Differential Equations (North Holland Mathematics Studies,92)

DIFFERENTIAL EQUATIONS

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NORTH-HOLLAND MATHEMATICS STUDIES

DIF'F'ERENTIAL EQUAnONS Proceedings of the Conference held at The University of Alabama in Birmingham, Birmingham, Alabama, U.S.A. 21-26 March, 1983 Edited by

Ian W. KNOWLES and Roger T. LEWIS The University ofAlabama in Birmingham Birmingham Alabama U.S.A.

IS84 NORTH-HOLLAND - AMSTERDAM. NEW YORK. OXFORD

92

Elsevier Science Publishers B . V . 1984 AN rights reserved. N o part of this publication may be reproduced. stored in a retrieval system, @

~

or transmitted, in any form o r by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

I S B N : 0 444 84875 5

Publishers: ELSEVIER SCIENCE PUBLISHERS B . V . P.O. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U . S .A . and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY. INC 52 Vanderbilt Avenue NewY0rk.N.Y. 10017 U.S.A.

Library of Congmi Cataloging in Publication Data

Main entry under title: Differential equations. (North-Holland mathematics studies ; 9 2 ) Lectures given at the International Conference on Differential Equations, held at the University of Alabama in Bi&ngham during March 21-26, 1983”--Pref. Bibliography: p. Includes index. 1. Differential equations--Congresses. I. Knowles, Ian W. 11. Lewis, Roger T. 111. International Conference on Differential Equations (1983 : University of Alabama) IV. Series.

~ 3 7 0 D53 . 1984 ISBN 0-444-86a75-5

515 -3 5

84-4060

PRINTED IN THE NETHERLANDS

V

PREFACE

T h i s volume forms a permanent r e c o r d o f l e c t u r e s q i v e n a t t h e I n t e r n a t i o n a l Conference on D i f f e r e n t i a l Equations h e l d a t t h e U n i v e r s i t y o f Alabama i n Birmingham d u r i n g March 21-26,

1983.

The c o n f e r e n c e was supported by about 150 mathematicians from t h e f o l l o w i n g countries:

Belgium, Canada, China, Eqypt, Enqland, Federal R e p u b l i c o f Germany,

France, I n d i a , I r e l a n d , I s r a e l , Japan, Mexico, N i g e r i a , Norway, R e p u b l i c o f South

U. S. S. R. and t h e U. S . A . I t s main t o p r o v i d e a forum f o r t h e d i s c u s s i o n o f r e c e n t work i n t h e t h e o r y o f purpose was

A f r i c a , Scotland, Sweden, S w i t z e r l a n d , t h e

o r d i n a r y and p a r t i a l d i f f e r e n t i a l e q u a t i o n s , b o t h l i n e a r and n o n - l i n e a r , under t h e g e n e r a l heading o f t h e e q u a t i o n s o f mathematical p h y s i c s . hour l e c t u r e s were g i v e n by

s.

unified

I n v i t e d one-

Agmon, F. V. A t k i n s o n , H. B r e z i s , R. C a r r o l l ,

M. C r a n d a l l , I. Ekeland, V . Enss, J . F r o h l i c h ,

T. Kato,

P. Lax,

E. L i e b ,

C . Morawetz, P. R a b i n o w i t z , M. Reed, P. Sarnak, M. Schechter, B. Simon, J . Smoller,

R. Temam, and K. Yajima.

The remainder o f t h e proqramme c o n s i s t e d o f i n v i t e d one-

ha1 f hour 1ec t u r e s . On b e h a l f o f t h e p a r t i c i p a n t s , t h e c o n f e r e n c e d i r e c t o r s acknowledge, w i t h g r a t i t u d e , t h e qenerous f i n a n c i a l s u p p o r t p r o v i d e d b y t h e N a t i o n a l Science Foundat i o n , under g r a n t number MCS-8214420, and b y t h e School o f N a t u r a l Sciences and Mathematics, U n i v e r s i t y o f Alabama i n Birmingham.

We a r e e s p e c i a l l y g r a t e f u l t o

P r o f e s s o r P e t e r O ' N e i l , Dean o f t h e School, and P r o f e s s o r L o u i s Dale, Chairman o f t h e Department o f Mathematics, f o r t h e i r s u p p o r t and encouragement.

We acknow-

l e d g e a l s o t h e v a l u a b l e s u p p o r t p r o v i d e d b y t h e f a c u l t y and s t a f f o f t h e Department o f Mathematics; here, we a r e p a r t i c u l a r l y g r a t e f u l t o P r o f e s s o r Fred Martens f o r h i s c o n s i d e r a b l e c o n t r i b u t i o n i n d i r e c t i n a t h e l o c a l arrangements, and t o Mrs. E i l e e n Schauer f o r u n d e r t a k i n g t h e enormous t a s k o f t y p i n g much o f t h e c o n f e r e n c e m a t e r i a l , i n c l u d i n g many o f t h e papers a p p e a r i n g i n t h i s volume. F i n a l l y , i t i s a p l e a s u r e once a g a i n t o acknowledge t h e f r i e n d l y and p a t i e n t a s s i s t a n c e o f Drs. A r j e n Sevenster, E d i t o r o f t h e Mathematics S t u d i e s S e r i e s o f N o r t h - H o l l a n d , d u r i n q t h e p r e p a r a t i o n o f t h e s e proceedinqs. I a n W . Knowles Roqer T. Lewis Conference D i r e c t o r s

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vii

CONTENTS

Preface L e c t u r e s Not Appearing i n t h e Proceedings Address L i s t o f C o n t r i b u t o r s

V

xiii xv

U n i f o r m D i s s i p a t i v e S o l u t i o n s f o r a T h i r d Order Non-Linear D i f f e r e n t i a1 E q u a t i o n A.U. Afuwape

1

On P o s i t i v e S o l u t i o n s o f E l l i p t i c Equations w i t h P e r i o d i c C o e f f i c i e n t s i n R n , S p e c t r a l R e s u l t s and Extensions t o E l l i p t i c Operators on Riemannian M a n i f o l d s S. Agmon

7

C l a s s i f i c a t i o n o f I n i t i a l Data f o r t h e Porous Medium E q u a t i o n i n R N N.D. A l i k a k o s and R. Rostamian

19

S t a b i l i t y o f Quantum Mechanical Shape Resonances v i a R i c c a t i ' s Equation M. Ashbaugh and C. Sundberg

25

Remarks on t h e I n v e r s e Square P o t e n t i a l i n Quantum Mechanics P. Baras and J.A. G o l d s t e i n

31

J u l i a Sets and Autonomous D i f f e r e n t i a l Equations M.F. B a r n s l e y and A.N. H a r r i n g t o n

37

A L i m i t i n g A b s o r p t i o n P r i n c i p l e f o r a Sum o f Tensor Products M. B e n - A r t z i and A. D e v i n a t z

43

An A l g e b r a i c G e n e r a l i z a t i o n o f S t o c h a s t i c I n t e g r a t i o n M. Berger and A. Sloan

49

V a r i a t i o n a l Problems i n v o l v i n g Lack o f Compactness and R e l l i c h ' s Conjecture H. B r e z i s

53

A F a c t o r i z a t i o n Method f o r Symmetric D i f f e r e n t i a l Operators and i t s A p p l i c a t i o n s t o D i r i c h l e t I n e q u a l i t i e s and t o t h e D i r i c h l e t Index R . C . Brown

61

S o l u t i o n s w i t h Asymptotic C o n d i t i o n s o f a N o n l i n e a r Boundary Value Problem N.P. CLc

71

A b s o l u t e Continuous Spectrum o f One-Dimensional Schrodinger Operators R. Carmona

77

viii

Contents

Some Topics i n Transmutation R. C a r r o l l

87

An Equation Modeling t h e E l e c t r i c B a l l a s t R e s i s t o r N. Chafee

105

Nonexistence o f P o s i t i v e S o l u t i o n s f o r S i n g u l a r Hyperbolic D i f f e r e n t i a l Inequalities C.Y. Chan

111

Almost P e r i o d i c i t y o f Bounded S o l u t i o n s t o Nonlinear A b s t r a c t Equations C. Corduneanu and J.A. G o l d s t e i n

115

A P r i o r i Estimates i n Nonlinear Eigenvalue Problems f o r E l l i p t i c Systems C. Cosner

123

Developments i n t h e Theory o f Nonlinear F i r s t - O r d e r P a r t i a l D i f f e r e n t ia1 Equations M.G. Crandall and P.E. Souganidis

131

R e l a t i v i s t i c Molecules w i t h Coulomb I n t e r a c t i o n I . Daubechies and E.H. L i e b

143

Non-Linear Delay D i f f e r e n t i a l Equations and F u n c t i o n Algebras L.D. Drager and W. Layton

149

Transformations o f D i f f e r e n t i a l Equations, t h e Levinson Asymptotic Theorem and D e f i c i e n c y I n d i c e s M.S.P. Eastham

155

A Morse Theorem f o r Hamiltonian Systems I.Ekeland

165

S c a t t e r i n g and Spectral Theory f o r Three P a r t i c l e Systems V . Enss

173

A L e f t D e f i n i t e Two-Parameter Eigenvalue Problem M. Faierman

205

A Semigroup Approach t o Burgers' System W.E. F i t z g i b b o n

213

On t h e Eigenvalues o f Non-Definite E l l i p t i c Operators J . F l e c k i n g e r and A.B. M i n g a r e l l i

219

Existence o f Generators and D i f f e r e n t i a b i l i t y o f E v o l u t i o n s M.A. Freedman

229

On t h e Asymptotic Behavior o f t h e P o s i t i v e S o l u t i o n s o f a D i f f e r e n t i a l Equation w i t h a Discontinuous Nonlinear Term J.R. Graef, P.W. Spikes, and M.K. Grammatikopoulos

237

Energy Estimates f o r Symmetric Hyperbolic I n t e g r o - D i f f e r e n t i a l Equations R. Grimmer and M. Zeman

241

Resolvent and Heat Kernels f o r Operators o f Schrodinger Type w i t h A p p l i c a t i o n s t o Spectral Theory 0. Gurarie

249

Contents

ix

V o r t i c i t y , I n c o m p r e s s i b i l i t y , and Boundary Conditions i n t h e Numerical S o l u t i o n of t h e Navier-Stokes Equations K. Gustafson and K. H a l a s i

257

Asymptotic Completeness f o r Few Body Schrodinger Operators G.A. Hagedorn and P.A. Perry

265

Asymptotics o f t h e Titchmarsh-Weyl m-Function, a Bessel-Approximative Case S.G. Halvorsen

271

L a t t i c e M u l t i s c a l e S i n g u l a r P e r t u r b a t i o n Theory C.R. Handy

279

S o l u t i o n s f o r Model Boltzmann Equations proposed by Z i f f D.P. Hardin and J.V. Herod

285

Asymptotic Behavior o f S o l u t i o n s o f Disconjugate D i f f e r e n t i a l Equations D. H i n t o n

293

Boundary Conditions f o r D i f f e r e n t i a l Systems i n I n t e r m e d i a t e L i m i t Situations A.M. K r a l l , J.K. Shaw, and D.B. H i n t o n

301

On t h e Spectrum o f a Hamiltonian System w i t h Two S i n g u l a r Endpoints J.K. Shaw and D.B. Hinton

307

A Product Formula f o r C e r t a i n Q u a d r a t i c Form P e r t u r b a t i o n s R.J. Hughes

313

On t h e Existence o f Resonant States K. I n g B l f s s o n

32 1

Asymptotic Behavior o f t h e S c a t t e r i n g Amplitude a t High Energies H. I s o z a k i and H. Kitada

327

F u l l - and Half-Range Theory o f an I n d e f i n i t e S t u r m - L i o u v i l l e Problem H.G. Kaper

335

Remarks on Holomorphic F a m i l i e s o f Schrodinger and D i r a c Operators T. Kato

341

Necessary and S u f f i c i e n t Conditions f o r S o l v a b i l i t y o f Non-Solvable L i n e a r P a r t i a l D i f f e r e n t i a l Equations S. K i r o

353

R e g u l a r i t y P r o p e r t i e s o f Schrodinger Operators on Domains of Rn M.A. Kon

359

R e l a t i v e Symmetries o f D i f f e r e n t i a l Equations B.A. Kupershmidt

367

Necessary and S u f f i c i e n t Conditions f o r O s c i l l a t i o n s o f Higher Order Del ay D i f f eren t ia 1 Equations G. Ladas, Y.G. Sficas, and I . P . S t a v r o u l a k i s

373

Boundary Behavior o f S o l u t i o n s o f Degenerate E l l i p t i c Equations and Generation o f Semigroups M. L a n g l a i s

381

X

Contents

The Zero D i s p e r s i o n L i m i t o f t h e KdV Equation P.D. Lax

387

Recent M u l t i p l i c i t y R e s u l t s f o r N o n l i n e a r Boundary Value Problems A.C. Lazer and P.J. McKenna

391

Nonstopping I t e r a t i o n f o r O r d i n a r y D i f f e r e n t i a l O p e r a t o r E q u a t i o n S.J. Lee

397

Some Vector F i e l d Equations E.H. L i e b

403

Weight D i s t r i b u t i o n s and Moments f o r a C e r t a i n Class o f Orthogonal Polynomials L.L. L i t t l e j o h n

413

On t h e L i m i t - P o i n t C l a s s i f i c a t i o n o f a Class o f N o n - S e l f - A d j o i n t O r d i n a r y D i f f e r e n t i a l Operators J.-L. L i u

42 1

A Moving Boundary Problem d e s c r i b i n g Oxygen Consumption i n S o i l R.C. McCann and P.K. McConnaughey

427

Converse BVP f o r A s s o c i a t e d E l l i p t i c and P a r a b o l i c F r a c t i o n a l P a r t i a l D i f f e r e n t i a 1 Operators P.A. McCoy

431

Bounds f o r C o n s t r u c t e d S o l u t i o n s o f Second and F o u r t h Order I n v e r s e Eigenvalue Problems J.R. McLaughlin

437

D i f f e r e n c e E q u a t i o n Models o f D i f f e r e n t i a l Equations h a v i n g Zero Local Truncation Errors R.E. Mickens

445

A Remark on Continuum E i g e n f u n c t i o n s o f N-Body Schrodinger Operators P.A. P e r r y

451

A C u r i o u s S i n g u l a r P e r t u r b a t i o n Problem P.H. Rabinowitz

455

A S i m p l i f i e d C h a r a c t e r i z a t i o n o f t h e Boundary C o n d i t i o n s which determine J - S e l f a d j o i n t E x t e n s i o n s o f J-Symmetric ( D i f f e r e n t i a l ) Operators D. Race

465

E s s e n t i a l S e l f - A d j o i n t n e s s f o r Powers o f Schrodinger Operators T.T. Read

47 1

Geometry and D i s c r e t e V e l o c i t y Approximations t o t h e Boltzmann E q u a t i o n M.C. Reed

477

Domains i n H y p e r b o l i c Space and L i m i t Sets o f K l e i n i a n Groups P. Sarnak

485

S e l f a d j o i n t Operators, S p e c t r a l and S c a t t e r i n g Theory, V a r i a t i o n a l Techniques, Non-Linear Phenomena, L i n e a r and Non-Linear P a r t i a l D i f f e r e n t i a l Equations, and R e l a t e d T o p i c s M. Schechter

501

Contents

xi

On Quasiuniqueness V . Schuchman

511

m-Functions and t h e A b s o l u t e l y Continuous Spectrum o f One Dimensional Almost P e r i o d i c Schrodinger Operators B. Simon

519

The Complete S o l u t i o n Space f o r a System o f R e a c t i o n - D i f f u s i o n Equations J.A. S m o l l e r

521

V a r i a t i o n a l Problems w i t h S i n g u l a r S o l u t i o n s R. Temam

537

B i f u r c a t i o n o f Subharmonic S o l u t i o n s : A Generic Approach A . Vanderbauwhede

545

A Class o f i s o p e r i m e t r i c V a r i a t i o n a l Problems on C e r t a i n Or1 icz-Sobolev Spaces P.A. V u i l l e r m o t

553

The R e l a t i o n o f S o l u t i o n s o f D i f f e r e n t ODES i s a Commutation R e l a t i o n B.F. W h i t i n g

561

P o i n t w i s e I n i t i a l - V a l u e Problems f o r F u n c t i o n a l D i f f e r e n t i a l Equations

J . Wiener

571

Asymptotics and S p e c t r a l Theory f o r High Order O r d i n a r y D i f f e r e n t i a l Equations w i t h Power C o e f f i c i e n t s A.D. Wood

581

Large Time Behaviors o f T i m e - P e r i o d i c Quantum Systems K. Yajima

589

Eigenvalues o f t h e L a p l a c i a n : An E x t e n s i o n t o H i g h e r Dimensions (11) E.M.E. Zayed

599

A u t h o r Index

607

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xiii

LECTURES NOT APPEARING I N THE PROCEEDINGS

Y. A k y i l d i z V.

Alexiades

S y m p l e c t i c geometry o f s h a l l o w w a t e r waves On b i n a r y a1 l o y s o l i d i f i c a t i o n

F. V. A t k i n s o n

Asymptotics o f the g e n e r a l i z e d Embden-Fowler e q u a t i o n

A. Azzam

S i n g u l a r i t i e s n e a r c o r n e r s o f s o l u t i o n s o f mixed boundary v a l u e problems f o r e l l i p t i c e q u a t i o n s i n s e c t i o n a l l y smooth domains

C . Bandle

Free boundaries i n n o n l i n e a r plasma problems

Edward Be1bruno

A f a m i l y o f p e r i o d i c o r b i t s i n t h e three-dimensional r e s t r i c t e d three-body problem

K. J . Brown

Steady s t a t e s o l u t i o n s o f systems o f r e a c t i o n d i f f u s i o n equations modelling c o - e x i s t i n g populations

Saber Elayd

N e g a t i v e a t t r a c t i o n and s t a b i l i t y i n semiflows

M. Hossam E -Din and Chen-Han Sung

Asymptotic b e h a v i o r on s o l u t i o n s o f a p e r t u r b e d 1i n e a r system o f o r d i n a r y d i f f e r e n t i a l equations

L . S. Frank

On a c l a s s o f s e m i l i n e a r e l l i p t i c s i n g u l a r perturbations

J. Frohlich

Unbounded, symmetric semigroups and v i r t u a l r e p r e s e n t a t i o n s o f symmetric spaces

I. M. G a l i and H. A. E l - S a i f y C o n t r o l o f systems governed by i n f i n i t e order equation o f hyperbolic type B a s i l i s Gidas

Free boundary problems f o r degenerate n o n l i n e a r e l l i p t i c d i f f e r e n t i a l e q u a t i o n s and a p p l i c a t i o n s t o physics

Louis G r i m

A n a l y t i c solutions o f d i f f e r e n c e equations

John R . Haddock

Converqence p r o p e r t i e s o f a semigroup on t h e memory space C y

Evans M. H a r r e l l , I 1

S p e c t r a l t h e o r y as a t o o l f o r n o n l i n e a r POEnonexistence o f s o l u t i o n s

I r a Herbst

Absence o f p o s i t i v e e i g e n v a l u e s f o r t h e Schr-odinger equation

x iv

Lectures Not Appearing in the Proceedings

R. Jensen

Domain of dependence f o r v i s c o s i t y solutions of f i r s t order p a r t i a l d i f f e r e n t i a l equations

Robert M. Kauffman

Limit-point c r i t e r i a f o r a c l a s s of p a r t i a l d i f f e r e n t i a l operators with l a r g e positive higher order coefficients

Ronald A. Knight

Fundamental dynamic r e l a t i o n a l rninimality

Tibor Krisztin

On the convergence of solutions of functional d i f f e r e n t i a l equations with i n f i n i t e delay

R . Mennicken

Eigenvalue problems non-linear in the parameter

Pedro Morales

Topological properties of the s e t of solutions f o r some Cauchy problems in l o c a l l y convex spaces

C. Morawetz

Some new phenomena f o r a nonlinear wave equation

J . W . Neuberger

Steepest descent f o r systems of nonlinear p a r t i a l di fferen t ia1 equations

Bill Patula

Riccati type transformations f o r second order l i n e a r difference equations

R . Vittal Rao and K . B. Athreya F i n i t e section integral operators spectrum and Kac-Akhiezer formula

-

V . Sree Hari Rao

Existence of solutions f o r a system of second order periodic boundary value problems

Louise A. Raphael

C r i t e r i a f o r equisummability under a n a l y t i c m u l t i p l i e r s

Eric Schechter

Gihman’s convergence c r i t e r i o n and compact perturbations of m-dissipative operators

R . Schianchi and E. Mascolo

Recent existence r e s u l t s f o r non convex v a r i a t i o n a l problems

Penny Smith

Regularity f o r sinqular non-linear e l l i p t i c systems i n weighted Sobolev spaces

Clasine van Winter

Non-selfadjoint Schrodinger operators generating groups

W. Walter

A simple proof of t h e Cauchy-Kowal ewski theorem

James R. Ward, J r .

On the s o l v a b i l i t y of some weakly nonlinear boundary

Val ue problems R. Weder

Spectral a n a l y s i s s c a t t e r i n q theory and eigenfunctions expansions f o r strongly propagative systems

xv

ADDRESS LIST OF CONTRIBUTORS

Anthony U. Afuwape

Department o f Mathematics, U n i v e r s i t y o f I f e I l e - I f e , NIGERIA

Shrnuel Agrnon

I n s t i t u t e o f Mathematics, Hebrew U n i v e r s i t y o f Jerusalem, Jerusalem, ISRAEL

Nicholas

D. A l i k a k o s

Department o f Mathematics, Purdue U n i v e r s i t y , West L a f a y e t t e , I n d i a n a 47907

Mark Ashbaugh

Department o f Mathematics, U n i v e r s i t y o f M i s s o u r i , Columbia, M i s s o u r i 65211

P i e r r e Baras

L a b o r a t o i r e IMAG, Tour des Mathgrnatics, Analyse Numerique, BP 68, 38402 S t . M a r t i n d'Heres Cedex, FRANCE

Michael F. B a r n s l e y

School o f Mathematics, Georgia I n s t i t u t e o f Techn o l o g y , A t l a n t a , Georgia 30332

Matani a B e n - A r t z i

Department o f Mathematics, Technion ITT, H a i f a 32000, ISRAEL

Marc B e r g e r

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

Haim B r e z i s

U n i v e r s i t e P. e t M. C u r i e , 4, p l . J u s s i e u , 75230 P a r i s Cedex 05 FRANCE

R i c h a r d C. Brown

Department o f Mathematics, U n i v e r s i t y o f Alabama, U n i v e r s i t y , Alabama 35486

Nguyen Phuong Cac

Department o f Mathematics, The U n i v e r s i t y o f Iowa, Iowa C i t y , Iowa 52242

Ren6 Carmona

Department o f Mathematics, U n i v e r s i t y o f C a l i f o r n i a a t I r v i n e , I r v i n e , C a l i f o r n i a 92717

Robert C a r r o l l

Department o f Mathematics, U n i v e r s i t y o f I l l i n o i s , Urbana, I l l i n o i s 61801

N a t h a n i e l Chafee

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

C. Y.

Department o f Mathematics and S t a t i s t i c s , U n i v e r s i t y o f Southwestern L o u i s i a n a , L a f a y e t t e , L o u i s i a n a

Chan

Address Lisf of Contributors

xvi

C. Corduneanu

Department of Mathematics, University o f Texas a t Arlington, Arlington, Texas 76019

Chris Cosner

Department of Mathematics and Computer Science, University o f Miami, Coral Gables, Florida 33124

Michael G. Crandall

Department of Mathematics and Mathematics Research Center, University of Wisconsin, Madison, Wisconsin

Ingrid Daubechies

TENA, Fab. WE, V U B , Pleinlaan 2, B-1050 Brussels, BELGIUM

Allen Devinatz

Department of Mathematics, Northwestern University, Evanston, I l l i n o i s 60201

Lance D. Drager

School of Mathematics, Georgia I n s t i t u t e o f Technology, Atlanta, Georgia 30332

M. S. P. Eastham

Department o f Mathematics, Chelsea College, (University o f London), London SWlO OUA, U. K.

Ivar Ekeland

CEREMADE, Universite Paris-9 Dauphine, 75775 Paris Cedex 16, FRANCE

Volker Enss

I n s t i t u t e f u r Mathematik I , Freie U n i v e r s i t a t , D-100 Berlin 33 GERMANY

Melvin Faierman

Department of Mathematics, University of the Witwatersrand, Johannesburg, SOUTH AFRICA

W. E. Fitzgibbon

Department of Mathematics, University of Houston, Houston, Texas 77004

J . Fleckinger

Department Math. Universite Paul S a b a t i e r , 118, Route de Narbonne, 31062 Toulouse CEDEX FRANCE

Michael A. Freedman

Department of Mathematics, Vanderbil t University Nashville, Tennessee 37235

Jerome A. Goldstein

Department of Mathematics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118

John R. Graef

Department o f Mathematics and S t a t i s t i c s , Mississippi S t a t e University , Mississippi S t a t e , Mississippi

Myron K . Grammatikopoulos

Department o f Mathematics, University of Ioannina, Ioannina, Greece

Ronald Grimmer

Department of Mathematics, Southern I l l i n o i s University, Carbondale, I11 i n o i s 62901

David Gurarie

Mathematics Department, Case Western Reserve University, Cleveland, Ohio 44106

Kar

Gustafson

Ge o ge A. Hagedorn

Department of Mathematics, University o f Colorado, Boulder, Colorado Department o f Mathematics, Virginia Polytechnic I n s t i t u t e and S t a t e University, Blacksburg, Virginia 24061

Address L i s t of Contributors

xvii

Kadosa H a l a s i

Department o f Mathematics, U n i v e r s ty o f Colorado, Boulder, Colorado

S. G. H a l v o r s e n

Department o f Mathematics, U n i v e r s t y o f Trondheim, 7000 Trondheim, NORWAY

C a r l o s R. Handy

Department o f Physics, A t l a n t a U n i v e r s i t y , A t l a n t a , Georgia 30314

D. P. H a r d i n

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

Andrew N. H a r r i ngton

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

J . V. Herod

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

Don H i n t o n

Mathematics Department, U n i v e r s i t y o f Tennessee, Knoxvi 1l e , Tennessee 37996

Rhonda J. Hughes

Department o f Mathematics, Bryn Mawr College, Bryn Mawr, Pennsylvania 19010

K e t i l l Ingolfsson

Department o f Mathematics, U n i v e r s i t y o f Alabama, U n i v e r s i t y , Alabama 35486

Hiroshi Isozaki

Department o f Mathematics, Kyoto U n i v e r s i t y , Sakyo, Kyoto, JAPAN

Hans G. Kaper

Mathematics and Computer Science D i v i s i o n , Argonne N a t i o n a l L a b o r a t o r y , Argonne, I 1 1 i n o i s 60439

T o s i o Kato

Department o f Mathematics, U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , C a l i f o r n i a 94720

Shmuel K i r o

Department o f Mahtematics, Rice U n i v e r s i t y , Houston, Texas

H i t o s h i K i tada

Department o f Mathematics, U n i v e r s i t y o f Tokyo, Meguro , Tokyo, JAPAN

Mark A. Kon

Department o f Mathematics, Boston U n i v e r s i t y , Boston, Massachusetts, 02215

A l l a n M. K r a l l

Mathematics Department, Pennsylvania S t a t e U n i v e r s i t y , U n i v e r s i t y Park, Pennsylvania 16802

B. A . Kupershmidt

U n i v e r s i t y o f Tennessee Space I n s t i t u t e , T u l lahoma, Tennessee

G. Ladas

Department o f Mathematics, U n i v e r s i t y o f Rhode I s l a n d K i n g s t o n , Rhode I s l a n d 02881

M i c h e l Langl a i s

Department o f Mathematics, Purdue U n i v e r s i t y , West L a f a y e t t e , I n d i a n a

P e t e r D. Lax

Courant I n s t i t u t e o f Mathematical Sciences, Mercer S t r e e t , New York, New York 10012

Address List of Contributors

xviii

W i l l am Layton

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

A. C

Lazer

Department o f Mathematics, U n i v e r s i t y o f Miami, C o r a l Gables, F l o r i d a

Sung J. Lee

Department o f Mathematics, U n i v e r s i t y o f South F l o r i d a , Tampa, F l o r i d a 33620

Elliott

H. L i e b

Departments o f Mathematics and P h y s i c s , P r i n c e t o n U n i v e r s i t y , P r i n c e t o n , New J e r s e y 08544

Lance L. L i t t l e j o h n

Department o f Mathematics, Computer Science and Systems Design, U n i v e r s i t y o f Texas a t San Antonio, San Antonio, Texas 78255

Jing-lin Liu

Department o f Mathematics, U n i v e r s i t y o f I n n e r Mongolia, Huhehot, Inner-Mongol i a , PEOPLE'S REPUBLIC OF CHINA

Roger C. McCann

Department o f Mathematics and S t a t i s t i c s , M i s s i s s i p p i S t a t e U n i v e r s i t y , M i s s i s s i p p i State, M i s s i s s i p p i

Paul K. McConnaughey

Department o f Agronomy, M i s s i s s i p p i S t a t e U n i v e r s i t y , M i s s i s s i p p i State, M i s s i s s i p p i

P e t e r A. McCoy

Mathematics Department, U n i t e d S t a t e s Naval Academy, Annapol i s , Mary1 and 21402

P. J. McKenna

Department of Mathematics, U n i v e r s i t y o f F l o r i d a , Gainesville, Florida

Joyce R. McLaughlin

Department o f Mathematical Sciences, Rensselaer P o l y t e c h n i c I n s t i t u t e , Troy, New York, 12181

Ronald E. Mickens

Department o f Physics, A t l a n t a U n i v e r s i t y , A t l a n t a , Georgia 30314

A. B. M i n g a r e l l i

Department o f Mathematics, U n i v e r s i t y of Ottawa, Ottawa, CANADA, K1N 984

P e t e r A. P e r r y

Department o f Mathematics, C a l i f o r n i a I n s t i t u t e o f Technology, Pasadena, C a l i f o r n i a 91125

Paul H. Rabinowitz

Department of Mathematics, U n i v e r s i t y o f WisconsinMadison, Madison, Wisconsin

David Race

Mathematics Department, U n i v e r s i t y o f t h e Witwatersrand Jan Smuts Avenue, Johannesburg, SOUTH AFRICA

Thomas T. Read

Department o f Mathematics, Western Washington U n i v e r s i t y , B e l l ingham, l l a s h i n p t o n 98225

Michael C. Reed

Department o f Mathematics, Duke U n i v e r s i t y , Durham, N o r t h C a r o l i n a 27706

Rouben Rostamian

Department o f Mathematics, Pennsylvania S t a t e U n i v e r s i t y , U n i v e r s i t y Park, Pennsylvania 16802

Address List of Contributors

xix

P. Sarnak

Courant I n s t i t u t e , New York U n i v e r s i t y , Mercer S t r e e t , New York, New York 10012

M a r t i n Schechter

Courant I n s t i t u t e o f Mathematical Sciences, New York U n i v e r s i t y , New York, New York 10012

V l a d i m i r Schuchman

Department o f Mathematics, Texas Tech U n i v e r s i t y , Lubbock, Texas 79409

Y. G. S f i c a s

Department o f Mathematics, U n i v e r s i t y o f Ioannina, Ioannina, GREECE

J. K. Shaw

Department o f Mathematics, V i r g i n i a Tech, Blacksburg, V i r g i n i a 24061

B a r r y Simon

Departments o f Mathematics and Physics, C a l i f o r n i a I n s t i t u t e o f Technology, Pasadena, C a l i f o r n i a 91125

A l a n Sloan

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

J o e l A. S m o l l e r

Department o f Mathematics, U n i v e r s i t y o f Michigan, Ann Arbor, M i c h i g a n

P a n a g i o t i s E. Souganidis

D i v i s i o n o f A p p l i e d Mathematics, Brown U n i v e r s i t y , Providence, Rhode I s l a n d

Paul W . Spikes

Department o f Mathematics and S t a t i s t i c s , M i s s i s s i p p i S t a t e U n i v e r s i t y , M i s s i s s i p p i State, M i s s i s s i p p i

I . P. S t a v r o u l a k i s

Department o f Mathematics, U n i v e r s i t y o f Rhode I s l a n d , Kingston, Rhode I s l a n d 02881

C a r l Sundberg

Department o f Mathematics, U n i v e r s i t y o f Tennessee, K n o x v i l l e , Tennessee 37996-1300

R. Temam

L a b o r a t o r i e D'Analyse Numerique, U n i v e r s i t e de P a r i s Sud, B a t i m e n t 425, 91405 Orsay, Cedex FRANCE

A. Vanderbauwhede

I n s t i t u u t voor T h e o r e t i s c h e Mechanica, R i j k s u n i v e r s i t e i t Gent, K r i j g s l a a n 281, B-9000

Gent, BELGIUM P i e r r e A. V u i l l e r m o t

Department o f Mathematics, U n i v e r s i t y o f Texas, A r l i n g t o n , Texas 76019

Bernard F. I J h i t i n g

Groupe d ' A s t r o p h y s i q u e , O b s e r v a t o i r e de P a r i s , 92 Meudon, P a r i s FRANCE

Joseph Wiener

Department o f Mathematics, Pan American U n i v e r s i t y , Edinburg, Texas

A l a s t a i r D. Wood

School o f Mathematical Sciences, N a t i o n a l I n s t i t u t e f o r H i g h e r Education, D u b l i n 9, IRELAND

K e n j i Yajima

Department o f Pure and A p p l i e d Sciences, U n i v e r s i t y o f Tokyo, 3-8-1, Komaba, Meguroku, Tokyo 153 JAPAN

xx

Address List of Contributors

Elsayed M. E. Zayed

Mathematics Department, Faculty o f Science, Zagazig University, Zagazig, EGYPT

Marvin Zeman

Department o f Mathematics, Southern Illinois University, Carbondale, Illinois 62901

DIFFERENTIAL EQUATIONS LW. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

1

UNIFORM DISSIPATIVE SOLUTIONS F O R A T H I R D O R D E R N O N - L I N E A R DIFFERENTIAL EQUATION A n t h o n y Uyi Afuwape Department o f Mathematics U n i v e r s i t y of I f e Ile-1fe.NIGERIA.

The f r e q u e n c y - d o m a i n t e c h n i q u e , a r i s i n g f r o m p r o b l e m s o f c o n t r o l , h a s b e e n v e r y useful i n d i s c u s s i n g t h e qualitative properties of solutions f o r non-linear d i f f e r e n t i a l equations. Earlier r e s u l t s are g e n e r a l i zed i n t h i s p a p e r t o e q u a t i o n s o f t h e form:

xr11+ fix") + g(x’l + hlxl = p(t,x,x’,x") where t h e f u n c t i o n s f , g , h and p a r e c o n t i n u ous i n t h e i r r e s p e c t i v e a r g u m e n t s . Uniform d i s s i p a t i v i t y is discussed for t h i s equation, using the frequency-domain t e c h n i q u e . 1.INTRODUCTION I n a n e a r l i e r work, B a r b s l a t and Halanay [ 4 1 o b t a i n e d a g e n e r a l r e s u l t , w h i c h was m o t i v a t e d by t h e w o r k s o f Popov V . M . , Kalman R . E . and Yacubovich V . A . T h i s r e s u l t i s g e n e r a l l y r e f e r r e d t o as t h e ) turned out t o frequency-domain t e c h n i q u e . The t e c h n i q u e ( c f [4-6] b e v e r y u s e f u l i n d i s c u s s i n g p r o b l e m s o f oscillations,dissipativity, u n i f o r m d i s s i p a t i v i t y a n d many o t h e r q u a l i t a t i v e p r o p e r t i e s o f solutions f o r non-linear d i f f e r e n t i a l equations. The a u t h o r i n [1,2] h a s used t h i s technique t o g e n e r a l i z e e a r l i e r results in [ 4 , 9 t o e q u a t i o n s o f t h e t h i r d - o r d e r w i t h two nonl i n e a r f u n c t i o n s . I n t h e p r e s e n t p a p e r , t h e frequency-domain t e c h n i que is u s e d t o g e n e r a l i z e t h e s e r e s u l t s f u r t h e r t o a t h i r d - o r d e r d i f f e r e n t i a l e q u a t i o n w i t h t h r e e n o n - l i n e a r terms. P r e c i s e l y , we c o n s i d e r e q u a t i o n s o f t h e form:

xPr1+ flx") + g l x ' ) + hlx)

=

p(t,x,x’,x"J

(1.1)

where f , g , h and p a r e c o n t i n u o u s i n t h e i r r e s p e c t i v e a r g u m e n t s . A s r e q u i r e d by t h e f r e q u e n c y - d o m a i n t e c h n i q u e , w e s h a l l a s s u m e t h a t t h e r e e x i s t p o s i t i v e c o n s t a n t s a , b , c w i t h ab > c a n d n o n - n e g a t i v e c o n s t a n t s p1 , l.~? , p 3 s u c h t h a t

hold

for

a

Q

f(z)/z b G glz)/z

Q

c

< hfz)/z

Q

A o l

a

< f’(zl

Q

;

a + p, b G g ' ( z ) G b + p*

;

c

< h"z)

Q

a + l . ~ ;~ c + p,

The c o n d i t i o n s a > 0 , b > o , ab > c > o Hurwitztr c o n d i t i o n s f o r t h e t h i r d - o r d e r 5"’

c + p3

X, l a r g e e n o u g h .

+ m'r + bx’ + cx

represent the l i n e a r equation =

0

which a r e s u f f i c i e n t f o r uniform d i s s i p a t i v e e q u a t i o n s .

It

Rout h

-

A.U. Afuwape

2

2.MAIN RESULTS The f o l l o w i n g t h e o r e m h o l d s f o r e q u a t i o n Theorem ( 2 . 1 ) : S u p p o s e t h a t i n ( 1 . 1 1, f(0) Po > O some c o n s t a n t I p ( t , X I , x 2 , 2 , ) I Q p, f o r a l l t , x ,x2, x 3 . Suppose f u r t h e r ties (2.11, junctions f I g and h

l i m (1/z2){&zf(a)da

-

(1.1).

=

= 910)

0

= h(01

a nd f o r

(2.1)

t h a t i n addition toinequalisatisfy

%z;Zflz)

1

0

2

Iz l-

Then, t h e s o l u t i o n s of e q u a t i o n ( 1 . 1 ) are u n i f o r m l y d i s s i p a t i v e .

p

Remark ( 2 . 2 ) . C o n d i t i o n (2.1) i m p o s e d on t h e f u n c t i o n c a n b e weakened t o I p1t,x1,x2,xJl Po f p, ( l X , l f I x21 f I x 3 II for > 0 1 j =o,r) a n d 4 > 0 s u f f i c i e n t l y s m a l l . Remark ( 2 . 3 ) . and

4

If

0, then equation

( 1 . 1 ) be c om e s

xrrrf f g1x’) + h ( x ) = p(t,x,x’,x”) Theorem ( 2 . 1 ) r e d u c e s t o a n e a r l i e r r e s u l t i n

If

Remark ( 2 . 4 ) .

,

$ .!

(2.4)

I11

If

Y

:0

,

equation

12.5)

(1.1) reduces t o

f f x ” ) f g f x ’ ) f- c x = p f t , x , x ’ , x ” ) I n t h i s case, t h e f o l l o w i n g i n d e p e n d e n t r e s u l t h o l d s : 5 ) ) )f

Theorem ( 2 . 6 ) . S u p p o s e t h a t i n e q u a t i o n ( 2 . 6 1 , c > 0 p o s i t i v e numbers a b a n d n o n - n e g a t i v e n u m b e r s ?.+ a b > c a n d f o r I z I 2 X, , X, l a r g e e n o u g h , a G f(z)/z

< a + p, ; a G

b C gfzl/z with

f(0)

= 0

g(0).

b

f

p2 ; b

Q

all

t, xi , z 2 , x 3

f’lzl g‘(zl

a

f

b +

(2.61

.

and t h e r e e x i s t , p2 s u c h t h a t

p,

p2

(2.7)

S u p p o s e f u r t h e r t h a t f o r some c o n s t a n t Po > 0

I p f t , x , ,z2’ X 3 )I G for

.

t h e n e q u a t i o n ( 1 . 1 ) be c om e s

x”‘ f f f x ” ) f bx’ f h ( x l = p ( t , z , x ’ , x ” ) [21. a n d Theorem ( 2 . 1 ) r e d u c e s t o a r e s u l t i n

Remark ( 2 . 5 ) .

12.31

12.81

P,

and

g f d d a - % zg(z) 1 > 0 I z b t h e s o l u t i s n s o f e q u a t i o n (2.6) a r e u n i f o r m l y d i s s i p a t i v e . -l fi 1m/ z 2 ) f c

Then

Theorem ( 2 . 6 ) g e n e r a l i z e s e a r l i e r r e s u l t s o f B a r b g l a t Remark ( 2 . 7 ) . [ 4 ] a n d o f H a l a n a y i n [51 and H al anay i n

.

3

Dissipative Solutions for a Non-Linear Equation

3.PROOF OF THEOREM ( 2 . 1 ) . h Let ffx") a ax" +F(Yh g & f ) = b x ' + $ ( x l J and h l x ) ex + h ( x l Then, equation ( 1 . 1 ) can be written in the vector form: X’ = AX - B Q ( u ) i- P(t,X), U = C*X (3.11 where 0 0 0 0

0

A = -a

-c -b

0

0

Hith

Thus a function

=

C*X

=

C(iw) =

X

and C*(iwI

o ( 0 )

- A)-’B

= is equivalent to

C(iw) where A = ( c - a w ' ) + iwIb - w 2 ) . , we have that matrix A in (3.1) is stable. ' The frequency-domain inequality (cf. 141 is ~ ( w )E D, + Re { Dldiag( cl.1 iL,,D 2 G(i9 1 + J + 02[ Re ID3 I + diag( l J j ) G(iw)>] > 0 (3.2) for all w 'R , where D . (j 1,2,3) are diagonal matrices with D > 0, D , D > 0 dnd pj are bounds for the elements o f vectar @ ( o f , sasisfying Choos T. >

J

Ti

(w

where

A. L! Afuwape

4

) ( c

r

=

{ ( T~

and

s

(

W{

-

el +

( c

W ~ T ,

W2e3

)

(c

-

aw2 )

+

au2 )

-

w2(

-

I

ad

d))

T 3 ) ( b el-W2ej ( b

-

( T ~

+

W~T,)

(b

)I

-

u2 )I.

F o r (3.3) to be true for all & g , Sylvester s criteria for positive definiteness of matrices can be applied. That is, we must show that for all 6 fl, (3.3) is valid if:

>

11

o

’:y2 )

(3.4)

u + iv det

-

u

iv

(3.5)

0

>

and > 0 (3.6). det n(w1 As in [ 1 ] , inequality (3.4) is valid if ( e l / T~ and w * e , + w ‘ ( % - b e , ) --c -1 > p1 ( c - a w ~ +) W’ ~ ( b - w 2 ) ’ On evaluating inequality (3.51, we have Till 7122 (u2 + v 2 ) / 4 1 A I 4 > 0

-

(

> (a/b)

.

This will be true, for all w and -1 -1

T ~ / T ~ ) (+ Lc ~ ~)

Finallv. detv ( w )

)

IR , if p2

a-’


will be positive for

+

(40 all

1 -


0 (3.8).

Considering the degrees of p , q , r! s , u, v and ( ~ 1 as~ polynomials in w , we observe that for inequality (3.8) to be true, it suffices to have 1 ‘2’3 -(-+

4C2

1)

(3.9)

‘3’2

a situation obtained in the limiting case with O 3 0. Equivalent to this is to have 0J-I + c- ) bounded below by a positive linear function in p 2 and p 3 . The supplementary conditions (2.2) follow since 2,3). Hence the conclusion of Theorem (2.1) follows generalized Theorem of Yacubovich ( [4] 1 .

0 . 2 0 (j J

from the

4.PROOF OF THEOREM ( 2 . 6 1 A In order to prove Theorem Let f(xff)=aff + f(x )and g(3:f)=b3.r+g(zf) (2.6), we first write equation (2.6) in vector form X = AX - B @ ( a ) + P(t,X), u C*X (4.1)

.

1,

5

Dissipative Solutions for a Non-Linear Equation

where

A = ( :

y)

0 1 --c

; B = [ y 0

C0 l ) ; c = [ O 1 l

0

0

0

-a

-b

For this system, the transfer function given by --a - i-cw 1 G(iw) -

(

A

C(iw) -a

C*(iwI

- i-cw --c

-e

-

A)-’B

is

1

where as before, A = (c - aw2J + iw(b -w2). The frequency-domain condition (3.2) € o r this becomes

/

iRz

dl

\

-cT2

d2 R,

(e,d

=

[ (a2

- hie,+ C ~ , I W +~

+ fb2e, and

L,

=

-

W{T,W"

+

-cTL )1w2

[ ( a 2 - b ) ~ ,i-c(e,+

ra-c(T2

I

-

+ c(h, ae, - T ,

8, - b e , m + - cr,) 1 ~ ] w ~+

+ [ f b 2 - a d T , + bclT, - el - c e,/bi] 1 Clearly, if we choose 6, > 0, we have for all w G E , d, > 0 and if ( 8 , / T,) > ( a / b ) we have d2 > 0 with (I/uJ > 1. and e 2 we have for all Equally, for these choices of wt?R det n ( w ) d,d2 - (R: + &: )/(4 ( 4 " ) > 0 (4.3). The supplementary conditions (2.8) follow by the choice of e . J (j = 1 , Z ) . The conclusion of Theorem (2.6) thus follows from the generalized Theorem of Yacubovich. REFERENCES [I] Afuwape, A.U., Frequency-Domain Criteria for Dissipativity of some Third-Order Differential Equations, Analele Sti. Univ, ’AL. I. CUZA’, xxiv (1978) 271-275. 1 2 1 Afuwape, A.U., An Application of the Frequency-Domain Criteria For Dissipativity of a Certain Third-Order Non-linear Differential Equation, Analysis 1 (1981) 211-216. I 3 1 Barbzlat, I.and Halanay, A., Nouvelles Applications de la Methodes Frequentielle dans la Theorie des Oscillations, Rev. Roum. Sci. Techn. Elect. et Energ., '16, 4(1971) 689-702. [ 4 I Barbzlat, I.and Halanay, A., Conditions de Comprtement

6

A. U.Afirwape

’presque lineaire’ dans la Theorie des Oscillations, Rev. Roum. Sci. Techn. Elect. et Energ., 29, 2(1974) 321-341. [5] Halanay, A., Frequency-Domain Criteria For Dissipativity, in: Weiss, L(ed.) Ordinary Differential Equations, MRL-MRC Conference (1972) 413-416. 161 Popov, V.M., Dischotomy and Stability by Frequency-Domain Methods, Pro. I.E.E.E., 62, 5 (1974) 547-562. [ T I Reissig, R. Sansone, G. and Conti, R.,Nonlinear Differential Equations of Higher Order (Noordhoff Int. Publ. 1974). [81 Yacubovich, V.A., The matrix method in the theory of the stability of non-linear control systems, But. Rem. Control, 25 (1964) 905-9 16. [g] Yacubovich, V.A., Frequency-Domain conditions for absolute stability and dissipativity of control systems withone differentiable non-linearity, Soviet Maths., 6 (1965) 98-101.

DIFFERENTIAL EQUATIONS

LW.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

7

ON POSITIVE SOLUTIONS OF ELLIPTIC EQUATIONS WITH PERIODIC COEFFICIENTS IN En,SPECTRAL RESULTS AND EXTENSIONS TO ELLIPTIC OPERATORS ON RIEMANNIAN MANIFOLDS Shmuel Agmon Institute of Mathematics Hebrew University of Jerusalem Jerusalem, Israel

1. Introduction.

It is well known that there are close connections between spectral properties of second order elliptic operators and properties of positive solutions of elliptic equations. Here are some examples (see also Allegretto [1,2,31, Piepenbrink 19,101, Moss-Piepenbrink 181 and Simon [ll]). Theorem 1.1. Let P =

-

A

+

q(x)

be a Schrodinger operator defined in a

domain R c En where q E Lioc(R) with r > n/2, q real. A necessary and sufficient condition for P to have a non-negative spectrum in the sense that (W,(P) =

for all cp

E Ci(R),

f(lWI2 + q1VJ12)dx 2 n

0

is that the equation Pu = 0

admits a positive solution in

n . Theorem 1.2.

Let P be as above and suppose that the form

bounded from below on of

P

in L 2 ( a ) .

C;(Q).

Let H

u

ess

(HI

is

Write 1 = inf u

where

(Pcp,cp)

be the Dirichlet self-adjoint realization

ess

(HI

denotes the essential spectrum of H.

Then

(i) For every A < C the equation (P-A)u = 0 admits a positive solution in some neighborhood of infinity in R (i.e., in Q \ K for some compact K C n). (ii) For every X > C the equation any neighborhood of infinity in n.

(P-A)u = 0

admits no positive solution in

Positive solutions are also useful as majorants for solutions of elliptic equations which do n o t grow too rapidly at infinity. For instance, we have the following result. Theorem 1.3. Under the same conditions as in Theorem 1.2 let $(XIbe an eigenfunction of H with eigenvalue X 3 C and let u(x) be a positive Then solution of cP-A)u = 0 in some neighborhood of infinity in R 5 Cu(x) in a neighborhood of infinity for some constant C. [$(x)l

.

In this lecture we shall discuss properties of positive solutions and related spectral results for elliptic operators with periodic coefficients on lRn and also discuss extensions of the results to a larger class of elliptic operators on certain non-compact Riemannian manifolds. FOK the sake of presentation we shall at first limit ourselves to the periodic case. The starting point here is the observation (proved in §2) that whenever the elliptic

8

S. Agmon

equation Pu = 0 admits a positive solution on Bn then it also admits a positive exponential type solution. The family of positive solutions of Pu = 0 which are "exponentials" has a major role in our discussion. Various properties of this family are presented in 52. Some results in spectral theory of elliptic operators with periodic coefficients which are based on knowledge of the family of positive exponential solutions are discussed briefly in 03. Finally in § 4 we describe extensions of the results to a more general class of elliptic operators defined on certain non-compact Riemannian manifolds. 2.

Positive exponential type solutions We consider a second order elliptic operator P

.

lRn

We assume that P n

ai

a/xi, x

=

n

=

(xl,

acting on functions u

on

is in a divergence form:

...,x,).

n

We assume that the coefficients of

P are real

measurable functions and that n

a

ij

a positive constant. We assume further that

E Lm(lRn) , bi E L~oc(l17n) , bi E L~occlRn) with p

with p1 > n/2. xl,

ni" , Y

and 5 in

for all x

...,xn

> n

and

Finally we assume that the coefficients of P

c E Lyl&lRn)

are periodic in

with period 1.

By a solution of the differential equation Pu = 0

in some open set

in Rn

we shall understand as usual a function u

(2.2) in

which verifies (2.2) in the weak (once integrated) sense. It follows

Hioc(a) from the results of Stampacchia 1121 that all solutions of (2.2) are Holder continuous functions. Furthermore, Harnack’s inequality holds for the class of non-negative solutions of (2.2) in n. We shall be interested in the structure of the family of positive solutions of (2.2) in Rn. The first basic result is the following Theorem 2.1. Suppose the equation Pu = 0 has a positive solution in Rn . Then the equation also has a positive solution of the form,

where

i s a periodic function

rp

in Rn

and 5 is some vector in En.

We give two proofs of the theorem. First Proof: Let

En

\\,

S

be the set of all solutions of the equation Pu = 0 in

and consider S as a locally convex linear space with a sequence of semi= max \u(x> I , u E S . Let K be norms ( \ , j = 1,2,. given by 11 <j Rn Ix Ithe set of positive solutions of Pu = I1 in normalized at the origin by u ( 0 ) = 1. Clearly K is a non-empty convex subset of S. Furthermore, using Harnack’s inequality and a-priori Holder norm estimates (see [12]) it is easy to see that K is a compact set in S.

-

. .,

.\\

Positive Solutions of Elliptic Equations ei, i

...,n, denote

=

1,

Rn and denote by

T~

Let

u(x) -+ u(x+ei).

Fi : u Since P

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

xi i n

t h e corresponding t r a n s l a t i o n o p e r a t o r a c t i n g on

Rn , -ri:

f u n c t i o n s on

9

c o m u t e s with

+ T

Introduce t h e n maps:

uEK.

u/u(ei),

i

i t follows t h a t

T.

(i = 1,. ..,n) i s a continuous

F.

map which t a k e s K i n t o i t s e l f . Hence, applying t h e Schauder-Tychonoff f i x e d p o i n t theorem [ 1 3 ] t o F1 we conclude t h a t F1 has a f i x e d p o i n t i n K which implies t h a t there e x i s t s a positive solution

X1.

some p o s i t i v e number subset of

N e x t denote by

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

K

commutes with

in K

u

TIU1 =

Xlul

for

t h e non-empty closed convex

K1

such t h a t

F2 maps

i t follows t h a t

T~

such t h a t

u1

X u. 1

T u =

1

K1 i n t o i t s e l f .

Since T2 Invoking again

t h e Schauder-Tychonoff f i x e d p o i n t theorem i t follows t h a t t h e r e e x i s t s a Continuing i n t h i s p o s i t i v e s o l u t i o n u2 i n K1 such t h a t T ~ =U A 2 ~u 2 . manner we conclude t h a t t h e r e e x i s t s a p o s i t i v e s o l u t i o n u of ( 2 . 2 ) i n Rn s a t i s f y i n g T . U = X iu f o r i = l,....,n, where t h e Xi are c e r t a i n p o s i t i v e constants.

The p o s i t i v e s o l u t i o n

u

thus obtained i s of t h e form ( 2 . 3 )

described i n t h e theorem. To s e e t h i s s e t ~ ( x )= u(x)e-Q'x> where 5 =(logX1, log An) and n o t e t h a t Ti'p = ( D f o r i = 1, n. This completes

...,

...,

t h e proof. This proof i s due t o Yehuda Pinczover ( p r i v a t e communication).

Second proof:

Let S and K be defined as i n t h e proof above. Then as was noted K i s a compact s e t i n t h e l o c a l l y convex l i n e a r space S . From t h e Krein-Milman theorem i t follows t h a t t h e set of extremal p o i n t s of K i s not-empty. Let u be any extremal p o i n t of K. We s h a l l show t h a t u must be of t h e form ( 2 . 3 ) . To t h i s end introduce as i n t h e proof above t h e t r a n s l a t e d f u n c t i o n s

...,

T u = u(x+e 1, i = 1, n. Then T u and u a r e p o s i t i v e s o l u t i o n s of i i i Pv = 0 i n R". From Harnack's i n e q u a l i t y applied t o u (see ( 2 . 4 ) ) i t

follows t h a t t h e r e e x i s t s a number i = 1,

...,n. v

Then vi

i

wi

w

i

belong t o u = a v

i i

with

> 0

such t h a t

u - ET u i

> 0

in

Rn f o r

Define

= .siu/u(ei),

and

E

ai

follows t h a t

vi

-

K

and

i i

=

u(0) - Eu(e.)

=

u(0)

=

1.

i = 1,

so t h a t

Since

must be a m u l t i p l e of Xi,

Eu(ei)).

+ B W

Bi

positive constants

-

ETiu)/(u(0)

+ Bi

ai = Eu(ei),

numbers with

= (u

...,n.

u

ai

and

Bi

are positive

i s an extremal p o i n t of K i t

u, that is: T u = X u for certain i i This implies t h a t u i s of t h e form

( 2 . 3 ) and completes t h e proof. Note t h a t t h e second proof of Theorem 2 . 1 y i e l d s a d d i t i o n a l information on t h e s t r u c t u r e of p o s i t i v e s o l u t i o n s of Pu = 0 v i a t h e Krein-Milman theorem. This a d d i t i o n a l information i s needed f o r t h e proof of t h e r e p r e s e n t a t i o n theorem f o r p o s i t i v e s o l u t i o n s which we d e s c r i b e l a t e r on.

10

S. Agmon

With some abuse of terminology we s h a l l r e f e r t o a s o l u t i o n u of t h e form ( 2 . 3 ) as an exponential s o l u t i o n of Pu = 0. The v e c t o r 5 i n t h e r e p r e s e n t a t i o n ( 2 . 3 ) w i l l be c a l l e d t h e exponent of U . The following two theorems d e s c r i b e p r o p e r t i e s of t h e manifold of exponents of a l l p o s i t i v e exponential s o l u t i o n s of (2.2). Theorem 2 . 2 . Under t h e c o n d i t i o n s of Theorem 2 . 1 any two p o s i t i v e exponential s o l u t i o n s of (2.2) with t h e same exponent 5 a r e l i n e a r l y dependent. Denote by

r t h e s e t of a l l 6 E IRn such t h a t 5 i s t h e exponent of some p o s i t i v e Then e i t h e r exponential s o l u t i o n of C2.2) i n IR"

.

(i)

r

c o n s i s t s of a s i n g l e p o i n t

(ii)

r r

=

or

i s a compact n-1 dimensional a n a l y t i c manifold embedded i n IRn such t h a t aK where K i s a s t r i c t l y convex n-dimensional s e t i n a n .

*

Theorem 2.3. Comider the o p e r a t o r P-X and P -1 w h e r e P * i s t h e formal a d j o i n t of P and A E R . There e x i s t s a r e a l number A = A@) such t h a t both equations: (P-X)u = 0

and

(P*-X)u = 0

possess p o s i t i v e s o l u t i o n s i n

possess no p o s i t i v e s o l u t i o n s i n

g E En

Ti) t h e s e t of

besp.

exponential s o l u t i o n i n (resp. K*)

an

( P - x ) ~=

be t h e convex body i n lRn

X

Theorem 2 . 2 ) .

Then

K

x

K;

=

-

o

(resp.

such t h a t

aK

.

c i n t KX f o r X < 1-1 5 A u o KA = { < 1 (a s i n g l e p o i n t ) and

(ii)

For any X 5 A denote by 'a 5 i s t h e exponent of some p o s i t i v e (P*-x)~= 0 ) . =

rh(resp.

Let

* * 3KX =Ti)

K~ (by

i s a continuous set valued f u n c t i o n of 1 such t h a t

(i) K (iii)

and

i f A > A.

such t h a t

I R ~ of

if A 5 A

Rn

K-_

=

IR"

KX.

. *

It follows from t h e l a s t theorem t h a t i f P = P then K, i s s m e t r i c It is c o n v e n i h t o introduce w i t h r e s p e c t t o t h e o r i g i n and t h a t KA = CO}. a t t h i s p o i n t t h e following n o t a t i o n : Definition. 0 E aK,.

We denote by

Ao(P)

t h e unique number X 5

A(P)

such t h a t

A(P)

if

n

It i s c l e a r t h a t self-adjoint. Note t h a t equation

A,(P)

=

no@*)

and t h a t

We s h a l l see l a t e r chat

A,@)

Ao(P)

=

P

i s formally

has a s p e c t r a l i n t e r p r e t a t i o n .

A (P) can a l s o be defined a s t h e unique r e a l 1 such t h a t t h e 0 (P-X)u = 0 has a p o s i t i v e p e r i o d i c s o l u t i o n i n R n .

We s h a l l not g i v e h e r e proofs of Theorem 2.2 and Theorem 2 . 3 . Instead we s h a l l e s t a b l i s h t h e following r e l a t e d theorem which i s used i n t h e proof of Theorem 2.2. Theorem 2 . 4 .

A s s u m e t h a t t h e c o n d i t i o n s of Theorem 2.2 hold and t h a t F i s

defined as i n Theorem 2 . 2 . Then r i s a compact s e t i n R n . L e t K(T) denote t h e convex h u l l of r and l e t TI be a p o i n t i n K(7) which i s n o t a n extremal point.

then t h e r e does not e x i s t a s o l u t i o n

change sign) such t h a t

uCx)e- 0 and l e t 0 < X 5 A _ . Then t h e equation ( A + X ) u = 0 does not admit a p o s i t i v e s o l u t i o n i n M which i s G-multiplicative.

+

It i s w e l l known Cseg I n t r o d u c t i o n ) t h a t the equation ( A A)u = 0 admits Thus Theorem 4 . 2 shows t h a t Theorem 4 . 1 a p o s i t i v e s o l u r i o n on M i f X 5 A i s f a l s e i f A > 0. This will be f o r i n s t a n c e t h e case i f M h a s a negative s e c t i o n a l curvature (see McKean 171). Recently Brooks [41 has shown t h a t A = 0 i f and o n l y i f wl(M) i s a n amenable group. (A group G i s amenable i f and only i f t h e r e i s a f i n i t e l y a d d i t i v e l e f t i n v a r i a n t measure on G.) Thus i t follows from Theorem 4 . 2 t h a t Theorem 4 . 1 f a i l s t o hold whenever nl(M) i s not amenable.

.

This i s t h e case, f o r i n s t a n c e , when group on two generators.

~i

CM) 1

c o n t a i n s a subgroup which i s a f r e e

(A + A)u = 0 admits We s h a l l show t h a t

[Sketch) Suppose t h a t t h e equation Proof of Theorem 4 . 2 . a p o s i t i v e s o l u t i o n uo i n M which i s G-multiplicative. t h i s leads t o a contradiction. o p e r a t o r s on

uo = eh

Set

i h +h A& = e Ae-

.

-

h

Since

and consider t h e two e l l i p t i c

M:

i s a G-additive f u n c t i o n i t follows t h a t

and

A+

a r e G-invariant

A-

e l l i p t i c o p e r a t o r s on 8. Thus we can consider A+ and A- a s being defined on 2 M. Since A i s formally s e l f - a d j o i n t i n L CM,dm) where dm i s the measure induced by t h e Riemannian m e t r i c i t follows r e a d i l y t h a t A- i s t h e formal a d j o i n t of

in

A+

2

L (H,dm). Now, i t i s n o t d i f f l l c u l t t o prove i n a g e n e r a l

s i t u a t i o n t h a t i f A i s a n e l l i p t i c operator i n M such t h a t t h e equation Au = 0 admits a p o s i t i v e s o l u t i o n i n M then t h e a d j o i n t equation A*u = 0 a l s o admits a p o s i t i v e s o l u t i o n i n M. We u s e t h i s r e s u l t i n our case. By X )u = 0 has t h e s o l u t i o n u E 1. Hence t h e c o n s t r u c t i o n the equation (A+

+

(A-

a d j o i n t equation denote by

+

A)u = 0

has a p o s i t i v e

C

2

solution in

Multiplying by a constant we s h a l l assume t h a t

I).

M

which we

min I)= 1. Next

M l i f t J,

to

and consider i t a s a G-invariant

s o l u t i o n of

(A-

+

A)$

=

0

i n 8 . Using ( 4 . 2 ) we thus f i n d t h a t (4.3) By a simple computation we o b t a i n from ( 4 . 3 ) t h a t U

U

(3) = 2 2

(A-x)

uO

J,

Since

(A+X)uo

=

0

in

I v(-$ l 2

U

2 1 on Recalling t h a t I) uo(l point

- 9-l) 0

(4.5).

G.

in

(4.4)

i t follows from ( 4 . 4 ) by s u b t r a c t i o n t h a t

I?,

d(u,

o

-

fi

0 T) 5-

while

i s a non-negative

C2

X(u,

+

uO 9

-)

< 0

in

0

M.

J,[x ) = 1 a t some p o i n t function i n

fi which

(4.5) xo, i t i s c l e a r

a t t a i n s i t s minimum a t a

.

x This i s however impossible i n view of t h e d i f f e r e n t i a l i n e q u a l i t y We have a r r i v e d a t a c o n t r a d i c t i o n which proves t h e theorem.

References

[l]

W. A l l e g r e t t o , On the equivalence of two t y p e s of o s c i l l a t i o n f o r e l l i p t i c

Positive Solutions of Elliptic Equations operators, Pac. J. Math. 55(1974),

319-328.

[2]

W. Allegretto, Spectral estimates and oscillation of singular differential operators, Proc. Amer. Math. SOC. 73(1979), 51.

[3]

W. Allegretto, Positive solutions and spectral properties of second order 15-25. elliptic operators, Pac. J. Math. 92(1981),

[4] R. Brooks, Amenability and the spectrum of the Laplacian, B u l l . h e r . Math. SOC. (New Series) 6(1982), 87-89. [5]

L. A. Caffarelli and W. Littman, Representation formulas for solutions to AU-U = o in R" , to appear.

[6] F. I. Karpelevic, The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces. Trudy Moskow. Mat. Obsc. 14 (1965), 48-185 = Trans. Moscow Math. SOC. 1965, 51-199, h e r . Math. SOC., Providence, R.I., 1967. [7]

A on a manifold of negative curvature, J. Diff. Geometry 4 (1970), 359-376.

H . P. McKean, An upper bound to the spectrum of

[8] W. Moss and J. Piepenbrink, Positive solutions of elliptic equations, Pac J. Math. 75 (1978), 219-226.

[9] J. Piepenbrink, Nonoscillatory elliptic equations, J. Diff. Eqn. 15 (1974 541-550. [lo] J. Piepenbrink, A conjecture of Glazman, J. Diff. eqn. 24 (1977), 173-177 [ll] B. Simon, Schrodinger semigroups, B u l l . Amer. Math. S O C . (New Series) 7 (1982), 447-526. [12] G. Stampacchia, Le probleme de Dirichlet pour les equations elliptlques du second ordre a coefficients discontinus, Ann. Inst. Fourier 16 (1965), 189-258. [13] A. Tychonoff, Ein Fixpunktsatz, Math. Ann. 111 (19351, 767-776.

17

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DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

19

CLASSIFICATION OF I N I T I A L DATA FOR THE POROUS MEDIUM EQUATION I N RN N i c h o l a s D. A l i k a k o s Department of Mathematics Purdue U n i v e r s i t y West L a f a y e t t e , I N 47907 U.S.A.

1.

Rouben Rostamian Department o f Mathematics Pennsylvania State University U n i v e r s i t y P a r k , PA 16802

U.S.A.

INTRODUCTION

In t h i s paper w e a r e concerned w i t h t h e Cauchy problem f o r t h e porous medium e q u a t i o n w i t h n o n i n t e g r a b l e non-negative i n i t i a l d a t a :

The i n t e r p l a y between t h e o r d e r s of magnitude of

uo(x)

at large x

and

t h a t of u ( x , t ) a t l a r g e t was s t u d i e d i n [l]. Here w e d i s c u s s t h e meaning and i m p l i c a t i o n s of some of t h e r e s u l t s t h e r e and add some e x t e n s i o n s and observations,

A u s e f u l measure of t h e "order o f magnitude" of

uo(x)

for large

is the

x

s-average which w e d e f i n e by

s 1 0

f o r any radius quantity

r

f o r which t h e l i m i t e x i s t s .

centered a t

c ERN

and

IB

1

Here

Br(c)

is i t s measure.

i s t h e b a l l in RN of

With

s = N

the

ag(N), i f d e f i n e d , is in some s e n s e q u i t e c l o s e t o t h e i n t u i t i v e

c o n c e p t of t h e a v e r a g e , however see S e c t i o n 3 f o r a n o t h e r p o i n t of view. The a p p a r e n t p r i v i l a g e d r o l e of t h e o r i g i n i n t h i s d e f i n i t i o n is s u p e r f i c i a l , as one can show u s i n g t h e n o n - n e g a t i v i t y of uo t h a t i f t h e l i m i t above e x i s t s t h e n t h e N replacement o f Br(0) by Br(c) w i l l produce t h e same r e s u l t f o r each c € R

.

A somewhat d i f f e r e n t s - a v e r a g e , and one more c l o s e l y a s s o c i a t e d w i t h (PM) i s t h e q u a n t i t y a D ( s ) d e f i n e d a s f o l l o w s . For any s > 0 l e t

fj - 1

A (x)

=

ilB1l

,

N x € R

-

{O}

20

N.U. Afikakos and R. Rostamian

and p a s s i n g

t o z e r o set

s

Ao(x) = 6(x) = D i r a c ' s d e l t a f u n c t i o n .

The s-average

i s d e f i n e d as t h e c o n s t a n t ( i f it e x i s t s ) f o r which t h e f o l l o w i n g

a,(s)

convergence

(S

-

rN-S u o ( r x )

AVD) :

+

aD(s)As(x)

D'(lRN).

h o l d s i n t h e s e n s e of d i s t r i b u t i o n s a,(s)

and

a,(s)

= aB(s),

as

r

If

+

e x i s t s , t h e n so does

a,(s)

but t h e converse i s n o t t r u e , [ l ] .

For t h e f u t u r e r e f e r e n c e , l e t ' s i n t r o d u c e h e r e one f i n a l d e f i n i t i o n . t h a t t h e a v e r a g e of uo i s uniformly e q u a l t o a i f

(UAV):

(x)dx = a

lim

uniformly i n

We s a y

N

c C R ,

r-

Almost p e r i o d i c f u n c t i o n s i n The q u a n t i t i e s

have t h i s p r o p e r t y , see Fink [ 4 , page 3 2 1 .

R1

aB(s),

aD(s),

Thus i f a n a v e r a g e o f

under (PM).

and uo

a,

whichever e x i s t , are i n v a r i a n t

e x i s t s t h e n t h e c o r r e s p o n d i n g a v e r a g e of

u ( - , t ) a l s o e x i s t s and remains c o n s t a n t f o r a l l t 3 0 ( s e e [l]). O f c o u r s e t h i s i s t u r e provided t h a t t h e s o l u t i o n of (PM) i s d e f i n e d f o r a l l t i m e . A r e s u l t of B e n i l a n , C r a n d a l l and P i e r r e [ 3 ] combined w i t h a theorem o f Aronscn and C a f a r e l l i [Z] shows t h a t a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r g l o b a l - i n - t i m e e x i s t e n c e of s o l u t i o n s of (PM) i s

2

a (B m-1

+

N)

= 0.

Thus throughout t h i s paper w e w i l l

assume t h a t

2

0 5 s < - + N m- 1

(*) :

t o e n s u r e t h e e x i s t e n c e o f s o l u t i o n s . We w i l l d i s c u s s t h e i m p l i c a t i o n s of ( s -AVD) i n r e l a t i o n t o s o l u t i o n s o f (PM) i n S e c t i o n 2 , and t h e "meaning" of aD(s)

2.

and i t s r e l a t i o n t o t h e more i n t u i t i v e q u a n t i t y

THE IMPLICATIONS OF

(S

i n Section 3 .

-AVD)

Recall t h e d e f i n i t i o n of q,(x,t;E)

aB(s)

hs(x)

i n S e c t i o n 1.

For any

E 2 0

d e n o t e t h e s o l u t i o n of (PM) w i t h t h e i n i t i a l c o n d i t i o n

let

cAs(x).

This

s o l u t i o n e x i s t s f o r a l l t > 0 provided t h a t s s a t i s f i e s (*). The f o l l o w i n g theorem from 111 s a y s t h a t i n some s e n s e any s o l u t i o n of (PM) e v e n t u a l l y l o o k s l i k e some * qs'

Theorem 2.1:

Suppose t h a t (*) h o l d s .

Then f o r any c o n s t a n t

c > 0

w e have

The Porous Medium Equation in RN

i f and only i f

aD(s> f a r

a =

2

+

uo

N - s (m-l)(N-s)

The asymptotic states

e x i s t s and equals

’

’ q,(x,t;C)

identical f a r a l l i n i t i a l data

uo

=

2

+

8.

21

Here w e have l e t

1 (m-l)(N-s)

a r e u n i v e r s a l , i n t h e sense t h a t they a r e

with a common s-average

aD(s).

The c a s e

s = 0 already occurs i n Kamin [ 6 ] and Friedman and Kamin [ 5 ] . The case s = N is p a r t i c u l a r l y i n t e r e s t i n g s i n c e q N ( x , t ; d ) z 8 . We s t a t e t h i s a s a c o r o l l a r y : Corollary 2 . 2 :

For any constant

i f and only i f

aD(N)

for

Thus t h e e x i s t e n c e of

uo

c

0

we have

e x i s t s and equals

aD(N)

8.

i s necessary and s u f f i c i e n t f o r t h e uniform

convergence of u ( x , t ) t o a constant on b a l l s expanding a t t h e r a t e of r a t e fi i s c r u c i a l a s i t can be seen i n t h e following example. N = L

The s o l u t i o n of (PM) corresponding t o

uo(x) =

(2.2) :

i

1

if

x > O

0

if

x c1 z 0.

Then f o r any

r z 0

uo

is

) and ( s - A V ) a r e equivalent and

n

( s -AVB)

holds.

is the

In f a c t we have

; g ( s ) = a,(s).

Proof:

uo 2 0

Fix

we have by a change o f v a r i a b l e s :

The Porous Medium Equation in RN

Jcc2

ITN-s

f(ra)aN-ldo

Thus a p p l y i n g ( s -AVB)

Now t a k e a in

cp

in

we obtain

N

CO(R )

and f o r s i m p l i c i t y assume t h a t it i s s u p p o r t e d

For a n a r b i t r a r y p o s i t i v e i n t e g e r

B1(0).

23

L

let

uk = k/L, k = 1,2, ...,L ,

and compute

Applying t h e mean v a l u e theorem r e p l a c e t h e i n n e r i n t e g r a l by f o r some :a r

-+

-

Taking

Isl(,)cp(o*,e)de k

E ( U ~ - ~ , U ~t h) e, n u s i n g t h e l i m i t i n g r e s u l t o b t a i n e d above l e t

to arrive a t

L

--t

=

t h e n w e o b t a i n from t h i s Riemann sum t h e f o l l o w i n g i n t e g r a l :

as r e q u i r e d . Another s p e c i a l case o f i n t e r e s t i s when ( s -AV B ) holds u n i f o r m l y w i t h r e s p e c t t o t r a n s l a t i o n s of c o o r d i n a t e s , t h a t i s

aB ( s ) = rlim

1 ~ -SIN ~ 'Br(c) 1

u (x)dx,

0

It can b e shown t h a t t h i s can happen o n l y i f

uniformly i n

ag(s)

=

0

c 6R

or

N

.

s = N.

The second

N.D. Alikakos and R. Rostamian

24

a l t e r n a t i v e i s t h e same as (UAV) of S e c t i o n 1. i m p l i e s t h e uniform v e r s i o n of ( s -AVD) w i t h Theorem 3 . 2 :

The n e x t theorem shows t h a t t h i s s = N:

(UAV) h o l d s i f and o n l y i f

+

u,(r(x-c)

uniformly i n

N c C R

c)

+

a

in

V'CR~)

.

The proof i s l o n g and t e c h n i c a l so w e omit it h e r e . by the uniform convergence here we mean t h a t f o r any

uniformly i n

c C R

N

c in

L e t ' s mention t h a t m N Co(R ) :

.

REFERENCES : N.D.

Alikakos, and R . Rostamian, On t h e u n i f o r m i z a t i o n o f s o l u t i o n s of t h e

porous medium e q u a t i o n i n

RN,

t o appear.

D . G . Aronson, and L.A. C a f f a r e l l i , The i n i t i a l t r a c e of a s o l u t i o n of t h e porous medium e q u a t i o n , m a n u s c r i p t . Ph. Bgnilan, M.G.

equation i n

RN

C r a n d a l l and M. P i e r r e , S o l u t i o n s o f t h e porous medium under o p t i m a l c o n d i t i o n s on i n i t i a l v a l u e s , m a n u s c r i p t .

A.M. Fink, A l m o s t P e r i o d i c D i f f e r e n t i a l E q u a t i o n s , L e c t u r e Notes i n Math., N o . 377, Springer-Verlag, 1 9 7 4 .

A. Friedman and S . Kamin, The a s y m p t o t i c b e h a v i o r of g a s i n an n-dimensional porous medium, T r a n s . h e r . Math. S O C . 262 (1980) 551-563. S . Kamin, S i m i l a r s o l u t i o n s and t h e a s y m p t o t i c s of f i l t r a t i o n e q u a t i o n , Arch. B a t l . Mech. Anal. (1976) 171-183.

DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T.Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

25

STABILITY OF QUANTUM MECHANICAL SHAPE RESONANCES VIA RICCATI’S EQUATION Mark AshbaughX Department of Mathematics University of Missouri Columbia, MO 65211

Carl Sundberg Department of Mathematics University of Tennessee Knoxville, TN 37996-1300

The problem of locating resonance energies of Schrgdinger’s equation with a barrier potential is considered in the limit where the barrier is sent to infinity., The sphericallysymmetric three-dimensional case is treated by the methods of ordinary differential equations using an outgoing wave boundary condition to define resonance energies.

A detailed

analysis of the associated Riccati equation, S(x) = 1

-

(Wn(x)

- E)S(Xl2,

plays a major role in our approach; here W

(XI

is the barri-

er potential which is assumed to approach infinity as n for almost all x in the barrier region. n +

m

-+m

We show that as

the resonance energies associated with the above prob-

lem approach the eigenvalues of a related "unperturbed" problem.

We consider the quantum mechanics problem, -$'I

i (U(X) i

Wn(X))$ = E$

on 0 2

X


a this is equi-

valent to the condition $ ' (a) = i&

$(a) so that our problem may be viewed as a

(non-self-adjoint) Sturm-Liouville problem on the interval [Oral with eigenvalue parameter appearing in the boundary conditions. For use in our subsequent discussion we define two functions, $ (x,E) and qrn(xIE), 0

as solutions to the differential equation (1) satisfying the initial conditions

$0 (0,E) = 0, $’(O,E) = 1 and $Jrn(a,E) = 1, $:(a,E) = ifi. These are essentially 0 the regular and Jost solutions, respectively, from the theory of potential scattIn terms of $,

ering [3,6].

and Qm, the condition for aresonanceenergy is is the Wronskian of $0 and

simply W{$Jor$rn))x=l= 0, where W{$Jo,$,)

$rn

and we have

used the fact that it is constant for two solutions to equation (1) in choosing to evaluate it at the convenient point x = 1.

Actually, to isolate $

0

and qrn we

prefer to find resonances using F (E) = 0 where JIO(lrE)

$rn(lrErn)

$A(lrE)

- Jbb(llEI")

F (E) !

W{$or’$rn~lx=l =

$;(1rE)$;(lrErn)

*

(In the course of our investigation it transpires that the extra factors introduced in the denominator here do not vanish atresonance energies if n is sufficiently large.) In the following section we will show that $m (1,Ern) 0 as n - + m $L(l,E,n) uniformly for E in certain compact subsets of C. -+

Armed with this information we

now conclude the argument showing stability of resonances. Fn(E)

+

First we note that

$O(lrE)/$~(lrE) E F(E) which has the values e such that $O(lre) = 0 as

its roots.

These will be called the"unperturbedeigenva1ues" and are in fact the 2

eigenvalues of the operator h = -d / d x conditions.

Since

9,

2

+

U(x) on [0,1] with Dirichlet boundary

is analytic in E and

$m

is analytic in E except at E = 0

we can use Hurwitz’s Theorem from complex variable theory 141 to conclude :

27

Stability of Resonances via Riccati's Equation Theorem 1. (stability of Resonances).

For any unpertured eigenvahe e > 0 and for

n sufficiently large there is a unique nearby resonance energy E of equation (1) with En

e as n

-+

-+ oJ.

THE RICCATI EQUATION -+ 0 as n

To show that $m(lrE)/q:(l,E) equation which it satisfies. from this equation.

we will have recourse to the Riccati

m

-

(Wn(x)

-

. E)y 2 with

We must now extract the behavior of $m(l,E)/$~(lrE)

where y = $-/$:.

y(a) = l/i&

-f

This equation is y’ = 1

In the remainder of this paper we content ourselves with

only outlining the essential steps of our argument; for detailed proofs see 121. As a preliminary step we consider the simpler equation S’(X) = 1 on [0,1] where A(x) t 0 and integrable. E

- A(x)S(x) 2

We prescribe S(0)

and examine the behavior of S(1).

[O,n/2]

(2) =

if3 ce c

5 Or

This problem should be viewed as

coming from the above problem concerning y by a reflection and a rescaling of the x-axis (so that a becomes 0 and 1 remains 1) and by dropping E. S(x) = f(x)

+

Letting

ig (x) where f (x) and gfx) are real-valued we have f’ (x) = 1

- A(x) [f(x)2 -

2

g(x) I

g’ (X) = -ZA(x)f (x)g(x). From this system of equations it is easy to show the following facts: (1) f > 0 on (0,ll; (2) g’ 5 0 on [0,1] and either g E 0 or g > 0;

most one point to the left of which f

(3) f = g a t

g

and to the right of which f > g; (4) f < 1

+

c on [0,11.

We now insert A (x) in place of A(x) in equation ( 2 ) where A (x) 2 0 and An(x) - + m n for a.e. x E [0,1] as n + m . For the family of differential equations so obtained iB we define S (x) as that solution which obeys S (0) = ce Then we have: n n

.

Theorem 2.

Sn(x) exists on [0,1] and Sn(x)

compact subset of [ O , - ) , e eventually arg Sn(x) Remark.

E

E

[O,IT/21,

[0,11/4)

-f

0 as n

+CO

uniformly for c in any

and x in compact subsets of (0,lI. Moreover,

for any x > 0.

This shows that by taking n sufficiently large fn will surely pass above

gn and, further, that the crossing point can be made as near as we like to x = 0. Here we have set S (x) = fn(x) + ign(xl. n

To deal with resonances we shall need to extend the above result to values of greater than T / 2 .

We take f3

E

[n/2,

5'T/8]

as a convenient choice.

The extended

result will follow from the above once we have shown that for sufficiently large n

M. Askbaugh and C Sundberg

28

fn has a zero z arbitrarily near x = 0 and that gn does not grow too rapidly while f is negztive; in particular, we show gn ( zn ) 5 c m These results

.

yield; Set B

Let S (x) and An (x) be as above.

Theorem 3 .

P

= {zlO 5

arg

z 5

x Then s (x) exists on [0,1] and S ( x ) 3 0 uniformly in S (0) E B n n n PI any p > 0 and x1 > 0. Also for any x1 > 0 there exists N such that 0 5 arg S (x) < 1T/4 for all n 2 N and all x

E

E

5n/8,lz/ p}.

[x,,ll

for

[xl,ll.

To finish off the argument showing that +m(l,E,n)/Jbb(l,EIn)

*0

as n

+ m

we put E

back into our equation and use a comparison technique. Passing back to second order linear equations for a moment we define u and v by u; = A (x)u and n n n n v" = (An(x) E)vn with initial conditions u ( 0 ) = 1 = v ( 0 ) and n -1 -iB - I v (x) = w (x)u (x) and fin((x) = wA(x)/wn(x) we uA(0) = c e - ~ ~ ( 0 )Setting . n n n

-

find that

From this equation and our previous results about Sn(x) that i2 (x) is bounded independently of n for large n.

+ w’fx)/wn(x)

= u;(x)/Unix) v’ix)/vn(x) n

it follows that v~(x)/vn(x) vn(l)/vA(l)

*o

as n

+a,

= l/SnfX)

*

0.

In particular,

or more precisely we have:

Theorem 4. With vn(x) as defined above, vn(l)/vA(l)

E

+ ,.(nif

for x > 0 or that vn(x)/vi(x)

+ m

all E in any compact subset of

iz

= u,(x)/u’ (x) one can show n Then from the relation

CIO

C

+

0 uniformly as n

for

and for all initial conditions in

< arg z 5 57~/8, p1

5

1.

5

~ ~ 1 .

When translated back into the notation of the resonance problem we began with, this shows that i/i,(l,E,n)/i/i~(l,E,n) E

E

{z

E

cI-r/4 5 arg z
0.

Let

6 > 0

and l e t (U6f)(X) for

f

c

2 N L (W )

and

x

E

IR

N

.

Then

=

U6

-N/2

6

f(6x)

is u n i t a r y on

L2(WN)

and a s i m p l e

c a l c u l a t i o n shows t h a t

U6HcUi’ so that

*

Hc

i s u n i t a r i l y equivalent t o

P a r t i a l l y s u p p o r t e d by an NSF g r a n t .

= 6 -2 Hc

6-

,

times i t s e l f .

But

H

2 -a1

Hc

P. Baras and J. A. Goldstein

32 2

implies 6- H

2

-6-2aI , and since

~ )O(H~) inf ~ ( U ~ H ~ U= -inf

(u

spectrum), we may let 6 + m to conclude H 2 0. can have no eigenvalues. A similar scaling argument shows that Hc 2 N Namely, suppose Hc@ = E+ with @ 6 L (W ) . For @,(x) = @(Ax) with A > 0 2 and x E WN we get Hc$A = A E$*. The orthogonality relation between eigen=

vectors corresponding to different eigenvalues gives ( f o r

as A

1. Thus $ E 0. With respect to semiboundedness C*(N)

X

f

1)

+

value of c.

is clearly the cutoff point for the

If Kc denotes a (semibounded if possible) self-adjoint extension

of Hc , then the initial value problem for the parabolic equation au/at 2 N is well-posed in L (R ) for c

5

+ K ~ = U

C*(N).

o

It is not well-posed if c > C*(N) ,

b u t it is not clear to which extent it is solvable o r not.

can be replaced by (Let us add parenthetically that the domain C:(RN\{O)) 2 N 2 5. This is because E Lloc(WN) if N 2 5.) Recently Baras and Goldstein [I] proved a result which, among other things,

XI-^

C i ( R N ) if

provides an alternate proof and strong sharpening of the second assertion of Theorem l(i). Of concern is the heat (rather than the Schrodinger) equation au/at

+

u + f(x,t) 1x1 T > 0 , c > 0 , and 0

= AU +

(1)

1 where S f E L (RN X ( 0 , T ) ) are given. A solution in the sense of distributions is sought whose initial value is a (nonnegative) finite Radon measure LI in the sense that

for

(x,t)

for each

E

RN x ( 0 , T )

@ E C;(W

N

)

.

We approximate (1) by

where

THEOREM 2 [l]. provided

(i)

If c

5

C*(N)

, then (l), (2) has a nonnegative solution u

33

The Inverse Square Potential in Quantum Mechanics

where cx i s the smallest root of

(N-Z-a)a=c. The solution i s given by solution o f (3,) constant

,

(2)

.

(4)

u(x,t) = lim un(x,t) where u i s the nonnegative n* For each E E (0,T) and R > 0 there i s a p o s i t i v e

C such t h a t u(x,t)

2

(u,f) #

i f 1x1 < R , t E (E,T) , and nonnegative solution, then

ClxJ-"

(51

ConuerseZy, i f

(0,O).

(11, (2) has a

f o r each R > 0 and each E E (0,t) where a is as above. (ii) If c > C*(N) , and i f a t Zeast one of p , f i s not zero, then (I), (2) has no nonnegative solution.

for each

t > 0 and each x

Moreover, f o r

E

lim u (x,t) = n-rm N

u

as i n (i) above,

m

w .

Note that a is given by ~1

= (N - 2)/2

- [((N

-

2)/2)'

- 4cI1"

Thus IxI-" E Lloc(R Let 1-I be absolutely 2 N ) i f c 5 C*(N). 1 N continuous, s o that u(dx) = u0 (x)dx with 0 2 uo E L ( R ) . If also 2 N uo E L (R 1 (and, say, f E 0), then ( 6 ) holds automatically by the Schwarz inequality. Thus the necessary condition (6) is consistent with the L2 theory. We shall give a heuristic argument to make the statement of Theorem 2 seem plausible. F o r a detailed proof see [l]. Let cx s (N - 2 ) / 2 . Then f o r k > 0 , $(x)

klxl-"

satisfies

where c is given by (4). By Maximum Principle and comparison arguments, we expect to have good existence results f o r c as in (4) corresponding to a < (N - 2 ) / 2 , i.e., f o r c s C * ( N ) . Now let c > C*(N). Rewrite

P. Baras and J.A. Goldstein

34

au/at

=

AU

+

C 2 u 1x1

as au/at

=

nu

-u

+ C*"Z'

+ f(x,t)

1x1

where

The n e c e s s a r y c o n d i t i o n s (5) and (6) imply

since

a = ( N - 2)/2

(which c o r r e s p o n d s t o t h e v a l u e

C*(N)

i n applying p a r t

T h i s c o n t r a d i c t i o n shows a s o l u t i o n cannot e x i s t .

( i ) o f t h e theorem).

A d i r e c t approach t o p a r t ( i i ) may b e based on a s t u d y of t h e k e r n e l

u(dy).

)1

, as b e f o r e , so t h a t uo(y)dy

can b e r e p l a c e d

An a l t e r n a t e r e p r e s e n t a t i o n i s g i v e n by t h e Feynman-Kac formula

N

lR ) is t h e s p a c e o f c o n t i n u o u s p a t h s i n IRN and Px Wiener measure s t a r t i n g a t x . T h i s c l e a r l y i m p l i e s t h a t u n ( x , t ) i s where

S = C([O,m);

nondecreasing

t > 0

l i m G (x,y,t) n-rm

x,y

An

Let

> 0

,

= m

IRN .

E

=

i n f o(H:)

be t h e ground s t a t e energy o f

c o r r e s p o n d i n g wave f u n c t i o n , i . e . $,(x)

for all

x ; Qn

@n

2

t

N

L (1R ) ,

II@nl(

An

+

-m

as

n

+a.

H: = 1

and l e t

, H:

9,

@n =

L

be t h e

Xn qn

and

e x i s t s and i s u n i q u e l y determined by t h e s e c o n d i t i o n s

by a s u i t a b l e v e r s i o n o f t h e Perron-Frobenius theorem. show

is

.

I t i s enough t o e s t a b l i s h t h a t , f o r

for all

Gn

,

can b e a measure

uo

Note t h a t by

= -A + V,(x)

H:

d e f i n e d by:

If

c > C *( N )

one can

The Inverse Square Potential in Quantum Mechanics

35

2

Since Vn(x) = n Vl(nx) , a scaling argument similar to those given above imp1ies =

nN’2

@l(nx)

,

xn

n

=

2

x1 .

Consequently -tX

exp(-tHn)

2

e

>

-tAn Gn(x,y;t) = LexP(-tHpxl(Y) 2 e -tn2A = n e b,(nx)@,(ny) But

A1
1, t a k i n g t h e A

extended complex p l a n e Ic =

(1: u Em}

o f F i s a s e t o f p o i n t s {z1,z2

into itself.

,...,z k )

For k

such t h a t F(zl)

6

{1,2,3,

= z2,

...1

F(z2)

=

a k-cycle

z 3 ,...,

and zi # z . f o r i # j . I f z1 # t h e k - c y c l e i s s a i d t o be a t t r a c J t i v e , i n d i f f e r e n t o r r e p u l s i v e a c c o r d i n g as l F ( k ) ( z l ) l < 1, =1 o r >1 respecF ( z k ) = z,,

tively.

If z1

=

l e t G(z) = l / F ( l / z ) and use G ( k ) ( 0 ) i n p l a c e o f F ( k ) ( z l )

t h e p r e v i o u s sentence.

The J u l i a set J o f F ( z ) can be d e f i n e d as t h e s e t of

p o i n t s z such t h a t z belongs t o a r e p u l s i v e k - c y c l e of F ( z ) f o r some k

E

in

M. F. Barnsley and A. N. Harrington

38

9

points a t t r a c t e d

/ to 0

@ ;;i;;s

attracted

unshaded points a t t r a c t e d t o +1

{1,2,3,...1, [2,3,4]. J i s completely i n v a r i a n t under F, namely F-lJ = J ; i t contains no i s o l a t e d points; and usually i t has no i n t e r i o r and has a complicated f r a c t a l s t r u c t u r e -- t h i s i s the case f o r example i f there a r e t h r e e d i s t i n c t points, each o f which belongs t o a n a t t r a c t i v e k-cycle of F f o r some k E {1,2,3, I. The J u l i a s e t f o r F(z) = z 2 - A , X E R , i s important t o the theory 2 o f t h e i t e r a t i o n s o f F(x) = x -1, X E IR , a s studied by Feigenbaum and o t h e r s ,

...

see [51. The J u l i a s e t J and i t s i n t e r p l a y with F can be characterized by t h e f o l lowing:

THEOREM [6,7]. Let F ( z ) be a r a t i o n a l function of degree N > 1. There e x i s t s a unique p r o b a b i l i t y measure p such t h a t p(E) = p(FY1E)/N whenever E i s a Bore1 J 1 subset o f $, where {F: ( z ) l ~ = ,i s a complete assignment of inverse branches of F. J

We c a l l p t h e balanced measure f o r F, [ 8 ] . F i s an ergodic transformation of J w i t h respect t o p. In order t h a t we may be able t o construct a s e l f a d j o i n t operator whose spectral density i s j ~ , we consider the moments o f u.

39

Julia Sets and Autonomous Differential Equations THEOREM [9].

L e t F ( z ) be a r a t i o n a l f u n c t i o n o f degree N > 1.

p o l e a t c, where a,B,y

and 6

L e t c belong t o

L e t $ ( z ) = (az+B)/(yz+6) have

an a t t r a c t i v e o r i n d i f f e r e n t k-cycle o f F(z).

A l l o f t h e moments

E (c.

f o r n E ~0,1,2y...}, f (@(z))'ddu(z), J e x i s t and can be c a l c u l a t e d r e c u r s i v e l y i n terms o f t h e c o e f f i c i e n t s i n F(z). Mn =

-

.

2 Then c = i s an a t t r a c t i v e 1-cycle f o r F(z) and L e t F(z) = (z-2) we can choose $ ( z ) = z. The theorem asserts we can c a l c u l a t e t h e moments EXAMPLE.

Mn =

I

We f i n d Mo = 1, M1 = 2, M3 = 6, M4 = 20, t h i s example t h a t J = [0,4]

for n

zndp(z)

J

E

{OY1,2,...1.

...; i n f a c t i t i s w e l l

known

[lo]

for

JX.(4-x)) f o r 0 < x < 4.

and dp(x) = l / ( n

The n e x t step i s t o consider t h e orthogonal monic polynomials { P , ( Z ) } ~ = ~ where Pn(z) i s o f degree n, has u n i t leading c o e f f i c i e n t , and

f

Pn(z)

dv(z) = 0

J f o r n # m, where the bar means the complex conjugate i s taken. When

F(z) i s a polynomial o f degree g r e a t e r than one, i t i s always p o s s i b l e

t o c a l c u l a t e r e c u r s i v e l y i n terms o f t h e c o e f f i c i e n t s i n F ( z ) an i n f i n i t e subsequence o f t h e associated orthogonal polynomials, [8, 17, 121.

F(z) i s any r a t i o n a l t r a n s f o r m a t i o n f o r which m belongs t o an a t t r a c t i v e o r i n d i f f e r e n t k-cycle and f o r which J c IR u Cm}, one can use t h e moments When

M,

=

f

xndp(x)

J for n

E

{Oyl,Z,

...I

t o c a l c u l a t e a l l o f t h e orthogonal polynomials associated w i t h

F, according t o

1

zn

These polynomials u n i q u e l y determine r e a l numbers an and bn, n where a.

= 0 and an > 0 f o r n

E

I0,l ,2,..

.I,

# 0, such t h a t

Pn+l ( 2 ) = (z-bn)Pn(z)

2 - anPn-l (z).

I n t u r n , t h e an's and bn's i n the three-term recurrence formula f i x a h a l f i n f i n i t e Jacobi m a t r i x

40

M.F Barnsley and A. N. Harrington

-

THEOREM [9]. which

{-I.

L e t F ( z ) be a r a t i o n a l f u n c t i o n , o f degree g r e a t e r t h a n one, f o r

belongs t o an a t t r a c t i v e o r i n d i f f e r e n t k - c y c l e and f o r which J

c

iR

u

The a s s o c i a t e d J a c o b i m a t r i x d i s s e l f - a d j o i n t w i t h spectrum J and s p e c t r a l

d e n s i t y p, t h e balanced measure f o r F(z). EXAMPLE.

L e t F ( z ) = z 2 -A

with

X > 2.

The c o n d i t i o n s o f t h e l a t t e r theorem a r e

met, J b e i n g a Cantor s e t i n R . In [131, i t i s shown t h a t bn = 0 f o r a l l n , 2 2 2 2 w h i l e a2m+, = h - a I n [14], i t i s and a2m+2 = X/ a2m+l f o r rn E {0,7,2, }. 2m sholvn t h a t t h e sequence {am} i s almost p e r i o d i c .

...

F i n a l l y , we d e s c r i b e an example which p r o v i d e s a c o n n e c t i o n between t h e t h e o r y of moments o f balanced measures on J u l i a s e t s and c e r t a i n autonomous o r d i n a r y d i f f e r e n t i a l equations.

Consider t h e e q u a t i o n

on E w i t h z(1) = P(eia)

and a / 2 ~any s u i t a b l e i r r a t i o n a l number. Here P(z) 2 3 denotes t h e W e i e r s t r a s s P - f u n c t i o n which obeys ( P ' ( z ) ) = 4P(z) -4P(z). The branch o f t h e square r o o t i s chosen t o ensure t h a t t h e m o t i o n proceeds " f o r w a r d " along the t r a j e c t o r y .

I n 191 i t i s shown t h a t t h e w - l i m i t s e t o f t h e t r a j e c t o r y

h

i s t h e whole o f t, and t h a t t h e c o r r e s p o n d i n g i n v a r i a n t measure i s e x a c t l y t h e balanced measure f o r t h e r a t i o n a l f u n c t i o n , f i r s t s t u d i e d by L a t t 2 s [13],

2

2

F ( z ) = ( z +1) /(4z

2

(z -l)),

whose J u l i a s e t i s t h e whole o f

-

be shown t o be d p ( z ) = ( c o n s t a n t ) d x d y / < l z /

n

E

{0,1,2,

...1 ,

2 I z -1

1).

?. T h i s

measure can

The moments J@(z)'dU(z),

do n o t a l l e x i s t , regardless o f t h e choice f o r t h e pole c o f

442).

Herman [15] has r e c e n t l y p r o v e d t h a t t h e r e e x i s t r a t i o n a l t r a n s f o r m a t i o n s A

which possess an i n d i f f e r e n t f i x e d p o i n t c =

m,

and have J = &. For t h e s e

examples a l l of t h e moments

J zndv(z) , J

n

t

{0,1,2

,...I,

do e x i s t and can be c a l c u l a t e d r e c u r s i v e l y . REFERENCES

[l]

Mandlebrot, B., 1982)

.

The F r a c t a l Geometry of N a t u r e

(W.

H. Freeman, San F r a n c i s c o

Julia Sets and Autoizornous Differential Equations

[Z]

Fatou, M. P., Sur l e s equations f o n c t i o n e l l e s , Bulletin de SOC. Math. de France 47 (1919), 161-271; ibid. 48, 33-94; i b i d . 48, 208-314.

[3]

J u l i a , G., Memoire sur L ' i t e r a t i o n des fonctions r a t i o n e l l e s , J r n l . de Math. Pures e t Appl. 4 (1918), 47-245.

41

141 Brolin, H., Invariant sets u n d e r i t e r a t i o n of r a t i o n a l functions, A r k i v f l r Matematik 6 (1965), 103-144. [5]

Collet, P. and Eckmann, J., I t e r a t e d Maps on t h e Interval a s Dynamical Systems (Birkhauser, Boston 1980).

[6]

Demko, S., In preparation.

[7]

Freire, A., Lopes A. and Ma;, R., An i n v a r i a n t measure f o r rational maps, Preprint, I.M.P.A., Rio d e daneiro (1982).

[8]

Barnsley, M., Geronimo, J. S. and Harrington, A. N., Orthogonal polynomials 7 (19821, associated with invariant measures on J u l i a s e t s , Bulletin A.M.S. 381 -384.

[9]

Barnsley, M. and Harrington, A. N., Moments o f balanced measures on J u l i a sets, P r e p r i n t , School of Mathematics, Georgia Tech., (Dec. 1982).

[lo]

Ulam, S. and von Neumann, J . , On combination of s t o c h a s t i c and deterministic processes, Bulletin A.M.S. 53 (1947), 1120.

[ll]

B e s s i s , D. and Moussa, P., Orthogonality properties o f i t e r a t e d polynomial mappings, Commun. Math. Phys. 88 (1983), 503-529.

[12]

Pitcher, T. S. and Kinney J. R., Some connections between ergodic theory and i t e r a t i o n o f polynomials, Arkiv f g r Matematik 8 (1968), 25-32.

[13]

L a t t k , S., Sur L ' i t e r a t i o n s des s u b s t i t u t i o n s r a t i o n e l l e s e t Les fonctions de Poincare, Note aux C. R. Acad. Sc. P a r i s 166 (1918), 26-28.

[14]

B e l l i s s a r d , J . , Bessis, D. and Moussa, P., Chaotic s t a t e s of almost periodic Schradinger operators, Phys. Rev. L e t t e r s 49 (1982), 701-704.

[15]

Herman, M. R., Examples de f r a c t i o n s r a t i o n e l l e s ayant une o r b i t e dense sur l a sphere de Riemann, Preprint, Centre de Mathematiques de L'Ecole Polytechnique, France, (March 1983).

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DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

43

A L I M I T I N G ABSORPTION PRINCIPLE FOR A SUM OF TENSOR PRODUCTS

Matania Ben-Artzi Department of Mathematics Technion I T T Haifa 32000, I s r a e l Allen Devinatzl Department of Mathematics Northwestern University Evanston, I L 60201, USA

A l i m i t i n g absorption p r i n c i p l e i s obtained f o r a sum of t e n s o r products of t h e form

H = HI 8 I2

+ I, 69

H2

under v a r i o u s hypotheses on t h e s e l f - a d j o i n t o p e r a t o r s H, and H 2 .

§ 1.

INTRODUCTION

In t h i s paper we s h a l l consider t h e problem of o b t a i n i n g a l i m i t i n g a b s o r p t i o n p r i n c i p l e f o r s e l f - a d j o i n t o p e r a t o r s of t h e form H = HI 69 I 2 + I, @ H 2 , (1.1) where H1 and Hp a r e s e l f - a d j o i n t o p e r a t o r s on H i l b e r t spaces Hiand &, and 11 and I 2 a r e t h e i d e n t i t y o p e r a t o r s on t h e s e spaces r e s p e c t i v e l y . A s i s w e l l known, o p e r a t o r s of t h e form (1.1) a r i s e q u i t e n a t u r a l l y i n c l a s s i c a l physics, i n quantum mechanical s e t t i n g s , a s well a s elsewhere. The most usual s i t u a t i o n s occur when HI and Hp a r e d i f f e r e n t i a l o p e r a t o r s a c t i n g i n L2 spaces. For example, t h e SchrEdinger o p e r a t o r s

a r e n a t u r a l examples. Limiting a b s o r p t i o n p r i n c i p l e s f o r t h e s e o p e r a t o r s are immediate consequences of t h e a b s t r a c t r e s u l t s we s h a l l p r e s e n t . Thus we supply a general framework i n t o which t o f i t known r e s u l t s (see e.g. [ 1 ] , [ 3 ] , [ 4 ] )a s w e l l as a general framework f o r o b t a i n i n g new r e s u l t s . The methods of proof of the theorems i n t h i s paper a r e elementary i n n a t u r e i n t h e sense t h a t w e use only well known c l a s s i c a l r e s u l t s and methods. These include t h e o p e r a t i o n a l c a l c u l u s i n H i l b e r t space, p r o p e r t i e s of t h e Poisson k e r n e l and t h e most elementary and c l a s s i c a l p r o p e r t i e s of H i l b e r t transform theory. We s h a l l o b t a i n a l i m i t i n g a b s o r p t i o n p r i n c i p l e f o r H under varying hypotheses on In t h e f i r s t s e c t i o n w e s h a l l g i v e t h e simplest of such r e s u l t s . In H I and H p . t h e n e x t s e c t i o n s w e s h a l l g i v e v a r i a t i o n s on t h e theme of t h e f i r s t s e c t i o n as w e l l as a f u r t h e r r e s u l t w i t h hypotheses of a somewhat d i f f e r e n t c h a r a c t e r .

44 52.

M. Ben-Artzi and A. Devinatz A LIMITING ABSORPTION PRINCIPLE

L e t us i n t r o d u c e some terminology which w e s h a l l u s e i n t h i s and i n succeeding s e c t i o n s . We s h a l l t a k e H1 and H 2 t o be s e l f - a d j o i n t o p e r a t o r s on t h e H i l b e r t I m z f 0, j = 1,2, Their r e s o l v e n t s a r e R j ( z ) = ( H . - .z)-', and H,. spaces and t h e r e s o l v e n t o f t h e o p e r a t o r (1.1) i s denoted $y R ( z ) . I n t h i s and i n t h e next s e c t i o n we s h a l l s t a t e our hypotheses i n terms of t h e d i f f e r e n c e of resolv e n t s , r a t h e r t h a n i n t e r m s of r e s o l v e n t s , s i n c e o u r hypotheses w i l l u s u a l l y b e e a s i e r to v e r i f y f o r t h e former t h a n f o r t h e l a t t e r . Toward t h i s end we set

We s h a l l a l s o suppose t h a t x l , y, a r e H i l b e r t s p a c e s w i t h

where t h e embeddings a r e d e n s e and continuous. The c o l l e c t i o n of bounded l i n e a r t r a n s f o r m a t i o n s from x t o y* w i l l be d e s i g n a t e d by B(x.,y?) and t h e i r norms by j J J J " 'I' B(xj ,yi). There a r e v a r i o u s t e n s o r product H i l b e r t s p a c e s which c a n be formed from t h e above and t h e s p a c e s . For example t h e norms on X I 8 H, w i l l be d e s i g n a t e d by 11 - 1 1 x 1 8 K2 norms i n t h e space of bounded l i n e a r t r a n s f o r m a t i o n s from x i 8 x 2 t o yp 8 H 2 by We t h i n k t h e s e i l l u s t r a t i o n s w i l l make c l e a r t h e n o t a t i o n s 11 -11 B(xl ~ ~ H,)., ~ f we s h a l l u s e i n s i m i l a r s i t u a t i o n s . ~

The assumptions we s h a l l make a r e as f o l l o w s : For every A f R

t h e r e exists an o p e r a t o r A1(A)

f B(xl.y:)

so t h a t

(2.3) There e x i s t s an M > 0 and an a , 0 < a < 1, s o t h a t f o r a l l A1,A2 w i t h I X g - All 5 1,

f 1R

F i n a l l y we s h a l l suppose t h a t (2.5a) and f o r every compact I

W,

(2.5b)

A s examples of o p e r a t o r s which s a t i s f y t h e p r e v i o u s hypotheses w e n o t e t h e following:

(2.6)

H1 = -A

(2.7)

H

1

=

acting i n L2(iRn),

d2 -7 -F dx

ex

n t 3,

a c t i n g i n L2(IR1),

x1 = y1 = L2"(Xn),

s > 1.

x1 = y1 = L z Y s ( R 1 ) , s > 114.

The r e l e v a n t p r o p e r t i e s f o r t h e example (2.6) a r e e a s i l y o b t a i n e d u s i n g t r a c e e s t i m a t e s i n Sobolev s p a c e s o r by t r a n s f o r m i n g t o p o l a r c o o r d i n a t e s and u s i n g w e l l known p r o p e r t i e s of Bessel f u n c t i o n s . They have a l s o been o b t a i n e d by s l i g h t l y d i f f e r e n t techniques [ 2 ] . For t h e example (2.7) t h e r e l e v a n t p r o p e r t i e s a r e e a s i l y shown because of t h e w e l l known p r o p e r t i e s of t h e Airy f u n c t i o n . A c t u a l l y t h e s e p r o p e r t i e s were obtained f o r t h e r e s o l v e n t i n [ 3 ] . However, s i n c e we r e q u i r e t h e s e

45

A Limiting Absorption Principle p r o p e r t i e s o n l y f o r t h e d i f f e r e n c e of r e s o l v e n t s , t h e proof is somewhat more direct. I t s e e m s h i h l y l i k e l y t h a t a wide v a r i e t y of o r d i n a r y d i f f e r e n t i a l o p e r a t o r s a c t i n g i n L ( R ) w i l l s a t i s f y t h e above hypotheses. For example, from t h e work i n [ l ] , i t would appear t h a t f o r 0 < 8 5 2 t h e o p e r a t o r

9

H~ = -

d2

-

e sgn x l x l '

x1

i n ~ 2 R( ) ,

= y1 =

L 2 , s - 8/4

,

s . 5 ,

But t h i s may r e q u i r e a f a i r amount of

w i l l a l s o s a t i s f y t h e g i v e n hypotheses. computation.

We now s t a t e t h e theorem of t h i s s e c t i o n . For s i m p l i c i t y of n o t a t i o n we s h a l l d e s i g n a t e t h e norm i n B(xl 0 f$, Y: 0 H2) by 111 * 111

-

THEOREM 2.1. Under t h e hypotheses (2.3) t o ( 2 . 5 ) , f o r every h ER, t h e r e e x i s t R'(A) f B(x, 0 H,, y: 8 SO t h a t

%)

l i m s u p 111 R ( A + i c ) - R'(X)III ESO ACE

(2.8)

Moreover, (2.9)

SUP

Ill R'(A)

111


0 and a n a , 0 < a < 1, so t h a t i f e i t h e r b o t h X 1 and X 2 a r e p o s i t i v e , o r i f b o t h h l and h 2 are n e g a t i v e , and j X 2 - A 1 5 1,

1

and f o r e v e r y compact s e t I cIR\{O)

M. Ben-Artzi and A. Devinatz

46 (3.3b)

SUP11 Al(Ln)llB(xl,y;)


0 THEOREM 3.1.

operators

Under the conditions (3.1) to (3.5), for every X C U there exist x2,y; B y;) s o that

P ( Ac) B ( X ~ B

R(X t i E )

(3.6)

-+

Ri(h)

in the norm topology of g ( x l B x2,yf Q y t ) , the convergence being uniform in every satisfy a uniform local H6lder condition of order a K c c U. Furthermore, R'(A) in K. REMARKS. A p r o t o t y p e example of an o p e r a t o r which s a t i s f i e s t h e hypotheses (3.1) t o (3.4) i s HI = -A a c t i n g i n L 2 ( a'), w i t h x1 = y1 = LzYs( l R 2 ) * s > 1. Thus we may apply t h i s theorem t o t h e o p e r a t o r

(3.7)

I

w i t h x2 = y2 = L 2 s s - B / 4 ( R ) , s > f . Indeed w e may r e p l a c e e sgn x1 x l I B by c o n s i d e r a b l y more g e n e r a l V(x1) s i n c e t h e c o n d i t i o n (3.5) can b e e a s i l y e s t a b l i s h e d once we have good asymptotic e s t i m a t e s f o r s o l u t i o n s t o t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n -d2/dx2 V(x) - A = 0 a t t m . Thus i f we a r e w i l l i n g t o a c c e p t a s t r o n g e r weight f u n c t i o n t h a t one could p o s s i b l y g e t w i t h a more p r e c i s e a n a l y s i s of t h e o r d i n a r y d i f f e r e n t i a l o p e r a t o r s involved, w e may apply Theorem 3.1 w i t h ease i n a wide v a r i e t y of c o n c r e t e s i t u a t i o n s .

+

A s a second s e t o f assumptions on H1 and H2 w e t a k e t h e following:

For t h e o p e r a t o r HI we make t h e f o l l o w i n g assumptions: For every A '2 R\{O} t h e r e e x i s t s a n o p e r a t o r A 1 ( X ) '2 B(xl,yT)

so t h a t

There exists a n a, 0 < a < 1, and f o r every 6 > 0, t h e r e exists an M6 > 0 so t h a t i f e i t h e r X1,X2 Z 6 o r A ~ , h 25 -6 and I X 2 - A l 1 2 1, t h e n (3.9) For every 6 > 0 (3.10a) and f o r every compact s e t I c _ I R \ C O l , (3.10b) n20

For t h e o p e r a t o r H2 w e make t h e f o l l o w i n g assumption:

A Liniititig Absorption Principle

There i s an open set U 533 so t h a t f o r every K t h e r e a r e o p e r a t o r s R*(A) f B(x,,y$) s o t h a t

47

cc

U and f o r every A f K,

2

THEOREM 3.2. Under the hypotheses (3.8) t o (3.11), operators R?(h) i n B(xl 63 x2,y: 4 y t ) s o that

(3.12)

R(A2 iE)

+

Rf(h),

f o r every h f U there e x i s t

EJ. 0 ,

i n the norm topoZogy of B(x1 8 x2,yT 8 y;), the convergence being uniform i n every K cc U. Further, i f R:(X) s a t i s f y a Z o c d uniform HttZder condition i n K, so do R?(A). REMARKS. A p r o t o t y p e example o f an o p e r a t o r which s a t i s f i e s c o n d i t i o n s ( 3 . 8 ) t o (3.10) i s H1 = -A i n L 2 ( R n ) , n 2 1, and X I = y1 = L2"(Bn), s > t . It i s e a s i l y shown t h a t a wide v a r i e t y of o r d i n a r y d i f f e r e n t i a l o p e r a t o r s s a t i s f y t h e c o n d i t i o n i n p a r t i c u l a r , t h e o r d i n a r y d i f f e r e n t i a l p a r t of t h e p u t on H 2 , namely (3.11): operator ( 3 . 7 ) . Thus Theorem 3.2 g i v e s a somewhat s h a r p e r r e s u l t i n t h e s e cases. F u r t h e r r e s u l t s a r e a v a i l a b l e when N1 ( a n d / o r H2) h a s more t h a n one s i n g u l a r p o i n t where l i m i t i n g a b s o r p t i o n d o e s n o t h o l d . The r e s u l t s , of c o u r s e , depend on what hypotheses are made concerning H1 and H2. 54.

A FURTHER RESULT

I n t h i s s e c t i o n we s h a l l make hypotheses of a d i f f e r e n t n a t u r e t h a n were made i n t h e previous section. I n some i n s t a n c e s t h e c r u c i a l h y p o t h e s i s below a p p e a r s t o b e e a s i e r t o v e r i f y t h a n t h e p r e v i o u s ones. For s i m p l i c i t y we s h a l l g i v e a set of hypotheses w i t h o u t g l o b a l u n i f o r m i t y c o n s i d e r a t i o n s , such a s uniform boundedness of t h e l i m i t i n g r e s o l v e n t s on unbounded sets, o r w i t h o u t c o n s i d e r a t i o n s of unbounded s e t s of e i g e n v a l u e s f o r e i t h e r o p e r a t o r HI o r H2. The assumptions we make on HI and H2 a r e a s f o l l o w s :

(4.1)

€I1 and H2 a r e bounded below.

(4.2)

A l i m i t i n g a b s o r p t i o n p r i n c i p l e i s v a l i d f o r H1 and H2 uniformly on comp a c t s e t s i n R+ = (O,-) i n t h e t o p o l o g i e s of B(x1,yT) and B(x2,yZ) resp e c t i v e l y . More p r e c i s e l y t h e r e e x i s t R*(A) C B(xj,y;) so t h a t f o r every I K c c R f and j = 1 , 2

(4.3)

For every C$ F C o ( IR),

m

4(H1)xI 5 x l -

+

Under the previous asswnptions, f o r every A f B , there are operTHEOREM 4 . 1 . ators R + _ ( x )c B ( X , 4 x2,yq 4 y*,) s o t h a t for every K ==R+,

REMARKS. The c o n d i t i o n (4.3) i s e a s i l y e s t a b l i s h e d f o r H = -A i n L 2 ( B n ) , n 2 1, and X I = y1 = L 2 , s ( R n ) , s > 3. Another example which c a n e a s i l y be shown t o f i t t h e p r e v i o u s assumptions i s g i v e n by H1 = Hg = -A + 1/ 1x1 i n L2 ( a n ) , n 2 2 and xj = y . = L 2 , S ( R " ) , s > 3. A l s o t h e hypotheses on Hi can b e shown t o hold f o r a I . wlde v a r l e t y of o r d i n a r y d i f f e r e n t i a l o p e r a t o r s of t h e form -d2/dx2 V(x) i n L 2 ( R ) , where x1 and y1 are t a k e n a s s u i t a b l e weighted spaces.

+

48

M. Ben-Artzi and A. Devinatz

REFERENCES [ I ] Ben-Artzi, M., U n i t a r y e q u i v a l e n c e and s c a t t e r i n g t h e o r y f o r S t a r k - l i k e Hamiltonians, ( t o appear).

[Z] G i n i b r e , J. and Moulin,

M., H i l b e r t s p a c e a p p r o a c h t o quantum mechanical three-body problem, Ann. I n s t . H e n r i P o i n c a r 6 , Sec. A, XXI (1974) 97-145.

[3] H e r b s t , I . W., (1977) 55-70.

U n i t a r y e q u i v a l e n c e of S t a r k H a m i l t o n i a n s . Math. Z.,

155

[ 4 ] Y a j i m a , K . , S p e c t r a l and s c a t t e r i n g t h e o r y f o r S c h r o d i n g e r o p e r a t o r s w i t h S t a r k e f f e c t , J. Fac. S c i . , Univ. Tokyo, Sec. l A , 2 6 (1979) 377-389.

IResearch p a r t i a l l y s u p p o r t e d by NSF G r a n t MCS-8200898.

DIFFERENTIAL EQUATlONS I.W.Knowles and R.T.Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

49

AN ALGEBRAIC GENERALIZATION OF STOCHASTIC INTEGRATION Marc Berger and Alan Sloan School of Mathematics Georgia I n s t i t u t e o f Technology Atlanta, Georgia 30332 U.S.A.

The authors use a generalization of s t o c h a s t i c i n t e g r a t i o n t o analyze real constant c o e f f i c i e n t e l l i p t i c operators of order n which generate strongly continuous semigroups on L2 (IR k ) . Let p n ( x , t ) be the fundamental solution t o u t = Au where

A

=

(-1)

(n/2)-1

c -an n

j = l ax

j

f o r n even. For real polynomials ql, ...,qk on Rm l e t q = (ql,...,qk) and define i n t e g r a l operators F ( t ) , f o r each n ,q t > 0 on functions on lRk by ( Fn ,q ( t) f ) Ix) =

.f

f(x+q ( 2 ))pn ( z , t )dz.

IRm

The authors determine when etQ = lim F ( t / j ) j strongly on 2 k as j a, f o r some q on some R". L (R -f

In (Berger and Sloan generalization of stochast application of t h i s genera c o e f f i c i e n t 1 i near p a r t i a l

1982) and March (1983)) the authors presented a Here, the authors will discuss an i z a t i o n t o the c l a s s i f i c a t i o n of real constant d i f f e r e n t i a l operators, Q , o f order n which generate strongly continuous semigroups on L 2 (IR k ) . T h e s e t of a l l such 4 ' s i s denoted by I ( n , k ) . The simplest example in I ( n , l ) , f o r n even, i s

c integration.

For t h i s Q , t h e solution t o t h e i n i t i a l value problem, u(x,O) corresponding evolution equation ut = Qu may be written as m

u(x,t)

=

I f(x+yhnb,t)dy -03

where

and p n ( y , t ) = t-"npn(yt-'/n).

=

f ( x ) for the

M.Berger and A. Sloan

50

The authors have investigated the following question: I n what sense and t o what extent i s the kernel pn s u f f i c i e n t f o r t h e description of solutions t o t h e i n i t i a l value problem formore general evolution equations? To be more s p e c i f i c , f o r z = (z ,zm) i n W m , l e t p n ( z , t ) = p , ( z l , t ) ... p n ( z m , t ) . For real polym nomials q l , . . . , q k on IR of degree no l a r g e r than n and with no constant term, k l e t q ( z ) = ( q l ( z ) ,...,qk( 2 ) ) . Define an integral operator on functions f on W

,,...

by

1 f(x+q(z))p,(z,t)dz. Rm For which Q i n I ( n , k ) i s t h e r e a q on some Rm such t h a t (Fn,,(t)f)(x)

Let n be even.

=

strongly on L 2 (IRk ) ? Such Q's a r e called pure. Before describing an answer t o t h e question posed above some examples will be presented which i l l u s t r a t e the nature of purity. In one dimension, k = 1 , or f o r second order operators, n = 2 , a l l generators in I ( n , k ) a r e pure. These values, k = 1 o r n = 2 , a r e the only ones f o r which every Q in I ( n , k ) i s pure so t h a t the c l a s s i f i c a t i o n of e l l i p t i c generators i n t o pure and not pure i s a multidimensional , higher order theory. I n p a r t i c u l a r , a4 + A a4 a4 -(2 2+$ ax ax ay i s i n I ( 4 , Z ) f o r -2 5 A b u t i s pure i f and only i f 0 5 A 5 6. While

a2

-(-7 ax

a 2 t 7) a2 2 t a - a3 axayaz

t

ay2 az i s pure f o r a l l real a i t turns o u t t h a t

a4 a4 +a4+ a -(-T ax ay4 t az

a3

+

i s pure only when a = 0.

Q.

Whether o r not a Q i n I ( n , k ) i s pure i s l a r g e l y determined by the symbol o f That i s , l e t Q =

...,a k ) ,

where a = ( a l y aa # 0 f o r some

10.

ai = 0,1,2,

c

O< I a 1 Ln

...,

aaaa

la1 = a1 +

... '

= n and

a'

= (

a,)a1 ax

... (--I a

axk The symbol f o r such a Q i s the polynomial on R k ,

Q ( Y )=

C aaya O 0, R(vE) = S + O ( E ) - c c ,

f o r V(X,Y) = @(x,Y) U,(x f u n c t i o n such t h a t 0 1

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

J

Step 2.

be a m i n i m i z i n g sequence f o r ( 1 3 ) , t h a t i s

Let

(vJ)




'

I*C

I + and Lemma 4 .

3.2 Remark.

=

N(7').

Since A i s a r b i t r a r y from t h e i m p l i c a t i o n s

S i m i l a r a r g u m e n t s w i l l show

f i r s t t h a t l : ' * = 1 and t h e n t h a t lA* = I

0

Proposition 1 implies t h a t 7 ,

densely defined o p e r a t o r s .

N(IA*)

.

IO’

I + and I:

are closed

Lemma 1, P r o p o s i t i o n 1, and t h e c l o s e d

r a n g e t h e o r e m [ 8 , T h e o r e m 5 . 1 3 , ~ . 2341 i m p l y t h a t t h e r a n g e s of l o , I + , and 1; a r e c l o s e d . Moreover s i n c e I and I a r e 1 - 1, 7' and 0

1: a r e o n t o .

Lemma 5 .

7+lo

and

1il

are s e l f - a d j o i n t

domains a r e c o r e s r e s p e c t i v e l y

and D

T

=

of

1,

of

extensions To. 1, F u r t h e r D

and

Their T = Do

D.

P r o o f . A t h e o r e m o f Von Neumann [8,Theorem 3 . 2 4 , ~ . 2 7 5 1 combined w l t h t h e Second R e p r e s e n t a t i o n Theorem [8,Theorem 2 . 2 3 , ~ . 3311. 3 . 3 Remark.

I+lo, l + I are 1 - 1 positive operators. 0

Therefore t h e i r

r a n g e s a r e c l o s e d and o n t o . 3.4 Remark.

I n c o n c r e t e t e r m s Lemma 5 s a y s t h a t t h e r e s t r i c t i o n s o f ( n - i ) I 2 < a, s a t i s f y i n g e i t h e r t h e boundary b n T s u c h t h a t Ja i $ o pi Iy conditions

R.C. Brown

66 (1) y ( i ) ( a )

= 0,

i =

D(y,y)(b-) = 0 ,

o ,...,n -

YZ E

1,

D(T) fl D ,

or ( 2 ) yLi1(a

= 0,

Dfu,7l b - ) are self-adjoint

4.

=

i = n, 0,

VLI

...,2n - 1 , E

D

.

APPLICATIONS TO DIRICHLET INEQUALITIES

Theorem 1. Let E be bounded below and A 2 h o l d f o r a o s i t i v e T I , i = 1 , . .&n d e f i n i t e t r a n s l a t i o n of t. Let 0 < Bi all y E U

.

where u > - m restriction T

andi s-t he least of 7'1

for

--

element in t h e spectrum of t h e

d e f i n e d by t h e boundary c o n d i t i o n s

..

i = 1,. , n , D[y,f] (b-) = 0 , s i n B . f pji-11 a Y 1 E q u a l i t y h o l d s f o r $ & D i f a n d o n l y if i s an v y E 21. f u n c t i o n of T B c o r r e s p o n d i n g 5 u 4 . O t h e r w i s e t h e r e i s sequence o f f u n c t i o n s & D s u c h t h a t -

==

Proof.

:-?-I

WLOG assume A 1 h o l d s .

Case 1.

A l l t h e Bi

+

a

=

5.

(4.1) and

t h e p r o p e r t i e s o f T ( i n c l u d i n g t h e f a c t t h a t u > - m ) a r e immediate B B c o n s e q u e n c e s of t h e F i r s t R e p r e s e n t a t i o n Theorem ( 4 . 2 ) f o l l o w s from [8,Theorems 2 . 1 o r 2 . 6 , ~ . 322-3233 and Lemma 5 . standard theory. 2. The form on t h e l e f t s i d e o f ( 4 . 1 ) i s c l o s e d and bounded below by A 2 , t h e i n e q u a l i t y o f 2 . 3 , example 4 , s e c t i o n 2 , and 18,Theorem 1 . 3 3 , ~ . 3201. L e t S be t h e s e l f - a d j o i n t o p e r a t o r guaranteed by t h e r e p r e s e n t a t i o n theorem. C a l c u l a t i o n shows t h a t t [ y , z ] = [T y , z J f o r a l l z E D. By [S,Theorem 2 . 1 , B T i s symmetric; a n d i f T B i s s e l f - a d j o i n t , iii., p . 3 2 2 1 , T B C S.

Case

B

2. I f n o t , we t r a n s l a t e t h e form and p r o v e t h e t h e o r e m i n a p o s i t i v e d e f i n i t e s e t t i n g . A s p e c t r a l mapping argument i s t h e n u s e d t o recover the "untranslated" inequality. For d e t a i l s see the argument i n [4,Theorem 2 1 .

A Factorization Method and Dirichlet Inequalities

61

T = S. Let S’ be the n dimensional extension of 7 A l with no B boundary conditions imposed at a. S’ is closed and S ' * is the n dimensional restriction of 7'1 having all zero boundary conditions 0 at a. So dimension (D(S’)/D(S’*)) = 2n;and T is an n dimensional B Therefore T is self-adjoint. The symmetric extension of S ' * . remainder of the argument follows case 1. 4.1 Remarks. 1. We note that T B is the Friedrichs extension of To representing the form in (4.1); in particular 1'1 are the Friedrichs and 1’7 0 extensions representing the forms 1 Ioyll2 and ~[y;). For n = 1, the boundary conditions at a of Ta are the most 2. general possible for self-adjoint extensions. F o r n > 1 such boundary conditions are described in terms of a hermitian matrix Q [ 7 , p . 2921. In this setting the cotangent terms in (4.1) become (n-1) t -(n-1) 1 (a). [ Y , ,Y 1 (a)Q [Y ,Y

--

---

3. Since minimal conditions, more general boundary conditions, no Dirichlet index hypothesis, and larger pi are assumed, Theorem 1 is more general than the inequalities established in [l].

Inequalities -for quasi-differential operators. The technique developed above works for quite general differential expressions provided the quadratic form is closed and bounded below. As aE illustration, we derive the simplest forms of Dirichlet inequalities for the ShinZettl quasi-differential expression M[y]:

=

[-sly’ +

We assume that a-’ 1

(ao

+

+

(i bo

alc

, ao, b o ,

2

IY

+

'

a,c)y]

+

(i bo

- alc)y’

.

1 c E LLoc(I) and are nonnegative.

As before T, T0 signify the maximal and minimal operators determined by M. (See [6] or [141 for accounts of the theory of symmetric quasi-differential operators.) We define the sesquilinear form t[u,vl: -

=

I,"

alu’v’

+

(ao

+ (ibo - alc)u

v'

+

alc2 )uv

- (ibo + alc)urv.

68

R.C Brown

Thus

+ 2 bo T E I ( Y

-

Lemma 6.

Let

holds where $ :

6 > 0 , e suppose

al

=

7’) -

bt

+

(alc)’.-

t

the

2 alcRc(yy’).

inequality

i s c l o s e d and bounded below.

P r o o f . The c o n c l u s i o n f o l l o w s from [8,Theorem 1 . 3 3 , ~ . 32C] and some routine integral estimates.

D[y,y] ( s ) : = y ( s ) [-a,:’ Again WLOG assume

+

( i bo

+

a,c)y] ( s ) .

t o be p o s i t i v e d e f i n i t e .

f,

I:,

Io, T

+

, To,+

+: fo a r e

d e f i n e d a s b e f o r e , b u t w i t h r e f e r e n c e t o t h e new c h o i c e s o f t, D[Y,F] ( s ) , and M . The same a r g u m e n t s m u t a t i s m u t a n d i s e s t a b l i s h t h e r e s u l t s o f s e c t i o n s 3 a n d 4 i n t h i s s e t t i n g . We a r r i v e a t t h e inequalities

uo b e i n g t h e i n f . o f t h e s p e c t r u m o f I;T

5.

(T+To)

i f y E i?(Uo).

THE D I R I C H L E T INDEX

I n t h i s s e c t i o n we s k e t c h how t h e D i r i c h l e t i n d e x t h e o r y o f R.M. Kauffman c a n be e x t e n d e d t o t h e n e g a t i v e c o e f f i c i e n t c a s e . We However t h e r e a d e r w i l l n o t f i n d i t d i f f i omit n e a r l y a l l p r o o f s . c u l t t o r e c o n s t r u c t them from [ 2 ] o r 131 s i n c e o u r l o g i c f o l l o w s t h e same p a t t e r n a s t h e e a r l i e r p o s i t i v e c o e f f i c i e n t t r e a t m e n t . D e f i n i t i o n 1.

z

€ D(T)

i s D i r i c h l e t (D) i f and o n l y i f z E H.

Definition 2. The D i r i c h l e t Index (DI) of M _ i s -t h e d i m e n s i o n o f t h e s ~ o fn t h e D s o l u t i o n s of T.

A Factorization Method and Dirichlet Inequalities

The

D e f i n i t i o n 3. (D ( T I / D ( T o ) 1 * Lemma 7 .

D e f i c i e n c y I n d e x (DFI)

D I = codimension R ( I )

1.

d i m e n s i o n ( D ( 1 ' 1 ) /D( I Z 0 )

=

0

of

M

69

1 / 2 dimension

d i m e n s i o n (Q/Q ) = 0

I t i s e a s y t o e s t a b l i s h t h a t 2n DI 2 n . I n g e n e r a l DFI - D I i s t h e d i m e n s i o n o f t h e s p a n o f t h e non-D s o l u t i o n s o f T . Thus DI = DFI T i s D. Following t h e terminology f o r o r d i n a r y d i f f e r e n t i a l o p e r a t o r s , w e c a l l I " l i m i t p o i n t " when DI = n . Hence I i s l i m i t p o i n t I 0

i s a n n d i m e n s i o n a l r e s t r i c t i o n o f I ; i . e . , D ( y , F ) ( b - ) = 0 , v y E: Q and z E D(T) r ) Q. I t c a n a l s o b e shown t h a t t h e i n d e x i s minimal D(T1) i s a c o r e o f t h e r e s t r i c t i o n o f Q s u c h t h a t 0

y ( i ) ( a ) = 0, i =

o ,...,n -

1.

The r e l e v a n t c o n j e c t u r e i s t h a t t h e i n d e x i s a l m o s t a l w a y s n, a t l e a s t when w = 1.

...,PA

a r e r e a l m e a s u r a b l e f u n c t i o n s on I s u c h t h a t 1 P 0 + P I , > 0, ( P o + P ; ) - l , P 1 + P ; , . . * , p , + p ; c L ~ o c ( I ) ' Define Suppose pA,p;,

t

E'

M, ?, 6 , ?, ?',

Define

< E).

{pi + p i 1 and Theorem 3.

Suppose

6

t1 Q.

t h e r e e x i s t s

/I

S i n c e (17 ( a i +

Then T and 7

(ai- a)i2 L

GA,

a , a. E:

+

2

/I I( a i - a111 2

L € 1 1 1 ~ ~ - a(Iw .

-E

1

2

h a v e t h e same DI.

1

E

and ? ( a i ) 2

/I a i - a /lW

+

fi

?(a) in

.

If Now

.

We c o n c l u d e from ( 5 . 1 ) t h a t

I ( a ) i n H , and s o a € Do. and 7 ( a . ) + I ( a ) i n H , t h e n I(ai)

1.

I t i s s u f f i c i e n t t o c h e c k t h a t Do = ,D . o.

a E

15.1)

(so that L1[y]

bounded.

Clearly

co,

+

r1 .

Proof.

=

t’

))yll, f o r ?0 w i t h r e s p e c t t o t h e c o e f f i c i e n t s e t e t c . , r e l a t i v e t o t h e form +

Assume t h a t A l , A 2 h o l d f o r

O n t h e o t h e r hand i f a .

1

-f

a in Do

s a t i s f y t h e h y p o t h e s e s o f [8,Theorems 1 . 3 1 , 1 . 3 3 , e . g . , t, I' 3. p. 319,320,l.

R. C Brown

70

REFERENCES

.

On the minimiBradley J.S., Hinton, D.B., and Kauffman R )I toy. SOC. zation -f singular quadratic functionals, ’roc Edinburg, 87A (1981) 193-208. Brown, R . C . , An approach to the Dirichlet index for operators satisfying minimal conditions, in: Everitt, W.N. and Sleeman, B.D. (eds.), Ordinary and Partial Differential Equations (Lecture Notes in Mathematics 964; Springer-Verlag, Berlin, Heidelberg and New Y o r k , 1982) 110-117. Brown, R . C . , The Dirichlet index under minimal conditions, submitted Brown, R . C . , A von Neumann factorization of some selfadjoint extensions of positive symmetric differential operators and its applications to inequalities, to appear in the Proceedings of the 1982 Dundee Symposium on Differential Equations. Brown, R . C . and Hinton, D.B., Sufficient conditions f o r weighted inequalities of sum and product form, in preparation. Everitt, W.N. and Zettl, A., Generalized symmetric ordinary differential expressions 1: The general theory, Nieuw Archief voor Wiskunde (3) 27 ( 1979) 363-397. Hinton, D.B., On the eigenfunction expansions of singular ordinary differential equations, J. Differential Equations 24 (1977) 282-308. Kato, T., Perturbation Theory f o r Linear Operators, (SpringerVerlag, New York, 1966). Kauffman, R.M., The number of Dirichlet solutions to a class of linear ordinary differential equations, J. Differential Equations 31 (1979) 117-129. [lo] Kauffman, R.M., Read, T.T., and Zettl, A., The Deficiency Index Problem for Powers of Ordinary Differential Expressions (Lecture Notes in Mathematics 621; Springer-Verlag, Berlin, Heidelberg and New Y o r k , 1977). [ll] Kwong, M.K. and Zettl, A., Weighted norm inequalities of sum form involving derivatives, P r o c . Roy. SOC. Edinburg, 88A (1981) 1 2 1 - 1 3 4 . [12] Kwong, M.K. and Zettl, A., Norm inequalities of product form in weighted Lp spaces, Proc. Roy. SOC. Edinburg, 89A (1981)293-307. [13] Naimark, M.A., Linear Differential Operators, Part 11, (Ungar, New York, 1968). [14] Zettl, A., Formally self-adjoint quasi-differential operators, Rocky Mountain J. of Math. 5 (1975) 453-474.

.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

71

SOLUTIONS WITH ASYMPTOTIC CONDITIONS OF A NONLINEAR BOUNDARY VALUE PROBLEM Nguygn Phuong Ca'c Department o f Ma thema t i c s The U n i v e r s i t y o f Iowa Iowa City, Iowa 52242 U.S.A.

INTRODUCTION L e t D be a bounded domain i n RN ( N 2 2 ) and l e t n = R N - E where E i s t h e e a r e conc l o s u r e o f D. We assume t h a t t h e boundary an o f n i s smooth. W cerned w i t h t h e boundary v a l u e problem ( a b b r e v i a t e d t o BVP h e r e a f t e r ) Au = p ( x , u , v u ) + f

in

n, u

= 0

on

an

(11

where A i s a n o n l i n e a r e l l i p t i c d i f f e r e n t i a l o p e r a t o r i n d i v e r g e n c e f o r m o f L e r a y - L i o n s t y p e , vu = g r a d u , f i s a d i s t r i b u t i o n on n. We s h a l l assume t h a t 1 the function p(x,t,q) d e f i n e d on n x R x R N has L -growth i n x and unres t r i c t e d growth i n t h e second v a r i a b l e t. I t w i l l be proved t h a t t h e BVP ( 1 ) i s s o l v a b l e i f i t has an upper s o l u t i o n Ji and a l o w e r s o l u t i o n rp w i t h rp i Ji and

-

c p , ~E ~ ~ ( nn ~) q ( n )

for some

E (I,-).

S i m i l a r problems a r e c o n s i d e r e d , among o t h e r s , by P. Hess i n [3],[4] and t h e a u t h o r i n [1],[2]. In [3] t h e growths o f t h e f u n c t i o n p a r e d i f f e r e n t from t h o s e assumed h e r e and i n [4] t h e domain i s bounded. Our r e s u l t seems t o p r o v i d e t h e answer t o a q u e s t i o n r a i s e d i n a remark a t t h e end o f [ 4 ] as t o whether i t s r e s u l t f o r bounded domains c o u l d be extended t o unbounded ones. In [l]t h e c o n d i t i o n s imposed on t h e upper and l o w e r s o l u t i o n s JI and cp a r e weaker t h a n t h o s e assumed i n t h i s paper: namely, i t i s assumed i n [I]t h a t JI and cp have o n l y l o c a l p r o p e r t i e s . Then we have t o r e s t r i c t o u r s e l v e s t o l i n e a r o p e r a t o r s A and t h e s o l u t i o n o b t a i n e d i s a l s o l o c a l i n n a t u r e . Furthermore, i t seems t o US t h a t t h e method o f [l]cannot be adapted t o n o n l i n e a r o p e r a t o r s . While [2] concerns m a i n l y w i t h t h e s o l v a b i l i t y o f t h e BVP (1) i n w e i g h t e d Sobolev's spaces u s i n g t h e r e s u l t o f [l] ( t h u s t h e e l l i p t i c o p e r a t o r s c o n s i d e r e d i n [2] a r e l i n e a r ) , b y t a k i n g t h e w e i g h t s equal t o 1, we have a l r e a d y o b t a i n e d i n [ Z ] ( c f . i t s Theorem 2 ) f o r l i n e a r o p e r a t o r s a r e s u l t s i m i l a r t o o u r Theorem I below. The method o f upper and l o w e r s o l u t i o n s i s c o n c e p t u a l l y s i m p l e and p a r t i c u l a r l y u s e f u l i n p r o v i n g t h e e x i s t e n c e o f a s o l u t i o n f o r n o n c o e r c i v e and p o s s i b l y s t r o n g l y n o n l i n e a r BVPs. However, i n r e a l i t y i t i s s e v e r e l y l i m i t e d by t h e d i f f i c u l t y encountered i n c o n s t r u c t i n g an upper s o l u t i o n Ji and a l o w e r s o l u t i o n cp w i t h cp i 4. T h i s d i f f i c u l t y i s g e n u i n e l y n o n t r i v i a l i f t h e domain i s unbounded, t h e o p e r a t o r i s n o n l i n e a r and we want, as i n o u r Theorem I below, t h e upper and l o w e r s o l u t i o n s t o be s i m u l t a n e o u s l y bounded and t o b e l o n g t o some space u

W''q(n) n L q ( n ) ( 1 c q , T < m) because c o n s t a n t s cannot t h e n serve as upper and 1oc l o w e r s o l u t i o n s . T h e r e f o r e , f o r i l l u s t r a t i v e purposes, we s h a l l g i v e an example f o r w h i c h o u r Theorem I a p p l i e s . We s h a l l e x p l i c i t l y c o n s t r u c t upper (and l o w e r ) s o l u t i o n s by " g l u i n g " t o g e t h e r upper (and l o w e r ) s o l u t i o n s on subsets of n. THE RESULT

Let

N.P. G i c

72

0. =

AU = - O i [ A i ( ~ , ~ , v ~ ) ] ,

1

-,a axi

w i t h t h e c o n v e n t i o n t h a t i f t h e i n d e x i i s r e p e a t e d t h e n summation o v e r t h a t i n d e x f r o m 1 t o N i s i m p l i e d . Throughout t h e paper we assume: (Hl)

For each

i = 1 ,.

.. ,N,

f o r each

(t,q)

E R x RN the function

A i : n x R x R N 3 R i s o f Caratheodory's t y p e , i.e.,

f o r a l m o s t a l l (a.a.) uous. tion

x E

(q* = &). k o ( x )

E Lq*(n)

f o r a.a.

ko(x)+co([tlq-l+

v (t,q)

n,

x E

S-

Ai (x,t,q)qi x E

n, b'

2

i s measurable and

( t , q ) -+ Ai(x,t,q)

0

a.a.

Iqlq-'),

-,

x E

c

0

i s contin2

0

and a f u n c -

n such t h a t

i = l , .,N, .-

E R x R ~ ;

(H3) f o r a.a.

+ Ai(x,t,q) q, 1 < q
0.

N + R i s o f C a r a t h e o d o r y ' s t y p e and t h e r e e x i s t s a c o n s t a n t E , 0 < E s q, a f u n c t i o n k l ( . ) E L1(n), x E R and a c o n t i n u o u s f u n c t i o n c ( - ) : [0,-)-+ [O,m) kl(x) 2 0 a.a.

The f u n c t i o n

p(x,t,q)

:R x R x R

such t h a t Ip(x,t,q) f o r a.a.

If

f E

E n

x

MiA;q*(n)

v (t,q)

for all where) i n

c(P)[kl(x)

E R x R~

then a f u n c t i o n

l o c a l sense o f t h e BVP ( 1 ) i f rp

JnAi

I

(x,rp,vcp)Oi

5

cp

0

v dx

with

E W;i:(Q) on

an,

+

it1

Irllq-El 5 p.

i s c a l l e d a lower s o l u t i o n i n the 1 E Lloc(n) and

p(x,rp,vcp)

. ~ ~ ( x , c p , v c p )v d x + (f,v)

v E WAyq(n) fl Lm(n) o f compact s u p p o r t and v 2 0 a.e. n, where (.,.) denotes t h e p a i r i n g between WA"(n)

(= almost e v e r y and i t s dual

w-1 * q * ( R ) . An upper s o l u t i o n i n t h e l o c a l sense i s d e f i n e d by r e v e r s i n g t h e i n e q u a l i t y s i g n i n t h e above d e f i n i t i o n . THEOREM I .

Let

f E W-lYq*(n).

and a l o w e r s o l u t i o n

cp

(T

cp i 0 s

= min(q,:))

1 u E Wo'q(R)

with

Suppose t h a t t h e BVP ( 1 ) has an upper s o l u t i o n

Lm(R) n Lq(cp) 0. Then t h e BVP ( 1 ) has s o l u t i o n w E WAyq(R) fl Lm(R) fl Lq"(R)

i n t h e l o c a l sense, b o t h b e l o n g i n g

4

on

a.e. i n t h e sense t h a t f o r e v e r y

jl

A Nonlinear Boundary Value Problem

the function

&

p(x,u,vu)w

73

i n t e g r a b l e and we have:

JL

A.(x,u,vu)D~ w dx =

n

p(x,u,vu)

w d x + (f,w).

PROOF. We s k e t c h t h e proof. I t i s hoped t h a t a d e t a i l e d p r o o f w i l l appear somewhere e l s e i n a more g e n e r a l c o n t e x t . For a number r > 0 l e t

6, We f i x

no > 0

R

= {x E

such t h a t

N

nr

: 1x1 < r ] ,

ac B

n n.

= B~

n > no

For each i n t e g e r

c o n s i d e r t h e BVP

“O’

Au = p ( x , u , v u ) + f By t h e r e s u l t of [ 4 ]

nn:

F o r each

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

n Lm(nn)

v E Wb’q(fln)

Ai (x,u,,vun)Di We deduce t h a t

in

llunll ,,q

u = 0

fin,

u E W:’q(62n)

ann.

on with

cp s un i li,

a.e.

in

we have

v dx =

i

wo CnJ

p(x,un.vun) v d x + ( f , v ) nn h e r e and i n t h e sequel Ki ( i = l , 2 , . . - )

K1,

de-

notes a p o s i t i v e c o n s t a n t independent o f n, n o t n e c e s s a r i l y always t h e same. We e x t e n d un t o t h e whole domain R b y d e f i n i n g u n ( x ) = 0 when x E f l - n n . We can e x t r a c t f r o m

{u,]

We s h a l l show t h a t

a subsequence, s t i l l denoted by [un]

converges weakly i n

[u,]

converges a.e.

u

and

rn > n

Vx E

0

a function

n, c,(x)

l v c m ( . ) \ i s bounded on

m > no

WAyq(0) t o

h2

=

1

for

such t h a t

u,

u.

For t h a t purpose, we con1

& ( - ) E C (n) w i t h t h e f o l l o w i n g x E

nm, crn(x)

=

by a c o n s t a n t independent o f

o m.

for

x

4

F o r any i n t e g e r

we can show t h a t lim p(x,un.vu,)&,(un-u)dx n-- R

Taking

to

i s a s o l u t i o n o f t h e BVP ( 1 ) .

s t r u c t f o r each i n t e g e r properties: ~ , ( x ) E [O,1]

on

tun],

v = &(un-u)

= 0

l i m (f.c,(un-u)) = 0. n-= i n ( 2 ) we t h e n o b t a i n lim

n--

n

A . ( ~ , u ~ , v ~ , ) ~ ~ D ~ ( u ~ - u )= d0. x 1

It f o l l o w s from t h i s equation t h a t lim [Ai(x,un,vun) - Ai(~,un,vu)]Di(un-u)dx = 0. nR We deduce f r o m t h i s ( c f . e.g. [5], P r o o f o f Lemma 2.2, page 184) t h a t we can exsuch t h a t [vu,] cont r a c t f r o m r u n ] a subsequence, s t i l l denoted by {u,],

verges a.e.

to

vu

on

nm.

Since t h i s i s t r u e f o r any i n t e g e r

d i a g o n a l process, we see t h a t we can e x t r a c t from n o t e d by

{u,],

i = 1 , ... , N ,

such t h a t

{vu,]

converges a.e.

rn > no,

using a

r u n ] a subsequence, s t i l l det o v u on n. Then f o r each

74

N.P. Cbc {Ai(x,un.vun)~

A . (x,u,vu)

converges weakly t o

w E Lq"(0)

We n e x t show t h a t f o r e v e r y lim

nu-

n

in

1

n WA'q(n)

r l Lm(n)

Lq*(n)

we have

w dx.

p(x,un,vun) w dx = .fp(x,u,vu)

(3)

I n f a c t , l e t E’ > 0 be a r b i t r a r i l y g i v e n . B e a r i n g i n mind t h a t see t h a t an i n t e g e r m > no can be chosen such t h a t

Furthermore we can f i n d

6

> 0 such t h a t i f

Eo

By E g o r o f f ' s theorem, t h e r e e x i s t s a subset

nm-Eo,

t h a t on

cp(x,un,vun)]

E cRm,

of

cp

mes E < 6

nm

converges u n i f o r m l y t o

with

5

un

i f

we

then

mes E < 6

0

p(x,u,vu).

such

Then i t i s n o t

d i f f i c u l t t o see t h a t ( 3 ) i s t r u e . Finally, i f i n (2) w i t h

n a m+l

m*we s h a l l see t h a t theorem.

u

we t a k e

v =

5mw and t h e n l e t n and t h e n

i s a s o l u t i o n o f t h e BVP ( 1 ) i n t h e sense o f t h e

AN EXAMPLE

As an a p p l i c a t i o n o f Theorem I we prove THEOREM 11. E R;

d : R+

Suppose t h a t

vt

Ip(x,t,ql)[

where

2 s q < -, 0


-Di[lDi~1q-2Di~]+d(u)

n

let

p

k(-) E

Lm(n),

Then the s t r o n g l y n o n l i n e a r in

n,

(4) (5)

I, i t s u f f i c e s t o c o n s t r u c t an 0 5 $ a.e. on n;

tp i

a(t)

2

At V t > 0

and

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

-Di[lDif,lq-2Di$lI+ 1x1 > p .

N.

and

F o r t h a t purpose we f i r s t f i x a number

$ , ( X I = M ~ ~ l x l - Since ~ .

can be shown t h a t i f

for

P,

an

on

of t h e BVP ( 4 ) , ( 5 ) .

rp,$ E LOD(n) L'(I1) M > 1 such t h a t

x # 0

5

A >0

u E W:¶~(Q).

1 PROOF ( s k e t c h ) . Since k ( . ) E L ( n ) , by Theorem upper s o l u t i o n $ and a l o w e r s o l u t i o n cp w i t h

For

d ( 0 ) = 0, d ' ( t ) z

x E 0

= p(x,u,vu)

u = 0

has 5 s o l u t i o n

Vltl

k(x)+c~p)1rllq-'

s q-1,

1x1 +

R

d(Jil)

2

p(x,$l,v$l)

We s e t

$ ( x ) = $l(x)

if 1x1 2 p ,

$(x) = M

if 1x1

p.

0
0 on n, $ ( - ) E Lm(n) n Lq(,). To prove t h a t JI i s an upper with s o lu t i o n of t h e BVP ( 4 ) , ( 5 ) i t s u f f i c e s t o show t h a t f o r each v E C;(n) v 2 0 we have

Then c l e a r l y

4’D i t l q - * D i t D i v d x + S d ( J I ) R

v dx

2

n p(x,$,v$)vdx.

This i s done by d i r e c t computation bearing i n mind t h a t the f i r s t i n t e g r a l on the l e f t hand s i d e can be wr i t t en as t h e sum of an inte gra l on R-C2 and a boundary i n t e g r a l on

aB . P

P

A negative lower s o l u t i o n

rp

can be constructed s i m i l a r l y .

REFERENCES Ca'c, Nguy& P., Nonlinear e l l i p t i c boundary value problems f o r unbounded domains, J . D i f f e r e n t i a l Equations 45(1982) 191-198. Ca'c, NguyGn P . , On some q u as i l i n ear e l l i p t i c boundary value problems with conditions a t i n f i n i t y . To appear i n J . D iffe re ntia l Equations. Hess, P . , Nonlinear e l l i p t i c problems i n unbounded domains. Inte rna tiona l Summer School on Nonlinear Operators, B erlin, Sept. 1975, i n "Abhandlung der Akademie der Wissenschafter der DDR." Hess, P . , A second o r d er nonlinear e l l i p t i c boundary value problem, i n "Nonlinear Analysis: A C o l l ect i o n of Papers i n Honor of Erich H. Rothe," Academic Pr e s s , New York, 1978. Lions, J.L., Quelques me'thodes de r 6 s o l u t ion des problemes aux l i m i t e s non l i n g a i r e s , Dunod, Gau t h i er - Vi l l ar s , P a r i s , 1969.

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

77

ABSOLUTE CONTINUOUS SPECTRUM OF ONE-DIMENSIONAL SCHRUDINGER OPERATORS Ren6 Carmona D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of C a l i f o r n i a a t I r v i n e I r v i n e , C a 92717

We 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 t h e a s y m p t o t i c behavior of t h e p o t e n t i a l f u n c t i o n q ( t ) f o r t g o i n g t o o n e of t h e i n f i n i t y f o r t h e s p e c t r u m of

--d2 + q ( t ) t o be p u r e l y a b s o l u t e l y c o n t i n u o u s i n dt some s u b s e t s o f IR .

I.

INTRODUCTION I n what follows,

q(t)

w i l l a l w a y s be a l o c a l l y s q u a r e i n -

t e g r a b l e r e a l v a l u e d f u n c t i o n which s a t i s f i e s :

f o r some

a > 0 , and w e i n v e s t i g a t e t h e s p e c t r a l p r o p e r t i e s of t h e

unique s e l f - a d j o i n t

If

q(t)

extension, say

H , of t h e o p e r a t o r d e f i n e d f o r

i s random, w e know i n some cases a n d w e b e l i e v e i n

g e n e r a l , t h a t i f i t i s random enough t h e s p e c t r u m w i l l b e a l m o s t s u r e l y p u r e p o i n t ( S e e [7] and [ 2 ]

f o r example).

On t h e o t h e r h a n d ,

w e know a l s o t h a t some c o n t r i b u t i o n s t o t h e p o t e n t i a l c a n a n i h i l a t e t h e t e n d a n c y t o l o c a l i z e and c r e a t e p u r e p o i n t s p e c t r u m d u e t o t h e randomness.

F o r e x a m p l e ( s e e [ S ] and m a i n l y [l])

t u r n i n g on a

c o n s t a n t e l e c t r i c f i e l d g i v e s i m m e d i a t e l y a p u r e l y a b s o l u t e l y cont i n u o u s s p e c t r u m e q u a l t o t h e w h o l e r e a l l i n e , i n d e p e n d e n t l y of t h e

R. Cannon0

I8

magnitude of the disorder parameters.

This is screeming for a

better understanding of what can force an absolutely continuous spectrum - Another motivation can be borrowed from the study of models like the following.

Lf.

n ,where ql(t)

E(t

w ) = F(Xt(w))

is a continuous periodic function and with

{Xt]telR

mot on on the unit circ e from

C

S1

H(w) = -

of

S1

.

[-e, fc]

onto

Let:

and

a stationary process ofBrownian

F

a smooth Morse function

Almost surely in

ci E

n the spectrum

d2

7+

q(t,w) is the set U[ai - e,b. + e ] where dt i is the spectrum of the (non random) operator

Cl = U[ai,bi] i H1 = d22 + ql(t) , C dt pure point in C\Cl

C1

is purely absolutely continuous in

~

.

and

A complete study of this example (and some

other ones) can be found in [ S )

.

Let us try to argue i t , at least at a heuristic leve1,by considering the asymptotics of the solutions of the eigenvalue equation : ( t , ) + q(t)y(t)

-y"

If

h e C\C,

, A

Schradinger operator

=

hy(t)

.

(E . V .E)

is in a gap of the spectrum of the periodic H1

so Floquet’s theory tells us that one of

the solutions of (E.V.E.)

decays exponentially as

t

-+

all the linearly independent ones explode exponentially. other hand, since

w

t

-+

m

while

On the

h e C , the Ljapunov exponent of ( E . V . E )

strictlypositivefor dichotomy.

-

is

and we may expect the same exponential

Moreover, since the noise process

ie(t);t)O]

is

random enough we may expect to be able to patch these random exponentially decaying solutions to the exponentially decaying

Spectrum of Schrijdinger Operators solution near A

4

.

T h i s c a n b e d o n e as i n [ Z ]

f o r a d e n s e s e t of

a n d t h i s t a k e s care of t h e claim o n t h e p u r e p o i n t s p e c -

C\Cl

trum.

-m

79

For

A

E

Xl

t h e random s o l u t i o n s on

are s t i l l f o r m i n g

IR,

t h e same e x p o n e n t i a l d i c h o t o m y w h i l e now, o n

a l l the

(-=,O]

s o l u t i o n s are bounded and ( a g a i n by F l o q u e t t h e o r y ) bounded u n i formly i n t h e i n i t i a l c o n d i t i o n a t t h e o r i g i n and c o m p a c t s u b s e t s of t h e i n t e r i o r o f

Cl

.

restricted t o

A

W e will see i n t h e f o l -

l o w i n g s e c t i o n t h a t t h i s is e n o u g h t o i n s u r e t h e p u r e a b s o l u t e Z1

c o n t i n u i t y o f t h e s p e c t r u m on

for

2

t

(E.V.E)

0

, i n d e p e n d e n t l y of w h a t

q(t)

is

and c o n s e q u e n t l y t h e b e h a v i o r of the s o l u t i o n s of

for

t

2

.

0

To b e c o m p l e t e w e m u s t s a y t h a t t h e a b s o l u t e c o n t i n u i t y of t h e s p e c t r u m of S c h r a d i n g e r o p e r a t o r s h a v i n g d i f f e r e n t a s y m p t o t i c s when

t

+ --m

and

t

-B

+-

h a s a l r e a d y b e i n g i n v e s t i g a t e d i n [5]

b u t i t seems t h a t o u r r e s u l t s a r e b e t t e r e v e n t h o u g h o u r m e t h o d i s

less g e n e r a l s i n c e r e s t r i c t e d t o t h e o n e d i m e n s i o n a l case.

11.

THE M A I N RESULT: I n o r d e r t o i n v e s t i g a t e t h e s p e c t r a l p r o p e r t i e s of

u s e t h e O.D.E.

(ordinary d i f f e r e n t i a l equation)

plained i n [4]

f o r example.

that

--

< a < 0 < b < +

w

Then w e c o n s i d e r t h e o p e r a t o r

HLY” [a,bl

LY

by :

0 =

a

and

and

p

in

b

such

[O,rr)

d e f i n e d on i t s domain:

Be” = i f E L 2 ( [ a , b ) , d t ) ; f is C 1 , f‘ a,b l u t e l y continuous, -f“+ qfsL2([a,b],dt)

f(a)cosCu+ f ‘ ( a ) s i n

we

a p p r o a c h a s ex-

F i r s t we p i c k reals and t w o a n g l e s

H

is a b s o and

f(b)cosp+ f‘(b)sinp=

01

.

R. Carmona

80 whenever

f

..

ho < h l
l / n , and

0'

E

lmSy?(x)p"(x>

Now l e t s n ( x ) b e an approximation t o t h e d e l t a f u n c t i o n 6 ( x ) i n

6" E C"

S:~(X)

(t)

f o r s u i t a b l e f , g of compact s u p p o r t where RQ = R E W’ can

Q

+

be w r i t t e n f o r m a l l y a s Ro

RZP. A

A

3 . BASIC FORMULAS I N THE Q THEORY. W e w i l l e x p r e s s t h e Q s p e c t r a l p a i r ing (

(

RQ,FG)X by

F,G)\, =

(

(

F,G)u.

Now l e t

?

and

6 be

of t y p e ( A ) o r (B) f o r example ( w i t h

4 P R ,FG)) and t h e fundamental t r a n s m u t a t i o n B: P

n -f

Q with

P

BpA = p y

can b e

o b t a i n e d a s Bf(y) = p ( 0 , y ) where IP i s t h e unique s o l u t i o n of t h e Cauchy problem n

P ( D )p = Q(D

Y

) q , p(x,O)

=

f ( x ) (extended t o

(-m,m)

as an even f u n c t i o n ) , and

Some Topics in Tvatisinutation

ip

18;19;53;551).

(x,O) = 0 ( c f .

Y

91

On t h e o t h e r hand i n [ 4 9 1 t h e t h e o r y of g e n e r a l A

A

U i z e d t r a n s l a t i o n s r e l a t e d t o a d i f f e r e n t i a l o p e r a t o r Q i s b a s e d on s o l v i n g Q(D ) X

=

$(D

Y

2

)U f o r y

0,

-m

< x
h

x under normal h y p o t h e s e s on Q. -

"

A

B v i a B.

2

Now t h e i d e a of t h e M e q u a t i o n i s t o r e l a t e B and

o u r development of t h i s m a t e r i a l i n [ 191 w a s based on [ 421

When P = D

Here w e can e s t a b l i s h analogous machinery f l 2 F i r s t following [48;71] f o r f o r P and the r e s u l t r e d u c e s t o [ 1 0 ; 1 9 ] when P = D

u s i n g t h e F o u r i e r t r a n s f o r m on

(-m,m).

.

6

iz

suitable f , @(f) =

P f ( x ) Q A ( x ) d x can be i n v e r t e d i n t h e form f ( x ) = (1/2a) i z @ ( f )

P p(A)Z: (x)dA ( n o t e o u r c o n f i r m a t i o n of t h i s s o r t of i n v e r s i o n i n [ 191, Chap. 2 , ReA mark 1 0 . 1 2 and a t t h e end of 1 1 2 1 w a s b a d l y p h r a s e d - a c o r r e c t v e r s i o n a p p e a r s i n One d e f i n e s a k i n d of g e n e r a l i z e d t r a n s l a t i o n C Y f ( x )

[ 271). P

X A (x)p(A)dX and a g e n e r a l i z e d c o n v o l u t i o n (g P

= ( 1 / 2 s ) lz@(f)@(g)BA(x)p(X)dhs o t h a t @ ( f

-P s i b l e t o express t h e r e s u l t N = P ': A

*

*

f ) ( x ) = (f

*

P

( 1 / 2 s ) Lz@(f)@ ( y )

=

A

g ) ( x ) = (g(y),E:f(x)) It i s t h e n pos-

g) = @(f)@(g)e t c . N

i n t h e form ( r e c a l l B(y,x) and t ( y , x ) v a n i s h

f o r y > x and we t a k e both k e r n e l s t o be 0 f o r a l l x < 0)

.

N

THEOREM 4.1

A

-

{B(y,-) (1/271)

.

H}(x) where @ ( H ) = M / c ( - A ) 1 Q

~ z @ ( H ) ~ ~( t() y P (1) P ~d i .

B(y,x) =

4

and w e can w r i t e t h e n B = B x w h e r e kerJC=

U N

One t a k e s t h e g e n e r a l i z e d G-L e q u a t i o n of [ l 0 ; 1 9 ] now i n t h e form B = BW A

Y

W

B;Kw where kerW

=

P P 42C2 W(x,y) = ( q ( x ) p , ( y ) , v / w U

=

' I

-

x

u

(u = vp, dw = GdX) and t h e i d e a Y-

t h e n i s t o s e l e c t a n o p e r a t o r JC such t h a t B X i s s u i t a b l y t r i a n g u l a r and (Jcwc) i s "nice".

-

A f t e r some c a l c u l a t i o n w e o b t a i n XWg =

Jm ( c /M-)(l

+

1)

is

-m

Q

[ 10;19;42

I

(p/p-))@i(t){@y(x) z

K(t,x) ,g(x)) with K(t,x) = ( 1 / 4 ~ ) N

For Jc a f i r s t c h o i c e ( f o l l o w i n g

ZP(x)}pdA. u

5

3(h

+

(

= (z(x,s) , h ( s ) ) w i t h X(x,s) = ( 1 / 2 1 ~ )

P P _/I y and With t h i s c h o i c e L ( y , x ) = ker(B%J = ( 1 / 2 1 ~ )L ~ ( l / c - ) @ Q ( y ) Q x ( x ) d A Q A

one h a s .

THEOREM 4 . 2 . With h y p o t h e s e s a s i n d i c a t e d above one h a s a g e n e r a l i z e d M

e q u a t i o n f o r k e r n e l s , namely, f o r x > y , 0 = S ( t , x ) = -(1/28)

D

2

$

P

@-x

=

P

+

P A@A).

r n h

Y

P P L ~ ~ QIm(MM1)@A(t)@A(x)dA PS and

@K(t)@!A(x)dA ( r e c a l l

the only

J

B(y,t)[S(t,x)

J(t,x) = (1/4n)

9

form one can w r i t e J ( t , x )

=

where

L~(~+(Q/P-))(Q/P-)

.

This is an i n t e g r a l equation f o r

i n p u t i n t o t h e k e r n e l s S and J comes from s

t h i s r e d u c e s t o [ 10;19 1.

+ J(t,x)}dt

= c /c-

Q Q

in S.

and For

^P

In o r d e r t o p u t t h e i n t e g r a l e q u a t i o n i n t o a s i m p l e r 6(x-t) - T(t,x) t o o b t a i n ( 6 ) i ( y , x )

=

J

m

Y

A

B(y,t)iT(t,x) , .+

- S(t,x)ldt.

=

A s l i g h t f u r t h e r s i m p l i f i c a t i o n can be a c h i e v e d by t a k i n g K w i t h

Some Topics iii Transmutation

-

95

P P 2 kernel x(x,s) = ( 1 / 2 ~ 1 )/ I ( M , / C ~ ) @(s)zA ~ (x)p dX and then we can w r i t e t h e i n t e g r a l A

equation

( r ) for B

i n t h e same form with s l i g h t l y d i f f e r e n t S and T. u

5. BASIC ?THEORY.

We t u r n now t o operators Q and t h e theory has a num-

ber of d i f f e r e n t f e a t u r e s ( c f . [ 3 ; 2 2 ; 2 3 ; 3 8 ; 4 0 - 4 3 ; 4 6 ; 5 1 ; 5 6 - 6 3 1 ) .

There i s f i r s t a

considerable l i t e r a t u r e on i n t e g r a l o p e r a t o r s which t r a n s f o m a n a l y t i c f u n c t i o n s or harmonic f u n c t i o n s i n t o s o l u t i o n s of e l l i p t i c equations ( i n p a r t i c u l a r t h e B-G operator does t h i s ) and i n a d d i t i o n c e r t a i n d i r e c t and i n v e r s e problems i n s c a t t e r N

ing theory have been i n v e s t i g a t e d using Q t y p e operators.

Let u s w r i t e A u

2 n- 1ur)r/rn-' F f r )u = 0, A u = ( r n

as depends only

or s p h e r i c a l v a r i a b l e s .

n 2 fonn Pu = P u = r urr N

+

+ Asu, n

AS, n

=

OSu/r2 where n

+ on angle

We express t h e r a d i a l p a r t of An m u l t i p l i e d by r 2 i n t h e @ n u ( s e t a l s o Qu = Q u = Pu

(n-l)rur

+ r -3F ( r 2 )u).

Gilbert's

+

method of ascent then shows t h a t s o l u t i o n s ( r e g u l a r around t h e o r i g i n ) of Anu =

0 for n

(xl,

22

...,xn)

+

r(Grr

s h a l l r e f e r t o t h i s a s t h e B-G

+ ( p"-3z(r,p),h(p,.))

tion.

+

FG) = 0; G(O,T) = 0 ; G(r,O) = -

i n t e g r a l operator.

r

I

0

2 F(p )pdp.

Now change v a r i a b l e s and w r i t e with Y t h e Heavyside func-

where K(r,p) = K(r,p)Y(r-p)

The d i f f e r e n t i a l equation s a t i s f i e d by G l e a d s t o a d i f f e r e n t i a l equation

This i s a l l embodied i n t h e d i s t r i b u t i o n context 2

-r2F(r )6(r-p)

(?$e

=

2 (p 8 ) ' ' - (n-l)(pg)').

t o r r e p r e s e n t s a transmutation u a-1

(pg(r,p),p

-4J0r F ( p 2 )pdp/rn-' a t P = r. -* n-3vKI = where one has (9, - P p ) { p

=

One determines then t h a t t h e B-G V

%(h) =

) =

(

B(r,p),h(p,')).

M{pg(r,p)I(o) (Mellin transform p T =

-a-n+2)

equation 68 = a(okn-2)9 with QQ(r)r-'

0(o+n-2)oT).

-

.

+

+

opera-

" a

Then w r i t i n g B f p ) - v

choosing $ Q ( r ) (= ILQ(r) with

=

-*v

a) we w i l l have Q B = P B and P

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

1 as r

+

0 we have (note :pT

=

~(~i-n-2)

"

THEOREM 5 . 1 . The extended B-G k e r n e l B(r,p) i s c h a r a c t e r i z e d by v

t h e s p e c t r a l formula (with i n t e g r a l s c-im + c+im)% B(r,p) = ( l / Z v i ) I p =

We

V

f o r K when p < r with a jump d i s c o n t i n u i t y K ( r , r ) =

=

where x =

and t h e Bergman k e r n e l G (which does not depend on n) s a t i s f i e s t h e

equation 2(1--r)GrT - Gr

u = h

+ J01 an-1 G ( r , l - c 2 )h(xa 2 ) d a

can be w r i t t e n a s u(x) = h(x)

Fu

(1/2iri) i ~ ~ - ~ p ' $ : ( r ) dand ~ N

g{p'}

=

$z(r).

-

.

-0-1 Q $o(r)do

The s p e c t r a l p a i r i n g i n Theorem 5 . 1 ,

corresponding t o t h e P spectrum, i s handled expeditiously v i a t h e Mellin transform and we w i l l s e e below how t o handle t h e Q spectrum more generally.

For n = 3 ,

=

96

R. Carroll

6 = x 2D 2 + 2xD + k 2x2

0,

and t h e n $:(x)

0+5

F u r t h e r i n t h e n o t a t i o n above one knows t h a t ( c f . [ 4 7 1 ) K ( r , p ) = -Sk

and z = k x .

(~/r)'~~{kr(l-(p/r))'}/(l-(p/r))~

-4

and i n p a s s i n g w e n o t e t h a t M{pK(r,p)j = -k-'

( k r ) where s d e n o t e s t h e s t a n d a r d Lommel f u n c t i o n .

Sa+3 / 2 , u+

(kr)

where v =

= 2a~k-4.(a+3/2)z-SJv(z)

Y

N

+

P = Pn = r2D2

2 Recall t h a t r A

(n-l)rD.

= P"

n

+ Q:

- -+

U

e x p l i c i t an i m p o r t a n t p r o p e r t y of t r a n s m u t a t i o n s B: P

L e t u s make

Q where Q = P

+

where

n:

2 2 r F ( r ) and

d o e s n o t depend on

2 n 2 It f o l l o w s t h a t f o r s u i t a b l e F , B{p A u} = B{Pnu +n>} = Q Bu +a> = r tan n 2 2 2 2 2 F ( r )}Bu and t h u s B{p An} = r {An F}B ( i . e . B: p A n + r2An + r F ) . Thus one

r.

+

+

wants m u l t i p l i c a t i o n by r L i n o r d e r t o e x t e n d t h e t r a n s m u t a t i o n t h e o r y from r a d i a l operators t o certain operators i n several variables. We w i l l t r e a t now t h e c a s e A

X

2 2 xu s o (x u ' ) ' = ~ ( x u ) " = x4" and Qu = X u b e c a n e s

2

(one w r i t e s a l s o A 2 = a ( u + l ) = v2

@

We d e n o t e by

Ip

(v,k,x) t h e "regular"

N

t i o n f(v,-k)

-

0

%. s o a

%

f(v,-k,x)

e

%

-

(ip

Q,

xv+'

eikx as x

(W(f,g) = f g '

+ x 2 { k2

2 xip"

(0)

r e a l , and set c

- q(x)b

= a n g u l a r momentum and

(0)

Y

= W(f(v,-k,x),ip(v,k,x))

and s a y fm x 2 ( T \ d x
0

( t h e range of a n a l y t i c i t y c a n b e e n l a r g e d w i t h s u i t a b l e h y p o t h e s e s on

q).

In order

t o d e a l w i t h s p e c t r a l q u e s t i o n s w i t h o u t i n t r o d u c i n g an u n n e c e s s a r y parameter deN

s c r i b i n g s e l f a d j o i n t r e a l i z a t i o n s of Q ( c f . 156 I) we f o l l o w f o r m a l l y t h e p r o c e d u r e of [ 31.

and l e t Z d e n o t e t h e s e t of z e r o s v . ( i f 3 m 2 i n Rev > 0 w i t h M ( v . , k ) = 1 g (v - k , r ) d r . With such s i m p l e J 0 j'

Thus s e t g(v,-k,x)

any) of f(v,-k)

= f(v,-k,x)/x

2

z e r o s v . one s e t s d p ( v ) = I s ( v - v . ) / M J J

2

(v

k ) f o r v E Z and f o r v E [O,i-) dp(v) =

j’

91

Some Topics in Transmutation 2 2iv dv/Tf (v,-k)f (-v,-k). = (

g(v,-k,r),g(v,-k,s))

,., an operator Q,

From 13 1 one has the formal completeness relation 6 ( r - s ) THEOREM 5.2. Let gl, P I , etc. refer to

and we show then.

P

based on yl. Define B(r,s)

= (

g(w,-k,r),gl(w,-k,s)) N

-4

(

g(v,-k,r),gl(u,-k,s))

P

1 with B f ( s )

= (

6(r,s),f(r))

and Bf(r)

-

P

and y(r,s)

= ( B(r,s),f(s))

suitable f. The r and s brackets refer to distribution pairings on [O,-) has triangularity B(r,s) = 0 for s > =

Im f(s)g(u,-k,s)ds +

r

r

Then B: Q,

-f

T

and B(r,s)

"B1): 6

+

(G

-Q,

and B{gl(v,-k,-)l(r)

and correspondingly 8

=

?(u)

(:(v),g(v,-k,r))

P

are transmutations with B{g(v,-k;)j(r) Set

B

-* = B

t 1f and b f

=

6,Bf

g(v,-k,r).

,-* = B ; then GBf

=

.

N

u

gl(v,-k,r)

=

for

and one

Set tf(v)

= 0 for r > s .

-1A = G f(r) = f(r)

that formally G?(r)

so

and 8

0

N

=

=

Bf(r)

(so

(with

B

=

.

B(r,s),f(s))) -1 B 1.

= (

=

Although the technique follows [ 31 there are significant variations and the results in Theorem 5.2 represent a considerable extension and refinement of [ 31.

Note also that

based on

4. m

=

x u"

+

= h(r, - )

tions of (An

)

represents Jm 0

in contrast to certain pairings in [ 31

. * 2 Thus one considers Qu = x u"

+

(n-1)xu’

+ ;I

,

It is appropriate at this point to mention an exterior transmuta-

tion of [ 4 0 ; 4 1 ] . 2

(

+

(n-1)su’

+ x2 {k2 - - q(x)lu

2 2 x k u which arise in working with formulas u(r,*)

~~-~K(r,s)h(s, -)ds linking solutions h of (An

+ F)u

with 2rn-2K(r,r)

= 0 for F =

=

transmutation P

the map Bef(r)

N

3

../-

s ) ) 1 (where Q, P

K(r,s)

Jr sq(s)ds and setting z(r,s)

THEOREM 5.3. For suitable U

2 k - q(r).

Q and for n

=

3 , 6(s-r)

=

satisfies =

f(r)

+ i(r,s)

+ k2)h $K

B(r,s)

=

P and p1 is the "free" measure arising when

1 has no zeros for Rew > 0 and d p (v) = -(v/rk)Sinvsdu).

0 with solu-

N

=

+ (g(r,s),f(s)) Q

Be{hj(r,*)

(

.

P K for s > r

one can show

K(r,s)Y(s-r)

-

=

=

and ?u

determines a g(u,-k,r),gl(u,-k, =

0 so that f(w,-k)

-

In the "free" case q

= 0

the inversion theory for b is the K-L theory which can be treated in various ways

-

(cf. 1501).

-q

The version which we obtain below in a general manner specializes for m 1 j m = 0 to G(v) = J G(s)Hv(ks)ds and rG(r) = 4 ,imwG(v)Jv(kr)dv. In order to arrive H

0

at a general form of this suppose f(v,-k) has no zeros for Rev > 0 From properties of f (tv,-k,x) and q(*v,k,x)

c(v)du. =

-(l/Zv){f(-v,-k)q(v,k,x)

rf(r)

=

(f(v),f(u,-k,r)>

P

- f(v,-k)lp(-v,k,x)) im * = % L,f(v)f(v,-k,r);(v)dv

so

that dp(v)

=

(note in particular f (v,-k,x)

and f(v,-k,x) is even in v) one has and from this (*) rf(r) =

R. Carroll

98 i m

A

= - ( i / n ) /imVf(v)@(u,k,r)dv

where @ ( v , k , r ) = I p ( v , k , r ) / f ( v , - k )

t h e f r e e c a s e Q 0 = (1rr/2k)'i-('-')

Jv(kr)).

=

?)

THEOREM 5.4.

w e have

$(v)du t h e i n v e r s i o n (*) h o l d s .

If

and in a d d i t i o n B t * l ( v , k , - ) > ( r )

= ( B(r,s),*l(v,k,s))

=

Given

5 and 5,

have continuous s p e c t r a t h e n B is c h a r a c t e r i z e d by B{g ( v , - k , - ) j ( r ) 1 -k,r)

in

Using t h e formal r e l a t i o n (*) - ( i u / n )

IOm ~ ( ~ , k , s ) g ( v , - k , s ) d s / s= 6(V-V) a r i s i n g from a b s o l u t e l y continuous & ( v )

-

( c f . Theorem 3.2

both

.

(6/sl)g(u,

=

(* = @ / r ) .

*(v,k,r)

One can c o n s t r u c t a f o r m a l proof of t h e l a s t r e l a t i o n f o l l o w i n g [ 1 9 1 (using analyt i c c o n t i n u a t i o n ) b u t a s i m p l e r f o r m a l v e r i f i c a t i o n can b e obtained ( c f . 53) by looking a t $(v,s)

(

B(r,s),g(u,-k,r))

= gl(v,-k,s),

=

a s an e x t e n s i o n of G t o B , so t h a t

gl(v,-k,s)

using the inversion

c"),

Further f o l l m i n g [ 191 t h e equations g(v,-k,r) t) =

(

B(u,t),g(v,-k,u))

and t h e n applying (*) f o r

and g

Q1

.,

1’

and gl(v,-k, * g i v e r i s e t o a G-L e q u a t i o n i n t h e form @ ( r , t ) = ( B ( r , s ) , = (

A ( s , t ) ) where A ( s , t ) = ( gl(v,-k,s),gl(v,-k,t))p’ v

-

t o r above where K arises from B: P

-

Y

B(r,s),gl(v,-k,s))

In connection w i t h t h e B-G opera-

Q w i t h P = x2D2

+

+

M

2xD and Q = P

+ k 2x 2

one

c a n modify t h e c o n s t r u c t i o n l e a d i n g t o Theorem 5.3 and c o n s t r u c t a n i n t e r i o r t r a n s v

, . . I

Q + P i n v e r t i n g B i n t h e form Biu = h = u

mutation Bi:

L i ( x , < ) = 4kx+(x-,Dg(v,-k,l)/g(v,-k,l))p

The "Cauchy" problem i n d i c a t e d i s t o b e c o n s i d e r e d i n two

and Cf(1) = f ' ( 1 ) ) . regions n


*,9 (0)dX

contains an unknown factor @:(O).

Another re-

sult in [ 371 allows one to construct an extended G-L equation (which looks promising for numerical computation) in the form H ( T ) - K(y where G(yo,~) = $tG(T+yo)

,T) =

G(yo,’)

+ QK(y,,-)

* G’

+ G(T-Y~)~. We remark that although connections between

autocorrelation ideas and spectral measures are known via the theory of Krein strings (see e.g. the book "Gaussian processes, function theory, and the inverse spectral problem" by H. Dym and H. McKead, the technique above based on the transmutation machine provides a new link directly connected with a geophysical problem, This appears promising in connecting various approaches in geophysics involving time series analysis and the inverse problem.

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Carroll, Osaka Jour. Math., 1 9 ( 1 9 8 2 ) , 833-861

10. R. Carroll, Rocky Mount. Jour. Math., 1 2 (19821, 393-427 11. R. Carroll, Jour. Math. Anal. Appl., 92 (1983), 410-426

1 2 . R. Carroll, Some inversion theorems of Fourier type, Rev. Roumaine Math. Pures Appl., to appear 13. R. Carroll, Proc. Royal SOC. Edinburgh, 91A (19821, 315-334 14. R. Carroll, Corn. Partial Diff. Eqs., 6 (1981), 1407-1427 1 5 . R. Carroll, Spectral Theory Diff. Operators, North-Holland, 1981, pp. 81-92 1 6 . R. Carroll, Anais Acad. Brasileira Ciencias, 54 (1982), 271-280 17. R. Carroll, Rend. Accad. Lincei, 72 (19821,

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32. R. C a r r o l l and F . Santosa, Applicable Anal., 1 3 (1982). 211-277 33. R. C a r r o l l and F. Santosa, On some s i n g u l a r i n v e r s e problems, t o appear

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37. R. C a r r o l l and F. Santosa, Some t r a n s m u t a t i o n methods i n geophysics. Proc. Conf. I n v e r s e S c a t t e r i n g , Univ. Tulsa, 1983, t o a p p e a r 38. K. Chadan and P . S a b a t i e r , I n v e r s e problems i n quantum s c a t t e r i n g t h e o r y , S p r i n g e r , N.Y., 1977 39. H. Chebli, J o u r . Math. P u r e s Appl.,

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42. L. Fadeev, Uspekhi Mat. Nauk, 14 (1959), 57-119 4 3 . L. Fadeev, Sov. Prob. Mat., 31 (1974), 93-180

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Ark. Mat.,

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1 (1970), 96-114

1 4 (1965), 807-819

49. V. Hutson and J. Pym, Proc. London Math. SOC., 24 (1972), 548-576 50. D. Jones, J o u r . I n s t . Math. Appl.,

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57. V. Marrenko, Sturm-Liouville o p e r a t o r s and t h e i r a p p l i c a t i o n s , I z d . Nauk. Dumka, Kiev, 1977 58. H. Moses, Jour. Math. P h y s i c s , 2 1 (1980), 83-89 59. M. Naimark, L i n e a r d i f f e r e n t i a l o p e r a t o r s , I z d . Nauka, Moscow, 1969 60. D. Naylor and P. Chang, SIAM J o u r . Math. Anal.,

13 (1982), 1053-1071

61. R. Newton, Geophys. J o u r . Royal A s t r . SOC., 65 (1981), 191-215 62. R. Newton, S c a t t e r i n g t h e o r y of waves and p a r t i c l e s , McGraw-Hill, N.Y.,

1966

63. R. Newton, J o u r . Math. P h y s i c s , 21 (1980), 493-505 and 1698-1715 64. P. S a b a t i e r , J o u r . Math. P h y s i c s , 8 (19671, 1957-1972 65. P. S a b a t i e r , Comptes Rendus Acad. S c i . P a r i s , 278 (1974), 545-547 and 603-606 66. P . S a b a t i e r , J o u r . Math. P h y s i c s , 1 2 (1971), 1303-1326 67. F. Santosa, T h e s i s , Univ. I l l i n o i s , 1980 68. F. Santosa, Geophys. J o u r . Royal A s t r . S O C . , 70 (1982),

229-243

69. F. Santosa and H. Schwetlick, Wave Motion, 4 (1982), 99-110 70. J. Siersma, T h e s i s , Univ. Groningen, 1979 71. M.

Stone, Math. Z e i t . , 28 (1928), 654-676

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72. W. Symes, Impedance p r o f i l e i n v e r s i o n v i a t h e f i r s t t r a n s p o r t e q u a t i o n , t o appear

7 3 . W. Symes, SIAM J o u r . Math. Anal.,

1 2 (1981), 421-453

7 4 . K. Trimeche, J o u r . Math. Pures A p p l . , 60 (1981), 51-98

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

105

AN EQUATION MODELING THE ELECTRIC BALLAST RESISTOR

Nathaniel Chafee School of Mathematics Georgia I n s t i t u t e of Technology Atlnata, Georgia 30332 U.S.A.

W e imagine a s t r a i g h t segment of very t h i n wire lying along the u n i t i n t e r v a l of an x-axis. This wire i s surrounded by a gas whose temperature has a c e r t a i n fixed value a > 0, and heat flows from t h e wire i n t o t h e g a s o r viceversa. A t the same time, an e l e c t r i c c u r r e n t I i s passing through t h e wire from one end t o the other, and, because of the e f f e c t s of e l e c t r i c a l r e s i s t a n c e , t h i s c u r r e n t I generates heat within the wire. We a r e i n t e r e s t e d i n the r e s u l t i n g d i s t r i b u t i o n of temperature u along t h a t wire. W e s h a l l regard u a s a function u ( x , t ) of the position x , 0 2 x 5 1 , along t h e wire and of the time t w i t h 0 ~t < +a. I n a l l t h a t follows we s h a l l work in a regime where u > 0. The device we have just described -- wire, c u r r e n t , and ambient gas -- i s sometimes called an e l e c t r i c b a l l a s t r e s i s t o r , and over the years i t has been t h e object of several s c i e n t i f i c i n v e s t i g a t i o n s , both experimental and t h e o r e t i c a l . Indeed, two e a r l y i n v e s t i g a t i o n s a r e reported i n [4] and 121, and several Much o f the t h e o r e t i c a l work on [5], and [3]. recent studies appear i n [1,6], t h e b a l l a s t r e s i s t o r concerns the nature of behavior of temperature d i s t r i b u t i o n s u ( x , t ) which might appear in t h e given wire. This present paper i s i n t h a t same vein. S p e c i f i c a l l y , we a r e going t o report here two new r e s u l t s concerning the boundedness of temperature d i s t r i b u t i o n s u ( x , t ) a s the time t approaches +m. Our f i r s t task i s t o formulate s u i t a b l e mathematical r e l a t i o n s governing u. To begin, we shall r e q u i r e t h a t u s a t i s f y the following equations:

Here, r ( u ) denotes the r e s i s t i v i t y of our given wire, t h a t i s , i t s e l e c t r i c a l r e s i s t a n c e per u n i t length. g ( u ) denotes the time-rate per u n i t length of wire a t w h i c h heat flows from the wire i n t o the gas. As t h e notation i n d i c a t e s , we a r e assuming t h a t both r ( u ) and g ( u ) a r e functions o f u and of u alone. Later, we shall impose precise conditions on these functions r ( u ) and g ( u ) .

N. Chafee

106

I n Eq. ( l c ) t h e q u a n t i t y + ( x ) s i g n i f i e s an i n i t i a l d i s t r i b u t i o n o f temperat u r e along t h e given w i r e , and as ( l c ) i n d i c a t e s we r e q u i r e t h a t $ ( x ) be p o s i t i v e a t a l l x i n [0,1]. Eqs. ( 1 ) a r e i n c o m p l e t e i n one v i t a l r e s p e c t : t h e manner i n which I i s r e l a t e d t o u, x, and t.

we have n o t y e t s p e c i f i e d We s h a l l now r e n d e r such a

r e l a t i o n s h i p i n t h e form o f the f o l l o w i n g hypothesis: c o n s t a n t V,

There e x i s t s a p o s i t i v e

independent o f x, t, o r u, such t h a t 1

V

=

I i

(21

r(u(x,t})dx.

0 W i t h r e g a r d t o o u r b a l l a s t r e s i s t o r , V i s t o be i n t e r p r e t e d as t h e v o l t a g e o r d i f f e r e n c e i n e l e c t r i c p o t e n t i a l g o i n g f r o m one end x = 0 o f t h e w i r e t o t h e o t h e r end x = 1. Eq. ( 2 ) i s a s t a t e m e n t o f Ohm's Law, t o g e t h e r w i t h an assumpt i o n t h a t I has no x-dependence. The h y p o t h e s i s we have j u s t i n t r o d u c e d in c o n n e c t i o n w i t h ( 2 ) embodies a r e q u i r e m e n t t h a t t h e v o l t a g e V be h e l d f i x e d w i t h r e s p e c t t o t h e t i m e t and t h e temperature d i s t r i b u t i o n u. A t t h i s j u n c t u r e , we f i n d i t c o n v e n i e n t t o i n t r o d u c e a f u n c t i o n a l p[u] setting

by

1 p [u] =

i r(u(x,t))dx.

(3)

0

C l e a r l y , ~ [ u ]r e p r e s e n t s t h e t o t a l e l e c t r i c a l r e s i s t a n c e o f f e r e d by t h e g i v e n w i r e i n t h e presence o f t h e t e m p e r a t u r e d i s t r i b u t i o n u ( x , t ) .

On t h e b a s i s o f ( Z ) , ( 3 ) , and o u r h y p o t h e s i s o f c o n s t a n t v o l t a g e V, we can now r e w r i t e ( 1 ) as

F o r l a c k o f a b e t t e r term, we c a l l ( 4 a ) an 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 o f p a r a b o l i c type.

Eq. ( 4 a ) , t o g e t h e r w i t h (4b,c), i s o u r mathematical model f o r t h e b a l l a s t r e s s t o r , and t h e r e m a i n i n g p o r t i o n o f t h i s paper i s an i n v e s t i g a t i o n o f Eqs. ( 4

.

H e n c e f o r t h , d u r i n g t h a t i n v e s t i g a t i o n , we s h a l l suppose t h a t

t h e f u n c t i o n s g and r i n Eq. (4a) s a t i s f y t h e f o l l o w i n g hypotheses.

fH1)

2 and r a r e C -smooth f u n c t i o n s mapping t h e i n t e r v a l [O,+m)

into

(-m,*t.

(H2) r i s p o s i t i v e on [O,+m)

and i t s d e r i v a t i v e r ' i s non-negative on

[O,+m).

(H3) The d e r i v a t i v e g ' i s p o s i t i v e on n e g a t i v e on

[O,+m).

[O,+m)

and t h e d e r i v a t i v e g" i s non-

At1

107

Equation Mudcliiig ihc Electric Ballast Resistor Indeed, a i s

0 such t h a t g ( a ) = 0.

(H4) There e x i s t s a u n i q u e number a

t o be regarded as t h e t e m p e r a t u r e o f t h e ambient gas mentioned s e v e r a l paragraphs above. We s h a l l l e t X denote t h e space o f a l l c o n t i n u o u s f u n c t i o n s @: [0,1]

+

and on X we s h a l l impose t h e usual Co-supremum norm, which we h e r e

(-m,+m),

denote by

[I 11.

Under

11 11

o f c o u r s e X becomes a Banach space. We s h a l l a l s o 1 impose an a u x i l i a r y norm on X, namely, t h e L - i n t e g r a l norm, and we h e r e d e n o t e sisting o f a l l @

+

.

11 11,

t h a t norm by

F i n a l l y , we s h a l l l e t Xo denote t h e h a l f - s p a c e i n X con-

X such t h a t @ ( X I > 0 on 0

t

5

x

5 1.

+

By a s o l u t i o n o f

Now w i t h r e f e r e n c e t o ( 4 c ) , l e t @ be any element i n Xo. Eqs. ( 4 ) we s h a l l mean a r e a l valued f u n c t i o n u = u ( x , t ) t h e f o r m [0,1]

x

[O,s),

with 0 < s

p o z i t i v e everywhere on [ O , l ]

x

(+m,

such t h a t :

d e f i n e d on a domain of

( i ) u i s c o n t i n u o u s and

[O,s); (ii) t h e p a r t i a l d e r i v a t i v e s ux, uXX, u t

x (0,s); ( i i i ) u s a t i s f i e s Eq. (4a) e x i s t and a r e c o n t i n u o u s everywhere on [ O , l ] x (0,s); ( i v ) u s a t i s f i e s Eqs. ( 4 b ) and ( 4 c ) on t h e i n t e r v a l s everywhere on [ O , l ]

5 x 5 1 respectively.

0 < t < s and 0

U s i n g arguments o f t h e s o r t a p p e a r i n g i n [7], element $

E

+ Xo,

one can p r o v e t h a t , f o r any

Eqs. ( 4 ) have a u n i q u e s o l u t i o n u = u ( x , t )

d e f i n e d on a domain o f t h e f o r m [0,1]

x

= u(x,t;@)

[ O , s ( @ ) ) w i t h 0 < s($)
0, t h e r e e x i s t s a c o n s t a n t M1 > 0 such t h a t , i f @ %then ]lu(*,t;@)]/l


0, t h e r e e x i s t s a subdomain Dt of D such t h a t ( t w ) 2~ 0. S i n c e tkw i s z e r o a t t = 0, we have t k w 0 i n Dt f o r t > 0. I t f o l l o w s from 2 w = -v ( u / v )

t

that

This c o n t r a d i c t s t h e hypothesis Because u/v = 0 a t t = 0, we have u/v 5 0 i n Dt. t h a t u > 0 i n R and t h e assumption t h a t v > 0 on X . Thus, v cannot be p o s i t i v e t h r o u g h o u t n-. From t h e p r o o f o f t h e theorem, we n o t e t h a t i f p = +m on S, t h e n t h e boundary c o n d i t i o n ( 2 . 2 ) on v, and t h e h y p o t h e s i s (2.7) can be o m i t t e d ; furthermore, if f o r i = 1, 2, 3, ..., n , b i - B i a r e i d e n t i c a l l y zero, t h e n t h e theorem i s v a l i d even when a i j E A j j . L e t us g i v e an example t o i l l u s t r a t e t h e above theorem and remarks. Example 1.

Let

Lu

=

Utt

Mv z v tt

- uxx - (1 + 2 t - Z ) u , 0 < x
R (20) 2 i s c o n t i n u o u s t o g e t h e r w i h i t s d e r i v a t i v e fu, and i s L -almost p e r i o d i c ( i . e . , as a map f r o m R i n t o L (a), u n i f o r m l y w i t h r e s p e c t t o u E R ) . Moreover, we assume t h a t

8 .

(21 1 where

fu 5 u < il,(t,x,u) X1

>

E

R

x

5x

R,

i s t h e s m a l l e s t e i g e n v a l u e o f t h e problem

0

AV

(22)

+ xv

= 0

in

R, v I S = 0.

Under t h e above mentioned assumptions, any C(2)-sol u t i o n u ( t ,x) o f t h e e q u a t i o n ( 1 8 ) , v a n i s h i n g on t h e boundary o f Q , i n w h i c h t h e r e e x i s t s M > 0 w i t h

L

u (t,x)dx

(23) 2

i s L -almost periodic. $ ( r ) = XlrZ

and

$(r) =

5M,

V t E

R,

Indeed t h i s f o l l o w s f r o m Theorem 1 b y l e t t i n g 2

.

pr

A c o m p l e t e l y s i m i l a r r e s u l t h o l d s i n t h e case o f t h e e l l i p t i c e q u a t i o n ( 1 9 ) . d e t a i l s see [ 8 ] . H,

We s h a l l now c o n s i d e r a system o f two e q u a t i o n s i n

(24) where u > 0 conditions:

U'

f

AU

BV + g(t,v)

f

-

= 0, V '

i s a c o n s t a n t , and t h e o p e r a t o r s

namely

BU + vv = 0,

A,B,g

s a t i s f y the following

b)

A i s a l i n e a r o p e r a t o r on H , w i t h A - a1 monotone f o r some B i s a n i n v e r t i b l e s e l f - a d j o i n t o p e r a t o r on H.

c)

g: R

a)

x

H

->

H

a > 0.

s a t i s f i e s a uniform L i p s c h i t z condition, w i t h constant

(25) and i s almost p e r i o d i c i n

IIg(t,x)-g(t,Y)ll t,

For

Imllwl

m > 0

I

u n i f o r m l y w i t h r e s p e c t t o t h e second argument.

Moreover, t h e f o l l o w i n g c o n d i t i o n w i l l be needed: d)

The c o n s t a n t s

(26 1

a , ~ , and

m are subject t o m2 < 4av.

L e t us p o i n t o u t t h a t t h e case v = 0 i n ( 2 4 ) l e a d s t o t h e second o r d e r e q u a t i o n 2 w" f A w ' f B w f g(t,Bw) = 0, (27) which can be viewed as d e s c r i b i n g n o n l i n e a r o s c i l l a t i o n s i n presence o f a f r i c t i o n

C Corduneanu and J.A. Goidstein

120

force with (nonlinear) damping.

The system ( 2 4 ) can be rewritten as a s i n g l e equation in t h e product space H w i t h usual s c a l a r product generated by t h e product o f H . Indeed, i f

U

=

col(u,v),

and A ( t =

[;

:I’

=

B1 + B2

B2 =

f

B3

f

G(t,-),

x

H

where

[-;’j! =I: 4’ B3

and G ( t , U ) = c o l ( g ( t , v ) , O ) , then 24) becomes (28) U’ + A(t)U = 0.

Summing up ( 2 9 ) (33)

-

(32) one obtains ( A ( t ) U - A ( t ) U ,U-i)

IJ

where =

Because o f ( 2 6 ) B

rnin{a

m - --, 28

( ( U - i 112,

2

v

- 2

>

0.

can be chosen i n ( 3 2 ) t o s a t i s f y

The inequality ( 3 3 ) gives the monotonicity of A(t) which allows f o r a d i r e c t application of Theorem 1. Therefore, under conditions a ) , b ) , c ) , and d ) , the system (24) s a t i s f i e s t h e Eohr-Neugebauer property: each bounded (on A ) solution i s almost periodic. REFERENCES [l]

Amerio, L. and Prouse, G . : Almost Periodic Functions and Functional Equat i o n s (Van Nostrand-Reinhold, New York, 1978).

[2]

B i r o l i , M. S u r l e s s o l u t i o n s born6es o u presque-p6riodiques des 6quations multivoques sur un espace de Hilbert, Ricerche Mat. 21 (1972) 17-47.

[31

Boles, B. and Tsend, L . , A generalized Bohr-Neugebauer theorem, Differential Equations 8 (1972) 1031-1035.

On the weak asymptotic almcst p e r i o d i c i t y of bounded solutions of u" E Au + f , f o r monotone A , J . Math. Analysis Appl. 37 (1980) 309-31 7. Corduneanu, C . Almost Periodic Functions, (John Wiley and Sons, New York, 1968).

c41 Bruck, R. E . , [5]

Almost Periodicity f o r Nonlinear Equations

121

[6]

Corduneanu, C. Bounded and almost p e r i o d i c s o l u t i o n s o f c e r t a i n n o n l i n e a r e l l i p t i c e q u a t i o n s , TGhoku Math. J o u r n a l 32 (1980) 265-278.

[7]

Corduneanu, C. Bounded and a l m o s t p e r i o d i c s o l u t i o n s o f c e r t a i n n o n l i n e a r p a r a b o l i c e q u a t i o n s , L i b e r t a s Mathematica I1 (1982) 131-139.

[8]

Corduneanu, C. Almost p e r i o d i c s o l u t i o n s t o n o n l i n e a r e l l i p t i c and p a r a b o l i c e q u a t i o n s , N o n l i n e a r A n a l y s i s , TMA 7 (1983) 357-363.

191 Corduneanu, C. and G o l d s t e i n , J . A. t o nonlinear abstract equations, ( i n preparation).

[lo]

P e r i o d i c and almost o e r i o d i c s o l u t i o n s

Dafermos, M. Almost p e r i o d i c processes and a l m o s t p e r i o d i c s o l u t i o n s o f e v o l u t i o n e q u a t i o n s , A. R. Bednarek and L. Cesari (eds), Dynamical Systems, (Academic Press, I n c . , New York, 1977).

[ll] F i n k , A. M. Almost P e r i o d i c D i f f e r e n t i a l Equations, L e c t u r e Notes i n Mathem a t i c s , No. 377, ( S p r i n g e r - V e r l a g , B e r l i n , 1974).

[12]

Haraux, A . N o n l i n e a r E v o l u t i o n Equations, L e c t u r e Notes i n Mathematics, No. 841 , ( S p r i n g e r - V e r l ag , B e r l i n , 1981 )

[13]

Haraux, A. G e n e r a l i z e d a l m o s t p e r i o d i c s o l u t i o n s and e r g o d i c p r o p e r t i e s o f quasi-autonomous d i s s i p a t i v e systems, J. D i f f . Equations 48 (1983) 269-279.

[14]

L e v i t a n , B. M. and Zhikov, V . V. Almost P e r i o d i c F u n c t i o n s and D i f f e r e n t i a l Equations, (Cambridge Univ. Press, Cambridge, 1982).

[15]

Pankov, A. A. Bounded and a l m o s t p e r i o d i c s o l u t i o n s o f e v o l u t i o n a r y v a r i a t i o n a l i n e q u a l i t i e s , Math. USSR S b o r n i k 36 (1980)519-533.

[16]

Zaidman, S. Remarks on d i f f e r e n t i a l e q u a t i o n s w i t h Bohr-Neugebauer p r o p e r t y , J . Math. Anal. Appl. 38 (1972) 167-173.

.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T.Lewis (Editors) QElsevier Science Publishers B.V. (North-Holland), 1984

123

A PRIOR1 ESTIMATES IN NONLINEAR EIGENVALUE PROBLEMS FOR ELLIPTIC SYSTEMS

Chris Cosner Department of Mathematics and Computer Science University of Miami Coral Gables, FL. 33124

The systems considered have the form A; = ):(?A in R , -+ u = 0 on an, where u : R + 7Rm, f ; !Rm + m m , R C 7Rn is a bounded domain, A is a matrix of second order elliptic operators, and X is a real parameter. For simplicity the results are stated for a single equation, but the range of validity for systems is discussed. The first type of a prior1 estimates give lower bounds for supl;l in terms of X and IRI when f is superlinear, upper bounds for supl 1, L e v i n s o n [lo] t i o n s f o r a l l X E (0,~).On t h e t h e r e may b e r e l a t i o n s between solution. The p r e s e n t work i s

When

showed t h a t ( 2 ) h a s p o s i t i v e s o l u o t h e r h a n d , f o r s o m e c h o i c e s of f , X and t h e s i z e of t h e a s s o c i a t e d concerned w i t h f i n d i n g such r e l a t i o n s .

F o r s i m p l i c i t y , t h e r e s u l t s of t h e n e x t s e c t i o n w i l l be s t a t e d f o r ( 2 ) w i t h n 2 3 ; however, most r e m a i n t r u e w i t h only minor m o d i f i c a t i o n s f o r t h e c a s e o f v e c t o r f u n c t i o n s u = ( u' , . . . , u r n ) s y s t e m s of t h e form (3)

q[ f

B=

( a q y ( x , u ) ux. R) x . i,]=1 1 J

+

+

n

+

bq'(x,u)u:

satisfying

caB(x,u)uR]

i

i=l

XfCl(x,u) = 0

i n R, a = l , . . . , m u = 0 on 2 8 , p r o v i d e d t h a t f o r ri = ( q q ) E ' I R ~ ~< , = either

€!Elm,

X E R and u € T R m ,

o r t h e n-volume of R i s s u f f i c i e n t l y s m a l l and

f o r some c o > O .

I n g e n e r a l , s t r o n g l y c o u p l e d s y s t e m s s u c h a s ( 3 ) do

n o t s a t i s f y a maximum p r i n c i p l e , so methods b a s e d on t h e maximum p r i n c i p l e d o n o t a p p l y . The d e t a i l s of t h e a n a l y s i s f o r t h e g e n e r a l c a s e ( 3 ) are g i v e n i n [21. Some i n e q u a l i t i e s a r e needed f o r t h e a n a l y s i s . t h e f i r s t e i g e n v a l u e f o r (1) on R; t h e n i f u

where p 0 i s t h e f i r s t e i g e n v a l u e f o r - A

E

First,

let u,(R)

be

W k J 2 (Sl) I

on t h e u n i t b a l l i n 'IR",

wn

t h e n-volume of t h e u n i t b a l l , and IRI i s t h e n-volume of R. T h i s combines t h e Faber-Krahn i n e q u a l i t y w i t h R a y l e i g h ' s c h a r a c t e r i z a t i o n Another i m p o r t a n t t o o l i s t h e S o b o l e v i n e q u a l t i t y : f o r of pl(R). p* = 2 n / ( n - 2 ) ,

125

A Priori Estimates in Nonlinear Eigenvalue Problems

b e t h e u s u a l norm on t h e Sobolev s p a c e W k r p ( Q ) , s t a n d a r d Lp i n t e r p o l a t i o n i n e q u a l i t i e s a p p l i e d t o ( 6 ) and ( 7 ) y i e l d

f o r u ~ W ~ ' ~ ( f i1 ) 0 as y varies, this information is contradictory. Thus ( 1 ) cannot have smooth solutions defined for all time except in extremely special circumstances. Simple examples also show that solutions of ( 1 ) with slightly less regularity than continuous first derivatives are not unique. For example, if H(p) = -p2 and 6 = 0, then u E 0 and v = max(t - I x l , O ) are distinct compactly supported (for bounded t) and piecewise linear solutions of ( 1 ) which satisfy the equation except on the lines 1x1 = t and x = 0, where v is n o t differentiable. The above remarks recall the classical observations that ( 1 ) does not have global smooth solutions in general and that the most natural weakenings of the

M. G. Crundull and P.E. Sougunidis

132

c l a s s i c a l n o t i o n of s o l u t i o n l e a d t o nonuniqueness. However, i n view of t h e way t h e s e problems a r i s e i n a p p l i c a t i o n s - i n p a r t i c u l a r , i n t h e c a l c u l u s of v a r i a t i o n s , c o n t r o l t h e o r y and d i f f e r e n t i a l games one e x p e c t s a n o t i o n of s o l u t i o n of (IBVPf and ( B V P ) f o r which t h e r e i s b o t h e x i s t e n c e and u n i q u e n e s s

-

.

The f i r s t d e m o n s t r a t i o n of u n i q u e n e s s f o r a n o t i o n o f g e n e r a l i z e d s o l u t i o n of ( I B V P ) o r (BVP) a d e q u a t e t o c o v e r a p p l i c a t i o n s t o , e . g . , d i f f e r e n t i a l games, w a s g i v e n i n M. G. C r a n d a l l and P. L. Lions [71. This n o t i o n of s o l u t i o n i s e x p l a i n e d

i n S e c t i o n 1 where u n i q u e n e s s r e s u l t s a r e a l s o d i s c u s s e d . The t o p i c of e x i s t e n c e i s t a k e n up i n s e c t i o n 2, w h i l e S e c t i o n 3 i s c o n c e r n e d w i t h t h e i n t e r p l a y between t h e s e t o p i c s , c o n t r o l t h e o r y and t h e t h e o r y of d i f f e r e n t i a l games. Approximation and r e p r e s e n t a t i o n of s o l u t i o n s a r e d i s c u s s e d i n S e c t i o n 4. As o u r g o a l is a b r i e f o u t l i n e of r e c e n t developments, t h e v e r y s u b s t a n t i a l l i t e r a t u r e which p r e d a t e s t h e work d i s c u s s e d h e r e i n w i l l n o t b e r e f e r r e d to. The book t281 of P. L. Lions w i l l p r o v i d e t h e i n t e r e s t e d r e a d e r w i t h an a p p r o p r i a t e h i s t o r i c a l view and r e f e r e n c e s t o t h e o l d e r l i t e r a t u r e .

There is a t h e o r y of second o r d e r e q u a t i o n s and t h e i r r e l a t i o n s t o s t o c h a s t i c c o n t r o l and games which c o r r e s p o n d s t o t h e f i r s t o r d e r t h e o r y reviewed i n t h i s paper. W e have n o t d i s c u s s e d t h i s t h e o r y h e r e i n and r e f e r t h e i n t e r e s t e d r e a d e r t o t h e p a p e r s [291, [301 of P. L. L i o n s i n t h i s r e g a r d . The t o p i c of q u a s i v a r i a t i o n a l i n e q u a l i t i e s and Hamilton-Jacobi e q u a t i o n s , which i s n o t o t h e r w i s e mentioned h e r e i n , is t a k e n up i n [ l l .

SECTION 1.

NOTIONS OF SOLUTION AND UNIQUENESS

I t w i l l be c o n v e n i e n t t o c o n s i d e r a g e n e r a l e q u a t i o n of t h e form

F ( y , u , D u ) = 0 i n 0,

(1.1) where 0

C fl i s open, y

= (yl,y2,.

.. , y m ) ,

Du = ( u Y1'

...,uYm

),

and F:OXRX#

+ R.

O f c o u r s e , t h i s g e n e r a l form i n c o r p o r a t e s the e q u a t i o n s i n b o t h (IBVP) and (BVP). For u e C ( 0 ) and z e 0 p u t

where a * b i s t h e s c a l a r p r o d u c t of a , b e #. E. g., i f 0 = ( - 1 , l ) and The r e l a t i o n p e D+u(z) can b e u ( y ) = l y l , t h e n D+u(O) = p and D-u(O) = [-1.11. w r i t t e n u ( y ) < u ( z ) + p - ( y - z ) + o(y - z ) , w i t h t h e u s u a l meaning of o ( y - z ) , and C l e a r l y u i s d i f f e r e n t i a b l e a t z e 0 e x a c t l y when a s i m i l a r remark a p p l i e s t o D-. b o t h D+u(z) and D-u(z) are b o t h nonempty and t h e n

D+u(z) = D-ulz)

=

{

Du(z)

}

where Du(z) d e n o t e s t h e u s u a l ( F r s c h e t ) d e r i v a t i v e of u c o n t i n u o u s f u n c t i o n s u which a r e nowhere d i f f e r e n t i a b l e , f u n c t i o n s u such t h a t a t l e a s t one of D+u(z) and D-u(z) z e 0. One way t o d e f i n e a v i s c o s i t y s o l u t i o n of ( 1 . 1 ) D e f i n i t i o n 1. (1.3)

Let u

e

C(0).

at Z. Since t h e r e are t h e r e are continuous i s empty a t e v e r y p o i n t is:

Then u i s a v i s c o s i t y s o l u t i o n of F

~ ( y , u ( y ) , p )6

o

for all y

e o

and p

e

~+u(y).

< 0 in

0 if

Nonlitreur Fmt-Order Equations Similarly, u is a v i s c o s i t y s o l u t i o n of 0 (1.4)

0 C ~ ( y , u ( y ) , p )f o r a l l y


0.

EtO

S i n c e A i s n o n d e c r e a s i n g i n i t s arguments a m o r e s e v e r e r e s t r i c t i o n is l i m A(R,p,E)

(H4 )

= 0 f o r R,

p

> 0.

EC 0

-

The c o n d i t i o n s (HI) (H4) are m e a n i n g f u l f o r t h e problem (BVP) as w e l l when H is i n t e r p r e t e d a s a f u n c t i o n of t which happens t o be i n d e p e n d e n t o f t . The u n i q u e n e s s r e s u l t o f 171 for ( B V P ) is:

Theorem "71). Let u, v be bounded c o n t i n u o u s f u n c t i o n s on Q which a r e v i s c o s i t y s o l u t i o n s of H = 0 i n Q. L e t $ b e c o n t i n u o u s on aSl and u ( x ) - $ ( z ) , and v ( x ) $(z) t e n d t o z e r o as x + z e a Q u n i f o r m l y i n Z . Set R = max(nuii

,lVn Lrn(Q)

) Lrn(Q)

and l e t [ H l ) and ( H 2 ) h o l d w i t h YR > 0 i n (H2). Then: ( i ) If (H4) h o l d s , t h e n u = v. ( i i )I f u and v are u n i f o r m l y c o n t i n u o u s and (H3) h o l d s , t h e n u = ( i i i ) I f u and v are L i p s c h i t z c o n t i n u o u s , t h e n u = V .

V.

T h i s r e s u l t i n f a c t f o l l o w s from m o r e g e n e r a l estimates comparing v i s c o s i t y sub-and-super s o l u t i o n s o f d i f f e r e n t problems. W e w i l l not formulate t h e s e r e s u l t s here. Observe t h a t a s t h e h y p o t h e s e s on u and v a r e s t r e n g t h e n e d , less i s r e q u i r e d of H. The c o r r e s p o n d i n g r e s u l t f o r (IBVP) is q u i t e s i m i l a r . The s t a t e m e n t f o r t h i s case arises upon r e p l a c i n g S l by QX[O,T], 352 by t h e u n i o n of anx[O,Tl and Qx{O}, and $ ( X I b y $ ( x , t ) on aQx[O,TI and by u o ( x ) when t = 0. Moreover, t h e r e q u i r e m e n t yR > 0 i n (H2) i s dropped. (The l i n e a r f u n c t i o n y R ( r 6 ) i n ( H 2 ) i s r e p l a c e d by a n o n l i n e a r f u n c t i o n i n t h e r e s u l t f o r (BVP) i n [ 7 ] . ) The n e c e s s i t y of c o n d i t i o n s l i k e (H3) o r (H4) i s shown v i a examples i n [71.

-

B e f o r e t h e r e s u l t s mentioned above, t h e main u n i q u e n e s s r e s u l t s which w e r e e s t a b l i s h e d i n a g e n e r a l i t y f o r which t h e r e w a s a c o r r e s p o n d i n g g l o b a l e x i s t e n c e t h e o r y concerned t h e case of convex H a m i l t o n i a n s . These r e s u l t s concern s o l u t i o n s o f t h e e q u a t i o n s i n an a l m o s t everywhere s e n s e which also s a t i s f y a t y p e of " s e m i concavity" condition. Concerning such r e s u l t s w e r e f e r to t h e book of P. L. L i o n s t281. The v i s c o s i t y n o t i o n i s u s e d i n C281, b u t t h e main emphasis i n t h i s book i s t h e i m p o r t a n t s p e c i a l case of v i s c o s i t y s o l u t i o n s which a r e L i p s c h i t z c o n t i n u o u s ( a n d hence s a t i s f y t h e e q u a t i o n a l m o s t e v e r y w h e r e ) . O t h e r u n i q u e n e s s r e s u l t s c o n c e r n domains of dependence ( e . g . , [ 7 1 ) , unbounded f u n c t i o n s , ( H . I s h i i [ 2 1 1 ) and H a m i l t o n i a n s which a r e n o t n e c e s s a r i l y c o n t i n u o u s i n t ( H . I s h i i [ 2 0 1 ) . With r e s p e c t t o domains o f dependence, o b s e r v e t h a t i f w e

135

Nonlin m r First-Order Equations

r e g a r d (IBVP) as a special case o f (BVP) by t h i n k i n g of t as as " s p a c e v a r i a b l e " , G e n e r a l r e s u l t s c o n c e r n i n g which p a r t t h e n w e have n o t p r e s c r i b e d d a t a a t t = T. o f t h e boundary of C2 i s i m p o r t a n t f o r u n i q u e n e s s i n (BVP) a r e t h e subject o f work of R. J e n s e n ( [ 2 2 1 ) i n p r o g r e s s a t t h e t i m e o f t h i s symposium.

SECTION 2. EXISTENCE The e x i s t e n c e t h e o r y f o r v i s c o s i t y s o l u t i o n s of (IBVP) and (BVP) is much m o r e a c o n t i n u a t i o n of t h e e x i s t e n c e t h e o r y which p r e d a t e s t h e n o t i o n o f v i s c o s i t y s o l u t i o n s t h a n t h e c o r r e s p o n d i n g u n i q u e n e s s t h e o r y (which is quite d i s t i n c t from what e x i s t e d b e f o r e ) i s a c o n t i n u a t i o n o f more c l a s s i c a l r e s u l t s . Roughly s p e a k i n g , known methods a d a p t t o p r o v i n g t h e e x i s t e n c e of v i s c o s i t y s o l u t i o n s and t h e f l e x i b i l i t y o f t h e n o t i o n allows one t o t a k e l i m i t s f r e e l y and o b t a i n new r e s u l t s . T h e r e are also new arguments which arose p a r t l y i n t r y i n g t o g o t t h e e x i s t e n c e t h e o r y i n harmony with t h e g e n e r a l i t y of t h e u n i q u e n e s s t h e o r y . W e are g o i n g t o d e s c r i b e , i n m o r e or less c h r o n o l o g i c a l order, r e s u l t s o b t a i n e d s i n c e t h e i n t r o d u c t i o n o f v i s c o s i t y s o l u t i o n s and a s k t h e r e a d e r t o b e aware t h a t t h i s d o e s n o t g i v e an accurate h i s t o r i c a l view. W e a g a i n r e f e r t o [281 f o r a m o r e b a l a n c e d view o f t h e earlier t h e o r y . The s o r t of d r a m a t i c e x i s t e n c e and u n i q u e n e s s theorems which are now p o s s i b l e may b e i l l u s t r a t e d b y t h e model problems u + H(Du) = v i n

(2.1)

R",

and

(2.2) I t w a s proven i n [71 t h a t if n is c o n t i n u o u s from If t o R and v and uo are bounded and u n i f o r m l y c o n t i n u o u s , t h e n ( 2 . 1 ) and ( 2 . 2 ) h a v e v i s c o s i t y s o l u t i o n s t o which t h e u n i q u e n e s s theorem a p p l i e s . The o n l y r e g u l a r i t y r e q u i r e d i s c o n t i n u i t y of H and u n i f o r m c o n t i n u i t y o f v and u o , and t h e n t h e r e i s a u n i q u e g l o b a l s o l u t i o n .

P. L. L i o n s , i n f271 and f281, c o n s i d e r s problems of the forms (IBVPf and Two t y p e s of a s s u m p t i o n s on H a r e i m p o r t a n t i n h i s work. One i s a c o n t i n u i t y a s s u m p t i o n of L i p s c h i t z t y p e which i s u s e d t o e s t a b l i s h u n i f o r m L i p s c h i t z estimates on s o l u t i o n s o f (IBVP) and (BVP) ( u s i n g r e s u l t s o f [261). A s i m p l e special c a s e o f t h i s h y p o t h e s i s r e a d s

(BVP).

For R

>

0 t h e r e i s a c o n s t a n t CR s u c h t h a t

for t e [O,TI,

X,

y

The o t h e r assumption r e a d s

(H6)

H(t,x,r,p)

+ m

as IpI +

e h,

p

e f

and Irl


2,

a-B

= 2

W e can a l s o consider higher-order

( p y ( " ) ) (k)

-

(an Euler equation).

e x t e n s i o n s of

( 2 . ) such a s

qy = 0 ,

a n e q u a t i o n which w a s i n t r o d u c e d by H i n t o n [ 1 4 ] , o r more g e n e r a l l y (2.6) The r e s u l t c o r r e s p o n d i n g t o (2.51, f o r example, i s t h a t t h e r e a r e solutions

where Q = W(P1

...

Pn-l)P'n

,

n

w .

7

= 1.

T h e r e a r e c o n d i t i o n s t o b e imposed on q a n d t h e pr a n d , when ar and q ( x ) = ( c o n s t . ) x ' , t h e c o n d i t i o n s become pr(x) = x

...

a + + a - B < n. W e r e f e r t o 1 n-1 a s y m p t o t i c t h e o r y of ( 2 . 6 ) .

[a]

f o r f u l l e r d e t a i l s of t h e

M. S.P. Eastham

160

3. HIGHER-ORDER EQUATIONS OF SELF-ADJOINT TYPE We consider, as an example, the odd-order equation

is nowhere zero on ( X , m ) . If q2r and iq where D=d/dx and q 2n+l 2r+l are real-valued, the left-hand side of (3.1) defines a symmetric differential expression, but we do not confine ourselves to this situation at this stage. Everitt and Zettl [lo] have shown that (3.1) can be written in the quasi-derivative form

Y

= AY,

where the first component of Y is y and A involves the q, but no The characteristic equation of A is derivatives of the 9,. 2n+1

c

qr flr = 0 .

r=O One set of conditions under which this equation can be solved (asymptotically) for the eigenvalues p . is 7

(a) qr are nowhere zero in (XI-) (b) qr-l/qr = o(qr/qr+l)

(1

5 r 5 2n)

(3.2)

Then we have

and p . = O ( W . ) . When the qr behave like powers of x as x -+ m , in 7 ]+I the sense that (x) = 0 {x- qr(x) } (s=1 2,), the transformed system ( 1 . 6 ) has the form

qis)

Z

=

{A, + O(x- )}Z

if T is chosen suitably. If, further, -1 x = O ( l . l , ) = 0tqo/q1), then x-l = o(Al) and we have Case (ii) of f l

.

The second diagonal-

Asymptotic Theory of Differential Equations

161

i z a t i o n , i n v o l v i n g a t r a n s f o r m a t i o n ( 1 . 8 ) , t h e n l e a d s t o a system of t h e form ( 1 . 2 ) .

Transforming back, w e o b t a i n t h e asymptotic n a t u r e

of s o l u t i o n s y . o f 3

( 3 . 1 ) i n t h e form (3.3)

Thus w e h a v e t r u e a s y m p t o t i c f o r m u l a e f o r

I t i s n o t c l e a r w h e t h e r t h e r e i s any s p e c i a l r e a s o n why a s y m p t o t i c

d o n o t a r i s e d i r e c t l y f r o m t h e method.

formulae f o r

F u l l d e t a i l s f o r ( 3 . 1 ) a r e g i v e n i n [ 3 ] , a n d t h e d e t a i l s f o r evenorder equations are i n [4,9,11].

A l s o i n [ 6 ] and [ l l ] i s a d i s c u s s -

§ I occur.

i o n o f s i t u a t i o n s where C a s e s ( i i i ) a n d ( i v ) of

4 . DEFICIENCY INDICES Here w e c o n s i d e r t h e case where ( 3 . 1 ) i s a s s o c i a t e d w i t h a symmetric o p e r a t o r i n t h e H i l b e r t space L 2 ( X I m ) .

Thus i q 2 r + 1 a n d q 2 r a r e r e a l -

v a l u e d , e x c e p t t h a t q o i s r e p l a c e d by qo+A,where t h e new q o i s r e a l v a l u e d and

i s t h e non-real spectral parameter.

i n d i c e s N + a n d N-

The d e f i c i e n c y

a r e t h e number o f l i n e a r l y i n d e p e n d e n t s o l u t i o n s of

(3.1) t h a t are L2(X,-)

for i m A

>

0 and i m A

< 0.

One d e f i c i e n c y

i n d e x l i e s i n t h e r a n g e ( n , 2 n + l ) and t h e o t h e r i n t h e r a n g e (n+1,2n+l).

Also,

N + = 2 n + l if a n d o n l y i f N-

= 2n+l.

The s i z e of t h e s o l u t i o n s y . i n ( 3 . 3 ) i s l a r g e l y d e t e r m i n e d by 7

A calculation gives

r e w 3. where

-

( i m A ) R I’ .

M.S.P. Eastham

162 W e have

R1 = -R2 =

4

,

i q -1 l

(4.1)

By by ( 3 . 2 ) , and R 2 r + l and R 2 r + 2 h a v e o p p o s i t e s i g n s ( 0 5 r 5 n - 1 ) . (4.1) a n d ( 4 . 2 ) , t h e r e i s a j, ( 0 L j 0 -< 2 n + l ) , j, # 1 ) s u c h t h a t

diverges f o r 1

5

j

5

j,

and c o n v e r g e s f o r j

0

+ 1 < j

5

Then w e

2n+l.

have (a)

L e t j, b e e v e n , j ,

(b)

L e t j, b e o d d , j ,

N-

These r e s u l t s g i v e v a l u e s of N + a n d N-

(0

= 2M

= 2M-1

= N+

-

5

(2

M

5

5

M

n)

5

.

Then

n+l).

1 = 2n+l-M

Then

(R2M-1 < 0 ) .

g e n e r a l c r i t e r i a on t h e q . f o r a l l p o s s i b l e 3 Note that, in t o o c c u r , s u b j e c t t o IN+-N-/(l.

(b), M#1 s i n c e j O # l .

Hence, i f one d e f i c i e n c y i n d e x i s 2 n + l , w e

must b e i n ( a ) and t h e n t h e o t h e r i s a l s o 2 n + l .

W e mention i n

p a r t i c u l a r t h e r e s u l t t h a t N + = N-

is L(X,-)

b e i n g t h e c a s e M=O of

(a).

= 2n+l i f

g-’ 1

,

this

W e r e f e r t o [ 3 ] f o r t h e p r o o f s of t h e s e

r e s u l t s , a n d t o [ 4 , 9 , 1 1 ] f o r t h e e v e n - o r d e r case. 5.DIRICHLET INDICES By i n t r o d u c i n g a s p e c t r a l p a r a m e t e r A i n t o o t h e r c o e f f i c i e n t s q . i n 1 ( 3 . 1 ) , w e can c o n s i d e r t h e d i f f e r e n t i a l e q u a t i o n

where

T~

and

T~

a r e symmetric d i f f e r e n t i a l e x p r e s s i o n s .

p o s i t i v e d e f i n i t e Dirichlet

i n t e g r a l D ( y ) , t h e Dirichlet

If

T~

has a

indices M+

Asymptotic Theory o f Difjerential Equations

163

and M- a r e d e f i n e d i n t h e same way a s N + and N- b u t now w i t h D(y)<m i n s t e a d of y

E

L2 ( X , m ) .

W e c o n c e n t r a t e h e r e on a p r o p e r t y of t h e

Dirichlet i n d i c e s which w a s n o t e d by K a r l s s o n [15] and w h e r e , i n c o n t r a s t t o N+ and N-, o n e of M+ and M-

c a n t a k e i t s maximum v a l u e

but not the other. Let

h a v e odd o r d e r 2n+l and l e a d i n g c o e f f i c i e n t i q ( x ) w i t h q ( x ) > O , w h i l e T~ h a s even o r d e r 2n and l e a d i n g c o e f f i c i e n t ( - 1 ) n p ( x ) w i t h T,

p(x)>O.

Then K a r l s s o n showed t h a t

(a)

L e t p / q be L ( X , m ) .

(b)

L e t p / q b e n o t L(X,-). M-

< 2n+l.

Then M+=2n+ i f a n d o n l y i f M-=2n+l.

If n is add a n d M + = 2 n + l , t h e n I f n i s even a n d M- = 2 n + l , t h e n M+ < 2 n + l .

The r e s u l t ( b ) raises t h e q u e s t i o n of e x a c t l y what v a l u e s a r e p o s s i b l e f o r o n e D i r i c h l e t i n d e x when t h e o t h e r D i r i c h l e t i n d e x h a s t h e maximum v a l u e 2 n + l .

The a s y m p t o t i c t h e o r y o f 13 h a s b e e n u s e d t o

show t h a t t h e v a l u e s 2n a n d 2n-1

c a n be r e a l i s e d b u t i s n o t known

whether any lower v a l u e s can b e r e a l i s e d .

W e r e f e r t o [5] f o r the

detailed analysis.

REFERENCES C a s s e l l , J.S., Q u a r t . J . Math.

(Oxford) 33 ( 1 9 8 2 ) 281-296.

Eastham, M.S.P., Theory of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s (Van N o s t r a n d R e i n h o l d , 1970) Eastham, M.S.P.,

Proc. Roy. SOC. Edinburgh 90A (1981) 263-279.

Eastham, M.S.P.,

J. London Math.

Eastham, M . S . P . ,

Proc. Roy.

Eastham, M.S.P.,

P r o c . Roy. SOC. London 383A (1982) 465-476.

SOC. 26 (1982) 113-116.

S O C . Edinburgh 9 1 A ( 1 9 8 2 ) 347-360.

Eastham, M.S.P., Proc. Dundee 1982 Symposium on D i f f e r e n t i a l E q u a t i o n s , S p r i n g e r L e c t u r e Notes i n M a t h e m a t i c s , t o a p p e a r . Eastham, N.S.P.,

J . London Math. S O C . , t o a p p e a r .

Eastham, M.S.P. and G r u d n i e w i c z , C.G.M., 2 4 (1981) 255-271. [ l o ] E v e r i t t , W.N. 363-397.

and Z e t t l ,

1111 G r u d n i e l i i c z , C . G . M . ,

A.,

J . London Math. SOC.

Nieuw Arch. Wisk. 2 7 (1979)

Proc. Roy.

SOC. Edinburgh 87A (1980)53-64.

164

M.S.P. Eastham

[ I 2 1 H a r r i s , W.A.

and Lutz, D.A.,

J. Math. Anal. Appl. 4 8 ( 1 9 7 4 ) l - 1 6 .

[ I 3 1 H a r r i s , W.A. 571-586.

and L u t z , D . A . ,

J . Math. Anal. Appl.

57 (1977)

E q u a t i o n s 4 (1968) 590-596.

[I41 Hinton, D.B.,

J. D i f f .

[15] Karlsson, B.,

J . London M a t h .

SOC. 9 ( 1 9 7 4 ) 1 3 1 - 1 4 1 .

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) OElsevier Science Publishers B.V. (North-Holland), 1984

165

A MORSE THEORY FOR HAMILTONIAN SYSTEMS Ivar Ekeland CEREMADE, Universit6 Paris-9 Dauphine 75775 P a r i s Cedex 16 France

I.

INTRODUCTION

Let us begin by r e c a l l i n g some well-known f a c t s about Hamiltonian systems. We a r e given a function H : R~~ + R , c a l l e d t h e Hamiltonian. The i n t e g e r n i s c a l l e d t h e number of degrees of freedom. Let J E L(RZn) be given by the

ern

We are i n t e r e s t e d in the d i f f e r e n t i a l equation:

x

(2)

I t has

JH'(x).

as a f i r s t integral: H(x(t)) = H(x(0)) ,

(3) T

H

=

all

t.

Assume we have a solution x of equation ( 2 ) which i s 0. Consider the l i n e a r i z e d equation around ST:

(4)

=

JH"(x(t))y.

The c o e f f i c i e n t s a r e Consider the resolvent

T-periodic in R

R = JH"(x(t))R,

(5)

T-periodic, f o r some

t.

of equation ( 4 ) : R(0) = I Z n .

The c h a r a c t e r i s t i c m u l t i p l i e r s of known t h a t :

x a r e the eigenvalues of

R(T).

It is

(a)

1 i s a c h a r a c t e r i s t i c m u l t i p l i e r , because the equation ( 2 ) i s autonomous. The associated eigenvector i s i ( 0 ) = i ( T ) .

(b)

1 i s again a c h a r a c t e r i s t i c m u l t i p l i e r , because t h e equation ( 2 ) has a f i r s t i n t e g r a l . So 1 i s a double eigenvalue of R ( T ) . The correspondi n g eigenspace, however, i s usually one-dimensional, generated by i ( 0 ) . For specjal systems, l i n e a r systems with constant c o e f f i c i e n t s , f o r instance, x = JAx, w i t h A* = A , i t can be two-dimensional.

(c)

i f i i s a c h a r a c t e r i s t i c m u l t i p l i e r , so i s real.

(d)

i f 1 i s a c h a r a c t e r i s t i c m u l t i p l i e r , so i s symplectic: R(T)*JR(T) = J .

-

A,

A

-1

because

R(T)

is

, because R(T) i s

I66

I. Ekeland

I t f o l l o w s from t h i s t h a t c h a r a c t e r i s t i c m u l t i p l i e r s come i n p a i r s (A,A-l) o r (A,A) i f A i s r e a l o r has modulus 1, o r i n quadruples

(A,T,X-~, A - ~ ) otherwise. L e t A # 1, 1x1 f 1 be a simple c h a r a c t e r i s t i c m u l t i p l i e r . The theory o f Krein ( [ 6 1, [ 7 1 , [ll];see also [ 9 1) assigns t o i t a d e f i n i t e sign, i n such a way t h a t i f A i s p o s i t i v e i n t h e sense o f Krein, then A i s negative. More generally, i f A # 1 i s an m-fold m u l t i p l i e r on t h e u n i t c i r c l e , we introduce the r o o t subspace

V

C

C2n:

V = Ker (R(T) -

(6)

hI)m.

I t i s m-dimensional and the r e s t r i c t i o n t o V o f t h e h e r m i t i a n form i s non-degenerate. L e t q be i t s index, i.e., the number o f n e g a t i v e squares i n a d i a g o n a l i z a t i o n . We then say t h a t A has Krein t y p e (m-9.9) ( q times n e g a t i v e and (m - q ) times p o s i t i v e ) .

-i(Js,c)

We now agree on a couple o f d e f i n i t i o n s : D e f i n i t i o n 1.

A T-periodic solution

(7)

dim Ker(1

-

x

of

x

= JH'(x)

i s non-deqenerate if

R ( T ) ) = 1.

0

D e f i n i t i o n 2. A T - p e r i o d i c s o l u t i o n x o f = JH'(x) i s non-resonant i f 1 has m u l t i p l i c i t y 2 as a c h a r a c t e r i s t i c m u l t i p l i e r , i f t h e remaining m u l t i p l i e r s on t h e u n i t c i r c l e can be w r i t t e n : A. =

(8) with

0


0 . Then doh ah? n E N

Proposition 2 . 4 . L e L g doh name vo E r := d i s t ( M ' ,

I I F ( E~ M I ) e-i ho

g(p)

F(X E

M)II 5

cn

(1

+ I ti

t t-)-n

.

(2.6)

.

The cavLltutclou2 Cn depend only an t h e nhape 0 6 g 16 a 6amZq 0 6 6uvzcLiavu g depe n h b t n O O t h & j (e.9. in t h e topology 06 .S(R")) on a paharn&m vmqing i n a compact s d , t h e n ,the c o n ~ t a m 2Cn can be chosen unLt(0tztdy 6vh t k i h ~m.2.q.

For the proof one can use a G a l i l e i transform t i o n and show t h a t the inverse Fourier transform $ ( u ) of @ ( p ) = exp(- i . t p h /2m) g ( p t m v o ) decays rapidly in I u I and I tl f o r I u I L v l t l by the s t a t i o n a r y phase estimate. Note t h a t t h e support condition on g says t h a t there i s an a > 0 such t h a t supp g c i p E Rw 1 I p - m voI < m ( v - a ) ] . The additional separation al tl takes care of t h e rapidly decaying "quantum t a i l s " . [ 2,4]. be any localized operator such t h a t - ( 1tE) IIF(Ixl > R) A II const ( 1 + R ) 9

Let A

and l e t

+

E

C:(R")

or +(P) = (ho

-

z ) - ~ .Then by Corollary 2.2

IIF(lxl > R ) g ( P ) All

R/2) All+IIF(I X I >R) @ ( P ) F ( I XI r t R ) g ( h ) F ( I XI< R ) i l < c o n s t ( l t r ) - (

the^ 1+E)

(2.16)

P r o o f . IIF(I X I >r+R) g ( h ) F ( I XI r t R ) [ g ( h ) t

-

g ( h o ) l 1I

!IF([ XI >r+R) g ( h o ) F(I XI O, R>1,

LeL

8E

C E ( R ) w L t h supp

llF(I XI >(1+2a)(RtvI tl ) ) e

< const(R+vl tl )

-iht

6c

(-

2

m,

m v / Z ) , v>O. the^ g o t any

o

g ( h ) F(1 XI ( 1 + 2 a ) ( R t v l t l ) ) [ g ( h )

-

(2.19)

g ( h o ) ] IIzconst(R+vI t l ) - ( 1 t E )

Therefore i t i s s u f f i c i e n t t o study 1I F( I XI > ( 1+2a)( R+vl t l ) ) g ( h o ) e-jht

< II F ( I XI > ( 1+2a) ( R t v l t l

+

IIF(I XI > ( l + Z a ) ( R t v l t l

d ( h ) F ( I X I (l+Ea)(R+vl t l ) ) g ( h o ) e-ihO(t-s)

8 ( h ) F ( I XI a(R+vl tl ) ) Vll

.

(2.22c)

The i n t e g r a n d in (2.22a) decays r a p i d l y i n ( R + v I t l ) u n i f o r m l y i n O i sld tl a g a i n by P r o p o s i t i o n 2.4. T h i s i m p l i e s f a s t decay o f t h e i n t e g r a l . S i m i l a r l y t h e i n t e grand i n t h e f i r s t l i n e o f (2.22~)decays r a p i d l y i n I t - s l . T h e r e f o r e t h e i n t e g r a l i s bounded n i f rmly i n R and t and t h e decay IIF(I XI >a(Rtvl tl ) ) Vll 5 < c ( l t a ( R t v 1 tl ) ) - r l + E y remains.

V. Enss

180

With the shorthands y(s)

= SF(1 XI =

>(1+2a)(R+vlsI ) ) e

(R+vl S I ) - ( I + € )z ( s )

-ihs o g ( h ) F ( I XI o

YII

T O

(3.13)

P r o o f . From t h e r e l a t i v e boundedness o f t h e H a m i l t o n i a n s i t f o l l o w s t h a t f o r g i and E t h e r e i s an E ' = E ' ( E , € ) such t h a t f o r a l l a

VCi-E

(3.14)

IIF(h">E') F ( H < E ) l l < c , l l F ( k ~ > E ' ) F(H<E)II<E. We keep t h i s E ' f i x e d i n t h e sequel. By Lemma 3.4 t h e r e i s a TI p,E, and E such t h a t f o r a l l CY 1 T+T1 SUP T E

!R

j

d t IIF(1 xCil < p ) Pcont(ha) F ( h " < E ' ) ,-iHat

~II<EII@II.

(3.15)

T

Keep T i f i x e d i n t h e sequel and determine R t h e f o l l o w i n g e x p r e s s i o n i s bounded by 3 ~ : Sup

depending on

IIF(I xal

O a g i v e n above,

P ~ U t llcCN(ltttR)-N,

(4.11)

F ( l yal < $ t 2 v t ) l l C N ( l t t t R ) - N .

(4.12)

P r o o f . I t i s s u f f i c i e n t t o t r e a t any o f t h e f i n i t e l y many summands

1 -ikgt IIF(IyaI < 9 t 2 v t ) e g.(q /v J

dist(M.+v.t,O)

J

a

=(R2t ( v . t ) ' ) l / '

J

J

a

1

(4.13)

F(ya E Mj)ll.

1. &(Rt

v. t )

R t3vt

J

(4.14)

h o l d s with vsl v . I / 5 as g i v e n above. Now P r o p o s i t i o n 2.5 w i t h r = R ( l J Z ( 4 . 1 1 ) . F o r (4.12) a n a l o g o u s l y .

-

1 / 2 ) shows 0

R e c a l l t h a t Ha=h"a+k: , and denote by Pq t h p r o j e c t i o n o n t o an e i g e n v e c t o r o f ha Pauf r e l a t e s t o t h e same p a i r i n g cx w i t h e i g e n v a l u e E j E &P(ha)c(- m , O ] . Operators on t h e f a c t o r spaces and t h e i r n a t u r a l e x t e n s i o n t o t h e f u l l t h r e e body H i l b e r t space a r e denoted by t h e same l e t t e r s .

.

(4.15) (4.16) l i m E(R)=O. R+ m P r o o f . There i s a

Then

Py =g(ha) P;

8E

Ct(R ) w i t h 8 ( w ) = l i f

w E

opp(ha), supp

8

.

c ( - m,pav2/2).

(4.17)

Next choose p ( 2 ) l a r g e enough such t h a t f o r g i v e n E ' > O , P L ~ ( E ' ) IIF(I XaI

'0)

PYII < E 1 / 4 .

(4.18)

To prove (4.15) i t i s t h e n s u f f i c i e n t t o show t h a t (4.19) F o l l o w i n g Cook's method t h i s i s bounded by t h e i n t e g r a l o f t h e norm o f t h e d e r i vative

7 dt

I1 (H-Ha)

< 1

7 d t IlVB(x3) e-ihat

0

-

emH i at

8(ha) F(I xaI

5 WaV

ga E C i ( R ) , Ozgazl, g a ( w ) = 1 i f w q -a Then f o r a#B:ga(hE) g'(h8) F(Ho>6) = 0. Moreover there i s a dinite s e t o f v e l o c i t i e s vg

There a r e

33

pal5pav

and H 0 s t o -

(7.11) 2 v / 4 and ga(u)=O f o r WLU,V 2/ 2 .

Rv .I v:l~5v,and

functions

C t ( W U ) , z[3g1*51,

E

210

g J. ( q a

=

o

z [ ?$(q ) I 2 j

J

if =

lqa

1 if

-

v

V?I>

a J -

6Ho<E

v

a ' '

and

hi 5

(7.12) pav 2 / 2 .

B

This decomposition i s e s s e n t i a l l y the same as ( 4 . 6 ) . On the range of F ( 6 < H o < E ) we then have (7.13) The g"(h;) ( a n d a l s o t h e i r squares) take care of t h a t p a r t of the 2u-dimensional momentum space, where t h e r e l a t i v e momenta of a p a i r might be close t o zero. On the remainder in Ran F(6 O towards i n f i n i t y ; ( i i i ) t h e r e a r e no zero-energy bound s t a t e s o r resonances i n any two-body subsystem. The f i n i t e n e s s o f t h e n e g a t i v e s p e c t r a o f subsystem H a m i l t o n i a n s , i . e . a f i n i t e number o f s c a t t e r i n g channels, i s i m p l i e d by (ii). Recently these l i m i t a t i o n s were overcome i n two d i r e c t i o n s . Loss and S i g a l 1121 c o u l d a v o i d t h e i m p l i c i t c o n d i t i o n ( i i i ) assuming f a s t e r decay i n ( i i ) ; and M e r k u r i e v [ 131 handles t h e a d d i t i o n o f Coulomb i n t e r a c t i o n s i n t h r e e dimensions as w e l l . Hard and v e r y d e t a i l e d e s t i m a t e s o f t h e s i n g u l a r two-body r e s o l v e n t s o r Coulomb r e s o l v e n t s , r e s p e c t i v e l y , a r e needed. Under t h e assumptions ( i ) - ( i i i ) Hagedorn and P e r r y [lo] use m a i n l y time-depend e n t methods. T h e i r p r o o f seems t o e x t e n d much e a s i e r t o h i g h e r p a r t i c l e numbers t h e n t h e r e s o l v e n t methods do. R e c e n t l y Mourre [ 151 a p p l i e h i method o f c o n j u g a t e o p e r a t o r s t o t r e a t p a i r p o t e n t i a l s decaying l i k e I XI towards i n f i n i t y w i t h o u t c o n d i t t o n ( i ) and ( i i i ) . Regarding a s y m p t o t i c completeness o u r g e o m e t r i c a l , time-dependent method covers a l l t h e s e r e s u l t s and i n g e n e r a l i t i s t e c h n i c a l l y l e s s demanding. I t p e r m i t s t o t r e a t more g e n e r a l long-range p o t e n t i a l s and t h e two body subsystems may have p o s i t i v e energy e i g e n v a l u e s (we t h e n need i n f o r m a t i o n about t h e decay o f these e i g e n f u n c t i o n s , w h i c h i s known). Moreover we a d m i t i n many cases s t r o n g e r l o c a l s i n g u l a r i t i e s and we can h a n d l e e a s i l y nonl o c a l p o t e n t i a l s , which d e s c r i b e v e l o c i t y dependent f o r c e s . The advantage o f t h e r e s o l v e n t e q u a t i o n method, however, i s t h a t i t y i e l d s a d d i t i o n a l r e s u l t s , e.g., t h e Faddeev e q u a t i o n s a r e a u s e f u l t o o l f o r c a l c u l a t i o n s .

-htef

V a r i o u s r e s u l t s about absence of s i n g u l a r continuous spectrum were c o n t a i n e d i n t h e papers on a s y m p t o t i c completeness. The most thorough s t u d y o f t h i s q u e s t i o n was g i v e n by Mourre [ 141 and extended i n [ 161 T h e i r assumptions a r e more r e s t r i c t i v e w i t h r e g a r d t o l o c a l s i n g u l a r i t i e s , b u t t h e y can a d m i t more general smooth l o n g - r a n g e f o r c e s t h a n we do.

.

Our method was f i r s t developed f o r p o t e n t i a l s c a t t e r i n g [ 1 , 2 and r e f e r e n c e s

203

Scattering Theor?,fur Three Particles

g i v e n t h e r e ] and t h e n extended t o three-body systems [3]. The f u l l y worked o u t p r o o f i n t h e g e n e r a l case w i l l be g i v e n i n [ 71. The a u x i l i a r y r e s u l t s g i v e n i n S e c t i o n I 1 have been shown under weak assumptions i n [4], and those o f S e c t i o n V i n [5] f o r t h e two-body case ( t h e g e n e r a l i z a t i o n t o t h e t h r e e body system f o r a l a r g e c l a s s o f p o t e n t i a l s i n s t r a i g h t f o r w a r d ) . The e x t e n s i o n t o h i g h e r p a r t i c l e numbers i s p r e s e n t l y under i n v e s t i g a t i o n . The e x i s t e n c e o f channel wave o p e r a t o r s w i t h o u t i m p l i c i t c o n d i t i o n s on t h e subsystems seems t o be new i n dimensions U = 1 and 2.

References

1. V . Enss: A s y m p t o t i c completeness f o r quantum mechanical p o t e n t i a l s c a t t e r i n g I , Commun. Math. Phys. 61, 285-291(1978). 2. --: Geometric methods i n s p e c t r a l and s c a t t e r i n g t h e o r y o f S c h r o d i n g e r operat o r s , i n : Rigorous Atomic and M o l e c u l a r Physics, G. V e l o and A.S. Wightman eds, Plenum, New York 1981 . (Proceedings Erl’ce 1980).

3. --: Completeness of t h r e e body quantum s c a t t e r T n g , i n : Dynamics, Algebras, Processes, P. Blanchard ed., L e c t u r e Notes i n Mathemati’==, p r e s s . (Proceedings B i e l e f e I d 1981).

S p r i n g e r , in

4. --: P r o p a g a t i o n p r o p e r t i e s o f quantum s c a t t e r i n g s t a t e s , J. Func. Anal. 52, 219-251 (1983). 5 . --: A s y m p t o t i c o b s e r v a b l e s on s c a t t e r i n g s t a t e s , Commun. Math. Phys.

E,

245-268(1983). 6. - - : Three body Coulomb s c a t t e r i n g t h e o r y , i n : Proceedings 11. 1nt.Conference on O p e r a t o r Algebras, I d e a l s , and t h e i r A p p l i c a t i o n s i n T h e o r e t i c a l Physics, L e i p z i g 1983, Teubner, L e i p z i g , t o appear.

7. --: Three body quantum s c a t t e r i n g t h e o r y , i n p r e p a r a t i o n . 8 . L.D. Faddeev: Mathematical Aspects o f t h e Three Body Problem i n t h e Quantum S c a t t e r l n g Theory, I s r a e l Program f o r S c i e n t i f i c T r a n s l a t i o n s , Jerusalem,

1965. 9. J. G i n i b r e and M. M o u l i n : H i l b e r t space approach t o t h e quantum mechanical three-body problem, Ann. I n s t . H. Poincari! A 21, 97-145(1974). ~

10. G.A. Hagedorn and P.A.

P e r r y : A s y m p t o t i c completeness f o r c e r t a i n t h r e e 36, 213-232(1983). body S c h r o d i n g e r o p e r a t o r s , Commun. Pure Appl Math. -

.

11. P. Hartmann: O r d i n a r y D i f f e r e n t i a l Equations, W i l e y , New York 1964; p. 24. 12. M. Loss and I . M . S i g a l : The t h r e e body problem w i t h t h r e s h o l d s i n g u l a r i t i e s , p r e p r i n t , ETH Z u r i c h , 1982. 13. S . P . M e r k u r i e v : Three-body Coulomb s c a t t e r i n g , i n : New Developments i n Mathem a t i c a l Physics, H. M i t t e r and L. P i t t n e r eds., A c t a Phys. A u s t r i a c a , Suppl. ’23, - 6 5 - T T o ( ) , (Proceedings Schladming 1981); and r e f e r e n c e s g i v e n t h e r e .

14. E. Mourre: Absence o f s i n g u l a r c o n t i n u o u s spectrum f o r certal’n s e l f - a d j o i n t o p e r a t o r s , Commun. Math. Phys. 78,391-408( 1981).

204

V. Enss

15. E. Mourre: Operateur conjugues e t p r o p r i e t e s de p r o p a g a t i o n 11, remarques s u r l a compl’etude asymptotique des systemes i t r o i s c o r p s , p r e p r i n t CNRS-Marseille, 1982. 16. P. Perry, I . M . S i g a l , B. Simon: S p e c t r a l a n a l y s i s o f N-body S c h r o d i n g e r operat o r s , Ann. Math. 114, 519-567(1981). 17. M. Reed and B. Simon: Methods o f Modern Mathematical P h y s i c s , V o l . I - I V , demic Press, New York, 19/2-19/9.

Aca-

18. J . G i n i b r e : S o e c t r a l and s c a t t e r i n q t h e o r y of t h e S c h r o d l n s e r e a u a t i o n f o r three-body systems, i n : The S c h r o d i n g e r G u a t i o n , W. T h i r r i n g and P. Urban eds., A c t a Phys. Austr., Suppl. 95-138(1977) (Proceedings Vienna 1976).

17,

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

205

A LEFT DEFINITE TWO-PARAMETER EIGENVALUE PROBLEM

Melvin Faierrnan Department of Mathematics University of t h e Witwatersrand Johannesburg South Africa

I n t h i s work we discuss t h e spectral p r o p e r t i e s , and i n p a r t i c u f a r the eigenfunction expansion problem, f o r a l e f t d e f i n i t e two-parameter system of ordinary d i f f e r e n t i a l equations o f the second order. INTRODUCTION W e c o n s i d e r t h e s i m u l t a n e o u s two p a r a m e t e r s y s t e m s

are r e a l - v a l u e d smooth f u n c t i o n s w i t h pi > 0 . We where p i , qi, A i , and Bi a l s o suppose t h a t : ( i ) A = A1B2 - A2B1 assumes b o t h p o s i t i v e and n e g a t i v e ( t h e p r o d u c t of t h e i n t e r v a l s 0 xr Q 1 , r = 1 , 2 ) , and v a l u e s i n I, (ii) t h e r e e x i s t r e a l numbers T l , T, s u c h t h a t T I A r ( x r ) + T2Br(xr) > 0 i n 0 Q xr 1 f o r r = 1, 2 . On a c c o u n t o f ( i i ) t h e r e i s no l o s s o f g e n e r a l i t y xr 1 f o r r = 1, 2 i n assuming h e n c e f o r t h t h a t A, > 0 i n 0

.

f o r (il, x 2 ) , w e c a l l h* an e i g e n v a l u e o f t h e s y s t e m (1-4) i f f o r ( 2 r - 1) h a s a n o n - t r i v i a l s o l u t i o n , s a y yr!xr, X*) , s a t i s f y i n g ( 2 r ) The p r o d u c t y1 ( X I , A * ) y2 ( x p , A * ) 1 s c a l l e d an e i g e n f u n c t i o n o f f o r r = 1, 2 (1-4) c o r r e s p o n d i n g t o A * . Then u n d e r t h e g i v e n a s s u m p t i o n s t h e o b j e c t o f t h i s paper w i l l accordingly be t o d i s c u s s t h e s p e c t r a l p r o p e r t i e s of t h e system (1-4) , and t h e n t o i n v e s t i g a t e t h e a s s o c i a t e d e i g e n f u n c t i o n e x p a n s i o n problem.

Writing

A

=

A* ,

.

I n o r d e r t o a r r i v e a t t h e r e q u i r e d r e s u l t s , it i s n e c e s s a r y t o i n v e s t i g a t e t h e s p e c t r a l p r o p e r t i e s o f a c e r t a i n e l l i p t i c boundary v a l u e problem a s s o c i a t e d w i t h The r e a s o n f o r t h i s w i l l b e made c l e a r i n t h e s e q u e l . t h e s y s t e m (1-4) denote t h e A c c o r d i n g l y , w r i t i n g x f o r ( X I , x2) and l e t t i n g R and t h e i n t e r i o r and b o u n d a r y , r e s p e c t i v e l y , of I2 , l e t us f i x o u r a t t e n t i o n upon t h e boundary v a l u e problem

.

LU

where

L

r

-

XA(x)u = 0

for

x E

R ,

u(x) = 0

for

x E

r,

(5)

denotes the e l l i p t i c o p erato r

D~ = a / a x r ,

a1 =

P1A2

,

a2 = p2A1

, and

q = qiA2 + qzA1

.

With

(

,

)

and

M. Faierman

206

11 11 denoting t h e i n n e r product and norm, r e s p e c t i v e l y , i n a l s o consider t h e D i r i c h l e t form a s s o c i a t e d with ( 5 ) , 2 B(v,u) = (Drv, arDru) + (v,qu) f o r v, u 6 V = r=l

1

,2? = L2

(R) ,

l e t us

;)(a) ,

where we r e f e r t o Agmon (1965) f o r terminology. W e note from Faierman (1978) t h a t V i s a closed subspace of H I (R) and t h a t an element u i n H I belongs t o V i f and only i f i t s t r a c e on is zero. I f y denotes t h e lower bound of B , then it is usual i n t h e i n v e s t i g a t i o n of t h e s p e c t r a l p r o p e r t i e s of t h e system (1-4) t o consider t h e c a s e s y 2 0 , y < 0 s e p a r a t e l y ; and s i n c e t h e f i r s t of t h e s e c a s e s has already been t h e s u b j e c t of some i n v e s t i g a t i o n by H i l b e r t (1953) and Faierman ( 1 9 8 1 ) , we s h a l l henceforth r e s t r i c t our i n v e s t i g a t i o n t o t h e case

(a)

r

I t w i l l always be supposed i n t h e sequel t h a t

ASSUMPTION 1.

y
y

I 1


0, we s e t

h

(Au)(x)

= vull(x)

with D ( A ) = cu

E

for

2

I

L (o,i)

I t i s v e r y w e l l known t h a t L ~ ( o , I ) . We now l e t

A

u

x E

E

[0,11

H ~ ( o , n~ HI 2 ( 0 ~ 1 ) )

generates an a n a l y t i c semigroup

{T(t)

1

t

2

01

on

A

and d e f i n e

= R 8 h

L2(0,1)

A

X

X

-+

by

;)

i =(-;

3.1 with

2

A:

D(A)

= R

e

D(A).

I t i s n o t d i t f i c u l t t o show t h a aemigroup { T ( t ) I t 2 O} on w i l l have norm

A

6

I / C ~ Y V I I l y = VYl"l Henceforth we s h a l l use y = 1/2 15 P ( ) : [O,T] { (t)

I

t

E

+

CO,Tl}

R

A i s t h e i n f i n i esimal g e n e r a t o r o f an a n a l y t i c and t h a t 0 p(A). The i n t e r p o l a t i o n spaces +

IIvlly

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 we d e f i n e by

We have t h e f o l l o w i n g g l o b a l e x i s t e n c e theorem Theorem 3.3. Let and r e s p e c t i v e l y ; suppose t h a t

t) I t (t) 1 t

[O,Tl} t

0}

be d e f i n e d v i a ( 3 . 1 ) and ( 3 . 2 ) i s t h e a n a l y t i c semigroup on 2

216

W.E. Fitzgibbon

A.

generated by If p f I U , vo( 11 E 2112 t h e n f o r any [O,TI -+ X s a t i s f y i n g o n l y one c o n t i n u o u s f u n f t i o n

(3.4) Moreover

(3.5)

v:

P ( t ) = ? ( t ) P o +Jt ? ( t - s)i(s)p(s)ds 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 on C0,TI

p( )

T

>

0

there exists

and s a t i s f i e s

p ’ ( t ) = ;iY”(t) f h y q t )

I n d i c a t i o n o f p r o o f The S a b o l e v s k i i Theorem p r o v i d e s a l o c a l r e s u l t . I f one can e s t a b l i s h t h e boundedness o f t h e e x p r e s s i o n 3 ( t ) p ( t ) one can use t h e methods of c l a s s i c a l o r d i n a r y i f f e r e n t i a l e q u a t i o n s t o e x t e n d o u r s o l u t i o n t o C0,Tl. i s e s t a b l i s h e d by showing t h a t I ( v x ( - , t ) I I i s The d e s i r e d bound on ! ( t ) v ( t ) bounded. M u l t i p l i c a t i o n o f t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n i n (1.1) by v and m u l t i p l i c a t i o n by v and i n t e g r a t i o n on C 0 , l l produces t h e energy e q u a l i t y

1/2 d / d t ( l l v l 1 2 ) + v ( l l v X 1 l 2 ) = U / \ V / / ~ T h i s shows t h a t 1 ( v ( - , t ) I and an bounded and r e l a t e s t h e boundedness The boundedness o f t h e t i m e d e r i v a of I I v x ( * , t ) I I t o t h a t o f I I v t i v e i s a consequence o f a u n i q u h e s s estimate. More s p e c i f i c a l l y we show t h a t i f (Ui, vi) i = 1 o r 2 a r e s o l u t i o n s t o ( 1 , l ) w i t h i n i t i a l c o n d i t i o n s (Ui(0), v i ( x ) ) and I ( t ) = {lU,(t) - U 2 ( t ) 1 2 + ( l v l ( * , t ) - v,(.,t)l(l then there

I

exists a

K(t)

so t h a t

dI(t)dt

2

I(t)

(!:;

1.

satisfies the differential inequality

K(t)I(t)

t h e desired r e s u l t follows. We immediately o b t a i n t h e f o l l o w i n g r e p r e s e n t a t i o n P r o p o s i i i o n 3.6.

Assume t h a t t h e c o n d i t i o n s o f Theorem 3.3.

il/* i n the

t h a t { F ( t ) I t 2 O} i s d e f i n e d o n and y( ) i s t h e s o l u t i o n t o (3.4) then

a r e s a t i s f i e d and

manner o f (2.4).

If

Po E &2

I n 171 t h e a u t h o r has u t i l i z e d t h e Hopf b i f u r c a t i o n Theory o f n o n l i n e a r semigroups o f Marsden C131 t o d i s c u s s t h e q u a l i t a t i v e b e h a v i o r o f a r e l a t e d autonomous model. Forthcoming work s h a l l e x p l o i t t h e s e m i l i n e a r s t r u c t u r e o f (1,l) t o examine t h e b e h a v i o r o f s o l u t i o n s t o modulated Burgers systems.

A Semigroup Approuck to Burger's System

211

REFERENCES

Ill

C.M. Burgers, " A mathematical model i l l u s t r a t i n g t h e t h e o r y o f t u r b u l e n c e , " Advances i n A p p l i e d Mechanics (R. von M i l e s and T. von KFirm'an, ed.) Academic Press, New York (1948), 171-199.

a,

r 2 1 J.D. Cole, "On a q u a s i l i n e a r p a r a b o l i c e q u a t i o n o c c u r i n g i n aerodynamics," Q u a r t , Appl. Math. 9 (19511, 225-236. L31

T. D o l t k o , "On t h e one dimensional Burgers e q u a t i o n e x i s t e n c e , uniqueness, s t a b i l i t y " , Zeszyty Nauk. Univ. J a g i e l l o n Prace Mat 23 ( i n p r e s s ) .

141

, "On c l a s s i c a l s o l u t i o n o f t h e one dimensional Burgers equation," Zeszyty Nauk Univ. J a g i e l l o n Prace Mat. 3 ( i n p r e s s ) .

I51

, "Some remarks c o n c e r n i n g t h e u n s t a b i l i t y o f t h e on dimensional Burgers equation," Ann. Polon, Math ( t o appear).

C 61

"The two d i m e n s i o n a l B u r g e r ' s t u r b u l e n c e model Univ.-2 1 ( i 9 8 1 ) , 809-823.

171

W. F i t z g i b b o n , " A two dimensional model f o r turbulence,"

C 81

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C 91

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,"J

Math. Kyoto

( t o appear).

C 101 J . G o l d s t e i n , "Semigroups o f o p e r a t o r s and a b s t r a c t Cauchy problems,"

lec-

t u r e notes, T u l a n e U n i v e r s i t y , 1970.

C 111 D. Henry Geometric Theory o f S e m i l i n e a r P a r a b o l i c Equations, L e c t u r e Notes i n Mathematics, v o l 840 S p r i n g e r - V e r l a g , B e r l i n , 1981.

C 121 I131

C 141

C.O. Horgan and W.E. Olmstead, " S t a b i l i t y and uniqueness f o r a t u r b u r l e n c e model o f Burgers," Q u a r t Appl Math, (1978), 121-127.

J. Marsden, "The Hopf b i f u r c a t i o n f o r n o n l i n e a r semigroups", B u l l .

Math. SOC. , 79 (1973) , 537-541.

w.

J . Marsden and M. McCracken, The B i f u r c a t i o n and I t s A p p l i c a t i o n s , A p p l i e d Mathematical S c i e n c e s x , S p r i n g e r - V e r l a g , B e r l i n , 1976.

C 151 W.E.

Olmstead and S.H. Davis. " S t a b i l i t y and b i f u r c a t i o n i n a Modulated Burgers system," A p p l i e d Mathematics T e c h n i c a l No. 7925, Northwestern Univ e r s i ty, 1980.

C 161

A. Pazy, "Semigroups o f 1 i n e a r o p e r a t o r s and a p p l i c a t i o n s t o p a r t i a l d i f f e r e n t i a l equations," L e c t u r e Notes No. lJ, U n i v e r s i t y o f Maryland, C o l l e g e P a r t , MD. 1974.

i 171

P.E. SobolevskJ;, "On e q u a t i o n s o f p a r a b o l i c t y p e i n a Banach space," Trud Moscor. -Mat Orsc 10 (1961), 297-350. T r a n s l . Amer. Math SOC. 240 ( 1 9 7 8 e 129-143.

1181 G.F. Webb, " E x p o n e n t i a l r e p r e s e n t a t i o n o f s o l u t i o n s t o an a b s t r a c t semiJ . Math. 70 (1977), 269-279. l i n e a r d i f f e r e n t i a l equation,"

w.

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DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

219

ON THE EIGENVALUES OF NON-DEFINITE ELLIPTIC OPERATORS

J. F l e c k i n g e r k and A.B.

Mingarelli

Department of Mathematics U n i v e r s i t y of O t t a w a Ottawa, Canada, K1N 9B4

I.

INTRODUCTION L e t R be a bounded open s e t i n IRn

w i t h a smooth boundary

r

.

Let A be a

l i n e a r d i f f e r e n t i a l o p e r a t o r of o r d e r 2m d e f i n e d on R which i s uniformly s t r o n g l y e l l i p t i c and f o r m a l l y s e l f - a d j o i n t .

We d e n o t e by A t h e p o s i t i v e r e a l i z a -

(1)

t i o n , s e l f - a d j o i n t and unbounded i n LL(Q) of a boundary v a l u e problem a s s o c i a t e d 2 w i t h A on R. W e suppose t h a t t h e imbedding of t h e domain D(A) i n t o L (R) i s compact w i t h dense r a n g e and t h a t t h e r e e x i s t s y > 0 s u c h t h a t f o r e v e r y U E D ( A ) , 2 (Au,u) 2 v l l ~ l l

.

Consider now c , g : R + I R

e a c h o f which i s c o n t i n u o u s on.

5

and g a t t a i n s b o t h

p o s i t i v e and n e g a t i v e v a l u e s : - m S g ( x ) 5 M and meas {x: g ( x ) = O } = 0 . W e s t u d y h e r e t h e e i g e n v a l u e s of t h e boundary v a l u e problem Lu: (') where B

1

( A + c ) u = Xgu

{BuI,

on

R ,

U E

D(A),

= 0

d e n o t e s t h e u s u a l homogeneous D i r i c h l e t o r Neumann boundary c o n d i t i o n s

Because of o u r hypotheses t h i s problem i s , i n g e n e r a l , non-definite i n t h e s e n s e t h a t L i s a n o t n e c e s s a r i l y p o s i t i v e o p e r a t o r and g a f u n c t i o n which may change its sign.

A t y p i c a l problem i s , f o r example,

W e o b t a i n h e r e some r e s u l t s on t h e e i g e n f u n c t i o n s and e i g e n v a l u e s (which w e

together c a l l eigenpairs, f o r brevity),

and on t h e e x i s t e n c e ( o r non-existence)

of p r i n c i p a l e i g e n v a l u e s ( i . e . ,

an e i g e n v a l u e whose a s s o c i a t e d ( r e a l ) eigenfunc-

t i o n d o e s n ' t change i t s s i g n i n

R).

The problem (P) h a s i t s o r i g i n s i n a p a p e r of R . G . D .

Richardson [ 8 ] i n t h e

(2)

J. Fleckinger and A. B. Mingarelli

220

one-dimensional case, and was studied more recently by Mingarelli [ 6 , 7 ] .

In the multi-dimensional case, when c is positive , results on eigenvalues and eigenfunctions can be found in Manes-Micheletti [5] and results on principal

...1 .

eigenvalues in [1,2,3,4,10, 11.

EXAMPLES IN THE ONE-DIMENSIONAL CASE. In his paper [ 8 ] Richardson mentions (without an example) that in such pro-

blems non-real eigenvalues may occur. We now list some examples of what may happens.

Consider the problem -y"+c(x)y y(a)

y(b)

2

-917 /16 on R

Let c(x)

Ex.1,

=

Xg(x)y

= =

0

=

(a,b),

.

( 0 , 2 ) and g(x)

=

R

on

=

1,

XE

(O,l), g(x) =-1, x t [1,2)

This problem has precisely one pair of non-real eigenvalues situated at about

.

+4.3628i Ex.2,

2

:1 - (9n /4),

Let c(x)

in ex.1 above,

R

Let R

Then h

=

=

In this case we note that (gy,y) 2

1 - (9n /4),

(0,4), c(x)

X E (CJ,1);

c(x)

[1.2) and g(x)

XE

Then X = l is an eigenvalue and y(x)

= (0,2).

associated eigenfunction. Ex.3,

2 (0,l); c ( x ) 5 -1- (917 / 4 ) ,

X E

2

=

f R

as

sin(3?rx/2) is an glyI2

=

0.

2 3 -1- (9T /4), X E (1,4)-

1 is an eigenvalue and yet its associated eigenfunction satisfies

f gy2 0 , A" > 0 ,

A*

If

>

-m

and i f t h e r e e x i s t s a p r i n c i p a l e i g e n v a l u e A.

then necessari-

. t h e n t h e r e d o e s n o t e x i s t any p r i n c i p a l e i g e n v a l u e .

= -m

By d e f i n i t i o n of A",

Proof.

{ ik::z; }

inf E D+

223

i f ho > A"

,

then there e x i s t s v

E

Df

such t h a t

.

T h e r e f o r e Qx (v) < 0 f o r some V E D(A) a n d h e n c e by lemma 1, ( L v , v ) / ( g v , v ) c Xo A. c a n n o t b e p r i n c i p a l . T h i s p r o v0 e s t h e f i r s t s t a t e m e n t . The p r o o f of t h e o t h e r i s similar.

COROLLARY 3 .

I f t h e r e exists u

E

D+ s u c h t h a t ( L u , ~ )5 0 t h e n t h e r e e x i s t s no po-

s i t i v e p r i n c i p a l eigenvalue.

COROLLARY 4 .

i, qi,

Let

If there exists

l S i S N ,

?A

JIn’

1 5n S

b e t h e e i g e n p a i r s of (Q) w i t h p. < 0 , (p. 5 0 ) .

N , s u c h t h a t (gQn,Un)

2

0 , ( (gqn,qn) > 0 ) , t h e n t h e r e

e x i s t s no p o s i t i v e p r i n c i p a l e i g e n v a l u e .

EXAMPLE.

L e t L = - A w i t h Neumann boundary c o n d i t i o n s .

If

1 g>O, n

then t h e r e is

no p o s i t i v e p r i n c i p a l e i g e n v a l u e , ( T h i s f o l l o w s from C o r o l l a r y 3 ) . THEOREM 4

I f A" >

--m

and i f Q

eigenvalue. Proof.

(u) t 0, f o r a l l u

Denote by s t h e smallest e i g e n v a l u e o f T

But by d e f i n i t i o n o f A",

11 vn 11

A*

A*

D(A), t h e n h* i s a p r i n c i p a l

= L-h*g

+

X*,

i.e.,

E D + such t h a t n Hence Qhx(vn) + 0 as n - t m

THE CASE p

1

A*

is p r i n c i p a l .

20.

The f o l l o w i n g r e s u l t s c a n now b e p r o v e n as i n [ 1 , 2 ] .

If

.

So s = 0 i s t h e smallest e i g e n v a l u e o f TA* ; a n a s s o c i a t e d e i g e n -

f u n c t i o n d o e s n ' t change s i g n and s o

LEMMA 2 .

: Then

t h e r e i s an i n f i n i t e sequence of v

= 1 and (Lvn,vn)/(gvn,vn)

sSQ,,~(V,)+O.

B.

E

vl>

0 t h e n Q,,(u)>-O,

for a l l UED(A).

J. Fleckinger and A. B. Mingarelli

224 LEMMA 3 .

> 0 and

0< A


0 as t + (1)

or w n-1

( t ) = z ( u - ' ) ( t ) + O as

t + m .

(11)

A t t h i s point we should point o u t t h a t i t i s possible t h a t equation ( E ) may have noncontinuable s o l u t i o n s . I n f a c t , Theorem 6 below gives s u f f i c i e n t conditions f o r (E) t o have no p o s i t i v e continuable s o l u t i o n s .

Next we w i l l s t a t e two lemmas which will be used i n proving some of our r e s u l t s . Lemma 1 . ([2,3; Lemma 11). Let u be a p o s i t i v e (n-u)-times continuously d i f f e r e n t i a b l e function on the interval [a,m) and l e t p be a p o s i t i v e continuous function on [a,m) such t h a t f[l/u(t)Idt = and the function w Moreover, l e t

uu

E

n-u)

-

i s u-times continuously d i f f e r e n t i a b l e on [a,-).

1

LI(~), i f 0 5 k

n-v-1

I f w n ( t ) E w ( " ) ( t ) i s of constant sign and n o t i d e n t i c a l l y zero f o r a l l l a r g e t , then there e x i s t t u ? a a n d a n i n t e g e r P , 0 5 P 5 n, with n + P even f o r on nonnegative o r n + L odd f o r on nonpositive, and such t h a t f o r every t tu L

>

0 implies w k ( t ) > 0 ( k = O,l,

. . . ,L - 1 )

and

P 5n

-

1 implies ( - 1 ) L+kw k ( t )

>

0 ( k = L , L + 1 , ..., n - 1 ) .

([2,3; Lemma 2 3 ) . If the functions u , U , w a n d wk a r e as i n Lemma 1 and f o r some k = 0 , 1 , ...,n - 2 w k ( t ) + c as t + m , then t ~ ~ + ~ ( t ) as + Ot-fm.

Lemma 2.

Our f i r s t theorem gives a g r o w t h r e s u l t f o r solutions o f ( E ) . Theorem 3. I f x ( t ) i s a positive continuable solution of ( E ) , then there e x i s t s a constant B 2 0 such t h a t x ( t ) / J ( t O , t )+ B as t + m .

239

Asymptotic BchaL'ior o f Positive Solutions

Proof. If x ( t ) > 0 i s a continuable solution o f ( E ) of type ( I ) , then successive integrations show t h a t w k ( t ) + m as t + - f o r k = O , l , ...,n - 2 . By L'HGpital's rule lim [ x ( t ) / J ( t o , t ) ] = lim z ( ' - ' ) ( t ) / ( v - l ) ! t + m

= B >

0.

t - f m

On the other hand, i f x ( t ) i s of type ( I I ) , then Lemmas 1 and 2 ensure t h a t t h e r e e x i s t s an i n t e g e r N with 0 5 N 5 n - 1 such t h a t U N ( t ) + E l 0 as t + - and

w k ( t ) + - as t + = f o r k Remark.




0 as

t + m

(11


0. We did n o t require ( 2 ) and were able t o show t h a t any solution of type ( I ) s a t i s f i e s ( 1 ) and any solution of type (11) s a t i s f i e s (1) with A 0. T a l i a f e r r o [5] proved a second order version of N a i t o ' s r e s u l t . Theorem 4.

I f there e x i s t s c

>

0 such t h a t

r f ( s , c J ( t O , g ( s ) ) ) d s=

(3)

m,

then every positive continuable solution of ( E ) i s of type ( I ) . Proof. If x ( t ) were o f type (11) then x ( g ( t ) ) 5 c J ( t O , g ( t ) ) f o r t 2 T f o r some T An i n t e g r a t i o n of ( E ) then y i e l d s a contradiction of ( 3 ) . to. Corollary 5. I f ( 3 ) holds f o r some c > 0 , then every p o s i t i v e continuable solut i o n x ( t ) of ( E ) s a t i s f i e s x ( t ) / J ( t O , t ) + B c as t + - . While Theorem 4 gives s u f f i c i e n t conditions f o r a l l positive continuable solutions o f ( E ) t o be cf type ( I ) , the following theorem gives conditions u n d e r which equation ( E ) has no p o s i t i v e continuable s o l u t i o n s . Theorem 6.

If

f o r every c

>

0 , then ( E ) has no positive continuable s o l u t i o n s .

For many functions f i f ( 4 ) holds f o r some c > 0 then i t holds f o r a l l c > 0. example where t h i s i s not the case i s the equation

( t x ' ) ' + k 2 e - l / t k/ t k e x

An

0, t > 0 where k > 0 i s a constant. Here ( 4 ) holds f o r any c > 0 such t h a t c + k 5 1 a n d f a i l s t o hold i f c + k > 1. Hence by Theorem 4 every p o s i t i v e continuable soluk t i o n i s of type ( I ) . Here x ( t ) = t n t - l / t i s such a solution. =

The equation

x" + t - 2 ( t 1 ' 2 + L n t

1/2

) x -1 ( t 1/2 )

=

0, t > 1 ,

s a t i s f i e s the hypotheses of Theorem 3. Note t h a t x ( t ) = t + Ln t i s a solution of type ( I ) ( u l ( t ) = x ' ( t ) = l + l / t ) which s a t i s f i e s x ( t ) / J ( l , t ) = ( t + L n t ) / ( t - l ) + l as t + = . Neither ( 3 ) nor ( 4 ) holds f o r t h i s equation.

240

J. R. Graef e t at.

DISCUSSION

F o r purposes o f comparison we now c o n s i d e r t h e e q u a t i o n ~ r ( t ) x ( ~ - ~ ) ( t ) -l ( f~( t), x ( g ( t ) ) ) = where r, g, f, and v a r e as b e f o r e .

(El)

o

In t h i s case a p o s i t i v e c o n t i n u a b l e s o l u t i o n

x ( t ) o f ( E ) w i l l be o f one o f t h e two t y p e s w n-1 ( t ) > 0

(111)

(t) < 0

(IV)

or w n-1

f o r sufficiently large t

to. i n g r e s u l t s f o r equation (E): i)

If x ( t )

>

o

I n [l]t h e a u t h o r s were a b l e t o prove t h e f o l l o w -

i s a solution o f

( i )o f

t y p e (111)

and

f o r every c ii) iii)

>

0, t h e n x ( t ) / J ( t O , t ) + A

If x(t)/J(tO,t)+A

>

0 as

t + m ,

> 0 as

t+m;

t h e n ( 5 ) h o l d s f o r some c

0;

If

f o r every c > 0, t h e n x ( t ) / J ( t O , t ) + m

as t + - .

While t h e r e i s n o t x a d i r e c t correspondence between s o l u t i o n s o f (E) o f t y p e ( I ) and s o l u t i ~ n so f (E) o f t y p e ( 1 1 1 ) , t h e r e i s an i n t e r e s t i n g s i m i l a r i t y between Theorem 3 (and N a i t o ' s r e s u l t ) and p a r t s i ) and i i ) above. One o f t h e b a s i c d i f f e r e n c e s i n t h e two e q u a t i o n s can be seen by comparing Theorem 6 and iii). REFERENCES

1.

Graef, J. R., Grammatikopoulos, M. K., and Spikes, P. W., On t h e p o s i t i v e s o l u t i o n s o f a h i g h e r o r d e r f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n w i t h a discont i n u i t y , I n t e r n a t . J. Math. Math. S c i . 5 (1982), 263-273.

2.

Gramrnatikopoulos, M. K., On t h e e f f e c t o f d e v i a t i n g arguments on t h e b e h a v i o r o f bounded s o l u t i o n s o f n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s , Ukrain, Mat. Z . 30 (1978), 462-473 (Russian).

3.

Grammatikopoulos, M. K . , A s y m p t o t i c and o s c i l l a t i o n c r i t e r i a f o r n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h d e v i a t i n g arguments, t o appear.

4.

N a i t o . M.. E x i s t e n c e and asvmDtotic b e h a v i o r o f o o s i t i v e s o l u t i o n s o f d i f f e r e n t i a l i n e q u a l i t i e s w i t h d & i a t i n g argument, F u n k c i a l . Ekvac. 22 (1979), 127-1 42.

5.

T a l i a f e r r o , S . , On t h e p o s i t i v e s o l u t i o n s o f y" + + ( t ) y - ' Anal. 2 (1978), 437-446.

= 0, N o n l i n e a r

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), I984

241

ENERGY ESTIMATES FOR SYMMETRIC HYPERBOLIC INTEGRO-DIFFERENTIAL EQUATIONS Ronald Grimmer* and M a r v i n Zeman Department o f Mathematics Southern I l l i n o i s U n i v e r s i t y Carbondale, I l l i n o i s 62901

1.

INTRODUCTION

We c o n s i d e r t h e Cauchy problem a s s o c i a t e d w i t h t h e l i n e a r i n t e g r o - d i f f e r e n t i a l equation au Ao(x,t)at(x,t)

= P(x,t,ax)u(x,t)

(I.E.)

+

.bt B ( x , t ; r

,ax)u(x,T ) & + f ( x , t )

u(x,O) = g ( x ) , x € R n , 0 5 t 5 T,

where t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n au AO(x, t )%(x , t ) = p ( x ,t ,ax)u(x, t 1 i s symmetric h y p e r b o l i c . We s h a l l a l s o assume t h a t B(x,t,T,ax) t and T , F and

can be f a c t o r e d as B = F P + K where f o r f i x e d

K a r e bounded o p e r a t o r s on Hs(Rn) and F i s s t r o n g l y d i f f e r e n t i a b l e

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

T.

One way i n which symmetric h y p e r b o l i c d i f f e r e n t i a l e q u a t i o n s have been s t u d i e d i s by t h e use of a - p r i o r i L2 e s t i m a t e s ( s o - c a l l e d energy e s t i m a t e s ) . l e a d t o t h e e x i s t e n c e o f u n i q u e weak s o l u t i o n s .

These e s t i m a t e s

Moreover t h e e s t i m a t e s p r o v i d e

r e g u l a r i t y r e s u l t s as w e l l as show t h a t t h e s o l u t i o n s propagate w i t h f i n i t e speed. The aim o f t h i s paper i s t o p r o v e energy e s t i m a t e s f o r

(I.E.) which a r e analogous

t o t h e e s t i m a t e s found f o r symmetric h y p e r b o l i c d i f f e r e n t i a l e q u a t i o n s . We w i l l p r e s e n t two examples a s s o c i a t e d w i t h continuum mechanics f o r m a t e r i a l s w i t h memory which s a t i s f y t h e c o n d i t i o n s we impose on ( I . E . ) .

The f i r s t example

d e a l s w i t h t h e g e n e r a l i z e d l i n e a r t h e o r y o f h e a t c o n d u c t i o n p o s t u l a t e d by G u r t i n and P i p k i n [ 4 ] .

Our second example d e a l s w i t h t h e e l e c t r o m a g n e t i c t h e o r y f o r

inhomogeneous a n i s o t r o p i c s t a b l e media w i t h memory.

We c o n s i d e r t h e c o n s t i t u t i v e

r e l a t i o n s proposed by V o l t e r r a [ 7 ] . 2.

PRELIMINARIES

We s h a l l w r i t e x = ( x n-tuple

*

ci =

,,..., x n )

o r 5 = (El

,..., En)

f o r a c o o r d i n a t e i n Rn.

( al,... ,an) o f n o n n e g a t i v e i n t e g e r s x" =

X ~ I

2x;

...

F o r an

x"n. n

Work by t h i s a u t h o r p a r t i a l l y supported by t h e N a t i o n a l Science Foundation under Grant No. MCS-8201322.

242

R. Grimmer and M. Zeman

Hs(Rn) i s t h e H i l b e r t space w i t h norm IluI/, d e f i n e d by

where u i s t h e F o u r i e r t r a n s f o r m o f u .

( U , V ) ~i s i t s i n n e r p r o d u c t .

By Q we s h a l l always mean an open s e t i n Rn.

We n o t e f i n a l l y t h a t C w i l l denote a

c o n s t a n t which may v a r y from l i n e t o l i n e w h i l e B ( X ) w i l l denote t h e Banach Algebra o f bounded o p e r a t o r s on t h e space X . I n o r d e r t o s t u d y ( I . E . ) we s h a l l impose a number o f c o n d i t i o n s .

( I ) The a s s o c i a t e d p a r t i a l d i f f e r e n t i a l e q u a t i o n

au

A0 ( X ,t )=( x ,t 1 = P ( X ,t ,a x 1u ( X ,t n au = C Ak(X,t)-(X,t) + A(x,t)u(X,t) k= 1 axk i s symmetric h y p e r b o l i c . By t h i s we mean t h a t Ao, Ak and A a r e N X A

A

where Ak,

k

Rnx[O,T]

= 0,

...,n,

a r e symmetric w h i l e A.

Friedrichs [2]).

(c.f.

N matrices

i s positive definite i n

F u r t h e r , we r e q u i r e t h a t A k ( x , t ) ,

k=O

,...,n,

la1 5 s , so t h a t Ak y i e l d bounded o p e r a t o r s on k = 1 , ...,n, a r e i n C([O,T],B(HS(Rn)))

have bounded d e r i v a t i v e a;Ak,

I n a d d i t i o n we ask t h a t Ak,

HS(Rn). w h i l e A.

i s i n C'([O,T),B(HS(Rn))).

( 1 1 ) The f u n c t i o n B(x,t,T,aX)

can be w r i t t e n as

= F(x,t,r,ax)P(x,t,ax)

B(x,t,r,ax)

05

+ K(X,t,r,aX),

T

5 t,

X

€Rn,

where F and K a r e z e r o o r d e r p s e u d o - d i f f e r e n t i a l o p e r a t o r s and t h e s e g i v e r i s e t o bounded o p e r a t o r s on Hs(IRn)). F u r t h e r F i s C 1 as a f u n c t i o n of T and f o r f i x e d t, 0

3.

5

t

5 T, F, K, FT E C([O,t],B(Hs(Rn)))

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

T.

THE B A S I C A - P R I O R 1 LL ESTIMATES

Before we p r e s e n t t h e L

2 e s t i m a t e s , we f i r s t t r a n s f o r m t h e e q u a t i o n

Ao$

= Au

n

a . c Ak ax

where A =

k=l

Since

is C

1 Ao$

1

in

*

oat

t ,b (FA + K')ud?+ f

= Au and

= Au

+ iu +

= Au

.

1Aog

= FAO and c o n s i d e r t h e system

$ F(x,t,T,aX);$h

+

,bt K ' u d? + f ( x , t )

we can i n t e g r a t e by p a r t s and g e t , a f t e r l e t t i n g v(x,O)

T , A

Ao$$

We l e t A

k

A + Au +

= Au + Au

-

+ F ( x ,t,t , a x ) v

-&

tFT(x,t

A*=Au.

oat

We n e x t c o n s i d e r t h e change o f v a r i a b l e s

so t h a t u = u ' and v = u ' + v ' .

,?

t

,ax)v(-r)dr +& K ' u d? + f ( x ,t)

= 0,

243

Energy Estimata .for Intcgro-Differential Equations

Then (1 ) becomes :

i

-

-&t

au' = A u ' + ;\u' + F u ' + F v ' A -

oat

A&'=

A % -

oat

n

, this

+,rt f O

T

- &t -FTv'd-r +,(, t K'u'd-r + f

3

oat

u'dT +,rt

O

FT v'd-r - &t K'u'd-c -

f.

system becomes

0 AJUt O

and

oat

-

- Fu' - F v '

[*o where

A

=A?!.-

oat

-

= -Au'

, .

A?!'-!

oat

-FTu'd.r

=

+ i U +,giUdT + F

:]U

a r e o p e r a t o r s o f o r d e r 0.

define

LU = A U

O t

-

AU

[t

Let A =

., - AU

-

:]

and A.

=

(Ao O

'1

and

A.

' .bt l3Ud-r.

We w i l l p r o v e t h e f o l l o w i n g e s t i m a t e f o r L : THEOREM 1 :

F o r e v e r y r e a l s, t h e r e i s a c o n s t a n t C ( s ) independent o f U ( b u t

depending on T) such t h a t (2)

llu(-J)lls

5 c(s)E(lU(-,O)llo

+

0 5 t 11

,$llLU(*,r)jlsdT)l,

f o r U E C' ([O,Tl,C;(n)). Proof: A

0

W e ' l l prove t h e r e s u l t i n a s e r i e s o f steps.

= I and s = 0.

We w i l l f i r s t assume t h a t

We w i l l t h e n p r o v e t h e e s t i m a t e f o r more general Ao, s t i l l under

t h e assumption t h a t s = 0. W e w i l l t h e n use t h e e s t i m a t e t o p r o v e t h e r e s u l t i n t h e g e n e r a l case. Step 1 : d 2 d ~ l l U ( t ) l I O= ( # t ) , U ( t ) ) O Using t h e f a c t t h a t Ak i s symmetric and k = 1,

...,n,

we have

[ ,a-:Vk U

d (U(t)&(t)),.

+

lo

=

,-k::[ - V

lo,

and l e t t i n g A k =

[? :]

R. Grimmer and M. Zeman

244

By Gronwal 1 ’ s inequal it y ,

liu(t)/lo f c

~ ~ u ( o ) l l+o

,t

/lLu(T)~lodTl.

Step 2: We now p r o v e e s t i m a t e ( Z ) , a g a i n f o r t h e case s = 0, f o r LU = A

(3) Since A

*

oat

- AU - AU

is p o s i t i v e - d e f i n i t e , d e f i n e Ail’‘

0 a positive

s p e c t r a o f A. V = Ao1/2U.

=

,[t B^ U ~ T . 0

-

1 6 X-+ 1 / 2 (XI 2 m ‘r

-AO)-’dA,

where l i s

p a t h o f i n t e g r a t i o n such t h a t Re X > 0 and i t c o n t a i n s a l l p o i n t i n i t s interior.

We n o t e t h a t Ail’‘

i s also positive-definite.

Then e q u a t i o n ( 3 ) becomes:

h

where A ‘ i n c l u d e s a l l o f t h e “bounded“ terms.

Hence

By G r o n w a l l ’ s i n e q u a l i t y , we t h e n have

Next, we o b t a i n a s i m i l a r e s t i m a t e f o r L*, t h e f o r m a l a d j o i n t o f L .

Taking

Let

Energj, Estimates for Iritegro-DifferentiulEquations

{U,V}

=

.(,T

245

(U(T),V(T))Od? as t h e L2 s c a l a r p r o d u c t on R n x [O,T],

L*

i s d e f i n e d by

the r e l a t i o n ILU,V}

i f U(x,O) = V(x,T) = 0.

{U,L*Vl

=

au - AU * au t c A av - A*V * cA - ,I' t B ( t , r ) U ( T ) d T , L*V = -A k = l kaXk 0 o a t k = l kaxk T * B * ( T , t ) V ( T ) d T , where A* i s t h e a d j o i n t o f t h e m a t r i x A .

au Since LU = A -

oat

.k

I t i s e v i d e n t t h a t L* i s s i m i l a r i n f o r m t o L and an argument s i m i l a r t o t h e p r o o f

o f Theorem 1 y i e l d s : COROLLARY 1:

F o r e v e r y r e a l s , t h e r e i s a c o n s t a n t C ( s ) independent o f V such t h a t Ilv(t)lls

f o r V E C1([O,Tl,

4.

5 c(s)

{ Ilv(T)(ls +

,kT ~

~ ~ * v ( ~ 0) 5~ t~ 5s T,d ~ ~ ,

C;(R)).

APPLICATIONS OF THE ENERGY ESTIMATES

The use o f energy e s t i m a t e s i s s t a n d a r d i n p r o v i n g well-posedness o f t h e Cauchy problem f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s .

(See L. Nirenberg, [6], f o r i n s t a n c e . )

One can prove, i n a s i m i l a r manner, t h e we1 1 -posedness o f ( I .E. ) : THEOREM 2: Consider ( I . E . ) . I f g E Hs(Rn) and f E Co([O,T],Hs(Rn)) are given, C 1 ([O,Tl,Hs-,(Rn)). t h e n t h e r e e x i s t s a u n i q u e s o l u t i o n u ( x , t ) E C 0 ([O,T],Hs(Rn))

n

I n a d d i t i o n , t h e energy e s t i m a t e s may be used t o show t h a t t h e s o l u t i o n o f ( I . E . ) has a f i n i t e domain o f dependence.

The domain o f dependence p r o p e r t y i s e q u i v a l e n t

t o t h e p r o p e r t y t h a t a s o l u t i o n propagates w i t h f i n i t e speed. Let

T =

X,,(x,t,E),

j=1,

...,N

n

be t h e r o o t s o f d e t ( T A 0 ( x , t ) -

c

E k A k ( x , t ) ) = 0.

k= 1

We d e f i n e Amax =

max

lXJ(x,t,c)l

( x , t Ed%O,TI 151=1 15j5 N THEOREM 3 : satisfying t

L e t D be t h e i n t e r i o r o f a backward cone i ( x , t ) : I x - x o l = X m a x ( t o - t ) } ,

2

0.

Suppose u i s d e f i n e d i n D s a t i s f y i n g

A au = P ( x , t , a x ) u ( x , t )

oat

i f u(x,t) = 0 f o r x E

then u ( x , t ) Proof:

131.

,(,t B u ( x , s ) d s ,

D

n

It = 01, t

0 in

and

5 0,

D.

The p r o o f i s s i m i l a r t o t h a t used t o p r o v e t h i s r e s u l t f o r p a r t i a l d i f f e r -

e n t i a l equations. Remark:

t

See

s.

Mizohata [5].

Theorem 3 i s a l s o proved by t h e use o f semi-groups by Grimmer and Zeman

R. Grimmer and M. Zeman

246 5.

EXAMPLES

We w i l l p r e s e n t two examples w h i c h can b e p u t i n t o a framework i n which t h e y s a t i s f y t h e c o n d i t i o n s o f Theorems 2 and 3. The f i r s t example d e a l s w i t h t h e g e n e r a l i z e d h e a t e q u a t i o n f o r m u l a t e d by G u r t i n and P i p k i n

[4].

Example 1 :

The g e n e r a l i z e d l i n e a r h e a t e q u a t i o n f o r m u l a t e d by G u r t i n and P i p k i n

is: B(0)Ut(x.t)

+

CUtt

(5)

f

,j_t_B'(t-~)U~(x,~)d~

2 t 2 = a ( o ) v u(x,t) + J m a ' ( t - r ) v U(X,T) + r ' ( x , t ) ,

where c > 0 i s t h e i n s t a n t a n e o u s s p e c i f i c h e a t , u ( x , t ) difference, r ( x , t )

i s t h e temperature

i s t h e h e a t s u p p l y and O ( s ) and a ( s ) a r e t h e energy r e l a t i o n

f u n c t i o n and t h e h e a t f l u x r e l a t i o n f u n c t i o n . It i s assumed t h a t B ( s ) and a f s ) 1 a r e o f c l a s s C and C2, r e s p e c t i v e l y , and t h a t a ( 0 ) > 0. We t r a n s f o r m t h e second o r d e r e q u a t i o n ( 5 ) i n t o a f i r s t o r d e r system. t h e change o f v a r i a b l e s ut

=

vo, u

x1 as e q u a t i o n s , t h e c o m p a t i b i l i t y c o n d i t i o n s

= v3 i n

= v2, u

u

= vl,

x2

We make ( 5 ) and add,

x3

- ( v ~ ) ~ , =] 0, i = 1,2,3.

a ( 0 ) [(viIt Assuming t h e h i s t o r y o f u ( x . t )

i s known, e q u a t i o n ( 5 ) i s then t r a n s f o r m e d i n t o

t h e system where

A0 =

('

0

0

O0

0 o a (00 ) a (00 ) 0 0 0 0 a(0)

1,

0

A1

0 0

0 0 0 0

0 a(0) 0

A 2 = ( a (00 ) 00 0 0

0 0

o0 ) ,

0

0

h

A and K a r e bounded m a t r i c e s , f i s a v e c t o r f u n c t i o n and U = (vo,v1,v2,v3) e a s i l y seen t h a t Ai, B = FA, where F = -

i

=

0,1,2,3,

a r e symmetric and A.

.

Remark:

.

i s positive-definite.

For t h i s example Xmax

t h e p r o p a g a t i o n speed o f t h e s o l u t i o n i s p +

T

It i s

Also

=e .

Hence

.

The f a c t t h a t t h e s o l u t i o n u has f i n i t e p r o p a g a t i o n speed was f i r s t proved

by F i n n and Wheeler [l]under t h e a d d i t i o n a l c o n d i t i o n B ( 0 )

1. 0.

E-nevgj, Estimates f o r It1 tegro-DifyerentialEquations

241

The second example deals with electromagnetic theory f o r an inhomogeneous anisotropic stab1 e di a1 e c t r i c . Example 2: The basic laws underlying the phenomena of electromagnetism a r e given by Maxwell's equations: f

where 6 i s the magnetic induction, H i s the magnetic f i e l d , 0 i s the e l e c t r i c induction, E i s the e l e c t r i c f i e l d , J i s the e l e c t r i c current and G i s an outside force representing charge. In a s t a b l e medium, Ohm's law a p p l i e s : J = oE, where a i s t h e r e s i s t i v i t y .

(7)

We consider t h e c o n s t i t u t i v e r e l a t i o n s which were f i r s t proposed by Volterra [ 7 ] : t 0 = E ( x , t ) E ( x , t ) + ,[,$(x,t,T)E(x,-r)d.r (8) B

(9)

=

p(x,t)H(x,t)

t

+ ,[,+(x,t,T)H(x,T)dT,

where C ( x , t ) , u ( x , t ) , $ ( x , t , - r ) , I ) ( X , t , T ) a r e tensors. e l e c t r i c and magnetic r e l a t i o n t e n s o r s , respectively. a r e p o s i t i v e - d e f i n i t e and t h a t 4 and IJJ a r e C1 in T .

@ and I) represent t h e We require t h a t E and p

Equations ( 6 ) - ( 9 ) lead t o t h e following system f o r U = ( E l ,E2,E3,H1 ,H2,H3)' :

where A.

1

=

[:

;I

'

0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0-1 0 0 0

0 0 0 0

, A =

A and K a r e bounded matrices.

[;: ;A) 0 0 0

0 0-1 0 0 0-1 0 0 0 0 0 0 0 0

1

0

0

0

0

0

[;::w) 0 0 0 0 0 0

=

0

1

0

0

0

0

- 1 0 0 0 0 0 0

In t h i s example B = FA + K, where F

0

0

0

0

0

0.

Remark: Vol t e r r a considered the q u a s i - s t a t i c problem when he proposed the c o n s t i t u t i v e r e l a t i o n s ( 8 ) and ( 9 ) . REFERENCES

[ l ] J.M. Finn and L . T . Wheeler, Wave propagation aspects o f the generalized theory of heat conduction, ZAMP, 23 (1972) 972-940.

[ Z ] K.O. Friedrichs, Symmetric hyperbolic l i n e a r d i f f e r e n t i a l equations, Comm. Pure and Appl. Math., 7 (1954) 345-392.

'

248

R. Grimmer und M. Zernun

[3] R.C. Grimmer and M. Zeman, Wave propagation for linear integrodifferential equations in Banach space, J. Diff. Equations, to appear. [4] M.E. Gurtin and A.C. Pipkin, A generalized theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968) 113-126. [5] S. Mizohata, The Theory of Partial Differential Equations, Cambridge

University Press, London, 1973. [6]

L. Nirenberg, Pseudo-differential operators, Global Analysis, Proc. Symp. Pure Math 16, Am. Math. SOC. Providence, 1970, 149-167.

[7] V. Volterra, Sur les gquations int6gro-differentielles et leurs applications, Acta. Math. 35 (1912) 295-355.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

249

RESOLVENT AND HEAT KERNELS FOR OPERATORS OF SCHRODINGER TYPE WITH APPLICATIONS TO SPECTRAL THEORY David

Gurarie

Mathematics Department Case Western Reserve University Cleveland, Ohio 44106. Dedicated to Yu. I. Ljubich on his 50-th birthday. INTRODUCTION In this paper we shall discuss properties of kernels 'it; = ( 5 - A1-l (resolvent) and U = e-tA (semigroup) and related problems of spectral theory for elriptic operators A . on En and certain of their singular perturbations A = A + B (order B 5 order A ) , modeled - A + V(x). Typical proglems that after Schrodinger operators: arise here are (I) construction of kernels ( 5

-

A) -1

and

e-tA

(11) estimates of kernels (111) asymptotic expansions of kernels in terms of parameters 5 and t. In solving problem I we first construct and study free (unperturbed) resolvent Ro = ( 5 - AO)-l using psedodifferential calculus. Then we 5 relate R = ( 5 - A)-1 to Ro via the perturbation series 5 5

The convergence of ( 1 ) as well as other properties of R depend on the operator BRo = B(5 - A 0 ) - ' . We give conditions for LP-boundedness or compactness of BRo ( in other words "A -boundedness" or "Ao0 compactness" of B 1 and derive explicite estimate of the operator 0

norm 1/BR 11 in terms of 5 E a: (Theorem l).This Theorem generalizes earlier known results [ R S ] ,[Schl , [We]. We also prove that in some cases the resolvent R_(x,y) and other related kernels admit radial convolution-type bouhd

Function H h a s local singularity of Green’s function and decay at 00 which depends on regularity properties of the leading symbol of A, a(x,5), at 5 = 0. Precisely,

D. Gurarie

250

Lr , hence LP-operator norm of R is dominated Er O )

tr

ut - r ( g

I

+1) t -n/m

m

.-tV(x)

P(X)

(16)

dx-

Observe t h a t i n t e g r a l ( 1 6 ) i s f i n i t e f o r a l l t > O by o u r a s s u m p t i o n ( 1 4 ) . To prove (16) w e c a n n o t a p p l y o u r c a l c u l u s d i r e c t l y t o A = A. + V , a s p e r t u r b a t i o n V i s n o t Ao-bounded.So w e " a p p r o x i m a t e " A by c u t - o f f ' s

But

Ut

and l e t c go t o

Ac = Ao+VC

i s no l o n g e r of t r a c e - c l a s s ,

spectrum [c;+m).

since

m.

For

Ac w e c a n w r i t e

Ac has a c o n t i n u o u s

W e s u b t r u c t t h e "continuous p a r t "

e

-t(Ao+c)

Of

and w r i t e t h e d i f f e r e n c e u s i n g p e r t u r b a t i o n series (1) a s f o l l o w s

ut

W e n e e d t o f i n d a s y m p t o t i c o f e a c h summond i n ( 1 7 ) . A s i n Theorem 1 0 t h e r e s o l v e n t R c a n b e r e p l a c e d by a YDO K = K = $ ( X , D ) , $ =

1

5

.

Then t h e k - t h t e r m o f ( 1 7 ) becomes t h e p r o d u c t KVK...VK. g i n {E, > , \ - I } p r o v i d e d f ( x ) L g ( x ) f o r each x i n X. And, t h e norm i s t o be

c o m p a t i b l e w i t h t h e o r d e r i n g i n t h e sense t h a t i f f ) g > 0 i n { E l >-, ( - / ) ,

If1 2

then

lgl. We suppose t h a t A i s a mapping f r o m E r E t o E which has t h e f o l l o w i n g

quadratic character: (i)A(f,g)

L

0 if f

1. 0

( i i ) A(f,g) = A(g,f), ( i i i ) A(af+g,h) = aA(f,h)

and g

+ A(g,h)

0,

f o r cx i n R, and

( i v ) t h e r e i s a convex s e t C c o n t a i n e d i n

E and c o n t a i n i n g t h e c o n s t a n t z e r o

f u n c t i o n such t h a t ( a ) i f f and g a r e i n C t h e n A ( f , g ) function A(f,f)

t

C,

and ( b ) t h e

f o r f i n E i s c o n t i n u o u s on C.

We now d e f i n e t h e b a s i c s e r i e s . Definition:

L e t f be i n E and I W In be a sequence w i t h values i n E g i v e n by P P=l n W1 = f and Wn+l -- 1- C A(b n -

n

L e t S n ( t ) = e-t Theorem 1 [3].

c (l-e-t)PW_ , T . . pfI p=o Suppose T > 0, f i s i n t h e convex s e t C, f

2

0 in

{E,L,[-[),

and l i m Sn(T) e x i s t s i n E. Then, t h e r e i s a continuous f u n c t i o n Y: [O,T) nsuch t h a t i f 0 5 t < T t h e n Y(t) = l i m Sn(t) nand

+

E

m

Y ( t ) = e-tf

+

i d s eS-tA(Y(s),Y(s)). 0

I n o r d e r t o a p p l y t h i s r e s u l t , we r e w r i t e (BEE).

F i r s t , n o t e t h i s standard

consequence o f t h e i n h e r e n t symmetry o f Pm which was i n d i c a t e d i n S e c t i o n 1 . Lemma 5 [l]. I f f i s a bounded s o l u t i o n o f (BEE) t h e n m

m

if(0,x)dx = if(t,x)dx 0 0 and

f o r a l l t > 0,

/mxf(O,x)dx = /mxf(t,x)dx f o r a l l t > 0. 0 0 T h i s lemma, t o g e t h e r w i t h Lemma 4, enables u s t o r e w r i t e (BEE) as

af (t,x) at

m

m

+ f ( t , x ) = / d y f ( t , y ) idzf(t,z)Pm(y,z;x) 0

f(0,x)

0 = fo(X).

Boltzmurin Equations

289

m

Here, i t i s assumed t h a t i f ( 0 , x ) d x = 1.

To w r i t e (BEE) i n t h e i n t e g r a t e d form,

0

identify

m

m

We seek s o l u t i o n s f o r

( IBEE) i n a f u n c t i o n space E. with

E

I n what f o l l o w s ,

llfll =

sup O<Xf u n c t i o n f such t h a t [O,m),

1.

If(x)

i s t h e space o f bounded f u n c t i o n s on

We t a k e C t o be t h e convex s e t of nonnegative UI

/f(x)dx 0 Lemma 6.

5

1.

I f f and g a r e i n C then m

u3

iA(f,g)(x)dx 0 ( i i ) for x > 0 ,

0

m

= if(x)dx.

0

llfll

5 A ( f , g ) ( x ) 5 4B,[

ig(x)dx 0

and

- l l g l l ll/*.

m

Proof.

Statement ( i ) f o l l o w s s i n c e /dxP(y,z;x) = 1. To e s t a b l i s h statement ( i i ) , 0 r e c a l l t h a t P(y,z;x) = 0 i f y+z < x and do a change o f v a r i a b l e i n t h e i n t e g r a t i o n

t o conclude t h a t m

m

i dy i dzf ( y ) g ( z ) P ( y , z ;x) 0

0 m

u

= J d u I d v f(v)g(U-v)P(v,u-v;x)

x

o m

u

< i d u i d v f(v)g(u-V)2Bm/u x o

The l a s t i n e q u a l i t y i s o b t a i n e d by r e a l i z i n g t h a t m

x

Lemma 7.

% o? d v f ( v ) g ( u - v )

0

m

and

du 7 i

0

0 and i f ( x ) d x

Suppose f

(4Bm)p/2 llfll

C

5 1

1 W (x)dx o

p

=

1.

dvf(v)g(u-v) +

1

0

i” d u i d v f ( v ) g ( u - v ) 0

=

1.

F o r p = 2,3,4

...

0

and x > 0, 0

2

W (x)’ P

D.P.Hardin and J. V. Herod

290

Proof.

m

That i W ( x ) d x = 1 f o l l o w s by i n d u c t i o n f r o m Lemma 6.

o

p

5 4Bm I l f

0 5 W,(x)

11 .

There i s a number T

Lemma 8.

(1-e t P

c p=o

and t h i s converges i f 1

-

rn

>

Then

wp+l

IBEE).

converges i n E and d e f i n e s a s o l u t i o n f o r Suppose t > 0 and

f o r 2 5 p < n.

0 such t h a t i f 0 5 t 5 T t h e n

>

e-t

Proof.

Ilf 11

Suppose W ( x ) (4Bm)p’2 P

Also,

n.

l/(%)

e


0 then any s o l u t i o n on [O,T)

ilv(t)II

2

e

< e

-t

llfll

+

J

0

-t

+

e

s-t

/” es-t

remains bounded.

I I A ( V ( s ) , v ( s ) ) l I ds

4Bm I l V ( s ) l l ds

0

Thus, by Gronwal 1 Is i n e q u a l i t y , IlV(t)\l 5

llfll

exp([4Bm-llt)

Hence, by s t a n d a r d arguments t h e s o l u t i o n may be c o n t i n u e d on [ O , - ) . m

Theorem 2.

I f f i s bounded on [ 0 , m ) ,

f ( x ) 2 0 f o r a l l x, and i f ( x ) d x = 1 then 0

converges i n E t o t h e o n l y f u n c t i o n Y: [O.m) -t E such t h a t Y s a t i f i e s t Y ( t ) = e - t f + J et-’A(Y(x),Y(s))ds.

0

Proof.

We know t h a t (IBEE) has a g l o b a l s o l u t i o n Y : [ O , - )

L e t uo(t,x) = e - t f ( x )

+

E f r o m Lemma 9.

and, f o r n 2 1 t

un( t , x ) = e - t f ( x ) + i et-sA(un-,

0

( s , ) ,un-, ( s ,-) ) ( x ) d s

Boitztnann Equations m

iun(t,.)ln=O

I n d u c t i v e l y , u n ( t , x ) 5 ~ , , + ~ ( t , x ) 5 Y(t,x). (see [3]).

29 1 converges i n Ll t o Y ( t , . )

Hence, l i m un(t,x) n-

By [3], t h e convergence o f I u n ( t , x ) l

converges t o Y ( t , x ) .

= Y(t,x).

t o Y(t,x)

implies

F o r each n and t,

i s a c o n t i n u o u s f u n c t i o n and v ( t , - )

-

e-tf(.)

i s a continuous function.

Hence,

B u t a l s o , as i n

t h e convergence i s u n i f o r m f o r x i n compact subsets o f [ O , - : . Lemma 6, A(f,g)(x)

x Hence, W

P+1

28

du

m

( x ) 5 2Bm i f x

0

o

1. 1 . .Thus, f o r t

converges u n i f o r m l y f o r x i n

[O,m).

m

I f ( x ) d x J g(x)dx.

5 2Bm I T I d v f ( v ) g ( u - v ) z $ >

0

0

T h i s completes t h e p r o o f .

REFERENCES:

M. F. Barnsley, J . V . Herod, V . V . J o r y , and G. B. Passty, The Tjon-Wu e q u a t i o n i n Banach space s e t t i n g s , J. F u n c t i o n a l Anal. 43 (1981), 32-51. A. V. Bobylev, Exact s o l u t i o n s o f t h e Boltzmann e q u a t i o n s , S o v i e t Phys. Dokl. 20 (1976), 822-824.

J . V. Herod, S e r i e s s o l u t i o n s f o r n o n l i n e a r Boltzmann e q u a t i o n s , J o u r n a l o f N o n l i n e a r A n a l y s i s : Theory, Methods, and A p p l i c a t i o n s , accepted f o r p u b l i c a tion. M. Krook and T. T. Wu, E x a c t s o l u t i o n s o f t h e Boltzmann e q u a t i o n , Phys. F l u i d s 20 (1977), 1589-1595.

J . A. T j o n and T. T. Wu, Numerical aspects o f t h e approach t o a M a x w e l l i a n d i s t r i b u t i o n , Phys. Rev. A 19 (1979), 883-888. G. T u r c h e t t i , On t h e s t r u c t u r e and a p p r o x i m a t i o n schemes o f t h e g e n e r a l i z e d Tjon-Wu e q u a t i o n , p r e p r i n t . R. M. Z i f f , Model Boltzmann e q u a t i o n s , Phys. Rev. A 23 (1981), 916-923. R. M. Z i f f , Model Boltzmann e q u a t i o n s 11. Rev. A 24 (1981), 509-513.

The M a x w e l l i a n Molecule, Phys.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS LW. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

293

ASYMPTOTIC BEHAVIOR OF SOLUTIONS

OF DISCONJUGATE

DIFFERENTIAL EQUATIONS Don H i n t o n Mat hema ti cs Department U n i v e r s i t y o f Tennessee K n o x v i l l e , TN 37996 U.S.A.

Asymptotic s o l u t i o n s a r e d e r i v e d f o r a c l a s s o f d i f f e r e n t i a l e q u a t i o n s which a r e d i s c o n j u g a t e . The method o f p r o o f r e q u i r e s a v a r i a t i o n of t h e c l a s s i c a l L e v i n s o n theorem. A p p l i c a t i o n s t o spectral theory a r e given.

1.

INTRODUCTION We c o n s i d e r here t h e d i f f e r e n t i a l e q u a t i o n

(1.1)

LY

LnY + Pn-1 L,-lY

=

where t h e o p e r a t o r s

L 0Y

Lk

+

...

+

Po LoY = 0

are defined by

= Y ; L p = rk(Lk-l

,

y)'

k = 1 ,..., n

-

1; Lny = (Ln-ly)'

The c o e f f i c i e n t f u n c t i o n s s a t i s f y

(1.2)

... '

pn-, , rl ,. . . , rn-1 a r e Lebesgue measurable, complex-valued a 5 x < b w i t h each o f p o , ..., Pn-1' l / r l ,. . . , l/rn-l Lebesgue i n t e g r a b l e on e v e r y compact s u b i n t e r v a l

PO'

f u n c t i o n s on i n t e r v a l of

[a, b)

The e q u a t i o n

2

.

Ly = 0

has t b e f o l l o w i n g m a t r i x f o r m u l a t i o n .

Define

6 and

by

(1.3)

ci

=

L 1-1 . Y , i = 1 ,..., n l/ri

, j = i + l

,

i

i = n, j = 1,

=

1,

...,

..., n , n ,

, otherwise. Then ( 1 . 1 ) i s e q u i v a l e n t t o (1.4)

C'

= d 6.

Standard e x i s t e n c e and uniqueness theorems a p p l y t o t h e system ( 1 . 4 ) .

Suppose

R

D. Hinton

294

i s a diagonal m a t r i x where t h e i t h d i a g o n a l e n t r y

(1.5)

i s a complex-valued,

Qi

s a t i s f i e s for

Qi

i = 1,

...,

n,

l o c a l l y a b s o l u t e l y c o n t i n u o u s f u n c t i o n which

never vanishes. D e f i n e now t h e t r a n s f o r m a t i o n

n

(1.6)

so t h a t

I-

=

R 5,

s a t i s f i e s the d i f f e r e n t i a l equation

I-’ =

(1.7)

[a’ a-1

+ R 4 n-']r-

We w i l l t r e a t t h e term R 4 f2-I as a p e r t u r b a t i o n term. For t h e theorem o f s e c t i o n 2 we d e f i n e a nonvanishing, complex-valued f u n c t i o n f on [a, b ) t o be e s s e n t i a l l y d e c r e a s i n g (ED) i f f o r some M > 0 , (1.8)

[f(x)/f(s)/

and we d e f i n e

f

M’

a 5 s 5 x < b

for

and

t o be e s s e n t i a l l y i n c r e a s i n q ( E I ) i f f o r some

Note t h a t if f i s (ED), t h e n l / f i s ( E I ) ; i f i f l i m sup I f ( x ) l = m as x -t b (which i n case l / f i s ( E I ) i f l i m s u p l f ( x ) l < m as x -t b

.

2.

l i m f(x) = 0 , x-tb

M > 0 ,

f i s ( E I ) , then l / f l i m I f ( x ) l = m as x

i s (ED) b ) and

+

AN ASYMPTOTIC THEOREM FOR SYSTEMS Our a s y m p t o t i c theorem f o r systems i s proved f o r a n o n l i n e a r v e r s i o n o f ( 1 . 7 ) ,

i.e., Q’ =

(2.1) where (2.2)

R

[a’ f2-l

+

i s as b e f o r e .

BIT-+

f(x,

I-)

F u r t h e r , assume

B, f a r e complex-valued, Lebesgue measurable f u n c t i o n s which a r e i n t e g r a b l e on compact s e t s , and I [ f ( x , II,)

- f ( x 3 I-1 )[I

f o r some measurable

We want t o t a k e advantage of s ume (2.3)

Sa",(x)=

:1

B ( s ) ds

y

B

satisfies

f

I r ( x ) \lo* - o,"

with

:j

y ( x ) dx
0 , (3.2) T i s a s e l f - a d j o i n t operator i n

which i s an e x t e n s i o n o f t h e minimal oper-

has a spectrum which i s bounded below and d i s c r e t e .

Proof. Since (3.2) h o l d s we have t h a t L - Xw i s e v e n t u a l l y d i s c o n j u g a t e f o r each A > 0 . Thus t h e r e a r e e v e n t u a l l y no n o n t r i v i a l s o l u t i o n s o f Ly = Xwy which have a p a i r o f z e r o s o f m u l t i p l i c i t y m . T h i s i s e q u i v a l e n t t o t h e spect r u m o f T b e i n g d i s c r e t e and bounded below ( c f . [ 1 3 ] ) . 4.

EXAMPLES Suppose i n (1.1 ) t h a t b = and f o r some a 1Iri(x)ILKx ,i=l,.. n. - , l , O < a < x < m .

Example 1 .

Let

E

and t a k e

> 0

.,

,I.-l-E

Qi+lcdx

1

Rn = 1

hence

, and f o r

IQi(x)/ri(x)

i = n - 1 ,..., 1

Q i + l ( ~ ) I 5 1/K

a.

satisfies

a

liCll


1

exists

, s a t i s f i e s these

conditions. Example 3. An e i g e n v a l u e problem a r i s i n g d e s c r i b i n g t h e i n t e r n a l f l o w i n a gas c e n t r i f u g e i s L(y)

(4.6)

=

,0 5

( e x ( e x y " ) " ) " + Xy = 0

x
0, a

where

*

c

denotes the conjugate transpose, when

Associated with

S

c

y

d < b, satisfies S

with

f

=

0.

is its transpose system

T : -z*'J Formal

i

premultiplication of

z"(XA(T)

=

*

by

S

z

+

B(t))

+

g*A(t).

and postmultiplication of

T by

y , and

subtracting yields Lagrange’s formula (z*Jy)'

DEFINITION. We denote by

2 L,(a,b)

llyll In

L2(a,b) A

z*Af

-

g*Ay.

those elements y(t) b *

=

=

(fa y

in Rn

satisfying

AY dt)li2
k

305

is equivalent t o

Hence (z*Jy) (b) = l i m ( S Z ) * ( V - ~ J V * - ~(Sy) ) x-tb

Similarly near by

6, ?,

x

=

a,

i f w e d e n o t e t h e a p p r o p r i a t e fundamental m a t r i c e s

etc., ( z * ~ y (a) )

=

lim x+a

(~~)*(V-'.JV*-')

(~y)

Consequently, THEOREM (GREEN’S FORMULA, WITH BOUNDARY CONDITIONS).

Ly

=

f, Lz

=

g.

Let

y

and

z

be i n

D,

Then < L y , z > - =

l i m ( s ~ ) * ( v - ~ J v ~( -s y ~ )) x-tb

- lim

(Sz)

* (V--I JV *_I) (Sy)

x-ta The l i m i t s above must e x i s t , b u t i n d i v i d u a l t e r m s i n t h e m a t r i x p r o d u c t s 2 Only i n t h e l i m i t - n c a s e , when a l l s o l u t i o n s of HS are i n LA(a,b)

may n o t .

w i l l a l l the individual l i m i t s exist.

S i n c e we wish t o e v e n t u a l l y conclude t h a t

t h e s e l i m i t s c a n be made t o b e z e r o , a n o t h e r approach i s r e q u i r e d .

306

A.M. Krall et al. REFERENCES

[l]

Atkinson, F. V., DISCRETE AND CONTINUOUS BOUNDARY VALUE PROBLEMS (Academic Press, New York, 1 9 6 4 ) .

[2]

Fulton, C . T. and Krall, A. M., "Self-adjoint 4-th order boundary value problems in the limit4 case," Symp. Ord. Diff. Eq. Ops. 1982, Lect. Notes Math. Spring Verlag, W. N. Everitt, R. T. Lewis Editors, 1983.

[3]

Hinton, D. B. and Shaw, J. K., "On the spectrum of a singular Hamiltonian system," Quaestions. Math, 5 (1982) 29-81.

[41

, "On boundary value Problems for Hamiltonian systems with two singular points." submitted.

[51

, "Well posed boundary problems for Hamiltonian systems of limit point or limit circle type," Ord. and Part. Diff. E q . Proc., Dundee 1982, Lect. Notes Math. 964, 614-631.

[61

, "Titchmarsh’s -dependent boundary conditions for Hamiltonian systems," Ord. and Part. Diff. Eq. Proc., Dundee 1982, Lect. Notes Math. 964- 298-317.

~ 7 1

, "On Titchmarsh-Weyl M( )-Functions for linear Hamiltonian systems," J. Diff. Eq., 40 (1981) 316-342.

[81

, "Titchmarsh-Weyl theory for Hamiltonian systems," Spec. Th. of Diff. Op., North-Holland, I. W. Knowles and R. T. Lewis, Eds., (1981) 219-231.

[gl

, "Parametrization of the MQ.) Hamiltonian system of limit-circle tape," submitted.

[I01

, "Hamiltonian systems of limit-point or limitcircle type with both endpoints singular," submitted.

function for a

[ll] Kimura, T. and Takahasi, M., " S u r l e s operateurs differentials ordinaires linears formellement autoadjoint I," Funkcial. Ekvac. 7 (1965) 35-90.

DIFFERENTIAL EQUATIONS 1.W. Knowles and R.T. Lewis (Editors) QElsevier Science Publishers B.V. (North-Holland), 1984

307

ON THE SPECTRUM OF A HAMILTONIAN SYSTEM WITH TWO SINGULAR ENDPOINTS J.K.

D.B.

Shaw

Department o f Mathematics V i r g i n i a Tech Blacksburg, VA 24061 U.S.A.

Hinton

Department o f Mathematics U n i v e r s i t y o f Tennessee K n o x v i l l e , TN 3 7 9 9 6 - 1 3 0 0

U.S.A.

I n t h i s paper we connect t h e p o l e s t r u c t u r e o f t h e TitchmarshWeyl m - c o e f f i c i e n t s w i t h t h e spectrum o f a H a m i l t o n i a n system w h i c h i s s i n g u l a r a t each end o f an i n t e r v a l . C h a r a c t e r i z a t i o n s a r e g i v e n f o r t h e r e s o l v e n t s e t , p o i n t spectrum, c o n t i n uous spectrum and p o i n t - c o n t i n u o u s spectrum. We a l l o w t h e system t o b e o f e i t h e r l i m i t p o i n t o r l i m i t c i r c l e t y p e a t each end. INTRODUCTION We c o n s i d e r a ( 2 n ) t h o r d e r l i n e a r H a m i l t o n i a n system ([1],[5])

,

J$I = [ h A ( t ) + B ( t ) ] ; + A ( t ) ? ( t )

- m

5

a

i

t < b

5

m

,

(1.1)

and t h e a s s o c i a t e d homogeneous e q u a t i o n

5;'

= [hA(t)+B(t)]$,

a < t < b,

(1.2)

where A ( t ) and B ( t ) a r e 2nx2n m a t r i c e s , h i s a complex parameter,

?

Lo -I1

w i t h A nonnegative d e f i n i t e (A r x r

=

$ ( t ) and

J = , where I = In i s t h e nxn i d e n I We t a k e A ( t ) and B ( t ) t o be l o c a l l y i n t e g r a b l e , complex H e r m i t i a n m a t r i c e s ,

= ? ( t ) a r e 2n*1 v e c t o r f u n c t i o n s and

tity.

$

and i n v e r t i b l e ( 1

"definiteness"

2

r s 2n).

5

hypothesis

:],

0) and o f t h e f o r m A =l : [

where A,(x)

is

F i n a l l y , suppose t h a t ( 1 . 2 ) s a t i s f i e s A t k i n s o n ' s

I," PA$

>

0

,a

5

c < d

5

b , f o r every n o n t r i v i a l solu-

t i o n y' o f ( 1 . 2 ) . T h i s i s an assumption on b o t h t h e m a t r i c e s A ( x ) and B ( x ) . D e f i where n i t e n e s s f a i l s f o r t h e system yl' = b ( t ) y , , - y i = ( X p ( t ) + b l , ( t ) ) y , + b y , , = ('l),y , ( t ) E O , y 2 ( t ) = y 2 ( 0 ) e - b t , Y ; p(t) y t ( t ) d t However, ( 2 n f t h o r d e r l i n e a r o r d i n a r y d i f has v a n i s h i n g norm 1 f e r e n t i a l e q u a t i o n s ([5] , [ 1 5 ] ) , as w e l l as D i r a c systems ( [ 1 2 ] ) , a r e imbeddable as " d e f in i t e " sys terns.

b i s a constant, f o r t h e n o n t r i v i a l s o l u t i o n

.

We p a r t i t i o n 2 n r l v e c t o r s i n t o f i r s t and second n x l components by w r i t i n g o r sometimes we w i l l use s u b s c r i p t s y

=

(Ek)

$

when i t seems c l e a r e r t o do so.

=

( Yx ) ,

We say t h a t a v e c t o r f u n c t i o n ? ( t ) i s o f " i n t e g r a b l e square towards b" i f

1 ; ?*A?


(x) 1 - I C 1 1 - E I (2) <x> = J l + / x l ' . Let S be the scattering operator for the pair (HOIH)defined by (14) 2 N in section 2. Then S is a partial isometry in L2 = L (R ) which commutes with Ho. (It is known that S is unitary in L2, but we need not use this fact in what follows.) Let 9 denote the Fourier transform = %S$-'. Then is decomposable with respect to in L2 and set Ho:


1 be large enough. Then for X t X o l d ( X ) has a smooth kernel A ( X , w , w ' ) for w # w ' . (d(X,w,w') is called the scattering amplitude.) The order of singularity of ~ ( X , w , w ' ) when ~ w - w l #~ o is small is at most ~ a - a ~ l - ( ~ - ~ ) / ~any - ~ f 6o r> 0.

Remark. For the Coulomb potential V(x) = C/lx( known that the singularity is of order [w--w'I -2 Our second result is Theorem 2. When N 2 2 and 5 # 0, one has

in

R3 ,

it is

.

N-2

/X(w-w*)

=c

where cN = ( 2 7 ) (Ni2)/2i. Furthermore, since T V our assumption, we have

E.

L1) ! R (

under

328

H. Isozaki and H. Kitada

(5)

Namely, the potential is uniquely reconstructed from the scattering amplitude when N 2 2 . Remark. For N = 1 , this is not true. It is known that there are infinitely many potentials which have the same scattering amplitude and give rise to arbitrary bound states when N = 1 . 2.

WAVE OPERATORS AND SCATTERING MATRIX. We first construct the (modified) wave operator in the form: t

(6)

W - = s-lim e J t-tfrn

itH -itHo Je

For this purpose we prepare the following lemma Lemma 1. Let -1 < u o < u < 1, d > 0 and 0 < 6 0 and Cm funct Om '9 x , C ) and a(x,() such that: i) For 151 2 d, cos(x,5) E [-l,oO] the solution of the eikonal equation

and for any (x,5) e R2N and

ii) equation

For any

and the estimates:

L 2 1

a , B,

y

and 151 2 d,

"

[ul,l]

and 1x1

2

R,

y

satisfies the estimate

a

satisfies the transport

is

329

Scattering Amplitude

and a, can be chosen arbitrarily as far as

In the above a. -1 < o o < o

1-


O),

(33)

5 Ca n

-E +

N’

nN+3/fi +

ca

n/fi +

cafL n-L

.

Therefore, we obtain

which proves Theorem 2. Remark. Throughout the paper, we have only considered the high energy part of all relevant operators. But it is also possible to treat the low energy part away from zero in a slightly different way. E.g. using this method, we can prove Theorem 1 for all h > 0. These will be discussed in subsequent publications. Acknowledgements. This work was refined during the stay of H. K. at California Institute of Technology in March and April, 1 9 8 3 . He wishes to express his sincere appreciation to Professors €3. Simon, V. Enss, F. Gesztesy, M. Murata and Dr. P. Perry for their interest in the work and stimulating discussions with him, to California Institute of Technology for its hospitality, and to USNSF under Grant N o . MCS-

H. Isozaki and H. Kitada

33 4

81-20833 f o r f i n a n c i a l s u p p o r t t o him d u r i n g h i s s t a y .

Reference

[I]

Kitada, H.,

Time-decay

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

f o r a Schrodinger e q u a t i o n , t o appear i n J . Fac. S c i . , Tokyo, sec. I A .

(1983).

Univ.

of

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

335

PIILL- AND EALF-BMIGE THEORY OF AN INDEFINITE STUBl-LIOOVILLE PROBLEM1 Hans G. Kaper Mathematics and Computer Science D i v i s i o n Argonne N a t i o n a l Laboratory Argonne, I L 60439

The Sturm-Liouville e i g e n v a l u e problem -u" = Xxu on (-l,l), u(-1) = u(1) = 0, admits a countably i n f i n i t e sequence of p o s i t i v e e i g e n v a l u e s and a countably i n f i n i t e sequence of n e g a t i v e e i g e n v a l u e s . The e i g e n f u n c t i o n s have f u l l - r a n g e , as w e l l as half-range completeness p r o p e r t i e s . 1.

Introduction

In [l] w e announced s e v e r a l r e s u l t s f o r t h e f o l l o w i n g Sturm-Liouville e i g e n v a l u e problem with i n d e f i n i t e weight:

lim

x4-1

u(x) and l i m u ( x ) e x i s t and a r e f i n i t e . x4 1

T h i s e i g e n v a l u e problem arises i n t h e s t u d y of c e r t a i n boundary v a l u e problems The r e s u l t s were p a r t i a l l y p r e l i m i n a r y ; in p a r t i c i n l i n e a r t r a n s p o r t theory. u l a r , t h e s o - c a l l e d h a l f - r a n g e theory w a s n o t w e l l understood by us a t t h e time of t h e announcement; c f . [ l , S e c t i o n VI]. R e c e n t l y , we proved a half-range completeness theorem f o r t h e e i g e n f u n c t i o n s of (l), (2), u s i n g a s y m p t o t i c estimates f o r t h e e i g e n v a l u e s . These r e s u l t s have been submitted f o r p u b l i c a t i o n e l s e w h e r e [2].

In t h i s n o t e w e i l l u s t r a t e t h e method of o u r proof on a problem t h a t is somewhat s i m p l e r , y e t t y p i c a l f o r t h i s class of Sturm-Liouville problems w i t h i n d e f i n i t e weights, v iz. , -u" = Xxu

(-1,l)

on

,

(3)

.

u(-1) = u(1) = 0

(4)

T h i s problem admits a countably i n f i n i t e sequence of e i g e n v a l u e s {An: f2, which can be ordered in such a way t h a t

...}


0 a r e monotone on [-1.01 and o s c i l l a t i n g on [ O , l ] ; f o r t h e e i g e n f u n c t i o n s $n w i t h n < 0 t h e s i t u a t i o n i s reversed. L e t P and Pm be t h e p r o j e c t i o n s which map HA o n t o t h e p o s i t i v e and P n e g a t i v e maximally S - i n v a r i a n t s u b s p a c e s ,

Then I S 1 = SP -SPm. P i n n e r product on HA:

T h i s o p e r a t o r i s p o s i t i v e d e f i n i t e on HA;

i t d e f i n e s a new

The corresponding norm i s II-lls. The completion of HA i n t h e topology d e f i n e d by The t h e S-inner product is a new H i l b e r t s p a c e H s , which e x t e n d s beyond HA. p r o j e c t i o n s Pp and Pm extend uniquely by c o n t i n u i t y t o o r t h o g o n a l p r o j e c t i o n s on The o p e r a t o r S e x t e n d s uniquely by c o n t i n u i t y t o a bounded l i n e a r o p e r a t o r Hs. on Hs; t h e extended o p e r a t o r i s s e l f a d j o i n t and compact i n 1 ( H S ) . TBEOBen 1.

Any u -

E

HS h a s a unique e x p a n s i o n ,

m

where an = ( U , $ ~ ) ~ / ( Q ~ , $T h ~ e)e~x p. a n s i o n converges i n t h e topology o f H S . The topology of HS is weaker t h a n t h a t of HA, and HA i s everywhere dense i n HS (in t h e topology of H s ) , so t h e set n=fl,f2, which i s a b a s i s of HA, i s c e r t a i n l y a fundamental s e t i n Hs. Moreover, t h e $ are m t u a l l y o r t h o g o n a l with r e s p e c t t o t h e S-inner product. 0

PROOF.

...},

I$,.,:

W e adopt t h e n o r m a l i z a t i o n I l + n l l ~ = 1 . Thus, HS i s t o p o l o g i c a l l y isomorphic with L2(Z\{O}). The isomorphism F which maps HS o n t o L2(Z\{O}) is g i v e n by

The t r a n s f o r m a t i o n F d i a g o n a l i z e s t h e o p e r a t o r S on HS. The s p a c e HS can, i n f a c t , be i d e n t i f i e d . Let P+ and P- be t h e p r o j e c t i o n s which map H onto t h e p o s i t i v e and n e g a t i v e maximally T-invariant s u b s p a c e s , P+u(x) = u ( x )

,

x

E

(0,l) ;

P-u(x)

,

x

E

(-1,O)

= u(x)

Then IT1 = TP+-TP-. i n n e r product on H:

;

P+u(x) = 0 P-u(x)

=

0

, ,

x

(-1,O)

E

x

E

(0.1)

;

(21-1)

.

(21-2)

This o p e r a t o r i s p o s i t i v e d e f i n i t e on H; i t d e f i n e s a new

338

H.G. Kaper (U,V)T = ( ) T l U , v )

9

U,V

6

H

(22)

The corresponding norm is 1 l - U ~ . The completion of H in t h e topology d e f i n e d by t h e T-inner product i s a new H i l b e r t s p a c e HT, which e x t e n d s beyond H. The p r o j e c t i o n s P+ and P- extend uniquely by c o n t i n u i t y t o o r t h o g o n a l p r o j e c t i o n s on HT. The o p e r a t o r T e x t e n d s uniquely by c o n t i n u i t y t o a bounded l i n e a r o p e r a t o r on HT; the e x t e n s i o n i s s e l f a d j o i n t in-C(%). TEEOREM 2.

The s p a c e s HS and % are t o p o l o g i c a l l y

isomorphic.

WOOF. The proof proceeds in a number of s t e p s and depends u l t i m a t e l y on t h e asymptotic estimate (15). (i)

From t h e i d e n t i t y 0

0

and t h e e x p r e s s i o n (14) one o b t a i n s t h e estimate

f o r some p o s i t i v e c o n s t a n t Y.

Hence, w i t h (15),

f o r m and n s u f f i c i e n t l y l a r g e .

(ii)

Given the estimate (23) one shows f i r s t t h a t t h e series 1T11 21:=lanQn

converges in t h e L2-sense

on (-1,O)

Then one shows t h a t t h e c o n d i t i o n

w

i f (a )

n n=l

m

E

I12(R),

with

2

9. (R) i s n e c e s s a r y and s u f f i c i e n t

IT11/211=lanQn

It f o l l o w s t h a t t h e in H. 0 m 2 mapping (an>nP1 ~ l ~ = i s~ bounded a ~ Qand~ boundedly i n v e r t i b l e from 9. ( 8 ) i n t o A s i m i l a r r e s u l t h o l d s f o r ne a t i v e i n d i c e s , so the mapping (a,) H l a n Q n is a continuous imbedding of I12(Z\fO}) i n t o %.

f o r t h e convergence of t h e series

3.

2

( i i i ) Because HS and II ( Z \ { O } ) a r e t o p o l o g i c a l l y isomorphic, t h e r e s u l t of (ii) implies t h a t HS can be imbedded c o n t i n u o u s l y i n %. Then, f o r any u 6 HA, uuu2 = ((P+-PJU,

S

< -

(P -P ) u & P m CR(P+-P-)ullTU(P -P )uII < 4CIlUllTUUllS p m S -

,

i 4ClluU~. The i n e q u a l i t y extends t o HS by c o n t i n u i t y , so HT can a l s o be 1 imbedded c o n t i n u o u s l y i n HS. This proves t h e theorem.

so llulls

COBOLLABY 3.

The s e t {Qn: n=fl,f2, ...} is a b a s i s of

339

An Indefinite Sturm-Liouville Problem T h i s r e s u l t is known as the f u l l - r a n g e completeness p r o p e r t y .

3.

Half-Eange Theory

w.r.t.

The p r o j e c t i o n s P+ and P- d e f i n e t h e d i r e c t sum decomposition ( o r t h o g o n a l t h e T-inner product):

HT = H+ 8 H-

,

H+ = P+(HT)

,

H- = P-(HT)

.

(25)

S i m i l a r l y , t h e p r o j e c t i o n s Pp and Pm d e f i n e the d i r e c t sum decomposition (orthogonal w . r . t . t h e S-inner p r o d u c t ) : HS = H

8 Hm

P

,

Hp = Pp(HS)

,

Hm = Pm(Hs)

.

(26)

The o b j e c t i v e of half-range t h e o r y is t o e s t a b l i s h p a i r w i s e connections between t h e components of (25) and ( 2 6 ) . The c o n n e c t i o n s are e s t a b l i s h e d by means of four connecting transformations, V = P P +P-P + P m ' W=P+P,+P-P

P '

v i/

wt

= P P + P P p + m - ’

(27)

= P P + P P m+’ p -

(28)

Because of Theorem 2, t h e s e are well-defined bounded l i n e a r t r a n s f o r m a t i o n s from HS t o and v i c e v e r s a .

9

TEEOREM 4. Each of t h e c o n n e c t i n g t r a n s f o r m a t i o n s V, W , Vi', b i j e c t i v e mapping of % o n t o i t s e l f .

PROOF.

and W"

defines a

The proof is based on t h e f o l l o w i n g i d e n t i t i e s :

These i d e n t i t i e s can be v e r i f i e d by d i r e c t computations.

0

The theorem i m p l i e s , in p a r t i c u l a r , t h a t P+: Hp + H+ and P-: Hm + H- are b i j e c t i o n s . The i n v e r s e s of t h e s e mappings d e f i n e e x t e n s i o n o p e r a t o r s E+ and E-,

which extend f u n c t i o n s d e f i n e d on t h e half-ranges i n Hp and Hm, r e s p e c t i v e l y .

( 0 , l ) and (-1,O)

..} is a b a s i s of

s e t {P+@n: n=l,2,. THHORPSI 5, {P-$-n: n = 1 , 2 , . . T i s a b a s i s of H-.

PROOF.

For any u

f

E

H*we

H+;

t o functions

the set

have

m

where B+n

= (E+U*,+*)~.

The expansion converges i n t h e topology of HT.

0

H G. Kaper

340

T h i s r e s u l t i s known as the half-range completeness p r o p e r t y . Of c o u r s e , t h e problem of c a l c u l a t i n g t h e expansion c o e f f i c i e n t s in (33) i s s t i l l open, because we d o n ' t have a n o r t h o g o n a l i t y r e l a t i o n f o r t h e f u n c t i o n s P+On and P-4,. In f a c t , a n i n t e r e s t i n g q u e s t i o n i s , whether E+ and E- can indeed be c h a r a c t e r i z e d d i r e c t l y i n terms of boundary v a l u e problems on t h e i n t e r v a l s (0.1) and (-1,O) separately.

References Kaper, H. G., Lekkerkerker, C. G., and Z e t t l , A., L i n e a r T r a n s p o r t Theory and a n I n d e f i n i t e Sturm-Liouville Problem, i n : E v e r i t t , W. N. and Sleeman, B. D. ( e d s . ) , Ordinary and P a r t i a l D i f f e r e n t i a l E q u a t i o n s , L e c t u r e Notes i n Mathematics, Vol. 964 (Springer-Verlag, B e r l i n , 1982). Kaper, H. G., Kwong, M. K., Lekkerkerker, C. G., Sturm-Liouville Problems, p r e p r i n t (1983).

and Z e t t l , A.,

Indefinite

Beals, R., Partial-Range Completeness and E x i s t e n c e of S o l u t i o n s of Two-way D i f f u s i o n E q u a t i o n s , J. Math. Phys. 22 (1981) 954-960. Beals, R., I n d e f i n i t e Sturm-Liouville Problems and Half-Range Completeness, p r e p r i n t (1983).

Handbook of Mathematical F u n c t i o n s , Abramowitz M. and Stegun, I. A. ( e d s . ) , Applied Math. S e r i e s , Vol. 55, Nat'l Bureau of S t a n d a r d s , Washington (1964).

' J o i n t work w i t h M. K. Kwong and A. Z e t t l (Northern I l l i n o i s U n i v e r s i t y ) . This work was supported by t h e Applied Mathematical Sciences Research Program (KC-0402) of t h e O f f i c e of Energy Research of t h e U.S. Department of Energy under Contract W-31-109-Eng-38.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

341

REMARKS ON HOLOMORPHIC FAMILIES

OF SCHRODINGER AND DIRAC OPERATORS T o s i o Kato Department o f Mathematics U n i v e r s i t y o f C a l i f o r n i a , Berkeley

Holomorphic f a m i l i e s T ( K ) = T + K A o f l i n e a r o p e r a t o r s i n a H i l b e r t space a r e considered, where T and A a r e m-accretive. D e t a i l e d r e s u l t s a r e g i v e n when A i s p o s i t i v e s e l f a d j o i n t . ble a r e i n t e r e s t e d i n t h e case i n which T ( K ) forms a f a m i l y o f t y p e (A) f o r K i n a r e g i o n o f t h e complex p l a n e and o f a weaker t y p e (say t y p e ( B ) ) i n a l a r g e r r e a i o n . D i r a c and Schrodinger o p e r a t o r s w i t h s i n g u l a r p o t e n t i a l s a r e discussed. The a b s t r a c t p a r t c o n s i s t s o f two theorems which a e n e r a l i z e a theorem o f H. Sohr. 1.

INTRODUCTION

I n t h i s paper we a r e concerned w i t h holomorphic f a m i l i e s o f l i n e a r o p e r a t o r s T ( K ) = T + K A i n a H i l b e r t space H, which may have d i f f e r e n t t y p e s a t d i f f e r e n t v a l u e s o f t h e parameter K . We a r e p a r t i c u l a r l y i n t e r e s t e d i n t h e cases i n which T ( K ) i s o f t y p e (A) i n a r e p i o n o f K and o f a more general t y p e i n a l a r g e r r e g i o n . ( F o r d e f i n i t i o n s c f . Kato [2, Chapter V I I ] . )

A s i m p l e example o f t h i s k i n d j s mentioned i n [Z, Example VII-4.151, n o t f u l l y discussed, where t h e Schrodinaer o p e r a t o r (1.1)

T ( K ) = -d2 +

KX-

2,

0


- 1/4. See Example 1.1 below f o r complex v a l u e s o f K . Another example i s g i v e n i n Kato [3], where t h e D i r a c o p e r a t o r (1.2)

H(K) = i-’a v a r a d + rnB + K V ( X ) ,

3 x € R ,

i s c o n s i d e r e d i n H = (L 2 (R3 ) ) 4 , where m i s r e a l , CY = ( a ,a2,a3) and B a r e t h e usual 4.4 h e r m i t i a n m a t r i c e s , and V ( x ) i s a ( n o t n e c e s i a r i l y h e r m i t i a n ) I f ( V ( x ) I i ( x l - 1 ( t h e m a t r i x norm), i t i s shown t h a t H ( K ) matrix-function. i s holomorphic of t y p e (A) f o r I K I < 1 / 2 and can be c o n t i n u e d t o a f a m i l y of a t y p e analoaous t o t y p e (C) f o r ~ K 0, T+!K) i s m-accretive. F o r Re K > 1/2, r e s i d u a l spectrum C+. T+(K) i s a l s o equal t o T m a x ( ~ ) ( t h e maximal r e a l i z a t i o n o f ( 4 . 1 ) ) . ( i i ) T - ( K ) i s d e f i n e d f o r Re h < 1/2 and e q u a l s Tmax(;]keALt has r e s o l v e n t s e t C, and p o i n t spectrum C-, w i t h eigenfunctions with 0. For Re h < 0, T - ( k ) i s i n - d i s s i p a t i v e . For Re K < -1/2, T - ( K ) Re i 1 a l s o equals Tmin(") and forms d f a m i l y o f t y p e ( A ) w i t h domain Ho.

T. Kato

348

REMARKS. ( a ) We have T m a X ( ~ =) Tmin(") f o r Re K < -1/2 and Both T+(K) a r e d e f i n e d on t h e s t r i p -1/2 < Re K < 1/2, where = T, C T- = Tmax. Tmni

Re

K

> 1/2.

generates a c o n t r a c t i o n semigroup f o r Re K > 0, and T - ( K ) does Each o f them i s holomorphic i n K i n i t s domain. I t appears t h a t n o r T - ( K ) generates a Co-semigroup f o r o t h e r values o f K .

( b ) -T,(K) f o r Re K < 0. n e i t h e r T+(K)

( c ) The f a m i l y T - ( K ) i s n o t o f t y p e (A) o r o f any f a m i l i a r t y p e f o r - K x x -1/2 2 Re K < 1/2, as i s seen f r o m t h e b e h a v i o r o f i t s e i g e n f u n c t i o n s x e (which become l e s s smooth as Re K grows). 5.

THE DIRAC OPERATOR Let T = a.grad + i m B ,

(5.1) where

a = ( a ,a ,a

)

and

B

A =

XI-^,

( x E R3),

a r e as i n I n t r o d u c t i o n . T i s i - t i m e s t h e f r e e T and -T a r e b o t h m - a c c r e t i v e .

(~elfadjoint)~Di6ac~operator.Thus

A s i m p l e computation analogous t o (4.2) g i v e s (2.2) w i t h a = 1/2. (Note I t f o l l o w s f r o m Theorem 2.1 t h a t t h a t t h e aj and B a r e h e r m i t i a n m a t r i c e s ) . T + K A i s a holomorphic f a m i l y o f t y p e (A) f o r Re K > 1 / 2 and f o r Re K < -1/2. Combined w i t h t h e r e s u l t o f [3] g i v e n i n I n t r o d u c t i o n , we have THEOREM 5.1. There i s a holomorphic, skew a d j o i n t f a m i l y T ( K ) 3 T + KA d e f i n e d on t h e u n i o n o f t h e d i s k < 1 and t h e two h a l f - p l a n e s Re K > 1/2, T ( K ) i s o f t y p e (A) i n these two h a l f - p l a n e s and i n t h e d i s k Re K < -1/2. ( K I < 1/2, w i t h D(T(")) = H1(R3)4 and T ( K ) = T + KA. The r e s o l v e n t s e t o f T(K) c o n t a i n s C- i f Re K > 1/2, C, i f Re K < -1/2, b o t h C, if ' ~ 1: 1/2, dnd a t l e a s t t h e r e a l a x i s ( e x c e p t t h e o r i g i n i f m = 0 ) i f 1

-

REMARK. The s e l f a d j o i n t D i r a c f a m i l y i s g i v e n b y H ( K ) = - i T ( K ) . The r e s u l t s o f Theorem 5.1 a r e n o t sharp. For s t r o n g e r r e s u l t s , see t h e end o f I n t r o d u c t i o n . 6.

A CLASS OF SINGULAR POTENTIALS

I n o r d e r t o a p p l y Theorem 2.2 t o S c h r o d i n g e r o p e r a t o r s , i t i s c o n v e n i e n t t o i n t r o d u c e a c l a s s o f p o t e n t i a l s . We say a p o t e n t i a l q d e f i n e d on Rm i s i n c l a s s ( V ) i f e i t h e r q = 0 i d e n t i c a l l y o r q ( x ) > 0 and

h e r e i t i s understood t h a t measure zero. If q e i ' c a l l y as c

=

+-

i s a l l o w e d on a c l o s e d s e t

Q

C

Rm

of

(V), -+

grad(q + c ) - ' l 2 n umbe r s (6.2)

q(x)

b

l l ( q + c)-1/21, i s f i n i t e i f c > 0 and decreases monoton( T h i s i s seenLiPby n o t i n g t h a t -1 3/2 = ( 1 + cq ) grad q - l l 2 . ) For l a t e r use, we i n t r o d u c e t h e

-.

= b[q]

= l i m II(q + c)-1/211Lip

5 llq'1/211Lip

= b[q]

= b.

C"

The c l a s s ( V ) has s e v e r a l i n t e r e s t i n g p r o p e r t i e s . ( a ) A p o t e n t i a l q E ( V ) may be r e g u l a r o r s i n g u l a r . I f q ( x ) < m everywhere so t h a t Q i s empty, (6.1) i m p l i e s t h a t q i s l o c a l l y L i p s c h i t z i a n on R". If Q i s n o t empty and i f x 9 Q approaches Q. q ( x ) must blow up l i k e , o r

.

3 49

Holomorphic Fundies of Operators

f a s t e r than, c o n s t ( d i s t ( x , Q ) ) - ’ t o s a t i s f y (6.1). I n thij,2ense, potentials i n should look l i k e c l a s s ( V ) a r e i n general h i g h l y s i n g u l a r . The graph o f qa quilt. q , q2 E ( V ) ( b ) The c l a s s ( V ) i s a d d i t i v e and m u l t i p l i c a t i v e , i . e . , I n f a c t , i t i s easy t o l e e t h a t i m p l i e s t h a t q1 + q2 and q1q2 E ( V ) .

( c ) Most p o t e n t i a l s t h a t i n c r e a s e ( e v e n t u a l l y ) m o n o t o n i c a l l y as b e l o n g t o c l a s s ( V ) ; f o r example

(6.4)

q(x) = P(x),

exp[p(x)l,

1x1

+

m

ex~[ex~[~(x)ll,~...~.~

where p ( x ) i s a p o s i t i v e - v a l u e d p o l y n o m i a l . The s i n g u l a r p o t e n t i a l q ( x ) = ! x J - ~ , k > 0, belongs t o c l a s s ( V ) i f and o n l y i f k > 2. Thus modera t e l y s i n g u l a r p o t e n t i a l s (such as Coulomb) a r e excluded. I n Z l l t h e e5amples g i v e n above, i t - i s e a s i l y v e r i f i e d t h a t b = b[q] = 0 e x c e p t f o r 1x1- , f o r which we have b = b = 1. 7.

SCHRODINGER

OPERATORS WITH SINGULAR

POTENTIALS

I n t h i s s e c t i o n we c o n s i d e r S c h r o d i n g e r o p e r a t o r s o f t h e form (7.1)

T

where

T

+

x E Rm,

i s i t s e l f a S c h r o d i n g e r o p e r a t o r g i v e n by T = -A + qo(X),

(7.2)

.ql(x),

40 2 0

( f o r m sum).

By t h i s we mean t h a t T i s a nonnegative s e l f a d j o i n t o p e r a t o r such t h a t and D(T) c H1(Rm) n D(qA/2) = D(T’/‘) (u,Tv)

(7.3)

u

for

E

D(T”2)

and

= ( g r a d u, g r a d v )

+ (qb/2u,qb/2v)

v 6 D(T).

THEOREM 7.1. L e t T be as above and l e t ql b e l o n g t o c l a s s ( V ) i n t r o d u c e d i n t h e p r e c e d i n g s e c t i o n . Then T + K q , forms a holomorphic f a m i l y o f t y p e (A), defined f o r K i n the e x t e r i o r o f the parabolic w i t h domain D(T) n D(q,), region S given by (7.4) (see ( 6 . 2 ) ) . Moreover, T + Kq can be c o n t i n u e d t o a h o l o where bl = b[ql] of t y p e ( B ) d e f i d d on a l l o f t h e K-plane c u t morphic f a m i l y T ( K ) 3 T + Kq a l o n g t h e r a y (-m,-l/k], wheie k i s t h e r e l a t i v e bound o f t h e f o r m q1 w i t h respect t o t h e form T ( s e t k = i f q1 i s n o t f o r m T-bounded).

-

REMARKS. ( a ) The assumption on T i s s a t i s f i e d , a f o r t i o r i , i f qo 0 and This i s true, i n p a r t i c u l a r , - A + qo i s s e l f a d j o i n t w i t h domain H2(Rm) n D(qo). i f qo i s i n c l a s s ( V ) w i t h b[qo] < 1. Indeed, t h e theorem i t s e l f then shows t h a t - A + qo i s s e l f a d j o i n t ( a p p l y t h e theorem w i t h k: = 1, qo = 0, and q1 replaced w i t h qo). ( b ) If bl = 0, which occurs i n many i n t e r e s t i n g cases (see p r e c e d i n g s e c t i o n , ( c ) ) , S reduced t o t h e ( c l o s u r e o f ) n e g a t i v e r e a l a x i s . Thus (7.1) i s o f t y p e (A) on a l l of t h e K-plane c u t a l o n g (-m,O].

(c)

R e s u l t s r e l a t e d t o (7.4) a r e found i n E v e r i t t and G i e r t z [l].

T. Kato

350

EXAMPLES 7.2. (a) -A + 1 x I 2 + ~ 1 x 1i s ~holomorphic o f t y p e (A) f n r a l l K a l o n g (--,O]. Note t h a t 6[q] = 0 f o r b o t h q = 1x1‘ and q = 1xI4 (see section 6,(c)). ( b ) -A + vided k > 2

K [ x ~ - ~i s

of t y p e (A) l i k e w i s e f o r a l l see l o c . c i t ) .

cut

on t h e c u t plane, p r o -

K

-2 ( c ) -A + ~ 1 x 1 i s o f t y p e ( A ) f o r K C $ S, where S i s g i v e n by (7.4) w i t h b = 1. T h i s r e s u l t i s n o t sharp, however. F o r a sharp r e s u l t , see Example 7.4 bilow.

I

P r o o f o f Theorem 7.1. _c > 0 i s a c o n s t a n t . bl i s r e p l a c e d by

We a p p l y Theorem 2.2 t o o u r T and A = c + ql, where L e t S, be t h e p a r a b o l i c r e g i o n o f t h e f o r m ( 7 . 4 ) , where

= 1I ( c + q1 )-1/zil

(7.5)

bl ,c We s h a l l show t h a t f o r

(lim b = bl). ctm 1 - c

Lip v E D(T*) = D(T),

l i m ( - ( A + ~ ) - ’ v , T * v ) = - [ ( c + ql)-’v,Tv)

E Sc, €40 where we have assumed IIvII = 1 w i t h o u t l o s s o f g e n e r a l i t y . Then i t w i l l f o l l o w hence T + Kql t o o , i s h o l o f r o m Theorem 2.2 t h a t T + KA = T + K ( C + ql), m o r p h i c - o f t y p e (A) f o r K $ S., Since, however, S S as c = by we then conclude t h a t T + Kql i s o f t y b e (A) f o r K f! S. bl ,c + bl ,

(7.6)

-f

-f

To p r o v e ( 7 . 6 ) , we s e t (7.7) Since v E D(T) c H1(Rm) n O(qA/‘), t h e same i s t r u e o f ( c + ql)-’v = f % v by fc E Lip(Rm). Thus a s i m p l e computation based on i n t e g r a t i o n b y p a r t s g i v e s (see ( 7 . 3 ) ) 2 -1 v,Tv) 5 - ( g r a d f c v , g r a d v ) = 5, + i q c , - ( ( c + ql) (7.8) where t h e n o t a t i o n 5 means t h a t t h e d i f f e r e n c e i s a n o n p o s i t i v e where = -1lgrad fcvl12

+ Ilv g r a d fcll 2 ,

(7.9)

6,

(7.10)

nc = -2 I m ( g r a d f c v , v g r a d f c ) .

Since (7.11) Hence

= bl,c

IIfcllLip 5,

:n 5 4b:,c(b:,c


0, 5 = - y ~ l be t h e charac(1 i s isomorphic as a s y m p l e c t i c m a n i f o l d t e r i s t i c v a r i e t y o f t h e system ( 2 . 1 ) . to

T+IR~\C.~

M. A. Kon

360

the rotation invariance of the Laplacian restricted to w . This symmetry is broken to the extent that w is non-flat, and uniform bounds on the curvature are required for the series to converge. A second perturbation series will be used to represent and bound the full resolvent. The deiailed analysis of resolvent kernels has been used bolh on R" (see, e.g., [l], [2]) and manifolds to analyze spectra of elliptic operators. We remark that the present results have a wide range of applications - including study of semigroup generation and corrcsponding description of boundary behavior of solutions of the heat equation on cylinders, uniform bounding of LP spectrum of elliptic operators, and study of self-adjointness of singular elliptic operators on domains. See [5] for "classical" results on essential self-adjointness of Schrodinger operators; Simon [7] has a thorough survey of Schrodinger semigroup theory on R". Our regularity results on semigroups generalize some of those in [8] on so-called symmetric heat diffusion semigroups. The result in Theorem 1 is new for Schrodinger operators with smooth coefficients, but is profitably expressed in the context of singular (LP) coefficients, given thc forms of Schrodinger operators which often arise in physics. For smooth coefficients, the point of departure of this result from known theory (see [4) for such results on R") is the uniformity in < of the bounds of Theorem 1, which cannot be obtained using standard techniques. b(z) . V V(z) be a We now develop some notation and definitions. Let A = -A Lm Schrodinger operator with a "vector potential" term b(s) = ( b l ( z ) ,. . .,b,(z)) E L"+' and potential V(z) E L%+' L", for some 6 > 0. For notation purposes we also identify bo = V. By L' Lm we mean {f = f 1 + fz : f~ E L', fz E Loo}. Let SZ R" be a domain with C" boundary w . We define H"(R") to be the 15’Sobolev space of order s. Let H"(R) consist of those u E Lz(n)which have extensions 5 E

+

c

+

+

+

HS(R"), ii In= u. If u E H"(R),define

llull, = max

{ IIiiIIH.(R,,)

: ii In= u

+

1

. It is well-

known that if u € HS(R),then the k-th normal derivative b'ku on w restricts to a function E H s - k - - l a(w). We define H " i 2 = H"-?(w) @ H"-s(w), so that ( u ,a,u) €,I H2*' when

fk

u

E H2(n). $1. Bounds on the Laplacian Resolvent

Consider the Laplacian in R,with boundary conditions (1). We will assume that tanO(s1) is bounded in C1norm on w ; this does not exclude Neumann boundary conditions from our results, since they can be handled similarly. We emphasize that the following arguments generalize to higher order constant coefficient homogeneous elliptic operators, and the following should be viewed as an illustration of a more general theory. DEFINiTlON 1: The symbol class S" is defined by

Sm(R") = {f E P ( R " ) : lb'"f(z)l

5 C,(1

+ 1~l)"'-'~' 1 1

where represents a partial derivative of order a,with cy a multiindcx. The constants C, are the symbol class seminorms of f . The rcsolvcnt R: = (< A)-' on all of Rn has a kernel satisfying

+-

( ~ : ( z- d j l

5 ~ h , , , ( . z- d),

(2)

Regularit), Properties of Schr6dinger Operators

36 1

where

with s = n - 2, and t any number greater than n (with suitable adjustment of C). The boiind in (2) m a r z = 5’ follows from the following well-known result. PROPOSITION 1: I f f E S-"(R"), for m > 0 then the Fourier transform o f f satisfies

DEFINITION 2: The boundary w is sparse if (2) its principal cu:vatures are uniformly bounded by a constant K ; (i) its two dimensional measure has bounded concentration in space, i.e., there is a constant M > 0 such that if 7 is n - 1 dimensional Lebesgue measure restricted to w and &(z) is the unit sphere at z, then -y(S u B l ( z ) )5 M (z E R"). We will assume that the boundary w is sparse in order to avoid pathologies. Let R,(z,y) denote the kernel of (c A)--' in R , with boundary conditions (1). We now proceed to estimate R,(zl, y) by assuming that it restricts in z1 on w to a function in H2r2 satisfying (l), leaving the proof of this fact to the end. Formally using Green's identities (in higher order cases certain generalizations of these apply; see [3]), we have the integral equation

+

where a; will henceforth represent outward normal differentiation with respect to the variable labeled i. Initially, we impose (1) on (4) to obtain

W,u(z) =

J,

u ( z l ) { a ~ R ~ (y)z ~ 4-, tan@(zl)R;(zl,y ) } k .

Define the operation S on E12(R) to be rest,riction to N $ ( w ) @ Hf(w);specifically, if E H2(R),then Su = (u,anu)JwEH 2 s 2 . The map S is continuous and onto (see [3]). We note that if w , is that part of R whose distance from w is lcss than E, then 121, is well-defined on functions in I i z ( w c ) . Uy our assuriiption R,(z, 51) is in the domain cf S in this sense, and u

( 5 ) can be written

M. A. Kon

362

where

* denotes complcx conjugate,

c=(

0 -I

I

)

0

is the operator matrix whose blocks correspond to the direct sum decomposition H ~ ( w@) H i ( w ) , and the inner product is component-wise sesquilinear multiplication followed by summation and integration in 21 over w with respect to its inheritcd measure. The operator S is always assumed to operate in the variable 5 1 . If T denotes the projection on EI2v2(w) defined by WO(Z),

fib)) = (fO(4, tano(4fofz)),

then (6) holds with CT replacing C. Note that T'SRf(z,zl)E H2s2,while CTSR,(z, 21) E H f @ H j C H-3 @ H - f ; we thus regard the second component of the inner product as being in the dual space of the first. We now define &(z, 51) to coincide with R,(z, 51) on we, but to be smooth in z1 on the interior of R; k: is deEned similarly. Note that the adjoint

1

(

S' is bounded from li-; @ H - f ( w ) to H-*(R"). Taking adjoints,

(CTSR,(z,zi),SR,O’(zi,y))= (S'CTSR,(z,zi), R,O*(zi,Y)) R" where the objects on the right are in H - 2 and H 2 , respectively. Since the support of S*CTSR,(z,51) is outside n, k:(q,y) may be replaced by (< - A)-'6,, where the delta distribution 6, is in H-f-' for every c > 0. Again taking adjoints, we have

(CTSR,(z, zi),SR ; *( z i,Y),

= (S - A)-'S'CTSR&,

Y),

(7)

where all operators act on the y variable. The operator ( n 1. C depends only on the sparseness parameters K and M in Def. 2. I m n m a 2 bounds ;dl but the extreme terms of (lo), namely those whose arguments lie on w . Thcse bounds fail to hold uriiforiiily f o r T,fz,x l ) whcn z is near hill riot in w . The

M.A. Kon

364

problem is the term a!,R,(z -- zl); bounding it on w quickly reduces to the same problcm when w is flat; in this case the only bound on a!,R,(z - z1) which is uniform in z and scales correctly with p (as in (13)) is singular in z1 (i.e., not'in L'(w)). Correspondingly, its Fourier transform in 51 on w is uniformly bounded only in the So symbol class. Fourier transforming the integrand of (10) in 5 1 (correct bounds on the integration in the remaining variables exist at this point by Lemmas 1 and 2) and analyzing the symbol classes (in the dual variable to zl)and seminorms of the transform of the first term in (10) and of the remainder, and transforming back with the help of Proposition 1, we have the following bound for k 2. 1 and p 2 1:

where henceforth C denotes a generic constant. We can sum (9) if C p i obtaining

5

lsin

!In+',

where C(s) is bounded outside of a domain of the form (14). Equation (15) shows that (9) converges in the uniform operator topology to R;. At this point we prove that R: is the resolvent R, by showing that the former satisfies the appropriate boundary conditions. To do this one must first verify that, fixing z,the series (9) converges in H'(R); this can be done by termwise differentiation in y, with use of similar estimates to the above. Let P = R:S*C. By (4) and ( 5 )

so that if P+

R: = R:

+ P SR;

R: = R:

+ PTSRY ,

= SP, P+(TSR; - SR;) = 0.

(17)

It can be shown using the identity of lV{ and the integration in (4) (see [6] for details when R is compact) that P+ and its complement P- = I - P+ decompose R3v2(u)= H";(w) @ H " - f ( w ) into bouiidary values of interior and exterior solutions of

365

Regularity Properties of' Sclirodinger Operutors

Namely,

p+H"'2= {f E Ha" : 3~ E H"(R) s.t. (I

+ A).

= 0, S u = f},

and the range of P- is the corresponding set of boundary values of exterior solutions. Equation (17) therefore implies that (TSRY - SR:) is the boundary value of an exterior solution of (18). Since this solution satisfies a self-adjoint Dirichlet boundary condition on w , it is an eigenfunction of a self-adjoint operator corresponding to a complex eigenvalue, and hence vanishes. Therefore RT satisfies (1) in z1. Since it satisfies the resolvent equation ( 3,

r,

> n, i 2 1, and

d r m a x sup--1, (a2l:

): .

-

Using $1together with bounds and estimates typical of those in [2], we have THEOREM 1: (i) The operator A with boundary conditions (1) can be closed in all LP spaces (1 2 p 5 mint-,). (22) The spectrum of A is contained in the parabolic domain

unifordy in all LP spaces in (i). (iii) If ( 4 R, then the kernel of (I -A)-'

is bounded by

where s = n - 1, t = n 4-1. From above we note that A is bounded below. Let be thc funct,ions in Ifz(n) satisfying (1). Somc irivest,igatiori of thc domain of the leading tcrm and the relative boundedness of t h e perturbation gives the following: C o r t o L L A R Y 1: If A is fornia,ff.v self-ml,joir:t, thcn it is rssentia/ly self-adjoint on fft.c.(n), or on any essential domain of the L a p h i a n -A,,.-..

Hi,,.(n)

M.A. Kon

366

The following extends results on Schrodinger semigroups (see [7], 121) to domains: COROLLARY2: (2) A generates an analytic semigroup e-'A for Re t > 0, in all Lp (1 p 5 minr,); (zi) the semigroup is LP-continuous a t t = 0, for p < 00; (iii) i f f E LP(n) (1 t

2's

- and is a Lie

is denoted by&nm)

-

1

The annihilator of 23(nm) in A (K,) algebra and a Kn-module (Theorem I 3 . 6 [3]). 1 is nothing but the Cartan submodule I . [This is the definition of the Cartan

n

submodule; the fact that the corresponding distribution is spanned by the tangent planes of graphs of jets of sections of n is a corollary (Theorem I 4.4 [31).1 If X (

B(M)

then the lifts -L

such that

gu

and

Xn

Obviously, if Xta(nm),

2.13 [3].)

2

are A-related: i U A k = A*gn (Lemma I1

- -

then again there exists a unique

4-

-

2 ea(um)

A" = A Xn; the resulting map a(nm) + .Xj(u,.) is a Lie algebra homo-

morphism. Lemma 2 . 1 .

Let @

:

K1

+

K2 be a homomorphism of commutative rings K1 and

let XI& B(K1) and X2 B(K 2 ) be two @-related derivations. Let%(@) a K -module of derivations of K1 into K2 along @. Then for any Zra(@), 2 (X2Z - ZX,) eE?4(@).

K2,

be

Proof. Obvious. 1 Recall that if w&A (K), X,Z ES(K), then the Lie derivative of w with respect = Z(w(X))-w([Z,X]). to Z is defined by the formula [Z(w)l(X) Lemma 2.2. In the notations of lemma 2.1,3(@) acts by derivations 1 1 1 along @ on A (K1) with values in A (K2). In particular, for w&A (K1) [Z(W)l(X2)

= Z(W(Xl))

-

w(ZX1-X2Z)

,

(2.3) 1

where on the right hand side the pairing between A (K1) andB(@) ,\df,g&K1. naturally : (fdg)(Z) = @(f)Z(g)

is understood

369

Relative Symmetries of Differentid Equations Again, t h e p r o o f i s o b v i o u s .

-

Now w e can h a n d l e t h e problem o f c l a s s i f i c a t i o n o f e l e m e n t s o f Dqev(A). L e t 1 1 1 Z t B q e V ( A ) , t h a t i s , Z ( I ) C Iv. Take any UJ & In = Ann@(nm)). Then Z(u) & _Tr_ 1 t I v = Ann(a(u,)) = Ann@(M),,) i f f , y X & a ( M ) , [Z(UJ)](?,,) = 0. By formula ( 2 . 3 ) , c _

t h i s is equivalent t o 0 = Z(UJ(~ )) 77

-

~ ~ ( 2-i2 2 ) . X f L

But w ( i i % ) = 0 s i n c e w & I

1

n’

1

n -2u 2 ) must b e l o n g t o t h e k e r n e l of In, t h a t i s , w e must have

Thus (2%

7:

(Zin-ivZ)

&

-

KvA$,(M)n, VXEBM).

(2.4)

Every Z&aqeV(A) i s u n i q u e l y d e f i n e d by i t s v a l u e Z e n k Conm,o' &a(n A) i s u n i q u e l y l i f t e d h a ( @ ) t o become

Theorem 2 . 5 .

v e r s e l y , any d e r i v a t i o n

"&O

-L

Z&Bqe"(A),

=

s u c h t h a t 2.n"

2.

- 9 0

__ P r o o f . To s t u d y ( 2 . 4 1 , f i r s t n o t i c e t h a t , l i k e i n t h e a b s o l u t e c a s e (n = V , A = i d ) , one h a s a d i r e c t sum decomposition

-.A’

a ( A ) = B(v,)

@ B(A)vert

,

(2.5)

12-11; = 01, and decomposition ( 2 . 6 ) i s p r o v i d e d by -.7 (Z-IT..yOA'* + [ Z - (Z-n,)U ,*A"]. S i n c e Z - n L ~ I D=)

: *a(p)

where%(A)vert

-L

-%

t h e formula Z =

,

I Cm(M)

satisfied. only

-

t h e n Z1 : = ( Z . r r ~ ) , & ~ v , ) and ( 2 . 4 ) f o r Z = ZIA*

is obviously

vert T h e r e f o r e w e s h a l l r e s t r i c t o u r s e l v e s t o v e r t i c a l Z ' s &%(A)

m { q a l a = 1 , . . . dim E - dim M , a d + ] ID b' b e s t a n d a r d l o c a l c o o r d i n a t e s on J n , and ( p J b = 1, ..., dim F-dim M, U & z y ] b e

Let (x~,. . . ,x ) b e l o c a l c o o r d i n a t e s i n M , m

U

m l o c a l c o o r d i n a t e s on J V .

" -,a as;

Let, locally, Z = UaA"

t o check ( 2 . 4 ) f o r t h e b a s i s v e c t o r f i e l d s X =

a

a Au&K

a &B(M). ax.

.

-

Since

1

a

a +

q:+i

I t i s enough

( u s i n g summation o v e r r e p e a t e d i n d i c e s ) , w e have

(-1a ax .

IT[

=

B. A. Kupershmidt

370

A

This last expression must belong to K A’b(M)n. V

along M, it must vanish, and this happens iff A i + :

Since there are no components

= (Di),

stands for (a/axi),,.

As’s

= (DU),(Aa),

Thus, :A

where (Di)v

( A : ) , 0

(DUIv: = (Di )v 1

1

...

m’ (Di )v m

,

and

are arbitrary.

ITRAJECTORIES Ordinary differential equations are equations of trajectories of vector fields on manifolds. Analogously, evolution equations are equations of trajectories of vertical evolution derivations (Theorem 1 5 . 6 [3]). (The reason for considering only vertical fields is explained in 51 5.3 [3]- for nonvertical. fields, equations become overdetermined.) Now let 2 cBqev(A), and consider 2 to be vertical. A tra’ector of$ is a one-parameler (t) family of sections y = y(t):M + F such tha~[j(v))iy)]~Z = &-[j(n)(Ay)] Let us find a coordinate

.

version of the last equation.

[j(v)(y)leZ

4

-

~-[j(n)(A~)l*

- 0

where DO: = (a/axi)

Let locally 2 = (D’)v(Aa)*A*

a/aq:.

Then 0 =

=

- a .--(a/axi ) m.

Since [a/at,DU] = 0, the above equality

m is reduced to

Thus we obtain the coordinate form of quasievolution equations. Remark 3.2. In contrast to the evolution equations, quasievolution ones need not be formally integrable. Obviously, integrability of a generic 2 depends only upon A. I conjecture that this integrability depends only upon dimensions and codimensions of the finite number of prolongations of the map A : J'V

-*

E.

ACKNOWLEDGMENTS The main theorem 2.5 of this paper was announced in the 1979 preprint IC/79/94 (ICTP, Trieste, Italy). I wish to thank F . Pirani for helpful discussions 1. there. Thanks are due to A . Greenspoon for reading this paper, and to Knowles and R . Lewis for hospitality in Birmingham.

Relative Symmetries o f Differential Equations

37 1

BIBLIOGRAPHY [I] V. G. Drinfel’d and V. V. Sokolov. "Equations of KdV type and simple Lie Algebras," Dokl. Akad. Nauk SSSR 258 (1981), 11-16 (Soviet Math. Dokl. 2 (1981), 457-462). [2] B. A. Kupershmidt. "On geometry of jet manifolds," Uspekhi Mathematicheskikh Nauk XXX:5(1975), 211-212 (in Russian). [3] B. A. Kupershmidt. "Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms," Lect. Notes Math. 775 (19801, 162-218, Springer.

[4] B. A. Kupershmidt and G. Wilson.

"Conservation laws and symmetries of generalized sine-Gordon equations," Comm. Math. Phys. 81 (1981), 189-202.

[5]

A. M. Vinogradov and I. S. Krasil’shchik. "A method of computing higher symmetries of nonlinear evolution equations, and nonlocal symmetries," Dokl. Akad. Nauk SSSR 253 (1980), 1289-1293 (Soviet Math. Dokl. 11 (19801, 235-239).

[6] G. Wilson. "The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras," Ergod. Th. & Dynam. Sys. 1(1981), 361-380.

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

373

NECESSARY AND SUFFICIENT CONDITIONS FOR OSCILLATIONS OF HIGHER ORDER DELAY DIFFERENTIAL EQUATIONS G. Ladas',

Y .G. Sficas2, and I.P. Stavroulakis' ,2

Department o f Mat hemat ics U n i v e r s i t y o f Rhode I s l a n d Kingston, R I 02881

*Department o f Mathematics U n i v e r s i t y o f Ioannina Ioannina, GREECE

We o b t a i n necessary and s u f f i c i e n t conditions under which a l l s o l u t i o n s o f c e r t a i n n t h order delay d i f f e r e n t i a l equations o s c i l l a t e when n i s odd, and necessary and s u f f i c i e n t cond i t i o n s under which a l l bounded s o l u t i o n s o s c i l l a t e when n i s even. The conditions derived i n v o l v e t h e c h a r a c t e r i s t i c equation o f the delay d i f f e r e n t i a l equation. INTRODUCTION Our aim i n t h i s paper i s t o o b t a i n necessary and s u f f i c i e n t conditions under which a l l s o l u t i o n s o f the n t h order delay d i f f e r e n t i a l equation (1 1 o s c i l l a t e when n i s odd, and necessary and s u f f i c i e n t conditions under which a l l bounded s o l u t i o n s o s c i l l a t e when n i s even. The c o e f f i c i e n t s and t h e delays of the d i f f e r e n t i a l equation are assumed t o be constants such t h a t 0, pi > 0 f o r i = 1,2 ,...,k w i t h k 2 1 and n 2 1. o = T' 0 < T1 < < T ~ po ; * - a

As i s customary, a s o l u t i o n i s said t o o s c i l l a t e i f i t has a r b i t r a r i l y l a r g e zeros. Solutions of (1) are continuous functions x defined on [-T~,-) t h a t s a t i s f y ( 1 ) . As usual, we s h a l l use the term "eventually" t o mean " f o r s u f f i c i e n t l y l a r g e t " . The c h a r a c t e r i s t i c equation o f (1) i s (2) The main r e s u l t i s the f o l l o w i n g . THEOREM 1. ( i ) no r e a l roots.

For n odd, a l l

s o l u t i o n s o f (1) o s c i l l a t e i f and o n l y i f (2)

For

n eyen, a l l bounded s o l u t i o n s o f (1) o s c i l l a t e i f and o n l y i f (2) (ii) has no r e a l r o o t s i n (-m,O]. I t should be noted t h a t f o r n even

F(O)F(+m) < 0. and so equation (2) has always p o s i t i v e r o o t s . That i s , equation ( 1 ) has always unbounded n o n o s c i l l a t o r y s o l u t i o n s . However, the bounded s o l u t i o n s o f (1) may o r

G. Ladas e l al.

314

may n o t o s c i l l a t e .

-

x"(t)

For example, x(t-n) = 0

has the bounded o s c i l l a t o r y s o l u t i o n x ( t ) = s i n t, w h i l e the equation x"(t) x(t -)1 0

-

-

4s

has the bounded n o n o s c i l l a t o r y s o l u t i o n x ( t ) = emt&. PROOF OF THE M A I N RESULT The f o l l o w i n g l e m a w i l l enable us t o r e s t a t e Theorem 1 i n such a way t h a t the odd and even case o f n are proved simultaneously. LEMMA 1. For n odd, ( 1 ) has no unbounded n o n o s c i l l a t o r y s o l u t i o n s and ( 2 ) has no real r o o t s i n 1 0 3 .

Next assume, f o r the sake o f PROOF. Clearly, ( 2 ) has no r e a l r o o t s i n [ O , m ) . c o n t r a d i c t i o n , t h a t ( 1 ) has an unbounded n o n o s c i l l a t o r y s o l u t i o n x ( t ) . Without loss of g e n e r a l i t y we may (and do) assume t h a t x ( t ) i s eventually p o s i t i v e . Then, f o r tl s u f f i c i e n t l y large, x(")(t)

< 0,

x("-')(t)

> 0, and x ' ( t ) > 0 f o r t

2 tl.

I n t e g r a t i n g (1) over [tl,t] we f i n d

Deleting some p o s i t i v e terms and using t h e f a c t t h a t x ( t ) i s increasing, we o b t a i n

which, as t

-+ m ,

leads t o a contradiction.

I n view o f Lemma 1, f o r n odd ever unbounded s o l u t i o n o f (1) i s o s c i l l a t o r y . Hence i t s u f f i c e s t o prove p a r t ( i f o f Theorem 1 only f o r bounded s o l u t i o n s . Therefore, the p r o o f o f Theorem 1 has been reduced t o proving the f o l l o w i n g r e s u l t f o r a r b i t r a r y n. The f o l l o w i n s statements are equivalent.

THEOREM 2 . (a) (b) PROOF.

Every bounded s o l u t i o n o f ( 1 ) o s c i l l a t e s . Equation ( 2 ) has no r e a l roots i n ( - m , O ] . (a)

-

(b).

Otherwise ( 2 ) has a r e a l r o o t Xo 5 0 and so (1) has the bounded

n o n o s c i l l a t o r y s o l u t i o n x ( t ) = exot.

Contradiction. ( b ) -B (a). (We prove i t by c o n t r a d i c t i o n . ) Otherwise (1) has an eventually posit i v e bounded s o l u t i o n x ( t ) . The r e s t o f the p r o o f w i l l be d i v i d e d i n t o a s e r i e s o f s i x lemmas. Throughout the remainder o f t h i s paper x ( t ) stands f o r the p o s i t i v e s o l u t i o n whose existence we have j u s t assumed. LEMMA 2. number

Under the hypothesis t h a t equation ( 2 ) has no r o o t s i n k

rn = min ( * E pie 120 1=0

X T ~

-

ln)

(-m,O],

the (3)

375

e x i s t s and i s a p o s i t i v e c o n s t a n t . LEMMA 3.

For i =

0,1,2

(-l)ix(i)(t) LEMMA 4.

,...,n > 0,

(4)

There i s a c o n s t a n t B > 1 such t h a t x(t-rk) < Bx(t),

LEMMA 5.

eventually.

eventually.

(5)

F o r any

x 2-B

- 1

Tk where B i s as d e f i n e d i n Lemma 4, x ' ( t ) + Xx(t) > 0 and t h e r e f o r e l i m [x(t)e"] tLEMMA 6.

z

0.

Define

xo

= P1l / n ,

and f o r each j = 1,2,

xo(t) = x(t)

... set

n-1 n-1-i i (i) x.(t) = z h (-1) Xj_,(t). J i=o j-1

(9)

Then t h e f o l l o w i n g statements h o l d f o r each j = O y l , 2 , (i)

x j ( t ) i s an e v e n t u a l l y p o s i t i v e s o l u t i o n o f ( 1 ) such t h a t (-l)ixii)(t)

(ii)

>

o for

i = 0,1,2

,...,n

eventually.

(10)

There e x i s t s a p o s i t i v e c o n s t a n t 6 . such t h a t J xj(t-rk)

(iii)

... .

< Bjxj(t)

(11 1

eventually.

( - l ) n + + ' x ! n ) ( t ) + x y x j ( t ) = x! ( t ) J J+1

+

x . x . ( t ) < 0. J J+1

(12)

LEMMA 7. W i t h t h e n o t a t i o n o f Lemma 6 and under t h e h y p o t h e s i s t h a t e q u a t i o n ( 2 ) has no r o o t s i n (-m,O], t h e f o l l o w i n g statements a r e t r u e f o r each j = O , l , Z ¶...

376

G. Ladas et al. Ajt

(ii)

l i m [x(t)e

1

= 0.

t-

The d e s i r e d c o n t r a d i c t i o n i n t h e p r o o f o f Theorem 2 f o l l o w s from ( 7 ) and (14) by choosing A . > A , t h a t i s , by choosing j such t h a t J (pl+jm)l'n > B - 1

-

-

Tk

I n t h e case n = 1 , Lemma 7 i s unnecessary because (6) and (12) a r e c o n t r a d i c t o r y I n f a c t , f o r n = 1, f o r A . > A.

J -

x.(t) = x(t)

for all

J

j

and t h e l a s t i n e q u a l i t y i n (12) reduces t o x ' ( t ) + X x ( t ) < 0. We o m i t t h e p r o o f s o f t h e above Lemmas f o r

l a c k o f space.

REMARKS The case n = 1 was f i r s t i n v e s t i g a t e d by Tramov [ l o ] i n 1975. The same r e s u l t was rediscovered i n 1982 by t h e authors o f t h i s paper 151, by Hunt and Yorke [2] i n an unpublished paper communicated t o the authors, and by Arino, Gy6ri and Jawhari t11* I n the case o f d i f f e r e n t i a l equations with one delay, t h a t i s f o r equations o f t h e form x(n)(t)

+

(-l)ntlpx(t-T)

= 0.

p,T > 0;

n21

(15)

we o b t a i n a necessary and s u f f i c i e n t c o n d i t i o n i n terms o f the c o e f f i c i e n t p and t h e delay T o n l y . I n t h i s case, t h e c h a r a c t e r i s t i c equation o f (15) i s

G(A) :A' + (-l)"+lpe-AT

= 0.

(16)

We prove t h e f o l l o w i n g theorem. THEOREM 3. (a) (b) (c)

For n

[n

even]

the f o l l o w i n g statements a r e e q u i v a l e n t .

u-

A l l s o l u t i o n s o f (15) o s c i l l a t e [ A l l bounded s o l u t i o n s o f (15) late]. Equation (16) has no r e a l r o o t s [Q. (16) has no r e a l r o o t s i n (--,0]]. pl/n > n e

1.

An analogue t o Lemma 1 a l s o holds f o r equation (15). Therefore, the p r o o f o f Theorem 3 i s reduced t o p r o v i n g t h e f o l l o w i n g r e s u l t f o r a r b i t r a r y n. THEOREM 4. (a) (b) (c)

The f o l l o w i n g statements are e q u i v a l e n t . Ever bounded s o l u t i o n o f (15) o s c i l l a t e s . The : h a r a c t e r i s t i c equation (16) has no r e a l r o o t s i n (--,Ol > pl/n I n e

1.

Oscillations o.fDeiajs Diff'crcntiai Equations

PROOF. F i r s t we p r o v e t h a t ( a ) i m p l i e s ( b ) .

377

Otherwise t h e r e e x i s t s a r e a l Xo

such t h a t

50

-XoT

+ (-1)"'pe

= 0.

w h i c h i m p l i e s t h a t x ( t ) = e l o t i s a n o n o s c i l l a t o r y s o l u t i o n of (15), a c o n t r a d i c tion. Next we p r o v e t h a t ( b ) i m p l i e s ( c ) .

-

G ( X ) :(-1)"'

pe-"

Equation (16) i s equivalent t o

= 0.

Since

G(0) = -p < 0 and G(X) has no n e g a t i v e r o o t s i t f o l l o w s t h a t G ( X ) < 0 f o r e v e r y X E Setting

X

=

-

(--,O].

i n t o G(X) we f i n d

F i n a l l y we p r o v e t h a t ( c ) i m p l i e s ( a ) . Otherwise e q u a t i o n ( 1 5 ) has an e v e n t u a l l y n p o s i t i v e bounded s o l u t i o n x ( t ) . As i n Lemma 3, f o r i = 0,1,2,

...,

(-l)ix(i)(t)

>

o

(4)

eventually.

Set q = P''~,

T u = n and

Observe t h a t

= ,(n) ( t ) - q x ( n - - l

1( t - u )

+ q2x(n--2) ( t & )

-.. .+

( - 1 p - 1 q n-1 x ' ( t - ( n - l ) u )

and t h e r e f o r e y ' ( t ) + q y ( t - a ) = x q t ) + (-l)"'px(t-.r) But t h e c h a r a c t e r i s t i c equation o f y ' ( t ) + qy(t-u) = 0

= 0.

378

G. Ladas et al.

is = 0

A t qe-1'

and, i n view o f ( c ) , = ; 1 I n ( q o e ) = 1 ln(pl"'

min (A+qe-")

e ) > 0.

A

T h a t i s , e q u a t i o n (19) has no r e a l r o o t s and t h e r e f o r e ( s e e [ 5 ] ) e v e r y s o l u t i o n y ( t ) o f ( 1 8 ) i s o s c i l l a t o r y . T h i s c o n t r a d i c t s ( 1 7 ) . The p r o o f o f t h e theorem i s complete. The above r e s u l t s have s t r a i g h t f o r w a r d e x t e n s i o n s t o d i f f e r e n t i a l e q u a t i o n s w i t h advanced arguments o f t h e forms X("(t)

-

k

C pix(t+Ti)

= 0

i=O

and x("(t)

-

p x ( t t r ) = 0.

The c h a r a c t e r i s t i c e q u a t i o n s o f ( 1 ) ' and ( 1 5 ) ' a r e r e s p e c t i v e l y

k

f ( X ) E A"

- c

g(A) z A"

-

'=o

Xi.

pie

i=O

(2)'

and (16) '

= 0.

pe"

By s i m i l a r arguments we e s t a b l i s h t h e f o l l o w i n g dual r e s u l t s . THEOREM 1 ' . ( i ) W has no r e a l r o o t s .

n

odd, a l l

s o l u t i o n s o f ( 1 ) ' o s c i l l a t e i f and o n l y i f ( 2 ) '

For even,

(ii) n a l l unbounded s o l u t i o n s o f ( 1 ) ' o s c i l l a t e i f and o n l y i f ( 2 ) ' has no r e a l r o o t s i n [OF). THEOREM 3 ' . (a)

For n odd [ n

3-

t h e f o l l o w i n g statements a r e e q u i v a l e n t .

A l l s o l u t i o n s o f ( 1 5 ) ' o s c i l l a t e [ A l l unbounded s o l u t i o n s o f ( 1 5 ) ' 0shas no r e a l r o o t s [Q.

(c)

( 1 6 ) ' has no r e a l r o o t s i n

[O,m)l.

P

REFERENCES

[l] 0. A r i n o , I . G y o r i and A. Jawhari, O s c i l l a t i o n C r i t e r i a i n Delay Equations, J. D i f f e r e n t i a l Equations ( t o a p p e a r ) .

n

[2] 131

B. R. Hunt and J. A. Yorke, When a l l s o l u t i o n s o f x '

oscillate (preprint).

=

- C

i=l

qi(t)x(t-Ti(t))

T. Kusano, On even o r d e r f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s w i t h advanced and r e t a r d e d arguments, J . D i f f e r e n t i a l Equations 45 (1982) , 75-84.

Oscillutiotis

01’Drluj, Dijyerentiul Eytiutions

379

141 G. Ladas, Sharp Conditions f o r O s c i l l a t i o n s Caused by Delays, Applicable Anal. 9 (1979), 93-98. [5]

G. Ladas, Y . G. S f i c a s and I . P. Stavroulakis, Necessary and s u f f i c i e n t cond i t i o n s f o r o s c i l l a t i o n s , Amer. Math. Monthly ( t o appear).

[6]

6 . Ladas and I . P. Stavroulakis, On Delay Differential I n e q u a l i t i e s of

[7]

G. Ladas and I . P. Stavroulakis, O s c i l l a t i o n s Caused by Several Retarded and Advanced Arguments, J . Differential Equations 44 (1982), 134-152.

[8]

V. N. Sevelo and N. V. Vareh, Asymptotic Methods in the Theory of Nonlinear O s c i l l a t i o n s , Kiev "Naukova Dumka", 1979 (Russian).

[9]

Y . G. S f i c a s and V. A. Staikos, O s c i l l a t i o n s of Differential Equations with

Higher Order, Canad. Math. Bull. 25 ( 3 ) (1982), 348-354.

Deviating Arguments, Funkcial. Ekvac. 19 (1976), 35-43. [ l o ] M. I . Tramov, Conditions f o r Oscillatory Solutions of First Order Different i a l Equations with a Delayed Argument, Izv. Vys5. UEebn. Zaved., Matematika 19, NO. 3 (1975), 92-96.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) @ Elsevier Science Publishers B.V.(North-Holland), 1984

38 1

BOUNDARY BEHAVIOR OF SOLUTIONS OF DEGENERATE ELLIPTIC EQUATIONS AND GENERATION OF SEMIGROUPS Michel L a n g l a i s

Department of Mathematics Purdue U n i v e r s i t y West L a f a y e t t e , I n d i a n a U.S.A.

The g o a l of t h i s t a l k i s t o s u p p l y a n example of a d e g e n e r a t e e l l i p t i c e q u a t i o n f o r w h i c h the a n a l y t i c a p p r o a c h and t h e p r o b a b i l i s t i c a p p r o a c h y i e l d d i f f e r e n t b u t w e l l - p o s e d boundary v a l u e problems. We c o n s i d e r t h e l i n e a r e q u a t i o n w i t h smooth c o e f f i c i e n t s :

(E+ao)u =

(0.1)

R

herein

=

{x E R

N

The c o n d i t i o n on rp

-

, rp(x)

> 0)

on

2R

,

i

+ aou =

VCp (x) # 0

cp(x)”

means t h a t

R but

t h a t e q . (0.1) i s e l l i p t i c i n

i a ux

+

(Cpaijuxi)xj

f

an

on

fl ;

in

d i s t a n c e from

and:

x

to

.

aR

It f o l l o w s

it degenerates i n t o a f i r s t o r d e r equation

.

The c l a s s i c a l a n a l y t i c t h e o r y of second o r d e r and d e g e n e r a t e e l l i p t i c e q u a t i o n s (FICHERA [31,

OLEINIK and WKEVIC 171, KOHN and NIRENBERG [61) asserts t h a t

2R where t h e v e c t o r

boundary c o n d i t i o n s are t o b e p r e s c r i b e d a t t h o s e p o i n t s o f field

aiaxi

C2 = {x E a R

R:

is directed inside

,

01

.

6 N

.

abxi>

This leads t o

t h e boundary v a l u e problem: 0

(A)

(E+a ) u = f

R

in

;

u = g

on

C a r r y i n g t h e d i f f e r e n t i a t i o n o u t i n (0.1) = - Cpa

ij

i

u ~ . + ~ A. uxi 1 J

-

= ai

c2

*

yields:

+

aijCpxj

0 a u

= f

ij - Cpaxj

in

R

1 s i

From a p r o b a b i l i s t i c v i e w p o i n t (STROOCK and VARADHAN [ 9 ] ) boundary v a l u e s are t o b e p r e s c r i b e d on t h e p o r t i o n of

= CX

E a9

(B)

In g e n e r a l Oax) # 0

,~

Z2

51 , t h i s i s a t

2R where t h e v e c t o r f i e l d A axi i s d i r e c t e d i n s i d e R b > ,01~. We now have t h e boundary v a l u e problem

(Efa 0 ) u = f

on

where t h e d i f f u s i o n c a n e x i t i

t h o s e p o i n t s of.

z;

an

and

20

.

in

R

; u = g

on

c;

.

do n o t c o i n c i d e b e c a u s e

(ai’)

i s e l l i p t i c and

We i n t r o d u c e t h e g e o m e t r i c a l i n v a r i a n t :

:

M. Langlais

38 2 . (0.3)

abXi

z(x) =

2R

on

ij

a cpxpxj Clearly

C2 = tx

E aR , z ( x ) > 0)

1;

while

.

, Z(X) > 1 1

ix E a R

=

(B) a r e d i f f e r e n t i f and only if t h e range of

Hence (A)

intersects (0,l).

z

For sake of s i m p l i c i t y we assume i n t h i s n o t e t h a t :

(0.4)

0


in

C(aR)

on

u = g

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

u

in

.

aR

This s o l u t i o n s a t i s f i e s t h e e s t i m a t e : IgIL”(a;2)I

111111 5 M ~ X ( ~ ; ’ I I ~ I 9I

and i s nonnegative when Let

A

f

and

*

g

a r e nonnegative.

be t h e unbounded operator i n

D(A) = { u E C(n)

,

C(n)

,

on

u = 0

with domain and

Eu I$ C ( n ) }

Au = Eu

,

u E D(A)

.

W e can summarize t h e foregoing r e s u l t s , v i a t h e HILLE-PHILLIPS-YOSIDA theorem into: Corollary 1.

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 F e l l e r semigroup on

(-A)

Proof of Theorem 1.

Any s o l u t i o n

in

u

C(n)

of

(E+ao)u = 0

belongs t o

-

R

.

C2(R)

and uniqueness follows from t h e maximum p r i n c i p l e . To g e t e x i s t e n c e we f i r s t assume t h a t

g = 0

.

For any

E

> 0

let

u‘

be t h e

unique smooth s o l u t i o n o f : -(EST

)aijux.x 1 j

+ ~~u~~ +

0 a u = fE

(1.1) =

where

fE

i s smooth and

fE+f

in

s o l u t i o n of eq. (0.1) continuous i n

Co(n)

R

.

o

in on

When

aR

an E+O

,

uE

converges t o a

.

The problem i s now t o d e r i v e t h e c o n t i n u i t y up t o t h e boundary. follow from the r e s u l t s of [ 7 ] which would r e q u i r e on

C2 , namely

Ahxi>

0

on

I?= (E+ao)cp

Z2 , and t h i s i s p r o h i b i t e d by

This does not t o be p o s i t i v e

z(x) < 1 i n

(0.4).

383

Degericrate Elliptic Equations Lema 1.

Choose

M = M(m)

such t h a t

Define

vE

m

m < z(x)

I

M[cp(x)+Elm

luE(x)

5

l2

Bi

A0 =

+

(z(x)-m)

ij

There e x i s t s

R ,0

f


2, there exists E such that if An < 5 < then for large 6 , ( P ) has at least five solutions. Again, in the future, we expect it will be possible to prove that in this situation, there are at least solutions. Generalizations: Other Boundary Value Problems Two natural problems arise in conjunction with our earlier discussion. To what extent are these results dependent on selfadjointness, and to what extent are they dependent on compactness. We now discuss parabolic and hyperbolic cases, i n each of which, one of these properties is missing. First we consider solutions of (9) u t - uxx + f(u) = s Sinx + h(x,t) u(0.t) u(x,t+T)

In similar

=

u(n,t)

=

0

= u(x,T).

[13], the second author to that proved for the

and W. Walter elliptic case

Namely, they show that if s is large, a


.

Here, the second author, with W. Walter, showed that if 0 < a < 1 , f is monotone, and the interval [ a , B 1 contains an odd number of eigenvalues (counting multiplicities) then for large s, R has at least 3 (generically 4 ) solutions. It is natural to conjecture that a modification of the methods of Hofer could prove the existence of at least four solutions if g + m 2 - n 2. Generalizations: Crossing Higher Eigenvalues A second natural question is to ask what is the situation if X < a < Xj+k < B <j+k+l. This situation appears much j

trickier. There are some results however. The second author 1 1 4 1 showed that if X


, a l l z

E

N u l l T.

Thus yo

E

Range T* i f a n d o n l y i f

-L

u E (Null T) , i . e . , u i s t h e unique l e a s t - s q u a r e s o l u t i o n of T ( z ) = g w i t h minimum L2-norm. REMARK. The a b o v e t h e o r e m i s r a t h e r t h e o r e t i c a l t h a n p f a c t i c a l . T h i s i s d u e t o t h e i n v o l v e m e n t o f t h e c o m p u t a t i o n of A ( f ) , where A:

= T(T*IS

+

I.

o n t o L 2 , which i s p o s i T h i s o p e r a t o r i s o n e - t o - o n e f r o m Dom T ( T * ) t i v e and s e l f - a d j o i n t . Assumifg t h a t d # 8 , i t would b e i n t e r e s t i n g t o know w h e t h e r o r n o t ( i ) A i s an i n t e g r a l operator, (ii) t h e r e e x i s t s a nonstopping i t e r a t i o n f o r t h e s o l u t i o n of A(y) = f .

R e f e r e n ce s

D u n f o r d a n d J. T . S c h w a r t z , L i n e a r o p e r a t o r s , P a r t 11, I n t e r s c i e n c e ( 1 9 6 3 ) , N e w York.

[ D S 11

N.

[Ll]

S. J . L e e , Boundary c o n d i t i o n s f o r l i n e a r m a n i f o l d s I , J. Math. A n a l . Appl. 7 3 ( 1 9 8 0 ) , 366-380.

[LNl]

S. J. L e e a n d M. Z . N a s h e d , G r a d i e n t method f o r n o n d e n s e l y d e f i n e d c l o s e d unbounded o p e r a t o r , P r o c . Amer. Math. SOC. (to appear).

[LN 21

S. J . L e e a n d M. Z . N a s h e d , G e n e r a l i z e d i n v e r s e s f o r l i n e a r m a n i f o l d s and a p p l i c a t i o n s t o boundary v a l u e problems i n Banach s p a c e s , C. R. Math. Rep. A c a d . S c i . C a n a d a , Vol 4 , n o . 6 ( 1 9 8 2 ) , 347-352.

[Lue 1 1 D . L u e n b e r g e r , O p t i m i z a t i o n by V e c t o r S p a c e Method, W i l e y (1969). [ N 11

Z. Nashed, S t e e p e s t d e s c e n t f o r s i n g u l a r l i n e a r o p e r a t o r e q u a t i o n s , SIAM J . Numer. A n a l . , V o l 7 . N o . 3 ( 1 9 7 0 ) , 358-362.

M.

[LOC 11 J . L o c k e r , Weak s t e e p e s t d e s c e n t f o r l i n e a r b o u n d a r y v a l u e p r o b l e m s , I n d i a n a U n i v e r s i t y Math. J . 2 5 ( 1 9 7 6 ) , 525-530.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

403

SOME VECTOR FIELD EQUATIONS Elliott

H. L i e b *

Departments o f Mathematics and Physics Princeton U n i v e r s i t y P r i n c e t o n , N. J . 08544

Systems o f t h e t y p e i and g ( u ) = aG/aui

i

= g (u),

-aui

with

u

(ul,*--,un)

=

d

IR , d 1. 2 a r e s t u d i e d .

on

G

s u i t a b l e c o n d i t i o n s on

and

g

Under

i t i s shown t h a t t h e r e

i s a s o l u t i o n o f t h e system t h a t a l s o m i n i m i z e s t h e 2 S ( u ) = ]lVul -/G(u), within the class o f a l l

action,

n o n t r i v i a l f i n i t e a c t i o n s o l u t i o n s o f t h e system. I. INTRODUCTION

A problem t h a t a r i s e s i n v a r i o u s branches o f mathematical p h y s i c s i s t h e following:

1

n

. L e t G: R n + lR be c o n t i n u o u s and i n C ( R \ { O } ) w i t h d e r i v a t i v e g, g ' ( u ) = aG(u)/aui, w i t h d~ = ( u ...., u I , and G(0) = 0, g ( 0 ) E 0. The t o bA c o n s i a e r e d i s system o f e q u a t i o n s on IR where each

ui: R d

-t

IR.

i = 1,

= gi [ u ( x ) ) ,

-Aui(x)

We s h a l l assume

d 22

... ,n

and seek a s o l u t i o n

i.e.,

(1.1) u $ 0.

A s s o c i a t e d w i t h t h e system ( 1 . 1 ) i s t h e A c t i o n S(U)

=

K(u)

=

K(u) - V ( U ) l n 2 - 1 /IVU.I dx 1 2 i-1

1

I

]VU]

2

dx

(1.2)

] G ( u ( x ) ) dx.

(1.3)

I t i s obvious t h a t g e n e r a l l y speaking mum ( t a k e , f o r example G(u) = -u2 + u4). f o l l o w i n g c l a s s f o r which S i s f i n i t e :

S(u) has no f i n i t e minimum o r maxiWe can, however, c o n s i d e r t h e

V(u) =

C =

r u l u s L:oc ( R d ) , 11([iul

V u E L2 (lRd),

> a])


(1 -4)

0)

where u denotes Lebesque measure and where [ l u l > a] denotes e i t h e r t h e s e t { x [ \ u ( x ) I > 31 o r 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 o f t h i s s e t , a c c o r d i n g t o t h e context. (The l a s t c o n d i t i o n i n ( 1 . 4 ) i s a weak n o t i o n o f u ( x ) + 0 as 1x1 -. I t i s v e r y i m p o r t a n t because w i t h o u t i t we c o u l d e a s i l y have s o l u t i o n s w l t h u ( x ) + c o n s t a n t as 1x1 + m . ) Furthermore, we can c o n s i d e r E C C, where +

E = {u

1

u E

c,

u

s a t i s f i e s (1.1) i n

0'1,

(1.5)

E.H. Lieb

404 and

The t o t a l problem i s : Show t h a t t h e r e i s a Smin

=

-

uE E

such t h a t

S(ii)

(1.7)

O f course some c o n d i t i o n s w i l l have t o be imposed on G so t h a t t h e problem has a s o l u t i o n , b u t f i r s t l e t us b r i e f l y r e v i e w e a r l i e r work on t h e problem. W i t h one e x c e p t i o n n o t e d below a l l p r e v i o u s r e s u l t s f o r t h e problem have been f o r t h e s c a l a r case n = 1. Probablydthe e a r l i e s t general t r e a t m e n t o f e x i s t e n c e o f s o l u t i o n s o f (1.1) ( f o r a l l o f IR , as d i s t i n g u i s h e d f r o m bounded domains) was by Strauss [12]. Coleman, G l a s e r and M a r t i n [8] l a t e r made an i m p o r t a n t c o n t r i b u t i o n t o t h e t o t a l problem (n = 1, d 3) by a s c a l i n g argument, o r " c o n s t r a i n e d minimum method," t o be e x p l a i n e d below. B e r e s t y c k i and P. L. L i o n s [ 4 ] improved t h e r e s u l t s i n [8]. Both [12] and [4] c o n t a i n e x t e n s i v e b i b l i o g r a p h i e s o f r e l a t e d work, so t h e r e i s no need t o r e p e a t i t h e r e . I n [3], B e r e s t y c k i and L i o n s announced t h e e x i s t e n c e o f a r a d i a l s o l u t i o n o f (1.1) f o r n 5 1 which minimizes However, t h e r e i s no reason t o t h e a c t i o n among a l l r a d i a l s o l u t i o n s o f (1.1). b e l i e v e t h a t t h e minimum a c t i o n s o l u t i o n we seek i s n e c e s s a r i l y r a d i a l . I n t h e b e g i n n i n g o f 1983, I a n A f f l e c k encouraged me t o t r y t o e x t e n d these r e s u l t s t o d = 2 and t o n > 1 . He e s p e c i a l l y needed t h e d = 2 case [l]. I was a b l e t o make b o t h e x t e n s i o n s p r o v i d e d t h e a d d i t i o n a l assumption l i m sup G ( u ) / l u 1 2 < 0 was made. T h i s was r e p o r t e d a t t h e conference. However, u+o I d i s c o v e r e d t h a t B e r s t y c k i , G a l l o u e t and #avian ( u n p u b l i s h e d ) a l s o s o l v e d t h e d = 2, n = 1 problem more t h a n a y e a r ago by t h e same method as mine. F u r t h e r more P. L. L i o n s t o l d me i n a r e c e n t l e t t e r t h a t he was a b l e t o make t h e e x t e n s i o n t o n > 1 by u s i n g h i s t h e o r y o f c o n c e n t r a t i o n compactness [lo]. The e x t r a assumption mentioned above seemed u n n e c e s s a r i l y r e s t r i c t i v e . I n c o l l a b o r a t i o n w i t h H. B r e z i s we were a b l e n o t o n l y t o e l i m i n a t e i t , b u t a l s o t o make e s s e n t i a l changes i n t h e p r o o f r e n d e r i n g i t s i m p l e r and s t r o n g e r . Therefore, t h i s e n t i r e work i s now d e f i n i t e l y a j o i n t e f f o r t whose d e t a i l s w i l l be g i v e n i n Here, o n l y an o u t l i n e w i l l be given. a f o r t h c o m i n g a r t i c l e [7]. 11.

THE MAIN RESULT

The d i s t i n c t i o n between d = 2 and d 1. 3 i s v e r y i m p o r t a n t . The assumpt i o n s t o be made about G and g a r e t h e f o l l o w i n g : (The common symbol C w i l l be used throughout t h i s paper t o denote i n e s s e n t i a l p o s i t i v e c o n s t a n t s . ) 1 F i r s t , we assume t h a t G E Co(Rn) and G E C (IR\ {Ol) w i t h d e r i v a t i v e i s done t o a l l o w t h e p o s s i b i l i t y t h a t G(u) 21 - 1 ~ 1 , f o r example, g = vG. ( T h i s n e a r u = 0.) We assume t h a t G ( 0 ) = 0 and d e f i n e g ( 0 ) = 0. d 2 3: --(i) (ii)

G(uo) > 0

f o r some

uo

E

IRn .

l i m sup G(u) U I - ~ . 0 as luJ(;,p p = 2* = 2d/ d - 2 ) . (Note

i

and not

IuI

IG

+ m.

u ) 1 .)

Here (2.2) (2.3)

405

Some Vector Field Equations d = 2: (i) (ii)

G(u,)

>

o

f o r some

uo E IR"

(2.4)

l i m sup G ( ~ ) l u l -5~ 0 as / u 1 m. He e p i s any f i x e d p < m. ( 1 u l - P c o u l d be r e p l a c e d by exp[-C/ul 1 f o r some C s u f f i c i e n t l y

F

+

s m a l l , i f necessary.) (iii)

F o r some

(iv)

Ig(u)I 'C

Remark:

>

E

0,

(2.5)

w

+ clulp-l,

/uI

when

G(u) < 0

(2.6)

< E.

u E R".

(2.7)

(2.3) and (2.7) i m p l y t h a t /G(u)I

C + C/ulP


0 and l e t u,(x) = ~ ( X X ) . Then d S ( u A ) = 0 a t h = 1. But

K(u,)

2-d

=

V(U,)

-d

= A

K(u),

w

V(u),

u E

c.

(3.1)

T h e r e f o r e , we have t h e v i r i a l theorem:

(d If(3.2)

h o l d s then, f o r

d

- ~)K(u)

= dV(u).

2 d - 2

=

3,

S(u)

=

~

V(U)

2 7 K(u)*

I t t u r n s o u t t h a t (3.2) w i l l h o l d n o t o n l y f o r t h e m i n i m i z i n g u s o l u t i o n t o ( 1 . l ) under somewhat m i l d e r r e s t r i c t i o n s than ( 2 . 1 ) - ( 2 . 7 ) . i s Pohozaev's i d e n t i t y [ll].

Lemma 2.

Suppose

G E C1(IRn\\IOl), dL2.

g=VG

u

E

and

(3.2)

L:~~. g(0)

vu 5

E

0.

(3.3) b u t f o r any This

2, ~ ( u E ) L ~ . suppose G E c 0 ( ~ " ) , Let u s c l t i s f y (1.1) in D’ on IRd,

L

Then, under no f u r t h e r a s s u q t i o n s , (3.2) h o l d s. P r o o f : The f o r m a l i d e a i s t o m u l t i p l y (1.1) b y x . V u . sum on i and i n t e g r a t e . I n t e g r a t i n g by p a r t s y i e l d s ( 3 . 2 ) , f o r m a l l y . T h l s can be t u r n e d i n t o a p r o y e r z r o o f [4], assuming G E C1(lRn)). I t i s s l i g h t l y t r i c k i e r i f G i s o n l y See [ 7 l . i n C (W \ { O l ) . To connect Lemma 2 w i t h t h e p r e v i o u s assumptions on g, and t h e r e b y t o est a b l i s h t h e v a l i d i t y o f ( 3 . 2 ) f o r a l l s o l u t i o n s t o ( 1 . 1 ) i n C, t h e f o l l o w i n g i s needed. Only (2.3) and (2.7) w i l l be used.

E.H. Lieb

406

Suppose ( 1 . l ) hoZds i n 9’

Lemma 3.

In f a c t u

hoZd. Then u E Lyoc. for a l l CY < 1 .

Proof: d = 2: wq


I I U J I I ~< C. .P UJ E C.)

G(u) 5 y ! u I p

6 > 0

E,

>++~

( [ ~ U ~ ~ E ] ) S U P ~ ( G ( U ) ~ ~ E ~0 ~ U ~ < ~ / E ~

Given ( 4 . 1 ) o r ( 4 . 2 ) , t h e f o l l o w i n g lemma i s r e l e v a n t .

J

Lemma 5. .Let I U I 1 d > 1 w i t h uJ E Lloc, thzt

u([lujl

>

bs a sdquence of R" v a ~ ud f u n c t i o n s on R v d uniformly bounded i n L' and It vuJll 5 C .

€1) >

6

f o r some

6 ,

Then t h e r e e x i s t s a sequence of t r i i n s l u t i o n s , f i x e d constant CI(S,E,C) > 0 such t h a t u({X E

where

B =

{XI

Proof.

6 >

o

uJ(x)

and +

C~ZZ

5

wJ(x)

B / I W j ( X ) l > ~ / 2 } )> a ( 6 , ~ , c ) ,

and

u

iz

(4.3)

See [9].

E s t i m a t e s on

-f

n 2 I, Suppose

1x1 5 1 1 i s t h e u n i t baZZ.

f i x e d bounded s e t uj

,

j

uJ(x + y . ) J

0

T r a n s l a t i n g uJ as i n Lemma 5 does n o t a f f e c t K ( u j ) s h a l l h e n c e f o r t h assume t h a t ( 4 . 3 ) h o l d s w i t h WJ = uJ.

that

d

uJ:

n,

pointwise

d 3: II uJII ,I (n)

a.e.,

By t h e Sobolev i n e q u a l i t y

C. (Rd)

V(uJ).

or II

uJtI

P


0 some a d d i t i o n a l t e c h n i c a l i t i e s a r e r e q u i r e d . 3. The analogue o f (4.10), (4.11) i s consider d tLp($)

o ( t ) ,T"1

f

i I d J l ) 1-2/d -

tL1($)

-

ltlC

i I+l[u

=

01 5 b

f

First,

11

(4.16)

u=o

Therefore, I L 3 ( @ )I 5 b b = CT(l - 2/d)

and

-

T(l t m d s t o a bounded l i n e a E o p e r a t o r on L 1 L3($) = j h $ w i t h h E L Thus

with

L 3 = L2

.

-Au

-

-

T(l

-

i

2/d)L1.

.

Since

(4.17) (4.17) implies t h a t

Lm i s t h e dual o f

L3 ex1 L ,

2 / d ) g ( u ) = h E Lm.

(4.18) 1

.

BY (4.17) l j h $ l I I$l[u = 01 f o r 1 . E ,C; and hence f r $ E L Thus h ( x ) = 0 when u ( x # 0. B u t (4.18) i m p l i e s t h a t u E C p y a , which i m p l i e s t h a t

-.

u E WzYp, Vp < Then A u = 0 a.e on [u = 01. Since g ( 0 ) = 0, h = 0 a l s o when u = 0. (4.18) i s i d e n t i c a l t o 4.14) and t h e c o n c l u s i o n f o l l o w s as b e f o r e .

( = 2 a l l we know i s t h a t i $ l [ u = 01. The p r o o f t h

When >

-C

L 5

($1 > 0 i m p l i e s t h a t l i m e i s a b i t l o n g e r , b u t we a g a i n r e c o v e r

See [i] f o r details. V.

FURTHER PROPERTIES

OF SOLUTIONS TO (1.1)

The goal s t a t e d i n t h e i n t r o d u c t i o n has been reached-to f i n d a f u n c t i o n i n i n t h e c l a s s o f a l l such f u n c t i o n s . I n t h e process o f d y i n g so i t was shown t h t a l l s o l u t i o n s t o ( 1 . 1 ) i n C s a t i s f y (3.2) and a r e i n W1;2, V q < m and i n C,Y:; WLX < 1 .

C t h a t s a t i s f i e s ( 1 . 1 ) and t h a t m i n i m i z e s S(u)

Here we s h a l l prove t h a t s o l u t i o n s t o ( 1 . 1 ) i n C a c t u a l l y go t o z e r o as i n t n e usual sense and, i n c e r t a i n cases, have compact support. m

1x1

+

-_Lemma

8:

Y

Let

LL

(weak Lr) for some 1 < r < m , A E L r ' with ~ ( S U P A) P < m and f E Lt with t > p. Assume t h a t A, Y, f 2 0. Suppose t h a t f - 1 5 Y * ( A f ) . Then f E Lq(B) for a22 1 5 q < m and f o r a22 B of f i n i t e measure. !Note: t h i s i s stroriger than L?OC* ) See [7] f o r t h e p r o o f . (2.7).

Theorem 9: Let Then l u ( x ) I Let

-Af

0

as

f

f = luIL. f = -2u.A~

Then

- 2 1 0 ~ 1+ ~f 5 2 U - g ( u )

0,

where

S,

since

u E).:;:W

E.

-

+

A f E L:oc

f o r any

Y

s a t i s f y (1.1) in D’ w i t h t h e assumptions (2.1) 1x1 m i n the usuaZ sense.

u E C +

n

Proof:

that

E

>

f

f 5

E

+ C,fP"

i s t h e c h a r a c t e r i s t i c f u n c t i o n of Since

i s t h e Yukawa, o r Bessel, p o t e n t i a l . f

-

u

E

Since

CE 5 Y*A,f

C, f

2 Y*(E

(5.1)

S,

[lul

>

€1.

(Note

+ C E f p / 2 S E ) , where

Y E L1 ,

(5.2)

41 1

Some Vector Field Equations with

AE = CEfPi2-' Y E Lr

u E Lp.

W'

SE. I f d r = p/2, r ' = d/2, since 3, A E L r ' , as i s w e l l known. By Lemma 8, w i t h t = p, f E Lq(B! f o r a l l

B o f f i n i t e measure and 1 5 Y E Lt f o r some 1 < t < r. f < CE + h, and h = Y*AEf hoids f o r a l l E > so t h e

q < m. Therefore, A f E Ls, Ws 1. 1. However, I n $ e r t i n g t h i s i n ( 5 . 2 ) , w i t h s = t ' , we see t h a t i s a f u n c t i o n t h a t goes t o z e r o as 1x1 + m . T h i s theorem i s proved. I f d = 2 we have, by t h e 2 G a g l i a r d o - N i r e n b e r g i n e q u a l i t y and t h e f a c t t h a t v u E L , t h a t f E Lq(B) f o r a l l 1 5 q < m and a l l B o f f i n i t e measure. The remainder o f t h e p r o o f i s t h e same as f o r d 1. 3. 0

6,

F i n a l l y , we t u r n t o t h e q u e s t i o n o f compact s u p p o r t f o r s o l u t i o n s t o (1.1). L e t us assume t h a t g(u).u f o r 1u1 < 6, where 0 < t < 2 7 f o r 1u1

2 -Clulr,

C , 6 > 0. 6.)

with

1


2k+l,

m

$

i - 2 k - l ) L . i,i-u . u m-u = 0 ,

2n [i-~-1)F'(m-2k-l, i=2k+l u=O

k = 0 , 1,

. .. , ( n - l ) , and

*

=

0 , m = 0,1,

...

Vm Pm+l""Uzm

what i s t h e r e l a t i o n s h i p between T (m) and S (m)? k k and c a l c u l a t i o n s r e v e a l , f o r example, t h a t T (m) = Sn-l(m) It i s n a t u r a l t o a s k :

Tn-l(m)

= Sn-2(m)

where

A1 = 1 and

-

n(n-1) (m-2114-3) (m-Zn+2) 2

Sn-l(m).

Easy

In general,

P(m-2nf2k-2i-3,

2j-2i+2)

=

0 , j = 1, 2 ,...,k.

i= 1

Thus, i t i s c l e a r t h a t Tk (m) = 0 i f and o n l y i f Sk (m) = 0 . Assuming t h a t ( 7 ) has a unique s o l u t i o n i n t h e s e n s e t h a t once p o i s known t h e o t h e r urn's are u n i q u e 1 determined, w e can conclude t h a t A a c t s as an o r t h o g o n a l i z i n g weight f o r {$mcX)

7.

By s e t t i n g @ ( x ) = 1 i n ( 5 ) , w e have t h e f o l l o w i n g r a t h e r s u r p r i s i n g c o r o l l a r y . Corollary 6

( A , b2k-1(x))

= 0 , k = 1, 2 ,

..., n.

EXAMPLES i)

The Hermite d i f f e r e n t i a l e q u a t i o n :

-

y"

2xy'

+

2ny = 0 .

In t h i s case, A s a t i s f i e s t h e d i f f e r e n t i a l e q u a t i o n A ' 2

+

2xA = 0 .

.

This has

The r e c u r r e n c e r e l a t i o n t h a t w e o n l y the classical s o l u t i o n A(x) = e-x g e t from ( 6 ) is - 2 ~ ~ + ~ + m u , - ~ =0 . T h i s i s e a s i l y s o l v e d t o y i e l d U ~ ~ + ~ = O

ii)

The Bessel d i f f e r e n t i a l e q u a t i o n : Here,

A satisfies

x2y"

+

2 ( x + l ) y ' - n ( n + l ) y = 0.

417

Orthogonal Polynomials T h i s e q u a t i o n h a s n o t y e t been solved f o r its g e n e r a l d i s t r i b u t i o n a l s o l u t i o n . I t i s n o t even c l e a r what t h e a p p r o p r i a t e s e t t i n g f o r A s h o u l d be. The r e c u r r e n c e r e l a t i o n t h a t w e g e t from (6) i s (m+2)u + 2um = 0. Again, t h i s m+l

um

i s e a s i l y solved t o y i e l d

=

Morton [2] showed t h a t w(x) =

(-1)m2mu'2 (m+l) !

1

.

It s h o u l d be n o t e d t h a t K r a l l and

2m+16(m)(x) ml(m+l)l f o r m a l l y a c t s as a weight d i s t r i -

m= 0 b u t i o n f o r t h e Bessel polynomials. (8).

One can check t h a t w(x) f o r m a l l y s a t i s f i e s

i i i ) The Legendre t y p e d i f f e r e n t i a l e q u a t i o n :

( ~ ~ -2y(4) 1)

+ 8x(x2-l)y"'

+

(4a+12) (x2-l)y"

+ 8axy'

= X

my

For t h i s OPS, A s a t i s f i e s t h e two d i f f e r e n t i a l e q u a t i o n s a)

( x 2 -1)’~’

b)

(x2 - l ) ' d 3 '

=

o

+

+

12x(x2 -1)h"

The g e n e r a l s o l u t i o n t o ( a ) i s A(x) = c 1 t h i s i n t o (b) y i e l d s c 1 = a c 2 = a c 3 . A(x) =

+

$6(x+l)+$&(x-l).

are e a s i l y s o l v e d t o y i e l d u

-

[(24

4a)x'

+ 4aIA'

= 0

+ c26(x-1) + c36(x+1).

I f w e l e t c2 =

t,

S u b s t i t u t i o n of

we find

The moments from t h e two r e c u r r e n c e r e l a t i o n s i n (6) 2m+l

=

0

and

u~~

=

at2m+l 2m+l

A NEW OPS

We now show how t h e t h e o r y developed can b e used t o f i n d t h e d i f f e r e n t i a l e q u a t i o n once t h e w e i g h t d i s t r i b u t i o n i s known. The method t h a t w a s used p r i o r t o t h i s w a s a method c a l l e d S h o r e ' s t e c h n i q u e , which i n v o l v e d the Lagrange i d e n t i t y ( f o r example, see [ 5 1 ) . To i l l u s t r a t e t h i s new a p p r o a c h , w e w i l l f i n d t h e s i x t h o r d e r d i f f e r e n t i a l equat i o n h a v i n g a sequence o f polynomial s o l u t i o n s o r t h o g o n a l on (0,m) w i t h r e s p e c t

1

t o A(x) = - 6(x) +xe-X. A

Suppose, t h e n , t h a t t h i s OPS s a t i s f i e s t h e s i x t h o r d e r

6

e q u a t i o n L6(y) =

1

bi(x)y i=l

formally s e l f a d j o i n t .

(i)

(x) = Amy(x).

It follows t h e n t h a t

W e s h a l l assume t h a t xe-XL6(y) i s

L.L. Littlejohn

418




goes t o i n f i n i t y , hence when

y

+ e2(bo) + e3(bo,y)

bo

+ e3(bo,l) + e4(bo,l)

+ e2(bo)

w i l l be less t h a n

But Y 3a/2.

y > 1

such t h a t -l+ibo e4(bo,v) = 3 ~ / 2 . Therefore x i s i n domain

y/2 - l + i b o 2 T](M). B u t y - 2 > -1, x x i s n o t i n L ( I ) , by Lemma 1, M i s n o t l i m i t - p o i n t . T h i s Theorem shows a l t h o u g h t h e r e a l p a r t and t h e i m a g i n a r y p a r t o f M a r e l i m i t - p o i n t , b u t t h e whole t h i n g f a i l s t o be l i m i t - p o i n t . It i s a l s o i n t e r e s t i n g t h a t i n t h e r e a l case, Kauffman proved t h a t a l l f o u r t h - o r d e r symmetric expressions w i t h e v e n t u a l l y p o s i t i v e polynomial c o e f f i c i e n t s are l i m i t - p o i n t . T h i s i s n o t t r u e f o r t h e complex case. B u t we can prove i n t h e second-order case, a l l such non-symmetric e x p r e s s i o n s a r e l i m i t - p o i n t . Lemma 2. and

d,

Let

M =

non-negative.

1Nk=O ( - l ) k c k D

x n(k)Dk

Suppose t h a t

n(k)

-

+ 2k

i I s=o N( - l ) S d S D S x m ( s ) D S , w i t h
j

ck

and

m ( s ) - 2s < m ( t ) - 2 t f o r s > t. I f one o f t h e f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d : ( 1 ) co > 0 and do > 0; ( 2 ) co = 0 o r do = 0 and n ( k ) - 2k # m ( s ) - 2s f o r k # s, t h e n To(M) i s separated. Remark 1. 2.

See [l]f o r t h e p r e c i s e d e f i n i t i o n o f s e p a r a t i o n . The p r o o f o f Lemma 2 i s t o o c o m p l i c a t e d t o be i n d i c a t e d h e r e .

Using t h e same t e c h n i q u e i n [l], we can p r o v e t h e f o l l o w i n g conclusions. Lemma 3. Suppose M s a t i s f i e s t h e c o n d i t i o n s d e s c r i b e d i n Lemma 2. L e t rl = m ’ c k > O } (r2= min{s I d, > 01). Then f o r any k ( s ) such t h a t

k > r1 and ck > 0

5

(a 5

(s

>

r2 and

d,

>

0)

there i s a positive E E n ( k ) - a 2k-k D(To(M)) C D(To(x x D ) ) (D(To(M)) rl

2k

=

Lemma 4. -ck

r 2 = 0,

-

II,

such t h a t C

D(To(x E x m(S)-2D2S-!?1 1 ) .

Suppose M s a t i s f i e s t h e c o n d i t i o n s d e s c r i b e d i n Lemma 2. L e t Suppose one o f t h e f o l l o w i n g con> 01 and r 2 = m i n { s I ds > 01.

ditions i s satisfied: m(r2)

and f o r any n a t u r a l number

2s),

(1)

rl = r2 = 0

and

m(0) 5 0 and n ( r l ) - 2rl < m(0): Then x n f i s i n L2(I)

2r2 < n(0).

n(0) (3)

5 0 or r1 = 0,

f o r any

f

m ( 0 ) 5 0; r2 > 0,

( 2 ) rl

n(0) 5 0

i n k e r T1(M)

>

0,

and

and any

42 3

Limit-Po in t Classification of Non-Self-A djo in t Operators n a t u r a l number

n.

Lemma 5. L e t M = P + iQ, where P and Q a r e 2Nth-order symmetric d i f f e r e n t i a l e x p r e s s i o n s w i t h r e a l c o e f f i c i e n t s . Suppose range To(M) i s c l o s e d t h e Lagrange b i l i n e a r and f o r any f i n k e r T (M) and any g i n k e r T (M'), form [f,g](x) c o n v e r g e i t o z e r o as x approachei i n f i n i t y . Then M i s l i m i t point. Lemma 6. M

n(1) - 2

(2)

n(0)




and

+ doxm), co + do

>

where c1 , co, 0, t h e n M i s

i s separated.

P r o o f . ( I ) Case m .I0. L e t M1 = -D(cl + id,)xm+'D M2 = ( c o + i d o ) x m . F o r any f i n k e r T 1 ( M + l ) , ( M 2 + l ) f

and is in

Mlf. Since M1 Lemma 6, Tl(Ml)

M

(11)

and M 2 + 1 a r e l i m i t - p o i n t , b y Lemma 5, i s separated, so i s T1(M).

Case

m > 0.

Now,

L'(I),

so i s

i s limit-point.

By

has c l o s e d range, so

T1(M)

2d(M) = n u l l i t y T1(M) + n u l l i t y T,(M+) T1(M) = 1,

so

i s a l s o separated.

2 n u l l i t y T1(M). To show n u l l i t y Mf = 0. Mx x = 0 i f and o n l y i f

=

c o n s i d e r t h e power s o l u t i o n s o f

-(cl + i d l ) X ( h + m + 1) + ( c

0

+ i d ) = 0.

Since

0

c1 + dl

>

0,

co + do > 0,

we

always have two d i f f e r e n t s o l u t i o n s A+ and A _ . A s i m p l e computation shows t h a t one of A+ and h- has r e a l p a r t l e s s t h a n -1 and t h e o t h e r ' s r e a l p a r t i s p o s i t i v e , so n u l l i t y T1(M) = 1 . Hence we proved M i s l i m i t - p o i n t , by Lemma 2.33 i n [l], we a l s o o b t a i n e d T1(M) i s separated. Lemma 9 . cl,

co, dl,

Let

do

M = - c 1Dxn(')D

+

a r e non-negative.

+

coxn(o)

M

Then

i(-dlDxm(')D

+

i s l i m i t - p o i n t and

doxm(o)), T1(M)

where is

separated. Proof. (I)

I f n(1)

Suppose a l l t h e c o e f f i c i e n t s a r e p o s i t i v e .

-

2 # m(0)

and

m(1)

-

1.

When n ( 1 ) - 2 # n ( 0 ) f r o m Lemma 7.

2.

When n ( 1 ) - 2 = n ( 0 ) (case f o l l o w i n g t h r e e cases.

and

2 # n(0). m(1)

-

m(1)

2 # m(O),

-

2 = m(0)

the conclusion f o l l o w s i s t h e same), c o n s i d e r

( 1 ) F o r m(1) - 2 < n ( 0 ) and m(0) < n ( O ) , l e t M = - C DXn(0)+2D + coxn ( 0 ) , Mo i s l i m i t - p o i n t and T1(Mo) i s separated. I t 0 1 i s obvious t h a t Mo i s t h e main p a r t o f M. L e t W = i(-dlDxm(')D + doxm(0)) and

ME = Mo + EW,

domain

To(M,)

0

5

equals

E

5 1.

I t can be shown t h a t

domain To(Mo).

To(M,)

i s separated, so

By t h e s t a b i l i t y o f t h e i n d e x o f a

Fredholm o p e r a t o r under s m a l l r e l a t i v e l y bounded p e r t u r b t i o n s , range T (ME + 1 ) i s c l o s e d and t h e d e f i c i e n c y o f range T (M + 1 ) i n L ( I ) i s a c o n s t g n t independent o f E . But t h e d e f i c i e n c y o? r h g e T (M + 1) i n L 2 ( I ) e q u a l s so i s M. n u l l i t y T1(Mr + 1 ) . Hence M + 1 = Mo + 1 + W i! l:mit-point,

h

For a l l t h e cases t h a t we w i l l c o n s i d e r l a t e r , t h i s s o r t o f argument w i l l be used again, so we o m i t t h e d e t a i l s .

Limit-Point Classificutio n of No n-Self-Adjoin t Operators (2) M

m(1) - 2

For

= - C DXn(o)+2D

f

sgparated. 1




By Lemma 7,

i d xm('). 0

let

n(O),

Mo

425

i s l i m i t - p o i n t and

T1(Mo)

is

( 3 ) F o r m(1) - 2 > n(O), l e t Mo = i(-dlDXm(l)D + d x ~ ( ~ ) ) T. h i s i s a c t u a l l y t h e r e a l case, so Mo i s l i m i t - p o i n t and T1(Mo)O i s separated. (11) I f n(1) - 2 = m(0) f o l l o w i n g cases. 1.

m(0)

>

n(0).

M0 =

- C 1 DXm(o)+2D

(3)

For

(1)

For

+ i d0 xm(').

m(1) - 2 = m(O),

m(1) = 2 = n(0)

(case

m(1)

2


m(O),

(2) let

-

i s t h e same),

m(0) < n(0).

+

l e t M = -idlDxm(')D. i( -dl Dxm(o)+BD + d x m ( o ) ) . 0

let For

-

m(1)

2 = n(O),

let

0

3.

m(0) = n ( 0 ) .

M0 = -clDx m(o)f2D

(1)

+ c 0xm(')

For

m(1)

-

+ idoxm(o).

2




1

I
0 = Ip

Theorem 1 3 shows t h e s o l u t i o n that

F(u) -

u E U

s) .

F

If

when

for a l l


0

such t h a t

Moreover

u(~,E)

0.

It remains o n l y t o check 3 " , i . e . t o v e r i f y (10)-(1’2).

I n e q u a l i t y (12)

happens t o be v a l i d i n d e p e n d e n t l y of how we choose approximate s o l u t i o n s . S e t af A. A.(t,u) - (t,u,u',u",u"'), i = 0,1,2,3. We w i l l g e n e r a l l y s u p p r e s s t h e 1 1 ax. t dependence of' Ai and o f

Thus

46 1

A Curious Singular Perturbation Problem F'(u)v

+v +

-v"

=

EA(U)V

Now (26)

so (12) holds with

1 -. 4

c =

T h e r e a r e many ways i n w h i c h o n e m i g h t t r y t o f i n d smooth a p p r o x i m a t e s o l u t i o n s of

F'(u)v = g.

T h i s i s t h e most a d h o c s t e p i n t r y i n g t o v e r i f y t h e

h y p o t h e s e s of Theorem 1 3 .

See e.g.

[Z]

f o r a d i s c u s s i o n of t h i s q u e s t i o n .

w i l l f i n d e x a c t s o l u t i o n s of a n e l l i p t i c r e g u l a r i z a t i o n of

t h e r e s u l t i n g functions s a t i s f y (10)-(11).

where

y > 0 , v(m)

m = m(k)

2

>

0

y

(30)

f

If

5

1

E

u

Ck+',
khllxll f o r some p o s i t i v e number k, a l l x i n D ( A ) , t h e domain o f A . D e f i n i t i o n 1.4. ( [ l l ] ) Let A empty. For any h o E v ( A ) t h e t h e dimension o f t h e subspace, denotes t h e domain o f A* and

be a closed, J-symmetric o p e r a t o r w i t h n ( A ) n o t d e f e c t number o f d e f ( A ) , i s d e f i n e d t o be { x E D(A*): (A* - X,oI)x = @I, where D(A*) I i s t h e i d e n t i t y o p e r a t o r on H.

A,

I n [ll],Z h i k h a r o b t a i n e d a p a r t i a l c h a r a c t e r i z a t i o n o f t h e J - s e l f a d j o i n t e x t e n s i o n s o f any q i v e n J-symmetric o p e r a t o r , under t h e assumption t h a t i t s regul a r i t y f i e l d i s n o t t h e empty s e t . I t was a l s o shown t h e r e t h a t t h e above d e f i n i t i o n o f t h e d e f e c t number o f a closed, J-symmetric o p e r a t o r A, i s independent o f t h e c h o i c e o f l o i n n ( A ) . I t i s a f u r t h e r consequence o f t h i s work t h a t f o r a c l o s e d J-symmetric t h e dimension o f t h e q u o t i e n t space, D(JA*J)/D(A), i s t w i c e t h e d e f e c t number o f A, i.e., o p e r a t o r A,

D.Race

466 dim(D(JA*J)/D(A))

= 2

Y

d e f (A)

(1.1)

I n p r a c t i c e , however, i t i s d i f f i c u l t t o check whether v(A) i s empty and hence t o know whether Z h i k h a r ' s c h a r a c t e r i z a t i o n i s a p p l i c a b l e i n any q i v e n s i t u a t i o n . Subsequent work by I. Ld. Knowles [5] (see a l s o [7], [ l o ] ) has extended s i g n i f i c a n t l y t h e r e s u l t s i n [ll].However, t h e c o n d i t i o n t h a t a ( A ) be not empty i s I t i s t h e i n t e n t i o n o f t h e p r e s e n t work t o remove t h i s s t i l l r e q u i r e d i n [5]. c o n d i t i o n c o m p l e t e l y . The f i r s t key t o d o i n a so i s t h e f o l l o w i n g simple observation. Lemma 1.5. I f A i s a c l o s e d , 3-symmetric o p e r a t o r and extension o f A then dim (D(JA*J)/D(A'))

A'

i s any J - s e l f a d j o i n t

= dim ( D ( A ' ) / D ( A ) ) .

The e x i s t e n c e o f such an o p e r a t o r A ' i s auaranteed by a r e s u l t o f A. G a l i n d o [l]which says t h a t e v e r y J-symmetric o p e r a t o r has a J - s e l f a d j o i n t ext e n s i o n . The above lemna can be t h o u o h t o f , as s a y i n g t h a t t h e domain o f such an o p e r a t o r A ' i s "mid-way'' between t h e domains o f A and JA*J. I t now f o l l o w s that dim (D(JA*J)/D(A))

= 2

x

dim ( D ( A ' ) / D ( A ) ) .

(1.2)

Comparing (1.1) and (1.2) i t i s n a t u r a l t o make t h e f o l l o w i n g d e f i n i t i o n . D e f i n i t i o n 1.6. I f A i s any c l o s e d J-symmetric o p e r a t o r , t h e ( g e n e r a l i z e d ) d e f e c t number of A, d e f (A) i s d e f i n e d by def (A) =

x

dim (D(JA*J)/D(A)).

Lemma 1.5 guarantees t h a t even i f n ( A ) were empty, d e f ( A ) i s e i t h e r a non-negative i n t e g e r o r i n f i n i t e . Since ( 1 . l ) was a consequence o f D e f i n i t i o n 1.4, D e f i n i t i o n s 1.4 and 1.6 c o i n c i d e when v(A) i s n o t empty, b u t t h e l a t t e r a l s o covers t h e case when n ( A ) i s empty. I n o r d e r t o d e t e r m i n e t h e Js e l f a d j o i n t e x t e n s i o n s o f a J-symmetric o p e r a t o r A we a l s o need t h e f o l l o w i n q result. Lemma 1.7. I f A i s a c l o s e d , J-symmetric o p e r a t o r w i t h d e f (A) < and i s a J-symmetric e x t e n s i o n o f A t h e n A ' i s J - s e l f a d j o i n t i f and o n l y i f dim (D(A')/D(A)) = d e f ( A ) .

A'

Using U e f i n i t i o n 1.6, t h i s i s an e a s i l y proved consequence o f a r e s u l t of

I . W. Knowles [4] which s t a t e s t h a t t h e J - s e l f a d j o i n t e x t e n s i o n s o f a J-symmetric o p e r a t o r , a r e t h e e x t e n s i o n s which a r e maximal J-symmetric. We a l s o need an e x p l i c i t c o n n e c t i o n between t h e domains o f A and JA*J. F o r t h i s we c o n v e r t D(JA*J) i n t o a H i l b e r t space by d e f i n i n g t h e u s u a l i n n e r product as f o l l o w s : ( o r . ) *

( x , y ) * = (Jx,Jy) + (A*Jx,A*Jy)

= (y,x)

+ (JA*Jy,JA*Jx).

It then f o l l o w s t h a t

U(JA*J)/D(A)

2 D(JA*J)

8 D(A)

.

One may now use s t a n d a r d t e c h n i q u e s t o prove Lemma 1.8. Q(JA*J)

If

f) D(A)

A

i s any closed, J-symmetric o p e r a t o r i n

= Iy E D(A*JA*J):

A*JA*Jy

= -yi

.

ff

then

46 7

J-Sclfadjoint Operators

2.

APPLICATION TO DIFFERENTIAL OPERATORS

We now c o n s i d e r t h e a p p l i c a t i o n o f t h e above r e s u l t s t o o p e r a t o r s a s s o c i a t e d w i t h the d i f f e r e n t i a l expression,

on an i n t e r v a l I on t h e r e a l l i n e . We assume t h a t t h e f u n c t i o n s p-1 ,pl,--.,pn a r e complex-Val ued, Lebesgue measurable o v e r I and Lebesgue i n t e g r a g l e on be r e s p e c t i v e l y t h e minimal and compact subsets of I. We l e t To and

lmaX

maximal o p e r a t o r s i n t h e H i l b e r t space L ( I ) , generated by T (see [5]). It i s w e l l known (see f o r Fxample [5]) t h a t i f J denotes o r d i n a r y complex conjugat i o n o f f u n c t i o n s i n L ( I ) then To i s J-symmetric and T; = J Tmax J. Thus JT*J = Tmax and D e f i n i t i o n 1.6 a p p l i e d t o To g i v e s 0

T h i s d e f i n i t i o n of t h e g e n e r a l i z e d d e f e c t number o f To c o i n c i d e s w i t h t h e d e f i I n t h i s terminology, n i t i o n o f t h e mean d e f i c i e n c y i n d e x f o r T , g i v e n i n L3]. lemma 1.5 may be i n t e r p r e t e d as s a y i n g t h a t t h e mean d e f i c i e n c y i n d e x o f a f o r m a l l y J-symmetric d i f f e r e n t i a l e x p r e s s i o n i s an i n t e g e r . (The n e x t theorem shows t h a t i t cannot be i n f i n i t e . ) T h i s i s n o t t r u e f o r g e n e r a l d i f f e r e n t i a l express i o n s , as i s observed i n [3]. Setting

A = To

i n Lemma 1 . 8 g i v e s t h e f i r s t p a r t o f

Theorem 2.1. ( i ) d e f ( T o ) = h a l f t h e number o f l i n e a r l y independent s o l u t i o n s o f y, T Y E L 2 ( I ) . J T J T Y = -y f o r which (ii) (iii)

0 5 d e f (To) if

I = [a,b),

5 2n. a < b

-m

5" then

n 5 d e f (To) 5 2n.

The second p a r t f o l l o w s f r o m t h e f i r s t , when i t i s observed t h a t JTJT i s a l i n e a r d i f f e r e n t i a l e x p r e s s i o n o f 4nth o r d e r . Theorem 2.1 was p r o v e d i n [ll] under t h e assumption t h a t n ( T o ) i s n o t empty b u t o u r methods have enabled us t o remove t h a t c o n d i t i o n , by u s i n g D e f i n i t i o n 1 . 6 and t h e r e s u l t s g i v e n above. Lemma 1.7 may now be used t o c h a r a c t e r i z e t h e boundary c o n d i t i o n s which determine a l l t h e J - s e l f a d j o i n t e x t e n s i o n s o f To. We denote by [.,.I the form obtained from L a g r a n g e ' s i d e n t i t y f o r T (see [5]) and s t a t e t h e main r e s u l t of t h i s t y p e . I t i s hoped t h a t f u r t h e r d e t a i l s o f t h e s e c h a r a c t e r i z a t i o n s w i l l appear e l sewhere. be a r b i t r a r y f u n c t i o n s b e l o n g i n g Theorem 2.2. L e t m = d e f ( T o ) and w l , - - - , w m t o D(Tmax) which a r e l i n e a r l y independent modulo D(To) and which s a t i s f y t h e relations

[w.,w I ( b ) - [w.,; J k J The s e t o f a l l f u n c t i o n s y i n

j , k = 1;-. ,m. ](a) = 0, k D(Tmax) which s a t i s f y t h e c o n d i t i o n s

_.

1Y,Wkl(b) - [ Y ' i k l ( a )

=

0,

k = 1,

.... ,m

i s t h e domain o f d e f i n i t i o n o f a J - s e l f a d j o i n t e x t e n s i o n o f J - s e l f a d j o i n t e x t e n s i o n s o f To a r e o f t h i s form.

To.

Conversely a l l

The s t a t e m e n t o f these c h a r a c t e r i z a t i o n s i s i d e n t i c a l t o t h a t g i v e n i n [5] b u t w i t h o u t t h e r e q u i r e m e n t t h a t n ( T o ) be n o t empty. The p r o o f s employed here a r e a l s o s h o r t e r and s i m p l e r t h a n those g i v e n i n [5].

D. Race

468 3.

THE CASE

n(To) = fl

We conclude b y c o n s i d e r i n g what happens i n t h e case when T ( T ) i s indeed empty. I t i s p o s s i b l e t o show t h a t i f rr(T ) = @ then d e f (To) O f 2n. T h i s has p a r t i c u l a r s i g n i f i c a n c e when n = 1 an8 T i s r e g u l a r a t one e n d - p o i n t o f I:

Theorem 3.1. I f n = 1, I = [a,b), -- < a < b def(To) = 1 and t h e J - s e l f a d j o i n t e x t e n s i o n s T,

D ( T y ) = {Y TY , where

y = ( y 1 ,y2)

=

TY

5

,

Tmax: r , y ( a )

+

5

and IT(T,) = fl t h e n To a r e p r e c i s e l y given by

m

0;

Y ~ P ~ ( ~ ) Y= ' 01 ( ~ )

Y E m y )

i s an a r b i t r a r y non-zero element o f

(c

2.

T h i s completes t h e g e n e r a l i z a t i o n o f t h e l i m i t - p o i n t , l i m i t - c i r c l e dichotomy f o r S t u r m - L i o u v i l l e e x p r e s s i o n s h a v i n g complex-valued c o e f f i c i e n t s : Theorem 3.2A. (i) (ii) (iii)

(i)

(iii)

on

r

[a,b)

when

n = 1:

d e f (To) = 1. p r e c i s e l y one boundary c o n d i t i o n i s needed t o d e t e r m i n e J - s e l f a d j o i n t e x t e n s i o n s o f To. t h e r e i s a t most one independent s o l u t i o n o f any A E E.

Theorem 3.2B. (ii)

The f o l l o w i n g a r e e q u i v a l e n t f o r

The f o l l o w i n g a r e e q u i v a l e n t f o r

on

T

TY

in

L2[a,b)

when

n = 1:

= hy

[a,b)

for

def (To) = 2. p r e c i s e l y two boundary c o n d i t i o n s a r e needed t o determine J - s e l f a d j o i n t e x t e n s i o n s o f To. t h e r e a r e two independent s o l u t i o n s o f x E t.

TY

=

in

Ay

L 2[a,b)

f o r any

Since Theorems 3.2A, B c o v e r a l l p o s s i b i l i t i e s t h i s would seem t o j u s t i f y r e f e r r i n g t o these as t h e l i m i t - p o i n t and l i m i t - c i r c l e a l t e r n a t i v e s even when t h e c o e f f i c i e n t s a r e complex-valued. F i n a l l y , we g i v e an example i n which " ( T o ) i s empty.

8.

Example 3.3. The f o l l o w i n g example was c o n s i d e r e d by J. McLeod i n [ 8 ] , where i t was shown t h a t t h e r e i s no s o l u t i o n o f T Y = Xy i n L (0,~)f o r any 4 E t. .ry(x) = - y " ( x ) - 2 i e2(1+i)x

y(x)

on

[o,-).

It f o l l o w s (see [9]) t h a t n ( T ) - p1 and t h e c o n t i n u o u s spectrum o f a s s o c i a t e d o p e r a t o r s covers t h e complex pgane. From Theorem 3.1 we may deduce t h a t t h i s example f a l l s i n t o t h e l i m i t - p o i n t case as d e s c r i b e d i n Theorem 3.2A. We t h e r e f o r e now know what t h e a s s o c i a t e d J - s e l f a d j o i n t o p e r a t o r s a r e and t h a t t h e maximal domain i s a two-dimensional e x t e n s i o n o f t h e minimal domain.

J-Selfadjoirit Operators

469

REFERENCES

Galindo, A . , On t h e e x i s t e n c e o f J - s e l f a d j o i n t e x t e n s i o n s o f J-symmetric o p e r a t o r s w i t h a d j o i n t , Comm. Pure and A p p l i e d Math. 15 (1962), 423-425. Glazman, I . M., An analogue o f t h e e x t e n s i o n t h e o r y o f h e r m i t i a n o p e r a t o r s and a non-symmetric one-dimensional boundary-value problem on a h a l f - a x i s , D o k l . Akad. Nauk SSSR 115 (1957), 214-216. Kauffman, R. M., Read T. T. and Z e t t l , A., The d e f i c i e n c y i n d e x problem f o r powers o f o r d i n a r y d i f f e r e n t i a l e x p r e s s i o n s , L e c t u r e n o t e s i n math 621, S p r i n g e r - V e r l ag, B e r l i n , 1977. Knowles, I . W., On J - s e l f a d j o i n t e x t e n s i o n s o f J-symmetric o p e r a t o r s , Proc. Amer. Math. SOC. 79 (1980), 42-44. Knowles, I . W., On t h e boundary c o n d i t i o n s c h a r a c t e r i z i n g J - s e l f j o i n t e x t e n s i o n s o f J-symmetric o p e r a t o r s , J. D i f f . Equations 40 (1981), 193-216. Knowles, I . W. and Race, D., On t h e p o i n t s p e c t r a o f complex S t u r r n - L i o u v i l l e o p e r a t o r s , Proc. Roy. SOC. Edin. 85A (1980) 263-289. Makarova, A. D., On t h e J-symmetry o f o p e r a t o r s w i t h a nondense domain of d e f i n i t i o n , Volzhsk. Matem. Sb., Ser. Funkts. A n a l i z i Teor. Funkts. 10 (1969) 77-83. McLeod, J. B., S q u a r e - i n t e g r a b l e s o l u t i o n s o f a second-order d i f f e r e n t i a l e q u a t i o n w i t h complex c o e f f i c i e n t s , Q u a r t . J. Math. O x f o r d ( 2 ) 13 (1962) 129-1 33. Race, D., S p e c t r a l t h e o r y o f complex S t u r m - L i o u v i l l e o p e r a t o r s , Ph.D. t h e s i s , U n i v e r s i t y o f t h e Witwatersrand, 1980.

[lo]

Raikh, L. M., On t h e e x t e n s i o n o f a J - h e r m i t i a n o p e r a t o r w i t h nondense domain o f d e f i n i t i o n , Mathematical Notes 17 (1975), 439-442.

[ll]Z h i k h a r , N. A., The t h e o r y o f e x t e n s i o n s o f J-symmetric o p e r a t o r s , Ukrain. Mat. Z. 11 (1959), 352-364.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

47 1

ESSENTIAL SELF-ADJOINTNESS FOR POWERS OF SCHRdDINGER OPERATORS THOMAS T . READ

DEPARTMENT OF MATHEMATICS WESTERN WASHINGTON UNIVERSITY BELLINGHAM, WASHINGTON 98225

C o n d i t i o n s a r e g i v e n which a r e s u f f i c i e n t f o r t h e minimal o p e r a t o r To a s s o c i a t e d w i t h a second o r d e r e l l i p t i c d i f f e r e n t i a l

Rn t o be e s s e n t i a l l y s e l f a d j o i n t , and f o r a l l powers of T o t o be s e l f - a d j o i n t . The c o n d i t i o n s f o r TO which a r e s i m i l a r t o t h e most g e n e r a l cone x p r e s s i o n on

d i t i o n s from t h e one-dimensional c a s e , a l l o w some c o e f f i c i e n t s n o t covered by o t h e r c r i t e r i a and a v o i d some of t h e i r t e c h n i c a l assumptions.

The c o n d i t i o n s

f o r powers of T o a l l o w f a s t e r growth by t h e l e a d i n g terms t h a n p r e v i o u s r e s u l t s . Let

L

be t h e second o r d e r e l l i p t i c e x p r e s s i o n

L

=

( p ( x ) ) - ’ ( - I D . a . (x)Dk t q ( x ) ) , J Jk

D . = a/ax. + i b . d e f i n e d on Rn. Here A = ( a . ) i s symmetric J J J’ t C1(Rn) and i k 6 C1(Rn). Also and p o s i t i v e d e f i n i t e , w i t h a 2 jk J q t Lloc(Rn) and t h e p o s i t i v e weight f u n c t i o n p i s c o n t i n u o u s .

We have t h e f o l l o w i n g r e s u l t on e s s e n t i a l s e l f - a d j o i n t n e s s of d e f i n e d by Tof t h e o p e r a t o r To w i t h domain C c ( R n ) use x * y t o denote t h e usual s c a l a r product i n C n . THEOREM 1.

Let

non-negative,

M

{wm(r)lm=l

2 sequence

of

=

L[f I .

We

compactly s u p p o r t e d ,

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

v a r i a b l e r such t h a t f o r e a c h f i x e d d e c r e a s i n g & m & bounded above.

r,

{wm(r)l

is non-

T. T. Read

472

Suppose that

with

91 (i) (ii)

0 , 42

q3

=

5

q 0,

can be decomposed into a sum and each

q

=

2 n q j € Lloc(R )

so that

5 (1

+

q1 t q2 t q3

div Q a.e.,

A V W ~ - VtW C ~ A - ~ Q --Q q2)wi

for some constants --

6

i

0

- 6)qlwi

KO

K > 0,

(iii)

Then

To

REMARKS. 1. For the one-dimensional expression -1 (-DpD t q) on [ O , 03 ) the hypotheses above become = p

L

(i),

q3

=

Q’ a.e.,

This is very similar to, and in some respects stronger than, the limit-point result in Read [7, Theorem 31 which contains essentially all of the standard limit-point criteria for the interval [ O , D o ) . 2. We will see following Theorem 2 that in many situations the family {wml can be replaced by a single function w. However it should also be noted that a sequence {wml of compactly supported functions can arise very naturally, as in the Corollary below. 3. For p = 1 , the hypotheses are somewhat similar to those of Evans [3] who was also interested in the essential selfadjointness of powers of To. However the hypotheses in [3] imply that lop OD -’l2(r)dr = C o t while Theorem 1 does not put any fixed limit on the rate at which the leading terms can grow. If one supposes, for instance that p(r) 5 ’r and q(x) = ql(x) 2 Cr’, then by letting w = 1 and using a modification of the Proposition following Theorem 2 it can be seen that the hypotheses of Theorem 1 2 are satisfied whenever ,' a - 2 or B = a - 2 and C > ( a - 2) . The case 6 > a - 2 is also covered by a recent result of Kato [ 6 ] , but 13 = a - 2 is not.

Powers of Schrijdinger Operators

473

4 . The p r e s e n c e of t h e d i v e r g e n c e term q 3 a l l o w s q t o have f i n i t e s i n g u l a r i t i e s , s i n c e t h e s e can a r i s e a s t h e d i v e r g e n c e of a bounded o r slowly growing f u n c t i o n .

= ( n - 1)-' d i v [ ( x - x o ) / / x - x o I l . I n Evans' r e s u l t [ 3 ] and a l s o i n a number of e a r l i e r 1 9 1 , 1 1 0 1 , f i n i t e s i n g u l a r i t i e s a r e p r o v i d e d f o r by a

Ix - x 0 1-1

xo 6 Rn,

5. papers [ 4 1 , term

For i n s t a n c e , f o r f i x e d

qs

satisfying

1 I q s ( x ) Idx -< Kr2s, I x I" j / q s ( y - x)Iv(x)dx * 0

and

Ixl5r uniformly f o r y

Rn

in

Y(X) =

r

as

+

0

with

I

n>2,

1 - l o g l x l , n=2.

Such a term could be i n c o r p o r a t e d i n t o Theorem 1 i n much t h e same

twml

way a s i n [ 3 ] p r o v i d e d i t i s assumed t h a t bounded, and t h a t

2

A(x)

i s uniformly

c > 0.

One remarkable consequence of Theorem 1 i s t h a t

T o can b e made t o be e s s e n t i a l l y s e l f - a d j o i n t by r e s t r i c t i n g t h e c o e f f i c i e n t s

o n l y on a c o m p l e t e l y a r b i t r a r y sequence of c o n c e n t r i c a n n u l i . Other r e s u l t s of t h i s t y p e , f o r i n s t a n c e Eastham, Evans, and McLeod [ Z ] , impose, a t l e a s t i m p l i c i t l y , some r e s t r i c t i o n on t h e t h i c k n e s s of t h e a n n u l i , g e n e r a l l y something l i k e

2 1 ym/pm =

t h i c k n e s s o f t h e m-th on t h e m-th a n n u l u s .

p,

COROLLARY.

a n n u l u s , and

where

y,

i s the p(r)

Suppose t h e r e i s a d i s j o i n t sequence of a n n u l i

J,:am 5 r constants

5

then

& essentially self-adjoint.

To

00

i s t h e maximum of

bm, am

+

go

c < 1

Here components

y

Im:aA 5 r I

of

5

, on which

q

0.

I f there exist positive

such t h a t --

b& where

a

m

< a;

< bk

Jm\Im each have t h e p r o p e r t y

b,

and t h e two

T T Read

474

REMARKS.

1.

The proof depends i n an e s s e n t i a l way on t h e p o s s i -

b i l i t y t h a t Iwm} may f a i l t o be uniformly bounded. Thus t h i s r e s u l t does n o t f o l l o w from e s s e n t i a l s e l f - a d j o i n t n e s s theorems such a s t h o s e of Evans 131 o r Kato [ 6 ] . 2 . I f w e w r i t e x = rc, 151 = 1 and L = -A

+

f ( c ) r a s i n ( r b ) where

f(5)

1c

> 0

for

151

=

1,

then by

a n argument i n [ 7 1 t h e hypotheses of t h e C o r o l l a r y a r e s a t i s f i e d provided a > 2~ - 2 . We t u r n now t o e s s e n t i a l s e l f - a d j o i n t n e s s f o r powers of . TO Our r e s u l t i s a s f o l l o w s : THEOREM 2 . I f t h e c o e f f i c i e n t s of L a r e i n C o o ( R n ) , t h e hypoof Theorem 1 s a t i s f i e d , and i n a d d i t i o n ( i v ) wm 5 W f o r some c o n s t a n t W and a l l m , ( v ) f o r each ro > 0 , E > 0 , positive integer M there

=

-theses

is -

-then a l l

s

2 r 0 -and m 1. M

powers of

To

are

such t h a t

essentially self-adjoint.

REMARK. A s noted above, t h e theorem of Evans [ 3 ] which c o n t a i n s previous r e s u l t s of Chernoff [l] and Kato [ 5 ] on powers of To r e q u i r e s fi-’l2 = 00 f o r p = 1. Condition ( v ) does n o t i n 0

p r a c t i c e impose any r e s t r i c t i o n s on t h e r e l a t i v e s i z e s of p and beyond t h o s e a l r e a d y p r e s e n t i n Theorem 1. I n p a r t i c u l a r , f o r qil p = 1, i f p(x) < ra, q(x) 2 Cr’ f o r D > a - 2 , then a l l powers of To a r e e s s e n t i a l l y s e l f - a d j o i n t . The proof of Theorem 2 r e s t s on t h e f o l l o w i n g lemma which permits t h e c o n s t r u c t i o n of s u i t a b l e c u t o f f f u n c t i o n s .

If ( v ) h o l d s , then t h e r e i s m > M and a compactly supported f u n c t i o n v ( r ) [ O , go ) such t h a t

LEMMA.

on

0

5 v 5 1, v(r)

p1/2/v’ I

2

=

1

for

0 2 r 5 r0/2,

Ewm(q;/2wm t P 1 / 2 ) .

By a d a p t i n g t h e proof of t h i s lemma s l i g h t l y we can o b t a i n

Powers of Schriidinger Operators

PROPOSITION. Suppose that w(r) lutely continuous function 10,

g0

475

a non-negative locally abso) such that

(i) pw12 t ( A - ~ Q --Q q2)w2 5 (1 - 6)qlw2 t ~ p ,

or (b) E

< 1

s2(r)

-

E L1(O,

(pII/p)1’2

(1 - &)’I2 with s1

then there is such that Cw,} --

a

00

1,

w

is

bounded, and for some

-there exist increasing +

m

as

r

+

gr,

functions

sl(r)

such that

sequence Ivml of compactly supported functions Cwvm 1 satisfies the hypotheses of Theorem 1.

=

REFERENCES: Chernoff, P.R., Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Anal. 1 2 (19731, 401-414. Eastham, M.S.P., Evans, W.D., and McLeod, J.B., The essential self-adjointness of Schrbdinger-type operators, Arch. Rat. Mech. Anal. 60 (1976), 185-204. Evans, W.D., On the essential self-adjointness of powers of Schrbdinger-type operators, Proc. Royal SOC. Edinburgh 79A (1977), 61-77. Kato, T., Schrbdinger operators with singular potentials, Israel J. Math. 13 (1972), 135-148. Kato, T., A remark on the preceding paper by Chernoff, J. Functional Anal. 1 2 (1973), 415-417. Kato, T., Remarks on the self-adjointness and related problems for differential operators, in: Knowles, I.W. and Lewis. R.T. eds., Spectral Theory of Differential Operators (NorthHolland, Amsterdam, 1981).

476

T.T Read

[7]

Read, T.T., A limit-point criterion for expressions with oscillatory coefficients, Pacific J. Math., 6 6 ( 1 9 7 6 ) , 243-255.

[8]

Read, T.T., A limit-point criterion for expressions with intermittently positive coefficients, J. London Math. SOC. ( 2 ) 15 ( 1 9 7 7 ) , 2 7 1 - 2 7 6 .

[91

Stummel, F . , SingulHre elliptische Differentialoperatoren in Hilbertschen R a m e n , Math. Ann. 1 3 2 ( 1 9 5 6 1 , 1 5 0 - 1 7 6 .

[lo]

Wienholtz, E,, Halbbeschrznkte partielle Differentialoperatoren Zweiter Ordnung vom elliptischen Typus, Math. Ann. 135 ( 1 9 5 8 ) , 5 0 - 8 0 .

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) OElsevier Science Publishers B.V. (North-Holland), 1984

417

GEOMETRY AND DISCRETE VELOCITY APPROXIMATIONS TO THE BOLTZMANN EQUATION* Michael C. Reed Department of Mathematics Duke U n i v e r s i t y Durham, NC 27706

I would l i k e t o d e s c r i b e some c a l c u l a t i o n s which Reinhard I l l n e r and I have been making on t h e Carleman model, t h e s i m p l e s t d i s c r e t e v e l o c i t y model o f t h e Boltzmann e q u a t i o n . But, perhaps i t i s w o r t h w h i l e t o s t a r t by d e s c r i b i n g how these d i s c r e t e v e l o c i t y models a r i s e . The Boltzmann e q u a t i o n i s an 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 f o r t h e d e n s i t y i n c o n f i g u r a t i o n and momentum space, n!x,p,t), of a l a r g e number o f p a r t i c l e s ( f o r example, t h e molecules o f a gas). Since t h e Boltzmann e q u a t i o n i s n o t o r i o u s l y d i f f i c u l t t o s t u d y , one can i n s t e a d p e r m i t o n l y f i n i t e l y many f i x e d v e l o c i t i e s p1 ,. . . ,pk. One then wishes t o d e t e r mine t h e c o n f i g u r a t i o n space d e n s i t i e s

By analogy w i t h t h e pi(x,t) = n(x,pi,t). Boltzmann e q u a t i o n t h e s e s h o u l d s a t i s f y e q u a t i o n s o f t h e f o r m

The l e f t hand s i d e m e r e l y expresses t h e f a c t t h a t t h e p a r t i c l e s i n t h e d e n s i t y pi a l l have f i x e d v e l o c i t y pi. The r i g h t hand s i d e s a r e ( u s u a l l y ) taken t o be quadratic functions

o f the

pi

w h i c h mimic i n some way t h e i n t e r a c t i o n s i n t h e

Boltzmann e q u a t i o n ; i n p a r t i c u l a r , t h e

qi

s h o u l d be such t h a t t h e expected

q u a n t i t i e s a r e conserved. The Carleman model,

1 (Ut + u x ) = v2 - u -

u(x,O)

= uo(x)

20

1 (Vt -

v(x,O)

= v (x)

20

n

VP

-

v x ) = u2

-

v2

i s t h e s i m p l e s t d i s c r e t e v e l o c i t y model. p a r t i c l e s w i t h v e l o c i t y p l u s one; v ( x , t ) v e l o c i t y minus one. The t o t a l mass

0

u ( x , t ) i s the density a t time t i s the density o f p a r t i c l e s w i t h

of

m

m =

/

u(x,t) + v(x,t)dx

-m

i s conserved. The i n t e r a c t i n g terms on t h e r i g h t guarantee t h a t i f t h e r e i s ( l o c a l l y ) an excess o f u ' s o v e r v ' s , t h e n some u p a r t i c l e s w i l l be t u r n e d i n t o v's. Some y e a r s ago, I l l n e r [ 7 ] showed t h a t t h e s o l u t i o n o f ( 1 ) i s g l o b a l i n time. 1 Suppose t h e i n i t i a l d a t a a r e C , nonThen, a s t a n d a r d c o n t r a c t i o n argument proves

Here i s t h e i d e a i n a s i m p l e case. n e g a t i v e , and have compact s u p p o r t .

*Research s u p p o r t e d b y NSF G r a n t #MCS-8201258

478

M.C.Reed

the existence f o r s h o r t time o f a so u t i o n o f Let B ( t ) = max I u ( x , t ) , v ( x , t

1 ) w i t h t h e same t h r e e p r o p e r t i e s .

}

X

and l e t

xo

be a p o i n t where

u(xo,t) = B ( t )

and

u(xo,t)

>

v(xo,t).

Then

has a maximum a t xo so u x ( x o , t ) = 0. Thus, f r o m t h e d i f f e r e n t i a l au < 0 so t h e maximum w i l l decrease. equation, ( x o , t ) = v(x0,t)' - u(x0,t)' A au s i m i l a r b u t somewhat more t e c h n i c a l p r o o f shows t h a t 5 (x,,t) 5 0 a t points u(x,t)

where b o t h

u(xo,t) = B ( t ) = v(xo,t).

Thus,

B(t)

i s n o n - i n c r e a s i n g so t h e

l o c a l s o l u t i o n o f (1) i s g l o b a l . Two y e a r s ago, I l l n e r and I g o t i n t e r e s t e d i n t h e a s y m p t o t i c p r o p e r t i e s o f t h e Carleman model and proved t h a t [8]

0 2 v(x,t) 5 ;

t 1.1

where c i s a c o n s t a n t depending o n l y on t h e mass. I t i s easy t o see t h a t t h i s t i m e decay i s t h e b e s t one c o u l d e x p e c t ( j u s t observe t h a t 1 - (ut + u x ) 2 -u2 becomes u ' ( s ) -u(s)' on r i g h t w a r d c h a r a c t e r i s t i c s ) . The

Q

f a c t t h a t c depends i n i t i a l data v e r i f i e s mechanics: l o n g t e r m i n i t i a l data b u t o n l y m w i l l disperse very o f t h e same mass m.

o n l y on t h e mass and n o t on any o t h e r p r o p e r t i e s o f t h e ( i n t h i s model) one o f t h e b a s i c i d e a s o f s t a t i s t i c a l b e h a v i o r s h o u l d n o t depend on d e t a i l e d p r o p e r t i e s o f t h e on a few o v e r a l l parameters. Very peaked d e n s i t i e s o f mass r a p i d l y and w i l l have t h e same a s y m p t o t i c s as f l a t d e n s i t i e s

The c a l c u l a t i o n which I want t o s k e t c h today i s f o r t h e Carleman model ( 1 ) i n box. The p a r t i c l e s a r e c o n s t r a i n e d t o s t a y on t h e f i n i t e i n t e r v a l [O,e] by r e f l e c t i n g w a l l s a t z e r o and e. u(0,t)

= v(0,t)

u(e,t)

= V(Q,t)

(2) These boundary c o n d i t i o n s m e r e l y a s s e r t t h a t p a r t i c l e s t h a t a r r i v e a t e w i t h speed p l u s one i m m e d i a t e l y l e a v e w i t h speed minus one and v i c e v e r s a a t zero. Q

u + v dx i s again conserved. L o c a l l y , excesses o f 0 produce v ' s and excesses o f v ' s produce u ' s so t h i s suggests t h a t u s h o u l d approach t h e homogeneous s t a t e as t m. That i s , there should be d as t m. O f course, d = c o n s t a n t d so t h a t u ( x , t ) + d, v ( x , t ) by c o n s e r v a t i o n o f mass. Here i s t h e r e s u l t .

m =

The t o t a l mass

-f

-f

-f

Theorem [9]-

Suppose t h a t

conditions (2).

uo(x)

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

and u, v

vo(x) of

are

C1

u's and v a m/2e

and s a t i s f y t h e boundary

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

Approximations to the Boltzmann Equation

where

c

i s a constant depending o n l y on

I want t o sketch t h e main step i n completely elementary, i l l u s t r a t e s t h e t h e geometry o f t h e c h a r a c t e r i s t i c s i s on i n these d i s c r e t e v e l o c i t y models. max

8(t) = a(t) =

m

and

419

e.

t h e p r o o f s i n c e t h e technique, though main p o i n t which I want t o make. Namely, fundamental t o understanding what i s going Set

iu(x,t),v(x,t)l

OLXZP" min { u ( x , t ) , v ( x , t ) }

ozxze

By conservation o f mass, a ( t ) < d < 8 ( t ) and a s i m i l a r a r ument t o t h e one i s non-decreasing. sketched above shows t h a t B ( t ) - is-non-increasing and act! I w i l l sketch t h e main step i n t h e proof t h a t B ( t ) - d 5 c / t . Since B ( t ) i s n = 1,2,... non-increasing i t i s s u f f i c i e n t t o show t h a t B(ne) - d 5 c/n, What we want t o do i s t o show t h a t B((n + lie) - d i s s t r i c t l y s m a l l e r than B(niL) - d. That i s , a f t e r a time s t e p o f l e n g t h e the maximum a c t u a l l y decreases (and o f course we need an e s t i m a t e on the decrease).

.

Define 6 = ( B ( n k ) + d)/2. Then, by conservation of mass t h e r e must be a at s e t o f reasonagly l a r g e measure so t h a t e i t h e r u o r v i s l e s s than 6, t = ne. More p r e c i s e l y ,

where

u

i s Lebesgue measure on [ O , e ] . Thus one o f t h e two terms on t h e l e f t m 7 , suppose i t i s t h e second. Then

must be g r e a t e r than

I n o t h e r words, we have a lower bound on t h e s i z e o f the s e t N a t t = n& on Set N- = N n [O,pG), N+ = N n [ p o t e l where po i s chosen so which v < 8., C

t h a t uW-} ' 7 ( B n - d ) , p{N+> 2~ (B,, - d ) . What we have i s t h a t v i s small ( l e s s than B n ) on a f a i r l y l a r g e s e t a t t = na. We want t o use t h i s t o show t h a t b o t h u and v a r e s t r i c t l y l e s s than ~ ( n e ) a t This i s where the geometry of t h e c h a r a c t e r i s t i c s comes in.

Figure 1

t = ( n + 1)e.

480

M.C. Reed

Let

p

be a p o i n t i n N+ and c o n s i d e r t h e l e f t - w a r d c h a r a c t e r i s t i c f r o m 1 S e t t i n g D- = - ( a t - ax) and l e t t i n g s denote a r c l e n g t h we have

p.

6

and

Thus

v ( 0 ) < B,.

-

5 u2

(D-v)(s)

v(s)

v2 = ( u

*

2 w(s)

where

v ) ( u - v ) 5 2B(ne)(3(ne) - v(s)) w(s)

solves

(D-w)(s) = Z a ( n e ) ( B ( n e ) - w) w(0) = Bn S o l v i n g t h e comparison problem e x p l i c i t l y y i e l d s

-

v ( s ) 5 B(ne) - c(B(ne)

(4) where

c

depends o n l y on

e.

and

m

d)

~

be a p o i n t on AP1 and c o n s i d e r t h e r i g h t w a r d c h a r a c t e r i s t i c , r. L e t M denote t h e p o i n t s on R where v s a t i s f i e s ( 4 ) . 1 utM} 2- c(Bn - d ) . We want t o f i n d a p o i n t on R where u i s

Now l e t q from q t o

R,

We know t h a t small.

n

Suppose B(nL)

f o r a l l p o i n t s on From ( l ) ,

u

2b2

B(l1k)

-

5 (o(ne)

I w i l l e x p l a i n why

R.

u ( r ) = u(q) +

J,

2 u(r),

ufq)

M We know t h a t v2

-

u2

b

b

M

5 B(ni).

d) c a n ' t be t o o c l o s e t o

b

If

B(ne),

i s very close t o

M

i s large.

possible positive contribution o f

But i f

1

v2

-

b

i s very close t o

u2

i s small s i n c e

B(ne)

b < a(ne)

-

-

i n any

When one makes

c(B(ne) - d)'

depends o n l y on

u(wo) 5 b ( n a )

the

v 5 B(nk)

MC case. Thus we g e t a c o n t r a d i c t i o n if b i s t o o c l o s e t o B ( n e ) . t h e e x p l i c i t e s t i m a t e s i n t h e above argument, one f i n d s t h a t

c

then

makes a s u b s t a n t i a l n e g a t i v e c o n t r i b u t i o n because o f ( 4 ) and t h e f a c t

M t h a t t h e measure of

where

~(ne).

- u2.

1 v2

v2 - u2

-

and

m

c ( 0 ( n e ) - d)'.

e . Thus t h e r e e x i s t s wo

on

R

such t h a t

By u s i n g t h e comparison argument which was used

above one concludes f r o m t h i s t h a t (5)

u ( r ) 5 B ( n i ) - c(B(nil) - d)

2

.

~

Notice t h a t t h i s holds f o r

r

in

P2C.

We would l i k e t o show t h a t ( 5 ) h o l d s f o r

r

in

6 too.

Here i s t h e

geometric argument ( o m i t t i n g a l l t h e a n a l y s i s ) . v i s s m a l l on a l a r g e s e t i n ApO. T h e r e f o r e by t h e comparison argument, v i s s m a l l on a l a r g e s e t i n Apl. ~

By t h e boundary c o n d i t i o n s ,

t h e comparison argument

u

F.

u i s t h e r e f o r e small on a l a r g e s e t i n By is s m a l l on a l a r g e s e t o f r i g h t w a r d c h a r a c t e r i s t i c s

Approximations to the Boltzrnunn Equation

48 1

q.

from T h u s , every leftward c h a r a c t e r i s t i c from to must cross a large s e t of points where u i s small. By the argument above, v i s small a t a l l points of Thus, by the boundary conditions, u i s small a l l points of plB. Therefore, by the comparison argument, u i s small on Bp2. This i s how one concludes t h a t ( 5 ) holds f o r a l l r in BC. Similar methods show t h a t ( 5 ) holds i f u i s replaced by v. T h u s ,

p.

~

which imp1 i e s

Iterating t h i s inequality yields

which i s what we wanted t o prove. This gives a sketch of t h e main s t e p of the proof of the theorem. More There t h e theorem i s used t o prove, i n a d d i t i o n , d e t a i l s can be found in [9]. t h a t t h e decay t o d i s a c t u a l l y exponential in the L2 norm. The point of making these c a l c u l a t i o n s here i s t o emphasize t h a t t h e geometry o f the characteri s t i c s i s crucial to t h e Boltzmann-like properties of the Carleman model. This geometry becomes obscured i f one w r i t e s ( 1 ) in standard evolution equation form (by taking the x d e r i v a t i v e to the o t h e r s i d e ) :

Of course, A generates a nice l i n e a r semigroup etA and one can t r y to t r e a t the whole equation by using semigroup theory, Ouhamel's formula, and estimates on the nonlinear map F. B u t , t h i s seems t o me t o be the wrong approach because i t obscures t h e underlying geometry of the problem. This then i s the point o f my l e c t u r e . There has been a tremendous development of functional analysis over t h e l a s t 50 years. I t i s therefore tempting t o r e c a s t a l l i n i t i a l - v a l u e problems i n semi-group form and t o i n v e s t i g a t e properties of solutions by investigating prope r t i e s of the generator. I think i t i s a mistake in hyperbolic problems and other problems where the underlying geometry i s important.

I t i s appropriate t o end t h i s l e c t u r e by suggesting to you a nice unsolved problem. The next simplest d i s c r e t e velocity model i s the Broadwell model. 2

Vt

+ vx = z

Wt

- wX

=

22

Zt

=

2(vw -

- vw

- vw 2

2)

.

Suppose t h a t one s t a r t s with smooth i n i t i a l data (say of compact support) a t time t = 0. I t i s known from the work of Nishida [lo], Crandall-Tartar [ l l ] , t h a t the solution e x i s t s globally and will be smooth. What i s the asymptotic behavior of the solution as t m? Notice t h a t i f the i n i t i a l data i s zero outside of the interval [ - k , a ] , then z and v should be zero f o r x < - 2 and z a n d w should be zero f o r x > P. Thus the i n t e r a c t i o n should be confined t o the s t r i p t > 0. -QZ X 5 ,.P +

482

M.C. Reed

Figure 2 Thus v w i l l be c o n s t a n t on r i g h t w a r d c h a r a c t e r i s t i c s from p o i n t s p on t h e r i g h t edge o f t h e s t r i p and w w i l l be c o n s t a n t on l e f t w a r d c h a r a c t e r i s t i c s f r o m p o i n t s q on t h e l e f t s i d e o f t h e s t r i p . I n o t h e r words, v and w w i l l be t r a v e l l i n g waves t o t h e r i g h t and t h e l e f t of t h e s t r i p r e s p e c t i v e l y . The t o t a l mass,

m =

v + w + z dx,

leak out o f the s t r i p ?

w

i s conserved.

The q u e s t i o n i s , does a l l t h e mass

I b e l i e v e t h a t i t does and t h a t t h e r e a r e f u n c t i o n s

i,

such t h a t

(6)

-

v(x,t)

-+

i(X

t)

w(x,t)

+

i(X + t )

z(x,t)

-+

0

as t m. I a l s o b e l i e v e t h a t one should be a b l e t o c o n s t r u c t a p r o o f u s i n g o n l y c a l c u l u s and t h e geometry of t h e c h a r a c t e r i s t i c s . -f

I should m e n t i o n t h a t a s t u d e n t o f J o e l S m o l l e r (D. Chang) has some numeri c a l evidence f o r t h i s c o n j e c t u r e and t h a t Russ C a f l i s c h and I have a p r o o f i n t h e case where t h e d a t a i s s m a l l . For l a r g e smooth data, t h e problem i s c o m p l e t e l y open. I t i s n o t even known whether t h e s o l u t i o n i s bounded. References

[ll Broadwell, J . E.

Shock s t r u c t u r e i n a s i m p l e d i s c r e t e v e l o c i t y gas, Phys. o f F l u i d s 7 (1964), 1243-1247.

[2]

Cabannes, H. S o l u t i o n g l o b a l e du p r o b l h e de Cauchy en t h i o r i e c i n i t i q u e d i s c r G t e , J. de M6c. 17 (1978), 1-22.

[3]

Cabannes, H. The D i s c r e t e Boltzmann Equation, L e c t u r e Notes g i v e n a t U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , 1980.

[4]

Cabannes, H. P r o b l b e s mathbmatiques dans l a t h i o r i e c i n i t i q u e des gaz, Publ. s c i . de l ' i n s t . M i t t a g - L e f f l e r , Uppsala, 1957.

[5]

G a t i n g n o l , R. T h g o r i e c i n i t i q u e de qaz 2 r 6 p a r t i t i o n d i s c r G t e de v i t e s s e s , L e c t u r e Notes i n Phys. 36 ( S p r i n a e r - V e r l a g , New York, 1975).

Approximntions to the Boltzmann Equation

[6]

Godunov, K. and Sultangazin, U. M., On the d i s c r e t e models of k i n e t i c equation o f Boltzmann, Uspekhi Mat. Nauk 26 (1971), 3-51.

[7]

I l l n e r , R . , Global existence f o r two-velocity models o f the Boltzmann equation, Math. Meth. Appl. Sci. 1 (1979), 187-193.

[8]

I l l n e r , R . and Reed, M . The decay of solutions o f t h e Carleman Model, Math. Meth. Appl. Sci. 3 (1981), 121-127.

[9]

I l l n e r , R . and Reed, M. box, p r e p r i n t , 1983.

48 3

Decay t o equilibrium f o r t h e Carleman model in a

[lo] Nishida, T. and Mimura, M. On t h e Broadwell's Yodel f o r a Simple Discrete Velocity Gas, Proc. Japan Acad. 50 (1974), 812-817. [ l l ] T a r t a r , L. Existence globale pour u n systhrne hyperbolique semi-lin6aire de l a t h 6 o r i e c i n h a t i q u e des gaz, Ecole Polytechnique, Sgminaire GoulaouicSchwartz, 28 Octobre, 1975. [12] Temam,,R: S u r l a r6solution exacte e t approchse d ' u n probleme hyperbolique nonlineaire de T. Carleman, Arch. f o r Rational Mech. Anal. 35 (1969), 351 -362.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) @) Elsevier Science Publishers B.V. (North-Holland), 1984

485

DOMAINS IN HYPERBOLIC SPACE AND LIMIT SETS OF KLEINIAN GROUPS P. SARNAK COURANT INSTITUTE Lecture Given At The International Conference In Partial Differential Equations, Birmingham, Alabama March 1983

Though it may not be clear from the title, this talk is concerned with differential equations and in particular the Laplacian on domains in hyperbolic space. I will assume that the audience is more familiar with spectral theory rather than Kleinian groups. The work reported on this lecture is joint work with R. Phillips. We begin with a problem in geometry which seems first to have been posed by Beardon [l]. Consider a finite set of spheres in En, n 2 2, which are mutually exterior to one another. We are interested in the smallest closed subset of Bn which is invariant by inversions in these spheres. If we denote this set by A , the then spheres by S1,S2,...Sk and the inversions by R1,R2...%l clearly A is invariant under the group r = = the group A is called the limit or singular set of generated by R1,R 2...Rk. r. It is easy to see that r is free on R1,R2,...Rk except for the relation Ri2 = 1. One way of describing A is that A is the set of accumulation points of any fixed orbit I’x = {yxlyEr}. It is easy to verify that A so obtained is independent of x. To give an example consider the case of n = 2 and four circles, see figure 1 (a). The region exterior to C1,C2,C3,C4 clearly contains no limit points of I’x, since an application of an inversion in a circle will place one inside that circle. Thus 4 A C.I.’,INT(C-7’. If we now apply inversions in C C C C we 1 1’ 2’ 3’ 4 fini=that the images of EXT(C1,C2,CJ,C4) also contain no limit points. Thus A is contained in the union of the twelve circles shown in figure l.(b). Repeating this procedure we find that A is inside the 3 6 circles of figure l.(C). Continuing this indefinitely we fine that A looks something like figure l.(d). For more such pictures and constructions see [ 4 ] , from where these pictures are borrowed. As is apparent from figure l.d, A is a curve whose dimension should be close to 1, in fact being wiggly, we expect it to be

...

48 6

P. Sarnak

larger than 1. By dimension we mean Hansdorff dimension, we recall the definition: For K C En, E > 0 define 6 : K C U Bi , where Bi are hg (K,E) = inf{Iri i Balls of radius ri, and r < € 1 i -

c3 FIGURE 1

Limit Sets of Kleinian Groups

48 7

Let h (K) = lim h6(K,E), which gives the definition of Hansdorff outer E + O 6-measure. For 6=n this is nothing but Lebesque outer measure. From the definition it follows that there is a 6 such that

We call 6 = 6(K) the dimension of K, and it clearly generalizes our usual notion of dimension. For example, for figure 1 with r1 = r2 = r3 = l,r4 = m, we have found numerically that & ( A ) = 1.0015. We are now in the position to pose the question referred to at the beginning. It may at first seem a little ad-hoc, but its relevance will become clear as we go along. Question 1 [l] In En, can the dimension of A be made arbitrarily close to n by packing in enough spheres? For n 2 3 theorem 1 contains a solution to this problem. For n = 2 the problem remains unsolved, we will discuss it further later. Theorem 1: For n 2 3, there is a Cn < n such that 6 ( A ) 2 Cn independent of the configuration of the spheres. For example if n = 3 , & ( A ) 5 2.99

This, and a number of related such results, follows from an analysis of spectral properties of the Laplacian for domains in hyperbolic space. The point is that our group r is a group of conformal mappings of En, and so is a subgroup of the group G(n) generated by all inversions in hyperplanes and spheres of En. The group G(n) is also the group of isometries of hyperbolic n+l space. This allows one to phrase question 1 in terms of hyperbolic geometry. So we make a slight detour to discuss hyperbolic geometry. Let Hn+l = {(y,x): y > 0, x E En} and equip it with the line This turns Hn+l into element ds2 = (dx12 + dxn2 + dy2 )/y 2 a Riemannian space of constant negative curvature, -1. The The geodesis of this geometry are vertical rays or semi circles perpendicular to En. Every two points of Hn+l are joined by a unique geodesic. Geodesic submanifolds of dimension n (ie

...

.

488

P. Sarnak

submanifolds M of dimension n, for which the geodesic joining any two points of M I lies entirely within M) are simply the n-hemispheres with equator on En or vertical planes of dimensions n.

FIGURE 2

The group of conformal mappings of 1 may be extended to act on Hn+l. Indeed the generators which are the inversions in En, act on Hn+ by simply inversion in the sphere of Bn whose equator is the sphere in question, ie inversions in hyperplanes (geodesic). This action preserves Hn+l and it is an exercize to show that such motions are isometries of Hn+l Hn+l is a symmetric space of rank one. The Lclndamental differential operator for hyperbolic space is its Laplacian A , which generates all invariant differential operators (ie ones which commute with G(n)). In terms of the coordinates introduced

.

489

Limit Sets of Kleinian Groups

The Riemannian volume element in these co-ordinates is xi1 dv = (dxldx2. dxndy)/y

..

and

We are interested in the spectrum of for various self adjoint 2 n+l boundary conditions. For the free space, ie L (H ,dv) the ',=) and is absolutely continuous. This is well spectrum is [ known but it is instructive to see why the spectrum only begins at (n/212.

(t)

.

Let u (x1,x2,. .xn,y) E c;(H"+~) and let D(u) =

lVuI2 dV

, H(u)

=

1

u2 dV

be the Dirichlet and L2 forms respectively. For each fixed (x1,x2, xn) an integration by parts in y yields

...

Integrating ( 3 ) w.r,t.

x gives

Thus from the variational description of the bottom of the spectrum, we have inf = X o = inf (D(u)/H(u)) UEL2 VU L2

2

2

(n/2)

.

(3' 1

Let r be a discrete subgroup of G(n) acting discontinuously on Hn+l, that is r acts on Hn+I so that no orbit of r in Hn+’ has accumulation points in Hn+l. An example of such a r is the

490

P. Surnuk

group of reflections in spheres introduced at the beginning. Corresponding to rone may choose a fundamental domain F. Such a set F, is an open set which has the property that for xI y E F and x = yyI y # identity => x = yI and further more for any If T acts freely z E Hn+l there is a y E r such that YzfF on Hn , which is to say that y E I r y # identity => y has no fixed points in Hn , then one may form the quotient space Hn+ /I , to obtain a hyperbolic manifold M = Hn+l/r. One may think of M as F with its faces glued appropriately. In general F may be chosen to be a convex(in hyperbolic metric) domain bounded by geodesic hyperplanes. For example for r the group generated by inversions in the disjoint circles C1,C2, Ck in the plane, the corresponding F in hyperbolic three space may be chosen as the exterior to the hemispheres

...

FIGURE 3 For simplicity we will restrict ourselves (except for the very end) to geometrically finite groups r I ie ones for which F may be chosen to have only finitely many boundary hyperplanes. The limit set of a discontinuous group r , denoted A r , is defined as before, as the set of accumulation points of any fixed

49 1

Limit Sets of Kleinian Groups

orbit Tx, with x E Hn+l. Thus A r is a closed subset of Sn = 2 ( Hn+I). In the case of the inversion group r in the beginning, the limit set as defined before and now are clearly the same. We turn to the I’-spectral problem for A. Let L2 (Hn+l/r) be the Hilbert space of automorphic functions for which

lfI2

dv
O there is Q' such that A,(Q') -< -42263 + E here a ' is bounded by only finitely many F,of the circles

Limit Sets of Kleinian Groups

499

of the aboveAppolonianpacking. Now if we translate enough of F, by the translations z * z + 1, z + z + i, we obtain a Schottky domain " sounded by these circles and their translations, or which x o ( Q " ) 5 .42263 + 2E. The last follows from considerations of such periodic (ie Euclidean translations) configurations (see t61 for more) It follows that there are geometrically finite lchottky r for which Xo(r) 5 042263 + 2 E, with E arbitrarily small. When translated into the language of Hansdorff dimension we get theorem 3. Whether 6(r) for such can be made arbitrarily close to 2, for H3 groups, is as far as we know still unsettled. We end by remarking that an argument similar to that used in the proof theorem 3 above, together with Boyd’s efficient packing of 2 spheres in a 3-cube [ 9 ] , shows that there are Schottky groups r in dimension n = 3 for which 6(Ar) > 2=0078

.

P. Sarnak

5 00

REFERENCES Beardon A.F. Boyd D.W.

Amer JR. of Math’ 88 (1966) 722-736 Aequationes Math 9 (1973) 99-106

Lax. P. and Phillips R. 280-350

Jr. of Funt. Analysis, 46(1982)

Mandelbrodt B. "Fractals" Freeman and Co, San Fransisco 1977 S.J. Patterson Acta Math 136(1976) 241-273

Phillips R and Sarnak "?he Laplacian For Domains in Hyperbolic Space and Limit Sets of Kleinian Groups 'I To Appear D. Sullivan Bull. Am.S. 6 (1982) 57-73 Thurston W. Geometry and Topology of Three Manifolds" Notes from Princeton University 1978 D. Boyd "On the Exponent of An Osculatory Packing" Can J. Math, Vol XXIII, 355-363 (1971)

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), I984

50 1

SELFADJOINT OPERATORS, SPECTRAL AND SCATTERING THEORY, VARIATIONAL TECHNIOUES, NON-LINEAR PHENOMENA, LINEAR AND NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS, AND RELATED TOPICS M a r t i n Schechter Courant I n s t i t u t e of Mathematical Sciences New York U n i v e r s i t y New York, N. Y. 10012

ble d e s c r i b e some r e c e n t r e s u l t s c o n c e r n i n s t h e t o p i c s men t ioned

.

1.

SELFADJOINT REALIZATIONS I N ANOTHER HILBERT SPACE.

L e t H, V be o p e r a t o r s on a H i l b e r t space H, and assume t h a t H i s s e l f a d j o i n t . The o p e r a t o r H ' = H + V may n o t have a s e l f a d j o i n t r e a l i z a t i o n on H . However, i t may have a s e l f a d j o i n t r e a l i z a t i o n on some o t h e r space. By t h i s we mean t h a t t h e r e a r e a n o t h e r t l i l b e r t space H1, a bounded l i n e a r map J from H1 such t h a t H t o H1 and a s e l f a d j o i n t o p e r a t o r H1 on H J = JH'. (1.1) 1 I n t h i s case we may be a b l e t o i n v e s t i g a t e H ' by means o f H1. Even i f H ' does have a s e l f a d j o i n t r e a l i z a t i o n on H , t h e problem may c a l l f o r one t o compare H n o t w i t h H ' , b u t w i t h an o p e r a t o r H1 on H1 s a t i s f y i n g ( l . l ) , where til and J a r e g i v e n . Again t h e q u e s t i o n a r i s e s , does t h e r e e x i s t a s e l f a d j o i n t o p e r a t o r H1 s a t i s f y i n g ( 1 . 1 ) . When H1 i s s e l f a d j o i n t , we c a l l i t a s e l f a d j o i n t r e a l i z a t i o n o f H ' on H1 r e l a t i v e t o J.

We s h a l l p r e s e n t two r e s u l t s i n t h i s c o n n e c t i o n . They b o t h concern a s e l f a d j o i n t o p e r a t o r H on a H i l b e r t space H and a b i l i n e a r f o r m ( V U , ~ ) = (Au,Bv)~,

(1.2)

where K i s a H i l b e r t space and A, B a r e l i n e a r o p e r a t o r s from H t o K such t h a t D(H) C D ( A ) fl D(B). (The use o f a b i l i n e a r f o r m i n p l a c e o f t h e opera t o r V a l l o w s us t o t r e a t more p e n e r a l problems.) !Ale assume t h a t t h e r e i s a complex number z i n t h e r e s o l v e n t s e t p(1.I) o f H such t h a t t h e f o l l o w i n q h o l d f o r z = zo 'and z =

q:

(a)

AR(z), BR(z)

(b)

R([AR(z)]*)

(c)

G(z) z 1

-

B(H,K),

E C

D(B)

and

where

R(z)

=

( z - H)-'

Q ( r ) z B(AR(T))*

has a bounded i n v e r s e on

O(z)

is in

B(K)

k'.

Here B(H,K) denotes t h e bounded l i n e a r o p e r a t o r s f r o m denotes t h e ranse o f H.

H to K

Our f i r s t r e s u l t concerns t h e case when H1 and J a r e a i v e n . such t h a t t h e r e e x i s t a s e l f a d j o i n t o p e r a t o r H1 on H1 (1.3)

(Ju,H,v)

=

(JHu,v) + ( A U , B J * V ) ~ ,

u

E

D(H),

v E D(H1).

and

R(H)

Nhen does

M. Schech ter

502

We have

Theoretn 1.1. There i s a s e Z f a J 3 o i n t operator H on H, such t h a t Jo =lJ*J s a t i s f i e s u(H1) C u(BJ*) and (1.3) hoZds i f and onZy i f

z = z0’

JoT(z)* = T ( F ) J o ,

(1.4)

-

zo

where

(1.5) If (1.5) holds, t h e n (1.6) and

~ ( z =) ~ ( z +) (AR(T))* ~ ( Z 1 - lB R ( ~ ) . H1 s a t i s f i e s R1(z)J = JT(T)*, R1(z) = (z - H1)-'.

I n the second r e s u l t , we a r e not qiven H, such t h a t (1.3) holds. We have

H1

and

J.

We search f o r

H1,

J

Theorem 1.2. There e x i s t a HiZbert space H,, an operator J E B(H,H1) and a s e l f a d j o i n t operator HI on H1 such t h a t D ( H 1 ) C U(BJ*) and (1.3) holds i f and only if t h e r e e z z s t s an operator J o E B ( H ) such t h a t J o 2 0 am! (1.4) hoZds. The followina pives a s u f f i c i e n t condition f o r (1.4) t o hold.

D(BJ,)

and

ImI(Hu,Jou) + (AU,BJ~U),I= 0 ,

u

E

C

U(BJo)

Theorem 1.3. (1)

( 2 ) R(CBJoR(z)I*)

(3) P ( z ) *

=

If D(H)

C

U(B),

C

R([BR(z)l*)

D(H)

BJ~(BR(z))*

where P(Z) =

B(BJ~R(I))*

then ( 1 . 4 ) holds.

Exam l e 1 . H = L2(R) , H = -id/dx, V ( x ) i s a complex-valuer! function in L1(R)*s W(x) 5 Im V(x) = 0, H ' = H + V does not have a s e l f a d j o i n t extension. By Theorem 1.2 i t does have a s e l f a d j o i n t r e a l i z a t i o n in some Hi 1b e r t space. Example 2. With H t h e same, take V = a6(x) where 6 ( x ) i s the Dirac d e l t a "function" and ~1 i s a complex constant. I f CY i s not r e a l , H ' does not have a s e l f a d j o i n t extension on H. By Theorem 1.2, i t does have a s e l f a d j o i n t r e a l i z a t i o n in another Hilbert space provided a # - i . Once t h e existence of a s e l f a d j o i n t r e a l i z a t i o n has been e s t a b l i s h e d , i t i s For instance we have

n o t d i f f i c u l t t o e s t a b l i s h a spectral and s c a t t e r i n g theory.

R -

Theorem 1.4. f

has measure

Assume f u r t h e r t h a t t h e r e i s an open s e t T 0 and hypotheses ( a ) - ( c ) hold i n t h e s e t wT = i z

f o r some

co

>

1 Re

z

E

T,

0 < Im z


0. i s t h e r e s t r i c t i o n t o n, H i s t h e s e l f a d j o i n t r e a l i z a t i o n o f - A i n H, K = L2(an), A = -a/an (normal d e r i v a t i v e ) , B = 1, on an. The wave o p e r a t o r s e x i s t and a r e complete. J

Example 4. H = H = L 2 ( R ) , H = -d2/dx2, o p e r a t o r s e x i s t and a r 4 complete. 2.

V

= 6(x),

J = 1.

The wave

DIFFERENTIATION I N ABSTRACT SPACES.

L e t G be a mapping o f an a r b i t r a r y subset V o f a t o p o l o g i c a l v e c t o r space X i n t o R , and suppose we a r e i n t e r e s t e d i n f i n d i n g an element u E f o r which t h e v a r i a t i o n o f G vanishes, i . e . ,

V

h o l d s f o r a l l q i n some dense s e t 0. Many problems i n mathematical p h y s i c s a r e s o l v e d i n t h i s way. I f t h e s e t V i s a l i n e a r m a n i f o l d and G i s bounded f r o m above o r below, one can t r y t o f i n d a maximum o r minimum. However, i f V i s n o t l i n e a r and G i s unbounded b o t h f r o m above and below, s e r i o u s problems p r e s e n t themselves. A l o c a l maximum o r minimum may n o t e x i s t . One s u g g e s t i o n i n a t t a c k i n g such problems i s t o i n t r o d u c e c o n s t r a i n t s . i n s t a n c e , one can c o n s i d e r G on a s e t S o f t h e f o r m (2.2)

S = {v E V

I F(v)

For

= 0)

where F i s some mapping o f V i n t o a t o p o l o g i c a l v e c t o r space Y. One has t o p i c k F i n such a way t h a t one can f i n d an element u i n S such t h a t ( s a y ) (2.3)

G(u) = min G(v). S

T h i s i n i t s e l f i s f a r from a t r i v i a l problem. Moreover, once (2.3) i s achieved, In p a r t i c u l a r , one wishes t o use ( 2 . 3 ) t o o t h e r d i f f i c u l t i e s p r e s e n t themselves. c o n c l u d e t h a t ( 2 . 1 ) h o l d s . B u t i n o r d e r t o do so, one must r e q u i r e u + t q E S f o r t near 0 and q E 0. T h i s would mean F ( u + t q ) = 0 f o r such t and q. T h i s i s t o o much t o ask. The most t h a t one can hope t o o b t a i n i s t h a t f o r each q E Q t h e r e i s a mapping q ( t ) f r o m IR t o Q such t h a t (2.4)

F(u + t q ( t ) ) = 0,

q(t)

+

q

in

Q

and even t h i s i s e x t r e m e l y d i f f i c u l t t o achieve. Moreover, t h e sense i n which q ( t ) converges t o q w i l l depend on F and w i l l n o t n e c e s s a r i l y c o i n c i d e w i t h t h e t o p o l o g y of X. The l i m i t (2.1) i s c l o s e l y r e l a t e d t o t h e d e f i n i t i o n o f d i f f e r e n t i a t i o n i n a b s t r a c t spaces. When V i s a l i n e a r s e t and t h e l i m i t i s a c o n t i n u o u s l i n e a r

504

M. Sehechter

o p e r a t o r , i t i s c a l l e d t h e Gateaux d e r i v a t i v e . O f t h e a p p r o x i m a t e l y t w e n t y - f i v e d e f i n i t i o n s t h a t e x i s t i n t h e l i t e r a t u r e , none o f them i s s u i t a b l e f o r o u r s i t u a t i o n . I n a l l cases one i s a i v e n a mappina f f r o m a t o p o l o g i c a l v e c t o r space X t o another, say Y. The mappina f i s d e f i n e d everywhere and i s s a i d t o have a d e r i v a t i v e A a t x i f A i s a c o n t i n u o u s l i n e a r o p e r a t o r f r o m X t o Y and f ( x + h)

(2.5)

-

f ( x ) = Ah + r ( h ) ,

h € X

where r ( h ) belongs t o a s e t o f " s m a l l " o p e r a t o r s f r o m X t o Y. It i s i n the c h o i c e o f t h e s e t of " s m a l l " o p e r a t o r s t h a t t h e v a r i o u s d e f i n i t i o n s d i f f e r . (Some a t t e m p t s have been made t o c o n s i d e r f n o t d e f i n e d on t h e whole o f X. However, i t i s always t a k e n as t h e r e s t r i c t i o n o f a mappina d e f i n e d everywhere.) We t h e n t u r n e d t o t h e problem o f f i n d i n g a d e f i n i t i o n o f d i f f e r e n t i a t i o n which would be s u i t a b l e f o r o u r c o n s i d e r a t i o n s . llle wanted a d e f i n i t i o n t h a t would t o be a l i n e a r m a n i f o l d

(a)

not require

V

(b)

not require

G'

t o be c o n t i n u o u s

(c)

not require

G'

t o be d e f i n e d on t h e same space as

(d)

have a reasonable c a l c u l u s

(e)

be u s e f u l i n s o l v i n o problems of t h e t y p e mentioned

(f)

i n c l u d e a l l o t h e r known d e f i n i t i o n s .

G

The d e f i n i t i o n which we f i n a l l y developed i s as f o l l o w s . L e t X be a r e a l v e c t o r space, and l e t 0, Y be separated r e a l t o p o l o g i c a l v e c t o r spaces such t h a t Q C X. L e t G be a mappina from a subset V c X t o Y. The d e r i v a t i v e o f G a t a p o i n t u E V w i l l be a l i n e a r o p e r a t o r f r o m X t o Y . I t s domain i s d e t e r mined as f o l l o w s . L e t C(V,O,u) denote t h e s e t o f t h o s e q € 0 f o r which t h e r e e x i s t sequences [ q n ? C 0 , { t n lc lR such t h a t (2.6)

qn

+

q

in

0,

0 # tn

-f

0,

u + tnqn E V

C(V,Q,u) i s a double cone. I t r e p r e s e n t s t h e l i m i t s of t h o s e elements o f 0 f o r be t h e s e t o f a l l which one can form a d i f f e r e n c e q u o t i e n t i n V . L e t E(V,Q,u) It i s t h e s m a l l e s t l i n e a r m a n i f o l d conf i n i t e sums o f elements o f C(V,O,u). t a i n i n g C(V,Q,u). of

G

D e f i n i t i o n . A l i n e a r operator A from X t o Y i s c a l l e d the d e r i v a t i v e a t u w i t h r e s p e c t t o Q and denoted by G ' ( u ) i f

Q

(1)

u E V

(2)

F o r any sequences

(2.7)

and

D(A) = E(V,Q,u) {qnl,

1 t i [G(u + t n q n )

-

I t n } s a t i s f y i n q ( 2 . 6 ) we have

G(uj]

+

Aq

in

Y

as

n

+

m.

Both t h e d e r i v lnlhen i t e x i s t s , G ' ( u ) i s determined u n i q u e l y on E(V,Q,u). a t i v e and i t s domain degend on C! and i t s t o p o l o g y . A l l o f t h e u s u a l theorems o f c a l c u l u s hold, i n c l u d i n g t h e mean v a l u e theorem, t h e c h a i n r u l e , t h e i m p l i c i t f u n c t i o n theorem, e t c . They h o l d under hypotheses no s t r o n a e r (and u s u a l l y weaker) t h a n t h o s e assumed f o r o t h e r d e f i n i t i o n s . Moreover, t h e p r e s e n t d e f i n i t i o n cont a i n s a l l o t h e r s i n t h e sense t h a t a mapping d i f f e r e n t i a b l e i n any o t h e r sense i s a l s o d i f f e r e n t i a b l e w i t h r e s p e c t t o t h i s d e f i n i t i o n . However, t h e f o l l o w i n a has no c o u n t e r p a r t i n o t h e r d e f i n i t i o n s .

Let 11. be a topological vector space continuously embedded in Theorem 2.1. e x i s t s , then GI:,(u) exists and is equal, to t h e restriction of Gb(u) toQ(EIVybI,u).

Q. If 6 ' u

Spectrul und Scuttering Tlzeorj’

505

Returnins to our oriqinal problem (2.11, one can show that if S is given by (2.2),and (2.3) holds, then under suitable conditions (2.8)

G (u)q

=

0

F (u)q

=

0.

Q

for all

q

E

C(Y,Q,u)

such that

(2.9)

Q

It then follows that by pickina F properly we can deduce that (2.8)holds for all q. Example 5. To find a stationary point of the functional . 2 + F1 r- 2 (x2 -l) +r 2V ( Y ) 1+ ~2~9 2 +x2y2-$r2i2-x 2 z 2 Idr G(x,y,z) = 1= [x 0

where x(r),

y(r),

z(r)

defined in 0 y(-)

=

b, z ( m )


0 and x(0) = a, y(0) = b are given.

G(x,y)

=

./ ( i 2-

3. ONE SIDED DERIVATIVES.

one if the any

The use of the restriction ( 2 . 2 ) is only one possibility. As an alternative may wish to use sets defined by means of inequalities. This is easily done Y (the space into which F maps) is finite dimensional. I prefer to use followina procedure which allows Y to be infinite dimensional. Let B be subset of Y ‘ (the space of continuous linear functional5 on Y), and put

S

(3.1)

=

tv

E

V 1 y F(v)

0 VY

E

B}.

For this situation we shall need the definition of a one-sided derivative. Let C+(V,Q,u) denote the set of those q E 9 for which there exist sequences tqn} c Q, {tn} cR, such that (3.2)

q,,

-

-f

q in Q , 0


O , n > 2 and l < p < - .

In [3],

i t was shown t h a t

(4.3) p r o v i d e d 0 < a < sp ( t h e e x p r e s s i o n I x - 1 o g l x - y l when ~1 = n and by 1 when cx I n [4] i t was shown t h a t depend on V. (4.4)

C(V,s,p,q)

-

yja-n i s t o be r e p l a c e d by n . The c o n s t a n t i n ( 4 . 3 ) does n o t

1

5- C sup( I V ( x ) l q l ~ - ~ l a - dx)’/q n y IX-Yl O .

= (OIO),

A s f o r t h e f u n c t i o n s which a p p e a r i n t h e e q u a t i o n s (l), w e r e q u i r e

t h a t t h e i r g r a p h s have t h e q u a l i t a t i v e forms a s d e p i c t e d i n F i g u r e 1.

to of

W e require t h a t f o r the function

g2

,

b

i s s m a l l , and t h a t

f o r t h e function b, t h a t

O ~ Z ' 1 + 5 , 5 > 0 .

f

r

g, t h a t

g1

is close

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

i s c o n s t a n t on some i n t e r v a l

523

A System o f Reaction-Diffusion Equations

Figure 1 Now l e t

b e a g i v e n s m a l l p o s i t i v e number, and c o n s i d e r

> 0

E

the set

where

a

and

C o n c e r n i n g t h i s s e t , w e h a v e t h e f o l l o w i n g lemma.

respectively. If

Lemma 1.

$!

a r e a s d e p i c t e d i n F i g u r e l ( i )and l(iii)

6

/g1-g21

i s s m a l l , and

b

h a s s m a l l slope, t h e n

i s a g l o b a l a t t r a c t i n g region f o r a l l s o l u t i o n s of

,

i n t h e s e n s e t h a t g i v e n any n e i g h b o r h o o d N of

lies i n

(u(x,t),z(x,t)) Proof.

I

while i f

-

-E

-

t

sufficiently large.

F ( u , z ) = ( a ( u ) f ( z ) , b ( u ) g ( z )+ $ ( z ) ) ; t h e n i f

Let

G1 ( u , z ) =

for

N

(l), ( 2 ) ,

every solution

u,

-

VG1

a >

a,


>

y 0,

and t a k e

y > 0

“

s o s m a l l t h a t t h e above 9

steady-state solutions a l l exist. Lemma 2 .

as

t

+

a

If

s o l u t i o n of m

.

i s s u f f i c i e n t l y l a r g e , and

(1), ( 2 1 , t h e n

u

In particular,

if

tends t o i t s ct

(u,z)

i s any

p a t i a l average

i s l a r g e , t h e above 9

s o l u t i o n s are t h e o n l y s t a t i o n a r y o n e s . Proof.

Let

Then f o r

t

(u,z)

b e any s o l u t i o n of

large,

(u,z) €

e i g e n v a l u e of

on

D2

1x1 < L

,

(1),

X

and i f

2 ) , and s e t

i s t h e f i r s t non-zero

w i t h homogeneous

Neumann

boundary c o n d i t i o n s ,

where w e h a v e u s e d P o i n c a r g ’ s i n q u a l i t y . then

0

> 0

if

CY

> M/X

and w e have

at

T h u s , u s i n g a s t a n d a r d i n e q u a l i t y [ 4 , Th.

1 = aX 2 so t h a t

Now t h e n i f < -54

,

-5

11.111, w e f i n d

- M,

527

A System ofReaction-Diffusion Equations where

v(t)

l v(t) = 2L

Thus i f u(x,t) if

L

t e n d s t o a f u n c t i o n of u ( x ) : U.

,

a(u) = 0

and

a

as

t

so t h a t

y

L > 0 y
1, a n d t h a t w e h a v e

p r e c i s e l y 9 s t a t i o n a r y s o l u t i o n s of Po =

t

S i n c e o = aUxx + a ( K ) f ( z ) = a ( L ) f ( z ) , a n d t h e p r o o f i s c o m p l e t e . [I

Now l e t u s assume t h a t chosen

.

is a s t a t i o n a r y s o l u t i o n , then

(u(x),z(x))

constant, say

u(x,t)dx

i s a s o l u t i o n of ( I ) , ( 2 ) , then

( u ( x , t ), z ( x , t ) )

w e see t h a t

u; i.e.,

i s t h e s p a t i a l a v e r a g e of

(11, ( 2 ) .

P1 = ( 0 , Z l ( X ) )

W e d e n o t e t h e s e by

,

P2 = ( 0 , z 2 ( x ) )

Qo = ( 6 , s z o ( x ) )

, Ql

=

(b,zl(x))

I

Q2 = ( E l i 2 ( x ) )

Ro = ( a , y o ( x )

, R1

=

(a,T,(x))

,

R2 =

,

(a,z,(X)) .

These 9 s t a t i o n a r y s o l u t i o n s , t o g e t h e r w i t h t h e i r c o r r e s p o n d i n g c o n n e c t i n g o r b i t s c a n b e c o n v e n i e n t l y d e s c r i b e i n F i g u r e 4 below.

Fjgure 4

I n o r d e r t o o b t a i n more p r e c i s e i n f o r m a t i o n c o n c e r n i n g t h e u n s t a b l e m a n i f o l d s of t h e s e s t a t i o n a r y s o l u t i o n s , w e s h a l l l i n e a r i z e e q u a t i o n s ( 1 ) . ( 2 ) a b o u t them.

We c o n s i d e r f i r s t t h e

The C o r r e s p o n d i n g l i n e a r i z e d e q u a t i o n s a r e , f o r

P.’s.

i = 0,1,2,

A;

=

A;

= ~ ~ " + ~ ' ( O ) ~ ( Z ~ ( X ) ) ; + ~ ; ) ( Z ~ ( z(?L) X ) ) ~ = , 0

(7)

a;’! + a l ( o ) f ( z i ( x ) ) ;

Suppose f i r s t t h a t

,

; I ( ~ L )=

i = 0 ; then since

o

,

a ' (0) f ( z o ( x ) ) < 0

see from t h e f i r s t e q u a t i o n i n ( 7 ) t h a t

-

A < 0

if

;f

.

0

.

we

.

On t h e o t h e r h a n d , i f u : 0 , ( 6 ) shows t h a t A < 0 I t follows 0 that h(Po) = C I n a s i m i l a r way, w e c a n show t h a t P2,R0

.

and

R2

a r e a l s o non-degenerate

and t h a t

h(P2) = h(Ro) =

528

J. A . Smoller

.

Now consider ( 7 ) when i = 1, i.e., at h(R2) = ’1 pl' Again, if u 4 0 , then X < 0 , and if u 5 0 , then again using ( 6 ) , we find that P1 is non-degenerate and h(P1) = E l ; similarly R1 is non-degnerate, and h(R1) = Z 1

-

5

Now let’s turn our

attention to the

Qi’s

.

.

The relevant

equations are

If 5 0 , then from (6) we find that Qi is non-degenerate, and that the second equation in ( 8 ) has exactly one positive eigenvalue if i = 1, and no positive eigenvalues if i = 0, o r i = 2. But let us note that

L =

+ a’(b)f(O)$)dx

sup @

,

1-L

and that these two expressions are the variational characterizations of the operators aD2

+

a’(b)f(zi(x))

, and

aD2

+

a’(b)f(O)

,

together with homogeneous Neumann boundary conditions. Since is a positive eigenvalue of the second operator, a’ (b)f ( 0 ) (corresponding to the eigenvector

-

u

E

11, we see that

Hence the first equation in(8) has at least one positive eigenvalue Xi If u is the corresponding eigenvector, we shall show that the second equation in ( 8 ) can be solved for z To this end, consider the operator 2 (9) yD - Xi .t s’(ii(x)) ,

.

- .

with homogeneous Dirichlet boundary conditions. Since Xi > 0, this operator is invertible if i = 0, or i = 2, (by ( 6 ) ) . k It follows that h(Qo) = 1 , h(Q2) = Em, and h(Q1) = En , where k 2 1, m 2 1 and n 2 1. We summarize these results in the following proposition.

529

A System of' Reaction-Diffusion Equations

Proposition 3 .

A.

h(R1) = C1: h(Qo) m 2 1, and n 5 1.

that that operators.

h(R2) = Z o ; n h(Q1) = Z m , h(Q2) =I: , where

h(P ) = h(P2) = h(R ko

,

C

=

=

0

h(P1)

=

k 2 1,

B. The P i t s and R i g s are non-degenerate in the sense 0 is not in the spectrum of the corresponding linearized

We next show that the equations (l), (2) are "effectively" gradient-like; in fact we have the following proposition. Proposition 4 .

If

is sufficiently large, and

c1

y

is sufficiently

small, then all solutions of (l), (2) tend to staionary solutions as t + +m

.

Proof.

Let

(u,z) be any solution: then

exponentially to zero as of

u.

t

-f

m ,

1Iu

v

where

-

vII L

tends

(.- L ..L )

is the spatial average

Since v

utdx = l

= -

- -l -

L

cla(v)

+ a(v)k

a(u) -a(v)]dx

2L

=

j-La(~)f(z)dx L

+

1,

f(z)dx

0 e-Ot)

as t + m, where c1 > 0 , hence u , tends to a root x (see 121).

we can conclude from 131 that v , and u of a(u), as t m, uniformly in -f

To complete the proof, we must show that

z

tends to a solu-

tion of yz"

(9)

+

b(a)g(z)

+ @(z)

=

0,

1x1 < L,

z(+L)

=

0.

The problem w t = y w x x + b(u)g(w) + @(w), (10)

W(fL, t) =

o , t

1x1 < L,

0, W(X,O) = Z(X,O),

t > 0, 1x1 < L

w(x) as t + + m , has a unique solution w(x,t), and w(x,t) uniformly on 1x1 < L, where w satisfies (9). Choose E > 0 so small that if Ilw(x,O) - z(x,O) I / m < E , then the corresponding solution of (10) tends to w as t + + m . -f

Now we can find

TE > 0

such that

530

J.A. Smoller wE

Let

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

w

=

wxx + b ( o + ~ ) g ( w+)$ ( w ) = 0 ,

If

2

t

T

t > 0,

= 0,

W(fL,t)

= z(x

W(X,O)

then

E '

-

zt


0,

-

if

T~

1x1 < L, w ( + L )

(see [ 6 ] ) . S i n c e

< L;

2

tends t o a solution

y w " + b ( u + ~ ) g ( w+) @ ( z ) = 0 , u n i f o r m l y on

t

if

5 wE(x) 2

Z(x)

Replacing

(u

+ 6 +

E)

(u

by

- E)

in the

above argument g i v e s

These l a s t t w o i n e q u a l i t i e s imply t h a t

t

on

1x1 < L ,

4.

T o p o l o g i c a l Methods

as

l i m z ( x , t ) = Z ( x ) , uniformly

T h i s completes t h e proof.

+ m.

0

W e a r e now i n a p o s i t i o n t o a p p l y t o p o l o g i c a l t e c h n i q u e s t o o u r problem.

Our main t o o l w i l l b e t h e C o n l e y i n d e x , and w e assume

t h a t t h e r e a d e r i s f a m i l i a r w i t h t h e main r e s u l t s i n t h i s t h e o r y ;

see 113 o r [ 4 , P a r t IV] Let points

I.

(resp.1

P o , P1,

and

a

)

. d e n o t e t h e s e t c o n s i s t i n g of t h e t h r e e r e s t

P2 ( r e s p . Ro,

R1

and R 2 )

,

together with t h e i r

c o r r e s p o n d i n g c o n n e c t i n g o r b i t s : see F i g u r e 4 . Lemma 5 .

I.

h(I ) = h ( I - ) 0

a

and =

I-

o a C .

a r e i s o l a t e d i n v a r i a n t s e t s and

A System of Reactioii-Diffusion Equations

Proof.

531

Define L

Y(t)

=

Y’(t)

=

then

I-, I-,

u(x,t)dx

;

L

.

f(z(x,t)) a(u(x,t))dx

Then Y ' < 0 if u(x,t) < 5; 1x1 < L. Since small neighborhoods of I. lie in u < g , any invariant set in such a neighborhood would have to tend to rest points in both time directions. The Hence I. and only such sets are the depicted ones in I. similarly I- are isolated invariant sets. a To compute the index of I0 ' we deform the function a(u) as depicted in the figure below:

.

Figure 5 Under this deformation I. continues to the maximal invariant set in @ ; hence h(Io) = c o by the continuation theorern. In a 0 similar way, h(I-1 = 1 . I] a We can now prove the following result which gives the precise indices of the Qi’s. Lemma 6.

h(Qo)

=

h(Q2) =

c1 ,

and

h(Q1) = 1

2

.

Proof. If I (@ ) denotes the maximal invariant set in , then h(I(&)) = z 0 We now use the continuation theorem and deform @ ( z ) by "pushing down the hill; see Figure 6.

.

Figure 6

J.A. Smoller

532

t h e r e s t p o i n t s R1, R 2 , 0 1 1 Q2 and c o n t a i n s o n l y t h e r e s t p o i n t s R o , Qo

Under t h i s d e f o r m a t i o n , P1, P 2 c a n c e l , a n d @ and

Since

Po.

connecting

and

Ro

t o both

Qo

are a t t r a c t o r s , t h e r e e x i s t o r b i t s

Po Ro

,

and

( s e e [ 4 , Ch. 2 4 ,

Po,

§El), a s

d e p i c t e d below.

Figure 7 PJext, l e t of

M

denote t h e i s o l a t e d i n v a r i a n t set c o n s i s t i n g

a n d t h e o r b i t s c o n n e c t i n g them.

Q o , Po

Then

h(n) =

( t h e homotopy c l a s s of a p o i n t ) , s i n c e w e may d e f o r m " p u l l i n g u p t h e v a l l e y " , whereby

and

Qo

Po

cancel;

a(u)

by

see

F i g u r e 8.

Figure 8 W e now u s e t h e l o n g e x a c t s e q u e n c e o f cohomology g r o u p s (see [ 4 , Ch. 2 3 ,

-t

§D])

Hn-l(h(P4))

-+

Hn-'(h(P0))

-t

H n ( h ( Q o ) )+ Hn(h(P4))

-+

-.-,

or

... + 0

+

En-1

This implies t h a t Similarly

1x0)

+

Hn-l (Co)

h ( Q 2 ) = IL.

down t h e v a l l e y " i n

IIn (Zk)

I$

Hn(Ck)

Since I

+ 0 +

O1

,

and

... . for e a c h n : h e n c e k = 1. 0, c a n c e l wehn w e " p u s h

w e c a n u s e an argument s i m i l a r t o t h e

one j u s t given t o conclude t h a t

h(Q1) = C 2

.

A System o f Reaction-Diffusion Equations

533

At this stage it is useful to depict the rest points in

@

together with the known connecting orbits in Figure 9.

Figure 9 We can now show that there are orbits-running from Qo to and Po as well as orbits running from Q2 to both R2 and P2. We shall give the details only for Q o to R o . Thus, let denote the isolated invariant set containing as rest points I2 the Qi’s and R . ' s , i = 0,1,2, as depicted in Figure 9. By continuation (push down the hill in @ as in Figure 61, this set Ro

continues to the empty set so that h(12) = 5 - If S denotes the isolated invariant set containing Ro and 0 as its only -0 (This is easily seen by pushing rest points, then h ( S ) = 8 . down the hill in a ; see Figure 5. Under this deformation, R1 and R2] as well as Q , and Q2 cancel so S continues to I2 . ) Now we can invoke the connecting orbit theorem [4, Th. 22-23], to conclude that there is an orbit running from Q o to Ro. Similarly, there are orbits running from and P2

.

R2

Qo

to

Po

and from

Q2

to

Finally, we shall show that there are orbits running from Q, to both P1 and R 1 ; we shall give the details only for the case Q, R1. Thus, suppose on the contrary] that there is no orbit running from Q1 to R1. Let T be the isolated invariant set containing Ol and R1 as its only rest points. We claim . that Hn(h(T)) = 0, n = 0,1,2 To see this, let Ptl be the disjoint union of Q2,R2 and the orbit connecting them, together with Q o l R o and the orbit connecting these rest points; see Figure 10. +

,...

J. A. Smoller

5 34

Figure 10 Let fa2 = T; then ( H l , M 2 ) is a Morse decomposition of the following sequence is exact (see [ 4 , Ch. 23, S D I ) :

...

+

H"-’(h(M1))

+

Hn(h(M2))

.

Hn(h(12)) *

-r

12. Thus

Since H " - ~(h(lll)) = 0 = Hn(h(12)) for all n, it follows that Hn(h(M2)) = 0 for all n , thereby proving our claim. Now if there was no a f4orse decomposition of

orbit from T,

to

Q1

R1

then

(U1,R1) is

and using the exactness of the

sequence .-.+Hn-l(h(T)) * Hn-l(h(Q1))

-r'

Hn (h(R1))

+

Hn(h(T))

+

--*,

or 0

+

Hn-l(h(Ql))

--t

Hn(h(R,))

-f

we see that Hn-l (h(Q1)) = Hn(h(R1)) or all n. This is impossible, and thus R1 p1 and Q,) are connected by orbits.

0

+

, H ~ ( C ~ )for ,

Hn-l(C2)

2

and

(as well as

Ql

We may depict (schematically) the maximal invariant set in

@ in the following diagram. That is, there is an invariant 2 dimensional manifold in @ consisting of orbits connecting rest

R,

Figure 11

535

A System of Reaction-Diffusion Equations

p o i n t s as d e p i c t e d .

rest p o i n t s as a l l s o l u t i o n s of

t

-+

A l l o t h e r s o l u t i o n s t e n d t o one of t h e 9 +m

.

T h i s i s t h e complete g l o b a l p i c t u r e of

(1), ( 2 ) .

REFERENCES : C o n l e y , C . , I s o l a t e d I n v a r i a n t S e t s a n d t h e Morse I n d e x , CBMS R e g i o n a l C o n f e r e n c e S e r i e s i n M a t h . , N o . 38, Amer. Math. S O C . , Providence, R . I . , 1978. Conway, E . , D. H o f f , a n d J. S m o l l e r , L a r g e t i m e b e h a v i o r o f s o l u t i o n s of systems of n o n l i n e a r r e a c t i o n - d i f f u s i o n equat i o n s , SIAM J . Appl. M a t h . , 2 ( 1 9 7 8 ) , 1-16. Markus, L . , A s y m p t o t i c a l l y autonomous d i f f e r e n t i a l s y s t e m s , C o n t r i b u t i o n t o t h e Theory o f N o n l i n e a r O s c i l l a t i o n s , v o l . 3, A n n a l s of Math. S t u d i e s , N o . 3 6 , P r i n c e t o n Univ. P r e s s , P r i n c e t o n , 1 9 5 6 , 17-29. S m o l l e r , J . , Shock Waves a n d R e a c t i o n - D i f f u s i o n S p r i n g e r - V e r l a g , New York, 1983.

Equations,

S m o l l e r , J . , and A. Wasserman, G l o b a l b i f u r c a t i o n of s t e a d y s t a t e s o l u t i o n s , J. D i f f . E q u s . , 3 9 ( 1 9 8 1 ) , 269-290. Conway, E . , R . G a r d n e r , a n d J. S m o l l e r , S t a b i l i t y a n d b i f u r c a t i o n o f p r e d a t o r - p r e y s y s t e m s w i t h d i f f u s i o n , Adv. Appl. M a t h . , 3 , ( 1 9 8 2 1 , 288-334.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

5 31

VARIATIONAL PROBLEMS WITH SINGULAR SOLUTIONS

R. Temam

L a b o r a t o r i e D’ Analyse Numerique U n i v e r s i t e de P a r i s - Sud B a t imen t 425 91405 Orsay Cedex France INTRODUCTION V a r i a t i o n a l problems which a r e c o e r c i v e on a n o n r e f l e x i v e Banach space may possess s i n g u l a r s o l u t i o n s . S i n g u l a r v a r i a t i o n a l problems o f t h i s t y p e a r i s e i n mechanics and p h y s i c s , f o r i n s t a n c e i n s o l i d mechanics ( p l a s t i c i t y ) , o r i n problems r e l a t e d t o minimal area s u r f a c e s o r t o s u r f a c e s o f g i v e n mean c u r v a t u r e . The t y p i c a l s i t u a t i o n i s as f o l l o w s : we a r e g i v e n a n o n r e f l e x i v e Banach space V and a l o w e r semi-continuous f u n c t i o n a l 0 f r o m V i n t o IR which i s when IIvllv + + -. The f u n c t i o n i s t h e n bounded coercive, i . e . ~ ( v )+ +

-

from below on V b u t t h e minimum i s n o t n e c e s s a r i l y achieved. I n order t o obtain t h e e x i s t e n c e o f a g e n e r a l i z e d m i n i m i z e r i t i s necessary t o imbed V i n t o a l a r g e r space

i,

t o extend

@

l o o k f o r t h e minimum o f

on

on

as a f u n c t i o n

from

into

IR and then t o

i.

I n t h e a p p l i c a t i o n s mentioned above, V i s s i m i l a r t o a Sobolev space b u i l t 1 i s a space s i m i l a r t o t h e space o f f u n c t i o n s w i t h bounded L (n) w h i l e

v a r i a t i o n , BV(n). F o r such problems t h e e x t e n s i o n o f O t o necessitates t h e s t u d y o f f u n c t i o n a l s o f a measure: t h i s i s o u r aim i n S e c t i o n 1 which p r o v i d e s t h e b a s i c framework f o r S e c t i o n s 2 and 3. S e c t i o n s 2 and 3 g i v e two examples o f v a r i a t i o n a l problems i n t h e c a l c u l u s o f v a r i a t i o n s w i t h s i n g u l a r s o l u t i o n s : one of them i s r e l a t e d t o minimal s u r f a c e s , t h e o t h e r one i s a problem encountered i n p l a s t i c i t y . The p l a n i s as f o l l o w s : 1. 2. 3. 1.

Convex f u n c t i o n o f a measure. Time dependent minimal s u r f a c e s . A problem i n t h e p l a s t i c i t y o f p l a t e s .

W E X FUNCTION OF A HEASURE Let

Q

I f ( t ) i 'c a151

(1.1) where an

u

E

1ci

r and l e t f IR, which i s s u b l i n e a r , i.e., has a t most

be an open bounded s e t o f Rn w i t h a smooth boundary

be a c o n t i n u o u s f u n c t i o n f r o m IRE i n t o a l i n e a r growth a t i n f i n i t y : +

v 5 €Re,

b,

i s t h e e u c l i d e a n norm o f

5

i n Re,

and

a, b 2 0.

Then, i f

u

is

L1 f u n c t i o n ( f o r t h e Lebesque measure dx = dxl.. .dxn) f r o m R i n t o R', 1 L ( a ) ' , t h e c m p o s e d f u n c t i o n x t+ f ( u ( x ) ) i s (a.e.) w e l l d e f i n e d on Q and

R. Temam

538 due t o ( l . l ) , f o

u

f o

u t o b e i n L 1 (n),

t o t h e case where f o

E

u

1 L (n)

( a c t u a l l y (1.1) i s necessary and s u f f i c i e n t f o r b' u E L'(Q)'). We want t o e x t e n d t h e d e f i n i t i o n o f f o u a. i s a bounded measure on R, u = u E Ml(n) , i n which case

w i l l be a bounded measure on

1-1

n,

f, 1-1

E

fa o f

We assume t h a t t h e a s y m p t o t i c f u n c t i o n

Then

V

< ER'.

If

5 a\O), and s a t i s f i e s the following hypotheses:

,

vx

E

R~ ;

(b) M is equivariant, i.e. we have : M(Tx,X)

=

?M(x,A)

,

V(x,X)

E

Xxfl

.

(2.2)

(c) Lo = DxM(O,O) is a Fredholm operator, with dim N(LO) = codim R(LO) = n 2 1. I t follows from ( 2 . 2 ) that Lo i s equivariant, i . e . we have Lo E Lr,~(X,Z), where

5 47

Bifurcution of Subharmonic Solutions

Then (see [ 81) there e x i s t equivariant projections Po that N(LO) = R(Po) and R(LO) = N ( Q ) .

The action of

r

E

.Cr(X) and Qo E Li;(Z) such

leaves N(LO) invariant,

while R(QO) i s an n-dimensional complement of R(LO) in Z which i s invariant under r. MOreover, i f there is a second projection Q, E d:i;(Z) such that R(LO) = N(Q,), then

maps R(Q,) isomrphically on R(Q), w i t h inverse Q,, and on R(Q,)

cides with

This shows that the action of

Q1o?oQ.

?

? coin-

on an invariant complement of

R(LO) i s independent on the particular choice of t h i s complement. W e can use this

t o formulate a further condition on M : (d) The action of r on N(LO) i s equivalent t o the action of complement of R(LO) i n Z.

?

on a ?-invariant

This means t h a t there are linear isomorphisms q : N(LO) *Rn and 5 : R(Qo) +IF?

such that

We can even choose q and 5 such that

ro E

i s orthogonal (see [ 81). Our final

hypothesis w i l l be a transversality condition; i n order t o formulate it we need some results on Fredholm operators. W e w i l l say that two operators L,

E

there e x i s t linear automorphisms A

E

E(X,Z) and L2

E

% ( X , Z ) are eqlLiude&

if

L ( X ) and B E l ( Z ) such that Lz = BoL,oA. W e

denote by S(L) the equivalence class of L

E

d:(X,Z). Similarly, two operators L,

and L2 i n d: -(X,Z) are (r,?)-equiuaeutX i f they are equivalent via equivariant r,r automrphisms A 6 f y ( X ) and B E d:-,(Z). The corresponding equivalence classes are denoted by E

r , -(L). r

The following lemma's describe the structure of &(Lo) and

Zr,?(Lo) when Lo is a Fredholm operator.

Lemma 1 . Let Lo

E

d:(X,Z) be a Fredholm operator. Then 8(Lo) is the s e t of a l l

Fredholm operators L codim R(LO)

E

.C(X,Z) such that dim N(L) = dim N(LO) and codim R(L) =

.

In one direction the proof is t r i v i a l ; the other direction can be proved by actually constructing appropriate automrphisms A Lemma 2. Let Lo

E

6(X,Z)

E d: (X)

and B

be a Fredholm operator, and l e t Po

E

E d:

(Z)

.

S(X) and Q,

E

d:(Z)

be projections such that R(PO) = N(LO) and N(Qo) = R(LO). Then there e x i s t a neighborhood U of Lo i n d:(X,Z) , and smooth mappings A* : u+.C(x)

5 48

A. Vanderbauwhede

and B’ : U+f(Z), with A* (Lo) for a l l L

=

Ix and B* (Lo)

=

Iz, such t h a t the following holds

E U :

( i ) A*(L) = I ~ + A ( L ) P f o~r sorne smooth ( i i ) B* (L) ( i i i ) Y(L)

=

:=

X

: u+~:(N(L~),N(P ; ~))

I z + g(L) (I-Qo) for some smooth

;

: U+f(R(LO),R(Qo))

B*(L)oLoA*(L) maps NIPo) isomorphically on R(LO) ;

(iv) Q(L), the r e s t r i c t i o n of Y(L) t o N(LO), maps N(LO) i n t o R(Q)

.

This r e s u l t follows from a straightforward application of the implicit function U is a Fredholm operator, w i t h index L = index

theorem. I t implies t h a t each L

E

Lo and dim N(L) = dim N(Q(L))

dim N(LO).

Q

Corollary 1 . Under the assumptions of lemma 2 we have E(Lo) n U = { L E U I @(L)= O } , and &(Lo) i s a submanifold of f(X,Z) with codimension equal t o dim f(N(LO),R(QO)). P r o o f . This follows from the f a c t t h a t

D ~ Q ( .L L ~=) oJl

,

“LO)

sri: Ef(X,Z) ,

(2.5)

so t h a t %@(Lo) E L(f(X,Z),L(N(Lo) ,R(Qo))) is surjective. One can even show t h a t t h e equivalence classes 8 ( L ) form a s t r a t i f i c a t i o n of the open subset of f(X,Z) containing a l l Fredholm operators. For equivariant operators we have similar results :

Lemma 3. Let Lo L

E

Ed:

r,r-(X,Z)

be a Fredholm operator. Then gr,r(Lo) consists of a l l

&(Lo)nd: -(x,Z) such t h a t :

r,r

( i ) the action of

r

on N(L) is equivalent t o the action of

r

( i i ) the action of

?

on an ?-invariant complement of R(L) is equivalent t o the

on N(LO) ;

action of ? on an ?-invariant complement of R(LO). Lema 4. U d e r the conditions of lemma 2 , suppose t h a t Lo E fr,i;(X,Z), Po E f,(X)

ard

%E

Li;(Z). Then we have f o r each L

B*(L) E d:?{Z) and O(L)

E f

E

Unfr,i;(X,Z) t h a t A*(L)

E

fr(X),

r,r-(N(LO),R(Q)).

Corollary 2 . Under the conditions of lemma 4 we have that E r , ~ ( L o ) n U= ILEUnd:r,~(X,Z)I @(L)= O } , and gr,f(L0) is a submanifold of S,,i;(X,Z) w i t h codimension equal to dim d: -(N(LO) ,R(Q)).

r ,r

The foregoing r e s u l t s allow us t o formulate our last hypothesis, which w i l l be s a t i s f i e d f o r generic mappings M i n the c l a s s under consideration. We define

5 49

Bifurcation of Sub harmon ic Solutions

L : fl + 6:,,,(X,Z)

by

,

L(X) =DxM(O,X)

VXEX?

.

We have L(0) = L o , and we w i l l assume : (e) The mapping L defined by (2.6) is a t X Since

g

r,r-(L o )

=

0 transversal t o grr,i;(Lo).

is the s e t of a l l operators having the same "structure" as L o ,

this i s a natural condition t o impose. I n analytic form (e) requires t h a t ), considered as a linear operator from the parmeter space QDXL(0)

fl into

r , r-(N(Lo)

6:

m

2

,R(Qg)),

is surjective. A necessary condition for this i s that

dim 6:r ,r-(N(Lo),R(%)) = dim LrQ@In)

.

(2.7)

For given m and k , t h i s severely r e s t r i c t s the p o s s i b i l i t i e s for n = dim N(LO) and ~ , for ro. Since rok = IRn, the eigenvalues of ro must have the form ( v ~ ) with

..

,k-1 I ; moreover, each of these eigenva'?< = exp(iek) Ok = h / k , and r E {0,1,. lues is semi-simple. Writing ro i n i t s real normal form one can e a s i l y verify

the following :

Lemma 5 . L e t ro E emn) be such that rok = I. For each r E { O , l j...jk-ll l e t vr denote the multiplicity of (pk)r as an eigenvalue of ro. Then we have : k- 1 d i m l y mn) =

1

r=O

w

.

An immediate consequence of (2.8) is that n

< dim 6:r omn) < n'.

Remark also that

~ =- vr,~ since ro is a r e a l operator. In the example of the subharmonic solutions of ( I . 1 ) , vr is the dimnsion of the eigenspace corresponding t o the eigen-

v

value (1~)’ X = h

of the mncdromy matrix of ( 1 . 2 ) for the c r i t i c a l parameter value

0'

In the next section we w i l l discuss some of the bifurcation pictures which a r i s e under our hypotheses when m = 1 o r m = 2. The relations (2.7) and (2.8) w i l l give the corresponding p o s s i b i l i t i e s f o r n and

ro.

3. THX BIFURCATION EQUATIONS

Ender the hypotheses (a)-(d) one can apply a n equivariant Liapunov-Schmidt method ([ 81) and use the isomorphisms n and 5 appearing i n ( 2 . 4 ) t o reduce ( 2 . 1 ) i n a neighborhood of the origin t o a bifurcation equation

550

A. Vanderbauwhede

F(u,X) = 0

where F : Rn xC? ( i ) F(0,X) = 0

, +

Rn is of class Cp, and such t h a t :

,

YX ;

( i i ) DuF(O,O) = 0 ;

,

( i i i ) F(rOu,A) = TOF(u,X)

V(u,X)

.

The hypothesis (e) translates into : (iv) D ~ D ~ F ( O E , OL)@ , E ~ ~ Q R ~ ) is ) surjective. When m = 1 t h i s condition can only be s a t i s f i e d f o r n = 1 , i n which case L @) = To EN). Our hypotheses (except for the equivariance) reduce i n t h a t case t o those of the Crandall-Rabinowitz theorem on bifurcation from simple eigenvalues [-3]. In particular, (iv) becomes the condition on the mixed derivative appearing i n the CR-theorem, and we conclude that this l a s t condition i s i n f a c t a transversality condition of the form described i n section 2. As i n the CR-theorem, (3.1) has precise,ly one branch {(u,X' (u)) I uElR1 of nontrivial solutions. As f o r ro, there are two p o s s i b i l i t i e s : (1) ro = 1. In this case the bifurcating solutions of (2.1) w i l l s a t i s f y Tx = x (See [ 8 1 for d e t a i l s ) . (2) k i s even and r0 = -1. Then it follows from ( i i i ) t h a t A* (-u) = A* (u), and we get a pitchfork bifurcation. The bifurcating solutions remain invariant under Y2, while the two symmetric parts of the bifurcating branch can be obtained one from the other by application of the symmetry operator

r.

L e t us now consider 2 - p a r a t e r problems, i.e. we take m = 2. I t follows from (iv) t h a t we must have e i t h e r n = 1 , o r n = 2 and dim L QR2 ) = 2. For n = 1 we obtain ro the same results as we have j u s t described, except that there is an extra parameter. So we w i l l assume t h a t n = 2 and dim f (IR 2 ) = 2. This implies that e i t h e r To k is even and ro has the eigenvalues +1 and -1, o r that ro has the eigenvalues r = exp(ir'dk) and ikr= exp(-ir'dk), for some r with 0 < r < k/Z. In the f i r s t case a l l bifurcating solutions w i l l be invariant under Y z , and we obtain a situation which describes the interaction between the cases (1) and (2) obtained for m = 1. We w i l l not consider this case here any further. 2 , and that ro has the eigenvalues pkr and Gkr, with r ' / k ' , such that r' and k' have no c o m n divisors. Then k' = I , and a l l bifurcating solutions w i l l s a t i s f y r x = x; replacing k and r

Let us assume that n 0

=

< r < k/2. Write r/k

rk'

m =

=

55 1

Bifurcation of Subharmonic Solutions

by k' and r*, we may therefore assume that k and r have no c o m n divisors. W e w i l l also identify IR2 with the complex place O . backward c o n t i n u a t i o n on (4, 01 i s proved. Furthermore, a n i m p o r t a n t f a c t i s e s t a b l i s h e d t h a t t h e i n i t i a l c o n d i t i o n s may be posed a t any two p o i n t s to-1 and Necessary and s u f f i c i e n t c o n d i t i o n s f o r a s y m p t o t i c to, n o t n e c e s s a r i l y i n t e g r a l . s t a b i l i t y of t h e t r i v i a l s o l u t i o n a r e determined e x p l i c i t l y v i a c o e f f i c i e n t s of t h e g i v e n e q u a t i o n . Then, t h e f o r e g o i n g r e s u l t s are g e n e r a l i z e d f o r e q u a t i o n s w i t h many d e l a y s and systems of e q u a t i o n s . Some c l a s s e s of advanced and n e u t r a l e q u a t i o n s a r e a l s o s t u d i e d . Next, l i n e a r e q u a t i o n s w i t h v a r i a b l e c o e f f i c i e n t s and some n o n l i n e a r e q u a t i o n s a r e i n v e s t i g a t e d . E q u a t i o n s w i t h unbounded d e l a y [51 a r i s e i n c a s e s of s e v e r a l argument d e v i a t i o n s . I n such problems t h e i n i t i a l f u n c t i o n i s p r e s c r i b e d on [-m, 0 ) and t h e s o l u t i o n i s c o n s i d e r e d f o r t > O .

J. Wiener

572

EQUATIONS WITH CONSTANT COEFFICIENTS Consider the scalar initial-value problem x’(t)

= ax(t)+aox(

[tl )+alx( [t-ll), ~ ( - 1 )= cdl, x(0) = co

(1.1)

with constant coefficients. This equation is very closely related to impulse and loaded equations. Indeed, write E q . (1.1) as m

C

x’ (t) = ax(t)+

(agx( i)+alx(i-l) ) ( H ( t-i)-H( t-i-1) ),

i=-m where H(t)=l for t>O and H(t)=O for tO. For a symmetric differential operator L, the deficiency indices (N+,N-) are defined to be the maximal number of linearty independent solutions of Ly = Ay in LZ(X,m) for Im(A)>O and 3 are fully considered in [ 6 ] using the change of argument theorem: when q = 7, for instance, it is possible to have four zeros crossing over into the L*-region, leading to an extra four algebraic type L2 solutions.

3.

The odd order self-adjoint equation

If the coefficients of the differential expression L are allowed to take complex values, then it is known 12, p.2041 that if L is equal to its formal adjoint L+ then n Ly = rz = ~ry

A. D. Wood

584

7):)' ...

where L y = irqn-r(.!.(qn-r(q ) ' , the differentiation being performed r times: with q.E&"’J and (q.br -1 real-valued. Another form is given by Naimark (5,p.7)! but that is’less suitable for our method of comparison with a hypergeometric equation. For the operator of order n = 2m+l, the deficiency indices are known to satisfy either mlN+i2m+l and m+llN- 0, we t a k e an a u x i l i a r y H i l b e r t space K = L t F , f f ) o f H-valued square i n t e g r a b l e f u n c t i o n s o v e r t h e c i r c l e T = R/T Z w i t h t h e usual Lebesgue measure. Then we d e f i n e a one parameter f a m i l y o f opera t o r s { U ( U ) , -m < u < m } on K by t h e e q u a t i o n (2.1) Since (2.2)

U ( u ) f ( t ) = U(t,t H(t+T) = H(t), U(t+T,

- o)f(t - 0)

the propagator

s + T ) = U(t,s)

for

f E K.

{U(t,s)}

for all

(t,s).

satisfies

K. Yajima

592

Thus U(a) i s w e l l - d e f i n e d and i s a s t r o n g l y continuous u n i t a r y group on I t f o l l o w s by S t o n e ' s theorem t h a t 12.3)

K.

U(o) = e x p f - i a K )

w i t h a unique s e l f a d j o i n t o p e r a t o r . By d i f f e r e n t i a t i n g t h e d e f i n i n g e q u a t i o n (2.1) by a and s e t t i n g a = 0, we see K i s g i v e n by t h e e x p l i c i t e x p r e s s i o n K = - i a / a t + H ( t ) , a t l e a s t f o r m a l l y . The f o l l o w i n g s i m p l e lemma i s t h e c l u e t o e v e r y t h i n g i n t h i s method. be t h e u n i t a r y o p e r a t o r on

Lemma 2.1.

Let

(2.4)

(Usf)(t) = U(t,s)f(t)

Us

and b y t h e p e r i o d i c i t y elsewhere.

I n particular,


0) have t h e l i m i t s i n t h e space o f bounded oper-

and

M(K- A T i 6 ) - ’ V

+

{q

u (K) U ZZI and t h e y a r e compact P o p e r a t o r s i n K f o r 6 1. 0. A c c o r d i n g t o Kato-Kuroda’s a b s t r a c t t h e o r y [S], t h i s i s enough t o conclude t h a t t h e l i m i t s (4.2) e x i s t and t h e y a r e complete:

ators in K

as

6

0

for

A

outside

R(W+) = Kc(K). W+(t) = U ( t , s ) W + ( s ) e x p ( i ( t - s ) H g ) ,

(4.3) Since Kc(K) =

UsKc(l @

desired equation

U(T+T,s))

R(W+(s))

R(W+) = U s R [ l Q W+(s)).

= Us(l B Hc(U(T+sTs))).

By Lemma 2.1

Hence 7 4 . 3 ) i m p l i e s t h e

= Hc[U(T+s,s)).

-

55.

PROOF OF THEOREM 3.

I n t h i s s e c t i o n we a r e concerned w i t h t h e e q u a t i o n (1.7) o n l y and f o r p r o v i n g t h e theorem we assume t h a t t h e p o t e n t i a l V(x) i s v e r y n i c e . Co(IRn) = I f : f

(6). (1)

i s c o n t i n u o u s and

There e x i s t c o n s t a n t s I V ( x ) I 5 C(1 +

(2)

/XI)-’

If(x)I

C > 0 and

Ho

+

0 as 1x1 + - )

and + V

y > 2

with

IIfll = s u p / f ( x ) l .

such t h a t

has no zero energy resonances.

F o r some 0 < a < n / 4 , t h e Co(lRn)-valued f u n c t i o n V(eex + p E ) o f 8 E R can be extended t o a s t r i p C = { z : I I m zl < a3 as an a n a l y t i c funchion ( f o r each f i x e d p E R ) and f o r each f i x e d R E t a i t i s a Co(Rx)-valued C“-function o f P E R.

W e s h a l l prove t h e absence o f t h e e i g e n v a l u e s o f U ( T + s,s) f i r s t . I n f a c t t h e o t h e r statements can be p r o v e d as byproducts. Ry v i r t u e o f emma 2.1, t h i s f o l l o w s f r o m 0 (K) = @, K = - i a / a t - ( 1 / 2 ) ~+ V(x + UE cos w t / w ) . Since we r e g a r d K as a ’ p e r t u r b a t i o n o f - i a / a t + H, we w r i t e t h e p e r t u r b e d o p e r a t o r K as K ( y ) t o make t h e parameter dependence e x p l i c i t . U s i n g t h e F o u r i e r t r a n s f o r m

b

595

Large Time Behaviors of Time-Periodic Quantum Systems m

i n the t-variable,

o(K(0)) = n!-mtn~

i t i s easy t o see t h a t

+ u(H)}

and t h e

eigenvalues appear as embedded eigenvalues. Thus w i t h o u t s e p a r a t i n g o u t t h e eigenvalues from t h e continuous spectrum, t h e r e a r e p r a c t i c a l l y no ways t o t r e a t t h e problem. F o r t h i s reason, we apply t h e so c a l l e d "complex s c a l i n g technique" o f Aguilar-Balslev-Combes 111, [3] t o separate t h e p o i n t from t h e continuous s p e c t r a ( i f t h e p o t e n t i a l i s n o t as n i c e as i n ( B ) , one may s t i l l apply t h e e x t e r i o r comp l e x s c a l i n g o f B. Simon [13] and some o f t h e statements o f Theorem 3 remain v a l i d , see G r a f f i - Y a j i m a [4]): For e E IR, we d e f i n e 2 K(u,e) = - i a / a t - (em2'/2)a + V(eex + VE cos U t / w ) which i s obtained from K(p) by a change o f t h e v a r i a b l e s x + eex. O f course K(u,e) i s u n i t a r i l y equivalent t o K ( v ) f o r any e E lR and t h e s e p a r a t i o n o f u (K(0)) from uess(K(0)) does P n o t t a k e place. However, i f V(x) s a t i s f i e s the c o n d i t i o n ( B ) , K(p,e) can be defined f o r

e

E

:t

e

E

0;

such t h a t

(K(u,e)-

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

Lemma 5.1.

Suppose

V

z)-'

for

0 E (c:

u IR.

+Im z > 0

i s holomorphic i n

We then have

s a t i s f y t h e c o n d i t i o n ( B ) and l e t

H(e) = - ( e - 2 e / 2 ) a + V(eex)

for

e

E Ica.

Then

(1)

aes,(H(e))

(2)

The d i s c r e t e spectrum od(H(e)) = u (H(e)) and op(H(e)) n IR = up(H). P a r g z < 5 2 I m e l and up(H(e)) n ( 0 \ IR) i s contained i n t z : 0 < A E up(H(e)) i s e-stable. i . e . , i f A E U (H(e)) then A E op(H(e')) P f o r e l E ti s u f f i c i e n t l y c l o s e t o e .

(3)

o ( K ( O , e ) ) = oess(K(O,e)) u ess (K(0.e))

= e-2elR+.

u op(K(O,e))

= n=i,,{nu + e - 2 e R + } ,

and

op(K(O,e))

=

+ up(H(e))l.

We r e f e r t h e reader t o t h e c e l e b r a t e d Aguilar-Combes [l]f o r t h e proof o f ( l ) , ( 2 ) . ( 3 ) f o l l o w s from t h e F o u r i e r t r a n s f o r m i n t h e t - v a r i a b l e and an e t i m a t e o f t h e r e s o l v e n t ( H ( B ) - z ) - l o f H ( 0 ) along t h e l i n e s p a r a l l e l t o i s i s o l a t e d from i t s essenThus f o r e E .:0 t h e p o i n t spectrum of K(0,e) t i a l spectrum and we can a p p l y the standard p e r t u r b a t i o n t h e o r y t o l o c a t e t h e p o i n t spectrum o f K(u,e) f o r small LI. What i s i m p o r t a n t i s t h a t i t i s enough t o work w i t h Lemma 5.2.

e (1)

E : 0

t o o b t a i n the i n f o r m a t i o n about 2

V(eex + LIE cos o t / w )

(2)

up(K(p,e))

(3)

up(K(LI,e))

n D1

= op(K(u)),

i s e-stable f o r

is

(-ia/at

e

E

ti.

e

E

ti.

-

u (K(p)): P -2e (e /2)n)-compact.

Statement ( 1 ) can be proved b y a method s i m i l a r t o t h a t f o r Lemma 3.1 and t h e p r o o f s f o r ( 2 ) and ( 3 ) are s i m i l a r t o those f o r Lemma 5 1 ( 2 ) . Lemma 5.2.(1) 6 {nu + e-2e&. i m p l i e s i n p a r t i c u l a r t h a t ueSs(K(u,e)) = n=-m

Me now compute t h e p e r t u r b a t i o n s e r i e s (Kato ['I]) f o r t h e eigenvalues and t h e e i g e n f u n c t i o n s f o r K(,,,e). Since o(K(v,e)) i s o b v i o u s l y i n v a r i a n t under t h e t r a n s l a t i o n by LO and by Lemma 5.1.(3), i t i s enough t o i n v e s t i g a t e t h e unperturbed eigenvalues A E u (H). I f H$ = A$, H(e)$, = W , w i t h P and $,(x) can be regarded as the e i g e n f u n c t i o n of K(O,e) $,(x) = ene"+(eex)

K. Yajima

596 with the eigenvalue the asymptotic s e r i e s vanish:

a2j+l

Im a

that

= 0

= 0

By a s t a n d a r d computation we see t h a t i n A: K(O,e)+, = A$,. A ( w ) = X + alu + a2u2 + a3v3 + t h e odd o r d e r terms

j’s

2j f o r a discrete set o f

With a l i t t l e work ([15]),

X +

with

0




0)

except

2 )dllE(K)~aV/axl)$ll 2 /dKIK=A+W

which i s a p i e c e w i s e a n a l y t i c f u n c t i o n o f w and I m a2 = 0 f o r A + w < 0 and I m a2 < 0 e x c e p t f o r a d i s c r e t e s e t o f w i f X + w > 0 u n l e s s t h e RHS o f (5.1) vanishes i d e n t i c a l l y . T h i s proves t h e absence o f u,(U(T + s,s)).

To prove (1.9) we compute t h e p e r t u r b e d e i g e n f u n c t i o n $e,,(t,x) f i r s t o r d e r : ~ ~ , ~ =( $te )+ uf, ( t ) w i t h IIf, 1I = O(1) as w

,w

,P K

using the e x p l i c i t representation o f

t o the -+

0.

Here

and t h e r e l a t i o n

fe 31-I

D(K(u,e))

c H1( T,H) c C(

(5.2)

W,H)

for

Im

SyP”fe,u(t)llH = 0 ( 1 ) ,

e #

0, Ll

+

we have i n f a c t 0.

F o r I m 8 L 0, t h e e q u a t i o n K ( u , e ) u = 0 generates a p r o p a g a t o r {U(t,s,u,e): t sl. i n H which i s a n a l y t i c i n B E k+ and i s u n i f o r m l y bounded f o r t 2 s ([15], Lemma 3.7). Since K ( W , ~ ) $ ~( t ) = ? ~ ‘ ~ , ~ ( itm) p l i e s ,lJ

u(t,s,v,e)$e,p(0)

= e -iA(u)(t-s)$e

(t),

we see f r o m (5.2) and t h e boundedness o f

+

= ey ’ i A ( ~ ) ( t - s ) t j e U(t,s,u,e) t h a t U(t,s,u,e)$e Taking t h e i n n e r p r o d u c t s w i t h $-, we see

e

(5.3)

(u(t,s,u,e)$e,$-

By t h e i n v a r i a n c e i n

e

e

1=

e

-iA(p) ( t - s )

o(p)

uniformly i n

he,$-)+ e

t 2 s.

O(U).

o f t h e i n n e r p r o d u c t s , (5.3) i m p l i e s ( 1 . 9 ) .

F i n a l l y t o p r o v e Theorem 3 . ( 3 ) , we proceed as above b u t u s i n i n t h i s case t h e degenerated p e r t u r b a t i o n t h e o r y . S i n c e H ( e ) $ . ( x ) = A $ ( x s , t h e f u n c t i o n s -iwt Je j je $1 ,e ( t , x ) = +l,e(~) and $2,e(t,x) = e $2,e(x) . s a t i s f y K(O.e)$j,e= Al$j,e Using t h e degenerated p e r t u r b a t i o n t h e o r y , we see t h a t t h e e i g e n v a l u e ( j = 1,Z). A s p l i t s i n t o two l e v e l s 2 i l ( w ) = A + (u/2w)($l,(aV/ax1)$2) + O(, 1, (5.4) 2 (5.4)‘ A,(u) = A - ( u / 2 , ) ( * , , ( a V / a ~ ~ ) * ~ ) + O(, 1, (1 = h l ) w i t h corresponding e i g e n f u n c t i o n s (5.5)

$l,e(t,x3p)

= ( 1 / ~ ) ( + ~ , ~ (+x e-iwt$2,e(x)) )

+

~(v),

(5.5)’

$2,e(t,x,u)

= ( ~ / Z ) ( * ~ , . ( X )- e - i w t $2,e(x))

+

oh).

Here

u.

O ( P ) s t a n d f o r H-valued c o n t i n u o u s f u n c t i o n s w i t h supremum norm o f o r d e r We p l u g (5.5) and ( 5 . 5 ) ’ i n t o

(5.6)

U ( t , s ,u

,e

je ( s ,x,u

and s o l v e t h e e q u a t i o n (5.6) w i t h $ . ( x ) i n H, we have JO

1

= exp( - i h

(u 1 ( t - s)) 0 je (t,x,u 1,

j = 1 2

Taking the i n n e r products

Large Time Behuviors of Time-Periodic Quantum Systems

Since t h e LHS o f (5.7) and ( 5 . 7 ) ' t h e equations (1.10).

do n o t depend on

8,

591

(5.4) and ( 5 . 4 ) '

imply 0

REFERENCES Aguil,ar, J. and Combes, J. M., A c l a s s o f a n a l y t i c p e r t u r b a t i o n s f o r one body Schrodinger Hamiltonians, Commun. Math. Phys. 22 (1971) 269-279. Amrein, W. 0. and Georgescu, V., On the c h a r a c t e r i z a t i o n o f bound s t a t e and s c a t t e r i n g s t a t e i n quantum mechanics, Helv. Phys. Acta 46 (1973) 635-657. B a l s l e v , E. and Combes, J. M., S p e c t r a l p r o p e r t i e s o f many body Scht-odinger o p e r a t o r s w i t h d i l a t i o n a n a l y t i c i n t e r a c t i o n s , Commun. Math. Phys. 22 (1971) 280-294. G r a f f i , S. and Yajima, K., E x t e r i o r complex s c a l i n g and t h e AC-Stark e f f e c t i n a Coulomb f i e l d , Comnun. Math. Phys. 39 (1953) 277-301. Howland, J. S., S t a t i o n a r y t h e o r y f o r t i m e dependent Hamiltonians, Math. Ann. 207 (1974) 315-335. Howland, J. S., S c a t t e r i n g t h e o r y f o r Hamiltonians p e r i o d i c i n time, Ind. Univ. Math. J . 28 (1979) 471-494. Kato, T.,

P e r t u r b a t i o n t h e o r y f o r l i n e a r o p e r a t o r s (Springer, New York, 1966).

Kato, T. and Kuroda, S. T., Math. 1 (1971) 121-171.

The a b s t r a c t t h e o r y o f s c a t t e r i n g ,

Rocky M t . 3.

Kitada, H. and Yajima, K., A s c a t t e r i n g theory f o r time-dependent l o n g range p o t e n t i a l s , Duke Math. J. 49 (1982) 341-376.

[lo]

Kitada, H. and Yajima, K., Remarks on o u r paper ' A s c a t t e r i n g theory f o r timedependent l o n g range p o t e n t i a l s ' , Duke Math. J. 50 (1983).

[11] Kuroda, S. T., An i n t r o d u c t i o n t o s c a t t e r i n g theory, L e c t u r e notes. No. 51 Aarhus Univ. (1978).

Series

[12] Ruelle, D., A remark on bound s t a t e s i n p o t e n t i a l s c a t t e r i n g theory, Nuovo Cimento 59A( 1969) 655-662. [13] Simon, B., The d e f i n i t i o n o f molecular resonance curves by t h e method of e x t e r i o r complex s c a l i n g , Phy. L e t t . 71A (1979) 211-214. [14] Yajima, K., S c a t t e r i n g t h e o r y f o r Schrodinger equations w i t h p o t e n t i a l s p e r i o d i c i n time, J. Math. SOC. Jpn. 29 (1977) 729-743. [15] Yajima, K., 331 - 352.

Resonances f o r t h e AC-Stark e f f e c t , Comnun. Math. Phys. 87 (1982)

[16] Yajima, K . and Kitada, H., Bound s t a t e s and s c a t t e r i n g s t a t e s f o r time p e r i o d i c Hamiltonians, Annales I H P , Sec. A. 39 (1983) 145-157.

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DIFFERENTIAL EQUATIONS LW. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

599

EIGENVALUES OF THE LAPLACIAN: AN EXTENSION TO HIGHER DIMENSIONS (11) Elsayed M.E. Zayed Mathematics Department Faculty of Science Zagazig University Zagazig, Egypt

In the first two decades of this century, the underlying inverse eigenvalue problem of determining the shape of smooth and convex regions and the unknown attendent boundary conditions from a knowledge of the spectrum of the eigenvalues of the Laplacian was rigorously studied in Gottingen by D. Hilbert, R. Courant and H. Weyl.

1. INTRODUCTION IRn be a bounded region with a piecewise smooth boundary a R Let R sequence of eigenvalues 0 < X I Q X 2 Q h 3 Q ...< Am < . . . + m as m - t m

. Let the (1 -1)

be given, which is counted according to multiplicity of the following eigenvalue problem (A, +h)u = 0 U = O

on a n ,

where A, is the Laplace operator in IR" and u E C2(f2) nC(f2) of R as well as the unknown boundary condition. shape

. Determine the

At the beginning of this century, the principal problem was that of investigating the asymptotic behaviour of the eigenvalues (1.1). If N(X) is the number of these eigenvalues 4 A , then

"XI

-

and

"A)

-

-

VOlLJJTIeRn ’A

as

!(44'

+co

(H. Weyl 1912) ( 1 . 3 )

(9)

n

’A n -

n-1 + O ( 7

2og A) as A+-

(R. Courant 1920).(1.4)

We remark, at the outset that a direct study of these eigenvalues reflects only t h e volume of the region R . In order to obtain further information about the geometry of R , one studies certain functions of the spectrum. The most useful to date comes from the heat equation or the wave equation. Accordingly, let -tAn e denote the heat operator, then we can construct the trace function

E.M.E. Zayed

600

which converges f o r a l l positive

t

.

- i t A n4 Suppose t h a t e i s the wave operator, then an alternative t o (1.5) i s t o study the tempered distribution

i(t)=

- itAn'

)

tr(e

m

=

1 m=l

- i t X ms e

The application of (1.6) t o our problem and t o more general ones can be found i n [€I] and the references given there. See, f o r example, L. Hormander [ l ]

.

For t h i s paper, we s h a l l concentrate on a study of the trace function (1.5). I t is easily seen that the trace function O(t) i s j u s t the Laplace transform m

gtx dN(X) , and then Weyl's formula (1.3) transforms into 3

A. P l e i j e l [41 and M. Kac [21 took up the matter of finding corrections t o (1.7) for plane region s2 with a f i n i t e nmber of holes. Kac put things i n the follow-

'

.

ing amusing language: thinking of R 5 IR as a drum and the eigenvalues (1 1) as i t s fundamental tones, is it possible j u s t by listening with a perfect ear to hear the shape of R ? Weyl's estimate (1.7) when n=2 shows t h a t we can hear the area of R Kac proved t h a t f o r R bounded by a broken l i n e 20 area - length O(t) + the sm over the corners 47Tt 4(4lrt)4

.

- -'

of

lT2-a' + 24m

O(1)

as

,

t+O

where O < c c < 2 ~ being the inside-facing angle a t the corner. Therefore, we can hear the perimeter of s2 By making the broken l i n e aR approximate to a smooth curve, Kac was l e t t o conjecture k+l N area R - length 2 0 O(t) C k t 2 4?rt 4(4lTt)% k=O

.

--

+

+

c

.

for plane region R with smooth boundary a R The constant "C" has special s i g nificance. I f R is smooth and convex, then C = 1 , and i f R , has a f i n i t e C = (1-h)In other nmber of smooth convex holes "h" , then words, i f R is smooth and convex then we can hear i t s area, the length of its boundary and also i t s connectivity. The computation of the remaining terms of (1.9) i s quite formidable, but we do know t h a t they involve integrals of powers of the curvature and i t s derivatives. Furthermore, L. Smith [81 has recently calculated i n addition the coefficients C, ,C, and C, using a new method of constructing a parametrix for e l l i p t i c boundary value problems.

.

So f a r , our discussion has been confined t o the Dirichlet boundary condition;

$

= 0 on suppose we have a Nemann boundary condition has shown t h a t for plane region R with smooth boundary

aR , then P l e i j e l [41 2R

,

60 1

Eigenidues of the Luplacian

-

area n + length as2 O(t) 41Tt 4(4.lTt)$. !!+I +0(t2 as t - 0 . +

N k + l +

I: c i k=O

)

t

2

_ I_

Comparison of this expansion with (1.9) shows that the coefficient of t changed sign.

'

(1 .lo)

has

2. DETECTING SHAPES AND BOUNDARY CONDITIONS

Let us return to the question of Kac; namely, "can one hear the shape of a dnnn?". If we interpret this question in a literal sense, it suggests that the only things we know about the drum are its frequencies of vibrations. To explore this question, we look at the simple one-dimensional problem of asking whether the length of a uniform vibrating string and the unknown boundary conditions can be found from a knowledge of its frequencies of vibrations. Mathematically, we have the problem: Suppose the eigenvalues {Am), m = 1 , 2 , . .. are known exactly for the eigenvalue problem

Determine the &own

length "a" and the d o w n coefficients ai, Bi, i = 1,2

.

This problem is fairly easy to analyse by any of the methods described in the next section and gives the following results

(i)

u,=8,=1,

O(t)

(ii)

O(t)

(iii)

a -- -' (4lTt)L

a1 = B,

a2,

=o ,

(Dirichlet problem)

a2=f3,=0 t + o(t2)

as t+O

,

a2 = B, = 1

1 a -7 (4lTt)4 +

+

o ( t ’ )

(Neumann problem) as t-0

8, + 0 and for any a , , 8,

.

(2.3)

(Mixed problem)

602

E.M.E. Zayed

I t i s of i n t e r e s t t o pose a similar problem for the more general ordinary differ e n t i a l equation - (PY')' q(x) Y = XY Y (2.5) together with the boundary conditions of ( 2 . 1 ) and i n which p(x) and q(x) are a l s o unknown s t i l l an open problem. The problem (2.1) has been investigated by myself i n [ l o ] when a1 =cos a , a p =-sin CY , B, =cos B and B, = - s i n B , where Oeacn , O