = l i m (C(h) 2 h-0 h of

=

i s t h e s e t of all

A

speaking, ii

- 21 +

(3.1) yields C(t)

(note t h a t

c(<) is

The i n f i n i t e s i m d g e n e r a t o r o f

Proof:

D(A)

be a. c o s i n e f u n c t i o n s t r o n g l y continuous i n

Then t h e r e e x i s t s a unique c l o s e d , d e n s e l y defined o p e r a t o r

a.

C(-h))u C(-t)

u

=

2 l i m (C(h) - I)u h-0 h

setting

s = 0).

(3.3)

The domain

such t h a t t h e l i m i t e x i s t s (roughly

i s t h e "second d e r i v a t i v e of

A

c E, a > 0 .

defined by

C

a t the origin").

Let

We define

ua =

- t)C(t)u

-?-

["(a a2u 0

Using (3.1) we o b t a i n 2

- (@(h) - I)ua

=

dt

-

.

(3.4)

t

h

- t)@(t)u

h2 h

.-

4 -

-

- t ) @ ( t ) ud t

a2h2 + h -

2 -

-h-

2 -

+

h

- t)@(t)udt

a%*

-h

- t ) @ ( t ) u d t, (3.5)

a2h2 thus

ua

E

D(A)

with AU"

ua

Since that

A

f o r all

-

u

as

a

-

hence

$ (c(a>-

I)U

a

0

it r e s u l t s t h a t

i s closed we n o t e t h a t

s,t;

=

(3.1)

D(A)

i s dense i n

implies t h a t

C(s)C(t)

E. =

T o show

C(t)C(s)

34

SECOND ORDER EQUATIONS 2 2 ( @ ( h ) -I)@(t)ll = @ ( t7 ) ( @ ( h ) -I)U

h

h2 for

h

#

Accordingly,

0.

u

if

D(A)

E

@(t)u

then

D(A)

E

and

A@(t)II = @(t)Au

f o r any

u

E.

E

Hence

after (3.6). Let Aun v. Then

{u }

-

2

(1.7)

n

( @ ( h )- 1)u

be a sequence i n

= l i m

so t h a t t a k i n g l i m i t s as

h

2 ( @ ( h )2

h

n-m

h

+

D(A)

h

I ) u n = l i m (Au ) n n- m

u

we deduce t h a t

0

D(A)

E

-

un

such t h a t

=

h

v

u,

,

and

Au = v

a s claimed.

If u

E

D(A)

we have

r t ( t- s ) @ ( s ) A u d s

=

' 0 = A

rt(t-

s ) @ ( s ) u ds =

Lt t2

( t - s ) A C ( s ) u ds

Au

t

=

C(t)u

- u.

(3.9)

0

Hence

@(G)u i s twice continuously d i f f e r e n t i a b l e i n @"(t)u

so t h a t

=

@(<) i s a s o l u t i o n of

@(t)Au (3.2)

=

-m


m

A@(t)u,

in

-m

C

with

(3.10)

t C

and t h e e x i s t e n c e

p o s t u l a t e i n t h e d e f i n i t i o n of w e l l posed problem i s v e r i f i e d .

To show

continuous dependence, we proceed a s i n Theorem 1.4.1; f i r s t we prove

ll@(t)lli s bounded

with t h e help of t h e uniform boundedness theorem t h a t on compact subsets of

and t h e n define

(-m,m)

s(t)u

rt@(s)u

= $'

so t h a t

Ils(t)il

ds

i s as well bounded on compacts.

a r b i t r a r y s o l u t i o n of

(3.2)

,

(3.11)

0

If

t h e proof of Theorem 2 . 1 t h a t t h e f u n c t i o n

@(t - S)U(S)

i s constant, whence t h e formula

u'(t)

u(;)

i s an

we can show e s s e n t i a l l y a s i n t h e end of

=

@(t)u(O) + s ( t ) u f ( o )

+ S(t

- s)ul(s)

35

SECOND ORDER EQUATIONS

This completes t h e proof o f Theorem

results.

$11.4

3.1.

The inhomogeneous e q u a t i o n . We c o n s i d e r here t h e e q u a t i o n

u"(t)

=

Au(t) + f ( t ) .

(4.1)

S o l u t i o n s a r e defined i n t h e same way a s s o l u t i o n s o f A c

(1.1); a g a i n i f

g2 uniqueness of s o l u t i o n s of t h e i n i t i a l value problem f o r

(4.1)

implies t h a t t h e r e i s a t most one s o l u t i o n of

(1.1)

satisfying the

i n i t i a l conditions u(0)

Let

f(:)

u(:)

is

uo, u ' ( 0 ) = u1'

be a continuous f u n c t i o n defined i n

be a s o l u t i o n of

fixed

=

a d

(4.1)

satisfying

(4.2) 0

5t 5

(4.2).

t > 0

Integrating i n

0

0 5 s 5 t with 5 t 5 T we o b t a i n

=

@ ( t ) u o + W(t)ul

(4.3)

A s i n t h e f i r s t o r d e r case,

S(t - s)f(s) d s .

(4.3)

(4.1)

w i l l be a genuine s o l u t i o n of

i f a d d i t i o n a l c o n d i t i o n s a r e imposed on

s)

derivative

+L t

u(t)

(1.1)

@(t - i ) u ( ;) + s(t - ^ s ) u ' (

i s continuously d i f f e r e n t i a b l e i n

- s)f(s).

and l e t

that @ ( f ) , s(i) a r e t h e s o l u t i o n o p e r a t o r s of

we o b t a i n using t h e e q u a t i o n t h a t t h e f u n c t i o n

S(t

T,

Assuming t h a t

f(t).

LENMA 4.1. Assume t h a t one of t h e following two c o n d i t i o n s

is

satisfied: (a)

f(t)

(b)

f(:)

E

D(A)

&

f(t), Af(t)

a r e continuous i n

0

5t 5

T.

5t 5

T,

or i s continuously d i f f e r e n t i a b l e i n

Suppose i n a d d i t i o n t h a t s o l u t i o n of

Proof:

(4.1)

g

0

uo,ul

E

D(A).

0

Then

(4.3)

D(A),

D1 2 D(A)

i s a genuine

1. t 5 T.

We have a l r e a d y noted t h a t

Do

=

and following comments) t h u s we only c o n s i d e r t h e l a s t term i n Since

%(;)

@ ( t )u,( t )

i s d i f f e r e n t i a b l e i n t h e norm of

i s always

(E)

(see (2.8) (4.3).

with d e r i v a t i v e

continuously d i f f e r e n t i a b l e w i t h d e r i v a t i v e u'(t)

=

rt@(t-s ) f ( s ) d s . 0

(4.4)

36

SECOND ORDER EQUATIONS

To e s t a b l i s h t h e conclusion of Lemma 4 . 1 i n e i t h e r case it i s enough t o show t h a t

u(t)

E

i s continuous and

D(A), Au(t)

=L t

(Au(s) + f ( s ) ) d s

u'(t)

If ( a ) holds we have

=Lt(

Au(s) d s

Lt(

LtAS(B

=

=

t h u s proving

/gsl(s

- r ) f ( r j d s)

Lt(C(t

- r ) f ( r )- f ( r ) )

(4.5).

If we assume ( b )

5t 5

(0

.

T)

- r)Af(r)dr)

(4.5)

ds

dr

,

dr

(4.3)

i n s t e a d we i n t e g r a t e

by p a r t s :

u(t)

=

-

[t(&-t-sS(r)f(s)dr) 0

=ktS(s)f(0) ds I n view of

ds

+Lt(

ds

j:-ss(r)f'(s)dr)

.

(l.?),

1 t ( @ (-t I'

Au(t) = @ ( t ) f ( O ) - f ( 0 )+

s ) - I)f'(s)

ds

' 0

= @ ( t ) f ( O )- f ( t )

+I

-t @(t - s)f'(s) d s

.

0

We o b t a i n now

(4.5)

i n t e g r a t i n g and r e v e r s i n g t h e o r d e r of i n t e g r a t i o n

( s e e t h e proof of Lemma 1 . 5 . 1 ) .

(4.3)

J u s t a s i n t h e f i r s t order case, we s h a l l c a l l s o l u t i o n of

(4.1)

say, a f u n c t i o n i n LEMMA 4.2. continuous i n

(w-'(ewt-1)

if 1

uo,ul

E

and

f

is,

L (o,T;E).

Assume 0

a r e a r b i t r a r y elements of

a generalized

5t 5

uo,ul T

E

E, f

E

1 L (o,T;E).

u(i> g

with

i s replaced by

t

continuously d i f f e r e n t t a b l e and

when

W

= 0).

u

0

=

0

then

U(<)

37

SECOND ORDER EQUATIONS

The proof i m i t a t e s t h a . t

of

Lema 1 . 5 . 2 and i s t h e r e f o r e omitted.

Estimations by hyperbolic f u n c t i o n s .

911.5

It i s sometimes convenient t o r e p l a c e t h e e s t i m a t e ( 1 . 1 0 ) f o r a cosine f u n c t i o n by

Obviously, both are e q u i v a l e n t , b u t t h e value of t h e constant i n g e n e r a l be d i f f e r e n t ; p r e c i s e l y ,

Co

will

(5.1) i m p l i e s ( 1 . 1 0 ) with t h e same

c o n s t a n t , whereas t o p a s s from (1.10) t o ( 5 . 1 ) t h e c o n s t a n t must be -2 doubled. If we denote by Q (Co,w) t h e c l a s s of a l l i n f i n i t e s i m a l g e n e r a t o r s of s t r o n g l y continuous c o s i n e f u n c t i o n s s a t i s f y i n g

If

W

> 0,

(5.1)

implies t h e following e s t i m a t e f o r

(5.1),

d(i): (5.2)

If

= 0,

t h e i n e q u a l i t y i s (1.15).

We can e a s i l y o b t a i n a g e n e r a t i o n theorem based on

(5.1) r a t h e r t h a n

although t h e c o u n t e r p a r t s of i n e q u a l i t i e s ( 2 . 1 ) a r e l e s s simple.

on ( l . l O ) ,

THEOREM 5.1.

Let

A

be c l o s e d .

The Cauchy problem for (1.1)2

uniformly well posed i n

(-m,m)

i f and only i f

e x i s t s i n t h e h a l f space

R(h2;A)

[l(hR(h2;A))(n)ll

5

with propagator

@(:)

satisfying

Reh >

(5.1)

(1)

C o ( - l ) nn!(Reh((Reh)*-U 2 )-1) ( n )

where t h e i n d i c a t e d d e r i v a t i v e s on t h e r i g h t hand s i d e a r e t a k e n with respect t o t h e variable

Proof.

ReX.

Combining t h e b a s i c formula (2.11) (which i s obtained

e x a c t l y as i n Theorem 2.1) with i n e q u a l i t y

(5.1) w e

obtain

38

SECOND XDER EQUATIONS

W e use t h e n again (2.11), t h i s time f o r t h e s c a l a r cosine function cosh

wt

(whose i n f i n i t e s i m a l generator i s

sequence o f i n e q u a l i t i e s

To prove t h e converse, we only need t o

t h e r e s u l t i s the

make a few minor changes i n

Observe f i r s t t h a t t h e f i r s t i n e q u a l i t y (5.3)

t h e proof of Theorem 2.1.

implies t h e f i r s t i n e q u a l i t y ( 2 . 1 ) .

l(t;u)

W2);

(5.3).

Thus t h e c o n s t r u c t i o n of t h e function

i n (2.12) and t h e p r o o f of i t s p r o p e r t i e s proceeds i n e x a c t l y t h e

same way.

However, t h e e s t i m a t i o n ( 2 . 1 5 ) i s s l i g h t l y d i f f e r e n t .

W e use

again t h e Post i n v e r s i o n formula (1.3.14) obtaining

where w e use on t h e right s i d e Laplace transform i s

h( h2

(I.3.u)for t h e function cosh

- w2)-l).

lilt

(whose

The r e s t of t h e proof i s j u s t l i k e

t h a t of Theorem 2.1 and we omit t h e d e t a i l s .

J u s t a s ( 2 . 1 ) , i n e q u a l i t i e s ( 5 . 5 ) follow from t h e i r

REMARK 5.2. r e a l counterparts

( s e e Remarks

$11.6

1.3.5 and 2 . 4 ) .

This can be again proved using Taylor s e r i e s .

Miscellaneous comments. Strongly continuous cosine f u n c t i o n s were introduced by SOV‘A [1966:1],

who defined t h e i n f i n i t e s i m a l generator and proved t h e generation theorem 2.1.

Other p r o o f s of Theorem 2.1 were given by DA PRATO-GIUSTI

and t h e author

[1969:3 1 i n c e r t a i n l o c a l l y convex spaces.

proof i s t h e one we have employed here. t h e norm of ( E )

[1967:17

This l a s t

Cosine functions continuous i n

were considered e a r l i e r by KUREF’A [1962:1] who t r e a t e d

a s well t h e case where t h e cosine f u n c t i o n t a k e s values i n a Banach algebra; t h e end r e s u l t of t h i s v e r s i o n of t h e theory i s t h e r e p r e s e n t a t i o n

C(t)

= cos(tA1I2)

( s e e Exercise

2 below).

The d e f i n i t i o n of properly

posed Cauchy problems f o r higher order equations (of which (1.1))i s a p a r t i c u l a r case) i s due t o t h e author [1969:21, a s well a s t h e r e l a t i o n between s t r o n g l y continuous cosine functions and s o l u t i o n operators of second order equations.

Theorem 1.1 i s due t o t h e author [1969:21; a

39

SECOND ORDER EQUATIONS

r e s u l t of t h e same "measurability implies c o n t i n u i t y " type was proved

KUREPA

e a r l i e r by

[1962:1], where both measurability and c o n t i n u i t y a r e

understood i n t h e norm of ( E ) ( o r , more g e n e r a l l y , i n t h e norm of a Eanach a l g e b r a ) .

Theorem 2.3 i s due t o t h e author [1969:2].

EXERCISE 1. Let

A

be a bounded operator i n a Esnach space E.

Show

n

that

A

E

t h a t i s , t h a t t h e Cauchy problem f o r

Ed,

i s uniformly well posed i n generated by

A

-00


The cosine f u n c t i o n

m.

c(:)

i s given by t2n

m

@ ( t ) = cosh(tA1/2)

An,

=

n=O t h e s e r i e s (6.2) uniformly convergent on compact subsets of


-m

m.

We also have 00

8 ( t ) = A-1/2sinh(tA1'2)

t2n+l

(2n+l)1 An,

=

n=O the series

(6.3) converging i n t h e same sense a s ( 6 . 2 ) .

expressions cosh (tA1/2)

and

A-1/2sinh(tA1/2)

do not assume t h e existence of square r o o t s of i s given by t h e s e r i e s ( 6 . 2 ) and

(6.3)

.

Note t h a t t h e

a r e p u r e l y symbolic and

A;

t h e i r actuaJ meaning

@(;),

Prove t h a t

S(:)

can

a l s o be expressed a s

@(t)

where

r

=

&J;osh(th1/2)R(h;A)

,

(6.4)

i s a simple closed curve ( o r t h e union of a f i n i t e number of

t h e s e ) oriented counterclockwise and enclosing interior.

dh

(Note t h a t t h e choice of square r o o t

of (6.4) and h-1'2sinh(th1/2)

o(A)

in its (their)

h1/2 i n t h e integrands

(6.5) i s i r r e l e v a n t , since cosh(th1l2) = t2"ln/(2n>:, = 7 t2n+1hn/(2n + l)!). Show t h a t (6.4) and (6.5) imply

40

SECOND ORDER EQUATIONS

; A

0 = SuprRe A’/‘

(i)

Alternately,

E

.

~ ( A ) I

(6.8)

i s t h e l e a s t p o s i t i v e number such t h a t

(,lo

a(A)

i s con-

t a i n e d i n t h e closed r e g i o n t o t h e l e f t of t h e parabola

5

(d

>0

(with

$/4‘,’2

2

p a s s i n g through t h e p o i n t s with

(? -

=

, -+ S i O i2 .

(0

,

(6.9)

I n p a r t i c u l a r , (6.6)and(6.7)hold

n a t u r a l l y depending on

C

( 6 . 6 ) and (6.7) do not n e c e s s a r i l y hold w i t h 8(;)

5;

0’

A

s(5)

and

s(t^)i s

@(;)

Show t h a t

(1)

holomorphic

@(;)

i n particular,

continuous i n t h e norm of (E)(of course, t h e norm of ( E )

w =

@(i)

can be extended t o f u n c t i o n s

(as (E)-valued f u n c t i o n s ) f o r all

is

Produce a n example t o show t h a t

contained i n t h e negative r e a l a x i s . and

a(A)

if

(11)

is

always continuous i n

l/@(;)ll

due t o formula ( 1 . 5 ) and boundedness of

on

compact s u b s e t s ) . n

EXERCISE 2. topology of

@ ( t )be a c o s i n e f u n c t i o n continuous i n t h e

Let

(E)

t

(continuity a t

suffices).

= 0

A

use formula ( 2 . 1 1 )

n = 1 and c o n t i n u i t y of

for

- I I/

Ilh2R( h2;A)

show t h a t 2 2 h R(h ; A )

of

< 1 for

h

h a s a bounded i n v e r s e ) .

r e p r e s e n t a t i o n s ( 6 . 2 ) and

EXERCISE 3.

Let

-

1x1 -,

as

0

t

at

=

0

to

@(;)

admits the

(6.4).

E = C

m

@(;)

(Hint:

s u f f i c i e n t l y l a r g e so t h a t

Show t h a t

be t h e Banach space of all (complex-

(-m,m)

0 valued) continuous f u n c t i o n s u ( x ) u(x)

Show t h a t t h e

@(<) i s a bounded o p e r a t o r .

i n f i n i t e s i m a l generator

defined i n

endowed with t h e norm

-m

<x <

m

and such t h a t

(1.6.8).

Define @ ( t ) u ( x )= Show t h a t

,.

5

(U(X

+ t) +

u(x

-t))

(-m

.c t

<

(6.10)

m).

@ ( t )i s a s t r o n g l y continuous c o s i n e f u n c t i o n i n

i t s infinitesimal generator.

EXERCISE

4.

Identify

Show t h a t x +t

b(t)u(x) =

E.

1,

u(%) d5

(-m

Prove t h e r e s u l t of Exercise

i n f i n i t e s i m a l g e n e r a t o r ) i n t h e space

LP(-m,m).


m)

.

(6.11)

3 (and i d e n t i f y t h e Show t h a t (6.11) i s

n o t s t r o n g l y continuous (or Even s t r o n g l y measurable) i n

Lm(-m,m).

41

SECOND ORDER EQUATIONS

EXERCISE

Show t h a t (defined by

5.

A E 6

Let

A

6.8) i s f i n i t e .

Prove t h e formulas

C(t)u =

where

P(dh)

H.

be a normal operator i n a Hilbert space -2 ( o r , equivalently, t o Q ) i f and only i f b10

2

(u

E

H,

--a,


(6.12)

C m),

i s t h e r e s o l u t i o n of t h e i d e n t i t y associated with

A.

Using t h e f u n c t i o n a l c a l c u l u s f o r normal operators show t h a t

EXERCISE

6.

Let

@ ( t )i s normal.

be a s t r o n g l y continuous cosine f u n c t i o n

i t s i n f i n i t e s i m a l generator.

H, A

i n a H i l b e r t space each

@(;)

A

Show t h a t

i s normal (Hint:

Assume t h a t proceed a s i n

Exercise 1 . 9 ) EXERCISE

@(t)

7.

Under t h e hypoteses of Exercise

i s self adjoint.

Then

A

i s self adjoint.

6.6 assume t h a t each (Hint:

proceed a s i n

Exercise 1.10)

EXERCISE 8.

Show by means of an

with i n f i n i t e s i m a l generator

A

example, t h a t t h e r e e x i s t s a

c(i)

i n separable Hilbert space H such t h a t Irl > tclo, where hjo i s defined

s t r o n g l y continuous cosine function

by (6.8) and 1 w = l i m sup - logllC(t) t-+m

More g e n e r a l l y , we can construct

where

h > 0

EXERCISE

t

I[

.

(6.16)

@(<) i n such a way t h a t

i s a r b i t r a r i l y preassigned ( s e e Ekercise 1.13).

9.

Prove t h a t t h e "real" i n e q u a l i t i e s ( 2 . 2 2 ) imply t h e i r

complex counterparts ( 2 . 1 ) . imply i n e q u a l i t i e s

(5.3).

Likewise, show t h a t i n e q u a l i t i e s

(5.6)

42

SECOND ORDER EQUATIONS

EXERCISE 10. Let

c(;>

i t s i n f i n i t e s i m a l generator, Show t h a t i f

u

E

D(A)

a s t r o n g l y continuous cosine function, '1)

t h e number defined by (6.16),

EXERCISE 11. L e t

>

W,

0.

then

It I > 0 .

t h e l i m i t being uniform on compact subsets of

S(;)

uf

A

A

E

2. Show t h a t

A

E

."_,t h e

propagator

of t h e equation

(6.17)

u ' ( t ) = Au(t) given by t h e a b s t r a c t Weierstrass formula.

where

C(t)

i s t h e propagator of

u " ( t ) = Au(t)

.

FOOTNOTES TO CHAPTER I1

(1) See i n e q u a l i t i e s (2)

(1.3.25).

See footnote (2), p . 27; h e r e

transform of

hR( A;A)

u

i s meant t o be t h e inverse Laplace

with t h e adjustments needed t o produce conver-

gence and t o l e r a t e two d i f f e r e n t i a t i o n s with r e s p e c t t o

t.

43

CIUPTEK I11 REDUCTION OF A SECOND ORDER EQUATION TO A A FIRST ORDER SYSTEM.

PHASE SPACES.

5111.1 Phase spaces. It seems n a t u r a l t o a s k whether t h e Cauchy problem f o r t h e equation

E

i n t h e Banach space

can be reduced t o t h e Cauchy problem f o r a f i r s t

t h e obvious candidate i s of course t h e system

o r d e r equation:

ui(t)

q(t),

=

Auo(t),

(1.2)

i s considered i n t h e space

u,(t)

E X E

it may g i v e r i s e t o an improperly posed Cauchy problem due t o t h e u'(t),

fact that

However, i f

u'(t) 1

where

=

u(t).

=

(1.2)

u(t),

unlike

may f a i l t o depend continuously on

u(o), u'(0). EXAMPLE 1.1. We can c a s t t h e one-dimensional wave equation utt

=

uxx

l e t , say,

u

E

to

E = L

2 (-m,m),

(1.1) i n t h e following way ( s e e Exercise

Au(x)

=

u"(x)

where

D(A)

11.3)

i s t h e s e t of a l l

E such t h a t u" (understood i n t h e sense of d i s t r i b u t i o n s ) belongs 2 Then t h e Cauchy problem f o r (1.1) i s well posed with pro-

L

.

pagators ly,

i n t h e form

@(;),

@'(t)u

s ( < ) given

by formulas

(11.

6.10),

(11. 6.11).

Obvious-

i s g i v e n by 1 @'(t)u(x) = (ul(G 2

-

t h u s it i s not a bounded o p e r a t o r i n However, t h e r e d u c t i o n measured i n a d i f f e r e n t norm.

(1.2)

+

t) - u'(X-t)),

E.

w i l l always suceed i f

u,(t)

is

Since more t h a n one of t h e s e norms w i l l

be used i n t h e f u t u r e w e g i v e t h e following g e n e r a l d e f i n i t i o n , i n which

we do not p o s t u l a t e t h a t t h e Cauchy problem f o r

(1.1)i s n e c e s s a r i l y

44

PHASE SPACES

well posed i n t h e sense of $11.1; we assume, however, t h a t c o n d i t i o n i n t h e d e f i n i t i o n of w e l l posed problem holds t h e r e e x i s t s a dense subspace t h e r e e x i s t s a s o l u t i o n of

C2 = E

x El

0

of

E

2

uo, ul

such t h a t f o r every

E

D

(1.1) i s a product

(endowed with any of i t s product norms) where

Eo

El a r e Banach spaces s a t i s f y i n g t h e following assumptions:

and

% 5E

( a ) Eo,

(b) C2 = E

0

2

t

0

1

f o r any s o l u t i o n

u ( t ) with u ( 0 )

We note an obvious consequence of

E

El

6(;) 2

such t h a t

(1.1) with

s o l u t i o n of u'(t)

2

(-.

Eo

There e x i s t s a s t r o n g l y continuous semigroup

x E

n Eo

D

w i t h bounded i n c l u s i o n : moreover,

D n E ) i s dense i n Eo i n t h e topology of 1 dense i n El i n t h e topology of El). (-.

in

(a)

t h a t is, t h a t

0,

(1.1) s a t i s f y i n g t h e i n i t i a l c o n d i t i o n s

t 2 0 for t h e e q u a t i o n

A phase space i n space

D

t

in

for a l l

t

u(0)

2

0,

E

(b):

Eo, u ' ( 0 )

thus

Eo

E

and

Eo, u ' ( 0 )

E

u(<)

if

El t h e n

El

E

El.

i s an a r b i t r a r y

u(t)

Eo

E

and

a r e i n v a r i a n t subspaces

f o r t h e vaJues of a s o l u t i o n and fox i t s d e r i v a t i v e , r e s p e c t i v e l y .

Note

a l s o t h a t t h e r e might w e l l be s o l u t i o n s whose i n i t i a l values do not belong t o

Eo, El; ( b )

does not apply t o t h e s e s o l u t i o n s .

The n o t i o n of phase space i n defined; h e r e

all

t

u(0)

E

if

B(:)


m

i s correspondingly

i s required t o be a group a n d

u(<)

Eo, u'(0)

-m

i s any s o l u t i o n of E

(1.1) i n

-= t

<

happens f o r m

with

El.

W e show below t h a t i f t h e Cnuchy problem f o r i n t h e sense o f

-m

(1.4)

$11.1 t h e n a phase space

(1.1) i s well posed

Em (which i s maximal i n

a n obvious s e n s e ) always e x i s t s (of course, t h e t r i v i a l choice Eo

=

E

=

1

-

= {O] a l s o provides a phase space!) To c o n s t r u c t 1 E with i t s o r i g i n a l norm, while E = D1, where D1

E

0

-

C2

we t a k e m i s t h e space

45

PHASE SPACES

u such t h a t @ ( t ) u i s c o n t i n u o u s l y d i f f e r e n t i a b l e i n < t < co ( a s we s h a l l s e e l a t e r i n Theorem 1 . 2 , t h i s s p a c e i s none

of a l l -m

D1,

other than

t h e space of a d m i s s i b l e i n i t i a l v a l u e s f o r s o l u t i o n s

-

o f (1.1), d e f i n e d i n 511.1). To d e f i n e the norm i n

LEMMA C = C(u)

For every u c Eo

1.2.

=

-D1

t h e r e e x i s t s a constant

such t h a t ll@t(t)ull 5 C(1

(id

we use t h e a u x i l i a r y result below.

Eo = Dl

the constant i n

+

It

(11.1.10)).

Proof: We b e g i n b y renorrning t h e space llullt

e-O

SUP

=

as f o l l o w s :

E

I

11.

1s l l ~ ( s ) u

-m<s<m

/1u/1 5 I/u/I'

Obviously,

5

t h u s t h e new norm i s e q u i v a l e n t t o t h e

ColluI1,

o r i g i n a l one and it i s enough t o p r o v e

u

t

l l @ ( t ) u l / l5 el*lItlllul/t (-a<

D

1' v a l u e s of @(:)

Set

q(t)

=

t <

We have

(1.6)

m).

exp(-i~it)(l@'(t)ull' f o r

t

2

Since d i f f e r e n t

0.

commute ( a consequence of t h e c o s i n e f u n c t i o n a l e q u a t i o n

(11.1.9)) we c a n d i f f e r e n t i a t e

11'11'

II*/l'.

for

(11.1.9). Hence

by t h e cosine functional equation

Let

(1.5)

norms, m u l t i p l y i n g by

(11.1.7) w i t h r e s p e c t t o

exp(-rA)(s + t ) )

s.

Taking

(71.1.10) we

a n d using

obtain

(1.7) i m p l i e s

Obviously,

m

arbitrary let

-

q((t m) of

q(s)

since

+

m)

in

@I(t^)u

?(a)

5

mq(1)

be the g r e a t e s t i n t e g e r

5

q(t

- m)

0

5

51.

s

+

mq(1)

5

5 t. where

=

...

1,2,

Then C

q(t)

For

t

2

5

i s t h e supremum

(1.5) f o l l o w s f o r t 2 0; ( 1 . 5 ) h o l d s f o r all t . T h i s

Accordingly,

i s a n odd f u n c t i o n ,

completes the p r o o f .

C(1 + t )

for m

0

PHASE S P A C E S

46

.. We can now d e f i n e a norm i n

D

1 by

+ t)-'llet(t)ul!I.

1 1 u l 1=~ rnax{llull, sup e - " t ( l

t>o

A

-

D1

moment's c o n s i d e r a t i o n shows i h a t

(1.8)

i s a Eanach space equipped

11. 1l0.

with

THEORFM 1 . 3 .

The space

i s a phase space for

C?

5

x E 1 with = E 0 < t < ffi; moreover,

= E

m

(1.1)

-m

-

-

and

Eo = D1

(1.9)

I11 = D1'

6(<)i s given by

The group

(1.lo)

and i t s i n f i n i t e s i m a l g e n e r a t o r i s

(1.11)

w i t h domain

D(I1) = D ( A )

D1.

X

max ( j \ u \ / , \ \ v l \ ) i s used i n

(-m

where

c0

and -

are the constants i n

u

=

m

m

~tl)?w

1)*(1 +

(gmm>

/l[u,vl\\,

\/U!le =

Em we have

5 (co +

IlG(t)l!

I f , say, t h e norm


(1.12)

m).

(11.1.10).

The proof t h a t follows i s by no means t h e s h o r t e s t but can be adapted e a s i l y t o a more g e n e r a l o i t u a t i o n ( f o r an a l t e r n a t e proof s e e

As basis a r e equalities

Exercise 111.15).

(11.1.7) and

We have a l r e a d y noted t h a t t h e second h o l d s f o r every

u

f i r s t can be extended from with r e s p e c t t o

t

in

0

E

5t 5

D 7,

to

u

We prove now t h a t each

i s even and

t

1.

u

c

-

D1

E

E.

The

f o r t h i s we i n t e g r a t e

u

E

E

by c o n t i n u i t y ,

and d i f f e r e n t i a t e

z.

with r e s p e c t t o

t o work i n

61;

extend t o d l

apply t h e e q u a l i t y t h u s obtained t o any

@(:)

E

u

(11.1.8).

0.

c'(:),

6 ( t ) i s a bounded operator i n

8(:)

To show t h a t

%;

since

a r e odd, it i s obviously s u f f i c i e n t

@(<) i s a bounded operator from

47

PHASE SPACES

El

(11.1.7) d i f f e r e n t i a t e d w i t h r e s p e c t t o

we use

El

into

@'(s)@(t)u

t h e form

=

@'(s + t)u-@(s)@'(t)u.

in

s

It i m p l i e s t h a t

l l @ f ( s ) @ ( t ) u l l5 l l @ ' ( s + t ) U / l + Il@(s)ll ll@'(t)Ull

-< (1 + Since

(1 + s

+ t) 5

ll@(t)ul10 5

i n t h e form

Estimates

t ) e w(s+t)jlul10 +

(1 + s ) ( l + t )

(co

@'(s)s(t)u

1)(1

+

8 ( t ) w e use

To h a n d l e

and t h a t

+

s

(1.13) t o

bounded o p e r a t o r from

E

(uO

6

E0)

(1.131

d i f f e r e n t i a t e d with r e s p e c t t o

@(s+ t ) u

S t ( t ) = @ ( t ) maps

t ) e~ ( s + t ) l l u l l O .

we o b t a i n .

t)el*~tlluI/O

+

(1I.l.a) =

c0(1 +

- @(s ) @ ( t ) u ;

into

E

s

the result i s

with

(1.16) c l e a r l y show t h a t e a c h 6 ( t ) i s a E

with t h e norm i n d i c a t e d i n

E

into

(1.12).

6(;) s a t i s f i e s t h e f i r s t o f t h e semigroup e q u a t i o n s i s o b v i o u s ; t h e second f o l l o w s from (11.1.7) a n d (11.1.8). The n e x t t a s k i s t o show t h a t 6(<)i s s t r o n g l y c o n t i n u o u s : s i n c e The f a c t t h a t

G(tl)

-6(t)

=

G(t)(G(t'

- t ) - I)

it i s o b v i o u s l y s u f f i c i e n t to p r o v e

s t r o n g c o n t i n u i t y a t t h e o r i g i n and, due t o t h e symmetries o f t h e o p e r a t o r s i n the m a t r i x o f

6(t)

IIE(h)u-ull

+

Q*

we o n l y need t o show t h a t 0

( h - 0+)

Again, t h i s must be done i n f o u r s t e p s . (11.1.7)

with respect t o

s

.

(1- 1 7 )

For t h e f i r s t we d i f f e r e n t i a t e

and recombine it i n the form

48

PHASE SPACES

@ ' ( s ) ( @ ( h ) u - u )= @ ' ( s

+

h)u-@'(s)u-@(s)@'(h)u

(U

F

E o ) . (1.18)

W e d i f f e r e n t i a t e now t h e c o s i n e f u n c t i o n a l equation @(s

+ t ) + @ ( s - t ) = 2 @ ( s ) @ ( t ) with r e s p e c t t o t , o b t a i n i n g + t ) u - @ ' ( s - t ) u = 2 @ ( s ) @ ' ( t ) u for u E Eo. Replacing s

@'(s

s

+ h/2

t

and

by

h/2

@'(s

we o b t a i n

(1.18), we o b t a i n

- ~ 1 1 " -, 0

ll@(h)u

u

E

E

f o r every

u

t

E.

0'

(1.191

+ h ) u - @ ' ( s ) u = 2@(~+ h/2)@'(h/2).

Combining t h i s e q u a l i t y with

f o r every

by

( h * 0+)

(1.20)

The second s t e p i s t o prove t h a t

To do t h i s we use t h e d i f f e r e n t i a t e d v e r s i o n of

(11.1.8) @'(s)8(h)u = @(s + h ) u - @ ( s ) u + @(s)(@(h)U-u).

(1.22)

F i n a l l y , we m u s t show t h a t

for e v e r y u

E

Eo,

and IlS'(h)u

u

f o r every

(1.19)

to

E

E,

(1.23)

- u1i

-

-,

(1.24)

0+)

P u t t i n g now

(1.17) follows, completing t h e proof

together,

6(<).

It only remains t o i d e n t i f y

14,

t h e i n f i n i t e s i m a l g e n e r a t o r of

T o do t h i s we show f i r s t t h a t 2i

In fact, l e t h-'(@(h)u

- u)

(1.22)

-

0

v,

(1-25)

u c D(A)

and

v

E

9.

(1.19); t h e r e s u l t i s t h a t

i n t h e norm 'of

applied t o

c - %. so t h a t

LI = [ u , v ] E D ( A ) ,

(1.18) by h and use

Divide

with

(h

both of which r e l a t i o n s a r e obvious.

o f t h e s t r o n g c o n t i n u i t y of

6(t).

0

Eo.

We proceed i n t h e same way

obtaining t h a t

h-'S(h)v

-, v,

also

49

PHASE SPACES

i n t h e norm of h-'(@(h)u-u)

-

Since

EO'

@'(O)u

h-l(@'(h)u

0

=

E,

in

- u)

-

C " ( 0 ) u = Au

(1.25)

t h e proof of

and i s complete.

Two more a u x i l i a r y f a c t s a r e necessary f o r t h e proof t h a t a c t u a l l y coincide.

(i)

2' J

%

and

23

The f i r s t i s

i s closed.

{un) = { [ u n , v n ] 3 be a

To see t h i s , l e t

-

Bm a.nd 8 u n ( w , z ) E Em u and vn w i n Eo, vn v and i n t h e norm of Em so t h a t un A u n - z i n E. Then w E Eo and (by closedness of a), u E D(A) with Au = z , i . e . [ u , v ] E D(N) a n d % [ u , v ] = [v,Au] = [ w , ~ ] . sequence i n

(ii)

D(8)

u

For every

E

E

[u,v]

un

such t h a t

E

-

-

w e have

(1.26)

Lh

S(t)u d t

E

D(A)

(U E

E).

(1.28)

The f i r s t two have a l r e a d y been e s t a b l i s h e d a t t h e beginning of t h e proof.

To prove

(1.28) A

we note t h a t f o r

kh

-

a sequence E

D1

{un}

E

D(A)

we have

S ( t ) u d t = @(h)u- u,

which e q u a l i t y can be extended t o a l l

u

u

in D(A).

As f o r

(1.29)

E approximating u by (1.27), w r i t e (1.29) f o r

u

E

and d i f f e r e n t i a t e (using a g a i n closedness of

A).

The r e s u l t i s

A L h @ ( t ) u d t = @'(h)u.

We show f i n a l l y t h a t

Let

uh

u

be an a r b i t r a r y element of Em. Since elements of t h e form

a r e dense i n

Em, i f

(1.25)

holds we can s e l e c t a sequence

50

PHASE SPACES

D(%)

in

such t h a t

2(un h =

%, %uh

closedness of

D(B)

u

h -, 0

we l e t

= l i m

uh

Un

particular that

Then

- u).

-

( U n p

-

h'(B(h)u

+

h-l(6(h)u

=

D(%)

belongs t o

D(%)

C?

in un)

u

If

whereas

Uh

(1.4.5))

u)(see

% = lim h

with

%

-1

(6(h)u - u )

that This

= 'pu.

(1.30).

must be dense i n

Em (Theorem 1 . 4 . 1 )

we o b t a i n i n

i s dense i n E0 i n t h e topology of

D(%)

t h u s by

i n a d d i t i o n belongs t o

and o b t a i n s g a i n from closedness of

completes t h e proof of Since

u

+

h-'(G(h)un-

=

Eo.

-D1,

Let

u be an a r b i t r a r y element of E,, = {un] a sequence i n D(A) such t h a t u u i n EO. Then we have S ( t ) u n s ( t ) u and +

n As(t>un = 8"(t)un = @ ' ( t ) u n that

8(t)u

D(A)

E

s o l u t i o n of

@'(t)u

AS(t)u =

and

(1.1) i n

-

+


-W

whence it follows

t

f o r a31

s"(t)u,

so t h a t

proving t h a t

m,

(1.9)

opposite i n c l u s i o n i s obvious, t h u s

S(t)u

is a

5 D1.

D1

The

holds and t h e proof of

Theorem 1.3 i s complete. Theorem 1 . 2 might be taken t o imply t h a t "every well posed second o r d e r Cauchy problem can be reduced t o an e q u a l l y well posed f i r s t order system", t h u s t h e t h e o r y of go t o semigroup t h e o r y .

(1.1) could be reduced from t h e word

I n r e a l i t y , t h e s i t u a t i o n i s not s o simple;

on t h e one hand, t h e growth of

s(i) a t

match a c c u r a t e l y t h e growth of

@(;)

on t h e o t h e r hand t h e norm of

i n f i n i t y ( s e e 1.11) does not

or even t h a t of

S(:),

while

i s not easy t o i d e n t i f y (it depends

Eo

on complete knowledge of t h e s o l u t i o n s of

(1.1)). Taking a c l u e

from t h e wave and Klein-Gordon equations we may expect t h a t t h e reduction (1.2) t o f i r s t o r d e r can be made t o work i f one measures graph norm of a square root of

A.

u(t)

i n the

This p r e s c r i p t i o n w i l l be seen t o

work, a l b e i t with some r e s t r i c t i o n s , i n t h e next s e c t i o n s . 5111.2

F r a c t i o n a l powers of closed o p e r a t o r s .

A densely defined o p e r a t o r a(C)

(C > 0)

i s s a i d t o belong t o t h e c l a s s

A

H(A;A)

i f and only i f

IIR(A;A)JI 5 C/A Obviously we have inequalities

a(1)

(1.3.24)

w e only can a s s u r e t h a t

=

$(l,O)

with

since

5

0

Cn/An

(2.1)

(2.1) implies all t h e

If

= 0.

(n

and

.

( h> 0)

Co = 1,

IIR(h;A)"II

A > 0

exists for

=

C > 1, however,

0,1,. ..)

which does

PHASE SPACES

A

n o t imply t h a t

E

&

51

(for a counterexample see HILLE-PHILLIPS

[1957:1,P. 3711). We show below t h a t f r a c t i o n a l powers of

(or r a t h e r of

A

(-A))

c a n be c o n s t r u c t e d employing t h e formula

(-A)

cy

=

sinwr -

R(h;A)(-A) d h

9

which i s t h e v e c t o r c o u n t e r p a r t of a w e l l known s c a l a r formula

[1963:1, p . 3031).

(GRADSTEIN-RIDZYK

Although complex v a l u e s of

a r e handled j u s t a s e a s i l y , we s h a l l only u t i l i z e i n t h e range

5 cy 5 1.

0

KQ'

i s defined by t h e r i g h t hand s i d e of

KCYu = -s inQ'iT where t h e branch of the integral

r e a l , mostly

We r e s t r i c t t h e t r e a t m e n t accordingly.

0 < cy < 1. The o p e r a t o r

Let

a

( w i t h domain

D(KcU) = D ( A ) )

t h a t i s , by

(2.2),

R(A;A)(-A)u

d?

,

i s t h a t which i s r e a l when

(2.3) h

2

That

0.

i s convergent a t i n f ' i n i t y i s c l e a r from

(2.3)

w

(2.1);

near z e r o we use t h e second r e s o l v e n t e q u a t i o n i n t h e form R( A;A)( -A)U

LEMMA 2 . 1

u

n

-

Proof: 0, K u a n

The o p e r a t o r 0 <

Let

-

v

t

Q'

E.

Ka

< 1,

[u,)

Select

p

Yn =

u

=

- M ( A;A)U.

(2.4)

i s c l o s a b l e for each

a sequence i n

> 0

D(A)

Q',

0 <

< 1.

with

f i x e d and write

.

R(~;A)K@u~

It follows from i t s d e f i n i t i o n t h a t

Q'

(2.5)

KQ' commutes w i t h

R(p;A)

i n the

sense t h a t R ( P ; A ) K ~ U = K"R(P;A)U

for

u

E

D(KQ')

=

D(A).

Moreover,

KaR(p;A)

(2.6)

i s everywhere defined and

bounded s i n c e K~R(P;A)U =

sinq TT

(I

- hR( A;A))u

dh

52

PHASE SPACES

(2.5)

Hence we obtain, taking l i m i t s i n

R(p;A)v

thus

0,

=

vn- 0

that

so t h a t

v = 0, ending t h e proof.

Lemma 2 . 1 makes p o s s i b l e t h e following d e f i n i t i o n :

(-A)

C

Y

-

KCY ( 0 <

=

where t h e b a r i n d i c a t e s c l o s u r e .

1

(-A)

= -A.

CI

< l),

(2.8) (-A)

O f course, we d e f i n e

(-A)cy

The following r e s u l t s imply t h a t

0

I,

=

exhibits, a t

l e a s t i n p a r t , t h e behavior expected of a f r a c t i o n a l power.

u

LEMMA 2.2.

D(A).

E

l i m K u cy a.- 1-

Proof: 5

where

=

and

h = a > 0.

III~II 5

(2.2);

SUP

A? a

the result i s

u + R(A;A)Au

E

D(A);

we r e t u r n t o

(2.11)

u

(2.1).

Since

we have

-

u

as

h

-

by v i r t u e of t h e f a c t t h a t

be extended t o a l l

(2.11)

ll((h + ~ ) R ( A ; A ) - I ) A u ~ / .

hR(h;A)u

u

we o b t a i n

For t h e second i n t e g r a l we make use again

T o e s t i m a t e t h e r i g h t hand s i d e o f

for

(2.2)

I2 correspond t o t h e division of t h e domain of

o f t h e s c a l a r v e r s i o n of

=

(2.9)

Using t h e s c a l a r c o u n t e r p a r t of

integration at

hR(A;A)u

-Au.

E

E

(2.12)

w

l(hR(h;A)(\
(2.12)

and shows t h a t t h e r i g h t hand s i d e of

can be made a r b i t r a r i l y small for a l a r g e enough. e s t i m a t e i s independent of

CY.

To d e a l with

can (2.11)

Obviously t h e

I1 we o n l y have t o

observe t h a t t h e integrand can be bounded uniformly w i t h r e s p e c t t o ct

for

ct

t o zero a s

near

1 ( s e e ( 2 . 4 ) ) thus t h e f a c t o r

sinah

(/%/I

drags

a - 1. This completes t h e p r o o f .

The corresponding r e s u l t i n t h e neighborhood of

LEMMA 2.3

Let u

E

D(A)

be such t h a t

=

0

is

53

PHASE SPACES hR(A;A)u

-

as

0

A-

O+.

(2.13)

Then l i m K u CY-

=

(Y

o+

u.

(2.14)

Proof: Proceeding i n a way s i m i l a r t o t h a t used i n Lemma 2.2 (2.4)

b u t making use of

where

J1

J2 a r i s e once again from d i v i s i o n of t h e i n t e r v a l of

and

A

integration a t

+

that

(A

that

11J211 -, 0

we o b t a i n

=

a > 0. u

l)R(A;A)uas

CY

-

=

0.

To e s t i m a t e t h e second i n t e g r a l we observe

R(A;A)(Au

+ u)

a n d use

it f o l l o w s

(2.1);

The corresponding bound for

J1

follows

from t h e i n e q u a l i t y

I n f a c t , the p o r t i o n of t h e bound corresponding t o t h e c o n s t a n t obviously tends t o z e r o when

t h u s we can make

llJ,II

(Y

-

0.

C

On t h e o t h e r hand,

a r b i t r a r i l y small by t a k i n g a small enough.

This completes the p r o o f . An obviously d e s i r a b l e p r o p e r t y of any d e f i n i t i o n o f f r a c t i o n a l power i s t h e a d d i t i v i t y r e l a t i o n

= (-A)"(-A)'.

I n t h e present

l e v e l of g e n e r a l i t y only a s l i g h t l y weaker statement can be proved (Theorem 2.5 below).

I t s key i n g r e d i e n t i s t h e following r e s u l t .

(2.16)

54

PHASE SPACES

u

Proof: Let

2 D(A ).

E

/'"/'"

s i n q si$n

KKu=--

ffB

Tr

7r

K u c D(A)

Since

D(K@u)

=

B

we may w r i t e

B-1 2 p R(A;A)R(p;A)A u dAdp

" O d O

.

( t h a t t h e i n t e g r a l i s a b s o l u t e l y convergent i n t h e quadrant

i s shown i n t h e same way a s for

(2.3)).

i n t e g r a t i o n i n t o t h e two t r i a n g l e s change of v a r i a b l e

A by

interchanging

@

5

2

A,p

0

Divide t h e domain of h

in the f i r s t ,

p = ?ur

(2.17)

A

5

A

and

and perform t h e

p

i n t h e second;

pa

=

i n t h e second i n t e g r a l we o b t a i n .

p

(2.18)

U s i n g t h e second r e s o l v e n t e q u a t i o n we deduce t h a t

2

R ( ;"o;A)R( A;A)A u

=

1 {R( h;A) - a R ( A a ; A ) ] (

1 - 0

and r e p l a c e t h i s expression i n t h e integrand of

(l-a)-l

of the factor

-

(2.19) On account

(2.18).

t h e two i n t e g r a l s r e s u l t i n g from

(2.19)

as t h e l i m i t when

(2.18)

cannot be separated, b u t we can w r i t e p

-A)u

1- of

s i n w s i M r Frn --

h"@-lR( h ; A ) ( - A ) u d a dh

TT

mJyp-

- -sinWJ lr

B h"+B-lR(b;A)(-A)~

7r

0

dadh.

(2.20)

0

W e compute t h e second term i n t e g r a t i n g f i r s t i n

A,

making t h e change

of v a r i a b l e

by

A.

)us = i-1

and t h e n chmging a g a i n

p

We o b t a i n (2.a)

where

C a,B

To e v a l u a t e t h e constant

E =

c,

A = 1,

so that

we apply t h e argument above t o

1

f o r all

ca,B --

sin(cu

KO

=

@

and it

tha.t

This completes t h e proof of

(2.16).

7T

-t

p )T

.

follows from

(2.21)

55

PKASE SPACES

where t h e b a r i n d i c a t e s c l o s u r e . Proof:

<

a+ B

1 and l e t

u E D((-A)'(-A)')

and of t h e f a c t t h a t

Assume f i r s t t h a t

>

p

and

commutes w i t h

(-A)'

i n t u r n follows from commutativity of

I n view of Lemma 2.4

0.

for a n y

R(p;A)

K,

and

w e have

R(p;A))

'O

= ( ~ R ( ~ ; A ) ) ~ ( - A ) , ( - A )+~ (-A)~(-A)'~ ~

-

(FR(~;A))~U u = ( -A)"( -A)'u,

(-A)""u

i s closed,

as

p

+

<

<1

y

Y

=

KY

Y n +

vn

+

.

If

(-A)'u.

m

then

v

u

and

K'v

E1u = Y

Y

u

n

{pn]

= K&u

Q

+ p.

D(A2)

= pnR(pn;A)un E = p

R(p ;A)K u

y n n n which shows t h a t

taking closures,

y =

with

(A ) ' +

-A)@ C - (-A)"@. K1 Y

for

(-A)'(

(2.23)

{u,]

Y

E D(A)

+

with

un

-

u,

u E D(K;+')

C

Y(-A)'(

on t h e o t h e r hand,

Y so t h a t

with Y and completes t h e p r o o f of ( 2 . 2 3 ) .

y n

K

= D(Kt);

(-A)'u

K1 Y

= (-A)

-A)'U

= D(A2 )

-A)@ 3 - (-A)"@

u E

'( -A)'u

w e have

D(K1)

=

K RKR u =

(-A)"(-A)B

3_ ; ' & K

b y v i r t u e of ( 2 . 2 3 ) for

This c o m p l e t e s t h e p r o o f o f (2.22) f o r

Assume now t h a t

u E D(K ) =

Assume t h a t

i s a sequence of p o s i t i v e numbers w i t h

S i n c e Lemma 2 . 4 i m p l i e s t h a t = K$;U

D((-A)"t') Since

.

K' C K t h u s K1 C K Y - Y' Y Y s o t h a t t h e r e e x i s t s a sequence

= D((-A)') pn

a.

+

To t h i s end, d e n o t e b y

= (-A)y

Obviously we have

+

p

we h a v e

KIY

K u

as

We p r o p o s e t o show t h a t f o r

D(A2).

restricted t o

E

( -A)"tB.

(-A)"(

i s c l o s a b l e and

We p r o v e n e x t t h e o p p o s i t e i n c l u s i o n ,

K

u

w e deduce t h a t

m

(-A)"( -A)@ C -

thus

-A)@

(-A)"(

t h e operator 0

(which

c1

( p ~ ( p ; ~ ) )= ~Ku K ( V R ( ~ ; A ) ) ~ ~= ( - A ) " ( - A ) B ( ~ R ( ~ ; A ) ) ~ ~=

K"+f3

Since

a or

( i n a n improved v e r s i o n ) i s obvious i f e i t h e r

(2.22)

p are z e r o , t h u s w e may suppose t h a t a , p > 0.

a + B = 1. Let

u E D(A)

y - K u Y

a + p < 1.

and c o n s i d e r t h e f u n c t i o n (2.24)

56 in

PHASE SPACES

5

0 < y

C o n t i n u i t y of

1.

y

t h e d e f i n i t i o n and c o n t i n u i t y a t

u

Accordingly, i f

-A

The p r o o f t h a t

E

2

D(A )

2 (-A)

0 < y < 1 i s obvious from

in

(2.24)

1 h a s b e e n proved i n Lemma 2 . 2 .

=

we have

o! (-A)@

i s e x a c t l y t h e same a s i n t h e p r e v i o u s

c a s e ; t h e o p p o s i t e i n c l u s i o n depends on t h e f a c t t h a t i f r e s t r i c t i o n of

A

to

D(A2)

argument employed above for

$111.3

Ti'

then

K;/

.

=

A

which i s shown u s i n g t h e

We omit t h e d e t a i l s .

R e s o l v e n t s o f f r a c t i o n a l powers.

A s s e e n i n t h e next result, c o n d i t i o n of

R(A;A)

The s e c t o r

c (cp-)

Obviously,

A

=

0

t h e s e t of a l l

h

implies existence

cp',

Proof:

0 <

Let

exists in

0 -c Cp <

(cp)

0

rn'

C

0;

lare A /

with

A

E

5

5

h0 > 0 .

Since

C/lAl

Ih- h0 1 < %/Cl

<.

The

c+(a) ( r e s p . y-(p)) i s larg A / < u?).

3(C1)

f o r some

a

(A

E

C1 > 0.

Then

a r c s i n (l/C1)

=

e x i s t s a constant

lIR(h;A)(/

replaced by (cp-).

or

Cp ( r e s p .

c+(cp-)w i t h

a, t h e r e

we w r i t e

T,

5

i s d e f i n e d i n t h e same way w i t h

exists i n the sector

f o r every

Cp,

b e l o n g s t o any s e c t o r

THEORD4 3.1. Assume t h a t

R(A;A)

Given

+ i n d i c a t e s e x c l u s i o n of

subindex

(2.1)

i n a sector containing t h e p o s i t i v e r e a l axis.

We i n t r o d u c e some n o t a t i o n s .

R(h;A)

is the

A'

C

=

C

Z+h' 1 ) .

Cp'

such t h a t

(3.1)

CLpO

]/R(hO;A)l] 5 5 l/llR(hO;A)ll

it f o l l o w s t h a t

and c a n be e x p r e s s e d

there by t h e power series m

R(A;A)

=

T j=0

Since

(pO

- A)~R(A@;A)~+'.

(3.2)

57

PHASE SPACES

with 0 <

u)

< cp, h

u)'

<_ ho

= arc sin

s i n cp'

E

t h e f i r s t statement stands proved. Let

(l/Cl),

C+(m'),ho

/ h l / c o s a r g A.

=

and we c a n e s t i m a t e

(3.2)

5

I h - hol

Then

los i n l a r g h /

as f o l l o w s :

This completes t h e p r o o f . The n e x t s t e p i s t h a t o f e s t a b l i s h i n g e x i s t e n c e and e s t i m a t e s similar t o

(3.1) f o r t h e r e s o l v e n t of t h e f r a c t i o n a l powers; we

r e s t r i c t ourselves here t o t h e case

since other values of

1/2,

D =

cy

are o f l e s s i n t e r e s t i n c o n n e c t i o n w i t h second o r d e r e q u a t i o n s .

p

b e a nonzero complex number i n t h e s e c t o r d e f i n e d by (Tr- v)/2

so t h a t

-k2 b e l o n g s t o t h e s e c t o r

exists.

Define Q(P)

=

5

(3.3)

cp < T r p ,

Z+(W-) where

-(PI- ( - A )

1/2

Let

2

)R(-P

the r e s o l v e n t of

A

(3.4)

;A).

D( ( 2 D(A), Q(p) i s everywhere d e f i n e d ; on t h e o t h e r hand, we observed i n t h e p r e v i o u s s e c t i o n ( s e e ( 2 . 7 ) ) t h a t K,R( h;A)

Since

i s bounded for

i s bounded.

0 <

-= 1,

o/

Since

and

(-A)cy

Q(p)u

for u u

E

E

thus

=

D(((-A)1/2)2) facts that that

Q(p)-

commute, we have

R(A;A)

2

-R(p ; A ) ( p I

- ( -A)1/2)u

(3.5)

then

(PI + (-A)1/2)Q(p)u ((-A)

fortiori

It a l s o f o l l o w s from c o m m u t a t i v i t y t h a t i f 1/2 2 R(-p2)u E D ( ( ( -A) ) ) and

D(A1/2).

D( ( ( -A)1/2)2)

since

-a

(-A)CYR( h;A)

1/2 2

5 -A

2

2

)

Q(p)

D(A )

=

- ( p 21

(Theorem 2 . 5 ) .

i s dense i n

+

A)R(-p 2 ;A)u

we c a n t h e n u s e

E;

(3.6)

(3.6)

and t h e

i s c l o s e d t o deduce

that is,

Q(p)E

(PI + (-A)1/2)Q(u)u : D ( (-A)"*) I n p a r t i c u l a r , p I + (-A)'/* PI- (-A)'12 i s one-to-one; f o r i f u

u

On t h e o t h e r hand,

i s bounded and t h a t

(3.6) h o l d s f o r e v e r y u,

=

-+

E

is

D((-A)1/2)

and

(3.7)

u.

=

E

5 D((-A) 1/2)

&.

On t h e o t h e r hand,

i s such t h a t

PHASE SPACES

58

( p I - (-A)I12)u

=

t h e n from (3.7)

(3.5)

t h e n it r e s u l t s from

0

u

that

that

111 + ( - A ) 1/2. This time we o b t a i n t h a t

i s onto, while

PI- (-A)”’

both -p

pI

belong t o

-

PI - (

+

pI

(3.4)

in

-

: D( (

(

E

The conclusion i s t h a t

are invertible, that is,

and

p

i n p a r t i c u l a r , we have

p((-A)’12;

R(p;

+

pI

i s one-to-one.

(-A)’/*

and

and

0.

=

We run now t h e e n t i r e argument r e p l a c i n g by

Q(p)u = 0

- (-A)’/~)

Although t h e mere f a c t

that

2

-(PI - ( - A ) ’ / ~ ) R ( - P

= ~ ( p = )

;A).

(3.8)

i s nonempty i s a l l what

p((-A)

i s needed f o r t h e moment, s i g n i f i c a n t information on t h e r e s o l v e n t can We s t a r t from

be obtained with a l i t t l e more work. b a s i c formula

(2.3),

(3.8)

using t h e

obtaining

Using t h e second r e s o l v e n t e q u a t i o n we deduce t h a t

Replacing i n t o

(3.9)

we o b t a i n s f t e r a simple c a l c u l a t i o n t h e formula

h1/2

$ 10 R(A;A) A + p -m

R(p;-(-A)’/?

=

(which has a well known s c a l a r c o i i n t e r p a r t ; see p . 303 I).

Up t o t h i s p o i n t ,

(3.3).

defined by

However,

(3.10)

dh

GRADSTEIN-RIDZYK [1963:1,

has been a complex number i n t h e s e c t o r

p

(3.10)

(E) - v a l u e d a n a l y t i c

defines a

f u n c t i o n i n t h e e n t i r e p l a n e minus t h e imaginary a x i s , t h u s it provides an a n a l y t i c e x t e n s i o n of

t o t h e r i g h t half plane

R(p;-(-A)’/*)

Rep > 0 .

Since t h e norm of t h e r e s o l v e n t

operator

E must tend t o i n f i n i t y a s

t h e resolvent s e t to

p(-(-A)’/’)

p(B)

with

approaches t h e boundary of

p

it follows t h a t e v e r y

R(p;-(-A)’/?)

of a n a r b i t r a r y

R(p;E)

g i v e n by

with

p

(3.10).

Rep > 0

belongs

This statement

c a n be improved a s f o l l o w s : THEOREM 3.2.

cp = arc s i n ( l / C l )

as i n Theorem 3.1.

R( P ; ( -A I/*) e x i s t s i n t h e s e c t o r

x+($(cp+ T ) - ) ;

0 < m’ C rp

C = CW ’

-

t h e r e e x i s t s a constant IIR(~;-(-A)’/’)II

5

c/lpl

(p

Then

moreover, i f

such t h a t

C+C

$ (TI

+

71))

(3.11)

59

PIMSE SPACES

< @' < cp.

cp'

Let

Proof:

of i n t e g r a t i o n i n ( 3 . 1 0 ) t o e i t h e r of t h a t i f , say,

w e have

0

p

3 . 1 we c a n deform t h e p a t h 0 5 A < m so t h e rays Aeki@',

Using Theorem

rind

1

A.1-/2~-i@'/2

m

R(V;-(-A)'/~)

h + ei V v 2

0

=

R( he-"

;A)dA

.

(3.13 )

The f i r s t i n t e g r a l f o r m u l a d e f i n e s a n a n a l y t i c f u n c t i o n i n t h e whole

e x t e n s i o n of

1

- (q' + T), t h u s p r o v i d e s a n analytic 2 t o t h e h a l f p l a n e R e e-iq"2p > 0. I n a

arg p =

p l a n e minus t h e l i n e

R(A;-(-A)1'2)

(3.13) y i e l d s a n a n a l y t i c e x t e n s i o n t o t h e h a l f p l a n e 1 t h u s R(k;-(-A)1/2) exists i n (q' + T I ) ) .

symmetric f a s h i o n ,

> 0,

R e ei"'2p

It remains t o estimate t h e r e s o l v e n t . so t h a t

with

For

c > 0,

p E

$

1 55 (n +

+ T ) ) , Im p

5

0

0

p

w e use (3.13) i n s t e a d o b t a i n i n g t h e

Gf

t h e mere f a c t t h a t

and

-p

p

b e a complex number s u c h t h a t

belong t o

p(

-( -A)1/2)

w e have

= ~ ( ~ ; - ( - A ) " ~ ) R ( - ~ ; - ( - A )l/2)E = ~ ( -2p; A ) E = D(A).

that

p((-A)'"

improvement of a ( p a r t i c u l a r c a s e ) of Theorem

Proof: L e t both

( ( -A)1'2)2

Im p 20

Then

T h i s completes t h e proof of Theorem 3.2.

An i m p o r t a n t consequence fGllOWing

IJ. E

TI).

so t h a t

x+( 2-1 (ql

same e s t i m a t e .

Let

5

p = (p\eiJi w i t h

c+( c+(12(ql + n)),

C

-A,

-p2

D( ( (

f

#

is the

2.5:

E

p(A).

Since =

Since (2.22) i m p l i e s

(3.14) s t a n d s proved.

5111.4 T r a n s l a t i o n of g e n e r a t o r s o f c o s i n e f u n c t i o n s . If A i s t h e i n f i n i t e s i m a l g e n e r a t o r of a s t r o n g l y c o n t i n u o u s

60

PIIASE SPACES

semigroup

then

S(:)

A-bI

any complex number) i s th:

(b

infinitesimal

The S ( G ) = e-btS(f). b corresponding r e s u l t for cosine functions i s somewhat harder to prove.

generator of t h e s t r o n g l y continuous semigroup

LEMMA 4.1.

be t h e i n f i n i t e s i m a l generator of a strongly

A

~ ( ts a t i s f y i n g

continuous cosine f u n c t i o n

Il@(t111 5 coeb It1 and l e t

b

(-m

be a n a r b i t r a r y complex number.


m>

Then

% = A - b 2I

(4.1) i s the

i n f i n i t e s i m a l generator of a s t r o n g l y continuous cosine function cb(Z)

satisfying (2)

/Eb(t)ll1. Coe Proof:

Define, i n d u c t i v e l y ,

C 0 ( t ) = @ ( t ) , @,(t)u

(u

-m


=

E,

E

m.

=J’

(S*@n-l)(t)u -m

.. .

el({),

Co(<),

Obviously

functions i n

(4.2)

< t <

m,

n

=

t 8(t 0

1,2

- s)@n-l(s)u

ds

(4.3)

,... ) .

are all s t r o n g l y continuous operator valued

Denote by

dn)

t h e s e t of a l l n-tuples

...,

(el,e2, en) with e = + 1; W(n) has 2n elements. It i s easy t o j see by induction s t a r t i n g from t h e cosine f u n c t i o n a l equation (11.3.1) t h a t f o r a r b i t r a r y r e a l numbers

...,sn

to’s1,

... @(sl)@(tO) = +

we have

~ @ (+ elsl t ~

+ a * .

2

where t h e sum i s extended over all 0

5 to f tl 5

Since

It

0

(el,.

.., e n

... 5 t n = t > o , o z s j 5 t j - t j - 1 ( j = 1, ...,n; tn = t). s + ... + e s I 5 t we obtain from ( 4 . 4 ) that 11 n n wt (4.5) II~(sn>@(sn-l) @(Sl)@(tO)I/ 5 Coe

+ e

7

We i n t e g r a t e now t h e l e f t hand s i d e of parallelepipedon

(4.5)

(4.4)

~ ( ~ 1Let.

*

of

+ en s n )

we g e t

0

5

s

j

5 t j - t j m 1( j

(4.4) =

1,

i n t h e n-dimensional

...,n;

tn

=

t).

Making use

61

PHASE SPACES We note next t h a t

- t n-1 ) ... S ( t1- t O ) @ ( t o ) u d t O .. . dtnml,

C ( t ) u =JS(t n

t h e i n t e g r a l t a k e n on t h e r e g i o n

it follows from

(4.5)

0

5 to 5 tl 5

... 5 tn-lz t ,

thus

that

(4.6) Consequently, t h e s e r i e s m

7(-b2lncn(t)

Cb(t) =

(4.7)

n=O

Cb({)

2

t

converges uniformly on compacts of

0.

This p l a i n l y i m p l i e s t h a t @ (0) =

(E)-valued f u n c t i o n with

i s a s t r o n g l y continuous

b

I.

Moreover

We o b t a i n from

(4.3)

(11.2.11),

and ( a v e c t o r valued v a r i a n t )

of t h e convolution formula for Laplace transforms t h a t

for Reh >

Hence, a f t e r a c l e a r l y p e r m i s s i b l e term-by-term i n t e g r a t i o n ,

[d.

m

emht@b(t)u d t

( -b2)"R( h2;A)n+1u

h

=

n=O = AR(A

for Reh >

(1)

+ Ib I.

2

=

2 hR(h ;A&

(U

Ab

@b( [ E l )

is

a s i n f i n i t e s i m a l g e n e r a t o r , completing t h e

b

w i l l be a r e a l number with

b

we have, i n view of t h e f i r s t i n e q u a l i t y

b

(4.10)

E E)

4.1.

I n t h e sequel, v a l u e s of

2 b ;A)u

It follows t h e n from Lemma 11.2.3 t h a t

a cosine f u n c t i o n with proof of Lemma

+

2 W.

For t h o s e

(11.2.11),

llR(h2;%)ll = /IR(A2 + b2;A)ll

'

cO

( A2 + b 2 ) l P ( ( h2

(where

C1

+

b2)1/2

depends of course on

b).

- u)

5-

(A> 0)

(4.11)

A2

Accordingly,

Ab

belongs t o t h e

62

PHASE SPACES

3(C )

class

2

= (b I - A )

k

defined i n 4111.2 and t h e f r a c t i o n a l powers

<

0

CY

i s a bounded o p e r a t o r i n

Proof:

K e,b

%.

t h e operator

u

If

E

- A)a - (b21 - A)ff

(4.12)

as a. consequence,

D(A);

Denote by

( b "1

Then

2 (A.

<1, b, b '

(-A,)"-( -%)CY=

ff

t h e operator

D(A)

(2.3)

corresponding t o

we have

CY

-(-%I = (Ko(,bl - Ko(,b)'

=

-

-lorn -

=

-

hff-l(R(h;%,)A,,

yL

R(A;%)A,)u

dh

1

h f f ( R ( h ; % r ) - R ( h ; % ) ) ~ dh

-J

m

+ ( b t 2 - b2 ) s i n f f r Tr

ABR(?;%,)R(h;%)u

dh

I

Both i n t e g r a l s d e f i n e bounded o p e r a t o r s , t h u s boundedness of

follows.

=

can be defined.

LEMMA 4.2. Let

(-%I)

(-%)"

If

u

-

E

D((-%)O)

t h e n t h e r e e x i s t s a sequence

ff

ff

un u, (-A,) un * (-Ab) u . i s convergent as w e l l t h u s u E D((-%,)").

By v i r t u e of

e q u a l l y well with t h e r o l e s of

b'

such t h a t

b

and

(4.12)

{un]

E

D(A)

(4.12), (-%,)"un

Since t h e argument works

(4.13)

reversed,

It remains t o d e f i n e t h e f r a c t i o n a l powers of

%

follows.

itself.

We

s e l e c t a r b i t r a r i l y one of t h e p o s s i b l e values:

In particular,

$111.5

~t/~

= i(-%) 1/2

.

The p r i n c i p a l value square root r e d u c t i o n .

Throughout t h e r e s t of t h i s c h a p t e r A w i l l be t h e i n f i n i t e s i m a l generator o f a s t r o n g l y continuous c o s i n e f u n c t i o n

(4.1),

t h a t is,

A

E

s(Co,m).

c(<)

satisfying

A s seen in Section 1 1 . 3 , A ~ g2 i s

e q u i v a l e n t t o t h e f a c t t h a t t h e Cauchy problem f o r

63

PHASE SPACES

i s ( u n i f o r m l y ) w e l l posed i n t 2 0 ( e q u i v a l e n t l y , it i s uniformly well We s h a l l a l s o r e q u i r e t h e following ad hoe posed i n - m < t < m ) . p o s t u l a t e , which w i l l be shown t o hold a u t o m a t i c a l l y i n c e r t a i n Eanach spaces i n t h e following s e c t i o n .

Let

5.1.

ASSUMPTION

2

b

(1;.

Then 8 ( t ) E t

i s a s t r o n g l y continuous f u n c t i o n of

for

E

-m

and <12s(t)

D(A;/')


m.

It f o l l o w s from Lemma 4.2 t h a t Assumption 5.1 i s independent of the particular

b

2w

The f o l l o w i n g result e s t a b l i s h e s another

used.

t r a n s l a t i o n invariance property.

LEMMA 5.2.

&&

Sb(t)E

..;'," 5

t

- w < t < m .

S (t)u b

@,(s)u d s .

=

and

D(Ai/2)

Proof:

2

b

%,

w,

(.,(<)

as i n Lemma 4.1,

Then i f Assumption

5.1 i s s a t i s f i e d ,

i s a s t r o n g l y continuous f u n c t i o n of

$/$(t)

Term-by-term i n t e g r a t i o n of t h e s e r i e s

(4.7) shows t h a t

where

On t h e o t h e r hand, c o n s i d e r t h e s e r i e s

where

2,(t) =

%1/2S ( t ) ,

2,(t)u

(u

E

=

E,

(S*2 -m

n-1


)(t)u

=it

8 ( t - s ) ~ ~ - , ( s ) ud s

0

W,

n = 1 , 2 ,...)

.

An argument v e r y s i m i l a r t o t h a t used i n Lemma 4 . 1 shows t h a t each term

in

(5.2)

and

(5.3)

i s s t r o n g l y continuous and t h a t both series a r e

uniformly convergent on compact s u b s e t s of

-m


m,

thus the l i m i t

64

PHASE SPACES

(5.3)

2b(c) o f

%'I2

i s a strongly continuous function.

commutes w i t h

S(t)

$/*

%

AtPsb(t)u

THEOREM 5.3.

=

R(h;A)

( t h i s f o l l o w s from t h e f a c t t h a t

commutes w i t h s(t) and from t h e d e f i n i t i o n o f R(?\;A)), hence a n ( t ) = 1 / 2 S n ( t ) ; s i n c e and

On the o t h e r hand,

At/2 i n t e r m s of is closed, Sb(t)E

-

C

2b(t), which completes t h e argument.

Let Assumption 5.1 h o l d .

Then for each

2

b

(A),

b

i s t h e i n f i n i t e s i m a l g e n e r a t o r of t h e s t r o n g l y c o n t i n u o u s group

(5.4)

(5.5) we renorm t h e space E i n t h e same way a s i n Lemma 1.2, so t h a t (1.6) h o l d s i n t h e new norm; t h e n Theorem 4.1 To prove

imp1i e s t h a t

ll~b(t)llt 5 e (r61+b)ltI Define for

(-m<

q ( t ) = e x p ( - ( h ) + b ) t ) / l %1/2 g k ( t ) l l

Cb, Sb

result i s

This shows

a f t e r multiplication by

(1.7), (5.5)

for t

t <

2

0

30

and it f o l l o w s f o r a l l

q(t)

t

The g r o u p e q u a t i o n s a r e a n immediate consequence o f and

(11.1.8)

a n d use

e x p ( - ( w + b ) ( s + t ))$/'.

which h a s b e e n s e e n t o imply t h a t

for t

(5.6)

m).

5

The

C(l + t).

by symmetry.

(11.1.7)

(11.1.8). Finally, if

Reh >! , f

=

we have

R(h;412)u

t h u s it r e s u l t s from Theorem g e n e r a t o r of

(Reh >

W

1.3.4 t h a t $/2

+

b)

,

(5.7)

i s the infinitesimal

% ( t ) a s claimed.

Assumption 5 . 1 and t h e p r e v i o u s results p r o v i d e a phase space for t h e e q u a t i o n (5.1), where E = E and E = D(%1 / 2 ) endowed wi.th i t s 1 0 g r a p h norm ( n o t e t h a t , by v i r t u e o f Lemma 4 . 2 t h e g r a p h norms of a w two


are equivalent f o r

b , b'

2

w).

This r e d u c t i o n of (5.1) t o

a first o r d e r Cauchy problem w i l l be r e s t a t e d i n a d i f f e r e n t form a t t h e

65

PHASE SPACES

end of t h e s e c t i o n .

THEOREM 5.4.

The space if=D(.$, 1 /2) x E (D(% 1 / 2 )

g r a p h norm) i s a p h a s e space f o r @

t a i n e d i n t h e s t a t e space

(5.1)

1/2

D(%

generator

)

c -D1

Proof:

D(A);

(5.8)

(1.11) r e s p e c t i v e l y and a r e t h e


(-m

We show f i r s t

(5.8). 0

5

s

Note t h a t

@ ' ( s ) u = Ab(s)u

=

~ ( s ) ud s

AJ(,

.

S i n c e t h e l e f t hand s i d e i s a bounded o p e r a t o r o f

E

E.

8 ( s ) maps

Assume now t h a t

u

E

for

5 t, t

@(t)u

u

(5.9)

m)

C > 0.

integrating i n

t h e i n t e g r a l of

which i s con-

6(;) and i t s i n f i n i t e s i m a l

The group

a r e given by (1.10)

f o r a n adequate co n s t an t

E

m

D1,

=

IIE(t)l/(3) 5 ~ ( +1 It I)e[lllt 1

u


o f Theorem 1 . 2 : p r e c i s e l y ,

m

t h e i n c l u s i o n b e i n g bounded.

-m

equipped with i t s

into

D(A) :

E

and

(5 . l o ) u

it f o l l o w s t h a t

(5.10)

h o l d s for a l l

u s i n g Assumption 5 . 1 we c a n w r i t e

t @ ( t ) u = (Ab + b 2 1 ) L

which o b v i o u s l y i m p l i e s t h a t

@ ()u t

8(s)u d s

i s continuously d i f f e r e n t i a b l e

with derivative

This e s t a b l i s h e s t h e i n c l u s i o n r e l a t i o n

(5.8).

To show t h a t t h e

i n c l u s i o n i s bounded it i s s u f f i c i e n t t o show t h a t

for a n adequate co n s t an t

estimate f o r

Ai/2bb(t);

C.

T h i s i s proved j u s t a s t h e c o r r e s p o n d i n g

a f t e r renorming of t h e s p a c e we w r i t e

(11.1.8)

66 for

PHfiSE SPACES

@

and

(1.7) f o r

p r e m u l t i p l y by 2/:A

8,

q(t>

and work with t h e i n e q u a l i t y

exp(-rnt)IlAt/2s(t)l\.

=

(5.12)

Once i n p o s s e s s i o n of

(5.11) t h a t

we o b t a i n from

(5.13) ( w i t h t h e obvious m o d i f i c a t i o n i f = 0 ) which shows t h e i d e n t i t y t o be bounded. o p e r a t o r from D(% 1/2 ) i n t o D1 f*)

6 ( t ) i s a group h a s a l r e a d y been p r o ed i n Theorem

The f a c t t h a t 1.2.

To show t h a t each

6(t): 3

-

R

i s a s t r o n g l y continuous f u n c t i o n i n

-

-

i s a bounded o p e r a t o r and

6(t)

we o n l y have t o show t h a t

-

@ ( t:)D($12) D($/*), S ( t ) :E D($/2), @ ' ( t:)D(
obeying e s t i m a t e s of t h e form (5.12). This i s obvious f o r s ' ( t )= @ ( t ) a s an operator i n E; i n D ( %1 / 2 ) we use t h e f a c t t h a t @ ( t )commutes with

@ ' ( t )i s based on

The treatment of


and t h e d e s i r e d p r o p e r t y of

(5.12).

by i n e q u a l i t y g e n e r a t o r of

6(t)

Reh >

u

and

8 ( t ) i s j u s t Assumption

1.5

and

(5.13)

complemented

It only remains t o i d e n t i f y t h e i n f i n i t e s i m a l

i n t h e space [u,v]

=

(5.11)

E

i3

3.

This i s done a s f o l l o w s :

if

t h e n we have

( i m p l i c i t i n (5.14) i s t h e e a s i l y v e r i f i a b l e f a c t t h a t t h e Laplace transforms of @ ( t ) u and @ ' ( t ) u e x i s t i n t h e norm of D(% 1/2 )). A r o u t i n e computation shows t h a t

thus

91

i s t h e i n f i n i t e s i m a l g e n e r a t o r of

6(;) by v i r t u e of Theorem

1.3.4. Let

3

=

E

X

E

endowed with any of i t s product norms.

ing r e s u l t i s e s s e n t i a l l y e a u i v a l e n t t h a t t h e Cauchy problem f o r

(5.1)

and t h a t Assumption 5.1 h o l d s .

The follow-

t o Theorem 5.4; we s t i l l assume

i s uniformly well posed i n

(-m,m)

67

PHASE SPACES

TiXOREM

in

9

=

E x E

5.5.

Let

b

2

w i t h domain

W.

Consider t h e o p e r a t o r

D(Ab)

=

Then

x D(A:/*).

D(A;/')

i n f i n i t e s i m a l g e n e r a t o r of a s t r o n g l y continuous group

A

is the

q D ( t ) such t h a t

(4) l l ~ ~ ( t ) I5 l (c(1 q

+

W 2 e ( "l+2b)ItI

(-m


(5.17)

m). h

of

There i s a one-to-one correspondence between s o l u t i o n s u ( t )

with

u'(t)

=

of

u(;)

and solutions

~ ' ( 0 )E

(5.1)

(5.18)

2ibU(t)

g i v e n by

(d/' + u(;)

ibI)u(i)

w Ll'({)

Proof:

Let

w i t h domain with

D($)

=

D($/*)

x D(A;/*)

=

0

il

-i I

0

Finally, l e t

In view of Theorem 5.3 we have

I)(%/'),

I

(5.19)

SO

that

9+,

=

+

by

68

PHASE SPACES

moreover, it follows a g a i n from that

u

(11.1.7),

(11.1.8) a n d Assumption 1 . 5

If Reh >

Pb(<) i s a s t r o n g l y continuous group.

= [u,v] E

W

+

b

and

3 we have

and we check e a s i l y enough t h a t from meorern

1.3.4

The f a c t t h a t

26

that

= % ,,

+

\

%(A) = R(h ; \ )

s o it follows again

i s t h e i n f i n i t e s i m a l generator of

bl) g e n e r a t e s a group

%(t)

Pb(t).

is (a particular

case o f ) a c l a s s i c a l r e s u l t i n p e r t u r b a t i o n t h e o r y ( s e e HILLE-PHILLIPS

(5.17)

[1957: 1,p. 3891) b u t t h e e s t i m a t e

i s s l i g h t l y nonstandard,

The main ingredient w i l l be t h e

t h u s we s t a r t t h e proof from s c r a t c h .

following r e s u l t , which i s a s o r t of complement t o t h e Hille-Yosida Theorem 1.3.1.

W e s t a t e and prove it i n a r a t h e r general version.

LEMMA 5.6. Banach space

L A S(i) be a s t r o n g l y continuous semigroup i n t h e E, A i t s i n f i n i t e s i m a l g e n e r a t o r . Then S ( t ) s a t i s f i e s

Ils(t)ll 5 m

f o r a.n i n t e g e r lIR(h;A)"II

5

21

cO(l

S(i)

talewt

(t

L

(5.22)

0)

if and only i f

Co((Reh-U)-"

+

(Reh >

If

-1-

n(n W,

n

... ( n + m - l ) ( R e h - w ) - ( n + m ) 0,1, ... ). (5.23)

+ a) =

i s a s t r o n g l y continuous group, t h e n Ils(t>ll

5 co(l +

Itlm)eWltl

(-m


m)

(5.24)

i f and o n l y if

IlR(h;A)"II.

5

... ( n + b l - l ) ( n = 0,1, ... ).

Co(( lRehl -m)-n + n(n

(IReh( >

0,

+

a)

[Rehl -h))-(n+m)) (5.25)

69

PHASE SPACES

Proof:

Using

(5.22)

i n formula

(1.3.8) a l l

result instantly.

The corresponding formula f o r

estimates

when

(5.25)

is a group.

S(t)

a r e a consequence of Lemma 1.3.2. =

( - l ) n n ! R(h;A)n

-A

(5.23)

t a k e s care of

The opposite i m p l i c a t i o n s

In f a c t , since

we o b t a i n from formula

inequalities

R(A;A)(~)

=

(1.3.14) ( s e e a l s o

(1.3.15))

that

Ils(t>llIc0 Colimtmn-n(n + 1)

... (n + m > ( l - - wt )-(n+m+l) n 5

t

The corresponding e s t i m a t e i n way.

w t -(n+l) + l i m (1- F )

assumed for

h >

co(l + tm)ewt. (t 2

0)

for groups follows i n t h e same and

(5.25)

need o n l y be

real.

End of proof of Theorem

%(.)

0

(5.23)

We observe i n passing t h a t

=

satisfying

(5.20).

IIR(~;%)~II(~)

5.5.

g e n e r a t e s a group

5.6 we o b t a i n

5 c ( ( I h l - r ~ - b ) -+~n ( l h I (111

%

The o p e r a t o r

Applying Lemma

- w - b)-(”+l))

,...

+ b , n = 0, 1

).

(5.26)

Consider t h e s e r i e s

.. . R( h;\)(R(

R( h ; $ ) ( R ( A;\)b’B+

for

kl,k

*,...

=

t h u s each term of

where

of

k =

(5.26)

O,l,Z

,..., Ihl

(5.27)

>

W

h;%)bp?”

(5.27)

+ 2b. It i s e a s y t o see t h a t

can be w r i t t e n i n t h e form

and p + q = k + n. We make use of t h i s r e l a t i o n and j t o deduce t h a t t h e g e n e r i c term i n t h e series (5.27) i s k

bounded i n norm by a n expression of t h e form

PHASE SPACES

70

Cbk

( Ih( - fIi

1 - b)k+n

+

Cb

k

k + n (/Al-fIl-b)

(k + n ) ( k + n + 1)

+ mk

( [ A ] - 0 1 - b)k+n+2

k+n+l

' (5.30)

*

We observe next t h a t

111 >

for

U

..

+

where it must be remembered t h a t

2b,

k =

..

cki

and t h a t

.,k assume independently all t h e v a l u e s 0,1,. We d i f f e r e n t i a t e n next (5.31) r e p e a t e d l y w i t h r e s p e c t t o Ihl o b t a i n i n g t h e e q u a l i t i e s

kl,

1

+

k

n

n)b ( , h / - u - b ) k+n+l

1 Accordingly,

k

k

+

n)(k

+n +

( / h i - u - 2 b ) n+l

-

l)b

k

- (

( ( ] A / -td-b)k+n+2

the series

(5.27)

n(n

(5.32)

'

+ 1)

I A l - u - 2b)n+2

(5.33) '

i s convergent i n t h e norm of t h e space

( 3 ) and we can e s t i m a t e t h e norm o f t h e sum by ( a constant t i m e s ) t h e

sum of

(5.31), (5.32) and (5.33).

We observe f i n a l l y t h a t

(2.57)

is

nothing b u t

and check (by d i r e c t a p p l i c a t i o n of t h e d e f i n i t i o n ) t h a t R(A;%)

(R(h;\)bp)j

=

R(A;%

(5.34)

+ bB) = R(X;%).

The end r e s u l t i s t h e sequence o f i n e q u a l i t i e s

l ] ~ ( X ; % ) ~ l / ( ~y) C ( l h l - ~ - 2 b ) -+ ~Cn(1Al - u - 2 b ) - ( n + 1 )

+

Cn(n

+ 1)( I A / -

U s i n g (a s l i g h t Il'$(t)ll

5 c(1

+

- 2b)-(n+2)

( / A 1 > w + 2b, n

=

... )

0,1,

(5.35)

71

PHASE SPACES

-00

C

t

C

(5.17).

which completes t h e proof of

m,

W e a t t e n d f i n a l l y t o t h e l a s t statement i n t h e proof o f Theorem

5.5. with u(0)

u(<)

Let u'(0) E

I)($/*).

E

D(A)

easily that

Since

u(t)

@ ( t ) u ( O ) + S(t)u'(O)

=

it f o l l o w s t h a t

u'(t)

( t ) , g i v e n by

(5.19),

LI

(5.1)

be a s o l u t i o n of t h e second order e q u a t i o n

0(4I2)

for a l l t

E

with and we check

(5.18).

i s a s o l u t i o n of

That

t h e correspondence i s one-to-one i s obvious.

§r11.6

Second order equations i n

Lp

spaces.

All t h e r e s u l t s i n t h e previous s e c t i o n ( i n p a r t i c u l a r t h e p r i n c i p a l value square r o o t r e d u c t i o n i n Theorem 5 . 5 ) depend on t h e v a l i d i t y of Assumption 5.1.

A s seen i n t h i s

s e c t i o n , Assumption

5.1

i s a u t o m a t i c a l l y v e r i f i e d i n a c l a s s of spaces i d e n t i f i e d below. A triple

(X,y,p)

where

i s an a r b i t r a r y s e t i s a measure

X

i s a u - f i e l d of s u b s e t s of

space i f

a d d i t i v e measure i n

7.All

positive, i.e.

2

p(e)

0

X

and

i s a countably

p

measure spaces considered here w i l l be

f o r any

e c

( f o r d e t a i l s on t h i s d e f i n i t i o n

and connected f a c t s see DUNFORD-SCHWARTZ

[l958:1, Chapter 1111). 'The

(complex) Lebesgue spaces

L r ( X ) (1 5 r

i n t h e customary manner.

Lr(X,F,p)

=

5

m)

a r e defined

A l l r e s u l t s below a r e based on t h e following

theorem on H i l b e r t transform of v e c t o r valued f u n c t i o n s .

THEOREM

1< r C

where

for

6.1. Let

f

E

For -

m.

Lp(-m,m;E),

depending on

p

be a measure space and l e t

(X,y,p)

E

>

0,

1< p <

b u t not on

define

Then t h e r e e x i s t s a c o n s t a n t

m.

f

or

C

such t h a t

E

(6.3) where

11. \ I p

i n d i c a t e s t h e norm of

Hf

=

l i m &+

e x i s t s ( i n t h e norm of

Lp(

LP(-m,m;E))

0

-a,

Moreover,

m;E).

(6.4)

HEf f o r every

f

E

LP(-m,m;E)

so t h a t ,

72

PHASE SPACES

with

( 6 . 3 ) , H i s a bounded o p e r a t o r i n LP(-m,m;E)

i n view of norm < C .

For a proof see DUNFORD-SCHWARTZ [1963:1, p , 11731. Theorem p;

Actually

6.1 w i l l be only needed for a n a r b i t r a r i l y f i x e d value of p = r,

for

t h e r e s u l t i s nothing b u t a n i n t e g r a t e d form of

M. R i e s z ’ s w e l l known result on Lp boundedness of t h e ordinary H i l b e r t transform (DUNFORD-SCHWARTZ [1961:1, p . 10591). We s h a l l need i n t h e sequel a c o r r o l l a r y of Theorem W

2

denote by

0

functions

Su(E)

f ( < ) defined i n

Ilfl/m,o

-a


mPA( E )

<m

-m


(E) f o r a l l z > 0; P family of semi-norms THEOREM 6.2.

-wltl

llf(t)ll,

(6.5) 15 p <

m,

E-valued s t r o n g l y measurable f u n c t i o n s

i s a l o c a l l y convex space endowed with t h e

{/l*l/p,T;~

Let

c > w,

>

0).

1< p <

m,

E

a Banach space s a t i s f y i n g

men -

m.

exists i n the t o p o l o g u

hp(E)

-+

- valued

such t h a t

m

(6.1) for some r , 1 < r <

Hc :gu

e

~ ~ * ~ ~ m ,O on. t h e other hand, f o r

denotes t h e space of a l l

f ( t ) defined i n

For

such t h a t

m

= e s s . SUP -m
endowed with t h e norm

6.1.

t h e space of a l l s t r o n g l y measurable E

and d e f i n e s a continuous operator

mp(E).

Proof: the interval

For any

It I < a.

a > 0 Let

let T

xa

3 1,

be t h e c h a r a c t e r i s t i c f u n c t i o n of

0 < E

5 1, t 5

IT].

We have

f(s) ds ISI?T+l

= (H:’lf)(t)

+ (HE’ C Pf ) ( t ) +

(H:’jf)(t)

.

(6.8)

73

PHASE SPACES

It i s obvious t h a t

HC”

=

E

E

( e - c l t l -l)/<

c o n v o l u t i o n of

(6.9)

holds f o r

r e s u l t s from Theorem LP(-m,m;E)

.

5 T>

(6.9)

by q+l(;)f(i) thus an e s t i m a t e of c 2 H C J 2 f ( t ) = l i m HE’ f ( t ) . F i n a l l y it c l H f = l i m HcJ1f e x i s t s i n t h e norm of

’

6.1 t h a t

E

and

where a g a i n

C

depends only on

THEORFM 6.3. r, 1 < r <

( It1

and t h a t

E

t e n d s uniformly on compacts t o t h e

HC’:f(t)

On t h e o t h e r hand,

i s independent of

f CllflIm,i)

ilHE’3f(t)ll

t h e type of

’

Hc’

E

and l e t

W,

A

T.

This ends t h e proof of Theorem

6.2.

(6.1) f o r some Then Assumption 5.1 h o l d s :

be a Banach space obeying E

2

(S

(W).

e qu iv a l e n t l y ,

qt)

=

(6.11)

@Jt> + Ay2Sb(t)

i s a s t r o n g l y continuous group ( w i t h i n f i n i t e s i m a l g e n e r a t o r

( 5 . 5 ) ) for b -5

satisfying Proof:

Let

b

2 W.

w.

u

If

E

D(A)

we have (6.12)

We have a l r e a d y proved t h a t

%

generates a cosine function

$(:)

satisfying

Let

c >

A = c2 of

W

+ b.

We d i v i d e t h e i n t e r v a l of i n t e g r a t i o n i n

(6.12)

and use i n t h e second i n t e g r a l t h e i n t e g r a t e d - b y - p a r t s

(11.2.11)

for n

=

1:

R ( h2 ;%)u v a l i d for

Reh >

W

+ b.

=Jim

e-7\Ssb(s)u d s

The r e s u l t i s

,

at

version

(6.14)

74

PHASE SPACES

The integrand i n t h e second i n t e g r a l i s now transformed a s follows. Write

(11.1.7)

the fact that

for

@'(t)u

Cb, Sb, u =

D(A)

F

f o r both

t

-t.

and

Using

and s u b t r a c t i n g t h e e q u a l i t i e s so

S(t)Au

obtained we prove t h a t

-sb(s)sb(t)A$ = Since

u

E

D(A),

(@,(s

(observe t h a t

C

(Cb(S-t)-Cb(S +

- t ) - Cb( s

t h e r e e x i s t s a constant Il(c$(s

5

+ t))u

= O( l s l )

- t ) - @b(s '

w i l l depend on

t))UIl

u

5

and

deduce t h a t

11 ( Cb( s - t ) - Cb( s + t ) h /I d s 5 C

Since

h-1/2(

as

s

+

(6.16) 0

SO

that

such t h a t

C

=

.

t))U

C ( A'/'

h1/2 - w - b ) -2

Cse

(w+b)s

(s

L

t ) . Accordingly,

dm

s e -(h1/'-W-b)sds

0)

we

=

- - b)-* . 0

.

i s summable i n

2 ( c ,m)

it follows from t h e

Lebesgue dominated convergence theorem t h a t

By (a vector-valued v e r s i o n o f ) T o n e l l i ' s theorem t h e order of i n t e gration i n

t h e i n t e g r a l on t h e r i g h t hand s i d e of

(6.17)

reversed. Once t h i s i s done, the a t t r a c t i v e formula r)

CC

4/28b(t)u

[

= f lr 8b ( t )d o

$l m

+

l i m E+o

-CS

n

h-1/2R(X;Pg)(-A,)~

dh

( @ b ( s - t )- C b ( s + t ) ) u d s

can be

75

PHASE SPACES

(6.18) So f a r , so good:

results.

with r e s p e c t t o

s)

(6.18)

but

(specifically, the integral

s t i l l o n l y makes sense for u

does not provide enosgh information on connection, of course, t h a t Theorem

E

i s an a r b i t r a r y element of

-

-

d12Sb(t).

D(A)

E

6.2 proves u s e f u l . { un 3

and l e t

and t h u s

It i s i n t h i s Assume t h a t

be a sequence i n

u

D(A)

u. Then C b ( s ) u n cb,( s ) u uniformly on compact s u b s e t s of n (-m,m). I n view of (6.13), Cb(s)u,+ C b ( s ) u i n XU,(E) and it t h e n r e s u l t s from (6.18) and Theorem 6.2 t h a t , for 1 < p < m, $12Sb(f)un u

with

converges i n T

> 0.

Lp( (-T,T);E)

t h a t i s , converges i n

hp(E),

f o r every

Passing if necessary t o a subsequence we can t h e n i n s u r e t h a t


g(t)

for t

o u t s i d e of a s e t

depending of course on

(-CO,CO)

(belonging t o

L’((-T,T);E)

u)

where

f o r every

closed o p e r a t o r , it follows t h a t i f <12Sb(t)u

=

T

t

(t

E

SF

i s a E-valued f u n c t i o n

g(t) > 0).

Since

d, Sb(t)u

E

g(t)

of f u l l measure i n

d

E

D(%

)

i s a and

(6.19)

d).

Let

so t h a t

e

has a complement of n u l l measure i n e = -e,

e

+

W e have

(-co,M).

-

e c e.

The f i r s t e q u a l i t y f o l l o w s from t h e f a c t t h a t f u n c t i o n ; t o show t h e second we apply

$/2

(6.20)

$j2Sb(t)

i s an even

t o both s i d e s of

(11.1.8)

obtaining

+ t)u

L+(s

= @(s)%1/2Sb(t)U

+ C(t)Ay‘&Js)u.

The two e q u d i t i e s ( 6 . 2 0 ) combined imply t h a t c l a s s i c a L r e s u l t i n measure theory, a > 0. that

Applying t h e n r e p e a t e d l y e =

(-co,~),

graph theorem

e

- e 5 e,

t h u s by a

c o n t a i n s an i n t e r v a l

t h e second r e l a t i o n

S b ( t ) E _C D(A;l2)

so t h a t

$12Sb(t)

e

for a l l

(6.20)

t;

(-a,a),

we deduce

by t h e closed

i s a bounded o p e r a t o r .

Consider now t h e group

l$(<). It f o l l o w s from

(6.19) t h a t

76

PHASE SPACES n

bb(t)u

i s s t r o n g l y measurable for each

u

( i i i ) shows t h a t

yielding Theorem 1.2.1.

so t h a t t h e argument

%(<)

i s s t r o n g l y continuous.

This ends t h e proof of Theorem 6.3.

9111.5 - including t h e construction of t h e s t a t e space i n Theorem 5.4 and t h e p r i n c i p a l Theorem

6.3 a s s u r e s t h a t

a L t the r e s u l t s i n

value square root reduction i n Theorem 5.5-hold.

More can be proved;

for instance, a s t h e following r e s u l t shows, t h e phase spaces constructed i n Theorems 1 . 2 and

5.4

a r e a c t u a l l y t h e same (except f o r

a readjustment of t h e norm). THEOF3M

6.4.

E

where, a s i n Theorem 1.2,

be as i n Theorem 6.3.

Then i f

a

u

Dl

i s t h e s e t of a l l

@(G)u i s continuously d i f f e r e n t i a b l e i n

norm (1.8) Proof:

of


-m

2W,

such t h a t Moreover t h e

m.

i s equivalent t o t h e graph norm of

D1

b

We have already shown i n Theorem 5.4 t h a t

1 2

D(Ab/ ). D(At/2)

5 %l

with bounded inclusion, thus a s soon a s we prove t h a t (6.22)

t h e closed graph theorem w i l l t a k e e w e of equivalence of t h e norms. To show ( 6 . 2 2 ) we start with formula (6.18)

always for

u

E

i n i t s second version,

and check t h a t t h e p r i n c i p a l value i n t e g r a l can

D(A)

be d i f f e r e n t i a t e d with respect t o

t

under t h e i n t e g r a l sign.

i s obvious enough i f we note t h a t (always f o r

c >

W

+ b)

This

we have

which, a s an ordinary i n t e g r a l , can be d i f f e r e n t i a t e d under t h e i n t e g r a l sign.

rrn

The d e r i v a t i v e equals e-'lSe

-03

= l i m E+

0

J;

(q(t- s ) - q ( t ) ) u ) e-c:st

Is1 1"

T(t-

S)U

ds

ds

.

77

PHASE SPACES

Hence

M

u

Let

D so t h a t @,(;)u i s continuously d i f f e r e n t i a b l e i n 1' As a p a r t i c u l a r case of Theorem 1 . 3 we know t h a t D ( % ) i s

E

(-m,m).

N

dense i n

Em,

thus

sequence

{un)

5 D(A)

particular

D1

i s dense i n

D(A)

u

with

@ I j ( t ) u n .+ @ $ ( t ) u

thus, by v i r t u e of Theorem

-

and we may s e l e c t a N

u i n t h e norm of D1: i n n uniformly on compact subsets of

@i(t)un

6.2,

-

in

@;(<)u

Sc (E ).

again Theorem 6.2 we deduce t h a t t h e r e e x i s t s

%/2@,(t)un

-

h ( t ) i n hp(E), used i n Theorem 6.3 shows t h a t

e

of f u l l measure i n

D(A)

t h e proof of Theorem

(11.1.7). O r i g i n a l l y , t h i s e q u a l i t y was

i s dense i n

1.3). u

=

u

thus of Theorem

E

s =

F

u c D 1

using

D1

Dl

(see

-t we g e t

@ ( t ) @ ( t ) u+ S ( t ) @ ' ( t ) u

D(<j2).

u

i n t h e topology o f

D1

Setting

On account of t h e f a c t s t h a t deduce t h a t

h ( i ) t h (E) P and and argument s i m i l a r t o t h e one $ ( t ) u E D(% 1/2 ) f o r t i n a s e t

but we can extend it e a s i l y t o

u c D(A),

the fact that

Applying such t h a t

(-w,m).

We make use now o f proved f o r

(-m,m)

and t h a t

.

S(t)E

5 D($12)

This completes t h e proof of

we

( 6 . ~ )and

6.4.

W e reformulate below some of t h e r e s u l t s h i t h e r t o

A s t r o n g l y continuous cosine f u n c t i o n

obtained.

@ ( t )i s s a i d t o possess a

group decomposition if t h e r e e x i s t s a. s t r o n g l y continuous group such t h a t

U(<)

@ ( t=) Let

u

E

B

$ (lA(t)

(-w

c t <

be t h e i n f i n i t e s i m a l generator of t h e group

2

D(B )

(L(t)u)"

then =

lA(t)u

B%(t)u,

(6.24)

m).

U(g).

If

i s twice continuously d i f f e r e n t i a b l e with

thus

i n f i n i t e s i m a l generator A 2 = B u; i n other words,

AU

+ U(-t))

u

belongs t o t h e domain

of t h e cosine f u n c t i o n

D(A)

of t h e

@ ( t )and

PHASE SPACES

78

2

E CA

16.25)

h be so l a r g e t h a t A, -1E p(B) and h2 2 2 ( w i t h R ( h ;B ) = - R ( A ; E ) R ( - h ; E ) . The e q u a l i t y

Let

E

p(A).

2

h I - B

2

h2

Then 2 2

E

p(E )

C_ h I - A

( t h a t follows from ( 6 . 2 5 ) ) and t h e e x i s t e n c e of i n v e r s e s a r e t h e n e a s i l y seen t o imply t h a t E2

(6.26)

A,

=

Thus, t o e x h i b i t a n example of a cosine f u n c t i o n not having any group decomposition it i s s u f f i c i e n t t o t a k e

@ ( t =) c o s ( t A ) ,

i s a bounded o p e r a t o r having no square r o o t whatsoever.

A

where

(6.27)

An operator l i k e that can be found i n HALMOS [1967:1,p . 115 I: EXAMPLE 6.5.

5

(c0,

=

5,,

. ..

E

Let

=

fi2 be t h e H i l b e r t space of a l l sequences

115 11

such t h a t

)

=

2/1)

(715

C

Consider t h e

W.

unilateral shift

Then t h e r e e x i s t s no (bounded o r lmbounded) operator E such t h a t 2 B = A. I n f a c t , assume such a B e x i s t s . Then, s i n c e D(A) = E we must have as w e l l

A

is.

Let

N(A)

d i m N(A)

Since

D(B)

=

E;

N(B)

5 N(A)

we must have

f

and N(A), N(B)

A

5 1.

d i m N ( B ) = 1. Then t h e r e e x i s t s

Bq

with

=

5.

If

q = pc

then

a r e l i n e a r l y independent.

which b e l i e s t h e f a c t t h a t =

103,

A ( r e s p . of B ) .

( r e s p . N ( B ) ) be t h e nullspace of

= 1 and

d i m N(B)

Assume

must be onto s i n c e

B

moreover,

so t h a t

B

5

=

0,

f

E

N(B),

#

f

0.

a contradiction.

B u t both

q

and

5

Take

q

Hence

q

belong t o

d i m N(A) = 1. We deduce t h e n t h a t

i s one-to-one.

E u t t h i s i s impossible s i n c e

i s not one-to one. I n view o f t h e f a i l u r e of t h e group decomposition (6.24) i n g e n e r a l

and of t h e r e s u l t s i n t h e l a s t t h r e e s e c t i o n s one may surmise t h a t a formula l i k e (6.24) may be a v a i l a b l e o n l y for a convenient t r a n s -

79

PHASE SPACES

l a t e of

cb(c).

This i s of course t h e case under t h e assumptions i n

vigor here.

6.6. Let E be a Bana.ch space s a t i s f y i n g t h e assumptions of Theorem 6.3 and l e t @(<) be a s t r o n g l y continuous cosine f u n c t i o n i n E such t h a t TKEOREN

Finally, l e t

b

2 W.

Then t h e s t r o n g l y continuous cosine f u n c t i o n

2

@,(t) generated by

A-b I

llc$(t)ll

5

(which s a t i s f i e s

(6.29)

Coe

possesses a group decomposition

$ (L4Jt)

cb ( t ) =

+

where t h e s t r o n g l y continuous group

(6.30)

03)

satisfies

(6.11)

(+1It )e

C > 0.

6.6 i s of 5.1 h o l d s :

Theorem Assumption

%(<)


I

h~(t)Il5 ~ f o r some c o n s t a n t

(-m

b.&-t))

course t r u e i n an a r b i t r a r y Banach space i f t h e only t h i n g t o be proved i s t h e e s t i m a t e

(6.311. The most s i g n i f i c a n t p a r t i c u l a r case of Theorem '*)

= 0,

where

C(c)

i s uniformly bounded:

ll@(t>ll 5 co Here

c(:)

(-02


(6.12)

m).

i t s e l f admits t h e group decomposition

(6.24)

It i s n a t u r a l t o a s k whether t h e growth of L L ( ~ ) a t b e t t e r matched t o t h a t of

(1 + itl) t i v e when

with

It] =

m

can be

t h a t i s whether t h e f a c t o r

@(;),

can be eliminated i n

E

(6.6) i s undoubtely

(6.33).

The answer i s i n t h e affirma-

i s a H i l b e r t space; t h i s w i l l b e shown i n Chapter 5 by

completely d i f f e r e n t methods. answer i s n o t known.

I n a g e n e r a l Banach space, however, t h e

PWSE SPACES

80

%(t).

A n a l y t i c i t y p r o p e r t i e s of

$111.7

There i s more t h a n meets t h e eye i n t h e semigroup s h a l l see i n t h i s s e c t i o n , each

a s we

%(t):

i s t h e boundary value of a

%(t)

The proof of t h i s r e s u l t w i l l

semigroup a n a l y t i c i n a h a l f p l a n e .

r e q u i r e a s h o r t d i g r e s s i o n i n t o t h e t h e o r y of a n a l y t i c semigroups. Let

S(i)

be an a r b i t r a r y s t r o n g l y continuous semigroup.

say t h a t

S(%)

i s analytic if there exists

) : ( S

admits an a n a l y t i c e x t e n s i o n

1%

5;

51 5

cp,

u

E

For every

S E

S(C)U

z+w

An argument very s i m i l a r t o t h a t l e a d i n g t o

A

5c

such t h a t

t o the sector

(7.1)

u.

=

(1.1.9)shows

t h e existence

e'l'

i s t h e i n f i n i t e s i m a l g e n e r a t o r of

t o indicate that

< r/2

such t h a t IIS(C)II

If

S(i)

u)

E, l i m

C,@

0 <

# o?,

5

151-0,

of c o n s t a n t s

Cp,

W e

satisfies

S(c)

S(t)

( a ) and

we w r i t e

A

E Ce(c0)

(b).

The following r e s u l t provides a f a i r l y complete c h a r a c t e r i z a t i o n of g e n e r a t o r s of a n a l y t i c semigroiips. TKEOREM CI

7.1 ( i ) Let

such t h a t

and f o r every

R(h;A)

Cp',

0

A

t

Cl(p).

Then t h e r e e x i s t s a r e a l constant

exists i n the sector

C

cp' < cp

t h e r e e x i s t s a constant

that

(ii) Assume t h a t

R(h;A)

Ih;(arg

(where a i s

r e a l and

exists i n a sector

( ~ - a ) Cl C D+ ~ / 2 , ~ # a }

0 < cp < ~ / 2 ) a n d s a t i s f i e s

C = C

CD'

such -

81

PHASE SPACES C > 0.

t h e r e f o r some c o n s t a n t

Assume t h a t

Proof: for

A

Then

a(a').

E

A

and of ( 7 . 2 ) , i f

Reh

if

a(co')

Consider formula

I n view o f t h e f a c t t h a t

R(A;A).

E

S(t)

0 < cp' < cp.

1.3.8 ( n

i s analytic i n

=

I)

F + (cp)

i s l a r g e enough, t h e p a t h of i n t e g r a t i o n can be

r+(a)

deformed t o e i t h e r of t h e r a y s

or

upper and lower boundaries of t h e s e c t o r

r-(a),

r+(a).

respectively the The f i r s t choice

y i e l d s t h e formula

Since

(7.7) provides an a n a l y t i c e x t e n s i o n of

the integral

t i l t e d half-plane

Re hi'

//R(A;A)II

r-

Using

results.

r+

i n s t e a d of

Rehe-icp > B

5

R(A;A)

t o the

> B and t h e e s t i m a t e

Bl

C/IReAeia-

(Rehim

> p).

(7.9)

t h e r e s o l v e n t can be extended t o

and t h e twin e s t i m a t e

Obviously, t h e union of t h e two s l a n t e d h a l f p l a n e s i s a

(7.3)

s e c t o r o f t h e type of

assembled on t h e b a s i s of The proof of

and t h e e s t i m a t e ,

(7.8) and

(7.4)

can be e a s i l y

(7.10).

(ii) i s l e s s t r i v i a l .

Assume t h a t

R( A;A)

exists

(7.5) and t h a t (7.6) h o l d s . Since A E a(@) + CIE a(q) for any c we may assume t h a t a: < 0 .

i n a s e c t o r of t h e form i f and only i f

Let

0 < %,cp2

two r a y s

A

< cp and l e t

arg A =

'4

+ 7/2,

r(y,cp2) ImA

2

0

be t h e contour c o n s i s t i n g of t h e

and

arg A =

-(a2 + ~ / 2 ) , Imh 5

t h e e n t i r e contour o r i e n t e d clockwise with r e s p e c t t o t h e r i g h t h a l f plane.

Define

Obviously t h e i n t e g r a l does not depend on t h e p a r t i c u l a r choice of

al,cp2 and d e f i n e s an i n f i n i t e l y d i f f e r e n t i a b l e E - valued f u n c t i o n i n 5 3 0 . Let now a' such t h a t 0 < cp' < a and a" with

0,

82

PHASE SPACES

cp' < cp" < m.

If we use i n t h e i n t e g r a l t h e contour

r(a" - cp',cp")

it

A

i s clear t h a t

S(5)

t o the h a l f s e c t o r

can be extended a s a n (E)-valued a n a l y t i c f u n c t i o n

5

0 f arg

5

f CI',

Symetrically, t h e contour

0.

r(co",cD"- a')

p r o v i d e s a (E)-valued a n a l y t i c e x t e n s i o n t o

-a' < arg 5 5

0,

5 f

t h u s t a k i n g c a r e of

0

of a n a l y t i c semigroups. first

0

5

axg

integration i n

(a)

It remains t o show ( b ) .

i n the definition To do t h i s we t a k e

5 5 CD' and use T(?" - m',c~") as t h e contour of (7.11); making l a t e r t h e change of v a r i a b l e z = hi

we o b t a i n

(7.12) Obviously, we can deform f u r t h e r t h e contour of i n t e g r a t i o n t o , say,

r(cD",a" - C D ' )

and t h e n modify t h i s contour i n such a way t h a t

z,

as

it t r a v e l s upwards, does not h i t t h e o r i g i n and p a s s e s t o it r i g h t . Call

r'

t h e contour t h u s obtained.

To e s t i m a t e t h e integrand we use

(7.6):

(7.12) t h a t

We o b t a i n t h e n from

A symmetric argument t a k e s c a r e of t h e range

-cp'

f Re

55

0.

(b)

for

u

Putting

t o g e t h e r t h e two e s t i m a t i o n s we o b t a i n

I n view o f t h i s i n e q u a l i t y , it i s enough t o show dense s u b s e t of

say, for

E,

5

0

we use t h e contour

q c p " -cp,a")

way a s

r ' ; we

R(h;A)u

=

call

h-'R(A;A)Au

r"

Re

55

parately the cases

u

:D(A);

Cp'

and

a s customary, we t r e a t se-

-

t ~ '5

Re 5 5 0 .

I n the first

modified near t h e o r i g i n i n t h e same

t h e r e s u l t i n g contour.

+ A-lu

in a

so t h a t

If

u

E

D(A)

we have

83

PHASE SPACES

5

(7.6) t h e f i r s t i n t e g r a l i s a continuous function of 5 a r g 5 5 m'; f o r 5 = 0 t h e i n t e g r a l vanishes, a s can be

I n view of in

0

e a s i l y seen t r a n s l a t i n g t o t h e r i g h t t h e domain of i n t e g r a t i o n . second i n t e g r a l evaluates t o

u

E

D(A)

1. This shows t h a t

(7.4),

and, i n view of

(b)

holds f o r

ends the proof of Theorem

Through t h e r e s t of t h i s s e c t i o n

A

The

7.1.

i s t h e i n f i n i t e s i m a l generator

(4.1).

satisoing

of a s t r o n g l y continuous cosine f u n c t i o n C ( < ) W e r e q u i r e Assumption 5.1 t o hold s o t h a t , i f

b

2 W,

i s a s t r o n g l y continuous group.

THEOREM 7.2. continuous i n

L$(;)

The group

t h e upper h a l f plane Im

52

function i n

Re

5>

Proof:

If

b

Im

5

0

(ii)

0

(iii)

0.

2W

a d m i t s a n extension

such t h a t

Lj(t)

(i)

i s an

l+(t)

to

(E)-valued a n a l y t i c

There e x i s t a constant

then t h e operator

%(i)

i s strongly C > 0

satisfies

-Ab

such t h a t

(2.1)

It follows t h e n from Theorem 3.2 t h a t t h e r e e x i s t s (see (4.11)). $ > 0 such t h a t R ( h ; (-%) 1/ 2 ) e x i s t s i n larg A \ < ~i + ~ r / 2 and satisfies

-

Thus Theorem 7.1 a p p l i e s t o show t h a t

-(-%)1/2

generator of a s t r o n g l y continuous semigroup

is the infinitesimal

bb(S)

analytic i n

larg 51 < q, 5 f 0. Since Theorem 3.2 does not provide d i r e c t i n f o r mation on t h e growth of b b ( t ) f o r t r e a l we s h a l l o b t a i n t h i s information by means o f an e x p l i c i t r e p r e s e n t a t i o n for b b ( t ) .

Define

To show t h a t t h e l i m i t e x i s t s we perform a n i n t e g r a t i o n by p a r t s , obtaining t h e equivalent express ion

rm

Tb(t) =

/

'- 0

h(t,h)R(h;-%)2

dh

,

(7.19)

84

PHASE SPACES

there e x i s t s a constant

-< c,h1/2

for t

2

(7.20)

C,

5

(t

+

2

may i n p r i n c i p l e depend on

C'

need only use

(7.20)

t >0

continuous i n

h> 0

in

b > w),

i s c e r t a i n l y the case i f

E

(7.20)

O),

(7.19)

h

at

= 1

(7.U) we o b t a i n t h e estimate

and

ll'b(t)ll where

2

Dividing t h e domain of i n t e g r a t i o n i n

6.

and using

(A

such t h a t

C6

' > '1, %'

(if

6.

so t h a t

(7.22) e x i s t s we

in

C' = 0

(7.22):

xb(t)

W e prove e a s i l y t h a t

i n t h e norm of

(E).

this

is

On t h e o t h e r hand, i f

D ( A ) we have Tb(t)u-u

=

f a s i n thlp(R(A;-%)u-$

l i m

a + ~0

and t h i s expression tends t o zero when

t

+

0.

Although

a c t u a l l y strongly continuous a t t h e o r i g i n ( i . e .

for every

u

E

E)

u ) dh

TJO

xb(t)u

xb(i) +

u

as

is

t

+

r e s u l t w i l l be obtained below.

If we t a k e

u

E

D(A)

then t h e previous

s t e p s show t h a t t h e following computation i s j u s t i f i e d :

A1/2

R(h;-%)u

=

d h = R(p;-(-%)1/2)~.

Since t h e same Laplace transform r e l a t i o n must of needs hold for bb(;)u,

-

where

0

we need not prove t h i s d i r e c t l y , a s a f a r stronger

bb(;)

we have

i s t h e a n a l y t i c semigroup generated by 3,(t)u

=

Irb(t)u

(by uniqueness of Laplace

(7.24)

PHASE SPACES

u

transforms) f o r We extend

u

a f o r t i o r i for

D(A),

E

85

E

l+( t oit) h e upper h a l f plane

E.

z2

by mea.ns of

0

t h e formula

%(C)

=

% ( t+

\(;)

Since

i'I) = % ( t ) b b ( z )

i s s t r o n g l y continuous i n

s t r o n g l y continuous i n upper h a l f plane

2

0,

l+(c)

group and

m

...

and t

S(c)

S(t)E

5 D(Am)

(7.26)

complex a s w e l l ) .

t > 0;

be an a r b i t r a r y element of

Lrb('c)u

E

D(-(-%)'12)

=

D(i(-%)'/*)

i n f i n i t e s i m a l g e n e r a t o r of with r e s p e c t t o

t

E.

t

-

- ( - A b ) 1/2*

T > 0,

Then, i f

is the

Since

=

\(:),

obviously,

W e apply t h i s obser-

v a t i o n (for m = 1) t o t h e a n a l y t i c semigroup generated by

u

for

R(A;A) dh

e

( 7 . 2 6 ) can be extended t o t Let

is

i s a n a n a l y t i c semi-

AmS(t) i s (E)-continuous i n

(so t h a t , i n c i d e n t a l l y ,

bb(q)

0 and

1

AmS(t) =

(7.25)

i s closed, t h e r e s o l v e n t e q u a t i o n

A

i t s i n f i n i t e s i m a l generator t h e n

1,2,

and

m

0).

i s s t r o n g l y continuous i n t h e

and Cauchy's formula t h a t i f A

=


-M

2

'c

m,

We observe next t h a t it follows from

0.

(7.11), t h e f a c t t h a t

formula

for R(A;A) each

'I

5 2

Im


(-m

%(t)Lrb('c)u

i s differentiable

with

Dt%(t)Irb(z)u =

%(t)%u=')'2(b'

-i %(t)(-(-%)"2)bb('I)u

= -i DTLj(t)bb(T)u

(-rn


03,

>

(7.27)

0).

B u t t h i s e q u a l i t y i s nothing b u t t h e Cauchy-Riemann e q u a t i o n for t h e (continuously d i f f e r e n t i a b l e ) function

Im

analytic i n

5>

\ ( g)u,

thus

w e l l known r e s u l t (HILLE-PHILLIPS [1957:1, p. 931) t h a t

i s analytic i n

Im

%( % ) u

is

as a E-valued f u n c t i o n ; it follows t h e n from a

0

5>

0

l+,(t)

itself

a s a (E)-valued f u n c t i o n .

To complete t h e p r o o f of Theorem 7.2 we only have t o observe t h a t the fact that b b ( 0 ) = I.

\(t)

i s an e x t e n s i o n of

bb(t)

The e s t i m a t e (7.16) follows from

from ( 5 . 5 ) f o r

i s obvious since

( 7 . 2 2 ) for b b ( ~ ) and

l+( i). Note, incidentally, t h a t the factor

can be a l t o g e t h e r eliminated i f c e r t a i n l y t h e case i f

b >

0.

%

has a bounded i n v e r s e :

1+

lIm5/

this is

F i n a l l y , it i s of i n t e r e s t t o observe

86

PHASE SPACES

L+(t) and

that

t

bb(‘C) commute f o r a r b i t r a r y v a l u e s of

t h e e a s i e s t way t o see t h i s i s by means of formula using t h e f a c t t h a t

%(t)

commutes w i t h

(7.18)

). b

R(’\;-A

and for

7 ;

bb(;),

A s a consequence,

we o b t a i n

For t h e sake of completenesn, we r e s t a t e Theorem 7.2 using the group decomposition language o f Theorem

THEOREM 7.3.

6.6.

Let t h e assumptions of Theorem

cb(i)

admits a group decomposition of t h e form

s t r o n g l y continuous group

to

Im

5 2

0

decomposition

(7.16).

(6.24),

$111.8

If

where

Im

s t r o n g l y continuous i n

Then, i f

(6.10),

b

2. L ,

where t h e

% ( t ) admits a s t r o n g l y continuous e x t e n s i o n Im

which i s a n a l y t i c i n

and t h e e s t i m a t e

6.6 be s a t i s f i e d

c(i).

f o r t h e s t r o n g l y continuous c o s i n e f u n c t i o n

’*

=

and s a t i s f i e s

0

0, @(%)

b(t)

5 >

5 >

(7.28)

i t s e l f admits t h e group

i s analytic i n

Im

52

0,

and s a t i s f i e s

0

Other square root r e d u c t i o n s .

The t h e o r y developed i n t h e l a s t s i x s e c t i o n s l a c k s s u b t l e t y i n one important r e s p e c t , namely i n t h e r i g i d choice of t h e p r i n c i p a l

%

v a l u e square r o o t of system,

i n t h e r e d u c t i o n of

(1.1) t o a f i r s t o r d e r

Although t h i s h a s been seen t o work s a t i s f a c t o r i l y i n

Lr

spaces, t h e following example i l l u s t r a t e s t h e drawbacks of t h e choice i n t h e general case. EXAMPLE 8.1. valued

Let

E = C

2lr

be t h e space of a l l complex

(-m,m)

2 ~ p e r i o d i ccontinuous f u n c t i o n s defined i n

with i t s u s u a l supremum norm.

(--m,

endowed

m)

Consider t h e s t r o n g l y continuous cosine

function =

(U(X

2

+ t)

Its i n f i n i t e s i m a l g e n e r a t o r i s c o n s i s t i r g of a l l

u

E

E

i

u(x

A

= D

such t h a t

-t)) 2

u

=

(-m

2

d /ds2


a)

.

with domain

i s twice continuously

(8.1) D(A)

87

PHASE SPACES

differentiable.

@ ( t ) i s u n i f o r d y bounded, we can apply t h e

Since with

b

2

be obtained s e t t i n g

t

= 0

theory i n

$5

w = 0.

An e x p l i c i t formula f o r

(6.23)

in

f o r any

t i v e e x p r e s s i o n r e s u l t s i f we t a k e limits a s

c

c > 0;

-

0

can a more a t t r a c -

( s e e Exercise ll

I n t h e p a r t i c u l a r case under c o n s i d e r a t i o n t h e

i n t h i s Chapter). formula r e a d s

(6.18)

S i m i l a r manipulations with

( s e e Exercise 10 ) y i e l d t h e

formula

where t h e i n t e g r a l i s understood a s t h e l i m i t when i n t e g r a l over

-tI 2

1s

E,

s

2

-

(8.3) 0

of t h e

However, t h e s i n g u l a r i n t e g r a l

0.

o p e r a t o r on t h e r i g h t hand s i d e of C 2lr( - m , m )

E

(8.3)

i s not bounded i n

so t h a t Assumption 5.1 does not hold and t h e p r i n c i p a l

v a l u e square r o o t r e d u c t i o n f a i l s u t t e r l y .

Nevertheless, a f a r more

elementary r e d u c t i o n works.

with domain

Let

B

=

d/dx

D(B)

consis-

t i n g of a l l continuously d i f f e r e n t i a b l e f u n c t i o n s i n E. Then 2 = A and we o b t a i n a group decomposition o f t h e form (6.24) with

E

L ( t ) u ( x ) = u(x + t ) .

(8.4)

I n o t h e r words, t h e second o r d e r e q u a t i o n t o t h e system

(

7 = ~ (u,),, ) ~

~

can be reduced

ut t = uxx ( u ' ) ~ = ( u ~ )as~ we l e a r n i n elementary

c o u r s e s on p a r t i a l d i f f e r e n t i a l equations. I n view of Example

8.1 one may wonder whether a d i f f e r e n t choice

of square r o o t may make t h e square r o o t r e d u c t i o n work f o r g e n e r a l c o s i n e f u n c t i o n s i n g e n e r a l Eanach spaces.

The following example

shows t h a t t h e answer t o t h i s q u e s t i o n i s i n g e n e r a l negative. EXAMPLE 8 . 2 . c o n s i s t i n g of

odd

(8.1) (note t h a t

(-m,m) be t h e subspace of C e T ( - w , m ) 2r f u n c t i o n s , @ ( t ) t h e c o s i n e f u n c t i o n defined by

Let

E

=

Co

h

@ ( t ) maps

t r a n s l a t i o n group (6.24), since L ( t )

(8.4)

C:lr(-m,m)

into itself).

However, t h e

does not provide a group decomposition l i k e

does not map

CElr(-m,m)

into itself.

As we s h a l l

88

PHASE SPACES

see below, t h e r e e x i s t s no s t r o n g l y making b

%

c,

E

A,,

(6.24) =

-

A

happen. 2 b I and

continuous group i n

I n f a c t , much more t h a n t h i s i s t r u e :

if

@b(t)i s t h e cosine f u n c t i o n generated by

L$,(<)

t h e r e e x i s t s no s t r o n g l y continuous group

such t h a t

I n view of t h e remarks i n §111.6,t h i s statement i s e q u i v a l e n t t o : if

b

E

%

C,

has no square r o o t

whatsoever t h a t g e n e r a t e s a

+/2

s t r o n g l y continuous group,

O u r claim i s proved a s follows.

Assume t h e - r e p r e s e n t a t i o n

%(<).

h o l d s for some s t r o n g l y continuous group let

En

be t h e one-dimensional subspace of

s i n nx.

@ ( ts)i n nx

Then

@ ( t )l e a v e s each

= cos n t s i n

nx.

(4.3)

series representation

En

n

=

..

1,2,.

generated

(-m,m)

277.

A

by

For

E = Co

(8.5)

i n v a r i a n t since

We deduce from t h i s and t h e p e r t u r b a t i o n

Cb

that

En

l e a v e s each

invariant

as well; p r e c i s e l y ,

@,(t)s i n nx b

2

0

u(t,x)

=

% ( t )s i n nx

for

%

Since

i s a s o l u t i o n of u(0,x)

=

F;”,,

utt(t,x)

En

n’

6

E

t h i s shows t h a t

We show next

E n

into

(4.3))

- b 2u ( t , x )

To prove t h i s we i s g i v e n by

@,(s) it a l s o commutes with

( a g a i n because of t h e p e r t u r b a t i o n s e r i e s commutes with

(8.6)

u xx ( t , x )

invariant.

t h e p r o j e c t i o n of

% ( t ) commutes with

=

s i n nx, ut(O,x) = 0 ) .

a l s o l e a v e s each subspace

observe f i r s t t h a t

2 1/2 b ) t s i n nx

(8.6) can a l s o be obtained noting t h a t

(formula

with i n i t i a l c o n d i t i o n s that

+

c o s (n2

=

@(s)

and a f o r t i o r i

% ( t ) maps E n

into itself.

Making u s e of t h e f a c t t h a t every semigroup i n a f i n i t e dimensional space must be an exponential, equation cos (n2

+ b2)”*t

=

$ (e

(8.7)

iTn(b)t

reduces t o

-izn(b)t + e

)

(-m<

t <

w)

i s a complex number; obviously t h e only choice i s 2 z n ( b ) = u ( n + b 2 ) l l 2 , where, for each n 2 1 on equals 1 or n and does not depend on t . We conclude t h a t \ ( t ) must be a

where

7

n

(8.8)

(b)

m u l t i p l i e r operator of t h e form

-1

89

PHASE SPACES

u

where

-

a

-

( n 2 + b2)1/2 in

(8.9)

n n

s i n m.

Ze

izn(b)t

a

n=l

as

n

..

)

s i n nx,

(8.9) b, T ( b ) - n = n Lj, t ) i s t h e operator

so that if

m

-+

there exist coefficients

(n = 1,2,.

n

Notice now t h a t , f o r f i x e d

O(n-l)

=

m

-

%(t)u

- C l t 1n-l /cn(t) <

cn(t),

such t h a t

(8.lo) % ( t ) i s a strongly continuous group i n E

It f o l l o w s t h a t only i f

i s ; each

b,(E)

i f and

i s g i v e n by

bo(t)

(8.11) Consider now t h e following operator i n

if

*

c27J *

u

-

~ neinx c

then

m

Qu(x)

- n=-m ancne i n x

(8.12)

where we have set

a.

D(Q)

The domain

=

Q

of

Since

Q s i n nx =

s i n nx

-Q

( n = 1,2

u

such t h a t Qu E C2r 2lr of a l l t r i g o n o m e t r i c polynomials i n

c o n s i s t s of a l l

and i n c l u d e s t h e subspace c2Tr.

,... ) .

0 , a-n = -on

-

PPT

icr

cos nx

n

s i n nx

=

C

E

we have

i a n ( c o s nx

- i an

sin m)

- inonx (8.131

= ia e

n

b(t)

On t h e o t h e r hand, i f

:kT

i s t h e t r a n s l a t i o n operator

(8.4) we have

b ( t ) b O ( t ) s i n nx d t

ina t

TT

2J

= 7r

e

i s a trigonometric

Hence, i f

u(x)

combining

(8.13) and Qu(x)

s i n n(x + t ) d t

= U(X)

=

ia e

-ina x n

n

polynomial i n

0

C21r

(8.14) w e have,

(8.14),

-as,”

b(t)bo(t)u(x) d t

,

(8.15)

90

PHASE SPACES

Q

so t h a t

i s a bounded o p e r a t o r i n t h e space

.

trigonometric polynomials t h e value of

Qu(x)

at

We d e f i n e now a

@u = 0

D includes P

@U(G) =

so t h a t , by c o n t i n u i t y of P

2T.

(8.16)

Q

p

Po

,

(8.17)

i s a continuous l i n e a r

@

2T’

Hence

- u(-.))}

a(; 1 (u(;o in

(8.12),

Moreover, i n view of

2T’

We can t h e n extend

t i n u i t y and by t h e theorem of Bore1 measure

taking

@

Qu(0).

f o r any even t r i g o n o m e t r i c polynomial.

functional i n

of all odd

0: @U

Again, t h e domain of

PgT

functional

@

to

Cg,

p r e s e r v i n g con-

F. Riesz t h e r e e x i s t s a p e r i o d i c f i n i t e

such t h a t

(8.18)

p - z o eni m . Since

2 n

a

=

a n

for a l l

n

and a theorem of HELSON

we have

[1953:1]a p p l i e s

t o show t h a t , a f t e r eventual

modification of a f i n i t e number of i t s Fourier c o e f f i c i e n t s

p

must

reduce t o a f i n i t e l i n e a r combination of p o i n t masses, t h a t i s ,

( i n a somewhat ad hoc n o t a t i o n ) .

Hence

(8.20) where

{gn]

i s t h e sequence

{an],

a f i n i t e number of i t s elements.

p o s s i b l y a f t e r modification of

Obviously, a c o n t r a d i c t i o n w i l l b e

obtained i f we show t h a t any sequence of t h e form

(8.20)

must e i t h e r

be p e r i o d i c o r t a k e an i n f i n i t e number of values, s i n c e t h e sequence

{Gn]

t a k e s only a f i n i t e number of v a l u e s (obvious) and cannot be =

-u

proved by i n d u c t i o n on

m

periodic (since

n’ i n f i n i t e number of changes).

t o make

{an]

p e r i o d i c would imply an

The underlined statement on

(8.20)

and we omit t h e d e t a i l s .

The absence of any group decomposition i n Example 8.2 m u s t of course be t r a c e d i n p a r t t o t h e l a c k o f d e s i r a b l e p r o p e r t i e s o f

the

is

91

PHASE SPACES

( f o r i n s t a n c e , E i s n o t r e f l e x i v e ) . One may r a i s e 27r t h e q u e s t i o n of whether some group decomposition can be guaranteed space

E =

CO

E

under assumptions on

r e f l e x i v i t y o r uniform convexity) t h a t

(say,

A s f a r as reflexivity

$111.5.

f a l l s h o r t of t h e very s t r i n g e n t ones i n

goes t h e answer i s i n t h e n e g a t i v e , a s we s h a l l see below. EXAMF'LE 8.3.

There e x i s t s a r e f l e x i v e space

densely defined o p e r a t o r

A

in

and and a closed,

F

F such t h a t ( i ) A g e n e r a t e s a

s t r o n g l y continuous cosine f u n c t i o n ( i i ) For any complex number b 2 t h e operator = A b I p o s s e s s e s no square r o o t whatsoever g e n e r a t i n g

-

%

This i s e a s i l y seen by modifying t h e

a s t r o n g l y continuous group.

space i n Example 8.2 i n a way suggested by F. LAUFBURSCHEN [ p r i v a t e communication].

( b ) for

k

2

m

E =

Let EE

(a) t h e subspace

c;~,

EZ

(8.1)

(obviously,

EE

b, m i s i n v a r i a n t by

Lj,

,m

% , & t =) 2 ( % , & t +)

cm(g),

(c)

;)

L

0, m

of

t.

Eo

of E genm, & @(t) be t h e cosine

Let

k).

t h e r e s t r i c t i o n of

@,(t)). EE

Consider

5 n3

@,(;)

to

Assume t h e r e e x i s t s a

such t h a t (-a

m(t>)

-


(8.Z)

a p p l i e s e q u a l l y well h e r e :

the

admitting t h e r e p r e s e n t a t i o n

(t)u(x)

m

-

C

e

ina t n

a s i n IIX

(8.22)

n

n=m

t, where each u equals 1 or -1 and i s independent n The r o l e o f t h e operator Q i s now played b y Q ~ U ( X >=

1. i n the

{ s i n nx; m

(8.2~) for c i m p l i e s a similar decomposition b, m t h e r e s t r i c t i o n of @(;) t o Em with a s t r o n g l y

continuous group

f o r each

\"b

%(t) i n E

The argument u s e d f o r

(i)

(t) in

1

for

<, n 5

and denote by

s t r o n g l y continuous group

decomposition

a fixed integer.

t h e ( f i n i t e dimensional) subspace

e r a t e d by t h e f u n c t i o n s { s i n nx; rn function

21

m

and

generated by t h e f u n c t i o n s

subspace

The operator

Q,

C

u c e inx

2m

n n

Em generated by a l l t h e f u n c t i o n s

{einx;

m

5 n].

a d m i t s the representation

(8.24) f o r every t r i g o n o m e t r i c polynomial i n

EE,

t h u s it i s bounded t h e r e

92

PHASE SPACES

and we can define a continuous l i n e a r f u n c t i o n a l (8.16)

(with

Q

and

= Qm)

@

m After extending

(8.17).

in

EE

by

@

to

Eo = C

1 2l.I by t h e Hahn-Banach theorem we deduce t h e existence of t h e measure p in

on

as before; since

(8.18)

argument applied t o

2

In1

= ~ t ; l for

i s unmindful

)L

and t h e

m

modification of a f i n i t e

Of

number of Fourier c o e f f i c i e n t s , t h e conclusion i s i d e n t i c a l :

Ei, t h u s L

cannot be a bounded operator i n strongly continuous group i n

to

in

Ir

(t)

O,m

0 _
I r o , m ( t O ) i s unbounded; for i f

t

i s bounded for d l

and

u

E

i s arbitrary, l e t

EE

Ei such t h a t

be a sequence of trigonometric polynomials i n Then L

0,m

(i)un

Qm

cannot be a

This implies t h a t t h e r e e x i s t s some

EE.

such t h a t

TT

O,m

(:)

i s continuous f o r a l l

n

Lc

and

O,m

(;)un-

{u,]

u u. n.. IrO,m(t)u +

li ( i ) u i s s t r o n g l y measurable for a l l u and 0,m i s a s t r o n g l y continuous applying Theorem 1.2.1 we o b t a i n t h a t IrO,LIl

pointwise, t h u s

(i,)

group, a c o n t r a d i c t i o n .

t h e choice of t h e sequence

t

For a r b i t r a r y

t o w i l l depend, of course, on

We note t h a t and

where t h e norm i s t h a t of

{an; n k

1m

2

in

m)

(8.22).

define

~ b , ~ ( t a)s an o>erator i n

i n f i m i m i s taken over a l l p o s s i b l e choices of

u = + 1. We claim t h a t , f o r any n (say, i n It] 5 1) such t h a t

-

as

k

-

m

I n f a c t , assume t h i s i s not t r u e f o r a c e r t a i n t h e r e e x i s t s a f i n i t e sequence and such t h a t

5

m,k

n

5

and t h e k ) with

t h e r e e x i s t s a sequence

m,

m

{an; rn

EO

{t,)

.

(8.25)

m.

Then f o r each

uk , m > ' k , m + l > " ' ~ o k , k

with

k,n

k

-

= +

(8.26) where

C

i s a constant independent of

_< ;n m5 k) { ~ ~ , ~

k

Note t h a t t h e (possibly unbounded) operator on t h e

on

with

and t h e sequence

has been used i n t h e d e f i n i t i o n

n

2k

Lc

0, m

(8.22) of (t)

Lc

O,m

(t).

a l s o depends

b u t t h e s e a r e i r r e l e v a n t when obtaining

~ ~ , ~ ( t It ) .i s now easy t o see t h a t t h e r e e x i s t s a subsequence k(1) < k(2) c

...

such t h a t

a k( j),m

=

'rn

(j

ak( j ) , m + l

=

am+l

93

PHASE SPACES

(j

2

2),

.. .

i.e.

Accordingly, i f

-

i s dense i n

(It 1

(8.22)

2

5

thus

(8.27)

seen t o be impossible.

(8.27) k).

But

implies t h a t t h e o p e r a t o r s

It] 5 1, t h u s f o r a l l

Em f o r

we o b t a i n

m>

i s a n i n c r e a s i n g f u n c t i o n of

Em,

a r e bounded i n

in

U ~ , U ~ + ~ ,

I L 0 , m (t)llm,k 5 II-(lm,k

O,m

. ..

corresponding t o t h e sequence

(note t h a t

o ( a 2 m ) becomes -k ( j >, 1 b ( t ) i s t h e group

such t h a t each sequence

s t a t i o n a r y a f t e r a wliile.

t,

This completes t h e proof of

A

as f o l l o w s .

m ( l ) < m(2) C

sequence

{t.], Itj 1 J

51

...

(t)

0,m

(8.25). F

and

we can produce a

of p o s i t i v e i n t e g e r s and a sequence

such t h a t

'm(j),m(j+l)

The space

(8 . 2 5 )

Using

L

which h a s a l r e a d y

With t h i s r e l a t i o n i n our hands w e c o n s t r u c t t h e space t h e operator

4 -> m Em,k

( t ) - m j

as

j

h

m

.

(8.28)

F i s t h e E l b e r t sum of a l l t h e f i n i t e dimensional spaces 2 t h u s i s a r e f l e x i v e Banach space; s i n c e t h e

F. = E J m ( j ,m(j+1)' norm i n 1x1 5 TT

i s dominated by

6

L

times t h e supremum norm i n t h e

same i n t e r v a l , F i s a subspace of t h e space Lo (-,a) of a l l odd, 2lr 2 ~ - p e r i o d i c f u n c t i o n s which a r e square i n t e g r a b l e i n 1x1 5 TT endowed The cosine f u n c t i o n @(;) i s defined by 2 i t s i n f i n i t e s i m a l g e n e r a t o r A i s d /ax2 with maximal domain.

with t h e corresponding norm.

(8.1);

Assume t h a t f o r some for

%(;).

shows t h a t

LL,(;)

i s bounded i n , say, defined b y (8.28)

b

we can f i n d a group decomposition

(8.9)

must obey (8.9) and t h a t i n case Ill.+(t)/lF It1 51 t h e same must be t r u e of ~ ~ ~ o (b,(t) t ) ~ ~ ,

with

b = 0.

( t . ) i s t h e sequence i n

But i f

J

then

.sup

3 21ym( j),m(

thus

(8.U)

An argument very s i m i l a r t o t h a t p e r t a i n i n g t o t h e space

j+l)(tk)

~ ~ b o ( t .+ k )m~ ~by v i r t u e of

ym(k),m(k+l)(tk)J

(8.28),

(8.29)

and a c o n t r a d i c t i o n i s

obtained. The following r e s u l t shows t h a t t h e problem of f i n d i n g a group

94

PHASE SPACES

decomposition of a cosine function becomes m d i c a l l y simpler

if one

i s allowed t o enlarge t h e underlying space.

THEOREM 8.4.

Let c ( ; )

i n t h e Banach space_ E

be a s t r o n g l y continuous cosine function

satisf'ying

wltl Then t h e r e e x i s t s a Banach space /Iu/IE5 (C

(u

E

E)

(8.30)

m).

5 F,

E

F such t h a t

l I ~ l5l ~Cllu/lE

(8.90))

t h e constant i n


(-..

,

(8.31)

a n d a s t r o n g l y continuous group

U(t)

such t h a t (8.32)

Moreover,

F

i s minimal i n t h e sense t h a t f i n i t e l i n e a r combinations

of elements of t h e form

Proof: L e t u(c) (0

defined i n

U(t)u

(-m


03,

u

E

E)

G

be t h e space of a l l continuous

-a


t h e constant i n

m

(8.30)).

such t h a t The space

axe dense i n

E-valued f u n c t i o n s

llu(s)ll = O(eWISI) as G

F.

(sI

becomes a Banach space

-

equipped with t h e norm 11u(i)11 =

sup

e-W~s~t~u(s)~~.

(8.34)

-fo<S<m n

E

The space

can be i d e n t i f i e d with a subspace

E

of

F through

t h e map

and we obviously have

(8.36) A

If

u(s)

h

E

E we have @ ( t ) u ( i ) = @(t)@(;)u =

L2 (@(t

+

i)u

-

+ @(ti ) u > =

03

95

PEIASE SPACES

1 (u($

=

- t)),

+ t ) + u(2

2

(8.37)

h

@(t)

thus

a c t s on elements of

i n t h e same way as t h e " t r a n s l a t i o n

1.,

(8.1), ( 8 . 2 ) and (8.3).

c o s i n e f u n c t i o n s " appearing i n Examples 1'

i s d e f i n e d as t h ? c l o s e d subspac? of

u ( z + t ) with

t h e form

10


g e n e r a t e d by a l l elements of

G

U(t^)

The semigroup

<m.

The space

F is

in

d e f i n e d by

U(t)u(S)

2

E with

Identifying

i s nothing b u t (8.30).

to

h

=

u(s + t )

(8.31) we n o t e t h a t i f

G,

$111.9

Miscellaneous comments

(8.38)

(8.33) follows; obviously (8.36)

( a s we may),

To check

I l u ( t ) u ( ~ ) l /=

.

SUP

e

-Mi

s

1 114.

u(G)

belongs

+ t)ll

.

Phase spaces f o r a b s t r a c t higher o r d e r equations more g e n e r a l t h a n

(1.1)were introduced by WEISS [1967:l] ( s e e Chapter 8 f o r a d d i t i o n a l information).

w a s introduced Em i n '$111.1

The maximal phase space

The t h e o r y of f r a c t i o n a l powers of c l o s e d

by KISYkKI [1970:1].

o p e r a t o r s i n $111.2 and $111.3 w a s developed b y B W I S H N A N [1960:1] although somewhat l e s s g e n e r a l d e f i n i t i o n s were around b e f o r e . results i n

of t h e a u t h o r [1969:2] and [1969:3]. t h e growth of t h e c o s i n e f u n c t i o n possible:

The

'$111.4, $111.5 and $111.6 a r e s l i g h t l y improved v e r s i o n s Some f u r t h e r improvements regarding

Cb(;)

s e e Chapter V I , Exercises

generated by

3 t o 8.

A

-

b21

are

Theorem 7.1 on a n a l y t i c

semigroups 2nd t h e i r i n f i n i t e s i m a l g e n e r a t o r s i s a c l a s s i c a l r e s u l t of H I L L E i n t h e f i r s t e d i t i o n ( p u b l i s h e d i n HILLF-PHILLIPS [1957:1].

1948) of

his treatise

The a p p l i c a t i o n t o second order equations

(Theorem 7.2) i s due t o t h e a u t h o r [1981;2], although t h e f a c t t h a t f r a c t i o n a l r o o t s of

-A

g e n e r a t e a n a l y t i c semigroups was proved much

b e f o r e by BALAKRISHNAN [1960:1]; t h e e s t i m a t i o n of i s there.

LA(;)

The problem of f i n d i n g g e n e r a l s q u a r e root

based on (7.19) reductions f o r t h e

e q u a t i o n (1.1)w a s s y s t e m a t i c a l l y examined by KISYGSKI [1972 : l ] , who provided Example

8.2 f o r b

= 0.

The r e f l e x i v e e x t e n s i o n (Example 8.3)

i s i n t h e a u t h o r [1981:2] and f o l l o w s an unpublished s u g g e s t i o n of

96

PHASE SPACES

F. LAUFBURSCHEN. EXEXCISE 1.

Let

A E QL(C,w)

s t r o n g l y continuous semigroup

A - bI E a(C)

Then A

>

for

b

R(A;A - b1)

(that i s ,

w

generate a

A

such t h a t

exists f o r

and s a t i s f i e s

0

jlR(h;A

For

2

(that is, l e t

S(t.)

> w,

b

(bI

-

A)a

-

( h > 0)).

b1)Il < - C/A

( a > 0 ) i s i n v e r t i b l e and

( s e e YOSIDA [1978:1, p. 2601). A s c a l a r counterpart of (9.2) i s i n

GRKDSTEIN-RIDZYK [1963 :1, p. 3311 EXERCISE 2. let

-

( t h e author, [1983:3]).

A

E

Il@(t)ll < - Cewltl

-

A

b21

E

for

S(C)

b

2

w.


(a

For

b

>

w,

(that is,

8(C,w)

generate a s t r o n g l y continuous cosine f u n c t i o n

A

Then

Let

such t h a t

.

(9.3

- A)'

( a >0) i s

< m ) )

(b21

@(t^)

1

i n v e r t i b l e and

KV denotes t h e Macdonald f u n c t i o n defined by

where

for

v

#

..

+3,~2,.

and extended lyi c o n t i n u i t y t o a l l values of

(WATSON [1944:1, p. 781).

of a well known i n t e g r a l formula (GRADSTEIN-RLDZYK [l963 :1, p.

EXFRCISE

3.

Let

A,

v

We note t h a t (9.4) i s a vector-valued analogue S(t)

b e as i n Exercise 1.

Given

u

763 ] ). E

E

we say

97

PHASE SPACES

that

S($)u

L

i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r

and o n l y i f t h e r e e x i s t s

p >

sal/fs(s)(l i n t e g r a b l e i n

s >_ 0

fB(g)

and a f u n c t i o n

w

t

2

0

if

continuous, w i t h

and such t h a t

i m

(t > - 0).

‘-1 f p ( s ) d s

=

e-%(t)u

&

0

(9.6)

h

The f u n c t i o n

of

c1

i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r

S(c)u

that

i s t h e d e r i v a t i v e of o r d e r

f,(:)

-

u E Ea = D ( ( b 1 ( n o t e t h a t b y Lemma 4.2,

D((b1

a

- A) )

e-@u. c1

Show

i f and o n l y i f

(9.7)

A)‘)

does n o t depend on

b).

The

r e s u l t shows, i n p a r t i c u l a r t h a t t h e d e f i n i t i o n of c o n t i n u o u s d i f f e r e n t i a b i l i t y of o r d e r EXERCISE 4.

2.

Given

E

a &

order

function at

u

t m

E -X

fe($)

does not depend on t h e

~1

[1966:1 3 or

KOMATSU

( t h e author,

[1983:3]).

we say t h a t

@(;)

<

t

<m

u

w

chosen.

See

Let

A, C(;)

be as i n E x e r c i s e

i s c o n t i n u o u s l y d i f f e r e n t i a b l e of

>

i f and o n l y i f there e x i s t s

continuous i n


-X

with


and a

w

(.)I1 B

integrable

sal]f

and such t h a t

<

(a

fp(g)

The f u n c t i o n t h a t if

p >

t h e a u t h o r [ 1983:3] f o r a d d i t i o n a l d e t a i l s .

t

< m).

(9.8) a of

i s t h e d e r i v a t i v e of o r d e r

@ ( t ) u i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r

..

a # n + 1/2, n = 0,1,.

E

E

5.

c1

( t h e author,

Show that i f

c1

>

6.

(the author,

(9.9)

the implication is i n general false.

[1983:3]). L e t > 0 and u

0, 6

i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r

EXERCISE

with

‘= D((b21 - A ) a ) .

For t h e e x c e p t i o n a l v a l u e s of EXERCISE

2 ‘

Show

then u

E x e r c i s e 2.

e-p%(t^)u.

[1983:3]).

A, @(;)

b e as i n

E E

then

‘+by

@(t)u

201

Let

E

b e a space s a t i s f y i n g

98

EVSE SPACES

@(z)

(6.1) w i t h 1 < r < m, A,

i's i n E x e r c i s e 2.

c o n t i n u o u s l y d i f f e r e n t i a b l e of order

2a w i t h

a

Then

>

0

@(t^)u i s i f and o n l y i f

(9.9) h o l d s . A

EXERCISE 7.

0

<

< -

1.

S(t)

Assume t h a t

t > - 0.

a in

Let

EXERCISE 0<

< - 1.

2

t

in

c1

8.

0.

h

@ ( t )b e a s t r o n g l y c o n t i n u o u s c o s i n e f u n c t i o n .

Let

h

@ ( t ) u i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r

-

Show t h a t

on compact s u b s e t s of EXERCISE 9.

i s Hdlder c o n t i n u o u s w i t h exponent

S(t^)u

>_

t

Assume t h a t 0.

i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r

S($)u

Show t h a t

on compact s u b s e t s of

b e a s t r o n g l y c o n t i n u o u s semigroup,

c(t^)u

i s Kdlder c o n t i n u o u s w i t h exponent c1

< t < m.

( t h e author,

[1369:2]).

Let

@($) a s t r o n g l y c o n t i n u o u s c o s i n e f u n c t i o n i n 2.

Banach s p a c e ,

t h a t Assumption 5 . 1 h o l d s i f and o n l y i f s t r o n g l y continuous f u n c t i o n i n

i

> - 0,

V(Z)E C - D(A)

=

/ ii

space,

~ ( 2 )a

AV(F)

Show

is a

-

log s ( S ( s + t ) - S(s

t))ds.

(9.10)

0

use t h e e x p r e s s i o n for

EXERCISE 10.

and

where

r l

V(t)

(Hint:

b e an a r b i t r a r y

E

(b21

-

o b t a i n e d i n E x e r c i s e 2).

( t h e a u t h o r [1169:1]).

Let

E

b e a n a r b i t r a r y Banach

strongly cosine function satisfying

Using f o r m u l a (6.18) show t h a t

@ ( s ) u ds

TI

<

t

<m

20

E

O+ of t h e i n t e g r a l o v e r

-

for e v e r y

u

in

E

D(A), (t-

SI

where

>_

E.

v.p.

i n d i c a t e s l i m i t as

Formula ( 9 . B ) i s a n o p e r a t o r

a n a l o g u e o f t h e scalar f o r m u l a

v a l i d for

a

>_

0

(9.Q)

(GWSTEIN-RIDZYK [1963:1, p. 4211)

99

PIUSE SPACES

EXERCISE 11.

Under t h e assumptions i n E x e r c i s e 10, show u s i n g

formula (6.23) t h a t

u

for e v e r y

D(A).

E

Formula

(9.14) i s a n o p e r a t o r a n a l o g u e of t h e

s c a l a r formula

valid for

a

>

0.

(GRADSTEIN-RIDZYK [I963:1,p. 4201).

EXERCISE l2. Using E x e r c i s e s 5 and 8 show t h a t f o r m u l a (?.l2), as < t < M for

w e l l as i t s more g e n e r a l v e r s i o n (6.18) h o l d i n

ucE,y>O. Y EXERCISE

13. Using E x e r c i s e s 5 and 8 show t h a t formula (9.14), as < t < m f o r u c: E (6.23), h o l d i n

w e l l as i t s more g e n e r a l v e r s i o n f o r any

y

> 1/2.

EXERCISE 14. n o t bounded i n

F?(t^)

equals

Y

Show t h a t t h e s i n g u l a r i n t e g r a l o p e r a t o r (8.3) i s

C271(-m,m).

EXERCISE 15. of

-M

Prove Theorem

R(h;U)

1.3 showing t h a t t h e Laplace t r a n s f o r m

and a p p l y i n g Theorem

1.3.4.

FOOTNOTES TO CHAPTER I11

(1) Elements of

Eo x El

and similar p r o d u c t s p a c e s w i l l b e d e n o t e d

as "row v e c t o r s " or "column v e c t o r s " a c c o r d i n g t o convenience. (2)

3

T h i s e s t i m a t e c a n b e c o n s i d e r a b l y improved (see Chapter VI, E x e r c i s e s

t o 8).

(3) (4) (5) (6) (7)

See f o o t n o t e ( 2 ) . See f o o t n o t e ( 2 ) . See f o o t n o t e ( 2 ) . See f o o t n o t e ( 2 ) . See f o o t n o t e ( 2 ) .

100

CHAFTER I V APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS

Wave equations:

5IV.l

t h e D i r i c l l e t boundary condition.

W e consider i n t h e f i r s t s i x s e c t i o n s of t h i s chapter t h e equation U"(t) = A ( @ ) u ( t ) .

(1.1)

Here

m

m

Au =

D j ( a . (x)Dju) Jk

with

x = (xl, . . . , x m ) , D ' = a/ax

j

~ ( x )a r e defined i n a domain 0 A(B)

denotes t h e r e s t r i c t i o n of

condition

@

r

a t t h e boundary

or

bj(x)Dju

-e

+

c(x)u

j =1

j=1 k = l

and t h e c o e f f i c i e n t s a . ( x ) , b . ( x ) , Jk J of m-dimensional Euclidean space Rm A

obtained by means of a boundary

of t h e form

D " u ( ~ ) = y ( ~ ) ~ ( ~( X)

-

E

r),

(1.4)

D" denotes t h e conormal d e r i v a t i v e t o be defined below ( s e e (4.1)). 1 Since t o r e p l a c e a j k ( x ) by ( a j k ( x ) + a k j ( x ) ) does not change t h e

where

a c t i o n of

A

on smooth f u n c t i o n s we s h a l l assume from now on t h a t

a

jk

We r e q u i r e t h e valued. A

If t h e

a

jk a

(x)

= a

kj

(x).

t o be real-valued; t h e

b. J

and

c

can b e complex-

have f i r s t order p a r t i a l d e r i v a t i v e s , we can w r i t e

jk

i n t h e more n a t u r a l form

m

Au =

m

m A

a . (x)DjDku + r b . ( x ) D j u j=1 k = l j=1 J

+

c(x)u

,

(1.5)

101

PARTIAL DIFFEREWIAL EQUATIONS

where

The passage from ( 1 . 2 ) t o (1.5) and v i c e v e r s a i s no longer p o s s i b l e i f

the

a

assume

(1.5) represent

a r e not d i f f e r e n t i a b l e ; i n t h i s case ( 1 . 2 ) and

jk

[1974:1]).

q u i t e d i f f e r e n t e n t i t i e s ( s e e PUCCI-TALENT1

We s h a l l always

i s w r i t t e n i n t h e form ( l . 2 ) , c a l l e d t h e divergence o r

A

v a r i a t i o n a l form.

The c o e f f i c i e n t s

a

b c w i l l be required t o be jk’ j’ merely measurable and bounded; we p o s t u l a t e i n a d d i t i o n t h a t A be uniformly e l l i p t i c i n t h e sense t h a t

(1.6) f o r some

ic

> 0.

Our f i r s t r e s u l t concerns t h e D i r i c h l e t boundary condition (1.3). No assumptions whatsoever w i l l be placed on t h e domain boundary

r.

or the

0

In t h i s high l e v e l of g e n e r a l i t y , it i s obvious t h a t

(as well as t h e boundary condition

u

=

0)

Au

will have t o be understood

i n a s u i t a b l y generalized sense; for instance, i n view of t h e l a c k of smoothness o f t h e

a

jk’

i t i s not c l e a r whether

can be applied t o

A

any nonzero function. The b a s i c space i n our treatment i s supporting r o l e w i l l be played by

a(n)

( o f Schwartz t e s t f b n c t i o n s )

+(O), in

H

2 = L (0).

An important

t h e c l o s u r e of t h e space

$(n);

m(X)

c o n s i s t s of all i n f i n i t e l y d i f f e r e n t i a b l e functions in

0, and t h a t t h e space

a;LI

functions

u

$(n)

(k

an i n t e g e r

and

+(n)

2

L2(n).

with support

1) c o n s i s t s o f

having p a r t i a l d e r i v a t i v e s of order

understood i n t h e sense of d i s t r i b u t i o n s ) i n

8(n)

we r e c a l l t h a t

5k

(derivatives

The spaces

( f o r all necessary f a c t s on t h e Sobolev spaces

ADAMS b975 :1I )

Hk(n), #(n)

consult

.

The f i r s t stage of o u r argument w i l l be t h e c o n s t r u c t i o n of operator

d(0)

a r e f i l b e r t spaces equipped with t h e s c a l a r product

Ao(B),

where

A0

i s the

self a d j o i n t p a r t ol

A,

the

102

PARTIAL DIFFEREmIAL EQUATIONS

With t h i s i n mind, we introduce a new s c a l a r product i n

#,(fi)

by t h e

formula

(u,v),

=

[(a- c ) c v dx

ifi

where

dx

=

... dxm

dxl

(u,~),

(and conjugate l i n e a r i n C > c

constants

h ( y y ajk&Dkv)

dx

(1.9)

and

cY>v It i s obvious t h a t

+

=

e s s . sup c

-

and t h a t

= (V,U)~

u).

.

(1.10) (U,V)~

i s linear i n

v

Moreover, we see e a s i l y t h a t t h e r e e x i s t

such t h a t

0

( w e u s e t h e uniform e l l i p t i c i t y assumption for t h e f i r s t i n e q u a l i t y ) . Accordingly, t h e norm

(1.12) corresponding t o t h e s c a l a r product norm of

4

defined by

(1.7);

(1.9) i s

equivdent t o the original

t h u s we s h a l l assume from now on

&(n)

endowed with (1.12) ( o f cotirse, t h e same arguments apply t o t h e

space

$(",

a f a c t t h a t w i l l be used i n 61V.4).

A function

u

E

$(n)

D(AO(f3)) i f and only i f

belongs t o

(1-13 1

i s continuous i n t h e norm of

L

2

(i)):

i f t h i s i s t h e case, w e extend

( s i n c e €$(a) i s dense L2(Q) 2 i n t h e topology of L (n) t h i s extension i s unique). L e t

t h e l i n e a r f u n c t i o n a l (1.1)) t o a l l of in v

L E

2

(Q)

2

L (0) be such t h a t

(1.14) Define A0(B)" (Motivation i s obvious:

= azI

-v .

i f the coefficients a

(1.15) jk

and t h e boundary

r

103

P A R T I A L DIFFEREIWIAL EQUATIONS

u

are smooth and

u

and

w

=

r,

on

0

=

r).

on

0

-A u = v in n 0 f o l l o w s f o r any smooth w such t h a t

i s a smooth f u n c t i o n such t h a t then

(1.14)

We check e a s i l y t h a t t h e d e f i n i t i o n of A (f3)

above does

0

a.

not depend on

We wish t o show t h a t t h e o p e r a t o r

j u s t defined i s s e l f a d j o i n t .

AO(B)

We b e g i n by proving t h a t

( U - AO(B))D(AO(f3)) h > v.

f o r any

In fact, l e t

(1.16)

= L2P)

be a n a y b i t r a r y element o f

v

L2(Q)>.

Define a l i n e a r f u n c t i o n a l by

w

(1.17) i s

Since

4(0),

L * ( ~ I it i s a s well continuous i n

continuous i n

u

thus there exists

E

$(n)

(1.16)

such t h a t =

( U , d A

hence

(1.17)

*

(V,.>X

-+

(1.18)

(v,w>, u

note t h a t o u r c o n s t r u c t i o n of

follows:

yields the

estimate

Rewriting ( 1 . 1 4 ) i n t h e form

w = u

and t a k i n g

we s e e t h a t

(1.19)we deduce t h a t from

(1.18) t h a t i f

R(h;Ao(p))

u, v

( h- AO(f3)~,v)= so t h a t

Ao(B)

defined.

A1

E

D(A,(B))

(U,V),

i s symmetric.

u

E

4(Q)i n

D(AO(@)).

X-D(A0(B))

To prove t h a t

result.

h > v.

combining with

It a l s o f o l l o w s

then =

=

( u , hv

- A(B)v) i s densely

AO(B)

We f i n a l l y prove t h a t

In o r d e r t o d o t h i s it i s sufficient t o show that

i s dense i n all

i s one-to-one;

exists for

t h e topology of

c a s e , t h e r e would e x i s t an element

to

- AO(B)

w

E

d(". $(."I

I f t h i s were not t h e with

( U , W ) ~=

I n view of (1.20) t h i s i m p l i e s t h a t which, d u e t o

A (B) 0

(1.16),

D(AO(B))

shows t h a t

w

0

for

i s orthogonal

w = 0.

i s s e l f a d j o i n t we make u s e of t h e following

104

PARTIAL DIFFERENTIAL EQUATIONS

Let

LEMMA 1.1. X l b e r t space

number

A.

Proof: -

Assume t h a t t h e resolvent

H.

Then

Let

be a. densely defined symmetric operator i n t h e

A

u,v

be two a r b i t r a r y elements of

( R( A;A)u,v) =

so t h a t

R()\;A)

( ( AI

p(A)

contains a r e a l

i s s e l f adjoint.

A

=

(R( A ; A ) U ,( AI

- A ) R (A;A)u,

R( A;A)v)

H.

Then

- A ) R ( A;A)V) =

(u, R( A, A ) V )

i s symmetric; t h u s

where t h e interchange of i n v e r s e s a n d a d j o i n t s i s e a s i l y j u s t i f i e d (see

RIESZ-SZ.-NAGY [1955:1I ) .

This ends t h e proof.

i s s e l f a d j o i n t and bounded above (by Ao(B) generates a strongly continuous cosine f u n c t i o n A0 ( @ ) @(<) given by Having shown t h a t

U)

we know t h a t

C(t)

=

cosh t A o (B)1/2

(1.22)

where t h e expression on t h e r i g h t i s defined by means o f t h e f u n c t i o n a l calculus for s e l f a d j o i n t operators; i f r e s o l u t i o n of

Ao(B)

P(dp)

i s the spectral

then

where t h e upper l i m i t of i n t e g r a t i o n i s due t o t h e f a c t t h a t

Ao(B)

Z V ' I I U ~for ~ ~ u E D(A)). A ( B ) we s h a l l construct a phase space for and t h e n incorporate t h e first order terms by means of a p e r t u r -

i s bounded below by U

((AO(B)u,u)

To handle t h e full operator

Ao(B) b a t i o n argument t o be proved i n next s e c t i o n .

The proof t h a t

A(B)

generates a s t r o n g l y continuous cosine f u n c t i o n w i l l be given i n P I V . 3 .

b . 2

The phase space.

Since t h e operator

Ao(@)

B

i s s e l f a d j o i n t , a square r o o t =

AO(P) 1/2

(2.1)

can be defined by means of t h e f u n c t i o n a l c a l c u l u s f o r unbounded f u n c t i o n s

105

PARTIAL DIFFERENTIAL EQUATIONS

of o p e r a t o r s (DUNFORD-SCHWARTZ r e s o l u t i o n of

Ao(B),

u

then

[1963:1I ) ; D(E)

E

if

i s the spectral

P(dp)

i f and only i f

and

There remains t h e problem of t h e choice of square r o o t

b1L-1/2 i n (2.3);

although (as we s h a l l see below) t h i s choice i s immaterial, we d e f i n e p1/2

w 2

for

= p2 ;j

nonnegative number

( r t / 2 t h e nonnegative square r o o t of t h e

0

r)

and

pl/*

i (-P)’/~ .t

=

i s not e a s i l y i d e n t i f i a b l e i n t e r m s of

D(B)

we b e g i n by noting t h a t i f

D(B)

p C 0.

Although

E

t u r n s out t o be

facilitate the identification

To

A > v then

.

= D((U-A~(@))’/~)

To see t h a t (2.4) h o l d s we n o t e t h a t

D(B)

Ao(f3),

a familiar space (Theorem 2.2 below). of

for

u

E

(2.4)

D((AI-AO(B))l / 2 )

i f and

only if

w)1/21iP(dw)uI/2 <

A(:-[

m

.

(2.5)

The f a c t t h a t ( 2 . 2 ) and ( 2 . 5 ) a r e e q u i v a l e n t f o l l o w s from t h e boundedness

of the function

1p

Ill2-

Ih-

p

1 /2.

LEMMA 2.1. Let Q be a densely defined, i n v e r t i b l e o p e r a t o r such -1 . that Q i s everywhere defined and bounded. Then D(Q2) i s dense i n

D(Q)

i n t h e Rraph norm o f Proof:

D(Q) w i t h

If u vn

-

T H E O E M 2.2.

6

v

D(Q)

D(Q). then

u

and note t h a t

Q-lv; -1 Q vn

=

-

s e l e c t a sequence

u,

Q(Q-lvn) = v

{vn] n

-

Qu.

in

106

PARTIAL DIFFEREWIAL EQUATIONS

It follows from Lemma 2.1 t h a t D ( A ( B ) ) i s dense i n D ( ( U - A 0 ( @ )1) /2 ) i n t h e graph norm of ( h r - A o ( B ) ) l / q ; on t h e o t h e r hand, we have shown i n 6 I V . l t h a t

$(n),

D(AO(B))

i s dense i n

%(n)

t h u s ( 2 . 7 ) i s e a s i l y seen t i imply t h a t

i n t h e norm of

D((U-Ao(!3))1/2)

=

*(o).

Combining with ( 2 . 4 ) , Theorem 2.2 f o l l o w s . We a r e now i n p o s i t i o n t o c o n s t r u c t a phase space f o r t h e equation

u“(t)

=

Ao(B)U(t)

.

(2.8)

I n f a c t , Theorem 111.5.4 a s s e r t s t h a t

A. ( @ )-ll2sinh t A. ( B )1/2

c o s h t Ao(B)1/2 r,(t)

r

=

1

; (2.U)

(2.12)

with domain

D(V0(B)) = D ( A O ( p ) ) ,<

J$(n).

T o t a k e c a r e of t h e f i r s t

o r d e r terms we introduce t h e p e r t u r b a t i o n o p e r a t o r

?=IF *I. 0

0

107

PARTIAL DIFFERENTIAL EQUATIONS

where

(2.14) Obviously,

p

i s a bounded o p e r a t o r i n

of t h e o p e r a t o r result

, where

A(B)

A0(B)

=

+

The necessary p r o p e r t i e s

@.

w i l l be a consequence of t h e following

P

does not have t h e same meaning as i n ( 2 . U ) .

P

Let

THEOREM 2 . 3 .

be t h e i n f i n i t e s i m a l g e n e r a t o r of a group

A

A

S(t)

i n a Banach space

E

Ils(t>li5 ithat is, l e t

A

t h e operator

gl(coJo

$-

Proof:

A

E

+

such t h a t COeultl

c1 ( C o , U ) ) , P

(-m

and l e t

P


m)

b e a bounded o p e r a t o r .

D(A + P ) = D ( A ) )

( w i t h domain

Then

belongs t o

cllplI)* ~y v i r t u e o f Theorem

1.1.3

R(A;A)

e x i s t s for

IAJ

>

~1

and s a t i s f i e s

I n p a r t i c u l a r , if

then

(2.16) hence t h e s e r i e s

i s convergent i n t h e uniform topology of o p e r a t o r s .

w e have Q(A)(~-A-P)U =

-

(PR( h;A))n( hl A

= R( A;A)

- P)u

n=O co

=

a3

T ( R ( A ; A ) P ) ~ T~ -( R ( A ; A ) P ) ~ ~= n=0

n=l

u.

If

u

E D(A)

108

PARTIAL DIFFEREIWIAL EQUATIONS

Similarly, i f

u

f

E, m

(AI-

A -P)Q(A)U

=

c

(XI - A - P)R(A;A)

(PR(A;A))"~

n= 0 m

1

=

W

( F R ( A . ; A ) ) ~ ~-

n=O

c

( P R ( A ; A ) ) ~ ~=

,

n=1

thus

Q(A) = H(A;A + P)

.. .

.

jl,. . , j m = 0,1,2,.

where

.

Due t o t h e f a c t t h a t some of t h e

jk

may

n PIS w i l l be of t h e form

b e zero, t h e t y p i c a l term containitlg

(2.17) where

s1

+

+

sn+l = n

+m

and

s

k

>

0.

Making use of (2.15) w e

s e e t h a t (2.17) can be bounded i n norm b y

Accordingly

Applying Theorem

1.3.3

( t h i s time backwards) as modified by Remark 1.3.5

we conclude t h e proof. P u t t i n g t o g e t h e r t h e r e s u l t s i n t h i s s e c t i o n we o b t a i n .

THEOREM 2.4.

Let A -

be t h e o p e r a t o r (1.2),

p the Dirichlet

109

PARTIAL DIFFEFJ"IAL EQUATIONS

(1.3), and l e t

boundary c o n d i t i o n

A(@) with domain

D(A(p))

D(AO(B)).

=

(2.18)

+ p

= A0(P)

Hi(CL) x L*(n)

Then t h e s p a c e

Q

phase s p a c e f o r t h e e q u a t i o n u"(t) = A(p)u(t).

(2.19)

srV.3 The Cauchy problem. The r e s u l t s i n t h e p r e v i o u s s e c t i o n do n o t show t h a t t h e Cauchy problem f o r t h e e q u a t i o n (1.1)i s w e l l posed i n t h e s e n s e of $11.1. We The b a s i c p r o d u c t s p a c e w i l l b e

s h a l l do so h e r e .

We d e f i n e a ( 3 ) - v a l u e d f u n c t i o n

3 (t) 0

I

=

30(z) b y

cosh t Ao(p)

1'

s i a i t Ao(p)l'2

sinh t Ao(p)

"'

cosh t AO(B)li2

=I where

B

cosh t B

sinh t B

sinh t B

cosh t B

i s t h e s q u a r e r o o t of

t h a t t h e group

X) = L2(n) x L*(n).

P.

B0(t)

AO(P)

I= (3.1)

c o n s t r u c t e d in fiyIV.2.

We n o t e

does depend on t h e c h o i c e of s q u a r e r o o t

B; i n

I\

"j,(t)

c o n t r a s t , t h e group

d e f i n e d b y (2,ll) i s independent of

s e e t h i s w e n o t e that, cosh t B

Ao(p)

To

c a n b e d e f i n e d d i r e c t l y i n t e r m s of

as f o l l o w s :

cosh t B = c(t;A

( B ) ) with c ( t , p )

0

The same o b s e r v a t i o n a p p l i e s to AO(f3)"'sinh

!L~(@)"~.

p-'/'sinh

t p1/2

= tp +

t3rJ.'/3! + t

contrast, P"~,

B.

thus

sinh t

=

:'

cosh

= I+ t2p/,2! + t

p1 2'

A o ( p ) -l/isinh . t Ao(f3)1'2

5! + =

+

-..

p1I2

+

t5p2/5!

+

depend on S3p3l2/3!

(3.1) depends on t h e v a l u e of

on t h e c h o i c e of

t

41~~/4! + .-.

and t o

To s e e t h i s w e s i m p l y n o t e t h a t the f u n c t i o n s

t + t3p/3! i-(

=

... p

and

a n d not on

+ t5p5/2/5! p1j2

t p"

pl"sinh

+

In

p. 2 ' 1

...

depends on

chosen ( e q u i v a l e n t l y ,

B).

Working as i n Theorem

=

111.5.5 we show t h a t t h e i n f i n i t e s i m a l

110

PARTIAL DIFFERENTIAL EQUATIONS

generator

8 of 0

W

0

is

To proceed f u r t h e r , we make t h e assumption t h a t

v

=

ess. s u p c ( x ) < 0 ;

(3.3)

( a s we s h a l l see l a t e r , t h i s can be dispensed w i t h ) . has a bounded i n v e r s e f o r

now t h e operator

F

it follows t h a t

A >V,

AO(B)-l,

bounded i n v e r s e

hence

B

X-AO(B)

Since

Ao(f3) i t s e l f h a s a -1 B Consider

.

h a s a bounded i n v e r s e

We claim t h a t

i n (2.14).

FE-’ i s a bounded o p e r a t o r i n

L2(n);

Theorem 2 . 2 ) i s continuous i f

(3.4)

since

D(E)

E-1 : L2(R) * D ( B )

i s c l e a r t h a t we o n l y have t o show t h a t t h e graph norm of

D ( B ) = +(?)

i s e q u i v a l e n t t o t h e norm of

$(Q).

consequence of t h e f a c t t h a t b o t h norms make

D(B)

(see

$(Q)

E

B,

it

in

This, i n turn, i s a complete and of

the

[1958:1 I ) .

open mapping p r i n c i p l e ( s e e DUNFGRD-SCHWARTZ Having proved t h a t

=

i s given t h e graph norm of

( 3 . 4 ) i s i n f a c t bounded, consider t h e o p e r a t o r

Applying Theorem 2.3 we deduce t h a t t h e o p e r a t o r

(4=%o+T g e n e r a t e s a s t r o n g l y continuous group

The r e s o l v e n t

R(A;B0 +

7)

(3.6)

S ( t ) i n t h e space

must e x i s t for

h

3.

Write

sufficiently large:

write

111

PARTIAL DI1’1’6PI IUIAL F~JUATIONS

(3.8)

Looking st the. first column of t h e r n s t r i x ? q u a t i o n

(3.9)

3

3,

the i d e n t i t y o p e r a t o r i n

(A’I

- B ‘

- P)R

(A)

11

we o b t - i i n

=

(kI

- AO(p)

- P)Rll(A)

=

hI.

(3.10)

Workini; i n t h e same way w i t h tlie ~ q u i ~ t i o i i

F i n a l l y , o b s e r v i n g t h e element i n t h e upper l e f t c o r n e r of the m a t r i x (24u a t i o n

rrn

R(h;% + 9 ) u =

e-htP(t)u

0

LJ

(which r e s u l t s from

(1.3.8))

dt

0

w e o b t a i n , making use of (3.L?), t h a t

U

which e q u a l i t y , i n view of Theorem 1 1 . 2 . 3 i m p l i e s t h a t

S(%) = V 11( 2 )

i s a s t r o n g l y continuous c o s i n c f u n c t i o n with i n f i n i t e s i m a l g e n e r a t o r

A(@)

(3.13)

= AO(P) + p,

D ( A ( B ) ) = D(AO(B)).

To get r i d of assumption

(3.4) it

suffices t o replace

c(x)

by

112 c(x)

PARTIAL DIFFERFWIAL EQUATIONS

-

- u 6 (6 >

operator

P

i n t h e definition of

0)

i n t h i s case, t h e

Ao(B);

i s defined by

m rb.(x)DJu + ( v + b)u

Pu =

j=1

(3.14)

J

i n s t e a d of (2.14). We have completed t h e proof of

THEOREM 3.1. Let A be t h e v p e r a t o r (1.21,

-boundary

the Dirichlet

c o n d i t i o n (1.31, and l e t

A(B) with domain

D(A(f3))

=

i s w e l l posed i n

-m

A;(@)


Wave e q u a t i o n s :

+

(3.15)

P

Then t h e Cauchy problem f o r t h e e q u a t i o n

D(AO(B)).

=

u"(t)

GIV.4

f?

=

(3.16)

A(B)u(t)

w.

o t h e r boundary c o n d i t i o n s .

We study i n t h e next two s e c t i m s t h e o p e r a t o r ( 1 . 2 ) with boundary

(1.4),

c o n d i t i o n s o f t h e form

where

v = (v

,..., vm)

i s t h e o u t e r normal v e c t o r on t h e boundary

1

t h e e x p r e s s i o n on t h e l e f t - h a n d s i d e of d e r i v a t i v e of

u

(1.1)w i l l be

L

$(a).

at

2

(a);

The o p e r a t o r

x

E

r.

(4.1) i s

Again, t h e b a s i c space i n our t r e a t m e n t of

t h e a u x i l i a r y space i s now Ao(B)

however, t h e s c a l a r product

r;

c a l l e d t h e conormal

d(n)

i n s t e a d of

w i l l be a g a i n defined by ( 1 . 1 4 ) and (1.15); (U,W)(,,

is different.

T o guess t h e c o r r e c t

d e f i n i t i o n we perform t h e f o l l o w i q ; formal computation using t h e divergence theorem, where t h e boundary enough.

r

and t h e i n t e r v e n i n g f u n c t i o n s a r e smooth

113

PARTIAL DIFFERENTIAL EQUATIONS

Accordingly, we should define

( u , ~ ) , = r ( a - c ) r v dx+L(rrajkD’%Dkv) dx-JiFv -0

dx.

u,v r

$(n)

However, t o give sense t o ( 4 . 2 ) for

arbitrary

(4.2) some

assumptions on Q w i l l be necessary, d u e t o t h e presence of t h e boundary integral.

A domain

0

5 Rm

i s said t o be of c l a s s

i f and only i f , given any point V

of

x

Rm and a map

in

t h e open u n i t sphere i n

(a)

q

(b)

The map

(c)

5k

in

(resp. i n

n r)

=

E

r ?

C(k) (k

an i n t e g e r

2

0)

t h e r e e x i s t s an open neighborhood (Sm = Sm(O,l) = {q E Rm;lql < 13

such t h a t

?

with

q(x) = 0 .

-1 ( r e s p . q ) possesses p a r t i a l d e r i v a t i v e s of

q

q ( v n n) = Q(V

q: V +

Rm)

i s one-to-one and onto

order

-v

x -

s

sm=

t so

(resp. i n

V 111

=

Sm) which a r e continuous i n

).

.

{q r

sD ;qm > 0 3

b-,

s ~m =; 0 3 ~ .

E

A l i t t l e use w i l l be made i n t h e following l i n e s of t h e Sobolev spaces

&’P(n)

c o n s i s t i n g of a l l f i n c t i o n s

LP(n);

p a r t i a l derivatives i n

t h e space

Also, we s h a l l employ t h e spaces u in

-

continuous i n

n,

C(l)(F)

having f i r s t

i s normed with

c o n s i s t i n g of a l l f u n c t i o n s

having continuous f i r s t p a r t i a l d e r i v a t i v e s

each d e r i v a t i v e admitting a continuous extension t o

THEOREM

1 f p c

u r LP(fi)

&”(n)

m.

4.1

& n

men

(a)

be a bounded domain of c l a s s

if

Dju

a. and l e t

114

PARTIAL DII’FERENTLAL EQUATIONS

t h e r e e x i s t s a constant

u

f o r everx

then

E

(depending only on 0, p , q ) s u c h t h a t

C

C(’)(F).

if

(b)

(4.3) holds f o r everx

q

2

1.

For t h e proof of a c o n s i d e r a b l y more g e n e r a l r e s u l t see ADAMS We n o t e t h a t Theorem 4.1 h o l d s a s w e l l f o r domains

[1975:1, p. 1141.

which a r e “piecewise of c l a s s c y l i n d e r s whose base i s a

(in

C (’”’

such a s , say, p a r a l l e l e p i p e d o n s or

- 1)- d i m e n s i o n d

also, t h e boundedness h y p o t e s i s i s not e s s e n t i a l : r e s u l t holds i f

THEOREM 4.2. 8

( b u t not

Let fl

n)

f o r instance, t h e

i s bounded.

be a domain o f c l a s s

, 15p

C( 0 )

(or, r a t h e r , t h e s e t of r e s t r i c t i o n s of f u n c t i o n s of

dense i n

<

Then -

M.

0) @

ds’p(o).

The proof can be seen i n ADAMS assumptions; r e c a l l t h a t t h e space t e s t functions i n Let

c(1);

domain of c l a s s

0

R

m

[1975: 1, p. 541 under =

l e s s stringent

i s t h e space o f Schwartz

.

be a bounded domain of c l a s s

domain of c l a s s

m @(R )

C(l)

w i t h a bounded boundary

(or, more g e n e r a l l y , a

r).

Assuming t h a t

E Lm(r), t h e following e s t i m a t i o n i s j u s t i f i e d by Theorem 4 . 1 (and t h e comments a f t e r i t ) : h e r e u,v a r e f u n c t i o n s i n a and we t a k e

y

p = q = 1 .

Now,

115

PARTIAL DIFFEPJ3I'dTIAL, EQUATIONS

We go back t o (4.2). with

CY

>

V = ess.

Assume t h a t t h e s c a l a r product

sup c

e l l i p t i c i t y condition

a s i n (1.10).

( U , V ) ~

i s chosen

Then, t a k i n g t h e uniform

(1.6) i n t o account we o b t a i n from (4.4) and ( 4 . 5 )

that

(4.6) t h u s it i s obvious t h a t , i f

o/

i s s u f f i c i e n t l y l a r g e , t h e f i r s t of t h e

two i n e q u a l i t i e s

w i l l hold f o r

u

E

8 ; that t h e second i s as w e l l t r u e follows from

(4.6) with no p a r t i c u l a r requirements on CY beyond cy > v . The f a c t 8 i s dense i n $(n) (Theorem 4.2) and t h e Schwartz i n e q u a l i t y

that

(u,~),

imply t h a t argument

for

can be defined, using a n obvious approximation

arbitrary

u

E

$(a).

Since t h e norm defined by ( 4 . 2 )

d(R), we

i s e q u i v a l e n t t o t h e o r i g i n a l norm of follows t h a t

$(n)

s h a l l assume i n what

Il-IIcy.

i s endowed w i t h

From t h i s p o i n t on, t h e c o n s t r u c t i o n of t h e o p e r a t o r corresponding t o t h e s e l f a d j o i n t p a r t (1.8) of condition

B

A

Ao(B) and t o t h e boundary

i n (4.1) proceeds e x a c t l y i n t h e same way as i n t h e c a s e

of t h e D i r i c h l e t boundary c o n d i t i o n :

u

E

D(A,(@))

(w

E

$(n))

i f and o n l y i f t h e

l i n e a r fbnctional

w

-

i s continuous i n t h e norm of

(u,v),

L2(R);

AO(B)u where

v

i s t h e unique element of

=

(4.9)

we d e f i n e cuu

L

2

- V, (n) s a t i s f y i n g

(4.10)

116

PARTIAL DIFFERENTIAL EQUATIONS

(4.11) A s i n sIV.1, coefficients

motivation f o r t h i s stems from t h e f a c t t h a t i f t h e a

jk function such t h a t

and t h e boundary

cm - A 0u

= v

a r e smzoth and

in

and

0

D"u = y,

i s a smooth

u

t h e n (4.11)

follows f o r any smooth w. Operating a s i n 6 I V . l we show t h a t

( U - AO(B))D(A,,(B)) t h i s time for any

h > a,

CY

=

so l a r g e t h a t (4.7) holds.

estimate of the type of (1.19) and prove t h a t

A,

i n t h e same range o f Finally,

Ao(B)

Ao(p)

W e o b t a i n an

U-AO(B) i s one-to-one

(AI-A0(p))-'

so t h a t

h >

exists i n

cy.

i s symmetric s o t h a t , using Lemma 1.1 we show t h a t

i s s e l f a d j o i n t and bounded above by

depending not only on t h e cosine f u n c t i o n

OIV.5

(4.12)

L2W,

a, where

v but also on t h e c o e f f i c i e n t

@,(t)

generated by

Ao(@)

ff

y.

i s a constant Accordingly,

i s t h i s time given by

The phase space.

The arguments i n s I V . 2 have an obvious counterpart h e r e . c o n s t r u c t i o n of t h e square r o o t

B of

Ao(B)

The

proceeds i n t h e same way,

as does t h e proof of THEORFM 5.1 D(E) =

d(n).

(5.1)

The phase space f o r t h e equation

u " ( t ) = Ao(B)u(t) i s now

(5.3)

El

=

$(".

(5.4)

117

PARTIAL DIFFEREWIAL EQUATIONS

Again, t h e phase space The group

Go(;)

( 5 . 3 ) i s t h e same one provided by Theorem 111.1.3.

propagating t h e s o l u t i o n s of ( 5 . 2 ) i s given by (2.11)

with i n f i n i t e s i m a l g e n e r a t o r D(210(f3)) = D ( A O ( f 3 ) )

(2.12), i t s domain being i d e n t i f i e d by

x €$(D).

To t a k e c a r e of t h e f i r s t order terms we

use Theorem 2.3 a p p l i e d t o t h e bounded p e r t u r b a t i o n o p e r a t o r (2.13).

I n t h i s way we o b t a i n ;

Let

THEOREM 5.1.

r,

A

0

be a domain of c l a s s

t h e operator ( l . 2 ) ,

(3

measurable and bounded on

I-.

with domain

D(A(f3))

=

CiLi

t h e boundary c o n d i t i o n

w i t h bounded boundary

( 1 . 4 ) with y

Let -

D(AO(f3)).

d(n)

Then t h e space

X L'(0)

is a

phase space f o r t h e e q u a t i o n

u"(t)

Q1v.6

=

.

A(B)u(t)

(5.6)

The Cauchy problem.

A l l t h e r e s u l t s i n S e c t i o n IV.3 have a n immediate c o u n t e r p a r t h e r e ; we

d e f i n e t h e semigroup B O ( i ) given by ( 3 . 1 ) i n t h e product space 2 2 = L (0) X L ( a ) ; again, depends on t h e p a r t i c u l a r square r o o t

z0(t)

of

Ao(B)

chosen.

B

However, we need

This can be achieved by r e p l a c i n g l a r g e i n t h e d e f i n i t i o n of

Ao(f3);

t o have a bounded i n v e r s e .

c ( x ) by

m Pu = C b . ( x ) D J u + j =1 J TmOREM boundary

y

2

6.1.

r, A

fi

t h e o p e r a t o r (1.2), B

D(A(B))

i s w e l l posed i n

-m

= D(Ao(f3)).

< t <

m

.

P

for

CY

sufficiently

i s t h e n defined by

(6.1) C(l)

with bounded

t h e boundary c o n d i t i o n

( 1 . 4 ) with

Let -

r.

u"(t)

-a

LXI

be a domain o f c l a s s

measurable and bounded on

with domain

c(x)

t h e operator

Then t h e Cauchy problem f o r t h e e q u a t i o n =

A(B)u(t)

(6.3)

118

PARTIAL DIFFERFNTIAL EQUATIONS

6IV.7

HXgher o r d e r equa.tions.

We consider b r i e f l y i n t h e r e s t of t h e c h a p t e r t h e e q u a t i o n

u"(t)

=

A(P)u(t)

(7.1)

an(x)Dnu

(7.2)

where

Au

=

l ( Y l 3

i s an a r b i t r a r y p a r t i a l d i f f e r e n t i a l o p e r a t o r of o r d e r p (Y

=

and

(a

1, D

,. . . , a m )

i s a m-gle of nonnegative i n t e g e r s ,

... (Dm) m,

= (D1)Qi

i n a domain

0

of

whose c o e f f i c i e n t s

m-dimensional Euclidean space

t h e r e s t r i c t i o n of

A

B

r.

a t t h e boundary

a,(.)

(here

la1 = a1 +

. .. + am

a r e defined

Rm; A ( B )

denotes

obtained by imposition of a boundary c o n d i t i o n Some i n s i g h t on t h e e q u a t i o n

(7.1)can

be

obtained examining t h e c o n s t a n t c o e f f i c i e n t case i n t h e whole space; we do t h i s for EXAMPLE

m

7.1.

=

1.

Consider t h e d i f f e r e n t i a l e q u a t i o n

u"(t] i n t h e space

w i t h ao, al,

2

L (Rx).

...,a

c o n s i s t s of all

Here

=

i s t h e d i f f e r e n t i a l operator

A

complex c o n s t a n t s ,

PA

u(x)

E

2 L (R)

o f d i s t r i b u t i o n s ) belongs t o

(7.3)

Au(t)

such t h a t

L*(R).

a 0; t h e domain o f A P Au (understood i n t h e sense

Through t h e Fourier-Plancherel

transform

( s e e STEIN-WEISS [l97l:11) the

equation

(7.3) i s e a s i l y seen t o be

equivalent t o t h e equation u"(t) where

=

Au(t),

i s t h e multiplication operator

(7.6)

119

PARTIAL DIFFWENTIAL EQUATIONS N

L2(RE).

in

We check t h a t

i s a normal o p e r a t o r , t h u s

A

2

E

d

( E x e r c i s e 11.5) i f and only i f w0 - sup{Re X1>’ ;X

a(x)) < m,

(7.8)

N

a(x), t h e spectrum of

where

E

i s e a s i l y i d e n t i f i e d as

A,

As proved i n E x e r c i s e 11.5, (7.8) i s e q u i v a l e n t t o t h e f a c t t h a t

~(x)

i s contained i n a r e g i o n of t h e form Re h < - w2 - ( I m h ) > / 4 3 . LEMMA 7.2.

-

A

2

i f and o n l y i f

E

p

(a)

(7.9) i s even ,(b)

a

is P -

r e a l with (-l)p’zap (c) j

is r e a l i f

aj

i s odd,

j

>

j

i s even

>

p/z,

,

(7.10)

(d) aj i s imaginary i f

p/2.

Assume t h a t ( a ) , ( b ) , ( c ) and ( d ) hold.

Proof:

P(t) = where

j

<0

q(c)

( - ~ ) P ’ ~ ~ , E P+

i s a polynomial of degree

r ( t ) + q(t),

5 p/z

Then we have

(7.11)

w i t h complex c o e f f i c i e n t s ,

and

with r e a l c o e f f i c i e n t s

b

3’

Obviously, f o r

1

sufficiently large

we s h a l l have

\El,

for l a r g e t h e form

which implies t h a t

(7.9), t h u s

E

i s odd.

Since

i s c o n t a i n e d i n a r e g i o n of

8.

We prove t h e converse. p

u(A)

Assume t h a t ( a ) i s v i o l a t e d , t h a t i s , t h a t

120

PARTIAL DIFFERENTIAL EQUATIONS

and

a.

f

0, it i s obvious t h a t

t e n d s t o i n f i n i t y along t h e

P(5)

-

5 2 a, hence o(i) cannot be contained i n a r e g i o n of t h e form (7.9). To show t h e n e c e s s i t y of (7.10) when p i s even we use t ! i e corresponding v e r s i o n of (7.14): imaginary axis i n opposite d i r e c t i o n s

p(5) if

(-l)’”a,

I ~ I - ~ ()1 5)1

+ 0(

(-1)p”ao5P(1

=

p(5)

i s not r e a l and negative,

-

(7.15)

m);

w i l l tend t o i n f i n i t y

i n t h e d i r e c t i o n of t h e p o s i t i v e r e a l axis. F i n a l l y , we show t h a t ( c ) and ( d ) a r e as well necessary.

Note f i r s t

t h a t both c o n d i t i o n s axe necessary and s u f f i c i e n t i n order t h a t t h e polynomid

r ( 5 ) i n (7.11)

have r e a l c o e f f i c i e n t s .

h a s a t l e a s t one complex c o e f f i c i e n t , and l e t

bk

Assume t h a t

r(5)

be t h e complex

c o e f f i c i e n t of h i g h e s t o r d e r ; we can w r i t e t h e n

where

p/2 < k

(7.13)w i l l

5

p

- 1.

2k > p,

Since

a n i n e q u a l i t y of t h e type of

51.

not hold i n t h i s c a s e f o r l a r g e

This ends t h e proof

of Lemma 7.1. We note t h e c u r i o u s consequences o f Lemma

belongs t o

2

,

d

-

A =

+

-

=

(-$I8

6

+(-&)5 (d/dx) 5

does not, i n s p i t e of t h e f a c t t h a t (d/dx)8

although t h e o p e r a t o r

t h e operator A

of

(dx)

7.1:

than

(d/dx)

6

i s a “tamer” p e r t u r b a t i o n

.

I n t h e following s e c t i o n we s h a l l attempt a t h e o r y of t h e equation

(7.l),

b u t only i n t h e c a s e where

t h e D i r i c h l e t boundary c o n d i t i o n . c o e f f i c i e n t s of

O1v.8

A

of o r d e r > p/2

B

i s t h e h i g h e r order v e r s i o n of

Lemma

7.1 i n d i c a t e s

that the

w i l l have t o be s u i t a b l y r e s t r i c t e d .

Higher o r d e r e q u a t i o n s ( c o n t i n u a t i o n )

We study here t h e e q u a t i o n

(7.1) w i t h

an operator

A

of t h e form

121

PARTIAL DIFFERFNTIAL EQUATIONS

c

7

Au =

(-l)Ial-'D"(a+(x)D

Bu ) +

I4 5 k

la1 5 k The c o e f f i c i e n t s

101 T

am, ba

k

a r e r e a l and defined i n a bounded domain

Rm.

in-dimensional Euclidean space

of

W e s h a l l assume t h a t t h e c o e f f i c i e n t s

of t h e p r i n c i p a l p a r t of t h e operator

a

(8.1)

bo/(x)Dau.

A,

OB

c

(-l)ial-lDw(a~D')

,

(8.2)

Ictl=k [BI=k

a r e continuous i n

-

n;

t h e r e s t of t h e

simply measurable and bounded i n r e s t r i c t i o n of

A

R.

a

*'

a s well as t h e

A(B)

The operator

obtained by imposition a t t h e boundary

b,

are

denotes t h e

r

of t h e

D i ric h l et b ound a ry cond it ion

... =

u = D"u =

(Dw)k-l~ = 0

(x

E

r)

(8.3)

(8.3) w i l l be s a t i s f i e d only i n a generalized sense t o be

(although

c l a r i f i e d l a t e r ) . We assume t h a t

and t h a t

A

f o r some

K

i s u n i f o r d y e l l i p t i c , which i n t h i s case means t h a t

> 0.

The following r e s u l t (Ggrding's i n e q u a l i t y ) w i l l be b a s i c .

To s t a t e

it w e introduce t h e Sobolev spaces wk'p(fi) (1 5 p < m ) c o n s i s t i n g of u defined i n fl and having p a r t i a l d e r i v a t i v e s of

all f u n c t i o n s

5k

order

For

p

(understood i n t h e sense of d i s t r i b u t i o n s ) i n

LP(R);

the

w k ~ p ( n >i s

norm of

=

2

wky2(n) =

( t h e only case of i n t e r e s t t o u s ) we s h a l l w r i t e

$(Q).

( i n t h e norm of

The space

Hk(n)).

$(n)

The statement t h a t

v e r s i o n o f t h e boundary conditions THEORail

8.1 L e t

L

i s t h e c l o s u r e of u E %(a)

(8.3).

be a d i f f e r e n t i a l operator;

d)(n)

in

$(n).

i s t h e weak

122

PARTIAL DIFFERENTIAL EQUATIONS

i n a bounded domain

7

c

bI5k

lPl5k

Q

5 Rm.

(-l)lN(-lDTY(aOIT;Dpu)

Assume t h a t a l l t h e c o e f f i c i e n t s

a r e measurable and bounded and t h a t

ICY~ =

=

k.

i s continuous i n

a$

Then t h e r e e x i s t constants

x

14 5 k Is I I

0

when

such t h a t

C,CY

JaaM(x)D?DBu

-

dx

2

k

For a proof see FRIEDMAN [1969:1,p.321. W e proceed t o t h e c o n s t r u c t i o n of a phase space f o r t h e equation

where

A.

i s t h e s e l f a d j o i n t p a r t of A, A~ =

7 Ao(B)

The d e f i n i t i o n of renorm t h e space

where

CY

$(n)

Y

(8.8)

(-i)lml-lDm(aOIT;Dpu).

IBl5k

l+k

follows t h a t f o r t h e second order case.

W e

by means of t h e s c a l a r product

i s t h e constant i n (8.6).

We have

(8.10) The second i n e q u a l i t y follows f r o m t h e boundedness of t h e c o e f f i c i e n t s

of A ; t h e first i s a consequence of Theorem 8.1.

u

E

4(.".)

belongs t o

D(AO(B))

i f and only i f t h e l i n e a r f u n c t i o n a l

w -, ( u , ~ ) , i s continuous i n t h e norm of element of

2

L (a)

L

2

(Q),

A,(@).

being t h e orily

that satisfies

W e show i n t h e same way as i n t h e case a d j o i n t and t h a t

An element

Ao(B)

k = 2

i s bounded above

g e n e r a t e s t h e cosine f u n c t i o n

(by

that o!),

Ao(B) so that

i s self Ao(B)

123

PARTIAL DIFFERENTIAL EQUATIONS

C(t) and a square r o o t

cash t A o ( B ) 1 / 2

=

=

(8.12)

can be defined as i n gIV.2: we have

B = A,(@)'/' D(B)

,

#(n)

=

D((U h >

t h e l a s t i n e q u a l i t y holding for

- A ~ ( B ) 1) /2 ),

(8.13)

Theorem 111.5.4, combined with

cy.

(8.13) i m p l i e s t h a t Q =

i s a state space for (8.7).

(8-14)

H p ) x L2(Q) To show t h a t

Gf

i s as well a s t a t e space

f o r the f u l l equation

we i n c o r p o r a t e t h e lower order terms i n

(8.1) through p e r t u r b a t i o n

(Theorem 2.3) d e f i n i n g

(8.16) and

:]

? = [ : We o b t a i n i n t h i s way: THEOREM 8.2.

Let A

(8.1), @

be t h e operator

the Dirichlet

boundary c o n d i t i o n (8.3), and l e t

(8.18)

A(B) = Ao(B) + P w i t h domain

Then t h e space

D(A(@)) = D(Ao(@)).

a phase space f o r t h e equation

$(Q)

x L2(n)

(8.15).

The t r e a t m e n t of t h e Cauchy problem f o r (8.15) f o l l o w s word by word t h a t f o r second order e q u a t i o n s i n pIV.3; THEOREM

8.3.

L A A

we only s t a t e t h e f i n a l r e s u l t .

be t h e operator (8.1),

boundary c o n d i t i o n ( 8 . 3 ) , and l e t

B the Dirichlet

124

PARTIAL DIFFERFNTIAL EQUATIONS

A ( B ) = Ao(B) with domain

D(A(B))

( 8 -19)

P

Then t h e Cauchy problem f o r t h e

= D(AO(B)).

equation (8.15) i s well posed i n

9IV.g

+-


-m

m.

Miscellaneous comments.

(7.1))

Wave equations l i k e (1.1) (and i t s higher order counterpart were t r e a t e d by semigroup methods i n GOLDSTEIN [1969 :1]. EXERCISE 1. A l i n e a r operator

in a

D(B) = E

with domain

i s said t o be compact ( i n older terminology, completely

E

Banach space

B

{Au

continuous) i f

3

n (un}.

c o n t a i n s a convergent subsequence f o r every

u(A)

of a compact

operator c o n s i s t s of a ( f i n i t e or countable) sequence

{\,$,... 3

bounded sequence

Show t h a t t h e spectrum

of

complex numbers having a n accumulation p o i n t at zero when t h e sequence

is infinite.

(See KATO [1976:1, Sec. 3.73).

EXERCISE 2.

u

E

E, u

#

0

A be a s i n Exercise 1, and l e t

Let

i s an eigenvalue of

h

Prove t h a t

A,

h

E

o(A), h

Show t h a t t h e space

(9.11

h

of all generalized eigenvectors with

Pg(A)

( i . e . t h e s e t of all v e c t o r s

u

such t h a t

(?J-A)mu = 0 f o r some i n t e g e r an i n t e g e r

m(h)

m

2 1)

i s f i n i t e dimensional.

such t h a t i f

(9.2)

Show t h a t t h e r e e x i s t s

(9.2) holds f o r any i n t e g e r

m

2

m(A)

then

(9.3)

(AI-A)m(h)u = 0 EXERCISE 3.

Show t h a t t h e r e e x i s t s a compact operator

separable Hilbert space such t h a t an eigenvalue of

A

(Hint:

(a)

o(A) = { O ]

(b)

0

A

in

i s not

t r y t h e Volterra operator

=k X

Vu(x> in

0.

such t h a t

(AI-A)u = 0

eigenvalue

#

that is, there exists

L*(O,~)).

U(5)d5

(9.4)

125

PARTIAL DIFFERENTIAL EQUATIONS

EXERCISE 4 . that

f o r all R(p;A)

Let

(Hint:

p E p(A) =

be a n o p e r a t o r i n a Eanach space

A

i s compact f o r some

R(h;A)

h

E

Then

@(A).

E

such

i s compact

R(p;A)

use t h e second r e s o l v e n t e q u a t i o n

R(A;A) + (A-p)R(p;A)R(A;A)

and t h e f a c t t h a t t h e sum o f two

compact o p e r a t o r s and t h e product of a compact o p e r a t o r and a bounded o p e r a t o r a r e compact; s e e KATO EXERCISE 5.

Let

A

[1976:11).

be as i n Exercise

empty o r c o n s i s t s of a sequence that

+

h

then

m

A

EXERCISE

6.

o(A)

is

and t h e space

Show t h a t i f

A

E

a(A)

EB(h) of g e n e r a l i z e d

-

enjoys t h e p r o p e r t i e s described i n E x e r c i s e 2. Show t h a t t h e r e e x i s t s a n o p e r a t o r as i n Exercise

(Hint:

a(A) =

A

Show t h a t

of complex numbers such

i f t h e sequence i s i n f i n i t e .

i s a n eigenvalue of

e i g e n v e c t o r s of

with

\, h2,. ..

4.

t r y t h e inverse of the Volterra operator

4

(9.4)).

n be a bounded domain i n m-dimensional into Ehclidean space Rm, B a l i n e a r bounded o p e r a t o r from L2(n) EXERCISE

7.

Let

$(".

Show t h a t

L2(Q),

i s compact (See MIHAILOV

E,

thought of a s a n o p e r a t o r from

L2(")

into

[1976:1]).

7 show t h a t t h e second o r d e r o p e r a t o r s i n (3.15) and (6.2) and t h e h i g h e r o r d e r o p e r a t o r s i n (8.19) enjoy t h e s p e c t r a l p r o p e r t i e s i n Exercise 5 ( H i n t : show t h a t R(h;AO(p)) i s compact u s i n g Exercise 7 and t h e n apply Ekercise 5 ) . EXERCISE 8.

Using Exercise

126

CHAPTER V UNIFORMLY BOUNDED GROUPS AND CmINE FUNCTIONS IN HILBERT SPACE

4 v.l

The Hahn-Baaach theorem: Let

E

Banach l i m i t s .

be a n a r b i t r a r y r e d l i n e a r space.

A functional

p :E

-W

i s c a l l e d sublinear i f

for

u, v

E

E

arbitrary.

THEOFEM 1.1. (Hahn-Banach). a linear functional.

Let F

be a subspace of

E, cp : F

J

R

Assume t h a t

Then t h e r e e x i s t s a l i n e a r f u n c t i o n a l

0 :E

-.

R

such t h a t

For a proof see BANACH [1932: 1, p . 281.

With t h e h e l p of Theorem

1.1 we can c o i s t r u c t a n i n t r i g u i n g

extension of t h e notion of l i m i t .

Let

bounded complex functions defined i n t

l i m i t in

B

i s a functional

p e r t i e s , where

f(t), g ( i )

E

B = B[O,m)

1. 0 .

be t h e space of d l

A Banach l i m i t or peneralized

LIM : B -t 6: enjoying t h e following pros +m B and C U , ~ a r e complex numbers.

127

I N HILBERT SPACE

5

lim i n f f ( s )

(d)

S-

(f)

LIM f ( s )

r e a l valued:

Banach l i m i t s i n

LIM f ( s )

s+

arbitrary.

B

for

f

S-

m

m

$( be t h e subspace of

Let

B

c o n s i s t i n g of

Define

t h e i n f i m u m taken over all p o s s i b l e f i n i t e sequences

BR,

of nonnegative numbers. Using Theorem 1.1 f o r a linear functional

We check i n s t a n t l y t h a t rp = 0

and

F = (0)

Q :BR

LIM = Q.

Obviously,

satisfies

p

(5,)

(1.1).

we deduce t h e existence of

R such t h a t

-+

(f

Q(f) < P ( f ) Set

LIM f ( s ) s +-m

LIM R e f ( s ) + i LIM I m f ( s )

=

r e a l valued f u n c t i o n s .

E

B exist.

once t h i s done we simply s e t

S-'W

f

is real.

f

m

Obviously, it i s enough t o construct

Pro3f:

for

if

if t h e l a t t e r e x i s t s .

l i m f(s)

=

s-

THEOFZM 1 . 2 .

E

sup f ( s )

S-m

S-m

m

L3-m

f

zlim

@z

~ L I Nf ( s ) [ I l i a sup]f(s)(.

(el

for

LIM f ( s ) s-

S-+W

(a)

holds.

E

%I.

(1.6)

Replacing

by

f

(1.6)

in

-f

S-.m

we o b t a i n

This y i e l d s (d)

(c).

-

p ( f ) < U r n sup f ( t ) and - p ( - f ) z

Since

l i m inf f ( t ) ,

(1.6) and (1.7); obviously, ( d ) implies we take 5 1 = h, 5 2 -- 2h, ...,cn = nh i n (1.5)

follows from

To check

(b) p(f(c

+

h)-f(i))

5:

(f). so t h a t

l i m sup ( f ( s + n h ) - f ( t ) ) . S+'X

Since

n

way t h a t that €3

is arbitrary,

p(f(i)

p(;'(t

- f ( f , + h ) ) 5 0,

Q(f(: + h ) ) = Q(f(;)).

be such t h a t

+

eie LIM f(s)

h)

- f ( ; ) ) 5 0.

t h u s it follows from

Finally, we show =

We deduce i n t h e same

ILIM f ( s ) l .

(e)

Then

(1.6)

and

a s follows.

(1.7) Let

128

I N HILBERT SPACE

This concludes t h e prsof of Theorem 1 . 2 . A Banach l i m i t

5

sequences

(co,Cl,

=

i n t h e space

LIM

n-

... )

is

p r o p e r t i e s corresponding t o

limits i n

l i m inf n- m

cn 5

I LIM

(el)

n(f')

LM n- m

Proof:

LIM n- m

5lim n-

m

5,

=

COROLLARY 1.3.

and

i n the d e f i n i t i o n of Eanach

(f)

lim n- rn

LIM

cn I'_ l i m

sup

cn

{Cn]

if

is real.

n-m

SUP IS,^. m

5,

if t h e l a t t e r e x i s t s .

am

Ba.nach l i m i t s i n

exist.

Define

LIM n- m s+

-

(a)

4" enjoying t h e

functional i n

R

B:

(d')

where

of complex bounded

Rm

m

cn

=

LIM f ( S ) , s-

m

i s one of t h e Banach l i m i t s constructed i n Theorem 1 . 2

m

f(s) =

5,

in

n

5

s < n + 1.

Uniformly bounded gro-ips i n Hilbert space.

8V.2

Throughout t h e r e s t of t h i s chapter (except i n Section V .3) we

shall assume t h a t Let

B

E

= H

i s a complex H i l b e r t space.

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

Then it follows e a s i l y

H.

from t h e f u n c t i o n a l c a l c u l u s f o r self ad j o i n t operators t h a t

U(;),

where U ( t ) = exp(itB)

(-a <

t <

m),

(2.1)

129

IN HILBERT SPACE

i s a s t r o n g l y continuous group i n

* exp(itE)

U(t)* =

exp(-itE)

=

H.

Moreover, since -1 U(-t) = U(t) , each U(t)

=

i s unitary;

i n particular

[1963:1, C h . XI11 f o r t h e necessary d e t a i l s on

( s e e DUNFORD-SCHWARTZ

It was f i r s t proved by Stone t h a t t h e

the functional calculus).

converse i s a s w e l l t r u e (See Exercise 1.11)

Let U(i)

THEORFM 2 . 1 .

t h a t each

E

= -iA,

U(t)

Then

t h e i n f i n i t e s i m a l generator of

E

Proof: that

be a s t r o n g l y continuous gro'ip.

i s a u n i t a r y operator.

Let

h > 1 real.

n = 1 and

/

(R(h;A)u,v) =

U(t).

be t h e i n f i n i t e s i m a l generator of

A

i s self adjoint.

-iA

(1.3.8) f o r

We have

=I

r m

7.m

emAt(U(t)u,v) d t

e-Xt(u,U(t)*v)

=

-

(u,R(-h;A)v)

where we have used i n t h e l a s t e q u a l i t y t h e f a c t t h a t R( A;A)* = -R(-$A).

=

N +A

If V(t)

=

exp(itE)

so that

A*

=

(2.3)

9

U(-t)

s t r o n g l y continuous semigroup with i n f i n i t e s i m a l generator Y

dt

0

=Lme-At(u,U(-t)v) d t

(XI-A)

We show

U(<).

To t h i s end we use formula

' 0

Accordingly,

Assume

holds f o r

(2.1)

is a -A.

Taking inverses, t h i s implies t h a t -A,

thus

3 = -iA

i s self adjoint.

(computed by t h e f u n c t i o n a l c a l c u l u s ) we show

easily that i

2"

-

e AtV(t)u d t

=

R(A;iE)u=R(h;A)u

( h > 1)

' 0

f o r each

~1 E

Since

H.

U(<) and

that

U(;)

s a t i s f i e s t h e same e q u a l i t y it follows

V(t) coincide by uniqueness of Laplace transforms.

The following result shows t h a t uniform boundedness of t h e group

S(:) space

implies that

S(t)

i s u n i t a r y a f t e r a '!change i n geometry" i n t h e

H. S u r p r i s i n g l y enough, t h e r e are no conditions whatsoever on t h e

t-dependence of

THEOREM 2.2

c ).

S(

S(i) be a group i n H (S(0)

= I,

S(S + t )

= S(s)S(t)

130

I N HILBERT SPACE

for

-W

< s, t <

Assume thai,

m).

‘ken t h e r e e x i s t s a bounded, self a.d,joint o p e r a t o r C - ’ I / U ~ ~ ~5 ( Q u , ~ ) 5 Cllul12

(u

t

Q

satisfying

E)

(2.5)

and such t h a t

U ( t ) = QS(t)Q-’

t.

i s u n i t a r y for every Proof:

9

:H x H

9

W e define a sesquilinear functional

c

--t

( t h a t i s , a map

l i n e a r i n t h e second v a r i a b l e , conjugate l i n e a r i n t h e

f i r s t ) by

LIM i s one of t h e Banach limits constructed i n Theorem 1 . 2

where

I ( S ( t ) u , S ( t ) v ) I 5 C211ul/

(note t h a t , since (s(i)u,S(i)v)

belongs t o I4(u,v)l

By

B[O,m)).

5c

2

l l ~ l lllvll

t h u s t h e r e e x i s t s a bomded o?erator L)(u,v) = Q(v,u),

Moreover, s i n c e and it follows from IlS(t)ull C-*1/Ul12

2

C-lllulI)

5

2

C /lull2

(e)

we have

(u,v

E

in

H

P

(Pl1,V)

HI,

(2.8)

such t h a t

.

t h e operator

(2.9) i s self adjoint,

p

(which i m p l i e s i n p a r t i c u l a r t h a t

and from p r o p e r t y

5

(PU,U)

(2.4)

=

I/v\I t h e f u n c t i o n

so t h a t

of Banach l i m i t s t h a t

(d)

i s p o s i t i v e a s well and hence

P

possesses a p o s i t i v e , s e l f a d j o i n t square root

Q

satisfying

(2.5).

Moreover, we have (PS(t)U,V)

=

LIM ( s ( s + t ) u , S-)

s(s)v)

for

u,v

E

H

=

m

by v i r t u e of

LIM ( s ( s ) u , S ( s - t ) v ) s-

LIM ( s ( s ) u , s ( s ) s ( - t ) v ) S-

=

m

(b).

Hence

m

(Pu,S(-t)v)

=

131

I N HILBEKT SPACE

S(t)*P

=

PS(-t)


(-m

m)

.

(2.10)

Qml l e f t and r i g h t , t h i s e q u a l i t y becomes

Multiplying by

Q-lS(t)*Q which i s n o t h i n g b u t

(QS(t)Q-')"

QS(-t)Q-l,

=

(2.11)

-1 -1 )

(QS(t)Q

=

.

This ends t h e p r o o f

of Theorem 2.2. The f a c t t h a t

=

Hence

S(t)*'

=

i s u n i t a r y d e s e r v e s some comment.

QS(t)Q-l

(u,PS(-t)u)

S(-t),

*Q

where

s c a l a r product

(2.12).

as stating that

each

=

(Qu,QS(-t)v)

=

(u,S(-t)v)

Define

Q '

denotes adjoint with respect t o the

Accordingly, Theorem 2.2 c a n be i n t e r p r e t e d

w i l l i n f a c t be unitary a f t e r replacing t h e

S(t)

o r i g i n a l s c a l a r product by

(2.12);

s i n c e b o t h norms a r e e q u i v a l e n t

t h e t o p o l o g y o f t h e s p a c e induced by a n y of t h e two norms i s t h e same. S(t^), no On t h e o t h e r hand, we have

I n t h e a b s e n c e o f h y p o t e s e s on t h e t-dependence of f u r t h e r c o n c l u s i o n s on

U(t^)

COROLLARY 2.3.

Assume t h a t

u

S(t)

s a t i s f y t h e assumptions o f Theorem 2.2.

i s c o n t i n u o u s (or, more g e n e r a l l y , s t r o n g l y rneasur-

S(t)u

a b l e ) f o r each

are possible.

E

E.

Then t h e r e e x i s t s a s e l f a d j o i n t o p e r a t o r

and a bounded s e l f a d j o i n t o p e r a t o r

S(t) Conversely, e v e r y group s a t i s f y i n g Proof:

=

Q-'exp(itE)Q

S(i)

o f t h e form

Q

satisfying (-m

(2.13)

t <

a).

and such t h a t

(2.13)

i s a strongly continuous

(2.4).

By v i r t u e o f Theorem I. 1.1

measurability of

C

(2.5)

E

s t r o n g c o n t i n u i t y and s t r o n g

S(t^)u f o r each u E E a r e e q u i v a l e n t .

Applying

132

I N HILEERT SPACE

Theorem 2.1 t o t h e s t r o n g l y c o n t i n u o u s group operators

(2.13)

-1 QS(t)Q of u n i t a r y

results instantly.

We c l o s e t h i s s e c t i o n w i t h a d i s c r e t e v e r s i o n of Theorem 2.2. THEOREM 2.4.

Let

operators s a t i s f y i q

-

{S,;

-m

< n <

S = I, S 0 m+n

a)

- 'DSn

be a sequence of bounded n so t h a t S = S1). Assume (n

that

Then t h e r e e x i s t s a bounded, s e l f a d j o i n t o p e r a t o r

Q

(2.5)

such t h a t

h o l d s and

-1

U =QSQ n n

i s u n i t a r y for every U

n.

(2.15)

Equivalently, t h e r e e x i s t s a unitary operator

such t h a t

sn

rz

Q-'$Q.

The p r o o f i m i t a t e s t h a t of ?lieorem 2.2. defined a s t h e positive,

(2.16) The o p e r a t o r

Q

is

s e l f a d j a i n t square r o o t of t h e o p e r a t o r

P

d e f i n e d by

i s one o f t h e Banach l i m i t s of sequences c o n s t r u c t e d i n -1 C o r o l l a r y 1.3. W e p r o v e a r g u i n g as i n Theorem 2 . 2 t h a t U = QS Q n n i s u n i t a r y for a l l n; since we have QSnQ-' = whence Un = (2.16) f o l l o w s for u = u 1' We n o t e t h e f o l l o w i n g r e s t a t e m e n t of C o r o l l a r y 2.3 i n t h e l a n g u a g e

where

LIM

p,

p

o f d i f f e r e n t i a l e q u a t i o n s i n Banach s p a c e s .

THEOREM 2.5.

the H i l b e r t s p a c e

Let A

H

be a closed, densely defined operator i n

such t h a t t h e Cauthy problem f o r u ' ( t ) = Au(t)

i s w e l l posed i n

-a C

e r a l i z e d s o l u t i o n of

t <

m.

(2.18)

(2.18)

Assume, moreover, t h a t for e v e r y gen-

w e have

IN HILBERT SPACE

(where

C

u(t)).

may depend on

133

Then t h e r e e x i s t s a s e l f a d j o i n t

E and a bounded s e l f a d j o i n t operator

operator

equalities(2.5)

D(A)

and such t h a t A

{u;Qu

=

Q

satisfying in-

D(B)]

E

@

-1

(2.20)

iQ BQ.

=

The converse i s a s w e l l t r u e . Theorem

2.4 has

an obvious formulation i n r e l a t i o n t o t h e

difference equation

where

i s a bounded i n v e r t i b l e operator i n

A

can be solved forward and backwards).

(so that

H

(2.21)

We omit t h e d e t a i l s .

It i s useful t o know whether t h e conclusions of

REMARK 2.6.

C o r o l l a r y 2.3 hold under weaker c o n d i t i o n s on t h e t-dependence of If t h e space

S(i).

H

i s separable it i s enough t o r e q u i r e t h a t

be weakly measurable ( t h a t i s , t h a t

S(t)

(S(t)u,v)

u,v

E

H)

i s s t r o n g l y measurable for every

u

a s a complex valued f u n c t i o n f o r every

be measurable

since t h i s implies

A

that

S(t)u

[1957:1, p . 73 I).

H)

To see t h i s we may use on t-dependence); s i n c e

S(c)

i m p l i e s strong c o n t i n u i t y .

(which does not r e q u i r e any assumption

(2.6)

w i l l be s t r o n g l y (weakly) continuous i f

S ( : )

S(f)

is, we may assume t h a t

U(i)

S(t)

i s continuous f o r each

(S(<)u,v)

since weak c o n t i n u i t y of

and only i f

( s e e HILLE-PHILLIPS

However, it i s enough t o assume t h a t

i s weakly continuous ( i . e . t h a t E

E

I n a g e n e r a l H i l b e r t space weak and s t r o n g measur-

a b i l i t y do not c o i n c i d e .

u,v

E

i t s e l f i s unitary.

Then

=

GV.3

2(u,u)

-

((s(s

-t) +

S(t

-

(2.22)

s))u,u).

Almost p e r i o d i c f u n c t i o n s . Let

R >

0.

A

set

e

(-m,m)

i f and only i f every (open) i n t e r v a l

i s s a i d t o be I

of l e n g t h

A-dense i n

2

k

(-m,m)

c o n t a i n s some

I N HILBERT SPACE

134

p o i n t of

e. *

f ( t ) defined i n

A function

E

space


with v a l u e s i n a Eanach

m

i s c a l l e d almost p e r i o d i c ( i n t h e sense of

i s continuous and f o r every

>

E

9, =

there exists

0

H. Eohr) i f it and a

R(E;f)

e = e ( & ; f ) such t h a t

1-dense s e t

W e s h a l l o r d i n a r i l y denote by

(3.1)

-m

t;

for a l l

t h e s e t of a l l

e(i;f)

every element of

& - t r a n s l a t i o n number of

which s a t i s f y

T

w i l l be denominated a

e(E;f)

f.

Obviously, a p e r i o d i c f u n c t i o n i s almost p e r i o d i c ( f o r every we may t a k e

e

{np,

=

-m


m]

where

i s t h e period of

p

>0

E

f(i)).

We prove below a number of elementary f a c t s on v e c t o r valued almost p e r i o d i c f u n c t i o n s which s u f f i c e

(Theorems

3.7, 3.8

3.9) w i l l be omitted.

and

f(i)

LEMMA 3.1.

uniformly continuous i n

-m

be almost p e r i o d i c .


(b)

m.

Rf(t^) = (f(t); i s precompact i n Proof: and a

for

To show


w

e

and

( a ) we s e l e c t =

e(E/3;f)

T

E

O z t ,

I l f ( t ) - f ( t ' ) / 1 5 & for

t,t'

It-ttlrs.


ml

> 0,

a number

E

number

\lf(t +

+ T)

1

= 1(E/3;f)

- f(t)ll 5 ~ f

/ 3

on

b , 0 < 6 < 1 such t h a t

t ' Z a + l

and

It-t'I58.

be a r b i t r a r y p o i n t s i n t h e r e a l l i n e such t h a t Choose

T

E

This completes t h e proof of

>0

(3.2)

/If(%

such t h a t

s ~u c h t h a t

O
llf(t)- f(t'>ll 2 llf(t> - f(t +

E

f ( t ) is f(t),

(a)

Usin5 uniform c o n t i n u i t y of

e.

compact i n t e r v a l s we s e l e c t next

Let

-m

Then

The range of

E.

A-dense s e t -m

t o i l l u m i n a t e somewhat t h e

On t h e o t h e r hand, t h e p r o o f s of t h e fundamental p r o p e r t i e s

definition.

(a).

To show

Then

T)ll

(b),

we s e l e c t a g a i n

and o b t a i n from t h e d e f i n i t i o n of almost p e r i o d i c f u n c t i o n a

1

=

1 ( & / 2 , f ) and a 1-dense s e t

z)- f(t)l/ 5 &/2

for

T E e

e = e(r/2;f)

and 611 t.

such t h a t

Since t h e image of t h e

135

I N HILBERT SPACE

interval

0

5t 5

4

tl,

f i n i t e sequence

denotes t h e sphere

8(u,p)

that

0 < t

f(t

-+

0

+

-T

T) E

5t 5

T =

< 4, t j

T(t)

4

we can s e l e c t a

E

such t h a t

... U8(f(tn);E/2)

s(f(t,);E/e)u

E

a r b i t r a r y r e a l number,

that

i s compact i n

. . .,t n i n

f(t)

where

f(i)

by

{v;\/v-uil

5

01.

Ilf(t)- f(tj)lI5 I l f ( t )

t

Let

be an

a t r a n s l a t i o n number i n

an element of t h e sequence

g(f(tj);E/2).

t

tl,

e

such

... , tn

such

Then

- f ( t '-TI11

+ Ilf(t + -TI

- f(tj)ll5 E

so t h a t

. . . u 8(f(tn);&)

5 s(f(tl);E)U

Rf(i) i s arbitrary,

Rf(i)

*

i s t o t a l l y bounded; e q u i v d e n t l y ,

Since

E

Rf(;)

i s precompact ( s e e DUNFOFD-SCHWARTZ A s a consequence of

(b)

[1958:1, p . 221).

we o b t a i n t h a t every almost p e r i o d i c

f u n c t i o n i s bounded.

LENMA 3.2.

&&.

(fn(c))

be a sequence of almost p e r i o d i c f u n c t i o n s

f n ( t ) f ( t ) uniformly i n

such that

-a

+


Then

m.

f(i)

&

&most p e r i o d i c .

Proof: Choose for

-m


E>O.

Once

m.

n

Let n be so l a r g e t h a t has been s e l e c t e d , l e t

e = e(E/3;f n ) a 4-dense s e t such t h a t -a < t < m and T E e. Then

- f(t)llI It(t

Ilf(t + +

f o r all

t

and

+

Ilfn(t

e,

=

2)

.>f n ( t+

l/fn(t+ -r> - fn(t)ll+ llfn(t) 2 E

+

which shows t h a t

Ilfn(t)- f ( t ) l l < E / 3 a(E/3;fn)

-fn(t)ll

and

5 ~ / 3for

-TI11 +

- f(t)lI5 E f(t)

i s almost p e r i o d i c a s

claimed. The following " t o p o l o g i c a l " c h a r a c t e r i z a t i o n of almost p e r i o d i c f u n c t i o n s w i l l simplify some of t h e p r o o f s t h a t follow. E-valued f u n c t i o n

f(g)

defined i n

if f o r every sequence of real numbers

-m

C

(h,]

t <

m

A

continuous

i s t r a n s l a t i o n precompact

t h e r e e x i s t s a subsequence

IN HILBERT SPACE

136 h

(f(t + h n ( k ) ) J

such t h a t

n(k)

THEORFSI 3.3.

A function

it i s t r a n s l a t i o n precompact

E:Assume

f(:)

i s convergent, uniformly i n

f(t)

.c t c

-m

i s almost p e r i o d i c i f and only i f

.

i s almost p e r i o d i c .

Let

be a countable

2

If s 0 i s an element o f we may t a k e advantage o f t h e precompactness o f t h e range E f ( t )

dense s e t i n (Lemma 3.1)

(say, t h e r a t i o n a l s ) .

(-m,m)

t o s e l e c t a subsequence

f ( s o + hn(k))

{hn(,)}

of

{hn}

2

such t h a t

Using t h e n t h e method of Jiagonal

i s convergent.

{kn3, such ( a t t h i s stage we cannot

sequences we can b e t t e r t h i s with a subsequence, denoted f ( s + k ) converges for s E 2 n a s s e r t t h a t t h e r e i s convergence f o r s k, 2 ,

that in

2

Select

E

> 0,

and l e t

1 - d e n s e s e t such t h a t 'G E

or even t h a t convergence

i s uniform).

e:

It - t '

choose then

I

5

1

+

lif(t

1 ( & / 5 ; f ) > 0, e

=

5 ~ / 5f o r

T )- f ( t ) l l

such t h a t

6 = F(E/?)

[O,aI

i n t e r v a l s of l e n g t h

56

with

f i n i t e number

R

and p i c k r a t i o n a l s tl,

t h e s e i n t e r v a l s ; once t h i s i s done, determine

'G = ' G ( t )

t

be an a r b i t r a r y r e a l number. e

E

t

and a p o i n t

(-t, -t

i n the interval

It +

such t h a t

j

b(t + kn) - f ( t

T -t

and a l l

5

E/5

if

.I

r

of sub-

i n each of

such t h a t

j

=

1,2

,..., r.

Pick a t r a n s l a t i o n number

+ a)

J

. ..,t r

no

+ km)ll 5 ~ / 5 f o r m,n 3 no,

l\f(t. + kn)-f(tj Let

a11 t

\if(t)-f(t')II

a

( t h a t t h i s i s p o s s i t l e follows from Lemma 3.1 ( a ) ) .

6

Finally, cover t h e i n t e r v a l

J

e(E/5;f)

=

(so t h a t

.c 6 .

+ km)ll 5 / k ( t + k n >

0 < t + T <

1)

We have

- f(t

+

T + kn)/I

+ Ilf(t + T + k n ) - f ( t . + kn)[I + l / f ( t .+ k n ) - f ( t . + km)II J J J

+ / I f ( t j + k m ) - f ( t + T + km)[I

+ llf(t + if

n,m

2 no.

T

+ km) - f ( t + km)II 5

This shows t h a t

E

(-m


m)

f ( < ) i s t r a n s l a t i o n precompact a s

claimed. We prove t h e converse. compact.

If

such t h a t f o r l a t i o n numbers

Assume t h a t

f ( t ^ ) i s t r a n s l a t i o n pre-

f ( t ^ ) i s not almost p e r i o d i c then t h e r e e x i s t s

no T.

>0 Let

there exists a a-dense set hl

be a r b i t r a r y and l e t

e

(al,bl)

E

of

>0

E-trans-

be an

m.

137

I N HILBERT SPACE

> 21%/

i n t e r v a l of l e n g h t of

Let

f(t).

Obviously,

h2

=

h2-hl

( 5 + bl)/2

of

1% 1 +

and d e f i n e

f(<)

(a2,b2),

belong t o

be t h e midpoint of

(y,bl), t h u s

belongs t o

f ( : ) .

& - t r a n s l a t i o n niimber of of l e n g t h > 2(

not containing any E - t r a n s l a t i o n number

Ih21) h

3

(a2,b2)

not containing any € - t r a n s l a t i o n number

= (a

2

+ b2)/2.

Then

h

3

-Ll and

h3-h;,

n - yh n - h 2 , . . . , h n - h n - l

{hn)

f(t).

which shows t h a t t h e sequence

Then, i f

(f(G

sequence uniformly convergent i n

+

-a

hn))


n > m

of r e a l numbers

a r e not

n, h

such t h a t , f o r each

we have

cannot c o n t a i n any subThis ends t h e proof of

00.

3.3.

COROLLARY @g($)

Let

3.4.

f(t), g(t)

be almost p e r i o d i c .

i s almost p e r i o d i c f o r a r b i t r a r y complex

Then

a,@.

(h ) be an a r b i t r a r y numerical seqJence. n Theorem 3.3 s e l e c t a subsequence { h n ( k ) ] such t h a t both Proof:

f(f

+

Let

hn(kl), g(c

Then af(t -W

f(t^).

hence cannot be & - t r a n s l a t i o n numbers o f

& - t r a n s l a t i o n numbers of

af(f) +

cannot be an

We s e l e c t next an i n t e r v a l

Arguing i n t h e same way we c o n s t r u c t a sequence

Theorem

(y,bl).

h2-hl


W,

+

hn(k))

h n ( k p + Bg(t thus

COROLLARY %complex

+

3.5.

+

a r e uniformly convergent i n hnik))

Using

-a <

t <

m.

i s uniformly conveFgent i n

a f ( < )+ @ g ( t ) i s almost p e r i o d i c by Theorem 3.3. Let

_.

f ( < ) be almost p e r i o d i c , and l e t

m l u e d almost p e r i o d i c f u n c t i o n .

Then

Q(f)

p(;)f(%) i s almost

periodic. The proof i s very s i m i l a r t o t h a t of C o r o l l a r y

3.4 and i s t h e r e f o r e

omitted. COROLLARY

where {un]

3.6.

~ e t

i s a sequence of elements of

r e a l numbers and t h e s e r i e s

Then f ( t )

(3.3)

i s almost p e r i o d i c .

E,

(An]

i s a sequence of

c o m e r g e s uniformly i n

-w

< t <

m.

138

I N HILBERT SPACE

P a r t i a l sums ot

Proof:

(3. 3)

are almost p e r i o d i c by C o r o l l a r y

3.4; t h e l i m i t of t h e p a r t i a l sums i s almost p e r i o d i c by Lemma 3.2. THEORESI 3.7.

exists

.

&J

f({)

be slmoat p e r i o d i c .

Then t h e mean value

Moreover,

&lT T

M(f)

l i m

=

T+ m f o r every

t,

-m


(3.5)

m.

For a proof see COIIDdNJ3ANU

[1971:1];we shall

f ( s + t ) ds

[l968:1,p . 145 1

o r AMERIO-PROUSE

only use t h e r e s u l t f o r complex-valued f u n c t i o n s .

The same holds t r u e of t h e following two theorems, where

f(c)

i s again

assumed t o be almost p e r i o d i c . TKEOREM 3.8.

The mean value M( emiA'f

(t) )

(-m


i s non null o n l y for a f i n i t e o r countable s e t of See CORDUNEANLT that

exp(-ihi)f($)

THEOREM 3.9.

-

m

< A <

m.

[l968:1,p. l k 9 ]

1's.

o r AMERIO-PROUSE [1971:1]. Note

i s d m o s t p e r t o d i c by Corollary 3.5. Assume t h a t

Then f(t) =

M(exp(-ihi)f(<)) = 0

for a l l

A,

0.

The proof i s i n COKDUNEANU

gV.4

(3.6)

m)

[1968:1,p. 1511.

Almost p e r i o d i c groups i n H i l b e r t space.

If Let B be a self a d j o i n t operator i n t h e H i l b e r t space H. B has p u r e p o i n t spectrum ( i . e . if all elements of a ( B ) a r e eigen-

v a l u e s of

B)

then t h e r e e x i s t s

R

{PA;-m < A <

family

orthogonal ( s e l f ad j o i n t ) p r o j e c t i o n s such t h a t

2' -m
and

PAu

= u

(U

E

H),

m}

of mutually

139

I N HILBERT SPACE

BU

Z

=

(U E

"P~U

D(B)),

(4.2)

-m
K with

both s e r i e s convergent i n t h e norm of'

IlulI2

/lPAu/l2 and

=

PAu = 0 except ~ ~ = ~ h211p,,Ul~2. U ~ ~ Of2 course t h i s i m p l i e s t h a t f o r h i n a f i n i t e or countable s e t ~ ( u ) , i n g e n e r a l depending on

u.

c o n s i s t s p r e c i s e l y of t h o s e

B

The domain of

s e r i e s on t h e r i g h t hand s i d e of

I n c a s e t h e space

m.

t o see t h i s , l e t

{un]

H ; c o n s i d e r t h e ( c l e a r l y countable) s e t

u

A

H

i s separable t h e f a m i l y of p r o j e c t i o n s

H

(PA] i s a c t u a l l y countable: sequence i n

for which t h e

c A ~ I ~ P<~ ~ I I 2

or, eqiivalently,

Then, i f

u

( 4 . 2 ) converges i n t h e norm of

u

and

i s a sequence i n

{un]

E

E

P u = l i m PAvn A

w e have

be a dense

o = a ( u )Ucs(u2) 1 0, where v

=

u. . .

n

u.

converging t o

It has a1rea.dy been observed i n 6V.2 t h a t U ( t ) = exp(itB)

(-m


(4.1)

m)

i s a s t r o n g l y continuous group with i n f i n i t e s i m a l g e n e r a t o r each

unitary).

U(t)

B

(with

Under t h e p r e s e n t assumptions we have

with

Condition (4.4)

(4.5)

obviously i m p l i e s t h a t convergence of t h e s e r i e s

i s unifozm i n

C o r o l l a r y 3.4 t h a t

(-m,m)

t h u s it f o l l o x s from Lemma 3.2 and

i s almost p e r i o d i c f o r every

U(i)u

u

E

E.

Obviously t h e same conclusion holds f o r a s t r o n g l y continuous group S ( t ) which admits t h e r e p r e s e n t a t i o n S ( t ) = Q-'exp(itB)Q where

Q

B

,

(4.6)

i s a s e l f a d j o i n t operator having pure p o i n t spectrum and

a bounded i n v e r t i b l e o p e r a t o r , o r , e q u i v a l e n t l y , for a s t r o n g l y

continuous group whose i n f i n i t e s i m a l g e n e r a t o r i s of t h e form

-1

A = iQ B Q , with

B

and

Q

a s before.

(4.7)

140

I N HILBERT SPACE

(4.6) with

We show below t h a t t h e r e p r e s e n t a t i o n

B

having

pure p o i n t spectrum i s a necessary c o n d i t i o n i n o r d e r t h a t each

S(t)u

be almost p e r i o d i c . THEOREM

u,v

E

H

Let

4.1.

be a group i n

S(<)

Assume t h a t , for each

i s almost p e r i o d i c .

(S(t)u,v)

the function

H.

Then

S(t)

( 4 . 6 ) with Q a bounded s e l f a d j o i n t operator s a t i s f y i n g i n e q u a l i t i e s of t h e type of ( 2 . 5 ) 4 B a self

admits t h e r e p r e s e n t a t i o n

a d j o i n t operator of t h e form

(PA]

i s separable t h e f a m i l y

H

(4.2);

i s countable. It follows from t h e f a c t t h a t almost p e r i o d i c f u n c t i o n s

Proof:


a r e bounded i n

-m

(HILLE-PHILLIPS

[1957:1, p .26 1)

For each r e a l

A

C

and from t h e uniform boundedness p r i n c i p l e

m

t,hat

Mh

d e f i n e a bounded o p e r a t o r

(MAuJv)

= l i m

T-

in

&J:e-ihs(S(s)u,v)

u,v

a.nd

except f o r a. f i n i t e o r countable s e t ) . E

T+m using Theorem

rT 2 T k

3.7:

-T

e-ih(s+t)(S(s

.

A

(and

(4.9)

For

-m

(M,,J,v) C

h

h, t <

+ t)u,v)

d t = e i A t (MhuJv)

so t h a t if w e m u l t i p l y T

m

(4.10)

ih t

MA

-

(4.11)

J

5 t <_

= 0

accordingly,

M S(t) = e

-T

ds

H we have

= e i At lim

Now

by t h e formula

rn

By v i r t u e of Theorem 3.7 t h e l i m i t e x i s t s for a.11 for all

H

we o b t a i n

(4.11)

by

exp(ipt)/2T

and i n t e g r a t e i n

141

I N HILBEKT SPACE

< A <

iMA;-m

Accordingly

i s a family of mutually orthogonal ( n o t

m}

necessarily self ad joint) projections i n Let

P

H.

be t h e s e l f a d j o i n t operator defined i n t h e proof of

(2.10) i n mind we deduce t h a t

Theorem 2.2. Keeping (PMAu , v )

( M Au,Pv) = lirn $L/I:e-ik( T- 02

=

S( s ) u , P v ) d s

T = l i m

$JT

e-ihs(u,S(s))tPv) d s

lirn T+ m

=

(MAv,Pu)

T -

= l i m

T+

e ihs(S(-s)v,P~) d s

s:

=

T+ m

e-ihs(Pu,S(-s)v)

(Pu,MAv).

=

ds

(4.14)

m

Acco-dingly, ic

PMA

so t h a t , i f

Q

=

MAP

,

(4.15)

i s t h e p o s i t i v e s e l f a d j o i n t square r o o t of

i s self adjoint f o r a l l

P, P A = QMAQ-l

B

a d j o i n t operator

by (4.2)

A.

a n d a group

V ( t ) = &-'exp(itE)Q

(-m

We d e f i n e t h e n a s e l f by

V(t)


m).

(4.16)

Noting t h a t

c

V(t)u = -m<

A<

m

w e deduce, usiw uniform convergence of l i m T + m

2T

"I

-T -T

for a l l real (V(s)u,v)

(4.17)

=

e-ihs(V(s)u,v)

A.

ds

=

-i

( M u,v) = l i m emihs( S( s ) u , v ) d s A T+ m 2T '-T

Application of Theorem

(S(s)u,v)

f o r all

that

u,v

t

H

3.8 t h e n y i e l d s t h e e q u a l i t y hence

S ( s ) = V(s),

establish-

i n g i n f u l l t h e claims of meorem 4.1. We r e s t a t e t h e r e s u l t i n t h e s t y l e of Theorem 2.5:

TKEORFM 4.2. t h e H i l b e r t space

Let H

A

be a closed, densely defined o p e r a t o r i n

such t h a t t h e Cauchy problem for

142

I N HILBERT SPACE

u'(t)

i s w e l l Esed in -

-m

=

(4.18)

Au(t)

< t < M. Assume, moreover, t h a t for every (4.18) and every v c H t h e f u n c t i o n

generalized s o l u t i o n of

i s almost p e r i o d i c .

Then t h e r e e x i s t s a s e l f a d j o i n t operator

pure p o i n t spectrum and a bounded s e l f a d j o i n t operator

(2.5)

i n e q u a l i t i e s of t h e type of

ic

( t h e sum understood elementwise) where

GV. 5

,

?Qq1PAQ

-mc A <

ad j o i n t p r o j e c t i o n s i n

satisfying

and such t h a t

= i Q -1BQ =

A

Q

E

co

{PA] i s t h e family of s e l f

(3.2).

Banach i n t e g r a l s . Consider t h e space

5

=

Bl(

-GO, m)

of bounded complex valued

1 defined i n

functions p e r i o d i c with period

< t<

-m

m.

We define

below anextension of t h e notion of' i n t e g r a l s i m i l a r t o t h e extension

.r

of l i m i t introduced i n $1. A Banach i n t e g r a l or generalized i n t e g r a l i n % i s a functional ( - ) a s : B1 + c such t h a t , f o r f,g E B1 and

au,B complex we have

+ Bg(s)) d s

( a ) [(nf(s) (b)

=

aJf(s) d s + B / g ( s ) ds

"

[f(s + t) d s = J f ( s ) d s

(c) >f(s)

ds

2

0

if

f

2

0

(-m


.

m).

*

( d ) l l d s = 1.

f i s l o c a l l y Riemann i n t e g r a b l e .

THEOREM 5.1.

Banach i n t e g r a l s i n

B1

exist. -

143

I N HILBERT SPACE

A s i n t h e case of Banach l i m i t s it i s obviously s u f f i c i e n t

Proof:

to define [(.)

f o r r e a l valued f u n c t i o n s .

ds

Let

Bll,R

be t h e

-1

B1,

corresponding subspace of p ( f ) = inf -m<

For

f

E define l,R

E

(5.11

sup s <m

t h e infimum t a k e n with r e s p e c t t o a l l f i n i t e s e t s Again we use Theorem 1.1 f o r

numbers. CD

F = {O]

@:%,R

obtaining a l i n e a r f u n c t i o n a l

= 0

@ ( f )I P ( f )

(f

@ ( f )1 - P ( - f )

(f

R

+

%,R)

E

...,

Sl, 5 n of r e d and t h e f u n c t i o n a l such t h a t

(5.2)

9

so t h a t

s

We define from

(-) ds

(5. 3).

@.Obviously

=

To show

a r b i t r a r y and t a k e

(5.3). = 2t

+ t )- f ( s ) ) 5

p(f(s

-m<

so t h a t

p(f(g

+

5

t)-f(;))

5o

(a)

(5.%) (c)

h o l d s and

it s u f f i c e s t o observe t h a t

(a)

(5.2) and 51 = t , 5 2

p ( - 1 ) = -1 and use

%,&

E

0;

To prove

,...,5 n

- f(s)),

we prove i n t h e same way t h a t

p(f(;)-f(i

+

L e t now

be a l o c a l l y Riemann i n t e g r a b l e f u n c t i o n i n

5,

f

= l/n,

52

t))

=

2/n,.

thus

..,
.s,

=

(b)

follows from

n

1 and l e t t i n g

so t h a t

(f)

such t h a t

follows.

eieJf(s)

//f(s) This shows

+

and

(5.3). Taking

we deduce t h a t

m

-J

El

be a r b i t r a r y .

f ( s ) ds, p(-f) I

0

Finally, l e t

i s positive.

ds

f

F

Then, by

d s ; = l e i e r { s ) d s =JRe(eief(s))

(e)

(5.2)

1 f ( s ) ds

1

P(f)

t

Then

(f(s + nt)

sup s <m

p ( 1 ) = 1,

we select

(b)

= nt.

follows

Choose

e

(c), ds

5 s

If(s)l d s

.

and ends t h e proof o f Theorem 5.1.

We s h a l l need i n t h e sequel a v e r s i o n of t h e Banach i n t e g r a l f o r t h e space

B

= B(-ca,m)

properties similar t o

of a l l bounded f u n c t i o n s i n

s

(-) ds

(-m,m)

satisfying

p l u s an i n t e r v a l a d d i t i v i t y p r o p e r t y

and a rudimentary change-of-variable formula.

144

IN HILBERT SPACE THEOREM 5.2.

There e x i s t s a f a m i l y of f u n c t i o n a l g

,/

r.b (.)ds:E-6:

‘ a defined f o r


-W

sal i s f y i n g t h e following p r o p e r t i e s :

m

b

( i ) Jb((l‘f(s) + B g ( s ) ) d s

1;;

f(s) d s = rbf(s

(ii)

(iii) (iv)

J

k

‘J a

-b f ( s ) ds a

2o

Ibl

t) d s

f(s)

if

f(s) d s + B

(l’Ja

(-m

Lb .

.

g(s) ds


m)

.

2o

.

ds = b-a

a

(v)

lLbf(s) ds)

I [blf(:)l

ds

*

b

( v i ) Ja

f(s) ds

rb/2

(vii)

+

y C f ( s ) ds =L[cf(s) “b

J a/2 Proof:

$l b

f(2s) d s

Let

c

=

if

ds

.

a
.

f(s) ds

If

0.

-c < 2a, 2b < c

(5.4)

define

f(s;a,b,c) = 0

& 5 s < ca , -bc <_ s < 1F )

(-

(E 5

f(s;a,b,c) = cf(cs) and extend

f(s;a,b,c)

to

-a

< s <

m

all

s.

(5.4)

b

< -)

i n such a way t h a t t h e r e s u l t i n g

f u n c t i o n i s 1 - p e r i o d i c ( t h a t i s , belongs t o the inequalities

s

El).

f a i l s t o hold we d e f i n e

I n c a s e one of f(s;a,b,c)

=

0

for

The Punctionals i n q u e s t i o n a r e defined by means of any of t h e

Banach i n t e g r a l s

I-

i n Theorem

Lbf(s)ds

=

5.1 by means of t h e formula

LIbl n- m

i

f(s;a,b,2n) d s ,

LIM i s any of t h e Banach l i m i t s c o n s t r u c t e d i n C o r o l l a r y 1 . 3 . n- m The checking of p r o p e r t i e s ( i ) Lo ( v i i ) i s f a i r l y r o u t i n e and i s

where

145

I N HILBERT SPACE

thus l e f t t o the reader. The i n t e g r a l s i n Theorem 5.2 w i l l a l s o be called Banach i n t e g r a l s .

Pv.6

Uniforaly bounded cosine functions i n HiLbert space. B be a s e l f a d j o i n t operator i n

Let is,

(Eh,u)

2

for

0

u

D(B))

E

S(t,A) = cos(th1/2) in

Efl2

(6.1)

=

S(t;e)

c (-1)nt 2nAn /(2n)!

h or of t h a t of

of square root o f

use of

=

such t h a t

B

10

(that

and l e t

@ ( t ) = cos(&) where

H

,

(6.1)

(note t h a t t h e choice

i s irrelevant; i n fact, the

B

i s purely symbolic).

continuous cosine function such t h a t each

Then

C(i) i s a strongly

@ ( t ) i s self adjoint;

more over,

We have s(t) where

7j(t,h)

t h e use of

=

=

s l / 2 s i n ( t I f / 2 ) = q(t;E)

A-1/2sin(tA1/z)

d/2

=

c (-l)nt2n+1A"/(2n

i s purely symbolic).

(6.3)

+ l)!

(again,

Since

we can o n l y assert t h a t

(which i n e q u a l i t y follows anyway from d i r e c t i n t e g r a t i o n of However, if

E

>

0,

Ik112sin so t h a t , i f

a(B)

-

i s contained i n

( h? E l ,

c h

2

E,

The converse of t h e s e observations holds a s w e l l .

(6.2)).

I N HTLBERT SPACE

146 *

THEOREM

Assume t h a t each

with

B = -A,

then

B

if

2

A

8(:)

2 EIIuII , u

Proof: n

=

with

D(B)

E

E

holds.

B 2 EI with

A

1 and

holds

@(:).

Inequality

>0

E

( t h a t is

>0).

It i m i t a t e s t h a t o f Theorem 2 . 1 .

i n f i n i t e s i m a l g e n e r a t o r of for

(6.2)

holds i f and o n l y if

2

(BU,U)

a s t r o n g l y continuous c o s i n e f u n c t i o n .

@ ( t )i s a s e l f a d j o i n t o p e r a t o r . Then (6.1) t h e i n f i n i t e s i m d g e n e r a t o r of @(;). rf

and t h e s t r o n g e r e s t i m a t e

0

for

(6.5)

Let @ ( t )be

6.1.

A

Let

be t h e

Making use of formula

(11.2.11)

r e a l and l a r g e enough we o b t a i n 2

( M ( h ;A)u,v) =

so t h a t

s e l f ad j o i n t . that

B(t)

o(A) =

=

Moreover,

R( A2;A)

h

e x i s t s for

i s contained i n a n i n t e r v a l

itself is

s u f f i c i e n t l y l a r g e so

1,

(-,a

A

a <

m.

Accordingly,

i s a s t r o n g l y continuous o p e r a t o r valued f u n c t i o n

5(t;-A)

satisfying

B(t)

i s self a d j o i n t , t h u s it f o l l o w s t h a t

R(h2;A)

Ili!J(t)ll 5 exp(Ult1)

with

'I2.

a

=

The f a c t t h a t

@ ( t )i s proved by uniqueness of Laplace transforms a s i n

Theorem 2 . 1 .

a(A)

Assume t h a t supremum of

cos

in

(6.5)

The f a c t t h a t

a > 0.

c o n t a i n s some

A

2

and

-:I

w i l l be t r u c when

Then

[l@(t)il i s the

(6.6) has no chance t o h o l d . a(A)

5 (-m,-~],

E

< 0

nas

been a l r e a d y e s t a b l i s h e d i n t h e remarks preceding t h e statement of Theorem

6.1.

( 6 . 5 ) holds.

Conversely, assume t h a t

p a r t s t h e f i r s t formula

(11.2.117

2 R(h ;A)u =

I n t e g r a t i n g by

we o b t a i n

r"

e-AtS(t)u d t

' 0 for

A > 0

and

u

E.

E

Taking norms, it r e s u l t s t h a t

I I R ( A ~ ; A ) I =I o(A-') which i m p l i e s t h a t hence

o(A)

5

(-m,-E]

0

f

p(A)

as

A-

(DUNFORD-SCHWARTZ

f o r some

E

>0

o+, [1958:1, p . 5671),

a s claimed.

This ends t h e proof

147

I N HILBERT SPACE

of Theorem

6.1.

The next r e s u l t i s a n exact c o u n t e r p a r t of Theoyem 2.2 for c o s i n e However, t h e method of proof i s somewhat d i f f e r e n t .

functions.

THEORZM 6.2. @(s

+ t) +

@(s

Let c(<)

- t) =

be a c o s i n e f u n c t i o n i n

2C",(s)@(t)

for

-m

c s, t <

m).

Then t h e r e e x i s t s a bounded, s e l f a d j o i n t o p e r a t o r

5

?-ll2(2C + 1)-111ul12

(Qu,~)

5

(u

C/IUI/~

H

Q E

(@(O) =

I,

Assume t h a t

such t h a t

(6.8)

E),

a nd Q ( t ) = Q@(t)Q-'

t.

i s self a d j o i n t for a l l

The proof uses some techniques a l r e a d y met i n connection with t h e t h e o r y of almost p e r i o d i c f u n c t i o n s , although nothing i s almost periodic here.

We b e g i n by s e l e c t i n g one of t h e Banach i n t e g r a l s

constructed i n Theorem 5.2 and one of t h e Banach l i m i t s i n Theorem 1.2, and observing t h a t

LIM T-m

e x i s t s f o r any

f

E

B;

4 FT f ( s ) d s

(6.10)

"0

i n f a c t , due t o p r o p e r t y

(v)

of Banach i n t e g r a l s

we have

where

i s a bound for

C

If(s)l

, so

t h a t t h e f u n c t i o n of

( 6 .lo) i s bounded i n T 2

t h e Banach l i m i t i n

LEMMA 6.3. ~ e f t

$L

E

f o r any

t

Proof:

&

inside

B.

T

LIM T 4 rn

T

0.

f(s + t ) ds

=

LIM

$

(6.11)

T-m

(-m,m).

Assume f o r t h e moment t h a t

t

2

0.

By v i r t u e of

(ii)

and

148

I N HILBERT SPACE

(vi)

we have

$LT

f(s + t )

which expression i s seen t o tend t o zero as

-

T

i t s Ba.nach l i m i t m u s t be zero as well, proving LEMMA

6.4. e

xe

Let T >

=

e(E,T,u)

0, E

=

5

Ct;o 5 t 5

t h e c h a r a c t e r i s t i c f u n c t i o n of

t

Proof: Set equation

=

s

= u/2

cr

e

E

E

H,

(6 -12)

ll@(t)uII < ~ l l u 1 1 3 .

e.

i n t h e (second) cosine f u n c t i o n a l

0/2

=

@(u) + I

-

we have

Accordingly, i f

so t h a t

hence

(11.1.9). m e r e s u l t i s 2qu/2)2

Hence, i f

T,

(v);

using

(6.11).

+ l), u

1/(2C

m

&

f! e.

5 1/(2C + 1)

we deduce t h a t

It follows t h a t i f

shows t h a t t h e f u n c t i o n s

x (i)

and

u

x

E

f2;)

e

then

2a

#

e,

which

have d i s j o i n t support.

Hence

by

(i)

and

t i o n we o b t a i n

(v).

Taking t h e change-of-variable property i n considera-

(6.13),

thus ending t h e proof of Lemma 6.4.

149

I N HILBERT SPACE

Proof of Theorem 6.2.

m e operator

i s t h i s time defined by

P

ds

E y v i r t u e of Lemma

6.4

with

E =

1/(2C m

t h u s it f o l l o w s from t h e d e f i n i t i o n of

+ 1)

P

.

(6.14)

we have

that

On t h e o t h e r hand, it i s obvious t h a t

Accordirgly, i f inequalities Let now

i s t h e p o s i t i v e , s e l f a d j o i n t square r o o t of P,

Q

(6.8) h o l d . t

be a real number,

u,v

elements of

H.

Using t h e

c o s i n e f u n c t i o n a l e q u a t i o n s and Theorem 5.2 we deduce t h a t ( P @ ( t ) u , v ) = LIM

L T ( @ ( s ) C ( t ) u , @ ( s ) v d) s

T-‘M T L =

1LIM 2 T-m

$lo -T

( @ ( s + t)u,C(s)v) ds

$k

T

+ 1_

*

LIM T-m

+ 1_

LIM T-m

(@(s - t)u,@(s)v) d s

PT

‘

for

u,v

E

H.

$ j o ( @ ( s ) u , @ ( s+ t ) v ) d s

Accordingly,

(6.18) P r e - and p o s t - m u l t i p l y i n g by QC(t)Q-l

Q

-1

= Q-’@(t)*Q

we obta.in =

(Q@(t)Q-’)*.

(6.19)

150

I N HILBERT SPACE

This completes t h e proof of Theorem

6.2.

The coriiments following Theorem 2.2 apply h e r e a s w e l l :

replacing

t h e o r i g i n a l s c a l a r product by ttie ( t o p o l o g i c a l l y e q u i v a l e d c ) s c a l a r product

(2.12)

-

@ ( t )s e l f a d j o i n t .

r e n d e r s each

COROLLARY 6.5.

Assume i n a d d i t i o n t h a t

C(;)

Q

B

u

B

2

t

E.

Then t h e r e e x i s t s a s e l f ad-

tnd a bounded s e l f a d j o i n t o p e r a t o r

0

(6.8)

s a t i s f y i n g i n e q u a l i t i e s of t h e form

@ ( t=) Q - l c o s (tB1/')Q @(;)

Conversely, e v e r y

(-a

E 2 EI f o r some

(6.7).

and such t h a t


(6.20)

of t h e form

cosine function s a t i s f y i n g only i f

6.2.

s a t i s f y t h e hypoteses of Theorem

@ ( t ^ ) u i s continuoas ( o r , more g e n e r a l l y

s t r o n g l y measurable) f o r each j o i n t operator

We omit t h e d e t a i l s .

m)

.

(6.20)

i s a s t r o n g l y continuous

Inequality

(6.5)

h o l d s i f and

> 0.

E

The following d i s c r e t e v e r s i o n of Theorem

6.2 corresponds t o

Theorem 2.4.

THEOREM tors in

2C C mn

H

Let {cn;-m < n c

6.6.

m)

be a sequence of bo,mded opera-

s a t i s f y i n g t h e " d i s c r e t e c o s i n e equations"

+ Cm-,

= @,+,

m,n.

f o r all

=

I

'

Assume t h a t

Then t h e r e e x i s t s a bounded, s e l f a d j o i n t o p e r a t o r

2-l(2c +

0

l)-'hIl2 5

(Qu,.)

Q

5 Cllul12

satisf'ying

(6.22)

and such t h a t

fin

i s s e l f ad,joint f o r a l l operator

U

n.

=

QCnQ

-1

Equivalently, t h e r e e x i s t s a u n i t a r g

such t h a t

@n

=

2

Q-l(V" +

U-n)Q

.

(6.24)

The proof i s r a t h e r s i m i l a r t o t h a t f o r t h e continuous v e r s i o n . The o;lerator

P

i s now defined by

151

I N HILBEHT SPACE

(Pu,v) where

1.3.

Il@2muII 2

then

P r o c e e d i n g as i n t h e p r o o f o f Lemma 6.4 we c a n show

IICmuII < Ellull

t h a t if

-

LM n- m

i s one o f t h e Banach l i m i t s o f sequences c o n s t r u c t e d i n

LIM

Corollary

=

11

f o r which

f o r an arbitrary integer

EIIU// i s

PmuII 5

a t least e q u a l t o

5

indicates the largest integer

[s]

m

and

5

E

hence t h e n m b e r o f i n t e g e r s between

Ellu/l;

s.

[ ( n - 5)/41,

Taking

E = 1/(2C

(2C +l)-l

0

and

n

where

+ 1) we

obtain

and it i s o b v i o u s t h a t

Q,

thus

(6.22).

t h e p g s i t i v e s e lf a d j o i n t s q u a r e r o o t of

hence e a c h

(6.23)

in

=

C*P, n

(6.27)

is self adjoint.

Consider now t h e sequence of o p e r a t o r s {AS,) m = n

satisfies

(6.17) shows t h a t

A computation e n t i r e l y s i m i l a r t o

P@n

P

{fin;-m


a].

Obviously

s a t i s f i e s a s well t h e d i s c r e t e c o s i n e e q u a t i o n s so t h a t , t a k i n g

we o b t a i n =

2 2an

-

I

.

Making use of t h e s p e c t r a l mapping theorem f o r bounded o p e r a t o r s (DUNFORD-SCHWAF3Z [1958:1,

p.

(An]

-

An

E

5691) w e deduce t h a t i f

i s d e f i n e d i n d u c t i v e l y by a(2Bn- 1). B u t

I An]

bounded i n d e p e n d e n t l y o f

We c a n t h u s d e f i n e inverse t o

cos A

n,

\ if

=

A

E

~ ( 8 , )and

A , A = 2A2 - 1 t h e n n n-1 > 1; s i n c e u(fin) must b e

I A1

w e deduce t h a t

E = arc cos

a1, where

i n the interval &

a r c cos A

i s the function

[ - ~ / 2 , ~ / 2 ] . Let

n = c o s (nB)

.

(6.28)

152

I N HILBERT SPACE

Then it follows from t h e f b n c t i o n ? l c a l c u l - u s f o r s e l f a d j o i n t o p e r a t o r s

{&

that

s a t i s f i e s a s w e l l t h e d i s c r e t e c o s i n e f u n c t i o n a l equation;

n

i n particular,

&

which shows i n d u c t i v e l y t h a t

(6.23)

with

n

(6.28)

We only have t o combine

(6.24),

t o obtain

8 f o r all n s i n c e & = 8 1 1' n 1 ( i n t h e f*orm & = f e x p ( i n E ) + e x p ( -inB)}) n 2 where LJ = e x p ( i E ) .

1

-

Theorem 6.1 i s obviously e q u i v a l e n t to t h e following r e s u l t for second order a b s t r a c t d i f f e r e n t i a l e q u a t i o n s .

Let

THEOREM 6.7.

be a c l o s e d , densely defined o p e r a t o r i n t h e

A

H such t h a t t h e Cnuchy problem f o r

H i l b e r t space

u"(t)

-
i s w e l l posed i n

(6.29)

Au(t)

=

(eq'iivalently, i n

moreover, t h a t f o r every g e n e r a l i z e d s o l u t i o n of u'(0)

=

(where

0

we have

C

may depend on

operator

with

B

B

1. 0

~ ( t ) ) Then .

=

LU;QU

E

D(E))

Moreover, all s o l u t i o n s of

-

-W


a)

0).

Assume,

(6.29) with

there e x i s t s a s e l f adjoint Q

(6.11) and such t h a t

and A=-&

in

2

and a bounded s e l f a d j o i n t o p e r a t o r

s a t i s f y i n g i n e q u a l i t i e s of t h e fo-m

D(A)

t

-1

a.

(6.31)

(6.29) a r e bounded i n t 1 0 ( e q u i v a l e n t l y , 12 2 &I f o r some & > 0 . The converse

i f and only i f

i s a s well t r u e . REMARK

6.8.

A s i n t h e group c a s e it i s of i n t e r e s t t o know

whether t h e conclusions of Corol1:try c o n d i t i o n s on t h e t-dependence of t h e same. function

Weak m e a s u r a b i l i t y of (@(t)u,v) f o r a l l

6.5 hold under l e s s s t r i n g e n t

@(;).

The answers a r e e s s e n t i a l l y

@ ( t ) ( t h a t i s , m e a s u r a b i l i t y of t h e

u,v e H)

s u f f i c e s when

H

i s separable:

i n t h e g e n e r a l c a s e , it i s enough t o assume weak c o n t i n u i t y , s i n c e it

153

I N HILRERT SPACE

implies strong continuity.

J!J(f)

(6.9) with

I n view of t h e r e p r e s e n t a t i o n

@(;)

s e l f a d j o i n t , it i s obviously s u f f i c i e n t t o prove t h i s when

i s s e l f a d j o i n t , i n which case we have

=

((C(2t)

+

I ) ~ J , u ) - 2 ( ( @ ( s+ t ) + @ ( s - t ) ) u , u ) + ( ( @ ( 2 s ) + I ) u , u )

@ ( t ) i s s t r o n g l y continuous i f it i s weakly continuous.

thus

pV.7

Almost p e r i o d i c cosine f u n c t i o n s i n H i l b e r t s , . Let

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

B

H.

Assume t h a t

B

2

0

and t h a t it has pure p o i n t spectrum; t h e n t h e r e e x i s t s a family {PA;A

2

0J

of mutually orthogonal p r o j e c t i o n s such t h a t PAu= u

(u

E

H)

(7.1)

h l0 and

for

u

E

D(R)

( s e e t h e comments following

results from t h e i n i t i a l remarks i n

(4.1)

and

It

(4.2)).

Pv.6 that

(7.3)

@ ( t=) cos (tIY2) i s a s t r o n g l y continuous, uniformly bounded cosine f u n c t i o n with i n f i n i t e s i m a l generator

@(t)u

=

A

A 20

=

-B.

We have

(u

cos thli2P,u

t h e s e r i e s convergent i n t h e norm o f

H

o n l y a coxntable number of t h e

P u

be non n u l l (as pointed out i n

8V.4,

h

if t h e space

ing p r o p e r t y f o r

s ( ; ) ~may

(6.4)


H

3.2 and C o r o l l a r y

not hold s i n c e

and following comments).

a;

u)

may

i s separable

P A must of needs v a n i s h ) .

@ ( < ) u i s an almost p e r i o d i c f u n c t i o n f o r every u bounded ( s e e

-m

which a l s o implies t h a t

( i n g e n e r a l depending on

a l l b u t a countable number of t h e p r o j e c t i o n s

(7.4), Lemma

(7.4)

E),

uniformly i n 2

llPAuli = Ilui12

t h i s f o l l o w s from t h e e q u d i t y

Uniform convergence of

E

E

H.

3.4

imply t h a t

The correspond-

llS(<)ll may not be However, t h e c o n d i t i o n

154 on

IN IIILBERT SPACE a( B)

11s;:) 11

t h a t guarantees boundeciness of

p e r i o d i c i t y a s well; i n f a c t , i f

a(B)

which i s a s well almost p e r i o d i c .

guarantees almost A

i s contained i n

2

we have

E,

Under t h e s e conditions, a n almost

p e r i o d i c i t y property holds a s well i n phase spaces; we l o o k only a t t h e “ p r i n c i p a l value square r o o t ” phase space Theorem 111.5.4 f o r constructed i n r o o t of B.

constructed i n

J

Observe f i r s t t h a t t h e square r o o t

= 0.

W

6111.3 coincides with t h e p o s i t i v e , s e l f ad j o i n t square

Since

@ ( t )we have

commutes with each

This, combined with (7.4) shows t h a t c(i) i s almost p e r i o d i c i n t h e (endowed with t h e graph norm of (-A)1/ 2 ). space Eo = D((-A)1/2) Since

8(<)

i s almost p e r i o d i c i n

almost p e r i o d i c i n I3

for a l l

11

E, €3

,

it follows t h a t 6(:)li i s 6(i) i s t h e group

where

(111.1.10) corresponding t o t h e phase space 3

. Noting

that

(7.6) we see t h a t

(-A)1/28(t)u

Accordingly, t h e group

i s almost p e r i o d i c i n

E

b o ( i ) i n t h e group decomposition

corresponding t o t h e p r i n c i p a l value square r o o t that

bO(t)

i s almost p e r i o d i c f o r any

i n t h e product space

5) =

E x E

u

’=

H

u

f o r every

E E.

(111.6.24) i s such

uo(t)

and t h e group

used i n t h e reduction t o a f i r s t order

system i n Theorem 111.5.5 enjoys t h e same p r i v i l e g e s . The following r e s u l t i s a s o r t of converse of t h e preceding observations. THEOREM

f o r each

7.1. Let @(:)

be a cosine function i n

u,v z H t h e f u n c t i o n

@(<) a d m i t s t h e r e p r e s e n t a t i o n

Assume t h a t

( @ ( t ) u , v ) i s almost p e r i o d i c .

(6.20) with

operator s a t i s f y i n g i n e q u a l i t i e s of t h e type of a d j o i n t operator of t h e form

H.

(7.2): if

H

Q

Then

a bounded s e l f a d j o i n t

(6.8) and

B

a self

i s separable, t h e family

{PA] i s countable.

Proof:

It follows from t h e f a c t t h a t almost p e r i o d i c functions a r e

155

I N HILBEHT SPACE

bounded i n

-m

< i; <

m

and from t h e uniform boundedness p r i n c i p l e t h a t

@ ( t )i s weakly continuous and t a k i n g heed of Remark 6.8 we n o t i c e t h a t @(<) f i t s i n t o t h e assumptions of Coj-ollary 6.5, thiis we might adopt i t s conclusion as Noting t h a t t h e assumptions imply t h a t

However, a s i n t h e grotip case i n Theorem 4.1 we

a starting point.

s h a l l p r e f e r a d i r e c t c o n s t r u c t i o n f e a t u r i n g a f a i r l y explicit formula

B.

f o r t h e s p e c t r a l family of operator

N,,

in

H

A

For each

2

by t h e formula rT

_I

( N ~ u , v ) = l i m IT-m

/

cos A t ( @ ( s ) u , v ) d t

T ~ J O

e

iA t

(@(t)u,v)d t

W e see applying Theorem 3.7 t h a t t h e l i m i t i n

(Nhu,')

(incidentally, of

A's.)

For a n y

d e f i n e a bounded

0

A

= 0

3 0,

-m

.

(7.8)

(7.8) e x i s t s for a l l

h

for a l l but a f i n i t e or countable number


m,

u,v

E

H

we have

or

C"(t)N,, by v i r t u e of

(3.5);

from t h e f a c t t h a t Multiply

(7.9)

Nx@(t)

the fact that @(s)

by

=

and

Nh

(7.9)

ht)NA

= (COS

and

@ ( t ) commute f o l l o w s

@ ( t )commute f o r a r b i t r a r y

exp(ipt)/2T

and i n t e g r a t e i n

-T

Since fl if

A = ~ = o

5

s,t.

t

1. T.

I N HILBERT SPACE

156 we have

Nh

if

0

if

N N = A +

Accordingly t h e o p e r a t o r s

a r e a family of mutually orthogonal ( n o t n e c e s s a r i l y s e l f ad j o i n t ) projections i n Let

be t h e s e l f a d j o i n t operator defined i n t h e p r o o f of

P

Theorem

3.1:

and l e t

Q

use of

H.

(7.9)

(7.8) and

T+ =

w

$LT

cos As(N,,u,@(s)v)

(N,,u,N~v)= l i m T+m

=

so t h a t

Making

we o b t a i n

(PNAu,v) = l i m

which i s

P.

be t h e p o s i t i v e s e l f a d j o i n t square r o o t of

'

ds

cos h(@(s)u,N,,v) d s

TL 0

(Pu,NAV)

*

PN,, = NAP;

PA = QMAQ-'

accordingly,

i s self adjoint f o r a l l

self a d j o i n t o p e r a t o r

B

from t h e family

A.

(PA;h

We define now a

2

01

of s e l f a d j o i n t

p r o j e c t i o n s by t h e formula

(7.14) ( t h e domain of

B

c o n s i s t s of a l l

convergent i n t h e norm of

E)

u which make t h e s e r i e s

and s e t

(7.14)

157

I N HILBEW SPACE

& ( t )= Q-lcos (tB1/2)Q

.

(7.15)

Since &(t)u

=

1/2 -1 cos t h Q PhQl = 2 cos tM,,u A? 0 A? 0

(7.16)

an obvious computation based on t h e uniform convergence of and on

(7.10) shows t h a t i f

- [T

l i m T+m

2Tb

T+

E

H

we have

eiAS(&(s)u,v) d s = ( N hU , V )

-T

= l i m

u,v

(7.16)

2T

(7.17)

e i h ( @ ( s > u , v )d s

m

(&( ; ) U , V ) and

so t h a t both almost p e r i o d i c f u n c t i o n s

By Theorem

have t h e same Fourier s e r i e s .

(@(;),u,v)

3.8 we must have

& ( s ) = @ ( s ) by a r b i t r a r i n e s s of u,v.

( & ( s ) ~ , v= ) ( @ ( s ) u , v ) , thus

This conclludes t h e p r o o f 3f Theorem 7.1. I n t h e language of a b s t r a c t d i f f e r e n t i a l equations, t h i s r e s u l t can be f o r n u l a t e d a s follows.

Let

THEOREM 7.2. t h e f i l b e r t space

H

A

be a closed, densely defined o p e r a t o r i n

such t h a t t h e Cauchy problem for

u”(t) i s w e l l posed i n

-m


=

(7.18)

Au(t)

t

(equivalently, i n

and every

i s almost p e r i o d i c .

v

E

H

E

D(B)l

with

Then t h e r e e x i s t s a s e l f a d j o i n t o p e r a t o r B

2

s a t i s f y i n g i n e q u d i t i e s of t h e f o r m

D ( A ) = 1u;Qu

Assume,

the function

pure p o i n t spectrum and such t h a t Q

0).

(7.18)

moreover, t h a t fo.r every generalized s o l u t i o n of u’(0) = 0

2

0

B

with

and a s e l f a d j o i n t operator

(6.8)

and such t h a t

and A

=

-Q

-1 BQ=

-c

-1

?Q PAQ

A? 0

( t h e sum understood elementwise) where a d j o i n t pro,jections i n

(7.2).

{PA] i s t h e f a m i l y of self

158

I N HILBERT SPACE

Miscellaneous comments .

Pv.8

Eanach l i m i t s and Banach i n t e g r a l s were introduced by BANACH

[1932:1]a s applications of t h e Hahn-Banach theorem.

As anfching based

on Theorem 1.1 (whose proof uses the axiom of choice or, equivalently, the Hausdorff-Zorn lemma o r t h e p r i n c i p l e of t r a n s f i n i t e induction), "construction1' of Banach l i m i t s o r i n t e g r a l s i s t o t d l y nonconstructive and vaguely incredible even t o mathematicians without a philosophical

Two of the most i n t r i g u i n g applications o f Banach l i m i t s

t u r n of mind.

t o a n a l y s i s a r e Theorems 2.2 and 2.4, both due t o SZ.-NAGY [1947:l]. These r e s u l t s a r e statements on representations of groups

G

(the r e a l

l i n e and t h e i n t e g e r s r e s p e c t i v e l y ) and have been generalized by DMMIER

[1950:l] t o uniformly bounded representations of c e r t a i n groups G

into

t h e d g e b r a of linear bounded operators i n a Hilbert space; for r e l a t e d

material see GREENLEAF [1969:1]. The corresponding generalization of Theorem 6.2 i s due t o K U " A [1972:1]. We note t h e results of LORCH [19L+l:11

on s p e c t r a l a n a l y s i s of operators s a t i s f y i n g t h e assumptions

of Theorem 2.4 i n r e f l e x i v e Banach spaces; the main result i s a s p e c t r a l resolution modelled on t h a t f o r

unitary operators i n Hilbert space.

Theorem 2.2 can be s t a t e d i n t h e following equivalent form: A

-

THEOREM 8.1. Let S ( t ) be a group i n a Hilbert space H s a t i s f y i n g (2.4). Then t h e r e e x i s t s a Hilbert norm II*I/' H of t h e form (2.12), equivalent t o the o r i g i n a l one and such t h a t Ils(t>ll'

51

(-m
(8.I1

m)

on

In f a c t , Theorem 8.1 i s an obvious consequence of Theorem 2.2: the other hand, i f (8.1) holds then we must have

In f a c t , if

IlS(t)u/l'

contradiction.

C

lIul/'

Since each

then S(t)

h111' =

llS(-t)S(t)ull' <

S ( t ) i s unitary, hence Theorem 2.2 follows. Theorem 2.4 i s equivalent t o

THEOREM 8.2. satisfying (2.14).

Let S

kII',

a

i s i n v e r t i b l e , (8.2) implies t h a t each

The same argument shows t h a t

be a bounded operator i n t h e Hilbert space

Then there e x i s t s a Hilbert norm

t h e form (2.12), equivalent t o t h e

I\*llt 2

o r i g i n a l one and such t h a t

H

of

H

159

I N HILBERT SPACE

It i s n a t u r a l t o ask whether Theorem 8.2 has a counterpart for semigroups, i . e . whether a semigroup s a t i s f y i n g (2.4) i n

2

t

will

0

s a t i s f y (8.1) a f t e r a renorming of t h e space along t h e l i n e s of The answer i s i n the negative; see PACKEL

Theorem 8.1.

[1969:1].

The

answer f o r t h e d i s c r e t e version was known e a r l i e r t o be also i n t h e negative. (FOGUEL [1964: 11). W e note t h a t t h e problem o f renorming a Eanach space with an equivalent Banach norm t h a t improves t h e constant i n ( 2 . 4 ) t o

1 has a

simple a f f i r m a t i v e answer for semigroups, groups or cosine functions a l i k e ( s e e Exercise 5 ) a s well a s f o r t h e i r d i s c r e t e counterparts.

The theory of almost periodic groups i n Hilbert space i s due t o t h e author [1970:4] although t h e f a c t t h a t

U(i)u, u

given by ( 4 . 3 ) i s

almost p e r i o d i c was known before (see AMERIO-PROUSE [1971:1] f o r references and for numerous r e s u l t s on almost periodic s o l u t i o n s of a b s t r a c t d i f f e r e n t i a l equations).

Likewise, t h e r e s u l t s presented here on

uniformly bounded cosine functions (Theorem

6.2 and 6.5) as well a s t h e

r e s u l t s on almost p e r i o d i c cosine functions (meorem 7.1) a r e due t o t h e author [1970:3], although t h e remarks on almost p e r i o d i c i t y of functions

of the form (7.4) go back e s s e n t i a l l y t o MUCKENHOWPT [1928-1929:1].

The

6.6) i s also due t o t h e author [1970:3]. Remark 2.6 i s i n SZ.-NAGY [1938:1]; t h e analogue for cosine functions (Remark 6.6 i s i n the author [1974:31).

d i s c r e t e version (Theorem

The theory of periodic and almost periodic semigroups, groups and cosine functions i n Eanach space was i n i a t e d i n t h e l a t e s i x t i e s . DA PRATO

[1968:1] and BART [1977:1] t r e a t p e r i o d i c strongly continuous Y

semigroups and groups while CIORANESCU [1982:1] t r e a t s p e r i o d i c d i s t r i b u t i o n semigroups. GIUSTI [1967:1].

Periodic cosine functions were considered by

Finally, almost p e r i o d i c semigroups and cosine functions

were considered r e s p e c t i v e l y by EART-GOLDEERG

L1982:lI.

[1978 :1] and by PISKARFV

Some of t h e i r r e s u l t s w i l l be found below (Exercises

EXERCISE 1.

Let

E

be an a r b i t r a r y

vector space.

has a b a s i s , tha,t is, a l i n e a r l y independent subset

for each

u

E

E

we have

{u,]

6 to 15).

Show t h a t such t h a t ,

E

160

I N HILBERT SPACE

the scalars

all zero except f o r a f i n i t e number.

{Am]

See JACOESON

[1953:1, P . 2391 Using Exercise 1 show t h a t t h e r e e x i s t s a f u n c t i o n

EXERCISE 2. f :R

4

R

such t h a t f(s + t )

=

f(s)

+

f(t)

(8.4)

but

for any

c

(Hint:

W i s a vector space over

A n y nonzero l i n e a r f u n c t i o n a l

one, l e t

[t,}

CY

be a base of

i s fixed

EXERCISE 3.

group

t h e f i e l d of r a t i o n a l

f : R .+ Q w i l l do. W a s i n Exercise 1 and define

numbers.

where

0,

To construct

. U s i n g Exercise 2 show t h a t t h e r e e x i s t s a u n i t a r y

( i n one-dimensiond Eanach space) t h a t does not a d m i t t h e

U(<)

representation (2.1).

4.

EXERCISE

that (a) (Hint:

in

-m

a(B) =

Give an example of a s e l f - a d j o i n t operator (-m,~),

(b)

< h c

number of

A

Every

consider t h e H i l b e r t space

H

E

o(B)

B

such

i s an eigenvalue of

of a l l f u n c t i o n s u ( i )

B

defined

vanishing for all but for a f i n i t e o r countable

W,

and such t h a t

A’s

Define Au(A) = h ( A )

A<

(-m<

W)

,

(8.7)

n

t h e domain of to

A

c o n s i s t i n g of all

u(A)

E

H

such t h a t (8.7)

belongs

H). EXERCISE

such t h a t

5.

A

(a)

Let

S(t)

be a semigroup i n a Banach space

E

161

I N HlLBERT SPACE

Show t h a t t h e r e e x i s t s a n e q u i v a l e n t norm

(b)

Let

II-II'

b e a group i n a Ba.nach space

S(t)

Show t h a t t h e r e e x i s t s a n equivalent norm

(c)

S t a t e and prove a v e r s i o n of ( b )

c exp

(id

It I )

(Hint:

and i n

and with cosh

in

E

such t h a t

f o r c o s i n e f u n c t i o n s both with

on t h e right-hand s i d e .

fdt

(b),

6.

Banach space

E

Let

be a s t r o n g l y continuous semigroup i n a

S(t)

IIS(t)ul/ = llull S(t)E

Then ( a )

(t 2 0 ) for

Each S ( t )

isometry) ( b )

is

for

t <0

u

Each E

E,

S(t)

The group

i s one-to-one ( i n

t h u s each

onto ( i n view of ( a )

i s dense in E) ( c )

S ( t ) = S(-t)-'

i s almost p e r i o d i c ( t h a t i s , a s t r o n g l y

S(i)u

such t h a t

almost p e r i o d i c semigroup).

that

II-1/'

such t h a t

i n ( a ) use

EXERCISE

fact,

E

such t h a t

S(t)

it i s enough t o show

S(t) = S(t)

i s s t r o n g l y almost p e r i o d i c

of almost periodic f u n c t i o n i n

t

2

0

i s an

i s t h e same a s i n

for

t

2

0,

(The d e f i n i t i o n -m


m

;

see BAFT-GOLDBEFG [1978:1] ). A

EXERCISE Eanach space

7.

Let

S(t)

E.

For

-W

be a s t r o n g l y almost p e r i o d i c group i n a

< A

C

m

d e f i n e a bounded o p e r a t o r

MA i n

E

by

(8.8) ( a ) Show t h a t

[MA;

projections, that i s

-a

-z h <

m)

i s a f a m i l y of mutually orthogonal

I N HILBERT SPACE

162

(b)

Show t h a t

,

MAS(t) = S(t)MA = eiAtS(t)

(8.10)

and t h a t

E A = MAE

(8.11)

D(A),

with AMA= i h M where

i s t h e i n f i n i t e s i m a l genera.tor of

A

more p r e c i s e l y , f o r each

u

E

E

{'I,) i s

MAu # 0 periodic function t h a t if pole a t

A's

f o r which

use t h e approximation theorem f o r t h e E-valued almost

S(t)u;

see AMEHIO-PROUSE [l9v:1,p .

lo, t h e n

has an isolated eigenvdue

A

E;

{ank) such t h a t

t h e ( f i n i t e or c o u n t a b l e ) sequence o f

(Hint:

Show t h a t t h e

i s dense i n

m)

t h e r e e x i s t s a sequence of i n t e g e r s

and a double sequence of complex numbers

where

(c)

S(t").

EA(-m < A

subspace generated by t h e eigenspaces

{r,)

(8.12)

A'

241).

R(A;A)

Show

(d)

h a s a simple

AO *

EXERCISE

8.

S t a t e a n d prove a. s u i t a b l e converse of Exercise

t h a t i s , g i v e s u f f i c i e n t c o n d i t i o n s on an i n f i n i t e s i m a l g e n e r a t o r

7; A

to

generate a s t r o n g l y almost p e r i o d i c group. EXERCISE

9.

Let

S(t)

be a s t r o n g l y p e r i o d i c semigroup ( i . e . a

i s p e r i o d i c f o r each

s t r o n g l y continuous semigroup such t h a t

S(;)u

t h e period depending i n g e n e r a l on

( a ) Show t h a t

u)

extended t o a s t r o n g l y continuous group

[1977:1]).

(See BART

Let

p

S(;)

periodic i n

A

{ikp; k =

..., -l,O,l,. ..

has a pole of order one.

-a

u,

can be

< t < a. S(<).

be t h e l e a s t p o s i t i v e period of

i s t h e i n f i n i t e s i m a l g e n e r a t o r of

Then i f the set

R(A;A)

(b)

S(i)

1

with each

(c)

S ( i ) , a(A)

ikp

i s conta.ined i n

a n eigenvalue where

Show t h a t ( c ) i n Exercise

7 can

163

I N HILEERT SPACES

be strenghtened t o m

u

(8.14)

Mkpu

=

k=-m

u

if

E

D(A).

If

u

(8.14) h o l d s i n Cgsaro mean,

E,

E

(8.15) or i n Abel mean,

EXERCISE 1 0 .

S t a t e and prove a s u i t a b l e converse of h e r c i s e

t h a t i s , g i v e s u f f i c i e n t c o n d i t i o n s on a n i n f i n i t e s i m a l g e n e r a t o r

9; A

t o g e n e r a t e a ( s t r o n g l y ) p e r i o d i c group. EXERCISE 11. Using EXercise 111.3 show t h a t (8.14) h o l d s i f

u

F

-

D( ( I A)ru),

(Y

> 0;

see $111.2 f o r d e f i n i t i o n s and p r o p e r t i e s o f

a s shown i n E x e r c i s e 111.7, i f

f r a c t i o n a l powers ( H i n t : then

S(i)u

i s gdlder continuous with exponent

Dini-Lipschitz theorem (ZYGMUID

w;

[1959:1, p , 63 I),

valued f u n c t i o n s , for t h e Fourier s e r i e s o f

u E D ( ( I - A)ru)

use t h e n t h e

generalized t o vector

S(i)u).

The following e x e r c i s e s are analogues of E x e r c i s e s 7, 8, 9, 1 0 and 11 f o r c o s i n e f u n c t i o n s . EXERCISE 12. i n a Banach space

N

@(:)

Let E.

For

- lim A - T - m T =

Show t h a t {Mh;A

5

A <

m

d e f i n e a bounded o p e r a t o r

F T c o s At(@(s)u,v)dt 0

l i m gIIeiAt(c(t)u,v)at

T+ (a)

be a s t r o n g l y almost p e r i o d i c c o s i n e f u n c t i o n 0

2

=

.

m

01

i s a family of mutually orthogonal p r o j e c t i o n s ,

where

M~ (b)

=

NA f o r

A > 0 , Mo =

&/2a0

Show t h a t M,,C(t) = @ ( t ) M A

and t h a t

Nhu,

=

cos h t @ ( t )

.

I N HILBERT SPACE

164

EA

=

(8.18)

MAE _C D ( A ) ,

with AM

- -A A -

2

MA,

(8.19)

A i s t h e i n f i n i t e s i m a l g e n e r a t o r of S(:). ( c ) Show t h a t ( c ) 7 h o l d s for t h e eigenspaces EA; i n p a r t i c u l a r , t h e r e p r e s e n t a t i o n (8.13) i s v a l i d f o r each u E E.

where

of a t e r c i s e

EXERCISE

13.

S t a t e and prove a s u i t a b l e converse of Exercise 1 2 .

EXERCISE

14.

Let

h

@ ( t )be a s t r o n g l y p e r i o d i c cosine f u n c t i o n

(defined i n t h e same way as for semigroups i n Exercise

(a)

@(;)

Show t h a t

is periodic i n

@(t).

l e a s t p o s i t i v e period of g e n e r a t o r of with each

@(;),

-(kpj2

order one. ( c )


-m

Then, i f

A

m.

(b)

an eigenvalue of

Show t h a t ( c )

u =

A

where

Let

p

be t h e

i s the infinitesimal 2 { - ( k p ) ; k = 0,1,.

i s contained i n t h e s e t

a(A)

9).

R(?\;A)

.. 3

h a s a p o l e of

i n Exercise 1 2 can be strengthened t o

TM

2u

(8.20)

k=O - ( k p ) if u E D(A)

u

( o r , more g e n e r a l l y , i f

Exercise 111.5).

For

u

E

E,

(8.20)

E

D((I-A)")

f o r some

a > 0 ; cf.

holds i n Cgsaro mean,

o r i n Abel mean

u EXERCISE

=

-

m

l i m (1 A) F A ~ M 2u h-. 1k=O -(kp)

.

15. S t a t e and prove a s u i t a b l e converse

(8.22) of Exercise

14.

165

CHAPTER V I

THE PARAEOLIC SINGULAR PERTUREATION PROELEM

Vibrations of a membrane

4VI.l

i n a viscous medium.

r

Consider a uniform membrane fixed t o t h e boundary

of a two

R and immersed i n a viscous medium.

dimensional domain

The small

o s c i l a t i o n s of t h e membrane a r e described by t h e equation pvtt where brane

+ yt

UAV

=

(X

E

a) ,

(1.1)

i s t h e v e r t i c a l movement of t h e mem-

v = v ( x , t ) = v(xl,x2,t) and t h e c o n s t a n t s p,y,a

are, r e s p e c t i v e l y , t h e mass d e n s i t y

p e r u n i t a r e a of t h e membrane, t h e c o e f f i c i e n t of v i s c o s i t y of t h e medium and t h e t e n s i o n of t h e membrane.

v satisfies

The displacement

t h e boundary c o n d i t i o n

o

v ( x , t )=

r) ,

(.

(1.2)

and t h e i n i t i a l c o n d i t i o n s v(x,O) If we d e f i n e

=

v ( x , t ) = u(x, (u/y)t)

and t h e boundary c o n d i t i o n ( 1 . 2 ) ; u(x,0)

(~p)'/~/y

Setting

=

0

u

R)

.

(1.3)

s a t i s f i e s t h e equation

=

-Y 0u

JX)

(x

E

0)

.

(1.5)

w e can w r i t e (1.5) i n t h e form

= F

E U E

then

E

the i n i t i a l conditions are

u ( x ) , ut(x,O)

2

where

(x

u 0 (x),vt(x,O) = u,(x)

tt

+ u

t

= a u

(.

E

n>

9

w i l l be small i f the medium i s h i g h l y viscous

can write t h e second i n i t i a l c o n d i t i o n i n t h e form ut(x,O) = c-1(;)1k1(x)

.

(1.6) ( y >> 1). W e

166

PARABOLIC SINGULAR PERTWATION

This suggests t h e problem of studying t h e behavior of t h e s o l u t i o n of

(1.6) on

as

-

E

allowing f o r dependence of t h e i n i t i a l c o n d i t i o n s

0,

This w i l l be done i n t h e r e s t of t h e c h a p t e r for a n a b s t r a c t

E.

model encompassing e q u a t i o n s l i k e

E x p l i c i t s o l u t i o n of t h e perturbed

Singular perturbation.

sV1.2

(1.6).

equation.

w i l l be t h e i n f i n i t e s i m a l g e n e r a t o r

Throughout t h i s c h a p t e r A

c(:)

of a s t r o n g l y continuous c o s i n e f u n c t i o n space

(that is,

E

A

E

8;

i n t h e complex Banach

see: Exercise 2 ) .

We c o n s i d e r t h e

Cauchy problems

2

E U"(t;E)

+

=

U(0;E)

+

= AU(t;E)

U'(t;E)

f(t;E)

3 o),

(t

(2.1)

u

0

( t ) ,

U'(0,E)

= U1(E),

and

u'(t)

-

=

Au(t)

+

(t

f(t)

-

3 0),

~ ( 0 =) u 0'

-

(2.2)

Roughly speaking, t h e r e s u l t s i n t h e f o l l o w i n g s e c t i o n s e s t a b l i s h that

if

u(t;E)

u(t)

as

E

0

if

uO(E)

i s suitably restricted.

ul(E)

u,

f(t;E)

f(t)

and

R e s u l t s o f t h e same t y p e hold

f o r derivatives.

A s a f i r s t s t e p , we compute t h e e x p l i c i t s o l u t i o n of ( 2 . 1 ) using simple changes of dependent and independent v a r i a b l e . Writing -t/2&2 t/2E U(t;E) = e v(t/E;E) ( o r , equivalently, v(t;E) = e u(Et;E)) we e a s i l y show that

i s a s o l u t i o n of t h e i n i t i a l v d u e

v(t;E)

problem v(t;&) + e

V(O,E)

= UO(E),

Conversely, e v e r y s o l u t i o n u(;;E)

=

V'(0,E)

t/2E

1 U (E) 2E 0

f(&t;&),

(2.3)

+

EU1(E).

v ( t ^ ; ~ ) of ( 2 . 3 ) g i v e s r i s e t o a s o l u t i o n

of t h e i n i t i a l v a l u e proklem ( 2 . 1 ) .

The ( g e n e r a l i z e d )

s o l u t i o n of t h e i n i t i d value problem (2.3) i s g i v e n by V(t;E)

=

C(t;E)V(O;E)

+

s(t;E)v'(O;&)

+,JtS(t

-

s ; E)es/2E f ( E s ; E )

ds,

(2.4)

PARAEOLIC SINGULAR PERTURBATION

c(:p

where

S(t;E)u =

i s t h e cosine f u n c t i o n generated by

167

A + ( 2 ~ ) - ~and 1

W e know from Lemma 111.4.1 t h a t t h e

C(s;E)u d s .

Cauchy prtbpem f o r

However, t h e s e r i e s r e p r e s e n t a t i o n (111.4.7) i s not

i s well posed.

convenient here and we provide a d i f f e r e n t r e p r e s e n t a t i o n below which, A

i n c i d e n t a l l y , proves a f r e s h t h a t LEMMA 2.1.

A

kt

8, t h a t

E

be well posed, and l e t

b

(2E)-21

E

8.

i s , l e t t h e Cauchy problem f o r =

U"(t)

+

Au(t)

(2.6)

be an a r b i t r a r y complex number.

Then

t h e Cauchy problem f o r U"(t)

i s well posed; t h e propagators

=

(A

2

+

b I)u(t)

cb(;),

S b ( t ) of (2.6)

by Cb(t)u =

c(t)U

d t Il(b(t2

+

(2.7)

bt

(t2

-

-

('1

a r e given

s~)'/~) C(s)u d s , ( 2 . 8 )

s2)1/2

and

where

Io,

5

a r e t h e Bessel f u n c t i o n s defined by (2.10)

The proof can be c a r r i e d out i n ( a t l e a s t ) t h r e e ways, all of which we sketch b r i e f l y below.

In t h e f i r s t , one t a k e s advantage of

t h e d i f f e r e n t i a l equation

(2.11) s a t i s f i e d by if

u, v

then

E

u(f)

Iv(x)

D(A)

and of a l i t t l e i n t e g r a t i o n by p a r t s t o show t h a t

(so that

= cb(;)u

c(t*)u, S(t*)u a r e genuine s o l u t i o n s of ( 2 . 6 ) ) i s a genuine s o l u t i o n of (2.7): uniqueness

+ %(t*)u

of s o l u t i o n s of ( 2 . 7 ) i s proved transforming them back i n t o s o l u t i o n s of

(2.6) by means of formulas (2.8) and (2.9) with b replaced by i b .

168

PARAEOLIC SINGULAR PERTUREATION

For d e t a i l s on t h i s l i n e of approzch (&though t h e context t h e r e i s somewhat d i f f e r e n t ) see SOVA

[1970:4].

The second uses Theorem 11.2.3; w e

show t h a t t h e Laplace transform of C b ( c ) u given by ( 2 . 7 ) equdls 2 2 2 The t h i r d i s based on Lemma 111.4.1, h R ( A ; A + b I ) u = AR(A2 - b ;A)u.

en i n (111.4.7) combined

p l i c i t c o n s t r u c t i o n of each of t h e terms s e r i e s r e p r e s e n t a t i o n (2.10) f o r

I o ( x ) , I1(x).

ex-

with the

We omit t h e d e t a i l s .

(2.8), (2.9) t o t h e equation

Applying t h e t r a n s l a t i o n formulas (2.5) we o b t a i n C(tjE)U

+

= C(t)U

and

=L t

S(t;E)u

-

l o ( ( t 2 s')'/~/~E)C(S)U d s

both v a l i d f o r a r b i t r a r y

u

and

t.

c ( s ) u ds

(2.12)

,

(2.13)

Using t h e s e formulas i n ( 2 . 4 ) and

making a n obvious change of v a r i a b l e i n t h e i t e r a t e d i n t e g r a l r e s u l t i n g from t h e i n t e g r a l i n ( 2 . 4 ) we o b t a i n t h e f i n a l formula for u ( c ; & ) , we have used u(t;&) = e

+

(11.4.3) f o r t h e s o l l i t i o n of t h e nonhomogeneous equation:

-t/2E

2 ( t /& )uo ( E 1

-

t e - t / 2 E 2 ~ t / &T ( ( ( t / E ) 2 E2

((t/E)*

2 = e-t/2E

where

C(t/&)u,,(E)

-

s)ll2/2E)

c(s)(, u O ( E ) ) d s S2)1/2

1

+

cp(t,s;&)C(s)(~ uO(E)) d s ' 0

2 = e-t/2E

C(t/E)Uo(&)

+

R(t;E)($

U,(E))

+

qt;E)($

U O(E)

+

E2U,(&))

(2.14)

169

PARABOLIC SINGULAR PERTURBATION

where t h e d e f i n i t i o n s o f t h e s c a l a r f u n c t i o n s

Cp(t,s;&),

qt;&)a r e

R(t;&),

t h e o p e r a t o r valued f u n c t i o n s t h e formula (3).

$(t,s;E)

and

e a s i l y read off

The f o l l o w i n g r e s u l t i s a n immediate consequence of formula (2.14) and t h e correspondence between s o l u t i o n s of ( 2 . 1 ) and t h o s e of (2.3):

LEMMA 2.2.

Every g e n e r a l i z e d s o l u t i o n

t 10.

continuous i n

f u

0

then

(E) = O

of

u(t;E)

(1.1)

u ( t ; & ) i s continuously

differentiable. precisely, i t s

We examine now t h e i n i t i a l value problem (2.2),

r e l a t i o n with t h e second order i n i t i a l value problem f o r (2.6).

Let A

THEOREM 2.3.

(2.6) be w e l l posed.

8,

E

t h a t is, l e t t h e Cauchy problem f o r

Then t h e i n i t i a l value problem f o r

u ' ( t ) = Au(t)

i s w e l l posed i n t

2

(2.15)

t h e propagator of (2.15)

0,

g i v e n by t h e

a b s t r a c t W e i e r s t r a s s formula

S(t)u = The propagator in -

if

5 >

-

Re A

0;

m

2

e-'

/4tC(s)u

more p r e c i s e l y ,

then

A

E

G(rp)

(t > 0 ) .

ds

c a n be extended t o a ( E )

S(<)

8(c0;w)

E

,(d *L

for

- vdued 0

< ep <

(2.16)

analytic function Finally,

7/2.

<(c0;" 2 1.

A E

The proof of Theorem 2.3 (which was proposed as Exercise 11 i n

-

Chapter 11) goes as f o l l o w s .

A E g(Co;w)

l/c(t)ll Thus, i f

S(i)

5

Co cosh CLTt

(-rn


(2.17)

a).

i s t h e ( E ) - v a l u e d f u n c t i o n defined by (2.16) we have

We check e a s i l y t h a t

t > 0

Recall t h a t , by d e f i n i t i o n ,

i f and only i f

S(i)

i s continuous i n t h e norm of

due t o t h e fast t e l e s c o p i n g of t h e integrand as

(E) s

-+

s(0) = I , we check next t h a t

S(t)

only c o n t i n u i t y a t

t

t h at , i f

p

> 0,

=

must be proved.

Defining (2.191

i s s t r o n g l y continuous i n t >_ 0; 0

for

m.

trivially,

To do t h i s we note f i r s t

170

PARABOLIC SINGULAR PERTUREATI O N

m

e - U 2 cosh 2 ~ t ' / ~ udu

-

t

which e x p r e s s i o n t e n d s t o zero a s

For

O+.

,

(2.20)

t > 0

we have

The argument below is standard i n approximation theory.

In

0

5

s

Ilc(s)u

- uII

taking

0

Once

d i v i d e t h e i n t e r v a l of i n t e g r a t i o n i n (2.21) a t

given, we

5

w e use ( 2 . 1 8 ) , t a k i n g

p

C &/2

there; i n

<_ t <_ 6,

This shows t h a t

6

llS(t)u

s

2

p

p

we e s t i m a t e by

for 0 <_ t 5 6 ,

E

2C0

is

t i m e s (2.20)

so t h a t

d e f i n i t i o n completed by (2.19), i s s t r o n g l y continuous i n 2 Using t h e e q u a l i t y i n (2.18) we o b t a i n Let A > w

E/4C0.

its

S(i), t

.

--

> 0

so s m a l l t h a t we shall have

so small t h a t ( 2 . 2 0 ) does not surpass

- ull 5

E

s = p.

1

2

0.

(2.22)

2 ' A- w

By T o n e l l i ' s theorem, convergence of t h e i n t e g r a l (2.22) i m p l i e s t h a t t h e order of i n t e g r a t i o n can be i n v e r t e d .

This j u s t i f i e s t h e following

computation:

R(A;A)u

=

( A > w2)

by t h e f i r s t e q u d i t y (2.11).

S(f ) A.

Re

We apply t h e n Theorem 2.3 t o deduce t h a t

i s a s t r o n g l y continuous semigroup with i n f i n i t e s i m d g e n e r a t o r That S(i) can be extended t o a ( E ) - v a l u e d f u n c t i o n a n a l y t i c i n

5 > 0,

G(m)

(2.23)

a s well as t h e e s t i m a t e s corresponding t o t h e c l a s s e s

can be proved r e p l a c i n g

we have

t

by

6

i n (2.23).

If

Re

6>

0

171

PARABOLIC SINGULAR PERTUREATION

ds

(2.24)

which implies t h e d e s i r e d e s t i m a t e s .

BVI.3

The homogeneous equation:

convergence of

W e i n v e s t i g a t e here t h e convergence of *

E

-+

0

we o b t a i n uniform bounds on that

cp,$

2

/lu(t;E)/I

where

f(t)

i n t h e homogeneous case

C0

0

u(<;&).

u(;;E).

to

u(t^;E)

* = f(t;E)

=

u(:)

as

A s a f i r s t step

0.

Using formula ( 2 . 1 4 ) and noting

we o b t a i n from ( 2 . 1 7 ) t h a t -t/2E2

ICoe

cosh ( Wt/E

h0( E 11

i s t h e constant i n (2.17).

Since

i s a cosine

cosh

f u n c t i o n ( i n t h e sense o f t h e preceding s e c t i o n ) with i n f i n i t e s i m a l 2 generator w t h e f u n c t i o n on t h e r i g h t hand s i d e of (3.1) equals

,

COt(t;E),

where

5

i s t h e s o l u t i o n o f t h e s c a l a r i n i t i a l value

problem 2

E %"(t;E)

+

s'(t;E)

=

2

W

,

5(t;E)

(3.2) S(0,E)

with

So(&)

explicitly.

=

Iluo(&)II, f

Let

= %()(E),

El(&)

A (E) =

S'(O,E)

= t1;1(E),

Ilu,(~)~~W . e compute t h i s f u n c t i o n (-1 (1 + ~ w * E ~ ) ' / ~ ) / ~ E * be t h e two ( r e d . , =

d i f f e r e n t ) roots of t h e c h a r a c t e r i s t i c equation

2 2

E

h

+

A-

2

W

= 0.

1 72

PARABOLIC SINGULAR PERTUBAT I ON (1+

4w2&2)1/2

other hand,, k(E)eA+('.)t

Am(&)

Since

< 1 + 2w2E2 we have C 0,

0

t h u s we may r e p l a c e

< -

A+(O)

< w2.

On t h e

k(E)eh-(E)t

by

i n t h e f i r s t term of (3.3) f o r e s t i m a t i o n purposes.

t h e second, we simply d e l e t e

-e '-('It

noting t h a t

>?(E)

-

In

h-(E) =

(1 + 4 u ~ ~ & ~ )>~1/ /~ ~/ ~ &W .e~ o b t a i n i n t h i s way t h e following r e s u l t :

where

Co,w

a r e t h e c o n s t a n t s i n (2.17).

To study convergence of

u(t;E)

we s h a l l use t h e asymptotic

s e r i e s f o r t h e Bessel f u n c t i o n s ,

where

( s e e WATSON [1948:1, pp. 203 and 1981). The asymptotic s e r i e s ( 3 . 5 ) w i l l be used with

x =

2E

((:)*

-

both t o c a l c u l a t e limits (as i n (3.10)-(3.11),

(3.25)-(3.26),

(4.14)-

(4.15)) and f o r e s t i m a t i o n purposes (as i n (3.14)-(3.15), (3.27)-(3.28), (4.16)). The second a p p l i c a t i o n deserves some comment. Since t h e functions x = 0 , (3.5) w i l l not provide good bounds

t o be estimated are r e g u l a r a t

near zero. However, t h i s i s not very s i g n i f i c a n t , s i n c e t h e s e functions converge t o limits t h a t do have s i n g u l a r i t i e s a t

x = 0. To improve t h e

estimations, we s h a l l r e p l a c e remainders of t h e form

O(x%)

by

o ( ( x + a)%) which a r e r e g u l a r a t t h e o r i g i n . It i s p l a i n t h a t t h e observations above can be a p p l i e d a s w e l l t o t h e asymptotic s e r i e s obtained from t h e s e r i e s (3.5) f o r functions of t h e form

173

PARAEOLIC SINGULAR PERTUREATION

x-'Iu(x),

where

B

and

a r e a r b i t r a r y r e a l numbers. Likewise, we s h a l l

u

( 3 . 5 ) term by term t o provide asymptotic s e r i e s for d e r i v a -

differentiate

t i v e s of a r b i t r a r y o r d e r . This can be easily j u s t i f i e d . To t h i s end we We proceed t o t h e e s t i m a t i o n of u ( t ; & ) - u ( t ) . d i v i d e t h e i n t e r v a l of i n t e g r a t i o n i n ( 2 . 1 4 ) i n

0

5

5

s

s(E),

where t h e asymptotic development ( 2 . 5 ) c a n be u s e d , and t h e "small" interval

s(&)

5 t 5 t / & where

rougher bounds w i l l s u f f i c e .

The

d i v i s o r y p o i n t between t h e " i n n e r " and " o u t e r " i n t e g r a l s i s defined by

where

q

< 1/2 w i l l be s p e c i f i e d l a t e r .

The l e n g t h of t h e second

interval is

We use t h e asymptotic development ( 3 . 5 ) t o o r d e r

0

5

s

5

S(E):

* X

IJX)

=

(1 +

o($))

m

=

in

0

.

(3.9)

(2lrx)l

A f t e r d i v i d i n g by

x

and performing a few c a n c e l l a t i o n s we o b t a i n

where =

X(t,S;E)

To o b t a i n (3.11) from

Obviously,

(1.10)

-

- q ( F )2 ) -3/4(1 + o($)).

(3.9) we

T(x),

-1

m

=

0.

e x i s t s a constant

have used t h e i n e q u a l i t y

(3.11) i s uniform i n t h e i n t e r v a l

To o b t a i n a n e s t i m a t e f o r

x

Since C

(3.11)

(lrt)

cp

we use formula

xmlI (x) i s r e g u l a r f o r 1

such t h a t

0

5

s

5

s(E).

(3.5) f o r t h e f u n c t i o n x

=

0,

there

PARABOLIC SINGULAR PERTURBATION

174

where

x; t h e choice of 2 i n t h e denominator of increasing, since ( 2 + x) -3/2 ex

does n o t depend on

C

(3.13) makes t h e f b n c t i o n (a

+

x)"

ex

is increasing f o r

l a t i o n s leading t o (3.10)-(3.11)

holds i n

0

5 s 5 t,

and t h e c o n s t a n t

C

a>

CY. Essentially

t h e same manipu-

reveal that t h e e s t i m a t e

where

does not depend on

E,t.

I n e q u a l i t y (3.14)

l e a d s t o t h e estimates below.

holds, where

i s a constant t h a t does not depend on

C

s,t,E.

We make use of (3.14)-(3.15) observing t h a t 2 1/2 2 (Es/t) ) 5 1 ( E s / t ) /2 i n t h e exponent and t h a t (Es/t) 2 ) 1/2 >_ (1 (Es(E)/t)')1/2 = 2~ i n t h e denominator of

Proof.

(1 (1

-

-

(2.14);

t h e term

-

2 4E /t

i n s i d e t h e p a r e n t h e s i s i s p o s i t i v e and can

be dropped.

where t h e c o n s t a n t

C

does not depend of

Proof. We use a g a i n (3.14)-(:,.15)

hand s i d e of t h e i n e q u a l i t y

s,t,E.

keeping i n mind t h a t t h e r i g h t

(3.13) is a n increasing f i n c t i o n of

x.

Accordingly, we can e s t i m a t e t h e r i g h t hand s i d e of (3.14) by t h e value obtained i n s e r t i n g t h e highest p o s s i b l e value of (which i s t h e summand

(1

-

2

(Es(E)/t)2)1/2

4E /t

= 2q).

(1

-

(Es/t

2 1/2

)

Once t h i s is done we d i s c a r d

i n t h e o u t e r p a r e n t h e s i s o f (3.15).

The r e s u l t

i s (3.17). A s a n immediate consequence of (3.17) and of the estimation (3.8)

PARABOLIC SINGULAR PERTURBATION f o r the length of the interval

s(E)

t h e r e and i n o t h e r i n e q u a l i t i e s

C

s ,t ,&

<_

175

5 t we o b t a i n

s

denotes a c o n s t a n t independent o f

not n e c e s s a r i l y t h e same i n d i f f e r e n t i n e q u a l i t i e s . Let

g(t;&)

> 0 . We s a y t h a t a family of f'unctions converges uniformly i n t > - t ( E ) t o a f u n c t i o n g(:) if and t(E)

> 0

f o r each

E

only if sup

1 h

-

llg(t;E)

g(t)lI = 0 .

Ed0 t)t(E)

I f t h e supremum i s t a k e n i n s a y that

t

2

t ( E ) <_ t 5 a

g ( ; ; ~ ) converges t o

a > 0

for

a r b i t r a r y we

uniformly on compacts of

g(:)

t(E). We prove below that f o r every

on compacts of

t ,t(E)

s u b s e t s of

as long as

E,

u

E

E, R(t;E)u-r S(t)u

uniformly w i t h r e s p e c t t o

t(E)/E2

4

(E

m

.+ 0 )

uniformly

u on bounded

.

(3.20)

I n f a c t , assume t h i s i s f a l s e . Then t h e r e e x i s t s a bounded sequence

[u,]

sequence

{t,]

For each

n

C

E,

a sequence 2

such that

we choose

tn/En

'n

-21-

{En]

*

with and

m

End

0

and a bounded

lIR(tn;En)un

- S(t,)unll

2 6 > 0.

such that 'in

-

WE

n

n --

(3.21)

n (note t h a t zero,

'n

-

n 1/2

< 1/2: moreover, s i n c e both as

n

+ m).

En

and

E n t n-1/2

tend t o

We d i v i d e t h e i n t e r v a l of i n t e g r a t i o n i n

(2.14) according t o t h e e q u a l i t y (3.7) with

q = q n'

We have

176

PARABOLIC SINGUMR PERTURBATION

The f i r s t i n t e g r a l tends t o zero as convergence theorem: that

(note that

1

-

(Es/t)2

uniform estimate

due t o the dominated

m

+

9

-

cp(t,,s;E,)

n

i n f a c t , t e asymptotic r e l a t i o n (3.10) shows e-'

/4tn-+ 0

2 211, -,

1. hence

(3.16).

as

-

n-,

Es/t

m

0)

for

s

fixed

and we have t h e

The second i n t e g r a l tends t o zero by (3.18).

A s f o r t h e t h i r d it i s e a s i l y seen t o telescope making the change of variable

ti1/'s

=

(J

tn a r e bounded. I n f a c t ,

and r e c a l l i n g t h a t t h e

Now, it follows from (3.8) and (3.20) that S(E

n

)

=

tn 2 1/2 (1 - 4q ) n n

t

2

s(En) >_ 2 t 1/4&-1/2 n

Thus

8n

-

m

(1

-

,3/4

2'in)l/2 2 2

as

n

-

m

n -

-

(3.24) El/2 n s o t h a t (3.23) tends

t o zero (we note t h a t i f

w = 0 t h e i n t e g r a l (3.23) tends t o zero 2 under the s o l e assumption that t,,/En -,m , where t h e tn may be

unbounded; t h i s f a c t bears on a resuLt below). a contradiction and j u s t i f i e d

OUT

claim about

We have then obtained sf.

We prove next t h e corresponding statement f o r

5(t;E).

The

estimates a r e obtained i n a s i m i l a r fashion, thus we only s t a t e the final results.

Formula (3,10)-(3.11) has t h e following counterpart:

with X(t,S;E) = (77t)-1/2(1 The estimate is uniform i n The inequality

holds i n

0 5 s 5 t, where

-

2 (y))-1/4(l +

0 5 s 5 s(E).

.

(I(%))(3.26)

PARABOLIC SINGULAR PERTURBATION

177

p(t,S;E) = t and t h e constant

(3.28)

does not depend on

C

we use t h e asymptotic formula (3.5) f o r

m = 0

E,t.

To o b t a i n (3.25)-(3.26)

m = 1; t h e same formula with

yields the inequality

Using t h e i n e q u a l i t y (3.26)-(3.27) we e a s i l y o b t a i n t h e following counterparts of L e m 3.2 and Lemma 3.3 :

holds, where the constant

where t h e constant Using

C

does not depend on

does not depend on

C

s,t,E.

s,t,E.

(3.31) and (3.8) we obtain

We prove t h a t

6(t;E)u uniformly i n of

t >_

S( t ) u

uniformly with respect t o

t(E)

i n e x a c t l y t h e same way used f o r

E

(3.33)

R;

u

i n bounded s e t s

d e t a i l s a r e omitted.

A f t e r a n elementary estimation of t h e f i r s t term i n (2.5) t h e proof of t h e following r e s u l t i s complete:

Let

THEOREM 3.6. UO(E)

and l e t

u(t;E)

-b

v,

uO(E),ul(E) 2

E UJE)

E

E

-. uo

be such that

- v

(E

+

0)

be the generalized s o l u t i o n of ( 3 . l ) ,

number such t h a t (3.20) holds.

,

(3.34) t(E) > 0

Then

U(itjE) -. u(Z)

(3.35)

178

PARABOLIC S INGUMR PERTURBATION

unifwmly i n compacts of s o l u t i o n of ( 3 . 2 ) w i t h respect t o

uo,v

REMARK 3.7.

if.

2

t

t(E)

u(%)

where

u(0) = uo.

i s t h e generalized

The convergence i s uniform with

(Iuo(I, /(v(I a r e bounded.

does not converge t o

uO(E)

thus t h e r e i s a "boundary l a y e r " near zero where approximation t o

t >_ 0 u

Obviously, uniform convergence i n

expected s i n c e i n g e n e r a l

u(^t)

cannot be

as

0

-

E

0,

i s not a good

u(;;&)

[1981:1] f o r a thorough

( s e e KEVORKIAN-COLE

treatment of t h e one dimensional case). Note a l s o t h a t m i f o r m -opt -w2t e u(t;E) t o e u ( t ) i n t >_ t ( E ) cannot be

convergence of

assured even i n t h e s c a l a r case.

To s e e t h i s , l e t

s o l u t i o n of t h e i n i t i a l value problem

-w

< w2

since ?(&),A*(&) e-',2tew2t = 1,

we have

e

L3t -2 1 with

<'(O;E)

-, 0

<(t;E)

be t h e

<(i;E)

as

t

Then,

= 0. +

the s o l u t i o n of t h e same equation with

=,

whereas

E =

0.

However, a n examination of t h e proof of Theorem 3.2 shows t h a t u(%) uniformly i n

u(i;E)

t

t(E)

i n t h e p a r t i c u l a r case

( i n f a c t , t h e r e i s no need t o assume t h e sequence

w = 0

{tn] bounded).

We

state this special result. THEOREM a d d i t i o n that

3.8.

w = 0

&

(2.17) ( i . e . assume that

uniformly bounded cosine function

i s uniform i n REMARK

u(t;E)

-*

t >_

3.9.

u(t)

3.6

Let t h e assumptions of Theorem

C(1)).

A

hold.

Assume i n

generates a

Then t h e convergence i n

(3.35)

t(E).

Condition (3.20) i s necessary i n order that

uniformly i n

t2

a t l e a s t without a d d i t i o n a l

t(E),

assumptions on t h e i n i t i a l d a t a ( s e e 4VI.5).

This can be seen a s

Consider f i r s t t h e one dimensional c a s e (3.2) with, say, -2 = 0 , <'(O,E) = E For each E > 0 l e t t ( E ) be such that 2 -, a , where 0 < a < m . Then we seE from (3.3) that

follows. <(O,E) t(E)/E

.

t(t(E),E)

+ .-

1 - e-a,

although

<(t;E)

+

ew

i n t h e sense of Theorem

However, uniform convergence i n t 2 0 can be obtained i f we 2 assume that E - <'(O,E) 4 0. A s we shall s e e i n 4VI.5 t h i s r e s u l t has

3.6.

an i n f i n i t e dimensional counterpart (Theorem following example (where uniform i n

lluoll 5 1

t 2 0,

u1( E ) = 0 )

5.5).

However, t h e

shows t h a t convergence, although

may not be uniform i n

u = l i m uO(E) f o r 0

i f condition (3.20) i s violated.

EXAMPLE 3.10.

Consider t h e space

E = R2

of a l l complex sequences

179

PARABOLIC SINGULAR PER!RJREATION

u = [un ] = [un;n >_ 1) w i t h lluIl2=

We check that

IlC~,31l

2 =

c bnI2<

i s a H i l b e r t space.

E

*

The operator

2 A u = A [ u ] = { - n u,] n

i s defined by

A

,

(3.36)

and generates t h e s t r o n g l y continuous, uniformly bounded cosine function c ( t ) { u n ] = {(cos n t ) u n ] . The (generalized) s o l u t i o n of t h e i n i t i a l value problem (2.1) f o r w i t h i n i t i a l conditions U(t;E)

{Un(t;E)]

=

uO(E) =

{uOn(E)] and

(E)

1

= {u

In( E ) ]

Qin, Yin

+

Yin(t;E)(E

defined as i n (3.3) f o r

2

0 < b

(3.37)

Uln(E)),

w = in,

mandatory modifications when t h e two r o o t s ,)&(:A An(&) 2 2 2 c h a r a c t e r i s t i c polynomial E A + A + n = 0 . coalesce. Let

is

with

n ( t ; E ) = Qin(t;E)Uon(E) the finctions

u

A

with the

of t h e

< 1. Take (3.38)

= b/2m.

Em

Then

-

= (-1 -)/2&* Am(&) m

h+(E) m = (-1+ -)/2E2

m'

We solve t h e i n i t i a l value problem (2.1) f o r

A

with

m' u

= 0,

(E)

1 u ( E ) = u = {u ] independent of E . Note that t h e s o l u t i o n of (2.2), 0 n that i s , t h e semigroup generated by A , i s s ( t ) E u n l = CeLet

{tm ]

n2t

(3.39)

Unl

be a sequence of p o s i t i v e numbers such that

2

tm/Em

Then, passing i f necessary t o a subsequence w e m y assume t h a t 2

tm/Em

We have

u (tm;Em) = Q. ( t ;e )u n in m m

+

a >_

o

.

f.

m.

180

PARABOLIC SINGULAR PERTURBATION

= -1k (1 - b2)1/2,

where y’

convergence f o r if

llull

which inequality precludes uniform

bounded i f

i s adequately chosen, a t l e a s t

b

a f 0.

We note the following reformulation of Theorem 3.6: THEOREM

3.11.

Let t(E) > R(t;E)

uniformly on cmpacts of

-

S ( t ) , eJ(t;E)

t >_ t(E)

(2.17) (that is, i f

w = 0

be such that (3.20) holds.

0

A

(3.40)

S(t)

4

i n the topology of ( E ) .

Convergence of

u ’ (t;E)

If.

generates a uniformly bounded cosine

function C(%)) t h e convergence i n (3.40) i s uniform i n

QVI.4

Then

t >_ t(E).

and higher derivatives.

A s pointed out i n QVI.2, formula (2.14) does not necessarily

provide a genuine s o l u t i o n of (2.1): a r e arbitrary,

u

(E)

0 u(Z;E)

u(%;E)

i n fact, i f

uo(E)

may not even be d i f f e r e n t i a b l e .

and

u~(E)

However, if

it follows from (2.4) (or d i r e c t l y from (2.14)) that

= 0

i s continuously d i f f e r e n t i a b l e :

noting that

we obtain

I;)(x) = I1(x)

0

-t/2EC

U’(tjE) = W(t;E)(E 2u

(E)) =

1

2 e C(t/E)(E2U1(E)) E

-

%s,t” ZE

$(t,s;E)C(s)(E2u1(E))

ds =

181

PARABOLIC SINGULAR PERTURBElTION

-2 where t h e obvious cancelling out of E2 and E i s t o be frowned 2 upon, s i n c e E u (E) not u (E) w i l l tend t o a l i m i t l a t e r . I n view 1 1 of (3.34) t h e l a s t two terms of (4.1) a r e i n d i v i d u a l l y divergent as E

-

thus they w i l l have t o be j o i n t l y estimated.

0

i n t e g r a l s defining q < 1/2

with

R and B a t

s(E)

W e divide t h e

as i n t h e previous s e c t i o n

t o be f i x e d l a t e r . The outer i n t e g r a l s a r e estimated

i n d i v i d u a l l y using Lemma 3.3 and Lemma 3.5. We have

by v i r t u e of (3.17) and of t h e bound (3.8) f o r t h e l e n g t h of t h e i n t e r v a l of i n t e g r a t i o n :

i n t h e same way, but t h i s time using (3.31)

we obtain 1

t/E

2JE ji ( t) > s; E l IlC(S>~IIds 5c We t a k e

q < 1/2

+ II~II(') -.{3/2

3/2

f i x e d i n both i n e q u a l i t i e s .

combined i n t e g r a l i n

0 5 s 5 s(E)

t h e asymptotic s e r i e s (3.5).

[0,11= r'(3/2)/r(-l./2) [1,1]= T(5/2)/T(l/2) = 3/4, thus

It f o l l o w s from (4.5) t h a t Cp(t,s;E) = X ( t , s ; E )

7

- WE

)I -

exp

(4.3)

Estimation of the

w i l l r e q u i r e an a d d i t i o n a l term i n

Taking now m = 1 we obtain

noting that

with

(i-

=

-1/4. On

t h e other hand,

182

PAWBOLIC SINGULAR PERTURBATION

- 3 p 72 r -1/pt-3/2

(1

- (Y) ) 2

-5/4 (1

+

0

($));

(4.7)

On t h e o t h e r hand.,

with = (Tt)

X(t,SjE)

+ - -1 E , l T2

-1/2t-3/2

(4.9)

4 where due a t t e n t i o n has b e e n p a i d t o (3.l2) i n t h e r a n g e

0

5

s

5

t(E).

Formulas ( 4 . 6 ) and (4.7) w i l l be m o d i f i e d a s f o l l o w s . We u s e T a y l o r ' s 1 i n t h e f i r s t p a r e n t h e s i s of ( 4 . 7 ) :

formula of order

2

=

1+

t(7)

(1 +

o((g))

(4.10)

I n t h e second p a r e n t h e s i s it i s enough t o u s e T a y l o r ' s formula o n l y up . t o order zero:

(1 -

(7y4 =

1+

"r);

2

)

(4.11)

.The r e s u l t i s

The t r e a t m e n t of (4.9) i s similar: t h e a n a l o g u e s o f ( 4 . 1 0 ) a n d (4.11) a r e

il and

-

(gy4 1 (q(1o((y)2)) =

1+

4

t

+

183

PARABOLIC S I N G U I A R PERTURBATION

thus t h e f u n c t i o n y,

i n ( 4 . 9 ) can be expressed as follows:

(4.13) We combine now (4.6)-(4.7)

with ( 4 . 8 ) - ( 4 . 9 ) , making use of t h e asymptotic

expressions (4.l-2) and (4.13): t h e r e s u l t i s

with

The next t a s k i s t o o b t a i n an e s t i m a t e f o r t h e f u n c t i o n i n (4.14)-( 4.15) which i s uniform w i t h r e s p e c t t o

in

s

0

<_ s <_ t ( & ) and which i s a t t h e

same t i m e independent o f € , t > 0. This i s e a s i l y done keeping i n mind t h e comments a f t e r ( 3 . 6 ) ; a l l we have t o do i s t o r e p l a c e t h e remainder

by 1

+ o((1 +

(which amounts t o r e p l a c i n g

$I1)

O ( X - ~ )b y O ( ( 1

+ x ) ~ i)n ( 3 . 5 ) ) and t h e n

note t h a t t h i s l a s t expression remains bounded for a l l values of

E,

t > 0.

The r e s u l t of t h e s e manipulations i s t h e bound E -2

in

0

IvJ(t,SjE)

-

Jr(t,SjE)I < - c(t-3/2 + s 2 t -5/2)e-s2/4t

5 s <- t ( E ) , where

C

does not depend on

we t a k e advantage of t h e i n e q u a l i t y

E

or t ; i n t h e exponent

(1 - (Es/t)2)1/2

T o use (4.16) i n t h e e s t i m a t i o n of

(4.16)

5 1- (E~/t)~/2.

(4.1) we t a k e p r o f i t of t h e

w e l l known formula. m e-'

2 /4tcosh

ws ds = ( n t )l/2e(u2t

(4.17)

184

PARABOLIC SIIKULAR PERTURBATION

(which w a s a l r e a d y used i n QVI.2) and of i t s d i f f e r e n t i a t e d version

La

2

s2e"

/4t cosh

ws ds

(4.18) both v a l i d f o r

t > 0.

To complete t h e e s t i m a t i o n of (4.1) it only remains t o dispose of t h e f i r s t ( n o n i n t e g r a l ) term.

if 0 <

This i s done observing that

< 1/h. v < 1/2 f i x e d i n ( 4 . l 2 ) , (4.13) and t h e i r consequences (4.14), (4.15) and (4.16) we o b t a i n from (4.17) and (4.18): E

Taking

THEOREM 4.1. and of

(0

E,t

<

There e x i s t s a constant E

< 1/h, t > 0 )

independent of

C

2

J I B ~ ( ~ ; E5) Ic(w* J

+

l/t)ew

The following r e s u l t e s t a b l i s h e s that of t h e d e r i v a t i v e THEOREM 4.2.

of t h e semigroup

S'(%)

Let

u

0

for a l l

= 0

(E)

w

2

0

such t h a t

.

(4.20)

(5' 1%;~) i s an approximation S(^t). E

> 0,

y(E)

E

E

such

that 2 E U p )

4

uo

(E

+

(4.21)

o),

u(t;E)

t h e generalized s o l u t i o n of (2.1), t ( E ) > 0

holds.

Then t u ' ( t jE)

uniformly on compacts of s o l u t i o n of (2.2) w i t h respect t o

uo

t

t(E)

u(0) = uo.

if 1 1 ~ ~ 1 1is

4

,

such t h a t (3.20)

t u ' (t)

where

(4.22)

u(t)

is t h e g e n e r a l i z e d

The convergence i s uniform with

bounded.

The proof i s e s s e n t i a l l y similar t o t h a t of Theorem 3.6; assuming

that (4.22) f a i l s t o hold t h e r e e x i s t s a sequence sequence

[tn] and a bounded sequence

{un}

in

{En],

E

a bounded

such that

185

PARABOLIC SINGULAR PERTURBATION

tn/En

2 4

m

and

- S ' (tn)unl/ 2 6 > 0 . We choose 'n An obvious d i f f e r e n t i a t i o n under the i n t e g r a l

tn//B'(tn;En)un

according t o (3.21). s i g n shows that

2

-s2/4t

C(s)u ds.

(4.23)

We have

where

The f i r s t term in (4.24) i s immediately seen t o tend t o zero using

I n t h e f i r s t i n t e g r a l we use the asymptotic development (4.14)-(4.15) t o show t h a t q ( t , s j E ) + 0 i n (2.17) f o r

0

5

s

5

liC(t)ull.

s(E)

n n and t h e estimate (4.16); t h e i n t e g r a l tends t o zero

then by t h e dominated convergence theorem.

The second and t h i r d

i n t e g r a l s a r e t a k e n c a r e of by (4.2) and (4.3) respectively.

Finally,

t h e f o u r t h i n t e g r a l i s seen t o tend t o zero by means of t h e change o f variables bounded:

tm1l2s = u, n in f a c t ,

keeping i n mind that t h e sequence 2 as

[tn] is

PARABOLIC SINGULAR PERTURBATION

186

ti1/2s(E ) -+ m ( s e e (3.24)). We obtain then a c o n t r a d i c t i o n n and thus c m p l e t e t h e proof of Theorem 4.2.

where

I n t h e case

tn a r e unbounded.

the

THEOREM 4.3.

2

(4.3) tends t o zero a s

= 0,

u)

tn/En+

even i f

m

Hence we may improve Theorem 4.2 as follows:

Let t h e assumptions of Theorem 4.2 hold.

i n a d d i t i o n that

u)

= 0

&

(2.17) ( t h a t i s , t h a t

uniformly bounded cosine function (4.22) i s uniform i n

C(%)).

Assume

generates a

A

Then t h e convergence i n

t >_ t(E).

Results of t h e type of Lemma 4 . 1 and Theorems 4.2 and 4.3 can

w e l i m i t ourselves t o

be obtained f o r higher d e r i v a t i v e s ;

i s not twice continuously d i f f e r e n t i a b l e f o r a l l

Since 6 ( t ; E ) u u

E

E,

6"(t;E).

however, t h e estimates w i l l r e f e r not t o B"(t;E)

i t s e l f but

to t h e "mollified" operator 6"(t;E)(E-1R(E-1;A)). THEOREM 4.4. There e x i s t s a constant C such that t , E ( t > 0 , 0 c E c 1/&)

independent of

u)

> 0

and of

5

116"(t;E)(E-1R(&-1;A))ll

C(w4

f

2 w 2 / t + l/t 2 ) eu) t

The proof i s straightforward b u t tedious. that

U E

D(A),

B"(t;E)u

s o that

.

(4.26)

Assume f o r t h e moment

e x i s t s ; a n e x p l i c i t formula f o r it

can be obtained from (4.1): -t/2&2 G,"(t;E)u =

E3

-t/2E2 C'(t/E)u

-

4

-t/2E C(t/E)U

2E

+

te

2 C(t/E)U

8E6

C(s)u ds

te

2 -t/2E f

t e 4E7

C(s)u ds 2

-t/E

Jo

Ii(((t/E)2

-

2 1/2

s )

/2&) C(s)u ds

Il(((t/E)2

-

S2)l/'/2E) C(s)u ds

PARABOLIC SINGULAR PERTURBATION

187

e 4E5

where terms a r e grouped t o g e t h e r as t h e y appear i n d i f f e r e n t i a t i n g

(4.1).

Note a l s o that t h e t h i r d and f o u r t h i n t e g r a l s a r e i n d i v i d u a l l y

divergent and must be combined i n t o one.

We t a k e a look f i r s t a t t h e

terms that l a y o u t s i d e of i n t e g r a l s . For t h e second we have

and t h e same estimate o b t a i n s for t h e t h i r d and t h e f o u r t h ,

SO

that

t h e y s a t i s f y (4.26) even without t h e i n t e r c e s s i o n of t h e mollifying operator then

E-1R(E-1;A).(4)

For t h e f i r s t term we note that i f

C(^t)v i s continuously d i f f e r e n t i a b l e w i t h

hence

C' ( t ) v = d(t)Av

Since

w

7

v

E

D(A)

C " ( t ) v = C(t)Av,

and we have

0, l/S(t)ii 5 C e x p ( ~ t ) ( ~ and ) t h e r i g h t hand s i d e of (4.29)

can be estimated i n t h e same way as (4.28). To e s t i m a t e t h e six i n t e g r a l s i n (4.27) we d i v i d e t h e domain of integration a t specified l a t e r .

s = s(E)

given by

(3.7), with q < 1/2 t o be

For t h e f i r s t o u t e r i n t e g r a l we t a k e advantage of

(3.17) f o r cp(t,s;E), divided by t E 2 ; f o r t h e i n t e r v a l of i n t e g r a t i o n we use (3.8). The r e s u l t is a bound of t h e form t h e estimate

The s e c o n d , f i f i h and s i x t h i n t e g r a l s a r e t r e a t e d i n t h e sane way: i n a l l c a s e s , due t o t h e a d d i t i o n a l f a c t o r e s t i m a t e of t h e form

t/E2

we end up with a n

188

PARABOLIC SINGULAR PERTURBATION

A s pointed out a f t e r (4.27) the t h i r d and f o u r t h i n t e g r a l s must be

combined i n t o one t o a m i d divergence a t

s = t/E

w r i t t e n s e p a r a t e l y only f o r typographical reasons).

( i n f a c t , they a r e The basis of the

r e s u l t i n g estimation w i l l be t h e asymptotic s e r i e s f o r the h n c t i o n obtained from (3.6): Q(x) = X-~(X-~I~(X))'

we deduce from it that

The combined integrand of the f o u r t h and f i f t h i n t e g r a l (including f a c t o r s outside of the i n t e g r a l ) i s

-

2 -t/2&2 t e Q ( ((t/E)2

16E~

-

~~)'/~/2E)C(s)u.

I n view of (3.30) we have

where p(t,S;E) = t

(3

(4.34)

is increasing w e can bound the r i g h t hand side of (4.34) by i t s value a t s = s ( E ) subsequently deleting the f a c t o r 6$/t from t h e outer parenthesis. The r e s u l t i s an upper bound f o r

Since

-t

x)-5/2ex

t h e combined integrand of the form

Therefore, the i n t e g r a l can be bounded by the following expression:

This completes the consideration of the outer i n t e g r a l s .

We look a t t h e inner i n t e g r a l s . t h e i n t e g r a l belaw:

We begin by grouping them i n t o

PARABOLIC SINGUL4R PERTURBATION

189

O(t,s;E)C(s)u ds.

Using t h e asymptotic developments (3.5) f o r

Io, I1 and Ii of

m = 1 i n t h e f i r s t and f o u r t h i n t e g r a l s and of order

order

i n t h e r e s t we o b t a i n f o r

B

m = 2

a n expression of t h e form

a l i n e a r combination of terms of t h e form

with X

with

(4.36)

j = 2,1,0,C

f o r each t e r m

expression f o r

J

>

We then use T a y l o r ' s formula of order 2 ) 'j, ending up w i t h t h e following

U . , ~ .0.

J

(1 - ( E s / t ) X:

o(&)) 2

where each

i s independent of

X,(t,s)

t > 0

cosine f u n c t i o n

and apply formula C(^s) = cos og,

(in fact,

E

(4.27) i n

where

t h e space

(4.39)

is a f i n i t e

X,

>_ 0 ) . We

s@t-* w i t h ct:B

l i n e a r c o m b i d t i o n of terms of t h e form then f i x

2j

E = C

t o the

i s a r e a l parameter.

0

Naturally, t h e r e s u l t must be t h e second d e r i v a t i v e of t h e s o l u t i o n of E 2t " ( t j E )

4-

< ' ( t ; E ) = w 2< ( t j E )

,

(4.40)

w i t h i n i t i a l conditions <(t,O)

= 0 , < ' ( t ; O ) = E -2

,

(4.41)

hence

< ( t F )= y 4s ( t F ) where

Yw(t;E)

Y

is t h e f b n c t i o n defined i n (3.3). <"(t;E)

-

Q

4

e

-0

2

We check e a s i l y t h a t

t

(4.42)

t > 0 i s being kept f i x e d , it follows from (4.30), (4.31) and (4.35) t h a t t h e o u t e r i n t e g r a l s tend t o zero as E -, 0; we o b t a i n

Since

t h e n making use of (4.28) and following comments t h a t lim E+O"O

O(t,s;E) cos

QS

ds =

0

4

ems2'.

(4.43)

190

PAFABOLIC SINGULAR PERTURBATION

Observe t h a t

We t r y next t o i d e n t i f y t h e d i f f e r e n t terms i n (4.39). E2/r,t

+

and t h a t

0

-, 0 i n t h e i n t e r v a l

Es/t

0 < s < t ( & )O .n t h e o t h e r

hand, on account of (4.37) and (4.39), of t h e form of each X

3’

estimating t h e exponent i n (4.37) i n t h e same way used t o o b t a i n (3.16) and making use of t h e dominated convergence theorem we deduce from (4.39) that

By uniqueness of Fourier cosine transforms we obtain that X . ( t , s ) = 0 J

2

-s

(&)

$qZe

2

/4t

me - m e 3

-s2/4t

-s2/4t

4

t

3 s2

-s2/4t

(4.45)

Replacing back i n (4.39), we see that t h e sum reduces t o a unique

The e s t k t e derived from (4.46) Lemma 4.4 s i n c e

E2/t

i s not good enough f o r t h e proof of

need not remain bounded.

To o b t a i n t h e

required bound we workwith t h e asymptotic developnent (3.5) minding t h e comments a f t e r (3.6) on obtention of bounds, as i n t h e argument leading t o (4.16).

The estimate obtained i s of t h e form

) X ( t , S , & ) ) <_ C(t’5l2 + s2 t -7/2 in

0

5

s

5

inequality

t(E),

where

(4.47)

does not depend on E or t. Using t h e 5 1 - (Es/t) 2/ 2 i n t h e exponent as we

C

(1 - (Es/t)2)1/2

d i d i n obtaining

s4t-9/2)

(4.16) we deduce t h a t 2

J@(t,S;E)I 5 C(t - 5 / 2 in

0 <_ s 5 t ( E ) , where

C

+

&-7/2

+

s4t-9/2)e-s

is independent of

E and

To complete the proof of Lemnla 4.4 w e use (4.18),

/4t

(4.48)

t. differentiated

191

PARABOUC SINGULAR PERTURBATION

once a g a i n : L m s 4 e - ' 2/4tcosh

cus d s = l 2 ~ r ~ / ~2 t +~ 4/ ~81/2t7/2w2ecu2t e~ ~ 2

+ 1 61/2t9/2w4ecu ~ t RF,MARK 4.5.

The c a s e

w = 0,

excluded from Lemma cu > 0

incorporated noting t h a t t h e hypotesis deduce that

that

118(t)ll

5

Ct,

If

w = 0

4.4 can be

was only used t o

which i n e q u a l i t y appears i n t h e

IlS(t)II 5 C exp cut,

estimation (4.29).

(4.49)

t h e n we can only a s s e r t i n g e n e r a l

thus t h e r i g h t - h a n d s i d e of (4.29) i s bounded

as f o l l o w s :

Accordingly, t h e following e s t i m a t e holds f o r

6"(t;E):

We can state u s i n g t h e preceding arguments a convergence r e s u l t for

-1

;A))

5,"(t;E)(E-&(E

of Theorem 4.2. result for

whose proof i s very much t h e same as t h a t

We i n c l u d e i n t h e s t a t e m e n t below t h e corresponding

6'(t;E)

which is nothing b u t a r e f o r m u l a t i o n of Theorem

k.2:

THEOREM 4.6.

Let

t(E)

> 0

t6'(t;E)

be such t h a t (3.20) holds.

-

tS'(t),

t*E'l(t;E) (E-$(E-l;A>)

uniformly on compacts o f

t >_

t(E)

4

Then (4.52)

t2S"( t )

,

(4.53)

i n t h e topology o f

(E).

Obviously, r e s u l t s of t h e type o f Lemma 4.4 and Theorem 4.6 can be obtained f o r d e r i v a t i v e s of any o r d e r o f

6.

We omit t h e d e t a i l s .

I n a s e n s e , t h e r e s u l t s above do not t e l l t h e whole s t o r y about

5" s i n c e t h e smoothing o p e r a t o r

E-lR(E-l;A)

only plays a r o l e i n

t h e f i r s t term on t h e r i g h t hand s i d e of (4.27); b e s i d e s , t h e f i r s t

t e r m a l s o makes it necessary to s e p a r a t e t h e c a s e t i o n purposes.

is :

A s t a t e m e n t on

@'(t;E)

u)

= 0

f o r estima-

t h a t avoid t h e s e inconvenients

192

PARABOLIC SINGULAR PERTURBATION

THEOREM 4.7 w

2

and of

0

IlG"(t;E)U

(a)

-

E

.There e x i s t s a constant

( t > 0, 0 <

t,E

-

- j e t/2E2

c ' ( t / E 1UII 2

5 c(w4 + a2/t + i/t )\lull (b)

-

E -3e-t/2E

.

(4.54)

D(A)

QVI.5

n a,

2 C'(t/E)u)

t2S"(t)u

(4.55)

t >_ t ( E ) uniformly w i t h r e s p e c t t o i s any bounded s u b s e t i n E.

uniformly on compacts of E

(u E D(A))

We have t2(6''(t;E)U

u

independent of

C

< 1/40) such t h a t

E

where CB

The homogeneous equation.

Rates of convergence.

We show i n t h i s s e c t i o n t h a t i f t h e r e i s no ''crossover" of i n i t i a l conditions ( i . e . i f we have uo,

uO(&)

r a t h e r than (3.34)) then

t

5

2 E

u1(&) -.

E

respect t o

u

D(A)

D(A)

(5.1) u(%)

u0

E

uniformly i n

D(A)

or t o c e r t a i n

I n contrast with the not be uniform w i t h

(Iu/(is bounded.

be t h e o p e r a t o r a c t i n g on t h e i n i t i a l c o n d i t i o n

i n (2.14), i . e . &(t;E) = e 4 / 2 2 C(t/E) E

and E.

3vI.4, convergence w i l l

even i f

Let Q(t;E)

o

-t

converges t o

u(^t;E)

subspaces intermediate between

u

as

w i t h p r e c i s e r a t e s of convergence i f

0,

r e s u l t s of 3Vr.3 and

If

o

both

R(:;E)u

d i f f e r e n t i a b l e , t h u s s o is

+ 5 1R ( t ; E ) + 2 1& ( t j E ) .

(5.2)

a r e twice continuously

and B(t;;E)u u(;;E)

uO(E)

= Q(;;&)U.

The d e r i v a t i v e

r)

v(t;E) = u ' ( t ; E ) i s a generalized s o l u t i o n of (2.1) with i n i t i a l conditions v(0;E) = u ' ( 0 j E ) = 0 and v ' ( 0 ; E ) = u"(0;E) = E -2Au. Hence, by uniqueness, we must have ( t ; E ) = & ' (t;E)U = 6(t;E)Au.

(5.3) 2 On t h e other hand, we may w r i t e (4.1) i n the form W ( t ; E ) = E - Q ( t ; & ) -2 E 6 ( t ; E ) , hence U'

G(tjE)U = &(tjE)U

-

2 E

G'(tjE)U

.

Applying t h i s e q u a l i t y t o a n element of t h e form Au we o b t a i n

(5.4) and using (5.3)

-

PARABOLIC SIPU'GUWI PERTURBATION Q ' ( t j E ) u = AiS(t;E)U

-

193

.

2 6'(t;E)Au

E

(5.5)

s o t h a t Q ( i ; & ) u i s a genuine s o l u t i o n of t h e nonhomogeneous first order equatior? (2.2). Consequently, t h e variation-of-constants

(1.5.3)applies and we have

formula

-

Q(t;E)u

lb(t;E)ll <_ C O Q w ( W= where

C

- s ) ~ ' ( s ; E ) A uds

S(t)U = - E 2 L t S ( t

c0

jit ew2(t-s)y'(s;E) w

a r e the constants i n (2.17)

0'

=

and

-&

2Q ( t ; & ) Au.

(t

ds

Yw

2

0)

(5.6)

, (5.7)

is the function i n

(3.3). It w i l l be divided i n s e v e r a l steps.

Proof:

e-AtS(t)u d t (see

(1.3.8)).

=I

u

On the other hand, i f

(A

If

u

e-ht6(t;E)u

(5.8)

D(A),

E

e'htt51

dt =

(t;E)u

dt =

1 22 u & A

0

+

E,

> w)

,m

L(h)u

E

Adi(A)u - 1 C(A)U -u + 1 2 h2 &*A

+ L m e - A t 6 f ' ( t ; E ) u d t = &'A2

( h > w).

E

so t h a t

2 h2 I f A

(E

- A)L(A)u =

u and we deduce using denseness of

D(A) that ( A > w2 ).

X(A) = R(E2 A2 + A;A)

(5.9)

Accordingly,

W e use now (11.2.11):

=k m

hR(A2;A)u

e-At@(t)u d t

( A > w, u

f

E).

(5.11)

Making use o f (5.11) and of the cosine f u n c t i o n a l equation (11.3.1) f o r C(t)

we obtain

2 2pv R ( ~ ~ ; A ) ;A)U R(~ =dmkae-(Ps*t)(C(~

+

t ) + C(S

-

t ) ) u dsdt

(p,v

> w).

(5.12)

194

PARABOLIC SINGUIAR PERTURBATION

Taking advantage of t h e convolutiun theorem i n t h e d e f i n i t i o n of

i n (5.6) we deduce, making use o f (5.8) and (5.10) t h a t

Q(^t;E)

2 2 m(h;A)R(E A

+

A; A ) u =

( A > w2,

dt

E).

(5.13)

( A > w 2 ),

(5.14)

u

E

By v i r t u e of (5.12) we m y a l s o w r i t e

AR(A;A)R(E

2 2

A

+

A; A)U =

[mlmh(tys,h;E)(C(s

t ) + C(S

-t

-

t ) ) u dsdt

with

Consider t h e s c a l a r cosine f u n c t i o n A

C ( t ) = cosh w t

( 4

< t<

m)

.

(5.16)

Here we have w

s(Z>

2

= e

t

( t 2 0)

I

and G(t;E) = YW(t;E), Yw

as defined i n (3.3); accordingly it follows from (5.7) t h a t $(t;E)

Applying formulas

= Ow(t;E)

.

(5.13) and (5.14) we obtain

rm

Let now

u be a n a r b i t r a r y element of

of t h e d u a l space

E*

with

* IIu /[

=:

function

According t o t h e previous arguments,

*

E, u

a n a r b i t r a r y element

l[ull = 1, and consider t h e s c a l a r

195

PARABOLIC SINGULAR PERTURBATION

Lme-Atr(

t;E)

+

dt =L F ( t , s , h ; E ) ( k ( s

t)

-

-

k(s

(5.18)

t ) ) dsdt

where

Obviously , k(s)

Let

0

(-m

<

<

s

m)

.

(5.20)

be a flrnction defined and i n f i n i t e l y d i f f e r e n t i a b l e i n

R(^A)

A >_ 0.

2

R

We say that

is alternating

(-1) nR (n) (A)

>- o

(in

(A?

0,

t 1 0)

if

n = 0,1,...)

We define correspondingly a l t e r n a t i n g functions i n

.

(5.21)

t >_ a.

It i s obvious that the swn of two a l t e r n a t i n g f u n c t i o n s and t h e product of an a l t e r n a t i n g f u n c t i o n by a nonnegative c o n s t a n t i s alternating.

More g e n e r a l l y , i t follows from L e i b n i z ' s formula that

t h e product of two a l t e r n a t i n g f i n c t i o n s i s a l t e r n a t i n g .

LEMMA. 5.2. alternating.

m(i)

be a f i n c t i o n such that

rn'(%)

Then 6

R ( A ) = e -m(^A)

(5.22)

i s alternating. Proof:

Obviously, it is enough t o show that each summand i n

t h e d e r i v a t i v e of order

with

j,.

.., p

*

n 2 1 of

R ( A)

i s of t h e form

1 and n k+(j-1)+ ...+(p- 1) (-1) = (-1)

n = 1; assuming it is t r u e for n,

This statement is obvious f o r validity for

its

n i 1 follows from L e i b n i z ' s formula.

LEMMA 5.3. Let

E

> 0,

*

m(A) = ( E A Then m' (A)

(5.24)

+

A)

q2

(A,O)

is alternating.

The proof i s l e f t t o t h e reader (Exercise 1).

.

(5.25)

196

PARABOLIC SINGULAR PERTURBATION LEMMA 5.4.

t

+

Aza.

Let

f ( % ) be continuous i n

Assume t h e Laplace transform

m.

t >_ 0 , f ( t )

=

O(exp a t )

Pf(^A) i s a l t e r n a t i n g i n

Then f ( t ) >_ 0

( t >_ 0 ) .

(5.26)

The proof is an immediate consequence of Lemma 1.3.2 ( s e e (1.3.14)). End of proof of Theorem 5.1. We go back t o (5.18). The d e f i n i t i o n (5.15) of t h e flmction h ( t , s , A ; t ) , Lemma 5.3, Lemma 5.2 and the is comments preceding it show t h a t h, a s a function of A, a l t e r n a t i n g f o r any s , t 2 0 , E > 0. Since t h e f i n c t i o n k ( s ) defined i n (5.19) i s nonnegative, it follows from (5.18) t h a t the Laplace i s a l t e r n a t i n g . Thus, by Lemma 5.4, transform of r(ht;E) r ( t ; E ) >_ 0 ( t 2 0 , E > 0 ) . Taking i n t o account t h e a r b i t r a r i n e s s of

*

u and u

,

(5.7) follows, completing t h e proof of Theorem 5.1.

I n a l l of t h e r e s u l t s t h a t follow u(:;&)

u(z))

(resp.

is the

s o l u t i o n of t h e homogeneous i n i t i a l value problem (2.1) (resp. ( 2 . 2 ) ) .

and applying (5.6) and (5.7) to t h e f i r s t term on t h e r i g h t hand s i d e t o estimate t h e other summands we use (3.4) which implies

of (5.28): (taking u

0

(E)

= 0

or

Ilc(t;E)ll <_ C o l w ( t ; E ) ,

u (E) = 0 1 IlG(t;E)II

alternately)

5 COYw(t;E)

( t >_ 0,

E

’0)

*

(5.29)

W e obtain a simpler but l e s s p r e c i s e bound noting t h a t @,(t;&), u?t (Lemma 3.1) and i n t e g r a t i n g ( 5 . 7 ) by p a r t s ; it r e s u l t s YW(t,E) 5 e t h a t Ow(t;E) 5 (1+ w2t)eat so t h a t (5.27) becomes 2 lju(t;E) u(t>li 5 c O2 (1 ~ + w 2t ) e w2t / l ~ u +o ~coew ~ t ~ l u o ( ~ )~ , I I +

-

-

197

PARABOLIC SINGULAR PERTURBATION 2 wLt e I/ul(~)I(

( t 2 0,

+ c0E

Theorem 5.5 implies t h a t when

t

D(A)

E

0

-

/lU(t;E)

uniformly on compacts of

u

u(t)ll

E

> 0)

.

(5.30)

we have 2

= O(E

1

(5.31)

if

0

I1uO(~)- uoIl = O(E

2

and

llu,(~>Il =

o(1).

(5.32)

Estimates of t h e same s o r t can be e a s i l y obtained f o r t h e d e r i v a t i v e u'(t;E)

if

u

0

E

2

D(A )

and

uO(E)

In f a c t ,

D(A).

E

v(%;E) = u ' ( % j E )

i s t h e s o l u t i o n of t h e i n i t i a l value problem (2.1) w i t h v(O;E)=U'(O;E)=U~(E),

~ ( 2 =)

On t h e o t h e r hand,

= u"(0;E) = E

v'(O,E)

-2

(AuO(E)

-

ul(E)).

(5.33)

i s t h e s o l u t i o n of (2.2) w i t h

u'(t)

(5.34)

~ ( 0= ) ~ ' ( 0 =) AuO. Accordingly, we have

THEOREM 5.6.

Assume t h a t

u

2

0

E

and

D(A )

uO(E) E

2 2 l ( u ' ( t ; E ) - u ' ( t ) l l 5 COE @ w ( t ; E ) l ( A uolI + CO+w(t;E)lIU1(E)

-

+ COYW(t;E)/lU1(E)

2

D(A).

-

Then

AuolI

AU,(.)ll

2

2 w t 2 (1 + w t ) e IIA uolI

5

COE

+

c0 ew t(I/U1(E)

2

- AUoll

.

( t 2 0) It follows from t h i s r e s u l t that i f

- Auo(E)/I)

IlU,(E)

+

uo

(5.35) 2

D(A )

E

uO(E) E D(A)

and

then llu'(t;E) uniformly on compacts of Ilu,(E)

-

t 2 0 2

- u'(t)l/ =

2 O(E

(5.36)

)

if

- Au~(E)II =

2

)

and

IIu~(E)

= Of& )

and

/ I A u O ( ~ ) AuoII = O ( E ).

AuoI/ =

O(E

O(E

> , (5.37)

or,e q u i v a l e n t l y , i f Ilu,(E)

- Auoll

Theorems 5.5 and

2

-

2

(5.38)

5.6 a r e e a s i l y s e e n t o i n p l y convergence r e s u l t s

v a l i d f o r a r b i t r a r y i n i t i a l conditions.

PARABOLIC SINGULAR PZRTURBATION

198

Let

5.7.

THEOREM

(resp. ( 2 . 2 ) w i t h u

t i o n of (2.1)

E E

0

uO(E)

~ ( 2 ) ) be

(resp.

u(^tjE)

-. uo,

arbitrary).

-. 0 &s

E'u1(E)

t h e generalized solu-

E

+

Assume t h a t

(5.39)

0.

Then U(tjE)

uniformly on compacts o f Proof.

u.

U(0) =

6> 0

Pick

r(^t)be

Let

u(t)

+

E

(5.40)

0

+

t >_ 0 .

u

and choose

E

D(A)

with

;1

- uoI/ 5 & .

t h e s o l u t i o n of t h e i n i t i a l value problem (2.2) with

Applying Theorem 5.5 and i n e q u a l i t i e s

(5.29), ( 5 . 3 0 ) , we

obtain

-

IIU(tF)

5

COE

5

U(t>ll

2 (1+

2 (u

-

IIU(tF)

+

t ) e w t/lAiiI

5

COE

2

2 w t

e

l l ~ ~ l +l

i ~-

uOl/

2

coew t l l u o ( ~ ) 2 w t

11u1(@)11+ C06 e

-

uo1l

.

(5.41)

> 0

s u f f i c i e n t l y small we c a n obviously make t h e r i g h t hand 2 2C0 & ew a i n 0 5 t 5 a , a > 0. This ends the proof.

E

side <_

2

t ~ ~ u l +( ~coew ) ~ t ~l

2 cO& 2 (1 + w2t > e wt

+ Taking

c0E 2ew

U(t)lI

-

C o e " tlluo(E)

2

t

-

II3t)

:(t>lI +

2

2

Concerning d e r i v a t i v e s , we have

Let

THEOREM 5.8.

that

u(t)

u(tjE),

u~,u~(E E )D(A) AuO(E) a Au, u l ( f ~ )

-

b e its i n Theorem 5.7.

Au0 -as

E

+

Assume

0.

(5.42)

-

Then

U'(tjE) uniformly on compacts of

U'(t)

4

as

E

(5.43)

0

+

t >_ 0 .

The proof follows t h e l i n e s of t h a t of t h e previous r e s u l t . 6 > 0,

,(^t)

and choose

u

E

2

D(A )

such that

llAu

- AuoI/ 5 6 .

Let

Then, i f

is a g a i n t h e s o l u t i o n of t h e i n i t i a l value problem (2.2) w i t h

U(0) = uo we a p p l y (5.29) and (5.35),

obtaining

199

PARABOLIC SINGULAR PERTURBATION

2

+ c 08 e w t

(t

0,

E

(5.44)

> 0).

T h i s completes t h e proof.

5.9.

Convergence i n (5.31) and (5.36) is uniform i n t ? 0 ( r a t h e r t h a n j u s t uniform on compacts of t ? 0 ) i f w = 0 . O f REMARK

course, t h e same observation applies t o a l l t h e other r e s u l t s i n t h i s section. For easy reference l a t e r w e c o l l e c t t h e s e p a r t i c u l a r cases of Theorems 5.5, 5.6 5.7 and 5.8 under a single heading. THEOREM 5.10.

cosine m c t i o n

Assume that A

c(Z>

with I M t ) II 5

Let

u(ht;E)

(2.1)

,

u(%)

generates a s t r o n g l y continuous

co


(4

-1.

be t h e generalized s o l u t i o n of t h e i n i t i a l value problem t h e generalized s o l u t i o n of t h e i n i t i a l value problem

(5.45)

(5.46)

(5.47)

2 00

PARABOLIC S I N G W R PERTURBATION

-

7

UO(E) + uo, ELUl(E)

&s E

0

+

0.

(5.48)

(5.49) uniformly i n

t 2 0

if

AuO(E)

--t

Au,

uI(E)

+

&S

Auo

E

-

(5.50)

0.

REMARK 5.11. The r e s u l t s i n t h i s section do not supersede those i n QVI.4 ( i n p a r t i c u l a r , Theorem j.6 on convergence of u ( t j & ) and Theorem 4.2 on convergence of t h e derivative u'(t;E)}. One reason is that "crossover" of i n i t i a l conditions (see (3.34)) is ruled out here, although t h i s will be remedled i n a sense i n QvI.8 by means of correction terms t o s t r a i g h t e n out convergence near zero.

Another,

and far more hiportant reason, is that the convergence r e s u l t s i n

gVI.3 and sVI.4 a r e uniform with respect t o bounded i n i t i a l conditions (see the statement of Theorem 3.6); more precisely, t h e r e s u l t s a r e o n convergence of

i n t h e norm of ( E ) ,

Q ( Z ; & ) , G ( $ ; E ) , 6'($;E)

i s , i n t h e uniform topology of operators.

that

A s t h e theorems i n t h i s

section make c l e a r , the p r i c e we pay f o r convergence i n t h e norm of

i s loss of convergence i n an " i n i t i a l layer"

(E)

o5t5

t(E),

even

if crossover of i n i t i a l conditions is avoided (see Example 3.10).

We note f i n a l l y t h a t t h e arguments i n 4VI.4 y i e l d bounds f o r G ' ( t ; & ) and &"(t;&)

4.4) not amenable t o t h e

(see Theorems 4.1 and

methods of the present section. It is n a t u r a l t o a s k whether we may obtain convergence of order E

~

0 ,<

~y

< 1 for uo i n spaces intermediate between E and

2

D(A) (D(A ) f o r t h e derivative). This question has s e v e r a l equally n a t u r a l answers, one of which we present here; another can be found

i n the author [1983:4] and a t h i r d , more useful i n p r a c t i c e , w i l l be examined i n 4VI.9. We introduce the spaces

u

E E

b(,

(0

5 CY 5 1) consisting of a l l

such that llAS(t)ull = O ( t Y - l )

as

Equivalent characterizations of t h e spaces BERENS if

[1967:1, pp.

ll1 and 1151:

if

0

t -. O+. kt

ff

5 LY 5

(5.51)

a r e given i n BUTZER1, u

E

HCy i f and only

PARABOLIC SINGULElR PERTURBATION

-

IIS(t)u

S(%)u

(equivalently, i f

a ) . We have Ill

=

D(A)

UII

o(tCY> a s t

=

4

201

o+,

(5.52)

is l o c a l l y Hblder continuous with exponent E i s r e f l e x i v e ; i n t h e general case only

when

H1 2 D(A) can be assured. We use nuw (4.23) t o estimate

S'(t):

2

2

5 ~ ( w+ l / t > e w Let

u

f

Ha.

(t

2

0)

.

(5.53)

Define

IuI,

= ~~u~~ + sup ( w 2

+

l/t)a-le-W

2 tl/AS(t)ul/.

(5.54)

t>_o Obviously,

I *ICYi s

a norm i n

31

CY

(which, i n c i d e n t a l l y , makes

Ha

a

Banach space). The following two r e s u l t s a r e formal counterparts of Theorem 5.5 and 5.6.

The proof of both r e s u l t s i s based on a d i f f e r e n t estimation of t h e operator

a.

commute with A

Assume f i r s t t h a t

u

E

D(A);

since

S

and

6'

and with each other we can w r i t e

=L t

B(t;&)u

It follows from (4.1) that

V(t

-

s;E)AS(S)U d s .

(5.57)

202

PARABOLIC SINGULAR PERTURBATION

-

Take the

(1 cu)-th power of both s i d e s , take the

a-th

power of both

sides of (4.20) and multiply the i n e q u a l i t i e s thus obtained term by term.

The r e s u l t is ll5'(tF)Il

Hence, i f

u

(u)

2

+

2

l/tlffew

(t

2 01.

(5.59)

Za,

E

/b(tF)UII 5

<- CE -2(l*)

+

CE -2(1*)\ulffew2tLlt(w2

- s))"(w2

l/(t

+ l/s>l*

ds, (5.60)

whence the estimates (5.55) and (5.56) r e s u l t .

REMARK 5.u. Direct v e r i f i c a t i o n that an element u to

m y be d i f f i c u l t .

31,

f

E

belongs

S u f f i c i e n t conditions can be obtained using

the theory of f r a c t i o n a l parers of i n f i n i t e s i m a l generators touched upon i n Chapter 111. I n f a c t , we have

Let

LEMMA 5.15.

b

>

.

- A)a)

D( ( b 2 1 The

w

C_

Ua

u

continuously d i f f e r e n t i a b l e of order Hblder continuous with exponent

-

lIs(t)u

t

+

Qm.6.

(5.61)

<_ 1).

D((b1

E

i s continuously d i f f e r e n t i a b l e of order

S(%)u

other hand, it has been proved i n Exercise

as

(Y

r e s u l t i s a consequence of Exercises 3 and 7 i n Chapter 111.

I n f a c t , it follows from Exercise 3 that if

c

(0

O-t,

thus

u

E

UII

CY

= 1/S(t)u

-

CY

>_

0.

S(t)u

i f and only

On the

is

c a 5 1) then S(^t)u is

t>_ 0,

in

c(

(0

that i f

- A)(Y)

s o that

S(0)ull = O(tCu)

Uu.

Singular i n t e g r a l s of Hilbert space-valued f'unctions and

w l i c a t i o n s t o inhomogeneous f i r s t order equations. We prove i n t h i s s e c t i o n a r e s u l t on the i n i t i a l value problem U'(t) = Au(t) v a l i d whenever

A

+

f(t)

(t

2 0),

U(0) = 0 .

(6.1)

generates an a n a l y t i c semigroup i n a H i l b e r t space

The b a s i s of t h e argument (as well a s o f the corresponding theorem f o r t h e i n i t i a l value problem (2.1)) i s t h e following r e s u l t on singular

H.

203

PARABOLIC SINGULAR PERTURBATION

i n t e g r a l s of H i l b e r t space-valued f’unctions.

Let

THEOREM 6.1. values i n

(H)

H

-

defined i n K(Z)

< tc

2 L (-,m;H)

(a)

.

(6.2)

A

K(a)

The Fourier transform

a f u n c t i o n with

and such that

m

n

L (-,a;H) -

(b)

m

1

E

K(Z)

be a H i l b e r t space,

satisfies

-

(X;f)(t) = r m K ( t i’

Then t h e r e e x i s t s a constant

C

( 6.5)

s)f(s) d s .

-m

depending only on

p

and

B

that

II’I1p

where

i n d i c a t e s t h e norm of

LP(-m,m;H).

The proof (of a more g e n e r a l theorem) can be found i n STEIN

[1970:1];the resull; f o r s c a l a r funccions i s i n pp. 29 and 34, while t h e extension t o vector-valued function is i n pp. 46-48. since ) :(K

e L

1

We note t h a t ,

(6.6) follows from (6.5) and Young’s

(-,m;H),

.

Hcwever, what is s i g n i f i c a n t i n Theorem i n e q u a l i t y (STEIN [ 1970:1] ) 6 . 1 i s that C does not depend, among other t h i n g s , of t h e L1 norm of K(%) but only on p and B. This w i l l be d e c i s i v e below. THEOREM 6.2. A

Assume t h a t

A

E

G(cp),

< cp <

0

7r/2

( t h a t i s , that

g e n e r a t e s an a n a l y t i c semigroup) i n t h e H i l b e r t space

T > 0,

1c p <

m,

f(%)

E

L’(0,T;H)

u ( : )

&a

s o l u t i o n of t h e i n i t i a l value problem (6.1). derivative exists

~ ‘ ( tf )L’(0,T;H)

u’(i)

E

0

5t 5

T

t h e (generalized)

Then

( i n the sense that t h e r e

=l t

u(t)

u<^t) has a

such that

LP(O,T;H)

u’(s) ds

(0

H.

5t 5

T))

-

PARABOLIC SINGULAR PERTURBATION

204

Moreover.

u(t)

f

D(A)

i s s a t i s f i e d a.e.

a.e. i n

(and of course on A)

p

0

<_ t 5 T

and t h e equation (6.1)

Finally, t h e r e e x i s t s a constant

C

depending o n l x

such that A

IIU' (

t)

Ip

i Cl/fllp

(6.8)

*

The proof naturdLlY w i l l depend on Theorem 6.1 and ( a variant o f ) t h e representation u'(t) that follows from

LEMMA

space

#

t S1(t

-

s ) f ( s ) ds

(6.9)

The hypotesis l e s s easy t o v e r i f y w i l l be

For t h i s we s h a l l use t h e following r e s u l t .

(c).

t

(1.5.3).

=&

E.

6.3. Assume that

K(^t) be a function with values i n a Banach K(i) is continuously d i f f e r e n t i a b l e for

0 and that

llK'(t)\l 5 B'/tl-2

Then

K(t)

s a t i s f i e s (6.4) ( w s

(t

#

(6.10)

0).

B' = (2log 2)B').

s o that

We use a similar argument i n t h e range

s < t.

The estimation runs i n exactly t h e same way when

t < 0 and we

omit t h e d e t a i l s .

Proof of Theorem 6.2. We say that an H-valued function f ( i ) defined i n 0 <_ t 5 T belongs t o C(O,T;D(A)) i f f ( t ) E D(A) and f ( i ) , Af(^t) a r e continuous. We have already proved (Lemma 1.5.1 ( a ) ) t h a t if f ( z ) E C(O,T;D(A))

of (6.1)

then t h e generalized solution

defined by

=k t

u(t)

S ( t - s ) f ( s ) ds

(0

5 t 5 T)

u(i)

205

PARABOLIC S I N G U U R PERTURBATION

(6.1); i n p a r t i c u l a r ,

i s a c t u a l l y a genuine s o l u t i o n of

Lt

S ( t - s ) f ( s ) ds

u'(t) = L t S f ( t -s)f(s) d s + f ( t ) = A

= / g f S ( t -s)Af(s) d s + f ( t )

Let

( 0 <_ t

f ( t ) be a f u n c t i o n i n C ( O , T ; D ( A ) ) .

5

+

f(t)

(6.2)

T)

Assume t h a t

p

> w,

where

w

i s a c o n s t a n t such lls(t)ll f o r some

Co.

5

wt

f ( t )= kR(b;A)f(t)

Then

belongs t o

b

that t h e previous c o n s i d e r a t i o n s apply.

=k

(6.13)

( t 2 0)

Coe

C(O,T;D(A))

We w r i t e (6.12)f o r

SO

f = f : LL

t

ul(t)

S 1 ( t -s)pR(p;A)f(s)

P

= AltS(t

=k t S ( t

Let

p > w,w

d s + pR(p;A)f(t)

- s)pR(p;A)f(s

- s)kR(p;A)Af(s)

t h e constant i n (6.13).

Consider t h e (H)-valued

function K

P ,P

for t >_ 0 ;

( t ) = e-PtS'(t)pR(w;A)

for

t C 0

we s e t

is infinitely differentiable i n second expression f o r

It follows t h a t particular for

K

PYP

(t)

(6.15)

= emPtS(t)pAR(p;A)

K ( t ) = 0. Obviously, K (t) P>P PL,P t > 0 ( s e e 5111.7). Using t h e i n (6.15) we s e e that

(t)

K f Lr(-m,=;H) f o r a l l r, 15 r 5 m , P?P r = 1, 2 a s r e q u i r e d i n (a) of Theorem 6.1.

We have

(27~)-'/~?~,~(~) = L m e b t K PYP ( t ) d t

in

PARABOLIC S INGUIAR PERTURBATION

206

It follows from t h e first inequality (1.3.9) t h a t

5 PC0(IL

IlCLR(IL;A>ll

- w1-l

(P

(111.7.6) i n Theorem 111.7.1 t h a t ,

and from

> w),

(6.18)

if p i s s u f f i c i e n t l y large,

s o that (6.17) implies that (6.3) i n Theorem 6 . 1 is s a t i s f i e s with a

constant

B

t h a t does not depend on

p

(say, i n 1.1

1 p).

To handle

( c ) we use t h e following estimate f o r t h e derivatives of an a n a l y t i c semigroup, which i s a consequence of t h e r e s u l t s i n Chapter I11 (see formula (111.7.U)): -m w i t IIS(~)(= ~ )III IA % ( ~ 5 ) I cmt I e

where

Cm i s a positive constant and

( t > 01,

w1 >

w, w

(6.19)

t h e constant in

(6.13). A moment's r e f l e c t i o n shows t h a t it i s possible t o deduce from this inequality, from (6.18) and from t h e first expression i n

thus we can apply Lemma 6.3 t o show t h a t

K

PYP

s a t i s f i e s as well (6.4).

We have thus v e r i f i e d a l l t h e hypc'teses of Theorem 6 . 1 f o r t h e kernel K

P7P

(i)

w i t h a constant

t h e r e e x i s t s a constant

B

C

that does not depend on p .

Accordingly,

depending on p but not on p

such t h a t

if

for

f E LP(O,m;H),

then llull

Now, i f

M> m> 0

T > 0

and

CY

Lp (0 ,m ;H)

5

Cllfll

(6.21) LP(O,m;H)'

is a r b i t r a r y , t h e r e obviously e x i s t constants

(depending only on

p,cu,T)

such t h a t

(6.22) hence, i f

207

PARABOLIC SINGULAR PERTURBATION

-

S'(t for

f

E

s)pR(p;A)f(s) ds

L~(o,T;H>,

We r e t u r n t o

(6.14).

Keeping i n mind that f ( t ) E C(O,T;D(A))

and

making use of t h e f a c t that pR(p;A)u

-+

u as

(6.24)

p --.

( s e e Exercise I.l7), and of t h e uniform bound (6.18) we can use t h e dominated convergence theorem i n combination with (6.22) and deduce

i s t h e s o l u t i o n of (6.1) (given by (6.11)) then

u(i)

that if

Let, finally,

C(O,T;D(A))

llf,

{$(:)I

LP(O,T;H),

be an a r b i t r a r y function i n

f(z)

a sequence i n

such t h a t

- fll

- 0

as

nhm

L~(o,T;H)

and

u

(i)

v(t>

E

L~(o,T;H).

n applied t o

.

t h e s o l u t i o n of (6.1) with fn

-

fm that

Since

f = f It r e s u l t s from (6.25) n converges i n t h e Lp norm t o a

.A(;)

u n ( t > -, u ( t )

o <- t 5

uniformly i n

T,

I\

where

u(t)

is t h e weak s o l u t i o n of (6.1), we can t a k e limits i n t h e

equality

and conclude t h a t ( 6 . 7 ) holds with of t h e claims on

u(;)

u l ( t ) = v(t).

we w r i t e (6.12) f o r

fn

To prove t h e r e s t

i n t h e form

.

u ' ( t ) = Aun(t) + f n ( t ) n

(6.26)

-

uh(f)

u' i n L ~ ( O , T ; H > , passing i f necessary t o a subsequence, we may assume t h a t u h ( t ) + u ' ( t ) and f n ( t ) f ( t ) i n a s e t e of full measure i n 0 5 t 5 T. Using t h e f a c t that A is

Since

closed we obtain t a k i n g limits i n (6.26) t h a t u ' ( t ) = Au(t) i f ( t )

(t

u(t) E

E

D(A)

and

e).

This ends t h e proof of Theorem 6.2. The next r e s u l t on t h e nonhomogeneous equation (6.1) i s considerably simpler and not r e s t r i c t e d t o H i l b e r t spaces. Banach space

E,

t h e class

Ha(O,T;E)

Given an a r b i t r a r y

i s defined as follows:

PARABOLIC SINGUIAR PERTURBATION

208 for

0

5 CY 5 1, Ua(O,T;E)

a-Hblder continuous i n

The spaces

E

Let

6.4.

H ~ ( o , T ; E ) ,o <

semigroup S(i)

T,

defined and

f

and is endowed w i t h t h e norm

a r e Banach spaces.

aa(O,T;E)

THEOREM

f(^t)

c o n s i s t s of a l l functions

5 t <_

0

cy

E.

E

be an a r b i t r a r y Banach space,

5 1, and assume

Then

u(t),

A

generates an a n a l y t i c

t h e generalized s o l u t i o n of (6.1)

i s a genuine s o l u t i o n , t h a t is, u ( i ) i s continuously d i f f e r e n t i a b l e i n 5 t 5 T, u ( t ) E D(A) (6.1) i s s a t i s f i e d i n 0 5 t 5 T. Finally,

0

t h e r e e x i s t s a constant

Ca

such t h a t (0 5 t I T)

llu'(t)ll 5 CaIf(:)la

Proof: derivative

We use again i n e q u a l i t y S (t)

.

(6.28)

*

(6.19), t h i s time f o r t h e

first

I n view of (6.27 ) we have

hence t h e function (6.30) i s w e l l defined i n 0

0 5 t 5 T.

5 t < t ' 5 T. Ignoring ( a s we

We show that

) : ( v

i s continuous.

Let

may) t h e nonintegral term, we have

I n view of ( 6 . 2 9 , t h e f i r s t i n t e g r a l i s bounded by a constant times

The integrand i n t h e second i n t e g r a l tends t o zero a s

t'

-

t

-

0

and

can be estimated, using (6.29) again, by a constant times

thus the i n t e g r a l tends t o zero due t o t h e dominated convergence theorem. It follows from inequality (6.29) that t h e r e e x i s t s a constant

Ca

PAFtABOLIC SINGULAR PERTURBATION

209

such t h a t

Let

be t h e weak solution of (6.1).

u ( : )

-

S(t

-

s)(f(s)

Writing

f(t))ds +

we see using once again (6.29) that

(6.33)

u(t)

E

D(A) and

n t

= p ( t

-

s)(f(s)

- f(t))ds + S(t)f(t) - f ( t )

,

(6.34)

s o that

which shows t h a t

is continuous i n 0 <_ t 5 T. Thus, t o i s a genuine solution of (6.1) we only

Au(t)

complete t h e proof that

u(t)

have t o show that

=

u’(t)

v(t)

,.

a.e. or

which, i n view of (6.32), a l s o implies (6.27).

follows. Assuming that

integrating

by parts

f(t)

We can do t h i s as

i s continuously d i f f e r e n t i a b l e we have,

,

P t

v(t> = -

- j o(DsS(t - s))(f(S)

- f(t))dS + S(t)f(t)

n t

\IO

P t

S(t -s)f’(s)ds+S(t)f(O)=J

(6.37) S(s)fl(t -s)ds+S(t)f(O)

.

0

Interchanging t h e order of integration,

Lt

v(s)ds

= k t S ( r ) (J,”f’

+ktS(s)f(0)ds To prove (6.36) f o r

f

E

H,

=

Lt

- r)ds

(s

S(t

-

s ) f ( s ) ds

we s e l e c t a sequence

.

(6.38) {fn] of continuously

d i f f e r e n t i a b l e functions such th a t

Ilf - f n l H Cy

0

Once this i s donewrite (6.38) for fn and take limits: that t h i s i s j u s t i f i e d follows from (6.32) applied t o f - f n . as

nd

-

m.

210

PARABOLIC SINGUIAR PERTURBATION

QVI. 7

The inhomogeneous e q u a t i o n :

u(

convergence of

u'( t ; & )

t;E)

We examine i n t h i s s e c t i o n e s t i m a t e s and convergence r e s u l t s for

t h e generalized solution

u( t ; E )

of t h e nonhomogeneous i n i t i a l v a l u e

problem &'u'l(t;&)

+ u'( t ; E )

= Au(t;E)

= 0,

U(0;E)

+

( t >_ 0 ) ,

f(tjE)

(7.1)

.

= 0

UJE)

The l i m i t w i l l be t h e s o l u t i o n of

u ' ( t ) = Au(t) Roughly speaking,

-t

f(t)

( t >_ 0 ) ,

(7.2)

u(0) = 0 .

t h e e s t i m a t e s a r e "dynamic" ( t h a t i s ,

&-dependent) c o u n t e r p a r t s of t h e r e s u l t s i n

QVI.6; the main t o o l s

will be t h e s i n g u l a r i n t e g r a l methods t h e r e a n d t h e estimates on and 6" o b t a i n e d i n QVI.4. 'The f i r s t results (Theorems 7.2 a n d 7.4) are a c t u a l l y independent o f s i n g u l a r i n t e g r a l s and a r e v a l i d i n a g e n e r a l Banach s p a c e E; t h e o t h e r s (Theorem 7.6) a r e r e s t r i c t e d t o

6'

H i l b e r t space-valued f u n c t i o n s . Throughout t h i s s e c t i o n

(7.1) and u ( i ) u(tjE) =

J:

u(;;E)

is t h e generalized solution of

is t h e generalized s o l u t i o n of (7.2):

=lo -

we have

,t

5 ( t -s j E ) f ( s j E ) d s ,

u(t)

S(t

s ) f ( s ) ds.

(7.3)

The f i r s t r e s u l t (which w i l l n o t be u s e d i n t h e s e q u e l ) i s on c o n d i t i o n s on s o l u t i o n of

f(t;E)

that g u a r a n t e e t h a t

i s a genuine

u(tjE)

(7.1).

LEMMA 7.1.

Assume t h a t one of t h e f o l l o w i n g two c o n d i t i o n s is

satisfied:

(a)

f(t)

(b)

f(t)

D(A)

f

and

f(t), A f ( t )

are c o n t i n u o u s i n

0 5 t

5 T,

or Then

is continuously d i f f e r e n t i a b l e i n

u(t;E)

0

<_ t 5 T.

is a g e n u i n e s o l u t i o n of (7.1).

T h i s r e s u l t c a n be reduced t o Lemma 11.4 . 1 t h r o u g h t h e t r a n s f o r m a t i o n s u s e d t o t r a n s m u t e t h e s o l u t i o n of ( 2 . 1 ) a n d viceversa.

i n t o t h e solution of ( 2 . 3 )

211

PARABOLIC SINGULAR PERTURBATION

Assume t h a t

THEOFEM 7.2.

o

4

T

<

hen

m.

~(Z;E)

(a)

(resp. u(Z))

d i f f e r e n t i a b l e ( r e s p . continuous) i n 2 w T Coe lif(kaly

IIdt;&)II5 (c)

If

f(:;&)

f(i) &

+

1

i s continuously

5 t i T.

0

1idt)ii 5

Ll(0,T;E)

with

f ( % ) E L (0,T;E)

f(%;E),

2 coew

(13)

Tllf("tll,

(OFt5T)

u(ht;&) -+ u ( t )

then

*

(7.4)

uniformly i n

O
Proof:

I n e q u a l i t y (7.4) f o r

u(%)

was a l r e a d y shown i n Lemma

1.5.2; t h e corresponding e s t i m a t e f o r u ( t ; E ) way from (7.3) and t h e e s t i m a t e (3.4) f o r

2

(t 2 0)

I I ~ ( ~ ; E >5I ICoew

Continuity o f

0

If

5 t < t' 5

(7.5)

T

we have

-t

t' l i u ( t l ) -u(t)ll<_/;

.

was shown as w e l l i n Lemma 1.5.2, b u t we

u(t)

s k e t c h t h e proof anew.

follows i n t h e same

B(^t;E),

i i S ( t ' - s ) - S ( t -s)llllf(s)llds

llS(t' -s)llds+]

' t

0

. (7.6)

The f i r s t i n t e g r a l i s shown t o converge t o z e r o on account of t h e estimate 2 m t lls(t>llI Coe

(t 2 0 )

(7.7)

*

The integrand of t h e second tends t o z e r o due t o c o n t i n u i t y of and i s bounded i n norm by a c o n s t a n t times

llf(s)ll,

S(^t)

t h u s we show

t h e i n t e g r a l tends t o z e r o using t h e dominated convergence theorem.

To prove that

i s continuously d i f f e r e n t i a b l e we note that it

u(t;E)

follows from (2.14) t h a t t h e norm o f u'(tjE)

(E)

in

B(%;E) i s continuously d i f f e r e n t i a b l e i n

t >_ 0 ;

exists for a l l

t

t h i s i s e a s i l y s e e n t o imply that and

=I -'

u'(t,E)

t 6l(t

-

s;&)f(s)ds.

0

Continuity of of

u(%).

To show

ul(t;&) (c)

i s d e a l t w i t h i n t h e same way as c o n t i n u i t y

we note t h a t

(e(t-S;&) -S(t

-s))f(s)ds.

If uniform convergence does not p r e v a i l t h e n t h e r e e x i s t s a sequence

PARABOLIC S INGUIAR PERTURBATION

212

[tn] i n the i n t e r v a l 0 <_ t 5 T such t h a t llu(tn;En) - u(t)ll i s bounded away from zero f o r some sequence {En) such t h a t E n + 0. We may assume t h a t tn + t (0 5 t 5 T ) . If t = 0 a contradiction is obtained with t h e help of (7.5) and (7.7); i f t # 0 we deduce from Theorem

3.6 that B(tn

as

n

-t

- SjE)f(S)

4

S(t

-

SjE)f(S)

This time t h e contradiction r e s u l t s from t h e dominated

03.

c onvergence theorem.

REMARK 7.3.

When

UI =

0

we can take

T =

i n the statement of

m

Theorem 7.2; i n e q u a l i t i e s (7.4) become llu(t;E)Il

5 Col!f("t~)Il1, Ilu(t)Il 5 coIIf("tIl1

Moreover, we obtain uniform convergence of

(0

u(t;E)

5 t IT) *

in

t 2 0

(7.10) (this

i s a consequence of (7.9)).

The next r e s u l t establishes uniform boundedness and convergence of t h e d e r i v a t i v e

7.4.

THEOREM 0

5

(a) t 5 T.

g

Assume that

and

u(^t;&) (b)

-,

Proof: -

f(t;E), f ( t )

f

aW(O,T;E), 0 <

There e x i s t s

C ,

(0

5 t IT I * (7.11)

+

f(i)

in

The f a c t that

u(%)

i s a genuine s o l u t i o n of (7.2) i n

second estimate (7.11):

6.4 together with the

continuous d i f f e r e n t i a b i l i t y of

follows from Theorem 7.2, under t h e only assumption t h a t

u'(t;E)

5 1.

such that

0
continuous.

cy

a r e continuously d i f f e r e n t i a b l e i n

U(t), U ' ( ~ ; E )5 1 0 ( 0 , ~ ; ~ )then U(t;E) uniformly i n 0 5 t 5 T. ~ ' ( t )- S(t)f(O)

f(Z;E)

a(t,e)f(O)

u(2)

5 C c y I f ( b ) l W Yllu'(t>ll 5 CcyIf("tl,

llu'(t;E)ll (c)

under stronger assumptions on t h e functions

f("t.

f(Z;E),

Then

u'(t;E)

u'

(t;E)

f(%) i s

To show t h e f i r s t inequality (7.11) we w r i t e L

Lt

6'(t

-

s;E)(f(S;E)

and use the estimate (4.20) f o r B ' ,

-

f(tjE))

i-

B(t;E)f(t;E).

which implies t h a t

(7.12)

213

PARABOLIC SINGULClR PERTURBATION

(7.12) t h a t

Accordingly, it follows from

which y i e l d s (7.11). We prove f i n a l l y t h e convergence statement ( c ) .

(7.8)

for

u'(t;E)

and

(6.9)

for

u'(t)

Using formulas

we e a s i l y assemble t h e

following expression: u'(t;c)

-

u'(t)

= j- t

6'(t

-

s;E)(f(s;E)

-

s)(f(s)

-

f(tjE))

f

G(t;E)f(t;E)

0 n t

-Jo-S'(t

=

j o-6'( t -

s ;&)

( (f( s ;E)

f j 0 - ( W ( t -s;E)

-

f(t))ds

- f ( s ) ) - ( f ( t ;E)

f

S(t)f(t)

- f ( t )) )ds t 6(t ;E)

f (t ; E )

- S ' ( t - s ) ) ( f ( s ) - f ( t ) ) ds - S ( t ) f ( t ) .

(7.14)

The f i r s t i n t e g r a l can be estimated i n norm by a n expression of t h e f orrn

The integrand i n t h e second i n t e g r a l tends t o zero i n moreover, we have, using (6.19) and

Theorem 4.2;

0

5

s

< t

by

(7.13),

F i n a l l y , we have

+

11(6(t;E)

-

S(t))(f(t)

2 5 C0ew tllf(:;E) f

CIlt"(6(tF)

-

f(0))ll

-

f(;))la

-

s(t))llllf(~)lla

.

(7.16)

The f i r s t term obviously converges uniformly t o zero i n The same i s t r u e of t h e second due t o Theorem

3.6.

0

5t5

T

.

214

PARABOLIC S I N G W R PERTURBATION

REMARK 7.5. G(t;E)f(O)

We s e e e a s i l y t h a t t h e c o r r e c t i o n terms

and

S(t)f(O)

t h a t produce uniform convergence of t h e

d e r i v a t i v e s cannot be dropped; t h i s i s due t o t h e f a c t t h e r e cannot be uniform convergence of

unless

u'(t;C)

to

u'(t)

(since

f(0) = 0.

I n t h e r e s t of t h e chapter we s h a l l assume that

E

= H

is a

Hilbert space; t h e estimations and convergence r e s u l t s w i l l be based on Theorem 6.1 on s i n g u l a r i n t e g r a l s .

THEOREM 7.6. belong t o

Assume H

LP(O,T;H)

(0

i s a H i l b e r t space, and l e t


continuously d i f f e r e n t i a b l e and u ( t ) E D(A)

(c)

There e x i s t s a constant

Proof:

u(t)

v

i n t h e sense that t h e r e e x i s t s (b)

1< p

m,

E

<

a).

P

L'(0,T;H)

with a.e.

including t h e second i n e q u a l i t y

(7.17)' u'(t;E)

u(^t)

S'(t),

Lp(O,T;H)

of (7.2)'

were proved i n Theorem 6.2.

(7.8)

t o which Theorem

6.1 can be a p p l i e d i n t h e way used t o show Theorem 6.2. unlike

is

t Iov(s)ds = u ( t ) 0 5 t 5 T.

w i l l be formula

expressing it a s an i n t e g r a l with k e r n e l B ' ( t ; E ) Gf(t;E),

f(i)

f($;E),

u(;;E)

such that

The statements concerning t h e s o l u t i o n

The b a s i s of t h e treatment of

(a)

has a d e r i v a t i v e i n

u ' ( t ) = Au(t) f f ( t ) C

Then

Since

is not singular a t t h e o r i g i n , t h e argument

i s a c t u a l l y simpler and t h e smoothing operator pR(k;A)

need not be

used. Let

p

t 2 0, K

for

LEMMA

and -

> w

t

2

(w

(t;E) 0

7.7.

such t h a t

t h e constant i n (4.20) and (4.26)) and define

=

o

for

t c 0.

There e x i s t constants

E

0'

B > 0

independent of

E

215

PARABOLIC SINGULAR PERTURBATION

-

(7 19)

Proof:

Write 6"(sjE)U

for

u

E

D(h),

of (4.27) and

where

5

0

=

X0 ( S ; E ) U +

is t h e f i r s t term on t h e r i g h t hand side

Xl i s t h e sum of t h e r e s t .

r a t h e r , Theorem 4.7)

(7.20)

Zl(S;E)U

Using Theorem 4.4 ( o r ,

we deduce 2

~ ~ ~ ( s ;5&c(w2 ) ~ -t ~ w2s-'

kt

t > 0, u

Kp(S;E)U-Kp(S

By Theorem

E

D(A).

-

tjE)U

+

s-2)eW

5

C's-2e ps

( s > 0)

.

(7.21)

We have =

4.1 we have

Putting together (7.21) and (7.23) we can estimate t h e integrand of t h e

f i r s t i n t e g r a l i n (7.22) by

Cb-2, thus t h e i n t e g r a l i t s e l f is

bounded by a constant times

1 s - t

- -s1 '

(7.24)

The second i n t e g r a l i n (7.22), a f t e r i n t e g r a t i o n by p a r t s , becomes

(7.25) A look at t h e integrand i n (7.25) makes p l a i n that it can be estimated

by a constant times

216

PARABOLIC SINGULAR PERTURBATION

thus t h e i n t e g r a l c o n t r i b u t e s another serving of (7.24).

Putting

t o g e t h e r a l l estimations and taking advantage of t h e f a c t t h a t

D(A)

is dense i n E we deduce t h a t

-p

c

+

~

2 -t),-( S-t)/2E eW( S-t)/E e

(5

-

,

E

(7 27)

t h e last two summands o r i g i n a t i n g from estimation of t h e boundary terms On t h i s basis, we proceed t o estimate t h e i n t e g r a l

i n (7.25).

(7.28) The i n t e g r a l of t h e f i r s t t e r m i n (7.27) i s computed as i n Lemma 6.3. The i n t e g r a l of t h e second term i n (7.27) i s

To compute t h e i n t e g r a l of t h e last term we make t h e change of variables

s

-

t =

t h e domain of i n t e g r a t i o n i s then

0;

< s <_ -2t

The bounding of (7.28) f o r

Let

LEMMA 7.8. Then t h e r e e x i s t s

B

t < 0

KO(;;@)

P runs along t h e same l i n e s and is

be t h e Fourier transform of

t

independent of

I/"K~(Q;E)II

Proof:

K ( t ; E ) = 0 f o r t < 0.

i s of course unnecessary since

omitted.

2 t and

Estimation of t h e p a r t of (7.28) with domain

we obtain (7.29) again. -m

s

5B

<

(a

D

<

and 03,

Kp(t;E).

such that

E

Eo>

.

(7.30)

14);A)

.

(7.31)

o <E5

We use e q u a l i t y (5.10):

= (p

"

b)R(E2(p

-

w2

4- ( p

-

We make then use of t h e f i r s t i n e q u a l i t y i n t h e sequence (11.2.22) c h a r a c t e r i z i n g generators of cosine functions; s e t t i n g

h= p

- b,

PARABOLIC SINGULSlR PERTURBATION

C0lP

--

2

(Re(E2(p

- b >+

(P

- 4

- b))1/2 - w ) I E 2 ( p -id2i-

A/

1x1

V/ 11-11

=

-

+ l)1’2(P/lUl)1’2)

(Re((U

2

UlAl-1’2}\U

0 < Re 1-1

i s t h e unique multiple of

h

- kdl

by

i s bounded away from zero i n t h e s t r i p t l y small, where

(p

(7.32) 1/2

]A\-’, setting u = E A we see t h a t it i s enough t o show t h a t

Multiplying numerator and denominator noting t h a t

217

(7.33)

+ 111’2 2 with

1.

sufficien-

E~

111 = P.

on t h e l i n e

p

and

We check e a s i l y t h a t (7.33) never vanishes, thus we only have t o show t h a t it i s bounded away from zero for 1 ~ +1 m. Note t h a t , f o r 1U1 = r a t t a i n s i t s minimum a t Re((u + 1)1’2(~/1~])1’2)

u

=

+ir, thus

(7.34) On t h e o t h e r

I

EOlUl -

1x1

hand,

q u + 111’2,

>

lhl-1/21U

so t h a t

Ei21~1

thus our claim holds f o r

Proof of Theorem 7.6.

T h a t the kernel

independent of

< 1, 1/2w.

K (t;E) P

satisfies

(a)

in

was shown i n Lemma 7.7, while (6.3), with B

E

likewise independent of

was t h e s u b j e c t of Lemma 7.8.

E

t h e operator

-Lt

f(;) i s bounded i n

-<

The estimate (6.4) with

Theorem 6 . 1 i s obvious from i t s d e f i n i t i o n . B

E~

+

K (t

-

Accordingly,

s ; E ) f ( s ) ds

(7.35)

P

Using (6.22) we deduce t h a t (7.6) defines

LP(O,T;H).

a s w e l l a bounded operator i n

LP(O,T;H).

This y i e l d s t h e first

estimate (7.15). We prove f i n a l l y

(d).

The statement on convergence of

i s a consequence of Theorem 6.1. Lp

we t a k e

f,

say, i n

To show convergence of

H1(O,T;E)

u(^t;E)

u‘(;;E)

in

and w r i t e t h e d i f f e r e n t i a t e d

version of (7.9) as follows: u’(t;E)

-

U’(t) =

Lt

6 ’ ( t -SjE)(f(SjE)

+ k t ( G 1 ( t -s;E)

+

(6(tjE)

-

-

f ( s ) ) ds

- S ’ ( t - s ) ) ( f ( s ) - f ( t ) ) ds

S(t))f(t).

(7.36)

PARABOLIC SINGULdR PERTURBATION

218

Apply (7.11) t o t h e first i n t e g r a l , Theorem 7.4 t o t h e second and Theorem

3.6 t o t h e last term:

t h e conclusion i s

To show convergence f o r a r b i t r a r y

g(t;E),g(:) (7.1),

E

('7.2).

H1(0,T;E)

and

f(i)

u'(%;E)

-

u'(;)

LP(O,T;H),

€

v(t;E), v ( t )

5

LP(O,T;H).

let

t h e respective solutions of

W e have

The f i r s t and last terms on t h e r i g h t hand s i d e of (7.33) can be made small using t h e f a c t that

i s dense i n

al(O,T;E)

LP(O,T;H)

and both

i n e q u a l i t i e s (7.17); f o r t h e second term we use (7.32) and following comments.

iv1.8.

This ends t h e proof of Theorem 7.6.

Correctors a t t h e i n i t i a l layer.

Asymptotic s e r i e s .

We work i n t h i s s e c t i o n with t h e homogeneous i n i t i a l value problem

+

E2u"(t;E)

U

u ' ( t ; F ) = Au(t;E)

u'(0;E) = ul(E)

= uO(E),

0 (0;s)

(t

2

0),

,

(8.1)

and t h e equation u ' ( t ) = Au(t)

(t

with i n i t i a l condition t o be fixed below.

2

(8.2)

0),

A s pointed out before

(see Remark 5 . l l ) , i n the general conditions of Theorem 3.6 (where t h e r e may be crossover of i n i t i a l conditions), uniform convergence of

u(t;E)

t o u(t)

t = 0

near

cannot be expected since i n general

uo (E) f , uo. However, uniform convergence can be a t t a i n e d through addition of correctors (solutions of a d i f f e r e n t approximating equation) a t t h e boundary.

This method can be applied equally well t o the case

where the i n i t i a l conditions i n powers of

E,

uO(E),

ul(E)

have asymptotic expansions

as made c l e a r below.

We assume that

u

0

and

(E)

have asymptotic developments

u1(E)

of t h e form U (E)

0

=

U

0

+

EU

1

+

2

E U

2

+

F

2 u

3

t.

.-.+

E

% + O(E IW-1 ),

N

PARABOLIC SINGUIAR PERTURBATION The objective is t o show that

219

possesses a similar asymptotic

u(t;E) development, uniformly on compacts of

t 2 0 ; t o produce convergence t = 0 we shall need t o introduce correction terms a t each step.

near

We examine f i r s t t h e cases

by d e t a i l s .

N = 0 t h e c e n t r a l idea is t o approximate u(i) u(t;E) but by u(t;E) - vO(t;E), where v ( t j E )

For

t = 0

near

N = 0,l where t h e method i s unencumbered

not by

0

is t h e s o l u t i o n of E2vyt;E) 0

+

V'(tjE) = 0 ,

v'(0;E) = 0

0

vo(t;E)

as

0

-+

E

-. m

E -2 v

0 '

(8.4)

.

We r e f e r t h e reader t o KEVORKIAN-COLE [1981:1]f o r a thorough A

discussion of

t h e choice of

vo(t)

i n t h e one dimensional case only

pointing out that the i n i t i a l condition is t o eliminate t h e contribution of

y(E)

to

u

I n f a c t , it follows from (8.4)

(see Remark 5.11).

0

that u'(0jE)

+

.

= 0

V'(0;E)

On t h e other hand, since 2

v0 (t;E) =

v (t;E)

,

-e-t/EVo

(8.5)

t 5 t ( E ) outside The p r i c e t o w i l l not be a solution of

w i l l not d i s t u r b convergence i n t h e region

0 of t h e boundary layer (here

pay, of course, i s t h a t

t(E)

s a t i s f i e s (3.20)).

-

v0 (t;E) t h e homogeneous equation (8.1), thus a l l t h e r e s u l t s below w i l l use u(t;E)

t h e theory of t h e nonhomogeneous equation (only t o t h e extent of Theorem

7.2).

Throughout t h i s section,

u(t;E)

i n i t i a l value problem (8.1) w i t h

uo(E)

denotes t h e solution of t h e and

asymptotic developments of t h e form (8.3). asymptotic expansions (8.3) i s an element of

5

CEk

f o r some constant

C

as

E

4

0.

ul(E)

having

The term

O(Ek)

E

i n the

whose norm i s

Solutions of t h e equation

( 8 . 2 ) , with i n i t i a l conditions specified i n t h e following r e s u l t s

w i l l be usually w r i t t e n

t h e fbnctions

uo(t), ul(t);

u 2 ( t ) , u3(t),

...

e t c . a r e solutions of a d i f f e r e n t equation (see (8.21)).

THEOREM 8.1.

Assume that (8.3) holds for N = 0 ,

uO(&)= u0

and t h a t

uo, v

0

E

+

o(E),

D(A).

u ~ ( E )= E

Then -

-2

v + 0

O(E

-1

)

(E

that is 4

0)

,

(8.6)

220

PARABOLIC SINGULAR PERTURBATION U(tjE) = u o ( t )

t >_ 0 ,

uniformly on compacts of (8.2)

+

vo(t;E) +

u

where

0

(8.7)

O(E)

(i)

is t h e solution of

W B

(8.8)

u ( o ) = u0 + v 0 ' 0

If

w = 0,

(8.7) holds uniformly i n

Proof.

t >_ 0 .

The function w(;;E)

= u(;jE)

- vo(G;E)

i s a s o l u t i o n of t h e i n i t i a l value problem ~

+

E2w"(tjE)

~ ( 0 ; s )=

1

w = w

U (E)

0

-

w'(t;E) = A w ( t ; E )

+ vo,

2 e-t/E Avo,

~ ' ( 0 j E ) = Ill(&)

- E -2

(8.10) To.

i s t h e s o l u t i o n of t h e homogeneous equation with t h e assigned i n i t i a l conditions and w2 is t h e s o l u t i o n

Write

-k

w2

where

w1

of the inhomogeneous equation with zero i n i t i a l conditions and 2 f ( t ; E ) = -e-t/E

We apply t o w1 while

w2

(8.W

AV0 '

Theorem 5.5 (with t h e simplified estimate (5.30)),

i s handled by means of Theorem 7.2 ( s p e c i f i c a l l y , t h e f i r s t

i n e q u a l i t y (7.4)). The final estimate i s

with t h e obvious modification i n t h e last term i f

w = 0.

This ends

t h e proof. For

N = 1 an a d d i t i o n a l c o r r e c t o r must be used, namely

2 vl(tjE) = -e -t/@

THEOREM 8.2.

1'

Assume t h a t ( 8 . 3 ) holds f o r

N = 1, t h a t i s

221

PARABOLIC SINGULAR PERTURBATION

u

0

u

=

(E)

= E - 2v 0

1 u , u 1, vo, v1

-~ and t h a t

!I

U(tjE) = u ( t ) +

D(A

U

2

0

t

u n i f o r m l y on compacts o f

O(E ),

f

+ d v l i-

o(1)

(8-131

1. Then

V0(tj&) f

0

2

t Eul

(E)

E(ul(t) +

(resp.

uo(i)

where

0,

+ O(E 2

Vl(tjE))

1

(8.14)

&

u,(t))

the s o l u t i o n o f (8.2) w i t h

u0 ( 0 ) = uo If -

i.

vo

(resp.

w = 0 , (8.14) holds uniformly i n

u,(o)

= u1 +

(8.15)

t >_ 0 .

We c o n s i d e r t h i s t i m e t h e f u n c t i o n

Proof.

w(^tjE) = U(ntjE)

-

-

vo(i;E)

= Aw(tjE)

E2w"(t;E) i w ' ( t ; E ) =

U

As i n Theorem

0

(E)

+

vo

C EV1,

2

-

W'(0,E) =

(8.16)

Evl(t;E)

t h a t s o l v e s t h e i n i t i a l v a l u e problem

w(O,E)

q.

e-t/E

U (E)

1

-

Av

0

-

2 @e-t/E Avl,

E -2V

(8.17)

-1

- E

y

1'

0

8.1, we write w as t h e sum o f a s o l u t i o n w1

(8.18)

of t h e

homogeneous e q u a t i o n t a k i n g t h e a s s i g n e d i n i t i a l c o n d i t i o n s and a solution

w2

of t h e nonhomogeneous e q u a t i o n w i t h z e r o i n i t i a l c o n d i t i o n s .

We a p p l y a g a i n t h e s e c o n d i n e q u a l i t y (7.4) t o

and

w2

(5.30) t o

W1,

obtaining

Obviously, a d i f f e r e n t t a c k must b e a d o p t e d f o r N >_ 2 , s i n c e t h e f i r s t term o n t h e r i g h t hand s i d e s of ( 8 . 1 2 ) and (8.19) c a n n o t b e squeezed smaller t h a n

level.

O(E2).

We p r o c e e d at first o n a p u r e l y f o r m a l

The a p p r o x i m a t i n g h n c t i o n w i l l b e of t h e form

u

N

(tjE)

= u (t) + EUl(t)

0

N

f

*-.

-t E UJt)

,

(8.20)

222

PAMBOLIC S I N G L U R PERTURBATION

a r e defined a s before and t h e where u0 (t), u1(^t) s a t i s f y t h e d i f f e r e n t i a l equations un' ( t ) = Aun(t)

-

~:-~(t)

u

( t >_ 0 )

(t), n 2 2, .

(8.21)

Noting t h a t t h e c o r r e c t o r s 2 vo, v1

used i n t h e cases N = 0,l a r e of 2 t h e form v O ( t j E ) = v (t/E ), vl(t;E) = v (t/E ), we s h a l l use a 0 1 combination of c o r r e c t o r s of t h e form II ( t ; E ) = v (t/E N 0

The

v

n'

n >_ 2

2

)

+

tVl(t/E

2

+

) +

N 2 VN(t/E ).

E

(8.22)

w i l l s a t i s f y t h e d i f f e r e n t i a l equations vn" ( t )

+

vA(t) = Avn-,(t)

(t

2

0)

,

(8.23)

and t h e decay condition vn(t)

-, o

as

t

4

m

and

(8.24)

n = 1,2

Note that t h e equation s a t i s f i e d by , ) : ( u U'n (

. is

(8.25)

t ) = Aun(t),

vn, n = 1 , 2 s a t i s f i e s v p )

+ vA(t)

= 0

.

(8.26)

Consider now t h e f'unction

N

I1=0 p )

= ( E 2U

N

+ &-2

N

c &"Vi(t/E2) + E-2 c E"V;l(t/E2)

+ E-2

17;O

+ up)) +

E ( E 2 u;l(t)

c E"(yll(t/&2) t VA(t/E2)) -2

+ up))

223

PARABOLIC SINGULAR PERTURBATION

+

N-2

c

EnAvn(t/E2)

n=0 N

=

c

EnA(un(t)

+

2

vn(t/E ) )

ti=O

The i n i t i a l conditions on u

Il’

u0 ( 0 ) = uo

-

E N - 1 AvNml(t/E2)

n = 0,l

+ vo,

ENAvN(t/E 2 ).

(8.28)

a r e those i n Theorem 8.1: = u1

u,(o)

+ v1

.

(8.29)

On t h e other hand, t h e i n i t i a l conditions on v n = 0 , 1 must be n’ 2 2 those t h a t insure t h a t v (t;E) = v (t/E ), v (t;E) = v,(t/E ), 0 0 1

v0 ( t ; E ) , v1(t;E)

where

a r e t h e correctors used i n Theorem 8.2.

Accordingly, $0)

= vo, vi(0) = vl,

hence, taking (8.26) and (8.24) i n t o account,

v ( t ) = -e 0

For

n

2

2,

-t vo, v,(t)

t h e i n i t i a l conditions f o r

= -e

un(i)

-t

v

1’

and

vn(t)

are,

respectively un(o) = un

thus for

un(E) tU(t;E)

-

must be constructed a f t e r

(8.32)

vn(o)

vn(t).

The i n i t i a l conditions

are obtained from (8.29) (8.30) ( 8 . 3 2 ) and (8.33):

PARABOLIC SINGULAR PERTUBBATION

224 lo ( 0 ; E ) = N

cN Enun(O) + cN Envn(o) = =O

Il=O

= u

0

- v

+

v

-

0

-

-

vn(o>>

G 2

+

EV

cN Envn(o) = cN n=2

0

=

c

+ N E n(un

“(9 + vl)

f

E

nun

(8.34)

=O

N

N

n=0

n=0

c Enu’n( 0 ) 4- c Enm2vn

N-2

c

E=O

EnU’(0) = n

c

n - 2 v n + E N-1 U&l(O)

E

I1;O

N

f

E

up)

(8.35)

Hence, i n view of (8.3), IlU(0F)

- mN(o;E)I/

= O(EPst1)

-

=

(8.36)

and IlU’(0;E)

lo$OjE)I/

O(EN-l)

.

(8.37)

We face now t h e problem of making a l l t h e s e computations valid. Roughly speaking, t h i s amounts t o :

(a) showing t h a t every d e r i v a t i v e w r i t t e n ( a s i n (8.171, (8.21), (8.24), e t c . ) a c t u a l l y e x i s t s . (b)

etc.),

showing t h a t every time we w r i t e

(as i n (8.17), (8.21),

Au

u a c t u a l l y belongs t o t h e domain of

A.

This w i l l be done by r e q u i r i n g “smoothness” conditions of varying degree on t h e c o e f f i c i e n t s

un’ vn

u o ( t > = S(t>(U0 + v&

i n (8.3). u,W

We begin with

= S(t)(U1+

a r e made e x p l i c i t i n (8.31).

“J

(8.38)

while

vo(t), vl(t)

v,(t),

v3(t) we solve (8.23) with t h e i n i t i a l condition (8.33) at and t h e decay condition (8.24) as t -. m:

t = 0

To construct

225

PARABOLIC S TNGULAR PERTURBATION

v;(o)

v2

=

-

v"(t) 3 v ~ ( o= ) v3

3

v2(t) -,o

u$o), f

-

vl(t) 3

= -e -tA v ~

-, o

u~(o), v 3 ( t )

t

as

=,

4

( t 2 0),

-

t

as

(8.40) m

.

Solving e x p l i c i t l y these e q u a t i o n s , v2(t)

= te

-t Avo

-

-t v ( t ) = t e Av

3

-t

(v2

- e -t (v3

1

- Au0 -

2Avo)

9

(8.41)

- AU1

avo)

9

(8.42)

-

A(uo + v,), U i ( 0 ) = A(U1 + vl). u s i n g t h e equation (8.21) and t h e

where w e have used t h e fact t h a t We compute next

e

u 2 ( t ) , u3(t)

U'

0

(0) =

i n i t i a l c o n d i t i o n (8.32) :

u;(t)

2

AU2(t) - S(t)A (u0 + v0)

=

( t >_ 0 ) , (8.43)

u2(o) =

?(t)

Au3(t)

=

-

U2

+ To

7

2

( t 2 0),

S(t)A (ul + vl)

(8.44) u ( 0 ) = u + v1 3 3

9

2 ) where we have used t h e f a c t s t h a t u " ( t ) = S(t)A (uo t v,), ~ " ( t = 0 1 2 (see (8.29) and (8.37)). = S(t)A (ul -t v,), v;)(O) = vo, v i ( 0 ) = v1

Hence

U,(t)

=

S(t)(U2 + v0)

=

S(t)(U2

f

v0)

-

u ( t ) = S(t)(u3

3

With

S(t

Lt

-

s)S(S)A

2

(u0 + v0) dS

tS(t)A 2 (u0 + v0) ,

-k

2

vl) - tS(t)A (ul

(8.45) -k

v~).

(8.46)

up(%),u 3 ( t ) ,

see that

v4(i),

y2(i), v3(t) already manufactured, we can e a s i l y v 5 ( t ) w i l l have t h e form v4(t)

=

e-tP4(t), v5 ( t )

=

e-tP5(t)

,

(8.47)

PARABOLIC SINGULAR PERTURBATION

226

where

is a polynomial of degree 2 whose c o e f f i c i e n t s a r e l i n e a r

P4(%)

combinations of

AJu

0’

AJvo ( j 5 j), Au2

and

Av2

uo, vo 7u2yv2 replaced by On t h e other hand, we have

t h e same polynomial with respectively.

U4(t)

- P4(0))

2tS(t)A3(uo

u (t) = S(t)(u

5

+

u

(i)

(resp. u5($))

-

vo)

- P5(0))

5

42

is

2

-2tS(t)A3(ul + vl) thus

P (t)

+ tS(t)A (u2 + v0) -

= S(t)(~4

-

and

5 u1,v1,u3,v3

t2 S(t)A4 (uo + v,),

+ tS(t)A2 (u3 i- vl)

(8.48)

-

- t2 S(t)A4(ul + v,),

(8.49)

can be constructed i f uo, vo E D(A 4 ), 4 2 D(A ), u3 E D(A ). However, i f we wish (8.47)

u E D(A ) (resp. ul,vl E 2 t o be a genuine solutions of (8.23) we a c t u a l l y need that

vo E D(A 5 ) and u E D(A 3 ), u4, P4(0) E D ( A ) ; i n view of our 0’ 2 previous comments about P4, it is s u f f i c i e n t f o r t h i s t h a t 4 3 2 uo, vo E D(A ), u2 E D(A ) v2 E D(A ) and u,, E D(A). Likewise, i f u

we wish (8.48) t o be a genuine s o l u t i o n of (8.21) we must a s k t h a t

3

2

E D(A5), u3 E D(A ) v3 E D(A ), u5 E D(A). It w i l l be of ul’ “1 i n t e r e s t l a t e r t o a s c e r t a i n t h a t u 4 ( t ) , u ( t ) a r e twice continuously 5 6 d i f f e r e n t i a b l e . This w i l l be t h e case i f u0’ V0Y U1’ v1 E D(A 1, u2, u3 E D(A 4), v2, v3 E D(A 3) and u4”-15 E D(A). From t h e s e observations we surmise t h e following r u l e s , v a l i d f o r

arbitrary m

2

1. I n t h e f i r s t place, we have

v,(t> where

(resp. Pml(%))

P,(t)

a r e l i n e a r combinations of (j

5

2m.-

A L ~A , J

3),

= e -tP*,(t)

= e-tP&),v;w,(t)

-

(8.50)

i s a polynomial whose c o e f f i c i e n t s

AJu ,AJ, 0

... Ajua-4,A3uh-4.(j

(j ~ 5~an

,

0

(j <_ 2m

-

l), AJu2,AJv2

5 31, Aua-2y Ava-2 (resp. AjV 11, A J U ~ , AJ, (j <- ;im - 31,. . AJ, 2m-3’ 2m-3 3

.

( j 5 3 ) , A U ~ - , , A V ~ - ~ . Accordirigly: h

(a)

v,(t)

and

vwl(^t) E

can be constructed i f

D(A2m-1), u2,v2,u3,v3

E

D(A2m-3),.

..

PARABOLIC S I N G U I A R PERTURBATION

& v*,(;)

vh(i)

Moreover,

227

a r e always genuine solutions of (8.23), (8.33)

vh(t),vml(t)

w i t h Av&(t), Avwl

E D(A)

.

continuous

if

The statement about genuine solutions i s obvious since (8.23) i s e s s e n t i a l l y a s c a l a r equation.

u&(t)

can be constructed i f (8.51) holds (b) u&,(%) These f m c t i o n s a r e genuine solutions o f (8.21) ~f

u,(t)

.

a r e twice continuously d i f f e r e n t i a b l e i f

,um,(t)

U2mm-2’v&2’u&4’v&-l

u2m ’v2m’ u2mt 1’vwl E Obviously, conditions (8.52)’

4

D(A ) 9

(8.54)

D(A~).

(8.53) and (8.54) a r e excessive f o r

t h e various purposes a t hand; we have ”equalized indices” among t h e

u

n

and t h e

v

n

t o shplify.

Assume that (8.3) holds i n t h e norm of E for 2 and that ~ > ~ - l ’ ~ ~ v ND(A - l > Y ~-2’~-ZjYvN-2’vN-3 u ~ , u ~ , vE ~ D(A , ~ ~).

THEOm8.3. Odd’

-4-

D(A )

3

N

,...,

N

N

n=O

n=O

(8.55) uniformly on compacts of

t 2

o

(uniformly i n

t

o

if

U)

= 0).

g

PARABOLIC SINGULAR PERTURBATION

228

i s even t h e same r e s u l t obtains under t h e assumption t h a t 2 Nt2

N>_ 2

uN>vN E D(A

1,

Proof. N = 2m

+

%-17%-29vN-19v~q-2

f

We consider f i r s t t h e case

1 and apply r u l e ( a ) .

N

odd

Avo(t),

...,AvN(t)

>_ 3;

we s e t here

Since conditions (8.51) a r e s a t i s f i e d

...

(with something t o spare) we deduce t h a t with

1.

D ( A ~ ) , . * * > U ~ > U O0, V E ~D(A >V

continuous.

vo(t), , v N ( t ) E D(A) Taking (8.50) i n t o account we

deduce t h a t

(8.56) This w i l l be used t o estimate the l a s t two terms on t h e r i g h t hand s i d e of (8.28):

f o r t h e f i r s t two terms we simply use t h e f a c t ,

u~-~(t)and

assured by (b), t h a t differentiable.

%(t)

a r e twice continuously

Using t h e f i r s t inequality (7.10) i n (8.28) we

obtain

where, i n v i e w of (8.56), the contribution of t h e l a s t two terms i s O(EW1),

This ends the proof.

The case

N

even >_ 2

i s handled

much i n t h e same way and we omit t h e d e t a i l s .

@?I. 9 E l l i p t i c d i f f e r e n t i a l equations. We apply the theory i n t h e lust eight sections t o t h e d i f f e r e n t i a l operator

m

m

i n a bounded domain R

.

m

of m-dimensional space w i t h boundary

T; here

229

PARABOLIC SINGULAR PERTURBATION

A(p)

d e n o t e s t h e r e s t r i c t i o n of

o b t a i n e d by means of t h e D i r i c h l e t

A

boundary c o n d i t i o n

o

=

U(X)

r),

(X E

(9.2)

or b y means of t h e v a r i a t i o n a l boundary c o n d i t i o n N

D ~ ~ ( X= )

The c o n s t r u c t i o n of

(x

y(x)u(x)

E

r).

(9.3)

w a s c a r r i e d o u t i n Chapter IV i n c o n s i d e r a b l e

A(@)

9IV.3 ( f o r t h e D i r i c h l e t boundary c o n d i t i o n ) and i n SN.6 (for t h e boundary c o n d i t i o n ( 9 . 3 ) ) t h a t A ( B )

d e t a i l ; i n p a r t i c u l a r , it w a s shown i n

g e n e r a t e s a s t r o n g l y continuous c o s i n e f u n c t i o n , t h u s a l l t h e r e s u l t s i n t h i s chapter apply automatically.

SVI. and u 0

5 , for 5.13.

i n s t a n c e t h e “ i n t e r m e d i a t e ” e s t i m a t i o n s i n Theorems 5.12 Combining Theorem

D ( ( b 2 1 - A(@))‘)

E

u n i f o r m l y on compacts of D((b21

-

5.u w i t h Lemma 5.15 w e deduce t h a t

-

u ( t ) I I = 0(E2’)

t > - 0.

p

Hl(n).

E

-o

(9.4)

The most i n t e r e s t i n g c a s e i s

D((b21 - A ( B ) ) if

as

can b e identified.

A(f3))‘)

if

l a r g e enough) t h e n

(b

jlu(t;E)

where

Of s p e c i a l i n t e r e s t a r e t h o s e i n

c1 =

1/2,

I n f a c t , w e s h a l l show t h a t

(9.5)

= Hi(Cl)

i s t h e D i r i c h l e t boundary c o n d i t i o n and

bl,

...,bm

belong t o

To show ( 9 . 5 ) we n o t e t h a t it h a s a l r e a d y b e e n proved t h a t

D ( ( b 2 1 - AO(@))1/2) especially

=

(see ( N . 2 . k ) ) a n d r e c a l l Theorem IV.2.2, HO(R) 1

(IV.2.6)). Thus, D((b21

We s k e t c h t h e p r o o f of

w e o n l y h a v e t o shod t h a t

- Ao(p))lb2) (9.6).

cosine function generated by used t o c o n s t r u c t

Let

- A(p))”I2)

.

(9.6)

C ( t ) = c o s h t Ao(@)1’2

Ao(p).

b e the 0 It f o l l o w s from t h e p e r t u r b a t i o n

e0(t) (or

6(t) from

cosine function generated by

= D((b21

A(p)lb

d i r e c t l y ) that

C(t),

the

c a n b e e x p r e s s e d b y means of t h e

perturbation series C ( t ) U = C0(t)U where domain

+

gTJF*Co(t)u

+ qTJF*qTJF*Co(t)u +

d e n o t e s t h e ( o n l y ) bounded e x t e n s i o n o f

O1 H (Q))

t o a l l of

* * *

,

So(t)P

(9.7) (with

L2( Q ) ; t h a t t h i s e x t e n s i o n e x i s t s follows

230

PARABOLIC SINGULAR PER’IURBATION

S ( t ) P i s bounded ( i n t h e norm o f 0

from t h e f a c t t h a t

1

L2(n))

in

~ ~ ( n )s,i n c e s o ( t ) P = (sinh t Ao(B)1/2)Ac(B)-1/2P, and

Using (1.5) and t h e “ r e c i p r o c a l ” series Co(t)u = we show t h a t

@(t)U

-

qqF*C(t)u

+

f40P*rn*C(t)u +

(9.8)

@ ( t ) u i s continuou.;ly d i f f e r e n t i a b l e i f and o n l y i f (9.6) follows f r o m Theorem

@,(ti) i s c o n t i n u o u s l y d i f f e r e n t i i b l e , t h u s

111.6.4. However, i n t h e p r e s e n t s i t u i t i o n , estimates on rates o f convergence l i k e (1.4) c a n b e o b t a i n e d under weaker assumptions b y more e l e m e n t a r y methods.

We s k e t c h below t h i s theeory i n a s u i t a b l y ” a b s t r a c t ” v e r s i o n .

Let

E = H

be a H i l b e r t s p a - e and

A.

a s e l f adjoint operator

such t h a t

with

K

>

0.

We c o n s i d e r t h e o p e r a t o r A = A.

where

P

+ P,

(9.10)

i s such that

m-l

(9.11)

i s bounded, where B = ( - A )1/2 d e f i n e d as i n srV.3. Using essentially 0 t h e same methods i n srV.4 we show t h a t A g e n e r a t e s a s t r o n g l y

c o n t i n u o u s c o s i n e f u n c t i o n , t h u s ill r e s u l t s i n t h i s c h a p t e r apply, i n particular those i n

sVI.5. We e x p l o i t t h e s e below.

Using t h e f u n c t i o n a l c a l c u l u s f o r s e l f a d j o i n t o p e r a t o r s we can d e f i n e f r a c t i o n a l powers

( -Ac)‘

of

-Ao

where

<=

+ iT i s

231

PAFXBOLIC SINGULAR PERTURBATION

a n a r b i t r a r y complex number:

where

P(dp)

P(-dp)

we s e t

i s t h e r e s o l u t i o n of t h e i d e n t i t y for . A

i s t h e r e s o l u t i o n of t h e i d e n t i t y for

-AO).

(so that

Due t o w e l l

known p r o p e r t i e s of t h e f u n c t i o n a l c a l c u l u s ( s e e f o r i n s t a n c e DUNFORD-

SCHWARTZ [ 1963 :11 ) we have ~ / ( - A o ) s ~ ~ / =z I/(-Ao)afi.iu!l

=

row

p2allp(-dp)ul12

.

= /l(-Ao)ou/!2

19-13)

Q : H -, H b e a l i n e a r o p e r a t o r s u c h t h a t

Let

llQd<_ K O I / ~ I I

(u

E

H),

11Qull <_ K I I I A O ~ / /

(u

E

D(A))

To o b t a i n estimates f o r

Q

(9.14)

.

i n t h e intermediate spaces

D( ( -Ao

)O")

w e a p p l y t h e t h r e e l i n e s theorem below t o t h e H-valued holomorphic function Cp(0

= Cp(a

+ iz)

THREE LINES THEORFSI

-

=

9.1.

Q(-Ao)

-cx-iT

f(c)

u

(0 < - a <- 1).

A

Let

be a Banach s p a c e v a l u e d

Then

l o g M(a)

The proof c a n b e found i n DUNFORD-SCHWARTZ a complex-valued f u n c t i o n

f(<):

<

a < I b , - Re i s a convex f u n c t i o n of 8, where

a n a l y t i c f u n c t i o n d e f i n e d and bounded i n a s t r i p

< Irn 5 < m.

(9.15)

[1958:1, p. 5201 for

t h e proof of t h e g e n e r a l c a s e i s t h e

same. Using

(9.13) we deduce from (9.14) t h a t t h e f u n c t i o n W ( 5 )

i n (9.15) s a t i s f i e s

defined

PARABOLIC S I N G W PEBTURBATION

232

The preceding two estimates a r e a notable improvement over

(5.55)

and

(5.56) i n t h a t no ad-hoc assumptions on t h e b. have t o be made and J i n t h a t t h e constants on t h e right-hand s i d e s a r e far b e t t e r i d e n t i f i e d . W e show below t h a t t h e r a t e s of convergence provided by t h e estimate (9.22) ( o r by

(5.55))

a r e b e s t possible.

233

PkRABOLIC SINGULAR FERTURBATION

EXAMF'IL 9.2.

We consider the operator

A= in

R

=

(OJ) E R1

(9.24)

I??

a n a l y s i s shows t h a t

i s isomorphic t o

L2(1(R)

j u s t t h e operator i n Example 3.10. E ( t ; E ) cun3 = w

that, for

n

Qin(t;E)

Assume

{unj

>

0

1 2

4

t

f i x e d and = e E

-n2t

+

E

n te

-n2t

and t h a t

Q2

Fourier

A(B)

is

We have E)Un1

CQ&;

Q ( t ; & )has been defined i n (3.3).

where

p.

with t h e D i r i c h l e t boundary condition

,

(9.25)

Using Taylor s e r i e s we show

2

-

2 2 -n t

2~ n e

+

O(E)

(E

-+

01 (9.26)

Q2 i s such t h a t

lMt; €1b n 3 f o r small

E

and

t.

Then

c~

4 Q 1i n ( t ;

thus, by (9.261,

c I n4te-n2t f o r small

so t h a t

9vI. 10

t.

This implies t h a t

{un] E D(A). Miscellaneous comments.

The s i n g u l a r p e r t u r b a t i o n problem f o r t h e system ( 2 . 1 ) was considered by K I S Y f k K I [1963:1] i n t h e case where d e f i n i t e operator i n H i l b e r t space.

A

i s a s e l f adjoint, positive

Similar assumptions a r e used by

SMOLLER [1965:1], [1965:2], although i n [1965:2] t h e operator A

is

not n e c e s s a r i l y p o s i t i v e d e f i n i t e and approximations of (2.2) by equations of order

>_ 3

a r e s t u d i e d a s well.

For other r e s u l t s i n t h i s

s e t u p see LATIL [1968:1]. The next s t e p i n t h e d i r e c t i o n of increasing g e n e r a l i t y w a s taken

PARABOLIC SIJ5GULAR PERTURBATION

234

by KISYfiSKI [1970:1],

SOVA [1970:2],

[ 1 9 p : 1 ] and SCHOENE [1970:1].

The f i r s t two a u t h o r s work under t h e assumptions i n t h i s chapter, t h a t

is, require

A

t o be t h e i n f i n i t e s i m a l genprator of a s t r o n g l y

continuous c o s i n e function; SCHOE,NE assumes t h a t

2

A = B

,

where

i s t h e i n f i n i t e s i m a l g e n e r a t o r of a s t r o n g l y continuous group.

B Every

such o p e r a t o r i s t h e i n f i n i t e s i m . 1 g e n e r a t o r of a s t r o n g l y continuous cosine f u n c t i o n ( s e e 111.6.24 anc, comments t h e r e ) , b u t t h e converse

111.6.5) t h u s KISYIkKP s and SOVA's r e s u l t s have

i s not t r u e (Example

a wider range of a p p l i c a b i l i t y (on t h i s matter s e e a l s o NAGY [1976:1]). The most p r e c i s e r e s u l t s a r e thoze o f KISYfiSKI [1970:1] who proved

5.5 and Thecirem 5.6 f o r u o ( ~ ) ,u ~ ( E ) f i x e d ; t h e c r u c i a l s t e p i s t h e e s t i m a t e (5.7) using t h e t h e o r y of a l t e r n a t i n g functions. SOVA's r e s u l t s i n [15"p:2] and [1972:1]a r e somewhat l e s s

Theorem 5.1,

Theorem

p r e c i s e and obtained i n a d i f f e r e n t way; i n s t e a d of w r i t i n g t h e s o l u t i o n s of ( 2 . 1 ) e x p l i c i t l y , S@VAe s t i m a t e s t h e i r Laplace transforms and uses t h e Post-Widder formula

(1.3.14).

A treatment of t h e s i n g u l a r

p e r t u r b a t i o n problem i n another v e i n w a s given by GRIEGO-HERSH [1971:1]. Convergence of t h e s o l u t i o n of (2.1) using e x p l i c i t formulas of t h e type of (2.14) w a s s t u d i e d bj' DETTMAN [1973:1].

The r e s u l t s i n

t h e f i r s t f o u r s e c t i o n s i n t h i s chapter a r e i n t h e a u t h o r [1983:4]. The main novelty h e r e i s t h a t t h e y are uniform with r e s p e c t t o t h e i n i t i a l conditions

u o ( ~ ) ,u l ( ~ ) as long as

bounded, although t h e r e i s a n i n i t i a l l a y e r

convergence i s l o s t .

Section

uO(e), E2 ul(&)

0

< -t < - t(E)

remain

where

5 is a l s o t a k e n from [1983:4]; t h e r e s u l t s

are s l i g h t g e n e r a l i z a t i o n s of t h o s e i n KISYfiSKI [1970:1] mentioned above i n t h a t t h e i n i t i a l conditions a r e allowed t o depend on "intermediate" e s t i m a t e s in Theorems5.U and

5.13

The

are i n [1983:4].

The treatment of t h e nonhomogeneous equation i n Sections

i s a l s o i n t h e author [1983:4].

E.

5 and 6

Theorem 6.2 i s due t o DE SIMON [1964:1].

Section 8 on asymptotic development of t h e s o l u t i o n of (2.1) i s a l s o i n t h e author [1983:4].

Thr asymptotic s e r i e s i s a n a b s t r a c t

v e r s i o n of a w e l l known s e r i e s u:ed i n d i v e r s e s i n g u l a r p e r t u r b a t i o n problems for p a r t i a l d i f f e r e n t i a l equations ( s e e f o r i n s t a n c e GEEL [1978:1]).

The p a r t i c u l a r form used h e r e i s modelled on HSIAO-WEINACVT

[1979:1], whoccnsider t h e h e a t equation i n dimension

1 i n a space of

continuous f u n c t i o n s .

The "numerical perturbation" formula (2.8) (due t o SOVA [1970:4])

PARABOLIC SINGULAR PERTURBATION

235

i s a n abstract v e r s i o n o f a w e l l known f o r m u l a for t h e s o l u t i o n of t h e t e l e g r a p h i s t ’ s e q u a t i o n (or of t h e Klein-Gordon e q u a t i o n ) i n terms o f t h e s o l u t i o n of t h e wave e q u a t i o n .

[1970:4].

a r e i n SOVA

O t h e r e x p r e s s i o n s of t h e same t y p e

They a r e p a r t i c u l a r c a s e s o f t r a n s m u t a t i o n

f o r m u l a s , which can b e r o u g h l y d e s c r i b e d as t r a n s f o r m i n g t h e s o l u t i o n o f a d i f f e r e n t i a l e q u a t i o n i n t o t h e s o l u t i o n of a n o t h e r d i f f e r e n t i a l See t h e a u t h o r [ 1983:11

equation.

f o r a d d i t i o n a l i n f o r m a t i o n and

references. An i m p o r t a n t a p p l i c a t i o n of (2.8) i s t h a t of estimatirg t h e growth 2 of t h e s o l u t i o n s of u ” ( t ) = ( A + b I ) u ( t ) f o r a n y complex b i n

terms of t h e s o l u t i o n s of

u “ ( t ) = Au(t).

On t h i s s u b j e c t , we i n c l u d e

some r e s u l t s ( E x e r c i s e s 3 t o 8 ) t a k e n from t h e a u t h o r Let

EXERCISE 1.

a

>

0

be a r b i t r a r y ,

m ( A ) = (ah2

Show t h a t

m’ (A)

a

Let

8.

(A >_ 0)

(10.I)

>

a n a r b i t r a r y complex number

0,

A

an

Show t h a t t h e Cauchy problem f o r

i s w e l l posed i n

-X

<

t

< m.

s o l u t i o n of ( 1 0 . 2 ) i n terms of by

+ A) 1h

is alternating.

EXERCISE 2 . operator i n

[1985:2].

Write a n e x p l i c i t f o r m u l a f o r t h e

@(6),

t h e cosine function generated

A.

I n Exercises

3 t o 8,

@(t;A)

i s a s t r o n g l y continuous c o s i n e

f u n c t i o n with infinitesimal generator

EXERCISE A

3.

Let

A

@(t;A

+

Il@(t;A

(Hint:

b21),

+

b

i s a complex number

b e a normal o p e r a t o r i n a H i l b e r t s p a c e

generates a cosine function Il@(t;A)11 < - e o s h c*rt

Then

and

Re b > - 0.

i n t h e r i g h t h a l f plane

such t h a t

A

@(t;A)

< t < m).

u s e E x e r c i s e 11.5).

(10.3)

(a

t h e c o s i n e f u n c t i o n g e n e r a t e d by

b21)ll < - cosh(w2 + \ R e bl

)d 2 t

H

with

2 A + b I


(a

<m).

satisfies

(10.4)

236

PARABOLIC S I I J G W PERTURBATION

EXERCISE

4.

Let

b e t h e i d i n i t e s i m a l g e n e r a t o r of a s t r o n g l y

A

C( t; A )

continuous cos i n e f u n c t i o n

Show t h a t

+

@(t;A

IlC(t;A

+

b21)

satisfics

5 Co

b21)I/

< t < m ) , (10.6)

cosh wt + CoKw(b;t)

K ( b ; t ) = Ku(b; tl )

where

satisfying

w

(a

and

Kw(b;t) = [ b

for

t

(10.7)

c o s h ws ds

2 0.

EXERCISE 5. function

Show t h a t t h e r e e x i s t s a s t r o n g l y c o n t i n u o u s cosine satisfying

@(t;A)


C ( t ; A ) = c o s h Wt

(a


(SO.8 )

9

and

+

Il@(t;A

(Hint:

2

b I)ll = cosh

i*rt

+


Ku(b;t)

use t h e c o s i n e f u n c t i o n (11.6.10) i n t h e s p a c e

~ ( x )d e f i n s d i n

continuous f u n c t i o n s Iu(x)(ewx

-

0

EXERCISE

1x1

as

6.

4

m,

4

< m )

(a

<x<

endowed w i t h t h e norm

.

Ew of a l l

and such t h a t

llullw

=

rnax(u(x)leWL).

Using t h e a s y m p t o t i c development of t h e Bessel f u n c t i o n

(GRADSTEIN-RYDZYK [1963:1,p. 9751 o r WATSON [1948:1,p. I1(x) show t h a t t h e r e e x i s t a c o n s t a n t C s u c h t h a t , i f 0 > -0

Ko(Q;t)

<_

C$2(t11/2

( 4



Kw ( i r l ; t ) < - C q l / 2 1 ~ ~ 1 / 4 ~ ~ I t( -I m < t < Show t h a t , g i v e n

EXERCISE

(10.9)

7.

6

>

0

there exists

Using the f a c t t h a t

c

>

0

I1(x)

1991)

,

(10.10)

m).

(10.11)

such t h a t

>_

0,

show: if

b

>_

0

237

PARABOLIC SINGULRR PERTURBATION

and

@(t;A)

i s a c o s i n e f u n c t i o n s a t i s f y i n g (lo.?),

I/@(t;A

(Hint:

+

b21)Il

<_

then

(a

c o s h &).

a p p l y (2.8) t o t h e s c a l a r c o s i n e f u n c t i o n

EXERCISE 8. H

Let

Ko(b;t), Kw(b;t) @(t;A)

for b

(10.14)

Using

(10.6), (10. lo), (10.11)and (10.14) plus the Phragm6n-Lindel.cif o b t a i n upper bounds f o r

.

< t < m)

Co cosh(w2 + b 2 ) l I 2 t

theorem

complex.

be a c o s i n e f u n c t i o n i n a H i l b e r t space

satisfying Il@(t;A)II

5 co

Show t h a t t h e r e e x i s t s a c o n s t a n t Il@(t;A

(Hint:

(C

<

t

.

< m )

(10.15)

depending o n l y on

+ b*I)II < - CeIRe b 1

It(

Co

< t < m)

(a

such t h a t

.

use t h e t h e o r y i n C h a p t e r V, s p e c i f i c a l l y C o r o l l a r y

(10.16)

v.6.5).

FOOTNOTES TO CHAPTER V I

(1) We note a n i n c o n s i s t e n c e of n o t a t i o n w i t h $111.4, where w e use -b2 i n s t e a d o f b2.

(2) S t r i c t l y speaking, we s h o u l d w r i t e since

@($) i s o n l y s t r o n g l y c o n t i n u o u s .

@(0)f ( ~ sE)dU ; i n t h e integrand, W e s h a l l ignore t h i s here

and i n o t h e r p l a c e s .

( 3 ) The l e t t e r

4

i s used w i t h a d i f f e r e n t meaning i n C h a p t e r s 111

and V I I I .

( 4 ) R e c a l l t h a t , due t o t h e f i r s t i n e q u a l i t y (11.2.22) we have

( 5 ) T h i s i s t h e o n l y p l a c e where t h e f a c t t h a t ( s e e Remark 4.5 on t h e c a s e w = 0).

w

>

0

i s significant

238

CHAPTER VII

OTHER SINGULAR PERTUREATION PROBLEMS

A .

PvII.1

I n r e l a t i v i s t i c quantum mechanics ( s e e SCHOENE

[1970:1]) one

considers

f u n c t i o n s of t h e form

2

u ( x , t ) = v(x,t)exp(imc t / h ) where

v

i s a s o l u t i o n of t h e Klein-Gordon equation f o r a f r e e p a r t i c l e

(here m

= -h c Av

i s t h e mass of t h e p a r t i c l e ,

t h e speed of l i g h t ) .

h utt

-

h/&c2

<< 1 we expect

u

i s Planck's c o n s t a n t and

h

If f o l l o w s t h a t

;f nC

since

+ m2c 4v

22

2

h vtt

u(x,t)

iu

-

c

s a t i s f i e s the equation

-Au;

t-an

t o d i f f e r by l i t t l e from t h e s o l u t i o n

of t h e Schrb'dinger equa.tion

It i s t h e n natural t o c o n s i d e r t h e e q u a t i o n &

where

2

utt(x,t;E)

- i ut ( x , t ; & ) =

u ( x , t ; & ) = u ( x l J . . .,x,,t;&)

i n t h e "space v a r i a b l e s "

xl,.

. .,xD

,

(1.1)

i s a d i f f e r e n t i a l operator

A

like rn

m

AU =

and

Au(x,t;E)

5 a (x)DjDku + j=1 7 b.(x)Dju + j=1 k=l jk

C(X)U,

(1.2)

J

or a s u i t a b l e higher order v e r s i o n ; t h e problem a t hand i s t h a t of determining i n which way ( i f any) u(x,t;E)

+u(x,t)

(1.3)

is

239

OTHER PROELEMS

as

where

+O,

E

u(x,t)

i s t h e s o l u t i o n of t h e unperturbed equation

(1-41

ut ( x , t ) = iAu(x,t).

This w i l l be done i n t h e r e s t of t h e cha.pter for a n a b s t r a c t v e r s i o n of

(l.l), (1.4).

The Schrgdinger s i n g u l a r p e r t u r b a t i o n problem.

k I . 2

Throughout t h i s c h a p t e r A

w i l l be t h e i n f i n i t e s i m a l generator of

a s t r o n g l y continuous cosine f u n c t i o n

(that is, &

C(t)

8 ) . We consider t h e Cauchy problems u"(t;E) - iu'(t;E) = Au(t;E) + f ( t ; E ) A

E

i n t h e Banach space

E

2

(-m


m),

(2.1) U(0;F)

= U,(€),

U'(0;E) = UJE),

iAu(t)

+

u(0) = u

and

u'(t)

=

if(t),

0

(-m


(2.2)

m).

The Schrgdinger s i n g u l a r p e r t u r b a t i o n problem i s t h a t of showing t h a t +u(t)

U(t;E)

as

E

40,

where convergence i n (2.3) can be understood i n v a r i o u s senses.

A s u p e r f i c i a l k i n s h i p of ( 2 . 1 ) and (VT.2.1)

factor

-i

(2.3)

before

u'(t)

i s obvious, although t h e

causes r a d i c a l d i f f e r e n c e s between t h e s e

i n i t i a l v d u e problems, as put i n evidence a l l throughout t h i s c h a p t e r . An e x p l i c i t s o l u t i o n of ( 2 . 1 ) can be constructed by t h e methods of p1.2

( o r i t can be formally obtained from where

u(t;E)

(V1.2.14) considering

u(it;iE),

i s t h e f u n c t i o n i n (2.14), and keeping i n mind t h a t

T(-x)

= -I ( i x ) = -iJ1(x)). I o ( - x ) = I ( i x ) = J O ( x ) and t h a t 0 1 r e s u l t i s a n e x p l i c i t expression f o r t h e s o l u t i o n of (2.1):

The end

OTHER PROBLEMS

240

+ ’-

rt

Ci(t

-

s ; & ) f ( s j E ) ds

0

The obvious d i f f e r e n c e between t h e r e p r e s e n t a t i o n ( 2 . 4 ) f o r ( g e n e r a l i z e d ) s o l u t i o n s of t h e i n i t i a l value problem ( 2 . 1 ) and t h e similar r e p r e s e n t a t i o n (VI.2.14) f o r t h e i n i t i a l v a l u e problem (VI.2.1) l i e s i n t h e d i f f e r e n t asymptotic behavior of t h e i n t e g r a n d s and t h e exponential factors.

I n (VI.2.14) we can combine t h e asymptotic e s t i m a t e 1 1 ~ x ) =l o(x-1/2ex)

( s e e 9VI.2)

IlC( )

11

(2.5)

+ w

e -t/2e2

with t h e r a p i d l y d e c r e a s i n g f a c t o r

bound (VI.2.17) f o r E +O

x

as

and t h e

t o show t h a t we can t a k e limits d i r e c t l y as

using t h e dominated convergence theorem:

t h e r e s u l t s a r e uniform

convergence r e s u l t s of t h e type of Theorems VI.3.6 and

VI.3.8. I n t h e

same way w e o b t a i n uniform bounds f o r d e r i v a t i v e s (Theorem vI.4.1). I n c o n t r a s t , t h e asymptotic e s t i m a t e s =

IJ”(X>I

combined with t h e i n d i f f e r e n t f a c t o r

o( X W 2 ) e it/E2

(2.6) in

( 2 . 4 ) do not allow

d i r e c t passage t o t h e l i m i t ; i n f a c t , doing so formally, one c a n only hope f o r a r e l a t i o n similar t o (VI.3.40) Si(t)u = S(it)u

for

5,Gi

with l i m i t

=

which does not make sense even i n t h e most f a v o r a b l e case where i s uniformly bounded i n

-m


a.

llc(t)/l

Hence, computation of t h e l i m i t

( 2 . 3 ) w i l l have t o be c a r r i e d o u t by i n d i r e c t means.

I n particular

(and i n c o n t r a s t with t h e p a r a b o l i c s i n g u l a r p e r t u r b a t i o n problem) t h e e x i s t e n c e of t h e group

S i ( t ) = exp(iAt) = S ( i t )

i s not assured by t h e

existence-uniqueness assumption f o r ( 2 . 1 ) above but w i l l follow from t h e s t r o n g e r assumptions i n bVII.2.

(we note i n p a s s i n g t h a t S i ( t ) i s t h e ”boundary v a l u e ” of t h e a n a l y t i c semigroup c o n s t r u c t e d i n 4VI.2).

A s it may be expected, t h e convergpnce results for t h e o p e r a t o r s and

Gi(t;E)

Qi(t;E)

(which w i l l be c a l l e d i n what f o l l o w s t h e p r o p a g a t o r s o r

s o l u t i o n o p e r a t o r s of ( 1 . 1 ) - ( 1 . 2 ) ) a r e d i f f e r e n t i n c h a r a c t e r from t h o s e

241

OTHER PROBLENS

for

&(t;&), q t ;

E )

i n Chapter VI, e s p e c i a l l y those i n 5VI.2 t o 5VI.4.

The main d i f f e r e n c e s between t h e r e s u l t s i n t h i s chapter and those i n Chapter V I can be summarized a s follows: There a r e no uniform bounds ( t h a t i s , bounds i n t h e norm of

(a)

for

(E))

(Example (b)

@i(t;E)

f$(t;E),

,...

of t h e type of (VI.4.20),

4.9). There a r e no convergence r e s u l t s i n t h e norm of

‘&(t;s), e i ( t ; E ) ,

e;(t;E),

...

@i(t;E),

m.4.6 o r Theorem VI.4.7: convergent (Example 4 . 6 ) . Theorem

for

of t h e type of Theorem VI.3.11, Ei(t;&) i s not even s t r o n g l y

6i(t;F)

The nonexistence of uniform bounds f o r

inhomogeneous equation ( 2 . 1 ) :

(E)

i n fact

well a s t h e lack of convergence of

(c)

(m.4.54).

E.(t;E)

and

@i(t;&), as

h a s consequences f o r t h e

i n fact,

Vr.7.4 f o r HGlder

There a r e no analogues of Theorem

continuous f u n c t i o n s o r of Theorem

v1.7.6

f o r functions i n LP

.

Among t h e r e s u l t s on t h e p o s i t i v e side, w e have

(a)

There e x i s t s a n analogue of Theorem VI.7.2

c l a s s of o p e r a t o r s (e)

A

(Theorem

for a restricted

7.1).

Most of t h e r e s u l t s i n p V I . 5 on r a t e s of convergence can be e s p e c i a l l y Theorem 4.1).

proved i n t h e present s i t u a t i o n ( s e e OVII.4, We mention f i n a l l y t h a t t h e theory i n

h . 8 concerning asymptotic

s e r i e s does not seem t o extend t o t h e Schrodinger s i n g u l a r p e r t u r b a t i o n problem:

can be formally defined

although t h e asymptotic s e r i e s ( V I . 8 . 2 0 )

i n t h e present s i t u a t i o n , t h e

L1

norm of

v (t/E n

2

)

over any f i n i t e

i n t e r v a l cannot b e conveniently bounded.

4VII.3

Assumptions on t h e i n i t i a l value problem.

A

A s pointed out i n m I . 2 , t h e f a c t t h a t

t h a t t h e Cauchy problem f o r ( 2 . 1 ) i s well posed.

E

8

allows u s t o show

The key condition

on ( 2 . 1 ) t h a t w i l l allow u s t o prove (2.3) i s ASSUMPTION

3.1.

& e J

Q.(t;E),

ei(t;E)

be t h e propagators of ( 2 . 1 )

242

OTmR PROBLEMS

Then t h e r e e x i s t c0nstant.s

Ilgi(t;E)lI

Co C

'

5

1'

and

t

independent of

w

E

such t h a t

ll~i(t;&)ll 5 ClewltI

COewlt1,

Assumption 3.1 can be g i v e n n c o n s i d e r a b l y simpler form a s f o l l o w s . be a s o l u t i o n o f ( 2 . 1 ) . it / 2 ~ * equivalently, u ( t ) = e v(t/)).

Let

v"(t)

Then

v(t)

1 -I)v(t)

(A

=

v ( t ) = e-it/2Eu(Et)

Set

u(t;E)

(-m

or,

satisfies


m),

4E

V(0) =

(3.2) =

V'(0)

UO(E),

i -2E

U0(E)

+

€U1(E).

u ( t ) can be e x p l i c i t l y w r i t t e n i n terms of t h e p r o p a g a t o r s (2~)-~1),8(t;A (2E)-21) of t h e i n i t i a l value problem (3.2):

Accordingly,

C(t;A

-

-

2

u(t) 2

+

-

C(t,/E;A

=

8(t/&;A

-

(21

(2i)-*I)uO(E)

&

)-*I)(-

uO(E)

+

(3.3)

Eu1(E)).

(We note, i n c i d e n t a l l y , t h a t f o r m t l a ( 2 . 4 ) c a n be obtained from

(3.3)

although t h i s w i l l p l a y no r o l e i n what f o l l o w s ) .

by Lemma VI.2.1,

Using

( 3 . 3 ) we deduce t h a t 2 &.(t;E:) = e i t / : ' E

-

2 i(2&)-1eit/2E

6.(t;E)

C(t;A

-

(t/&;A

= E

(2E)-21) = e

8(t;A

-

S(t/E;A

-

-

(2E)-21),

-1 it/2E2 e 8(t/E;A

-it/2cG i

(2E)-21)

-

( t ; E ) + (i/2)e

(2E)-"I)=

(3.4)

,

(2E)-21)

-it/2EG

Ee-it/2EG i ( E t ; E )

i ( t;E) '

.

Accordingly, Assumption 3.1 v i l l hold i f and only i f

(3.5) (3.6)

(3-7)

243

OTHER PROBLEMS

We s h a l l examine i n pvII.5 a c l a s s of o p e r a t o r s s a t i s f y i n g Assumption 3 , l :

pv11.6, t h e s e o p e r a t o r s i n c l u d e t h e p a r t i a l d i f f e r e n t i a l

as s e e n i n

o p e r a t o r ( 1 . 2 ) ( w i t h boundary c o n d i t i o n s ) , t h e c o e f f i c i e n t s s u i t a b l y restricted. We p o i n t o u t below a consequence o f Assumptios 3.1. THEORJ%V 3 . 2 . i A

and

w

Proof:

Coeu:lt/

u

2 D(A )

E

Si(<)

such t h a t


(-m

(3.9)

m),

t h e n it f o l l o w s from f o r m u l a ( 2 . 4 ) ( o r from

t h e c o n s i d e r a t i o n s a t t h e b e g i n n i n g of t h i s s e c t i o n ) t h a t four times continuously d i f f e r e n t i a b l e , E

2

QI'"(t;E)u

-

Then

(3.1).

t h e constants i n If

3.1.

s a t i s f y Assumption

g e n e r a t e s a s t r o n g l y c o n t i n u o u s group

Ilsi(t)ll 5 C0

A

Let t h e operator

i&y(t;E)u

=

K'll(t;E)U

AQy(t;E)u

E

(-m

K,(t;&)u I

and

D(A)


m).

On t h e o t h e r hand, we have E;(o;E>~ Ey'

= E -2 ( A E ~ ( o ; E ) ~

= E

(0;E)U

+ iQ!(O;&)u)

-2

+

(AEi(0;E)u

= E-'A~,

iEy(0;E)u) = 1

i E

-4 Au,

t h u s we o b t a i n , a p p l y i n g (2.4),

+

Ej'(t;E)u = E - 2 Q . ( t ; f ) A u Accordingly, it f o l l o w s from Assumption

5

l/Qy(t;&)U/l

cE-*ewltl1lAu(l

We t a k e now a sequence

later.

For u

E

= A(Qi(t;En)U

Using a g a i n

(2.4),

n

1,

3.1 t h a t

(-m

1> El >

E2


C m,

>

.. . >

0 < E

0

9

Eo).

t o be s p e c i f i e d

we have

D(A)

E,(E;(tjEn)U 2

CE

i& -2 G i ( t ; f ) A u .

-

Ei(t;En+l)u)

-

-

i(Ei(t;En)u

a i ( t ; E n + l ) ~ )+ ( E 2 n

-

-

Ei(t;En+l)u)

E 2n + l ) ~ ~ ( t ; E n + l ) ~ .

=

is

244

OTHER PROBLEMS

In view of (2.12) and Assumption 7.1, i f

Selecting now a sequence (say,

En =

n-1/2)

we show t h a t

t

E

2 D(A )

we have

t h a t t e n d s t o zero s u f f i c i e n t l y slowly

[En)

uniformly with respect t o

u

i s a Cauchy sequence,

{PSi(t;En)u}

on compact subsets of

t h e uniform bound (3.1) and t h e denseness of

D(A2)

-

m

Using

m.

we deduce t h a t

converges strongly, uniformly on compacts of

{Qi(t;en)J t o a s t r o n g l y continuous operator valued f u n c t i o n


Si( s )

-

m


m

satisfying

t h e e s t i m a t e (3.9). Denote by

i.( \;E)u

equation (2.1) f o r

t h e Laplace transform of

Ki(t;u)

Qi(0;&)u = u, Q;(O;E)u

= 0

Writing

Ki(t;E)u.

D ( A ) ) and t a k i n g i n t o account t h a t we deduce, a f t e r i n t e g r a t i n g from 0 t o

(u

E

t

two times, ds

.

Taking Laplace transforms, we o b t a i n

where t h e i n t r o d u c t i o n of

A

under t h e i n t e g r a l s i g n i s e a s i l y j u s t i f i e d .

It follows t h a t c.(h;E)u = (E2h 1

for

A

>

w.

Putting

E = E

n with

-

2 2 i)R(E h

-

ih;A)u

ren 1 a sequence a s above and

t a k i n g l i m i t s , we o b t a i n Si(h)u = -iR(-h;A)u for

u

E

D(A)

and thus, by denseness of

follows t h e n from Theorem 1.3.4 t h a t group with i n f i n i t e s i m a l generator

= R(h;iA)u

D(A),

Si(i) A.

f o r all u

E

E.

It

i s a s t r o n g l y continuous

This ends t h e proof of Theorem

3.2. We note t h a t o u r proof i n c l u d e s a r e s u l t on convergence of to

Si(t).

However, i t ' s not worth formulating since it w i l l be

considerably improved i n next s e c t i o n .

gi(t;E)

245

OTHER PROBLEMS

h1.4

The homogeneous e q u a t i o n : be an element o f

u

Let

convergence r e s u l t s .

D(A).

u(t;&)

=

The f u n c t i o n

&!(t;e)u 1

i s a g e n e r a l i z e d s o l u t i o n of t h e homogeneous e q u a t i o n (2.1) w i t h u(0;r)

= 0,

= &;(o;E)u

U'(0;E)

-2

AQ.(o;€)~+ i E

= E

1

-2 ~ ( o ; & ) u= 1

E-2~u.

It f o l l ow s t h a t Ei(t

;E)u

=

6. ( t ; & ) A U

(4.11

*

1

On t h e o t h e r hand, v ( t ; & ) = G!(tjE)U 1

is a l s o a g e n e r a l i z e d s o l u t i o n of (2.1) w i t h = C2U,

v(0;E) = Q ( 0 ; E ) U

= v ( o ; E ) ~= F

-2

+

A ~1. ( o ; E ) ~

V'(0jE)

iE-2Gi(o;E)u

= iE

-4u,

so t h a t d!(t;E)u 1

Since a l l operat o r s i n

u

t o all

E

E.

= €

(4.2) a r e

-2

+

Ei(t;E)u

i E

-2

fj(t;E)U.

(4.2)

bounded, t h e e q u a l i t y c a n be extended

We combine (4.1) and (4.2), t h e l a t t e r i n e q u a l i t y w r i t t e n

for a n element o f t h e form Au.

The r e s u l t i s

& i ( t ; E ) u = iA&.(t;E)u

-

i& 2% ( t ; & ) A u .

(4.3)

1

iA

Since

semigroup

i s t h e i n f i n i t e s i r n a i l g e n e r a t o r of t h e s t r o n g l y c o n t i n u o u s Si(z),

we o b t a i n a p p l y i n g f o r m u l a

gi(t;E)u

-

Si(t)u

=

u

E

2

D(A )

Ei(t;E)u

-

-ii2ktSi(t

= -iE2Lt

If

(1.5.7)

E$(t

that

s)Gi(s;E)Au d s

-

s;&)Si(s)Au d s . ( 4 . 4 )

we can i n t e g r a t e b y p a r t s :

- Si(t)u=

- E ~ B(t;&)Au . 1

- iE

- s;&)Si(s)A 2u

ds.

E s t i m a t i n g t h e i n t e g r a l i n (4.5) i n a n o b v i o u s way, w e o b t a i n

(4.5)

246

OTHXR PROELEMS

THEOF34 4.1.

let

u(t;&)

L A A be a n o p e r a t o r s a t i s f y i n g Assumption 3.1 @ u(t)

be a s o l u t i o n of t h e homogeneous problem (2.1),

a s o l u t i o n of t h e homogeneous problem ( 2 . 2 )

with

uo

E

D(A2).

Then we

have iiu(t;E)

+

-

U(t)ll

5

C l & 2 e W I t I ( ~ ~ ~ u+g C~ i~

11)

I ~ I I I A * U ~

~ o e U : ~ t ~ ~ ~ u o ( ~ ) - u O ~ ( + ~ ~ ~ 2( -emU
The proof of ( 4 . 6 ) i s e s s e n t i a l l y t h e same a s t h a t of (VI.5.27): we o n l y have t o note t h a t

-

U(t;E)

U(t)

-

+ Ei(t;E)(Uo(E)

Theorem

4 . 1 i m p l i e s t h a t when u

0

-


-m

Si(t)uo

u o > + e i ( t ; E ) ( E 2U 1 ( E ) )

2 D(A )

E

Ib(t;E)

uniformly on compacts of

-

= gi(t;E)Uo

-

(4.7)

we have

(4.8)

u(t)/l = o(E2) if

m

Estimates of t h e same sort c a n be e a s i l y obtained f o r t h e d e r i v a t i v e u'(t;E)

if

uo E D ( A 3 )

and

uO(E)

E

In fact,

D(A).

v(t;&) = u'(t;&)

i s t h e generalized s o l u t i o n of t h e homogeneous equation ( 2 . 1 ) with

v(O;E)

=

u'(0;~)=

y ( ~v'(O;E) ), v(t) = u'(t)

On t h e o t h e r hand, equation (2.2)

=

u"(0;~)=

-2 E

(Au~(E)

+

iul(E)).

(4.10)

i s t h e s o l u t i o n of t h e homogeneous

with ~ ( 0 =) u ' ( 0 )

Applying Theorem 4 . 1 t o

THEOREM 4.2.

Let

v(t;&), v ( t )

A

=

iAu

(4.11)

0'

we o b t a i n

be a s i n Theorem 4.1 and l e t

s o l u t i o n o f t h e homogeneous problem ( 2 . 1 )

u(t;&)

with uO(E) E D ( A ) , with uo E D(A').

a s o l u t i o n of t h e homogeneous problem ( 2 . 2 )

u(t)

Then

we have IIu'(t;E)

+

-

u'(t)ll

ewltl(COllul(t)

5

-

+ C01t~I~A3uol~)

C1&2e"ltl(llA2uoll

iAuo/I

-t C1Ilu,(E)

-

iAuO(E)/I).

(4.12)

247

OTHER PROBLEMS

uo

It f o l l o w s from t h i s r e s u l t t h a t i f

D(A3)

E

and

uO(E) E

D(A)

then

- u f ( t ) / (=

IIU'(t;E)

uniformly on compacts of

\iul(

E)


-w

- i A u0 11 = O( E2 )

and

llul(

if

m

E)

E>II=O(E 2)

- iAuO

as

(4.14)

E

or, e q u i v a l e n t l y , i f

Theorems 4 . 1 a n d 4 . 2 a l l o w u s t o deduce convergence r e s u l t s f o r a r b i t r a r y i n i t i a l conditions.

Let

THEORDd 4.3.

u(t;E)

homogeneous problem ( 2 . 1 )

with

be a g e n e r a l i z e d s o l u t i o n of t h e uO(E),

ul(E)

with

s o l u t i o n of t h e homogeneous problem ( 2 . 2 )

2

uO(&) + u o ,

E u,(E)

E, u ( t )

E

as

-0

uo

E

E

-0.

a. g e n e r a l i z e d Assume t h a t

E.

(4.16)

Then u(t;E)

uniformly on compacts o f

Proof:

-u ( 0 ) u(t) = u. Let

Pick

+ >

E

2 D(A )

such t h a t

/lu-u0ll

5

6.

uO(E),

-

u(t)ll

(I

l[AFll +

4 . 1 we deduce t h a t

I: I I 4 t ; E ) -

U(t>ll

+

II3t) -

I

C o l t llA2'ull) + CoemJt lIuO(E)

u(t)ll

- ull

.

(4.18)

s u f f i c i e n t l y small we can obviously make t h e right-hand

0

5 2C06eUB i n

THEORFM 4.4. uo

E

It1 5 a > 0.

Let u ( t ; & ) , D(A)

u(t)

T h i s ends t h e p r o o f .

be as i n Theorem 4.3.

Assume

and t h a t

AuO(&)-Au, Then -

u

Clelult~E211u1(E)~/+ Cobemltl

side o f (4.18)

that -

(4.17)

4 0

E

m.

and choose

Applying Theorem

< C1E2ewlt

&

0


as

b e t h e s o l u t i o n of t h e i n i t i a l value problem ( 2 . 2 ) with

Il.(t;E)

Taking

>

6

-a

+u(t)

ul(&) + i A u 0

as

E -0.

(4.19)

248

OTHER PROBLEMS

+

u'(t;E)

uniformly on compacts of

Proof:

Given

Ilu'(t) -

so t h a t

6

>

-a

0


u

u ' ( t ) / l = IlSi(t)(u

u

E

(4.20)

-0

m.

-

choose

We use t h i s time (4.12) with

as

I'(t)

D(A3 )

E

-

such t h a t

llAu

- AFoII 5

i n s t e a d of

uo:

6

5 C06elUlt1.

uo)ll = IISi(t)(AE-Auo)II

d e t a i l s a r e omitted.

We have a l r e a d y pointed out t h a t no analogues of t h e uniform bounds i n Chapter VI o r of t h e uniform convergence r e s u l t s t h a t we e s t a b l i s h e d t h e r e e x i s t i n t h i s case.

To p u t t h i s i n evidence we u.se ( a s we d i d

a l r e a d y i n Chapter VI, f o r i n s t a n c e i n &le VI.3.10) t h e H i l b e r t E = A2

space

1IuII

that

( r1 ~ ~ 1 ~ <) " ~ and

=

m

A

fun; 15 n <

such

m 3

i s t h e ( s e l f adjoint) operator

{ F n ] i s a sequence o f r e d l numbers bounded above ( t h e domain of

where

A

o f all complex valued sequences

fun]

c o n s i s t s of all

belongs t o

a2).

such t h a t t h e r i g h t hand s i d e of (4.U)

We shall a l s o u s e t h e space

ordinary Euclidean norm and a n o p e r a t o r

E =C

m

with i t s

o f t h e form ( 4 . U ) .

A

We

check e a s i l y t h a t t h e s o l u t i o n o p e r a t o r s o f ( 2 . 1 ) corresponding t o t h e

(4.22)

(A:

where

E),

hi( E)

a r e t h e r o o t s of t h e c h a r a c t e r i s t i c polynomial

+ n

2E

2

2E

( n o t e t h a t , since t h e sequence

h i ( & ) w i l l be d i f f e r e n t f o r

h:(E),

EXAMPLE 4.5.

to

fwn3

uo

even i f

i s bounded above, t h e r o o t s E

s u f f i c i e n t l y small).

Convergence i n Theorem

lluoI/ i s bounded.

4.3 i s not uniform with r e s p e c t

We t a k e

E = A

2

, LLn

= -n

2

,

249

OTHER PROBLEMS

(4.24)

(4.25)

(4.26) I n c o n s t r a s t , t h e r e i s uniform convergence o u t s i d e of a n i n i t i a l l a y e r 0

5t 5t(E)

s e e Theorem VI.3.11.

i n t h e parabolic case:

EXAMPLE 4.6. convergence of

2

The c o n d i t i o n

u(tjE)

& ul(E)

i n Theorem 4.3

-0

cannot be weakened.

E = Q1

We t a k e

for and

r e w r i t e formula (4.23) a s follows:

6.(tjE)E 1

2

ul(&) = 2 i e

- 4 E2 & 9 / 2 & 2 ) t 2 1/2 (1 - 4 E w )

2 ul(&),

it/2e2 s i n ( ( 1

which does n o t have a l i m i t as

1-0 u n l e s s

E

E

2

E

(4.27)

ul(E) -0.

For t h e a b s t r a c t p a r a b o l i c case see Theorem VI.3.11: we n o t e that t h e r e i s a g a i n convergence o u t s i d e of a n i n i t i a l l a y e r .

EXAMPLE 4.7. possible.

The r a t e of convergence i n Theorem 4.1 i s b e s t

We use t h e space

E

=

3

and t h e o p e r a t o r

A

i n (4.U).

Write (4.22) i n t h e form Ei(tjE)

bn3 =

P,(t;E)Un

3

(4.28)

A f t e r some computation with Taylor s e r i e s we check t h a t

(4.29) where

as

r n ( & ) i s a r e a l number.

E 4 0 .

Assume t h a t

Rewrite t h i s i n e q u a l i t y a s

f u,)

E

Q 2 i s such t h a t

250

OTHER PROBLEMS

iVnt

~ ~ - ~ - pe ~ I (1un ~ -c ;c2.~ ) Taking (4.29) i n t o account, we obtain, a s s u m i n

that

p n

i n (4.2l),

+ m

(4.10) u

so t h a t

2

D ( A ).

E

We r e c a l l t h a t i n t h e a b s t r a c t p a r a b o l i c case,

convergence of order

u

that

D(A)

E

EXAMPLE even when

<

V

4.8.

m.5.5).

Convergence i n Theorem

i n (3.1).

0

0 , u O ( E ) = u,

eA+(E)t

can be obtained under t h e weaker assumption

( s e e Theorem

u1 =

uniformly t o

E2

u

1

We t a k e h e r e

the fact that

(E) = 0 ;

S i ( t ) = eiWt

4.9.

i f a bound of t h e type of

E +O;

e i't

uniformly i n e;(t;E).

2

t

0.

We o b t a i n

(4.28),

(VI.4.20) were v a l i d :

A

would be a bounded o p e r a t o r , absurd i f

VII.5

does not converge

Q.(t;E)u 1

There a r e no uniform bounds f o r

from (4.23) t h a t , w i t h t h e n o t a t i o n of

as

S'(t)

i s unbounded.

V e r i f i c a t i o n of t h e hy-poteses.

We examine i n t h i s s e c t i o n o p e r a t o r s t h a t s a t i s f y Assumption beginning with t h e case where normal o p e r a t o r .

is, that

0,

i s a n obvious consequence of t h e f a c t t h a t

does not converge uniformly t o

EXAMPLE

4.3 i s n o t uniform i n t > E = C1, A u = ku w i t h

A

E

= H

i s a H i l b e r t space and

It f o l l o w s from Fxercise 11.5 -that A

E

A

8(C,w)

g e n e r a t e s a s t r o n g l y continuous c o s i n e f u n c t i o n

3.1,

is a

(that

C(t)

satisfying

Ilc(t)ll 5 i f and only i f

cr(A),

C e U J l tI

t h e spectrum of

(-m

A,


m),

(5.1)

i s contained i n a r e g i o n o f

t h e form

( t h a t i s , t h e r e g i o n t o t h e l e f t o f t h e p a r a b o l a p a s s i n g through 2 2 It 2iw ). We note i n p a s s i n g t h a t C ( t ) c a n be computed using t h e

UJ

,

f u n c t i o n d c a l c u l u s f o r normal o p e r a t o r s :

251

OTHER PROELEMS

C ( t ) U = c(t;A)u =

where

P(dp)

i s t h e r e s o l u t i o n of t h e i d e n t i t y for A 2

c(t;p,) = cosh t$/2 = 1 + t ~/2! constant

+

...

+

and

Moreover, t h e

C i n (5.1) c a n be t a k e n e q u a l t o 1. ( I n f a c t , t h e e s t i m a t e

Ilc(t)II 5

can be improved t o THEOFSM and o n l y i f

Proof:

cosh U r t ) .

5.1. The normal o p e r a t o r o(A)

s a t i s f i e s Assumption

A

3.1

if

i s contained i n a h a l f - s t r i p . Rep

iA

4 2 t p /4!

5

IIm

a,

FI 5 b .

(5.3)

We have s e e n (Theorem 3.2) t h a t Assumption

g e n e r a t e s a s t r o n g l y continuous group.

3.1 i m p l i e s t h a t A

Since, on t h e o t h e r hand,

g e n e r a t e s a cosine f u n c t i o n i t follows t h a t

o(A)

i s contained i n t h e

i n t e r s e c t i o n of a h o r i z o n t d s t r i p with a r e g i o n defined by (5.2) which i n t e r s e c t i o n i s i t s e l f contained i n a h a l f - s t r i p of t h e form Conversely, l e t inequalities

(5.3).

i s a subset of

o(A)

be contained i n a h a l f - s t r i p defined by both

w(E)

-

(2E)-2

a s e t of t h e form (5.2) i f and only if

> -

W(E),

i s a s o l u t i o n of t h e e q u a t i o n a m - -

- w(E)2

-

E

5 (4a)-1/2

b2 , 4 4E)2

4E2 if

a ( & )= a

The h a l f - s t r i p corresponding t o

w

where

(5.3).

we have

and t h e f i r s t e s t i m a t e (3.8) i s v e r i f i e d , s a y for

i s a c t u a l l y s a t i s f i e d f o r all

(5.6) away from z e r o ) . S(t;A

E

2

0,

0

_<

E

5 (8a)-’l2

(it

as we s e e e a s i l y e s t i m a t i n g

W e n o t e next t h a t

-

(2~)-~1) = s(t;A

-

(2E)-*I),

(5.7)

252

OTHER PROBLEMS

where

s ( t ; p ) = p-1/2sinh

t h e norm

IlS(t;A

-

s i n tp1/21

lP-1’2

= t

tpl/*

-

t3p2/3!

- ... ,

+ t5p3/5!

thus

does not surpass t h e supremum of

(2E)-21)ll

i n t h e h a l f s t r i p defined by

5

Rek

a

1 -4E2 ’

IIm 1-115 b

.

If 1 belongs t o the region defined by (5.3) t h e n

On t h e

for

li.

l i m i t e d by (5.8)

118(t;A

-

w = w(E)).

( 2 ~ ) - ~ I ) 5l l 2 ( 1

-< 2&(1 -

w

(recall that

r e g i o n defined by (5.2) with

-

must be contained i n the Hence

La&2 ) -1/2& , w ( 4 l t l

4 a E 2 -1/2 e x p ( ( 1

which i s t h e second i n e q u a l i t y (3.8).

-

4a&2)-1/%EIt

1) ,

T h i s ends t h e proof of Theorem

5.1. We note t h e important p a r t i c u l a r case where with

-A

2

i n which case we can t a k e

0,

A

in

w = 0

i s self-adjoint

(3.8).

Another

case t h a t can be reduced t o t h i s i s covered by the following r e s u l t

THEOREM 5.2. function

Let

A

generate an uniformly bounded cosine

C(t),

IlC(t)II H.

i n a H i l b e r t space Proof:

Then

5 c A

(-,
<

(5.10)

co)

s a t i s f i e s Assumption 2.1

We have shown i n Theorem

v.6.7 t h a t

if

A

with

w = 0.

i s a n operator

t h a t s a t i s f i e s t h e assumptions i n Theorem 5.2 t h e n t h e r e e x i s t s a (self-ad j o i n t ) bounded, i n v e r t i b l e operator operator

B

with

B

5

0

Q

and a s e l f ad j o i n t

such t h a t A = $

-1

BQ.

(5.11)

I n obvious notation, C ( t ; A ) = Q-’C(t;B)Q,

8 ( t ; A ) = Qm18(t;B)Q,

(5.12)

253

OTHER PROBLEMS t h u s i t follows from Theorem claimed. constants

5.1 t h a t A

s a t i s f i e s Assumption

To compute e x p l i c i t l y t h e c o n s t a n t s

Ci,

l/Ql/, lIQ-'/l

(7.8)'

in

C1

Co,C1 i n (3.1)

or the

e x p l i c i t e s t i m a t i o n s f o r t h e norms

These a r e given i n Theorem

a r e needed.

3.1 a s

v.6.7 although it

i s not c l e a r whether t h e y a r e b e s t p o s s i b l e . Another c l a s s of o p e r a t o r s s a t i s f y i n g Assumption 3.1 i s i d e n t i f i e d below.

5.3.

THEOREM

Let

be a n a r b i t r a r y Banach space,

E

3.1, B a bounded o p e r a t o r . s a t i s f i e s Assumption 3.1 a s w e l l .

s a t i s f y i n g Assumption

A.

a n operator A = A

Then

+

0

B

The proof of Theorem 5.) i s based on t h e following r e s u l t on p e r t u r b a t i o n of c o s i n e f u n c t i o n s , where

E i s again a n a r b i t r a r y

Eanach space. THEOREM 5.4.

Let

be t h e i n f i n i t e s i m a l g e n e r a t o r of a s t r o n g l y

A.

continuous c o s i n e f u n c t i o n

Then

+B

A = A.

C(i) = C(t-A ) ' 0 '

B

a bounded o p e r a t o r .

g e n e r a t e s a s t r o n g l y continuous c o s i n e f u n c t i o n

C ( t ; A ) = C(t;AO + E).

5.4 i s a n obvious k i n of Theorem IV.2.3, where a s i m i l a r

Theorem

"perturbation-by- bounded -operator" r e s u l t i s shown f o r g e n e r a t o r s o f s t r o n g l y continuous groups.

Although Theorem

5.4 can be shown

by

means of e s t i m a t i o n s of t h e r e s o l v e n t a s i n Theorem IV.2.3, we s h a l l use here another method t h a t y i e l d s an e x p l i c i t formula f o r

C(t;AO

+ E).

Given two

(E)

-

valued f u n c t i o n s

s t r o n g l y continuous i n

-m


C

m

and

G(t^),

defined and

we d e f i n e t h e i r convolution by

t F(t

(F*G)(t)u=J

F(t")

-

s)G(s)u d s .

(5.13)

0

We check e a s i l y t h a t f u n c t i o n defined i n

(FxG)(<) -m


-= m.

theorem, any s t r o n g l y continuous bounded i n t h e norm of moreover,

llF(:)II

(E)

i s a s t r o n g l y continuous

(E)

-

valued

Note t h a t , by t h e uniform boundedness

(E)

-

valued f u n c t i o n must be

on compact s u b s e t s of

i s t h e supremum of t h e f a m i l y

-m


m;

{IIF(t)uII; IIuI(

513

of continuous f u n c t i o n s , t h u s i s lower semicontinuous as a f u n c t i o n of

t.

This g i v e s sense t o t h e following e s t i m a t i o n :

254

OTHER PROBLEMS

ll(Fx-G)(t)!

51

t l17(t - s)llllG(s)Il d s

(5.14)

' 0

for

t

2

0,

with a n obvious c o u n t e r p a r t f o r

t h a t t h e convolution of

(E)

-

t < 0.

We check e a s i l y

v,CLued s t r o n g l y continuous f u n c t i o n s

enjoys all t h e p r i v i l e g e s of i t s s c a l a r c o u n t e r p a r t (such as a s s o c i a t i v i t y ) that make sense i n t h e p r e s e n t c o n t e x t . function

Define a

Q ( t ) as t h e sum of t h e s e r i e s

C ( t *'A 0) ~ + C ( t ; A O ) + + B 8 ( t ; A O ) ~ + C ( t ; A o ) ~ E S ( t ; A O ) ~ B B 8 ( t ; A O ) U+ To show convergence o f (5.15) we pick c o n s t a n t s

and make use of (5.14) i n each of t h e terms.

C

0

,W>

(5.15)

a s .

such that

0

The f i n a l r e s u l t i s

It follows t h a t (5.15) converges uniformly on compact s u b s e t s of -W


m

to

s i n c e each term i n (5.15) i s a s t r o n g l y

Q(t)u;

continuous function,

Q(t)i s

a s t r o n g l y continuous ( E )

-

valued

function satisfying

l/Q(t>ll 5 Ce

(5.18)

(5.17) and (5.18) we see t h a t we can m u l t i p l y (5.15) by + CollBll and t h e n i n t e g r a t e term by term. By v i r t u e exp(-At) i f A >

Using

u)

of t h e convolution theorem f o r Laplace transforms i n each term we obtain, using t h e f i r s t e q u a l i t y (11.2.11),

where t h e j u s t i f i c a t i o n of t h e l a s t i n e q u a l i t y follows t h e l i n e s of t h e

255

OTHER PROBLEMS

We apply t h e n Theorem 11.2.3.

argument used i n Theorem IV.2.3.

This

completes t h e proof of Theorem 5.4. We note t h e following formula, which i s obtained i n t e g r a t i n g (5.15)

in

5

0

s

1. t :

S ( t ; A ) u = S(t;A 0 -t

-t

B)u = 8 ( t ; A

( 2 ~ ) - ~ 1 )and

)u

0

...

-t

(5.20)

We use t h e s e r i e s (5.15) t o express

+ B - ( 2 ~ ) - ~ 1=) C ( t ; A O 0

C(t;Ao-

+ S(t;AO) * W ( t ; A O ) u

)U

8(t;Ao)*E!S(t;Ao)*BS(t;A

Proof of Theorem 5.3: C(t;A

0

( 2 ~ ) - ~+ 1B)

-2

S(t;Ao- ( 2 ~ ) I)

i n terms of

and estimate t h e convolutions

i n an obvious way, using a s a b a s i s both i n e q u a l i t i e s

(3.8).

We

obtain

IIC(t;Ao -t

+

B

C;CfllBlI 2

-

& 2: ~e 2

(act<=,

WEJtl

8(t;Ao-

+

C;C1l/Bl(t(s

ewEltl

... = C'e (wtClllBI1)El tl 0 (5.21)

o < _ & I E O ) .

A t o t a l l y similar estimation of

of

+

( ~ E ) - ~5I C;ew'ltl )~I

8(t;A

0

-t

B

-

w rIi t)t e n i n terms (~E)-~

( 2 ~ ) - ~ 1 using ) (5.20) y i e l d s :

llS(t;Ao+ B

-

( 2 ~ ) - ~ 1 ) 1c -I ClEeW'ltl

+

CII/BII 2 Itl$ewFiti

This ends t h e proof of Theorem 5.3. We examine i n t h e r e s t of t h e s e c t i o n a complement t o Theorems and 4.2 obtained by means of i n t e r p o l a t i o n theory. E = H

We assume that

i s a H i l b e r t space and that

A = A0 + B ,

(5.23)

i s s e l f - a d j o i n t and bounded above (that 0 s a t i s f i e s Assumption 2 . l h a s been proved i n Theorem 5.3). To

where A

4.1

B

i s bounded and A

simplify, we a l s o assume that

a(Ao)

5

achieved by an obvious decomposition of

(-m,O),

Ao.

which can always be F i n a l l y , it i s required

256

OTHER PROBLEMS

that BD(AO)

c D(AO) -

(5.24)

We have already pointed out i n 8V1.8 how t o defined f r a c t i o n a l powers

5

where

(-Ao)',

= cy +

i z i s a n a r b i t r a r y complex number. The pertinent

formula i s (-A u')

where

=

[m p 5 P(-+)u

i s the resolution

P(-+)

,

Jo

O

(5.25)

of t h e i d e n t i t y for Ao.

It was already

proved i n Section v1.8 that lI(-Ao)

Let

5

2

Ull

Q :H * H

= Il(-Ao)

cy+izull2

be a l i n e a r operator such t h a t

IlQull IKolluil

(u

E

HI,

2 IlQul' 5 $l1AOull

(u

E

D(AE))

-

(5.27)

Consider t h e H-valued holomorphic function c p ( ~ )= cp(a + i.t) = Q(-A 0 )

for

u

E

H

-orti.tu

5

CY

5 21,

Making use of both i n e q u a l i t i e s (5.27) and applying

fixed.

Hadamard's t h r e e - l i n e s theorem t o cp lldQ

(0

+ i7)ll

(see again gVI.8) we deduce t h a t

5 KO(2*)/*Ig2llUIl

(0

5 a 5 2),

hence

(2*)/2g/2~~(-~O)cyu~~ (u

IlQull<_KO

E

D((-A 0 )~u,o 5

5

We apply t h i s argument t o t h e operator Ki(t;&) - S i ( t ) .

2).

(5.28)

Using (3.1)

and (3.9) we o b t a i n ligi(t;E)u

-

Si(t)u/j 5 *coew(tl

II~II.

(5.29)

The second estbnate i s somewhat l e s s t r i v i a l , since (4.5) provides 2 bounds i n terms of IlAul] and of 11A2ull, r a t h e r than llAOull a s needed. To perform t h e conversion we note that 2 -2 2 -2 2 = (Ao f A B + FAo = (Ao + B) A. A A. 0 =I+ABCI 0

-2 0

-1 2 -2 +BAo + B A o ,

f

2 -2 B )A

(5.30)

where t h e f i r s t , t h i r d and fourth operators a r e t r i v i a l l y bounded; t h e

257

OTHER PROBLEMS

second ( i n f a c t , even A BA-l) 0

closed graph theorem.

i s bounded because of (5.24) and t h e

0

On t h e other hand, we have

AAO' = ( A ~+ B > A-2~ =

-l+

.'0"

(5.31)

It follows from (5.30) and (5.31) that

.

2 2 2 //A uI/5 C / I A O ~ / I ,h/l5 C I I A O ~ / l

(5.32)

We combine (5.32) with (4.5), obtaining IIQi(t;E)U - s i ( t ) u l /

5

It1

clE2(i+

IIA;UII

(u

E

(5.33)

D(A2) = D ( A i ) ) .

(5.33) with t h e preceding remarks ( e s p e c i a l l y (5.28)) we deduce t h a t i f u E D((-A 0 )") 0 < 01 5 2 , we have

Combining (5.29) and

/IQi(t;E)u -si(t)ull 5 c ( a ) E 0 ( ( i + It1 )a/2eultl (U E

Let

THEOREM 5.5.

B

bounded and

A

0

-

D((-A)~), E = H

c t <

II(-A,)"UII

(5.34)

m).

be a Hilbert space,

A = A

0 s e l f - a d j o i n t and bounded above, and l e t

u(t)

be a s o l u t i o n of t h e homogeneous problem (2.1),

+ B with u(t;E)

a solution

of t h e homogeneous problem (2.2) w i t h u E D((-A )"), 0 < CY 5 2. 0 0 Then, i f (5.24) holds t h e r e e x i s t s a constant C(a) such t h a t llU(t;E)

-

u(t>il 5 c ( ~ ) E Q ( +~ I t l ) a / 2 e ~ l t JI I ( - A ~ ) ~ ~ ~ I I

+ coewltli~uo(E)-uo1l + cle"ltlE21/q(e)ll The proof follows that of Theorem vI.8.2;

(-m

< t c

m)

. (5.35)

we omit t h e d e t a i l s .

Theorem 5.5 implies that

uniformly on compacts of i~u,(E>

- u0ll

THEOREM 5.6.

-m

= o ( E CY

Let

E,A

< t <

1,

a

II%(E>II

if = ~ ( E c y - ~ )a s

E

4

o

be as i n Theorem 5.5, and l e t

.

(5.37) u(t;&)

be a s o l u t i o n of t h e homogeneous problem (2.1) with uO(&) E D @ ) , u ( t ) a s o l u t i o n of t h e homogeneous problem (2.2) with uo E D((-AO)1- ). Then, i f (5.24) holds t h e r e e x i s t s a constant

C(U)

such t h a t

~ l u l ( t ; E )- u * ( t > l l sc b > E a ( i + I t l ) a / 2 e " l t l ~ ~ ( - ~ o ) 1 ~ u o ~ ~ (5.38)

258

OTHER PROELEMS

The proof c o n s i s t s i n applying Theorem t h e proof of Theorem

u0

4.2).

u0 (E)

D((-AO)lw),

E

E

llul(E)

-

iAuOII

~ ' ( t )( s e e

u'(t;E),

A s a consequence, we o b t a i n e a s i l y t h a t , i f D(A)

then

- u'(t)ll

IlUYtjE)

uniformly on compacts of

5.5 t o


-co CY

= O(E

(5.19)

=

if

m

-

) and

IIul(E)

)

/IAuO(E)

cy

iAuo(E)II

= O(E

) ,

(5.40)

or, e q u i v a l e n t l y , i f IIu,(E)

Sv11.6

-

iAuOII =

CY

O(E

arid

-

Auo// =

O(Ecy)

.

(5.41)

E l l i p t i c d i f f e r e n t i a l operators.

We examine t h e o p e r a t o r (1.2) w r i t t e n i n divergence or v a r i a t i o n a l form,

i n an a r b i t r a r y domain

Rm; h e r e (6.1) i s

fl of

m-dimensional Euclidean space

D j = ?/ax

A s pointed out i n SIV.1, and x = ( x ,..,x,). j 1 equivalent t o (1.2) i f t h e c o e f f i c i e n t s a are d i f f e r e n t i a b l e . We jk assume a g a i n t h a t a = a The symbol A ( @ ) w i l l denote t h e jk kj' r e s t r i c t i o n of (6.1) obtained by imposition of a boundary c o n d i t i o n R, e i t h e r t h e D i r i c h l e t boundary c o n d i t i o n U(.)

=

o

(.

r),

F

(6.2)

or t h e v a r i a t i o n a l boundary c o n d i t i o n

-

D~U(X) =

r

where

F 7 ajk(x)vjD k u ( x )

i s t h e boundary of-

n

normal ( u n i t ) v e c t o r a t x; DVu

u

at

= y(x)u(x)

and

w

= (vl,.

(x

. .,v,)

E

r)

,

(6.3)

i s the exterior

i s c a l l e d t h e conormal d e r i v a t i v e o f

x.

The o p e r a t o r

A(B)

h a s a l r e a d y been c o n s t r u c t e d i n Chapter I V

under minimal assumptions on t h e c o e f f i c i e n t s .

However, a d d i t i o n a l

hypoteses w i l l have t o be placed on t h e f i r s t order c o e f f i c i e n t s t o force and

c

t o s a t i s f y Assumption

A(P)

a r e measurable and bounded i n

allowed; t h e

a

jk

3.1. We assume t h a t t h e

n.

Complex v a l u e s f o r

b

a c

j

jk are

a r e r e a l and s l t i s f y t h e uniform e l l i p t i c i t y c o n d i t i o n

259

OTHER PROBLEMS f o r some

K>

0.

F i n a l l y , we assume t h a t t h e f i r s t order c o e f f i c i e n t s

a r e imaginary, t h a t i s

Gj(x)

b.(x) J with

(6.5)

ib.(x) J

=

b

r e a l (we s h a l l see l a t e r t h a t t h i s requirement cannot be j e l i m i n a t e d ) and t h a t each b belongs t o $ y m ( R ) ( i . e . it has f i r s t j p a r t i a l derivatives i n Lm(R)).

A s we see below, t h e requirements t h a t t h e b

j

a

..ki

be r e a l and t h e

be imaginary cannot be omitted.

EXAMPLE

6.1.

m = 1, 0 = R, A

Let

t h e constant c o e f f i c i e n t

operator

+

A u = aull

bu'

+

cu

.

Using t h e Fourier-Plancherel transform we show t h a t

o(A)

so t h a t : if

a

(a)

o(A)

i s not r e a l

=

[-aa2

-

ibcr

+ c;

-a

< u <

,

m]

w i l l not be contained i n a r e g i o n of t h e form ( 5 . 2 ) (b)

if

a(A)

i s not imaginary,

b

contained i n a h a l f - s t r i p of t h e form The c o n s t r u c t i o n of t h e o p e r a t o r

w i l l not be

(5.3). h a s been c a r r i e d out i n

A(6)

Chapter I V without t h e s p e c i a l requirements on t h e

b

j

present here.

In t h e case of t h e D i r i c h l e t boundary c o n d i t i o n , no r e s t r i c t i o n s on R

t h e domain

were necessary:

it i s enough t o r e q u i r e t h a t ( o r a domain of c l a s s

y E Lm(r). Assumption

C(l)

f o r a boundary c o n d i t i o n of type

n

be a bounded domain of c l a s s

with bounded boundary

However, o u r job i s now t o show t h a t

r)

A( B)

3.1 under t h e r e i n f o r c e d assumptions on t h e

s h a l l do s o by proving t h a t

A(R)

AO(B)

with

s e l f - a d j o i n t and

=

A&B)

E

.t

where

g(n) d e f i n e CY

> 0

is linear i n

i s a parameter t o be f i x e d below. v,

conjugate l i n e a r i n

u

3'

and w e

This w i l l be achieved by

s l i g h t m o d i f i c a t i o n s of t h e arguments i n Chapter I V

E

satisfies b

(6.6) t h a t we o u t l i n e

below, beginning with t h e D i r i c h l e t boundary c o n d i t i o n . u,v

and t h a t

B ,

bounded.

(6.3)

C(l)

For

Obviously,

[u,v],

and we check e a s i l y t h a t

260

OTHER PROBLEMS

[v,u],

Using t h e uniform e l l i p t i c i t y assumption ( 6 . 4 ) , t h e

= [u,v&.

5

IDJ,),[

inequality

+

( ~ / 2 ID%[* )

and i t s counterpart

(1/2E) IvI2

[ Z J v I we e a s i l y show t h a t i f

for

i s l a r g e enough, t h e f i r s t

CY

ineq u a l i t y 2 c (u,.)

5

c > 0,

holds for some

5c

[U,Ul,

where

2

(u

(u,u>

#(W

E

(6.8)

i s t h e o r i g i n a l s c a l a r product of

(u,v)

t h e second i n e q u a l i t y ( 6 . P ) i s a consequence of t h e assumptions

#(O);

on t h e c o e f f i c i e n t s .

We s h a l l from now on assume

AO(R)

The operator

((@I

-

=

[u,wl,

(w

c o n s i s t i n g of all u

E

$w,

$(n)

E

right-hand s i d e of (6.9) continuous i n t h e norm of of t h e theory of (u,v),

has:

(6.9)

which make t h e

L2(Q).

The r e s t

unfolds e x a c t l y as i n Chapter I V , s i n c e it i s

Ao(B)

[u,v],

based on t h e p r o p e r t i e s of t h e s c a l a r product same

[U,U~/‘.

i s defined by

AO(B))U,W)

A (5) 0

the domain of

endowed with

$(Q)

(lullCY=

t h e s c a l a r product (6.7) and i t s associated norm

which a r e t h e

Ao(B)

we check i n t h e same way t h a t

i s symmetric

and densely defined, t h a t i t s c o n s t r u c t i o n does not depend on

(11 - AO(B))D(AO(B))

-

h

E

A1

-

if

p(AO(D))

A?

3 a> ,

h 2 a.

Ao( fi) i s one-to-one f o r byproduct of (6.8) t h a t ( A 1 A0(3))-’ and t h a t

(A

E

=

that

(6.10)

We a l s o o b t a i n as a

i s bounded, so t h a t

This i s known t o imply t h a t

(Y.

Q‘,

is

Ao(i3)

s e l f ad j o i n t ( see Lemma IV.l.1). The f u l l operator Bu =

2

( b .Dju J

The assumptions on t h e operator.

We define

A(f3)

b

i s constructed by p e r t u r b a t i o n .

+ cu

D’(bju))

3

and on

A(5)

=

c

=

imply t h a t

no(fi)

and i t follows from Theorem 5.1 t h a t

- 2 7 (Djbj)u

for

u,v

E

$(O).

cu

.

A(@)

(6.11)

i s a bounded

(6.12)

+ B

The case where t h e boundary c o n d i t i o n a g a i n a s i n Chapter I V .

B

+

Let

s a t i s f i e s Assumption 3.1.

(6.3) i s used

i s handled

This time, however, t h e f u n c t i o n d i s

The d e f i n i t i o n of t h e operator

A0(6)

is

261

OTHER PROBLEMS

-

((@I The o p e r a t o r

A ~ ( Q ) ) ~ , W ) = [U,~I:,

(6.11).

B

t h e f u l l operator

3.1.

A(B)

satisfies

Summarizing :

Let R

THEORFM 6.2.

be a domain

& Rm,

A

(6.1) with

t h e operator

dJrn(C2),

b.

a

8)

A(

i s t h e bounded o p e r a t o r defined

It f o l l o w s a g a i n from Theorem 5.1 t h a t

Assumption

(6.14)

$,(Q)).

E

Ao( 0) i s a g a i n s e l f a d j o i n t :

i s obtained by formula (6.12), where by

(w

c E Lm(Q), E Assume, moreover t h a t t h e a are real jk’ J jk and s a t i s f y t h e uniform e l l i p t i c i t y assumption (6.4) and t h a t t h e b j a r e p u r e l y imaginary. If 8 i s t h e D i r i c h l e t boundary c o n d i t i o n (6.2)

I

A ( B ) defined by (6.7) and (6.9) s a t i s f i e s Assumption R i s bounded and of c l a s s and R i s t h e boundary measurable and bounded i n r t h e n t h e c o n d i t i o n (6.3) w i t h o p e r a t o r A ( 8 ) defined by (6.13) (6.9) s a t i s f i e s Assumption 3.1.

t h e operator

7’)

If

3.1.

6.3. Theorem 5.5 h a s an i n t e r e s t i n g a p p l i c a t i o n i s not e a s i l y i d e n t i f i a b l e even f o r D( ( -Ao( B))cy)

REMARK Although

here. @

=

1

under t h e p r e s e n t smoothness assumptions, we have show i n Theorem I v . 2 . 2 and Theorem

Iv.5.l

that

D((-Ao(8))1’2) when

8 is

=

5.5

8

E

HbQ)

%(a)

(6.3).

Using Theorem

E)

-

u(t)ll

(6.17)

= O ( E1/2)

and

Ilu0(d

-

uoII =

o(E1/2),

IlU,(E)II

u

E

$(a).

0 c o n d i t i o n (5.24) holds.

(6.18)

= O(E - 3 F ) .

The same r e s u l t h o l d s for boundary c o n d i t i o n s we assume t h a t

(6.16)

B i s t h e D i r i c h l e t boundary c o n d i t i o n ,

we deduce t h a t i f

uo

=

i s t h e v a r i a t i o n a l boundary c o n d i t i o n

Ilu(t; if

(6.15)

t h e D i r i c h l e t boundary c o n d i t i o n ( 6 . 2 ) , and

D((-*o(B))1/2) when

$(a)

B of type (6.3) where

However, we c a n only guarantee

(6.17)

This i s e a s i l y seen t o be t h e case i f

r Djbj,

c

E

$’“(n).

if

262

OTHER PROBLEMS

Sv11.7

The inhomogeneous e q u a t i o n .

A s pointed o u t i n SVII.2, t h e e x p l i c i t ( g e n e r a l i z e d ) s o l u t i o n of t h e i n i t i a l value problem ( 2 . 1 ) with n u l l i n i t i a l c o n d i t i o n s

is

UJE)

U0(E),

=k t

-

Gi(t

u(t;E)

s;&)f(s;&)ds.

We have a l r e a d y noted ( i n Example 4.6) t h a t s t r o n g l y convergent a s

(7.1) of

E

+

0.

(7-1) i s n o t even

ei(t;E)

However (and somewhat s u r p r i s i n g l y )

t u r n s out t o t r a n s l a t e convergence of

i n t o convergence

f(t;E)

a t l e a s t f o r a c l a s s of o p e r a t o r s c o n t a i n i n g t h e d i f f e r e n t i a l

u(t;E)

o p e r a t o r s i n Sv11.6.

THEORFM

7.1. Let

E = H

be a H i l b e r t space,

(7.2)

A = A0 + E , where

A.

operator.

L ~ -T,T;H) (

i s a s e l f a d j o i n t o p e r a t o r bounded above, Let

such t h a t f(s;E)

in

o

T > 0, k ( s ; E);

1

L (-T,T;H).

-

E

f(s)

Finally, l e t

5

as

u(t;E)

E

F

~

-

)a

a bounded

B

f a m i l y of f u n c t i o n s i n

(7.1)

0

be t h e (weak) s o l u t i o n of t h e

i n i t i a l value problem 2

EU"(t;E)

-

iU'(t;E)

+ f(t;E)

= AU(t;E)

( I t ]5 T ) ,

-

(7 4) U(0;E)

uniformly i n

It I 5 T,

= 0, l ~ ' ( 0 ; E ) = 0 .

u(t;

where

E)

i s t h e weak s o l u t i o n of

u ' ( t ) = i A u ( t ) + i f ( t ) (It1 5 T)

, (7.6)

u(0) = 0 .

Proof:

We c a n obviously assume t h a t

incorporate i n t o

B

t h e " p a r t of

s h a l l show f i r s t Theorem 7.1 f o r E,

c o n s i d e r i n g f i r s t t h e case

P(dp)

A.

a(Ao)

5 ( 0 , ~ )( i f

with spectrum i n

IJ.

2

not we 0").

We

and t h e n mix t h e p e r t u r b a t i o n A. f ( t ; E ) = f ( t ) independent of E . Let

be t h e r e s o l u t i o n of t h e i d e n t i t y f o r

A.

and

Ei(t;E;Ao)

the

263

OTHER PROBLEM2

(7.3) w i t h E = 0 . The same c o n s i d e r a t i o n s l e a d i n g t o Examples 4.5 and 4.6 show t h a t

(second) propagator of t h e e q u a t i o n

PO

for

u

are

t h e r o o t s o f t h e c h a r a c t e r i s t i c polynomid

Let

0

E

5

E,

t

where

5

T.

We can w r i t e

t

u(t;E)=

[ ei(t - s ; € ; A 0 ) f ( s )

c, d s = [:P(dp)L[

e(t -

s ; & ; p ) f ( s ) ds

(7.10)

" 0

a f t e r an e a s i l y j u s t i f i e d interchange i n t h e order of i n t e g r a t i o n . note next t h a t

hence

On t h e o t h e r hand,

Since

we deduce t h a t , f o r

p

fixed,

12(t;p;

E)

-

ieipt -/ote'Psf(s)

ds

W e

264

OTHER PROBLEMS

uniformly i n

0

5 t 5 T.

To handle t h e f i r s t i n t e g r a l we note t h a t

-

+

h (p;~)

as

i m

-o

E

and use t h e following uniform v e r s i o n of t h e Riemann-Lebesgue lemma: if -

g(t)

i s a ( s c a l a r or v e c t o r - v a l u e d ) f u n c t i o n i n

lim

J'''eiosg(s)

the

L1

0

5

t

5 T;

t h e proof i s achieved approximating

g

in

Applying (7.15) t o t h e f i r s t

norm by smooth f u n c t i o n s .

integral i n

(7.15)

ds = 0

0

a-'m

uniformly i n

then

L1( 0 , T )

(7.13) we o b t a i n T(t;p;E)

uniformly i n

0

5

Assume t h a t

t

as

0

F

-

(7.16)

0

5 T.

u(t;&)f. u ( t )

itn],

e x i s t s a sequence

-+

0

uniformly i n

5 tn 5

T

0

5t 5

and a sequence

T. {En],

Then t h e r e E~

-+

0

such t h a t

llu(tn;tn) Making use of

(7.14), (7.16)

-

Il(t,)II

of t h e range

-T

0

5t 5

5t 5

0.

(7.17)

6 > 0.

and a v a r i a n t of t h e dominated convergence

theorem we o b t a i n a c o n t r a d i c t i o n with h o l d s uniformly i n

2

T.

(7.17).

This shows t h a t

(7.5)

An e n t i r e l y s i m i l a r argument t a k e s c a r e

The case where

f

depends on

E

i s handled

writing

+ktGi(t

- S;E;A)f(S)

ds

(7.1.8)

and making use of t h e uniform bound (7.11). We i n c o r p o r a t e f i n a l l y t h e p e r t u r b a t i o n

B.

It results from (3.7)

and from t h e p e r t u r b a t i o n formula (5.20) ( o r d i r e c t l y ) t h a t we have

+ 6. ( t * & ' A ) x BGi(t;&;A0)u 1 " O

265

OTHER PROBLEMS

hence U(t

;E)

ei(t;€ ; A O )

=

Y

+ S ( t ; &;Ao)

f (t ;&)

+ ei(t;&;Ao) * EEi(t;E;A0) Now, using

Y

E e i ( t ;€ ; A O ) x f ( t ; E)

*Eei(t;&;AO)

...

*f(t;&)+

(7.20)

(3.1) we show t h a t t h e n-th term of t h e s e r i e s ( 7 . 2 0 ) i s

bounded i n norm by

On t h e o t h e r hand, using r e p e a t e d l y t h e p r e v i o u s l y proved r e s u l t on convergence of that

e.(t;E-A

3 0

1

) * f ( t ; & ) i n each term of (7.20) we deduce

ei(t;&;Ao)*E6.(t;&;AO)*f(t;E), 1

E6i(t;E;Ao)*f(t;c),...

qt;E;Ao)

*Bei(t;&;Ao)x

all converge uniformly i n

It/

5

T;

the

l i m i t of t h e n-th term of ( 7 . 2 0 ) i s

. .. x E i S ( t ^ ; i A o ) * f ( t ) S(i;iAo) * iES(i;Ao) . .. * i B S ( f ; A o ) * i f ( ; ) * BiS(t";iAo) x

iS(;;iAo) =

t h u s t h e sum of t h e s e r i e s converges uniformly, as S(<;iAO) *

if(t) +

S(i;iAo)

+ S(t;iAo) = S(<;i(A0

where

S(t;iAo)

i A ( r e s p . by 0

,

Y

x

+

iBS(:;iAo)

-,

0,

+ B))

x if(;)

B)).

+

E))

to

x if(;)

iBS(<;iA0 ) x- iBS(t";iAo) x if(;) +

(resp. S(t;i(Ao

i(Ao

Y

E

...

, i s t h e group generated by

This completes t h e proof of Theorem

7.1.

SVII. 8 Miscellaneous comments The Schr'ddinger l i n e a r p e r t u r b a t i o n problem w a s discussed by

SCHOENE [1970:1] i n t h e c a s e where

E

i s a B i l b e r t space and

A

a

s e l f a d j o i n t operator; i n t h a t p a r t i c u l a r s e t t i n g , Theorem 4.1 w a s proved by Schoene with somewhat s t r o n g e r assumptions on t h e i n i t i a l conditions. author

The m a t e r i a l i n t h i s c h a p t e r i s e n t i r e l y t a k e n from t h e

[1985:1].

Problems of t r a f f i c flow ( s e e WfTCTHAM whose l i n e a r i z e d v e r s i o n i s

[1974:1]) l e a d

t o equations

266

GTHEE PROBLFMS

where

E

i s a small parameter; t h e main problem a b o u t (8.1) i s t h a t of

showing t h a t t h e s o l u t i o n t e n d s t o t h e s o l u t i o n of t h e l i m i t e q u a t i o n

w i t h due a t t e n t i o n b e i n g p a i d , among o t h e r t h i n g s , t o t h e l o s s of one i n i t i a l c o n d i t i o n i n c u r r e d i n g o i , i g from (8.1) t o (8.2).

For a c l a s s i -

c a l t r e a t m e n t of t h e problem see MECTHAM [1974: 11 o r ZAUDFBER [ 1983 :11. An a t t e m p t t o t r e a t t h i s problem I n a n o p e r a t o r t h e o r e t i c way w a s made

[1985:1];some

i n t h e author Exercises

3 to

u.

of tlie key r e s u l t s a r e g i v e n below i n

The methods u:,ed resemble b o t h those i n t h e S c h r g d i n g e r

s i n g u l a r p e r t u r b a t i o n problem ( s u c h as t h e uniform bounds i n Assumption

3.1) and t h o s e i n t h e p a r a b o l i c s i n g u l a r p e r t u r b a t i o n problem ( s u c h as t h e a s y m p t o t i c developments i n

sVI.8).

EXERCISE 1. Write and i n t e r p r e t f o r m u l a ( 2 . 4 ) for t h e o p e r a t o r

i n t h e spaces

LP(-,m)

EXEBCISE 2 . Assumption

and

Co(i.,m)

(see E x e r c i s e s

11.3 and 11.4 ).

Show t h a t t h e o p e r a t o r s i n E x e r c i s e l do not s a t i s f y

3.1 ( e x c e p t i n L2(-m,o,)).

EXERCISE 3.

Let

E = H

be e (complex) H i l b e r t s p a c e ,

A

a self

a d j o i n t o p e r a t o r such t h a t

with

K

>

0, Q

t h e ( o n l y ) p o s i t i v e self a d j o i n t s q u a r e r o o t o f

-A,

B

a c l o s e d , d e n s e l y d e f i n e d o p e r a t o i such t h a t

Show t h a t the Gauchy problem 2

E

u"(t)

+

u l ( t ) = (s2A. + B ) u ( t )

(-



(8.5) u ( 0 ) = uo, u ' ( 0 ) = ul,

267

OTHXR PROBLEMS

i s w e l l posed f o r a n y

E

>

(Hint:

0

t h i s f o l l o w s from a s t a n d a r d

theorem on p e r t u r b a t i o n of g e n e r a t o r s of c o s i n e f u n c t i o n s . a u t h o r [ 1971:11, SHIMIZU-MIYADNU

See t h e

[1978:I], TAKENAKA-OKAZAWA [1978:11

o r TMVIS-WEBB [1981:1].

4.

EXEBCISE

(Hint:

i s a s o l u t i o n of

(8.5) w e have

u(t^)

If

i s a s o l u t i o n of

6.

u(t)

and i n t e g r a t e ) .

(8.5) w e have

m u l t i p l y t h e e q u a t i o n (8.5) scalarly by

EXFXCISE

5;

u(t^)

m u l t i p l y t h e e q u a t i o n (8.5) scalarly b y

EXERCISE 5.

(Hint:

If

u' ( t ) and i n t e g r a t e ) .

The assumptions a r e t h e same i n E x e r c i s e s

3, 4, and

we r e q u i r e i n a d d i t i o n t h a t

(Hint:

m u l t i p l y (8.7) b y

and add t o (8.6).

Take r e a l p a r t s .

Combine t h e f i r s t t h r e e i n t e g r a l terms i n t o one and use t h e f a c t t h a t

for a l l

u

E

D(Q), v

EXERCISE

7.

E

H,

consequence of (8.8)).

Assume t h a t

268

OTHW PROBLEMS

Re(Bu,u)

<_

w\/u/12

(u

D(B))

E

.

(8.11)

Using t h e i n e q u a l i t i e s 21 ( u ( t ) , u1 (t))12<_ ~ 1 ~ 1 (t)l12 1 ~ 1 ~+ a - 2 ~ ~ u ( t ) ~ ~ 2 , 21(U0,u1)1 2 f p 2 1 ~ u l ~ ~+2 f3-211u0112 and (8.9)show t h a t

(8.U)

EXERCISE 9.

i n (8.u) l e t t i n g

cx = $2

Setting

f? +

0

and u s i n g

G r o n w a l l l s i n e q u a l i t y show t h a t , i f uo = 0, \Iu(t)!l*

EXERCISE 10.

Let

+ 4~

4

2,4& l~Q~(t)ll" <_ 464 \/ul/\

yl(t;,),

6,(t;E)

.

(8.14)

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

(8.5) ( d e f i n e d i n t h e same way K(t;&), B ( t ; & ) are d e f i n e d for t h e U s i n g (8.13) m d (8.14) show t h a t

e q u a t i o n (VI.2.1)).

\\(I+ ~ E ~ Q ) \ ( ~+ ;2E ~ ~)4 ( ) -I ' 1 1<_ 2e2&

EXERCISE 11.

9, and 10.

(t? 0)

(8.15)

9

The assumptions are t h e same i n E x e r c i s e s

We r e q u i r e i n a d d i t i o n :

if

u

E

D(A)

then

Bu

E

6, 7, 8, D(Q)

and

II(W - BQ)u/I <_ C I I Q d f o r some c o n s t a n t C.

EXERCISE12.

Show t h a t

(8.15) c a n b e " r e c t i f i e d " t o

Show that (8.18) (in fact, the improved estimate

(8-17)

269

O'I'HER PROBLEMS

can IJe shown without assuming (8.17); we r e q u i r e i n s t e a d t h a t as well

(8.4), (8.8)

7

> 0 and t h a t

for

II(AB

satisfy

and (8*11). D((ELA + 8)) 5 D(BA)

EXERCISE 13. Assume t h a t E

P*

u

E

D(FA),

-

BA) U I

w i t h t h e same assumptions on

i s a s t r o n g l y continuous

5 gW

5 D(AB)

for a l l

KIIAuII i n Exercise l2. Show t h a t X h ( t ; E ) A

-1

(H)-valued f u n c t i o n s a t i s f y i n g

EXERCISE 14. The hypotheses a r e t h e same i n Exercise 13; we assume i n addition that Re A >

(8.5).

C.

BD(A) C_ D(Q) and t h a t

(A1

-

B)D(A)

is dense i n

H

for

Prove a n analogue of Theorem 4 . 1 f o r t h e i n i t i a l value problem

270

CHAPTER VIII ME COMPLETE SECOND ORDER EQUATION

gVIII.1 Let

The Cauchr problem. be, as usual, a complex Banach space,

E

D(A)

with domains

in t >_ 0

and

D(B)

dense i n

E

A,B

and range i n

l i n e a r operators

E.

Solutions

of t h e a b s t r a c t d i f f e r e n t i a l equation u"(t)

+

Bu'(t)

+

(1.11

Au(t) = 0 (1)

(which, by obvious reasons, is c a l l e d t h e complete second order a b s t r a c t d i f f e r e n t i a l equation) are assumed. t o s a t i s f y t h e following conditions:

i s twice continuously d i f f e r e n t i a b l e ,

u(2)

Au(^t) and

Bu'(^t) a r e continuous and

u ( t ) E D(A),

~ ' ( t f) D(B),

(1.1) I s satisfied i n

Similar d e f i n i t i o n s a r e used i n i n t e r v a l s other t h a n

t

2

0.

[O,m).

The theory of (1.1) i s considerably more complicated than t h a t of i t s incomplete counterpart (11.1.1) and can hardly be s a i d t o be i n d e f i n i t i v e I n f a c t , a l l we shall do i n t h i s chapter is t o t r y t o decide which is t h e c o r r e c t notion of well posed Cauchy problem f o r (1.1). A s we s h a l l

form.

see i n gVIII.2 use of ( t h e obvious extension o f ) t h e d e f i n i t i o n i n $11.1 f o r t h e incomplete equation leads t o paradoxical s i t u a t i o n s e n t a i l i n g

loss of exponential growth of s o l u t i o n s and nonexistence of phase spaces; moreover, t h e Cauchy probl-em may be w e l l posed i n an i n t e r v a l 0

5t5T

without being w e l l posed i n

t >_ 0

(Example 2.2).

This

motivates t h e i n t r o d u c t i o n of an a d d i t i o n a l assumption i n $3 (Assumption

3.1).

I n t h i s s e t t i n g a phase space generalizing that

constructed i n $111.1f o r t h e incomplete equation (111.1.1) i s assembled

i n $4. Modifying s l i g h t l y t h e d e f i n i t i o n i n $11.1we say t h a t t h e Cauchy problem f o r

(1.1) i s w e l l posed or properly posed in t

5

0

i f and

only i f (a)

There e x i s t dense subspaces

uo E Do, u1 satisfying

E

D1

DO,D1

then t h e r e is a s o l u t i o n

of u(%)

u ( 0 ) = u0,u'(O) = u1

.

E

such that i f

of

(1.1) & t

5

0

(1-2)

THE COMPLETE EQ,UATION

(b)

t L0

27 1

C(%) defined i n

There e x i s t s a nonnegative, f i n i t e f u n c t i o n

such t h a t llu(t)ll

5

t >_ 0 .

f o r any s o l u t i o n of (1.1) I f the function

t >_ 0

(1.3)

C(t)(llu(O)lI + l l U ~ ( 0 ) l l )

C(t)

in

(1.3) can be chosen nondecreasing i n

( o r , more g e n e r a l l y , bounded on compacts of

t

0)

then we

say t h a t t h e Cauchy problem f o r (1.1) i s uniformly w e l l posed ( o r

t >_ 0 .

uniformly properly posed) i n

The propagators o r s o l u t i o n operators of (1.1)a r e defined by

u(2)

where

(resp.

u ( 0 ) = u, u ' ( 0 ) = 0

C ( t ) (resp. of

D

Since both C(t)

and

0

a(t)

v(%))

i s t h e s o l u t i o n of (1.1)with v(0) = 0, v ' ( 0 ) = u).

(resp.

The d e f i n i t i o n of

s(t)) makes and

sense f o r u E D ( r e s p . f o r u E D1). 0 Dl a r e dense i n E we can extend (using (1.3))

t o all of

E

a s bounded operators; t h e s e operator-

valued functions r e s u l t s t r o n g l y continuous i n Iic(t)li

5 C ( t > , ils(t>iI 5 C ( t >

Moreover, by d e f i n i t i o n ,

C ( 0 ) = I,

S ( 0 ) = 0.

t >_ 0

U(t) =

c(t)u(o) +

and s a t i s f y

( t >_ 0 ) .

(1.5)

F i n a l l y , we prove e a s i l y

u(%) i s a n a r b i t r a r y s o l u t i o n of (1.1)i n

t h a t if

(1-4)

S ( t ) u = v(t),

@ ( t ) u =u ( t ) ,

t 2 0

then

S(t)Ul(O).

(1.6)

The proof i s t h e same a s t h a t of (11.1.6). We s h a U assume from now on that t h e operators

A

and

B

are

closed. $vIII.2

Growth of s o l u t i o n s and existence of phase spaces.

The d e f i n i t i o n of phase space i s , except f o r small modifications, t h e same i n $111.1. A phase space i n

t

0

f o r t h e equation (1.1)

equipped with any of i t s product n o r m , @ = Eo x E 1' El a r e Banach spaces s a t i s f y i n g t h e following assumptions:

i s a product space where

E 0 (a)

(-.

D1

dense i n (b)

and

E ,E

6 E with bounded inclusion; moreover, 1Do Eo El) is dense i n E i n t h e topology of Eo ( r e s p . i s

0

n

0

El i n t h e topology of

El).

There e x i s t s a s t r o n g l y continuous semigroup G ( t )

272

THE COMPLETE EQUATION

E = E

in

0

X

t 2

E

o

1

such that

with

for any s o l u t i o n u ( i )

u(0)

E

E ~ u , l(0)

E

E ~ .

The comments a f t e r t h e d e f i n i t i o n of phase space i n $111.1apply here:

we omit the d e t a i l s . We examine i n the rest of t h e s e c t i o n t h e r e l a t i o n of t h i s notion

w i t h t h a t of w e l l posed Cauchy problem i n the case where

E = Q2 is

t h e set of a l l sequences with

2 ~ ~ { u n= ] ~ Iu ~n

c

l2

u = [un jn >_ 1)= {un] of complex numbers and A , B a r e the operators c

ACunI = lanun),

B{ un] = Fbnun],

(2.2)

rb ) sequences of complex numbers t o be determined l a t e r : n t h e domain of A c o n s i s t s of a l l {u ) E E w i t h {a,.,) E E. The n domain of B i s s i m i l a r l y defined; observe that both A and B a r e

{an]

and

normal operators commuting w i t h each other.

u(%) = [un(%)] i s a s o l u t i o n of (1.1)then each u (%) s a t i s f i e s the s c a l a r equation n u''(t> + b n u ' ( t ) + a u ( t > = 0. On t h i s b a s i s , we deduce that the n?? propagators C(%), b ( t ) of (1.1) must be given by

where

+

hn,A,

If

a r e the r o o t s of t h e c h a r a c t e r i s t i c equation h2 + b A + a n = O ,

(2.5)

h = h- (a case t h a t we w i l l n n Obviously, a necessary condition f o r t h e Cauchy problem

w i t h the modifications de rigueur when

avoid here).

f

f o r (1.1) t o be w e l l posed i n

a(t) =

Ilc(t)II

t

2

= SUP

n>_l

0

i s that

-

A+ e n

n

A$

THE COMPLETE EQUATION

273

and

be bounded on compacts of

0

5 t <

Conversely, t h e preceding conditions

w.

imply t h a t t h e Cauchy problem f o r (1.1)i s well posed: the Fourier c o e f f i c i e n t s of

u ,u 0

1

f o r , i f (say)

a r e a l l zero except f o r a f i n i t e

number, then 4 t ) = c(t)Uo + s(t)U, t 2 0

f'unishes a solution of (1.1)i n

u(0) = u u ' ( 0 ) = ul. 0' 1 Moreover we obtain taking coordinates t h a t any solution u(%) of (1.1) m u s t be of the form

u ( t ) = C(t)U(O)

f

with

( t 2 0)

S(t>U'( 0 )

,

(2.8)

then

I( SUP Q ( S ) ) l I ~ ( O ) I I o5sst

Il.(s)ll

(a)

(

SUP

~(~))llU'(O)Il.

o<_sst

Let w ( - ) be an a r b i t r a r y function i n t

EXAMPLE 2.1.

on compact subsets. that

+

T-hen there e x i s t A , B

The Cauchy problem f o r Ilc(t)ll L 4 t ) ,

I n fact, l e t

2

0,

(2.9) bounded

of the form (2.2) such

(1.1)i s w e l l posed i n

ils(t>II >_ 4 t )

( t >_ 1)

.

t 2 0.

(b) (2.10)

il = ( w n 1, n >_ 1 be a sequence of p o s i t i v e numbers

such that (2.11) but otherwise a r b i t r a r y .

for

Define

n 2 1, and l e t CY

m(t) = sup

t

-

n z l @n Noting t h a t

( t z o ) .

(2.14)

THE COPLETE EQUATION

274

for 1 - t/n 5 1/2 m(t) <

for a l l

n = n(t)

such t h a t

we s e e t h a t

t

2

as

a t = o(pn)

n

t

Moreover, f o r each

0.

-..

n

00

t;

for a l l

then

t h e r e e x i s t s an i n t e g e r

t

c?

n -

m(t) =

(2.15)

on

Let now t < t'; s i n c e

CY

> 1 for a l l

Also, a,

m(n> >_

t >_ 0 ,

i s increasing i n

m(^t)

accordingly t h e f u n c t i o n on compacts.

n

thus bounded

n = wn

'n

(n

2 1)

.

(2.17)

Define yn = l o g I n view of t h e i n e q a l i t y

Cyn

+ Lwl/". n n

= log w;/n

log x i x

5

2-1/2ex,

valid for

(2.18) x

5.

0, we

h.ave yn We s e l e c t now

a n,bn i n

n

-

n'

(2.19)

(2.5) i n such a way that (2.20)

We have i

( h I = p > e . n n -

(2.21)

On t h e o t h e r hand, i n view of (2.19),

2

(8, thus the sequence

A =

{A'-]n /A/

-

2 1/2

VJ

>_ Yn>

i s contained i n t h e region e,

Accordingly t h e r e e x i s t c o n s t a n t s

0 C_

Re h

0> 8

7

5 I m A. 0

independent of

fl

such that (2.22)

275

THE COMPLETE EQUATION

I n view of t h e d e f i n i t i o n (2.14) of

m(t)

we obtain from (2.23) and

(2.24) that

e(m(t> -et> 5 a(t>5

5

e(m(t) - e t > in

t 2 0.

T(t)

o(m(t>

+ e

t

1,

(2.25)

5 o ( m ( t > + et >,

(2.26)

The i n e q u a l i t i e s on t h e right-hand s i d e s of (2.25), (2.26)

imply that t h e Cauchy problem f o r (1.1) i s w e l l posed i n only remains t o choose t h e sequence i n e q u a l i t i e s (2.10) a r e s a t i s f i e d .

0

To do t h i s , we assume ( a s we

Keeping i n mind that t h e

constant i n (2.22) i s independent of t h e choice of

Both conditions (2.U) a r e obvious. the greatest integer

5 t.

It

0.

i n such a way t h a t t h e

u(%) i s nondecreasing.

obviously may) t h a t

t

we s e t

fl,

On the other hand l e t

t

2

1, n = [ t ] ,

Then, taking (2.16) and (2.27) i n t o account,

we o b t a i n

whence t h e f i r s t i n e q u a l i t y (2.10) r e s u l t s from (2.25); t h e second follows i n a similar way from (2.26).

EXAMPLE 2.2.

Let

a > 0.

Then t h e r e e x i s t A , B of t h e form

(2.2)such t h a t t h e Cauchy problem f o r (1.1)i s w e l l posed i n 0

b u t not w e l l posed i n any i n - t e r v a l 0 <_ t <_ a!, a’ > a.

5 t 5 a,

For t h i s example we s e t bn

A = 0

(so that

+

n

= h

n

= 0 ) ; we p i c k t h e

i n such a way t h a t

1

A-n = -al o g

(so that

b

n

=

-

-1n ).

i

n + -(n a

2

- (log

2 1/2 n) )

W e check immediately t h a t

thus we o n l y have t o check t h e boundedness have

a

of

(n

L

1)

C(t) = I T(;)

(2.29) for a l l

i n (2.7). We

t,

276

THE COMPLETE EQUATION

Example 2.1 has important consequences i n t h e theory of phase spaces f o r t h e equation (1.1). I n f a c t , l e t

(l.l), t h e choice of

= E

0

x El

be a phase space f o r

El e n t i r e l y a r b i t r a r y .

and

Eo

t?

The b a s i c exponential growth r e l a t i o n (1.1.9) f o r t h e semigroup

qi)

i n ( 2 . 1 ) implies t h a t t h e r e e x i s t constants

such t h a t

C,W

A

f o r any s o l u t i o n and

u(t)

of (1.1)with

do not depend on t h e s o l u t i o n

W

u(0)

E

u(t).

Eo, u ' ( 0 ) =

If Eo

El, where

C

includes all f i n i t e

sequences ( t h a t is, all sequences whose elements, except f o r a f i n i t e

(2.8), say, u ( 0 ) = {S,,], u ' ( 0 ) = 0 The f a c t t h a t (2.31) cannot hold f o r all

number vanish), we can take i n ('mn

m

t h e Kronecker d e l t a ) . i s obvious since

Re A:+

observation a p p l i e s t o

THEOREM 2.3

A similar

(see (2.3) and ( 2 . 4 ) ) .

~4

We have t h e n shown:

El. A, B

be t h e operators employed i n Example 2.1.

t 1 0 f o r t h e equation (1.1) but (1.1)does not admit any phase space is: = E0 x E 1 -where EO or contain a l l f i n i t e sequences i n E:. ' k e n t h e Cauchy problem i s w e l l posed i n

Example 2.2 i s even more i n t r i g u i n g ; i n f a c t , d t h o u g h we could e a s i l y define a notion of "phase space i n

0

5t 5

a"

f o r (l.l), t h e

Cauchy problem for a f i r s t order equation must be well posed i n i f it i s w e l l posed i n

0

5t 5

t 2 0 a, It then follows that an equivalence

between (1.1) and a f i r s t order equation i s u n l i k e l y i n any sense. Examples 2 . 1 and 2 . 2 seem t o i n d i c a t e t h a t " t h e r e i s something missing" from t h e d e f i n i t i o n of well posed Cauchy problem we have introduced i n gVIII.1. This holds in t h e sense that an a d d i t i o n a l assumption i n next s e c t i o n w i l l guarantee existence of s t a t e spaces. SVIII.3

Exp onentidl growth of s o l u t i o n s and e x i s t e n c e of Dhase spaces.

W e r e t u r n t o a general Banach space

E.

Assume

Cf

=

E

0

x El

is a

phase space f o r t h e equation u"(t)

+

Bu'(t)

+

Au(t) = 0

(3.11

such t h a t E = E with i t s o r i g i n a l norm (such was t h e case f o r dl1 1 phase spaces constructed i n Chapter I11 f o r t h e incomplete equation

277

THE COMPLETE EQUATION

(111.1.1). Then,by d e f i n i t i o n of phase space we m u s t have

S 1 ( t ) must be a bounded operator i n

It follows t h a t

E

(we check

This j u s t i f i e s

e a s i l y t h a t t h i s i s not t h e case i n fiamples 2.1 and 2 . 2 ) .

at l e a s t h a l f of t h e following assumption:

3.1.

ASSUMPTION

t

2

0

u

E

E.

u

f o r all 8(t)E

(b)

( a ) S(:)u

i s continuously d i f f e r e n t i a b l e i n

E.

E

5 D(B)

i s continuous i n

BS(t)u

t

2

f o r all

0

A s we s h a l l see below, Assumption 3.1 guarantees exponential growth

(3.1), as well as e x i s t e n c e o f a s t a t e space.

of t h e s o l u t i o n s THEOREM 3.2.

t

2

3.1 be s a t i s f i e d .

and l e t Assumption

0,

C, w

Let The Cauchy problem f o r

(3.1) be well posed i n Then t h e r e e x i s t c o n s t a n t s

such t h a t

Ils;(t)ll

Ilc(t)llJ

1. Ce

wt

(t 2

(3.2)

'

The proof of Theorem 3.2 i s l e n g h t y and w i l l be c a r r i e d out i n W e examine f i r s t some immediate consequences of

several steps. Assumption 3.1. operator

u

-

a >0

Let

C(0,a;E) be t h e Banach space of

and l e t

E-valued f u n c t i o n s with i t s usual supremum norm.

611 continuous

St(t)u

from

E

The

t o C(0,a;E) i s easily seen t o be closed;

since it i s everywhere defined, by t h e closed graph theorem it i s as well bounded.

Ils'(t)ll B8(t)

0

5 t 1. a .

st(;)

a r b i t r a r y we see t h a t

f u n c t i o n such t h a t

E8(:)

a

Taking i n t o account t h a t

t

i s s t r o n g l y continuous i n

bounded on compacts t h e r e i n .

shows t h a t

t

S t ( t ) i s a bounded operator f o r all

B u t then

i s bounded i n

llSt(t)ll

2

0

and

is with

The same argument applied t o

i s a s t r o n g l y continuous o p e r a t o r valued

IIBS,(t)ll

i s bounded on compacts of

t

2

0.

We s h a l l need l a t e r t o solve t h e inhomogeneous equation

u"(t) + Bu'(t)

+

Au(t) = f ( t )

.

(3.3)

Solutions of (3.3) a r e defined i n t h e same way as f o r t h e homogeneous equation (3.1).

THE COMPLETE EQUATION

278

W 3.3.

L

Then

f ( t " ) be continuously d i f f e r e n t i a b l e i n

2

0.

(a) u(t)

=

( S - ! + f ) ( t )=

i s a solution of

(3.3) w i t h u ( 0 )

solution of (3.3)

then

v(t) where

t

= u'(0)

=

0.

C(t)u(O) + S(t)u'(O) + u(t)

=

v(;)

(b)

(t

2 0) ,

(3.5)

i s defined b x (3.4).

u(t)

Integrating (3.4) by p a r t s we obtain

Proof:

Differentiating, rt

u'(t) = 8(t)f(0)

+J

8(t - s ) f ' ( s ) d s ,

0

u"(t) = S'(t)f(o) +

[t S ' ( t - s ) f ' ( s ) d s .

'0

Let now

u

E

W e have

D1.

AS(s)u

Integrating i n

0

5

s

=

-B~'(s)u-~"(~)u.

5 t,

A L t 8 ( s ) u d s = -Bb(t)u- S ' ( t ) u

+ u.

Since the right-hand side i s a continuous operator of from closedness of

A

and the f a c t t h a t

(3.9) u,

i s dense i n

D1

it follows E

that

(3.10)

L t S ( s ) u d s E D(A)

for all u E E and t h a t (3.9) a c t u a l l y holds f o r every u E E. These observations make c l e a r ( a f t e r t h e i n t e g r a t i o n by p a r t s (3.6)) t h a t i f

u(i)

i s t h e function defined by ( 3 . 4 ) then

i s continuous i n

u'(t)

E

D(B) with

t

2

0.

Bu(i)

u ( t ) c D(A)

and

Au(;)

Moreover, we obtain from (3.7) t h a t continuous i n

t 2 0

and a l s o that

u(t)

is

279

THE COMPLETE EQUATION

i n f a c t a solution o f v(i)

(3.1) a s

To show

claimed.

i s a n a r b i t r a r y s o l u t i o n o f ( 3 . 3 ) and

(b)

u(;)

we note t h a t i f

t h e s o l u t i o n provided

by (3.4), v ( t ^ ) - u ( t " ) i s a s o l u t i o n of t h e homogeneous equation (1.1) so t h a t formula ( 1 . 6 ) a p p l i e s .

3.4.

LEMMA

(a)

u

Let

D(A).

Then

C ' ( t ) u = -8(t)Au

(t

E

c(i)u

(a)

i s continuously

d i f f e r e n t i a b l e and we have

(b)

Let

u

E

Do

n D(B).

0)

.

(3.11)

Then

( a ) According t o Lemma

2

(t

S ' ( t ) u = C(t)u-S(t)Eu Proof:

2

0)

I

3.3 t h e f u n c t i o n

u(t) = -Lt8(s)Au ds

u(0) = u'(0) = 0.

+

u"(t)

i s a s o l u t i o n of t h e equation

E u ' ( t ) + Au(t)

+

v(t) = u(t)

Consequently,

u

(3.5) we have v ( t ) C(t)u-u

\c

= C(t)u, =

-

-

with

v(0) = u, v ' ( 0 ) = 0 .

t h a t is,

S(S)AU ds,

which i s t h e i n t e g r a t e d v e r s i o n o f (3.11).

u(t) =

-Au

satisfies the

homogeneous equation (1.1)with i n i t i a l c o n d i t i o n s By v i r t u e o f

=

To show

(b),

let

/;tS(s)Bu d s . "0

Applying a g a i n Lemma 3.3 we see t h a t u ( t ) satisfies f Au(t) = Bu. On the other hand, let

-

v(t) = We have v ' ( t ) = C(t),

s,' ;at

C(S)U d s

v " ( t ) = C'(t)u = thus

v(t)

conditions

satisfies v(0)

=

v"(t)

+

U,

jo'

Bv'(t)

0, v ' ( 0 ) = u .

.

C'(S)U d s +

=

u"(t) + Bu'(t) t

C"(S)U d s ,

+

Av(t) = Eu,

Accordingly,

with i n i t i a l

w(t) = v ( t )

- u(t)

2 80

THE COWLETE EQUATION

s a t i s f i e s (1.1)with i n i t i a l conditions w(t) = S(t)u

w(0) = 0,

w ' ( 0 ) = u,

SO

that

o r , equivalently, s ( t ) u = / o t ( C ( s ) u - b(s)Bu) d s ,

(3.13)

from which (3.12) i s deduced by d i f f e r e n t i a t i o n .

3.5.

COROLLARY

Do = D(A).

(a)

n D(B).

E. ( c ) D1 _3 D(A)

D

(b)

g

(d)

u

E

s " ( t ) u + S'(t)Bu + S(t)Au

D(A)

=

D(A)

n

D(B)

0)

.

i s dense i n

n D(E), (t

= 0

2

(3.14)

3.4 ( a ) it was proved t h a t C(;)u is a s o l u t i o n of (3.1) f o r a l l u E D ( A ) ; since it i s always t r u e t h a t Do CD(A), ( a ) follows. Similarly, one of t h e s t e p s i n t h e proof of Lemma 3.4 ( b ) was t o show t h a t f o r any Proof:

u

E

D~

n

A s a by-product of t h e proof of Lemma

D ( B ) = D ( A )n D ( B ) , s ( ; ) ~

n

D1 = D(A)

D(B)).

To show t h e denseness statement

Y

consider t h e subspace

of

E

rl,(

i)

(b)

we

c o n s i s t i n g of a l l elements of t h e form ds

\[rl,(S)S'(S)U

where

(c)

4.5 it i s not i n general t r u e

r e s u l t s ( a s we s h a l l see below i n Example that

thus

i s a s o l u t i o n o f (3.1),

i s any i n f i n i t e l y d i f f e r e n t i a b l e ( s c a l a r ) function with

t > 0.

support contained i n

Let

i s a continuous function with

u

E

E

be a r b i t r a r y .

S'(0)u = u.

Then

Accordingly, i f

,.

S'(s)u

{Qn] i s a

sequence of nonnegative i n f i n i t e l y d i f f e r e n t i a b l e functions, each rl, with support i n

0

5 t 1. 1

and such t h a t

srl,n(s)8'(s)u d s as

n

+

m.

Hence,

Y

i s dense i n

by p a r t s , any element of

-

E.

[Qnds

+

=

n

1, we have

u

Observe next t h a t , i n t e g r a t i n g

Y can be w r i t t e n i n t h e form

/'$'(s)S(s)u

)

d s =~rl,"(s)(L[Oss(r)u d r

W e deduce from the f i r s t expression

(3.15) t h a t

Y

ds

5 D(B);

.

(3.15) on t h e

other hand, t h e second combined with (3.10) and t h e comments preceding

i t , implies for

(d),

that

Y

5 D(A),

which completes the proof o f

(b).

As

it follows from d i f f e r e n t i a t i n g (3.12) and then expressing

c t ( t ) u by means of (3.11).

W e have t h e n completed t h e proof of

28 1

THE COMPLETE EQUATION

1.5.

Lemma

W e point out t h a t , as a consequence of e q u a l i t y (3.12), t h e operator

S(t)E

D(A) f l D(B)

(with

'w t o d l

extension

E,

of

a s domain) admits a bounded

namely

S(t)E = C ( t )

c(:), -

at(<>

Since is

- 8f(t).

(3.16)

are s t r o n g l y continuous functions i n t

2

0, so

S(t)E.

W e consider i n what follows t h e characteristic polynomial of

2 P ( h ) = h I + h e + A . For each

h, P(h)

3.6.

LEMMA

cy,p

constants

Proof:

C

P(h)

i s closable f o r all A.

such t h a t

0

Assume

{un]

sequence

(3.17)

i s a l i n e a r operator with domain

(a)

2

(3.1),

(a)

D(P(A))

P(h)

There e x i s t

i s one-to-one for

i s f a l s e for

such t h a t

(b)

un

some +

A.

Then t h e r e e x i s t s a

0 , vn = P(h)un

-

v

#

0.

Let

u ( t ) = eAtun (t 2 0 ) . Clearly u ( i ) s a t i s f i e s t h e inhomogeneous n equation (2.3) with f ( t ) = ehtvn. W e g e t a s a consequence of Lemma 3.3 (b)

that

Letting

n

-

w e obtain

D i f f e r e n t i a t i n g twice,

If w e s e t

t

= 0

i n t h i s l a s t expression w e o b t a i n v = 0,

a contra-

diction. Regarding

(b),

assume

P(A)

i s not one-to-one f o r some

A.

Then

282

THE COWLETE EQUATION

u

there exists

u(t) = e

At

u

D(P(A)),

E

such t h a t

0

P(A)u = 0 .

Obviously

i s a s o l u t i o n of (3.1); making use of t h e estimate (1.3)

u

t >0

f o r any f i x e d

we deduce t h a t (Re h ) t

5

C(t)(l

PI)

+

-

Taking logarithms, we o b t a i n t h e i n e q u a l i t y opposite t o (3.18) f o r

B

t-1 l o g C(t),

cy =

=

t-l.

End of t h e proof of Theorem 3.2:

Let

cp be a twice continuously

d i f f e r e n t i a b l e s c d a r valued f u n c t i o n with compact support and such that

m(t)

1 near zero.

Consider t h e ( p l a i n l y bounded) operator i n

t h e space E, R(A;m)u = defined f o r all complex

A.

.io, e

(3.19)

m(t>S(t)u d t ,

We have

R(h;m)E

5 D(E).

Moreover, we can

w r i t e , i n t e g r a t i n g by p a r t s ,

t h u s it follows from (3.10) t h a t

5 D(A)

R(A;m)E Assume of

u

E

D(A)

n D(B) 5 D1.

(3.1) (Corollary 3.5

+L

5 D(A).

R(h;cp)E

r~D(B) =

Hence

D(P(A)).

(3.a) S(i)u

U s i n g the fact that

(c))

i s a solution

we show, i n t e g r a t i n g by p a r t s , t h a t

m

P(A)R(h,m)u

=

u

e-At((coS;)"(t)u

+L

+

E(&)'(t)u

+

fo

=

where

u

emAtM(t,m)u d t = u + ;(h,K~)u

M(t,m) = 2 m t ( t ) 8 ' ( t ) + m"(t)8(t)

s t r o n g l y continuous support contained i n

(E)

- valued

+

( 0 , ~ ) . Let

W

>0

Re h

2

W

and t h u s

I

+

M( A,@)

t

2

0

,

(3.22)

is plainly a with compact

be so l a r g e t h a t

A

Then, indicating Laplace transforms with

for

cD'(t)B(t)

function i n

A(@)(t)u) d t

,

has a bounded inverse i n t h i s region.

283

THE COMPLETE EQUATION

Define a bounded o p e r a t o r

R(h)

b y means of t h e formula m

I + I? h,cp))-l ( = R( h,O)

R( A) = R( A,@)(

2 (-l)%(

h,cp)",

(3.25)

,

(3.26)

n=O We r e w r i t e ( 3 . 2 2 ) i n t h e form

+

$A,cp)u

P m denotes t h e c l o s u r e of

P( A).

mR(h,co)u = u where

u

must i n f a c t hold f o r a l l

may r e p l a c e

E.

It f o l l o w s immediately from

Then

R( h,m)E 5 D(P( A)), R( A)E

Observe now t h a t , s i n c e

P(A) by

E

P(h)

in

D)

E

h(h,a) i s bounded t h a t (3.26)

i s c l o s e d and

the facts that

(u

(3.27).

5 D(P( A));

hence we

The e q u d i t y t h u s obtained

implies t h a t

R(A)E

=

D(P(A))

A

a t l e a s t f o r t h e s e v a l u e s of

v

fact, l e t Then

P( h)(v

E

D(P(h))

- R( h ) u )

D

=

D(A)

v

p

n D(E), P(h)

f o r which

be such t h a t

= 0,

We show next t h a t observe f i r s t t h a t

=

(1.28)

i s one-to-one.

R(h)E,

u

and l e t

=

In

P(A)v.

a contradiction. i s one-to-one

P(h)

k(i,m)

R(^A;m),

in

ReA >

W.

To do t h i s ,

a r e e n t i r e f u n c t i o n s of

7

(as

Laplace t r a n s f o r m s of f u n c t i o n s with compact s u p p o r t ) . Note t h a t , by v i r t u e of t h e e s t i m a t e

(3.24),

t h e series on t h e right-hand s i d e of

(3.25) converges uniformly i n Reh 2 Let now

v

E

D(P(h))

=

r e g i o n defined by ( 3 . 1 8 ) . element of

E).

n D(B),

D(A)

W,

v

R(h)

i s analytic there. A

and l e t

be, say, i n t h e

By t h e preceding comments,

v

=

R(h)u

(u

some

Then R(h)P(h)v = R( A)P(A)R(

The lef't

hence 0

h ) =~ R( h ) =~ V .

hand s i d e of (3.28) i s a n a l y t i c i n

Reh >

in t h e r e g i o n defined by (3.18), it must e q u a l Re)\ > W, which shows t h a t P ( h ) v # 0 throughout

v

C o l l e c t i n g all t h e o b s e r v a t i o n s made about

W;

v

(3.29) s i n c e it e q u a l s

a s well i n

Reh > P(A)

and

a s claimed.

R(A)

we

can w r i t e

R( A)

= P( A)-1

(Reh > a).

(3.30)

284

THE COMPLETE EQUATION

W e o b t a i n now some rough e s t i m a t e s for

h a l f plane

Reh

2

Plainly

W.

R( A,cp),

R(h), E R ( h ) ,

ER(h,cp)

AR(h)

t h e other hand, it follows from t h e expression (3.20) f o r I/AR( h , U )

/I 5 C I hl .

m

r yn

5

R(h;A)

on that

(3,24),

F i n a l l y , i n view of II(1 + i ( h , U ) ) - l l l

i n the

a r e bounded t h e r e :

- y)-l.

= (1

n=O Accordingly,

in

Reh >

0

for some constant

I n view of t h e f i r s t estimate the

C.

-

For

W

>

W

and

u

E

E

define

(3.31), we can d i f f e r e n t i a t e twice under

i n t e g r a t i o n sign, (3.33)

(k

=

0,1,2)

with

u(k)

continuous i n

we deduce i n a d d i t i o n t h a t

t

2 0,

i n t r o d u c t i o n of

being p o s s i b l e .

Bur(;), and

A

F

t

Au(;)

2

0.

Using t h e o t h e r e s t i m a t e s

e x i s t and a r e continuous i n

under t h e corresponding i n t e g r a l s

Hence

P(h)R(h)u d)\

Pushing t h e contour of i n t e g r a t i o n i n (3.33) for k = 0,l

t o the right

we show t h a t u(0) = u'(0) = 0

.

(3.35)

We express t h e n t h e s o l u t i o n of t h e i n i t i a l value problem (3.34)-(3.35) by means of formula

(3.4),

a n e q u d l i t y t h a t suggests Laplace transform o r

obtaining

-

as w i l l be proved l a t e r

-

that

i s the

R(h)

F+(<).

We now t r y t o find a new r e p r e s e n t a t i o n f o r

R(h).

Let

u

E

D;

285

THE COMPLETE EQUATION

o p e r a t i n g as i n t h e l i n e s l e a d i n g t o (3.22) and making use of C o r o l l a r y

(a)

3.5

we show t h a t , i f

R(h,co)P(h)u = u

+

u

D(P(A))

E

"we-ht((&)'t(t)u

=

D(A)

n D(E), + (&)(t)Au)dt

+ (c&)'(t)Bu

J 0 ,W

=

where now

N(t;cO)

u

+ j oe-htN(t;co)udt +

= 2CD'(t)8'(t)

= u

+

N ( h ; c ~ ) u,

c o " ( t ) g ( t ) + u)'(t)

m.

(3.37) If

W'

30

i s such t h a t

Il$h;m)ll

then Let

Q(h)

= (I

5y

in

Reh

2

D,

E

and

+ ;V(h;CD))-lR(h;@).

Q(A)P(h)u for u

W'

I + $h;u))

It f o l l o w s from

i s invertible there.

(3.37) t h a t (3.39)

= u

which p l a i n l y shows t h a t

&(A) = R ( h )

in

Accordingly, m

R(?\) =(I + i ( h ; C ~ ) ) - ~ R ( h ;=@() (-l)n~(h;co)n)R(h;c). n=O

(3.41)

This formula suggests(by i n v e r s i o n of Laplace t r a n s f o r m s ) t h a t

where

*

denotes convolution and t h e exponent

convolution power.

We attempt t o j u s t i f y

of (3.38) and of Young's i n e q u a l i t y ,

and, i n g e n e r a l ,

If

C

i s a c o n s t a n t such t h a t

*n

indicates the

(3.42) d i r e c t l y .

n-th

By v i r t u e

286

THE COPPLETE EQUATION

we can combine (3.

I\N(t;q)*”\I

44) and (3. 43) *(n-1)

= \IN(t;a)

a s follows:

*N(t;w)/\

n-1 u r t e

5

(t

Cy

2

0,

n = 1, ...) (3.45)

Consequently, t h e s e r i e s W

(-l)nN(t;cq) n=l converges uniformly on compacts of h(t;cp)

t

2

*n

0

(3.46) t o a (E)-valued f u n c t i o n

such t h a t Ilh(t;cp)I/

-1 w ’ t

5 c(1- r)

(t

e

F

(3.47)

0).

Since t h e p a r t i a l sums of t h e s e r i e s (3.46) s a t i s f y t h e e s t i m a t e (3.47)

as well and each of i t s terms i s s t r o n g l y continuous, i t s Laplace transform i n t h e region -

integration.

Let

Reh

2

W’

can be computed by term-by-term

h

S(t)

be t h e f u n c t i o n defined by t h e r i g h t hand s i d e

of (3.42), t h a t i s

-S ( t ) u

=

= (8@I

(&)(t)u

.t

+Jtn(t

h(t;cp))* ( & ) ( t ) u

- s;ep)(cpS)(s)u

ds

.

(3.48)

0

Plainly,

n

for some constant

C.

Computing t h e Laplace transform of

S(t)

by

a p p l i c a t i o n of t h e convolution theorem and l i k e w i s e applying t h e convolution theorem t o each of t h e terms i n t h e s e r i e s of

h(t;cp)

we

e a s i l y see t h a t it e q u a l s m

( by (3.41).

(-l)%( ~ ; P ) ~ ) A;@) R(

=

R( A)

(3.50)

n=O

B u t then, by a well known r e s u l t on Laplace transforms of

a n t i d e r i v a t i v e s , we have, a f t e r inversion,

for

-w > W1,

u

E

E.

Compare t h i s with

(3.36)

and d i f f e r e n t i a t e t h r e e

t i m e s t h e e q u a l i t y obtained from uniqueness of Laplace transforms.

The

.

THE COMPLETE EQUATION

287

result i s

Z(t)

( t 1 0),

S(t)

=

(3.51)

t h u s , i n view of (3.49) w e have

118(t>ll5 Apply (3.48) t o an a r b i t r a r y

account t h a t

(@)(O)

=

0

u

E

E

cp

wt

(3.52)

and d i f f e r e n t i a t e :

taking into

w e obtain

S t ( t ) u = (8

@ I

+ n(t;ep))* ( @ ) ' ( t ) u .

(3.53)

S i m i l a r l y , applying b o t h s i d e s of (3.48) t o elements of t h e form

Eu

(with

u

E

w e obtain

D ( A ) l l D(B))

'qqEu

= (6 @ I

(3.54)

+ h ( t ; q ) ) * (&)(t)u;

it i s obvious t h a t t h e equality can be extended t o a l l

u

E

E.

We u s e

now (3.16) t o deduce t h a t

W e obtain from (3.55) and (3.53), i n conjunction w i t h (3.47), t h a t

llc(t>ll 5 Cew t t

7

118 ( t )

11 -< CeWtt

(t

10)

(3.56)

C . The estimate (3.52) and t h e first i n e q u a l i t y (3.56) comprise t h e claims i n Theorem 3.2, whose proof i s t h u s complete.

for some constant

COROLLARY

3.7.

Under t h e assumptions i n Theorem 3.2 we have

liS'(t)ll 5 Cewt, b ( t ) l i 5 Ce for some constants

(t

2

0)

(3.57)

C,W.

(3.57) i s t h e second inequality (3.56). show t h e second w e make u s e of t h e "left-handed" companion of (3.48), Proof:

To

wt

The f i r s t inequality

namely

8(t)u

=

( @ ) ( t ) * (6 @ I + h(t;v))u

,

(3.58)

THE COMPLETE EQUATION

288

where m

nqt;w)

=

F (-l)%(t;a)*n.

(3.59)

n=1

Formula (3.59) can be j u s t i f i e d d o n g t h e same l i n e s as t h e "right-handed" formula was.

On t h e b a s i s of

(3.23) w e

deduce t h a t

thus (3.59) converges u n i f o r d y on compacts of

t

2

0,

its l i m i t

h

sat i s f y i r g

Equality

(3.58) can be e s t a b l i s h e d , a s i n t h e case o f (3.48) by t a k i n g

t h e Laplace transform of both s i d e s and using (3.25) and (3.33). Using t h e f a c t that of

t

i s a (E)-valued s t r o n g l y continuous f u n c t i o n

B8(;)

we deduce from

(3.58) t h a t

.

ES(t)u = ( a B S ( t ) ) * ( 6 63 I + h ( t ; q ) ) u We o b t a i n t h e second i n e q u a l i t y (3.57) from

(3.62)

(3.58) and (3.61).

This

ends t h e proof of C o r o l l a r y 3.7. We note i n c l o s i n g t h a t o t h e r e q u a l i t i e s , l i k e have as w e l l left-handed

S'(t)u

(3.53)

and

(3.55)

companions; t h e s e are =

( @ ) ' ( t )(6 * @ I

+ h(t;a))u

(3.63)

and =

C(t)U

respectively.

( ( c p C ) ( t ) + ( c p ' S ) ( t ) ) * (6 @ I

+ rn(t;a))u

Nothing milch can be obtained from t h e s e i d e n t i t i e s t h a t

was not known p r e v i o u s l y on t h e b a s i s of (3.53) and (3.55), t h e exponential bounds emanating from exponent

W

(3.64)

in

improvement i f

(3.23) i n s t e a d of

<mT

W'

except t h a t

(3.63) and (3.64) employ t h e i n (3.38), which may be a n

( t h i s i s not v e r y e x c i t i n g , however s i n c e b o t h

exponents a r e probably v e r y f a r from o p t i m a l ) .

REMARK 3.8.

Using technique.;

similar t o t h o s e i n t h e proof of

Theorem 3.2 we can show t h a t i f t h e Cauchy problem f o r (3.1) i s well

289

THE COMPLETE EQUATION

posed i n

0

5t 5

a

t h e n a s u i t a b l e v e r s i o n of Theorem 3.2 holds.

Exercise 14 and t h e author [1970:2]) REMARK

3.9.

for details.

I n t e g r a t i n g by p a r t s i n

R(A;cp)u =

ik

See

(3.19) w e

obtain

C x

(3.65)

e-At(c@)'(t)u d t ,

thus

I n view of (3.25),

hence it follows from

(3.66) t h a t

Using t h e uniform bound (3.66)

and t h e denseness of

can extend (3.67) t o a r b i t r a r y

gVIII.4

u

E

D(A)

n D(B),

we

E.

Construction o f phase spaces.

Em f o r t h e equation

W e show i n t h i s s e c t i o n t h a t a phase space

u"(t) + Eu'(t) + Au(t)

=

0

(4.1)

can be constructed i f t h e Cauchy problem f o r ( 4 . 1 ) i s well posed i n

t

2

0

and Assumption 3.1 i s s a t i s f i e d ( a s noted i n Theorem 2 . 3 t h e

s o l e assumption t h a t t h e Cauchy problem f o r (4.1) i s well posed i n t h e sense of QVIII.~ i s insufficient). g e n e r a l i z a t i o n of t h e space

@

The space

am i s

the d i r e c t

i n $111.1 f o r t h e incomplete equation.

m For i t s d e f i n i t i o n , t h e following r e s u l t s h a l l be needed. LEMMA 4.1.

t

2

0

Let t h e Cauchy problem f o r ( 4 . 1 ) be well posed i n

and l e t Assumption

such t h a t

C(t)u

3.1 be s a t i s f i e d .

Assume t h a t

u

2

0.

i s continuously d i f f e r e n t i a b l e i n t

E

E

&

Then t h e r e

290

THE COMPLETE EQUATION

exist -

such t h a t

C,W

where C -

(but not

Proof:

W)

we use equal-ity

t 30

differentiable i n

( d ) ( i +)

may depend on

(cp'S)(i)u

(3.55);

u.

c(i)u

if

i s continuously

( i n f a c t , i n t h e support of

CD)

then

i s continuously d i f f e r e n t i a b l e as well:

t h e n d i f f e r e n t i a t e both s i d e s of

(3.55),

we can

obtaining

C ' ( t ) u = ( & @ hI (+ t;v))* ((U'C)(t)u+ ( d ' ) ( t ) U +

(U"s)(t)u+ (U'S')(t)u),

(4.3) hence ( 4 . 2 ) follows using

(3.47).

This ends t h e proof of Lemma

4.1.

The c o n s t r u c t i o n of t h e m a x i m a l phase space f o r ( 4 . 1 ) proceeds now much i n t h e same way a s f o r (111.:!.1).

The space

m'

I s defined as

follows : @

m

= E o X E

endowed w i t h any of i t s product norms.

u

E

E

c(i)u

such t h a t

The norm i n

W

>

2

0

0.

SUP

s20

e

-us

IFr( s ) u l l ,

(4.5)

Eo

=

(4.6)

D ( A ) _C E o .

i s a Banach space i s much t h e same as t h a t f o r t h e

(111.2.1) and w e omit i t .

THEORE3l 4.2.

t

2

W e obviously have Do

equation

c o n s i s t s of all

so l a r g e t h a t ( 3 . 2 ) , t h e f i r s t i n e q u a l i t y (3.57)

W'

and ( 4 . 2 ) h o l d .

The proof t h a t

Eo

is

Eo

W',

The space

i s continuously d i f f e r e n t i a b l e i n t

l I ~ 1 1=~ IIuII + where

(4.4)

Let t h e Cauchy problem f o r

and l e t Assumption 3.1 be s a t i s f i e d .

f o r (4.1). Proof:

We must show t h a t

( 4 . 1 ) be well posed i n Then

Em i s a phase space

29 1

THE COMPLETE EQUATION

We prove f i r s t t h a t each Em. q t ) i s a bounded operator i n Em. I n order t o do t h i s we t a k e u E D0 and f i x t > 0 . Due t o time invariance of ( 4 . 1 ) t h e f u n c t i o n i s a s t r o n g l y continuous semigroup i n

u(i)

=

C ( t + g)u

i s a s o l u t i o n of ( 4 . 1 ) t h u s it follows from formula

(1.5) that C(S

+

= C(s)C(t)u

t)U

This e q u a l i t y i s extended t o

+ S(s)C'(t)u

u

aJ_1

E

Eo

(s,t

as follows:

2

(4.8)

0).

integrate i n

0 5rzt,

-rote(. +

T)u dT = c ( s )

Lt

C(7)U dT

+

and extend (4.9) t o a r b i t r a r y

u E E by denseness of we d i f f e r e n t i a t e and o b t a i n (4.8). The analogue of ( 4 . 8 ) f o r S ( t ) i s S(S

+ t)u

= C(s)S(t)u

+

u(s) = 8(s

+

Do;

u

for

E

2

0),

(4.10)

t)u, u

E

D1;

since all

u

operators i n (4.10) are bounded we can extend t h e e q u a l i t y t o all We note i n passing t h a t (4.8) i t s e l f can be extended t o all

a( s ) C ' ( t )

modified form observing t h a t

Eo

(s,t

S(s)S'(t)u *I

and i s shown by applying (1.5) t o

(4.9)

S(s)(c(t)u-u),

u

f

E

E

E.

in a

must have a bounded extension.

We s h a l l not make use of t h i s i n what follows. We prove t h a t each

qt)

i s a bounded o p e r a t o r i n

Em. To do

t h i s , we m u s t show t h a t t h e o p e r a t o r s C ( t ) :Eo

-

c ' ( t ) : Eo

E 0 E

are bounded i n t h e spaces i n d i c a t e d .

8 ( t ) :E

-

8 ' ( t ) :E

Eo

-.

(4.11) E

This i s r a t h e r obvious f o r

c'(t)

E ) and f o r a t ( t ) (from Assumption 3.1). 0 Note a l s o t h a t it follows from Corollary 3.7 and Lemma 4.1 t h a t

(from t h e d e f i n i t i o n of

292

THE COMPLETE EQUATION

( h e r e and i n o t h e r i n e q u a l i t i e s

C

denotes an a r b i t r a r y constant, not

n e c e s s a r i l y t h e same i n d i f f e r e n t p l a c e s ) . Continuity of

C(t)

C(s)C(t)u = C ( s + t ) u - S ( s ) C ' ( t ) u j

form

and d i f f e r e n t i a t e with respect t o

apply t o an element

u

+ t ) u - 8'(s)CI(t)u.

i s a bounded operator from

C(t)

i n the of

E~

We obtain

s.

Cl(s)C(t)u = C l ( s If follows t h a t

Write ( 4 . 8 )

i s proved a s follows.

Eo

(4.13) into

and

Eo

(4.14) f o r some constant

Finally, boundedness of

C.

Write (4.10) i n t h e form

It follows t h a t

' 0

i s continuous i n

+ t ) u - 8t(s)Sl(t)u.

)

5

wt

(t

Ce

(4.15)

L

(4.16)

0)

W e have t h e n completed t h e proof t h a t each

C.

Em: moreover, t h e r e e x i s t s a constant

wt

l l ~ t I l l ~ 5~ Cme ) f o r some constant

W e then

The r e s u l t i s

i s a bounded operator and

8(t)

lls(t)ll(E.E f o r some constant

i s shown a s follows.

+ t)u-S(s)S'(t)u.

C(s>S(t)u = 8 ( s

d i f f e r e n t i a t e t h i s e q u a l i t y term by term. C f ( s ) S ( t ) u= S l ( s

S(t)

C,

t h e constant

(t

C

10 )

q t )

such t h a t (4.17)

being t h e same i n Corollary 3.7

w

and Lemma 4.1. The semigroup equation

follows from (4.8) and (4.10) and t h e i r d i f f e r e n t i a t e d versions (4.13) and (4.15). n

The next step i s t o show t h a t

q t )

i s s t r o n g l y continuous.

It

i s enough t o prove t h a t Ilqh)u as

h

qtk

-

O+.

- uII( Em)

+

(4.19)

0

However, we s h a l l skip t h i s step since we show below t h a t

has a derivative at t h e o r i g i n ( i n t h e norm of

s)

for u

in

293

THE COMPLETE EQUATION

Gm; t h i s , combined w i t h t h e uniform bound (4.28)

a dense subset of obviously i m plies

THEOREM 4.3.

(4.19),

q;)

since

i s a s t r o n g l y c o n t i n u o u s semigroup w i t h

8 given by

infinitesimal generator

8=

=

c l o s u r e of

71,

(4.20)

where

w i t h domain D(%) = D ( A )

The f u n c t i o n

i s a s o l u t i o n of

u(;)

u(t)

(4.22)

( D ( A ) fI D ( E ) ) .

X

(4.1) o n l y i f

= [u(t),u'(t)l

(4.2?)

i s a s o l u t i o n of

u'(t)

=

%u(t).

Proof: We b e g i n b y showing t h a t t o p o l o g y of

Eo.

i s dense i n

D(A)

To d o t h i s we s e l e c t a

(4.24)

"6-sequence"

Eo

{@,I

i n the

of scalar

f u n c t i o n s l i k e that used i n t h e proof o f C o r o l l a r y 3.5 ( b ) , and show that

u as

n

-

( f o r any

f o r each

m

u

E

E)

(4.13) we s e e that

u

n

= J$ , ( t ) c ( t ) u

E

E

0' i s obvious.

dt

That ( 4 . 2 6 )

-

(4.26)

u

h o l d s i n t h e t o p o l o g y of

Assume now t h a t

u

E

EO.

Then, using

E

294

THE COMPLETE EQUATION

and we check e a s i l y t h a t

e-WSC'(s)un converges uniformly i n

to

un

emWSCt(s)u, s o t h a t

-

u

in

20

t

EO'

W e show next t h a t

u

Em f o r each

in

l i m i t r e l a t i o n s as

E

h

-

D(3).

This i s equivalent t o t h e following f o u r

0+:

-

h'l(C(h)u for

for

u

u

u

for

E

E

E

u

E

h-lS(h)u

-

u

(4.28)

h-lC'(h)u

-

-Au

in

Eo

(4.29)

D(A) n D(B),

D(A),

D(A)

n D(B).

- u)

+

--. -Bu

in

0

(4.31)

E

To prove (14.28) we use (4.13) i n t h e form

s,

i s bounded i n norm by a constant

t h e constant described a f t e r

e W I S , m'

(4.32) as h

(4.30)

E

in

and

This expression, as a f u n c t i o n of

times

in Eo

D(A),

h-'(S1(h)u for

u ) -, 0

(4.5).

The l i m i t of

is

C"(~)U-S'(S)C"((~)U= C"(S)U + S'(S)AU = 0 after (3.11). C1(s)(h"S(h)u

To show (4.29) we w r i t e (4.10) in t h e form

- u ) = h''(St(s

+ h)u

- S t ( s ) u ) - g'(s)h''(8'(h)u - u ) - c ' ( s ) u , (4.33)

which i s bounded i n norm as well by a constant times

as

h

-

O+

eWts; i t s l i m i t

is

S"(S)U-S'(S)S"(O)-C'(S)U

=

S"(S)U + S'(S)BU

+

S(S)AU = 0

(4.34)

295

THE COMPLETE EQUATION

i n view of Corollary 3.5(d).

F i n a l l y , t h e two l i m i t r e l a t i o n s (4.30)

and (4.31) a r e obvious, since and

u

E

h-l(W'(h)u

- u) n D(E)

-D(A) 1

D

-

-

h-lC'(h)u

( s e e Corollary

3.5

%

=

=

-AC(O)u- EC'(0)u

= -Au

-Eu f o r

(c)).

q;)

Having proved (4.27), we know t h a t semigroup and t h a t , i f

C"(0)u

- BS'(0)u

S"(0)u = -A8(0)u

i s a s t r o n g l y continuous

i s i t s i n f i n i t e s i m a l generator, t h e n

(4.35)

U C B . T o improve (4.35) t o (4.20) it w i l l be s u f f i c i e n t t o prove t h a t Uh

f o r all u

E

D(9J)

s e l e c t a sequence

In f a c t , i f

{un]

(t)u d t

= kJhE

5 D(Z)

E

D(U)

(4.36)

(4.36) i s t r u e and

-

un

with

u

in

u Qm

E

(g, we may

(that

D(8)

is

dense i n ( u )h + 11

follows from (4.26) and f r o u Corollary 3.5 ( b ) ) . Then whereas u(un) h = 8 ( u n ) h = h -1(F(h)un- un) -+ h -1( S ( h ) u - u);

uhn€ D(E)

f o r any

u

E

Gh = Assume that tends t o 8 u

u

that

E

h > 0

Qm and any

and

-

h-'(5(h)u

.

u)

(4.37)

u E D(%). Taking i n t o account t h a t t h e r i g h t s i d e of (4.37) as h+O+ it follows from t h e fact t h a t i s closed

D(@

u

and

uu

= %u, which completes t h e proof of (4.20).

The i n c l u s i o n r e l a t i o n (4.36) c a n be reduced t o t h e f o u r r e l a t i o n s

(4.38)

(4.39) / g h C l ( t ) u d t = C(h)u

L h S t ( t ) u d t = S(h)u-u If

u

E

D(A)

E

-

u

D(A)

we have

E

D ( A ) fl D ( B )

n D(B)

(U E

(u

D(A)

=L

E

D(A)),

(4.40) (4.41)

D(E)).

h

AJOhC(t)u d t

so t h a t (4.38) h o l d s .

8(<)u

AC(t)u d t ,

On t h e o t h e r hand, i f

u

E

D(A)

i s a s o l u t i o n of ( 4 . 1 ) s o t h a t (4.39) holds and

(4.42)

n D ( B ) 5 D1

then

296

THE COMPLETE EQUATION

ALhs(t)u dt =/rAS(t)u

(4.43)

dt.

Obviously, (4.40) i s v e r i f i e d ; t o check ( 4 . 4 1 ) we note t h a t , since

u

E

n

D(A)

S(h)u

E

5 D1,

D(B)

D(A):

S(t^)u i s a s o l u t i o n of ( 4 . 1 ) so t h a t

on t h e o t h e r hand, h

1-

.Io

s'(t)u d t

=&

h

(4.36).

This ends t h e proof o f

We check t h e f i n a l statements i n Theorem 4.1.

i s t h e s e t of d l

D(@)

(4.25), we note t h a t

i s d i f f e r e n t i a b l e i n &!

qt)u

q;)

(4.44)

BS'(t)u d t .

. m

I n t h e matter of

u = [u,v]

such t h a t

I n view of t h e d e f i n i t i o n (4.7) of

t h i s means t h a t

(a) C(t)u + 8(t)v (b)

Cl(t)u

+

Statement ( b ) i s obvious i f again

(4.8) and ( 4 . 1 0 ) .

omit them.

i n Eo,

i s differentiable

S * ( t ) v i s differentiable i n

[u,v]

E

D(A) x

A s for ( a ) , we use

The d e t a i l s shoula be familiar by now and w e

We have concluded t h e proof of

EXAMPLE 4.4.

D1.

(4.45)

E.

Theorem

4.4.

We show below t h a t , i n g e n e r a l ,

% + a ;

(4.46)

(4. 25) cannot i n g e n e r a l be guaranteed, so t h a t t h e r e i s no t o t a l equivalence o f t h e equations ( 4 . 1 ) and (4.24). We

moreover, e q u a l i t y i n

E =

consider t h e space

we assume t h e elements o f We d e f i n e t h e o p e r a t o r s

so t h a t we have

-

A = -n, n

zc

i n Example2.1, b u t , f o r t e c h n i c a l reasons, E

t o be of t h e form

A, B

i n (2.2) thusly:

+

A n

=

-n

2

n2

It f o l l o w s t h a t

=

tun) = {un;n

2

21.

, 2

2 -nt

and

u

-

-Yu} n ' n

(4.48)

THE COMPLETE EQUATION

llc(t>ll5

o(t) =

297

= 3 . 1 nl2

n

-2'

- n

hence t h e Cauchy problem f o r ( 4 . 1 ) i s well posed i n

3.1 i s s a t i s f i e d .

e a s i l y t h a t Assumption

u = (un)

sequences

t

2

Eo

The space

We check

0.

c o n s i s t s o f all

such t h a t

(4-50)

n t h e norm

I 1

being equivalent t o t h e g e n e r d norm (4.5).

space

5E

c o n s i s t i n g of all sequences

*

E

1

Obviously,

D ( A ) C El

Take a sequence

C

Eo

C

=

{u ) n

such t h a t

each i n c l u s i o n being bounded and s t r i c t .

E,

{urn] = {u

u

Consider t h e

nm

)

of elements of

D(A)

such t h a t

um - + L I C 1 E

El,

i n t h e topology of

(4.52),

both

{nu

u

where

nm 1

{vm]

=

Qw

proving t h a t

does not belong t o D ( A ) . I n view of 2 converge i n E, so

{w ) = {n ),u ?

and

that

converges i n

But

u

#

D(U),

u

=

+

D ( A ) x D1.

[u,v]

belongs t o

D(g) = D(@).

so t h a t D(W

REMARK

(4.52)

4. 5.

The r o l e o f p a r t ( b ) of Assumption

3.1 i s somewhat more

obscure t h a t of ( a ) , which has been shown t o be e s s e n t i a l i n t h e c o n s t r u c t i o n of phase spaces.

It can be shown ( s e e Exercises 1 t o 11)

t h a t ( b ) can be given up, but some a d d i t i o n a l assumptions i n

D1

must be added

t h a t a r e not e s p e c i a l l y e a s y t o v e r i f y .

of Theorem 4.2 holds (Exercise

13).

Do

and

A version

298

THE COMPLETE EQUATION

QVIII .5.

Miscellaneous comments

.

The equation ( 4 . 1 ) has been extensively studied by semigroup methods since t h e paper of LIONS [1957:1

1.

Other e a r l y c o n t r i b u t i o n s a r e due

t o M I T J A G I N [1961:11 and SOEOLEVSKI? [1962:13, can i n f a c t depend on

t.

[1964:ll

where

A

and

E

For r e f e r e n c e s see KREiN [1967:1]; some of t h e

more recent l i t e r a t u r e i s i n t h e author

[1983:1].

In these papers, the

emphasis l i e s i n reducing ( 4 . 1 ) t o a f i r s t order system i n a product space (by means o f ad hoc assumptions on t h e c o e f f i c i e n t s

A,

E)

and

t h e n using semigroup theory. The d e f i n i t i o n s and r e s u l t s presented i n t h i s chapter a r e due t o t h e author; Sections VIII.1, VIII.2 and VIII.3 a r e taken from [1970:2]. contained ( i n a somewhat d i f f e r e n t formulation) i n

Section VIII.4 i s

A much e a r l i e r treatment of ( 4 . 1 ) i n t h e s p i r i t of 6v111.4

[1981:1].

i s due t o WEISS [1967:1], where t h e notion of phase space i s introduced i n a somewhat more general form:

i n t h i s formulation of t h e theory, t h e

( 4 . 1 ) i s not assumed t o be well posed i n t h e sense

Cauchy problem f o r of gvI11.1.

It i s obvious t h a t one could generalize t h e t h e o r y i n t h i s chapter t o more general equations (say, d i f f e r e n t i a l equations of order

n).

This has been done f o r t h e material i n t h e f i r s t t h r e e s e c t i o n s i n t h e author [1970:2].

A more i n t e r e s t i n g l i n e of approach i s suggested by

Assume t h e Cauchy problem f o r (4.1) i s well

t h e following argument.

posed and t h a t Assumption

n

X = D(A)

D(E)

3.1 i s s a t i s f i e d .

Consider t h e space

endowed with t h e " j o i n t graph norm"

\ \ u \ /+ l\Bul\ + \\AuI/. Since A and 13 a r e closed, X i s a Eanach space and 63 = 6" @ I + 6' B E + 6 '8 A (6 t h e Dirac d e l t a ) i s a

l\uIIx

=

d i s t r i b u t i o n i n t h e space -W


00,

&l((X;E))

with support i n

t

2

0

of a l l d i s t r i b u t i o n s defined i n and va.lues i n t h e space of operators

On t h e other hand, we check on t h e b a s i s of p a r t ( b ) of

(X;E). Assumption 3.1 t h a t

S(;)

(extended t o

t h e r e ) i s a d i s t r i b u t i o n defined i n and values i n we have

D(A)

(E;X).

-m

By d e f i n i t i o n of

n D ( B ) 5 Dr)

t < 0 by s e t t i n g 8 ( t ) = 0 t < m, with support i n t

C:

we prove t h a t L?*8=6@1.

On t h e other hand, (3.14) implies t h a t

2

0

S(i) (and due t o t h e f a c t t h a t

(5.1)

299

THE COMPLETE EQUATION

where

( r e s p . I) i s t h e i d e n t i t y o p e r a t o r i n

J

X (resp. E).

In this

s e t t i n g , t h e problem of c o n s t r u c t i n g t h e propagator S ( t ) of (4.1) of f i n d i n g a c o n v o l u t i o n i n v e r s e of (such as

2

t

having support i n

63

is that

s a t i s o i n g suitable properties

0, e t c . ) and we can pose t h i s problem

i n r e l a t i o n t o an a r b i t r a r y d i s t r i b u t i o n

63

E

t h u s being

&I+((X;E)),

a b l e t o t r e a t e q u a t i o n s more g e n e r a l t h a n d i f f e r e n t i a l ; f o r i n s t a n c e ,

63

€C I

= St(:)

-

+ 6(;) C% A + 6(t"

h) @

E corresponds t o t h e d i f f e r e n c e -

d i f f e r e n t i a l equation

u ' ( t ) + Au(t) + Eu(t - h )

st(;)

and t h e d i s t r i b u t i o n 63 =

@

i s t h e Heaviside f u n c t i o n

h(:)

t < 0)

I + 6(:)

=

C 3A

0

,

+ h(:) t

(h(t) = 1 for

@

2

where

E,

0, h ( t ) = 0

for

corresponds t o t h e i n t e g r o d i f f e r e n t i a l e q u a t i o n

~ ' ( t+) A u ( t ) + E

Lt

u(s)ds = 0.

For d e t a i l s on t h i s approach s e e t h e a u t h o r [1976:1], [1980:11, [1983:11,

b983 :2 1. I n Exercises 1 t o

i s well posed i n Assumption

t

2

4

we suppose t h a t t h e Cauchy problem f o r

a s defined i n $ V t I I . l ;

0

we d o not r e q u i r e

3.1.

EXERCISE 1. Let holds (Hint:

u

D(A)

E

EXERCISE 2.

E

D,I

.

Show t h a t (3.11)

Let

u

s,'

n D(E)

Do

F

-

~ ( s ) A ud s )

.

be such t h a t

(5.4) Bu

E

Dl. Show t h a t

(Hint : prove t h e i n t e g r a t e d v e r s i o n

8(t)u = EXERCISE 3.

i s such t h a t

Au

prove t h e i n t e g r a t e d v e r s i o n c(t)U = u

(3.12) h o l d s .

such t h a t

Au

Lt(

C(

s)U

-

8(s ) E u ) d s )

(5.5)

Combining Ekercises 1 and 2 show t h a t i f E

D1, Bu

E

Sl'(t)u

D 1

+

u

E

D

0

t h e n (3.14) h o l d s , t h a t i s

b'(t)Eu

+

S(t)Au

=

0

.

(5.6)

Dl

300

THE COMPLETE EQUATION

EXERCISE

4.

Show t h a t p a r t ( t ~ of ) Lemma 3.6 i s t r u e i n t h e present

l e v e l of g e n e r a l i t y , t h a t i s P ( h ) = h'I

+ hE + A

(5.7)

i s one-to-one i n a region of t h e f c i r m R e h > a + p l o g (1 + I h l ) I n Exercises ASSUMPTION

u

for all

of

E

{u

(5.8)

5 t o 11 we r e q u i r e p a r t ( a ) of Assumption 3.1, t h a t i s

5.1.

i s continuously d i f f e r e n t i a b l e i n t

S(i)u

E

5.

U s i n g Exercise 2 show t h a t t h e operator

0

Do Tl D(B); Bu

D1]

E

8(t)B

(with

h a s a bounded extension

t o all

given by = C(t)

EXERCISE 6.

Define

with

m(F)

-

Sf(t)

.

(5.9)

R(h;c~) as i n (3.19),

R( h;fn)u

Jbwe-htm(t)s(t)u

=

,

dt

a t e s t f u n c t i o n i d e n t i c d l y equal t o

(5.10)

1 near zero.

(a s l i g h t l y extended v e r s i o n o f ) (5.6) show t h a t if u such t h a t

Au, Bu

E

where

N(t;rp) = 2 v f ( t ) 3 ' ( t ) EXERCISE

Show t h a t for

7.

E

Do

Using

n D(B)

is

D1 t h e n (3.37) holds, that i s ,

+ $h;a)u

R(h;a)P(h)u = u

Define

-t

p"(t)s(t)

R(h)

as i n

+

,

(5.11)

a ' ( t ) W .

(3.41)

for

Reh

3 W,

w l a r g e enough.

u as i n Exercise 6 we have R(h)P(h)u = u

EXERCISE 8.

h(:;rp)

2

E E.

EXERCISE domain

.

For Reh

given by (3.46).

R( A). EXERCISE 9 .

Define

2

w,

W

.

l a r g e enough, d e f i n e

S(;)

Prove that t h e Laplace transform of

(5.12)

- (3.48),

by

8 equals

301

THE COMPLETE EQUATION

jta-'/r(a) ya ( c o n v o l u t i o n by

;lo

0)

(t < 0 )

Ya produces t h e " a n t i d e r i v a t i v e of order a " ) . Show

t h a t , m u l t i p l y i n g (5.12) by o b t a i n , u s i n g Exercise

for

2

(t

and i n v e r t i n g Laplace t r a n s f o r m s we

8,

as i n Exercise 6.

u

EXERCISE 10. Assume t h a t t h e s e t of d1 u

Au, Eu

E

D1 i s dense i n t h e space X

Do

E

n D(B)

= D(A)

n D(B)

such t h a t

endowed with t h e

norm

EXERCISE 11. Snow t h a t , i f

(Y1

€3 I

+ Y2

Combining (5.13) and

@ E

u

E

D1,

+ Y @A) * 3

SU = Y

€3 u

3

(t

2

(5.14)

0).

( 5 . 1 4 ) , prove t h a t , under t h e c o n d i t i o n s of

Exercise 10,

qt) = E(t). EXERCISE 12.

Under t h e c o n d i t i o n s of Exercise 1 0 , show t h a t

(3.53) ( r e s p . (3.54)) h o l d s for st(;)

formula

t h a t there exist constants

EXERCISE

u

E

E

SL(t)B).

c(t)u

Show

i s continuously d i f f e r e n t i a b l e i n

if

t 10

(4.3) h o l d s , so t h a t Ilcl(t)UII

C

(resp. f o r

such t h a t

C,W

13. Under t h e c o n d i t i o n s of Exercise 8 show t h a t ,

i s such t h a t

t h e n formula

with

(5.15)

( b u t not

w)

5

Ce

wt

may depend on

under t h e p r e s e n t hypotheses.

(t

u.

L

0)

-

(5.17)

Show t h a t Theorem 4 . 2 i s v a l i d

THE COMPLETE EQUATION

302

EXERCISE w e l l posed i n

14. 0

5t 5

E

D(A)

n D(B)

i s well posed i n

2

0

u E.

i s dense i n

t

t h a t Assumption

a (a > 0),

t h e r e and t h a t t h e s e t of all

Bu

(5.3) i s 3.1 i s s a t i s f i e d

We suppose h e r e t h a t t h e Cauchy problem f o r

f

D ( A ) I- D(E)

such t h a t

Then t h e Cauchy problem for

(5.3)

and Assumption 3.1 i s s a t i s f i e d . Note t h a t 611

t h e assumptions i n t h i s Exercise a r e s a t i s f i e d f o r t h e incomplete equation

u"(t) + Au(t) = 0

(5.18)

under t h e only assumption t h a t t h e Cauchy problem for

(5.18) i s w e l l

posed; of course, t h e r e s u l t for (5.18) can be proved i n a more elementary way by ad hoc methods. FOGTNOTES TO CHAPTER VIII

(1) We note t h e i n c o n s i s t e n c y of n o t a t i o n involved i n w r i t i n g t h e incomplete e q u a t i o n

u" + Au

=

0,

and not

u"

=

Au

as i n Chapters I1

and 111. (2)

Although t h e argument could be completed using (3.25), t h e " l e f t -

(3.41) s i m p l i f i e s some of t h e arguments. ( 3 ) We might s e t h e r e W1 = min(U,wl): f o r i f W ' < U, R(A) c a n be ana'Lytically continued t o Reh > W ' by means of Q ( h ) . ( 4 ) Convolution by Y i s employzd h e r e t o avoid using convolution of 3

handed" r e p r e s e n t a t i o n

distributions.

303

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Periodic strongly continuous semigroups. Ann. Mat. Pura

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