0444704434 Mathematical Physics

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MATHEMATICAL PHYSICS


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NORTH-HOLLAND MATHEMATICS STUDIES 152Notas de Matematica (121

Editor: Leopoldo NachbinCentro Brasileiro de Pesquisas FisicasRio de Janeiroand University of Rochester

NORTH-HOLLAND AMSTERDAM NEW YOAK OXFORD TOKYO


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MATHEMATICAL PHYSICSRobert CARROLLUniversity of I l l inoisUrbana, Illinois, U.S.A.

NORTH-HOLLAND -AMSTERDAM NEW YORK OXFORD TOKYO


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ELSEVIER SCIENCE PUBLISHERS B.V.Sara Burgerhartstraat 25p.0. Box 21 1,1000 AE Amsterdam, The NetherlandsDis tribu tors for the U.S.A. and Canada:ELSEVIER SCIENCE PUBLISHING COMPANY, INC.655 Avenue of the Am ericasNew York, N.Y. 10010, U.S.A.First edi t ion: 1988Second impression: 1991

LIBAARY OF CONGRESSLibrary o f Congress Cataloging-In-PublIcatlon Data

Carroll. Robert Way ne, 1930-M a t h e m a t i c s p h y s i c s / Robert Carroll.p . cm. -- (North-Holland mathematics stu die s ; 152) (Notasd e n a t e n i t i c a ; 121)Bibliography: p .

Includes index.1. Mathematical physics. I. Title. 11. Series. 111. S e r i e s .ISBN 0-444-70443-4N o t a s d e n a t e n i t i ca ( R i o d e Ja n ei r o . B r a z i l ) ; no. 121.O A l . N 8 6 no. 121[OC20

510 S--dcl9(530.1'51 88-11195C I P

ISBN: 0 444 70443 4Q ELSEVIER SCIENCE PUBLISHERS B.V.. 1988All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying,recording or otherwise, without the prior written permission of the publisher, ElsevierScience Publishers B.V. / Physical Sciences and Engineering Division, P.0.Box 103, 1000 ACAmsterdam, The Netherlands.Special regu lations fo r readers i n the U.S.A. - This publication has been registered with theCopyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtainedfrom the CCC about conditions under which photocopies of parts of this publication may bemade in the U.S.A. All other copyright questions, including photocop ying outside o f theU.S.A., sho uld be referred to the publishe r.No responsibility is assumed by the publisher for any injury and/or damage to persons orproperty as a matter of products liability, negligence or otherwise, or from any use oroperation of any methods , products, instruction s or ideas contained i n the ma terial herein.Printed in the Netherlands


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V

PREFACE

A g r e a t d e a l o f m at he m at ic s i s u se d i n s t u d y i n g p h y s i cs , as i s we l l known,and i t i s my b e l i e f t h a t a grea t dea l o f p h ys ic s i s u se d i n d e v e l op i n g math-emat ics (more than i s pe rhaps rea l i z ed ) . A t one t ime i t seemed convenient( f o r me a t l e a s t ) t o t h i n k o f an e qu a tio n p h ys ic s = geometry, bu t one m igh ta l s o make a case f o r phys i cs = p r o b a b i l i t y , o r p h y si cs = r e c u r s i o n , e t c .I t a l s o s e e m e d a t t r a c t i v e a t one t i m e ( t o m e) t o t h i n k o f t h e s tu d y o f p hy-s i c s ( an d pe rh ap s a l s o m at he m a ti cs ) i n t h e c o n t e x t o f " r e c o g n i z i n g G o d'shand iwork and p ra i s i ng it". Bu t one can a l s o ask o f course whether God hadany c h o ic e i n c r e a t i o n ( c f . h e re [ P l ] wh ich dea l s w i t h comp lex i ty , en t ropy ,i n fo r ma t i on , re cu rs i ve games, se l f - r ep ro duc ing mach ines , e t c . ) . I t i s a ls operhaps f i t t i n g t o t h i n k o f r e l a t i o n s b etween gods and c i v i l i z a t i o n s ( c f .[Tu l ] ) . F requ en t l y one makes ma themati ca l mode ls o f a ph ys i ca l s i t u a t i o nand i f t h e model i s a ny g ood i t s m a th e m at ic a l s t u d y w i l l l e a d t o in fo rm a -t i o n o f use i n p h ys ic s.p h y s i c a l i n t u i t i o n then so much th e be t t e r ; one w i l l be l o o k i n g t h e n a t p h y-s i c a l l y i n t e r e s t i n g f e a t u r e s and t h e m a th e m at ic a l q u e s t i o n s as ke d and i n v e s -t i g a t e d w i l l b e e n r i c h e d by t h e i n t e r a c t i o n w i t h p h y s i c s . Such an i n p u t cana l s o a r i s e f ro m n um e r ic a l o r co mp ute r s t u d y o f a mathemat ical model ; t h ecompu ta t iona l a l g o r i t h m ic t h i nk i n g toward so l va b l e numer ica l p rob lems canl ea d t o t h e o r e t i c a l i n s i g h t i n t o t h e m o d e l . One i s o f c o u rs e a d v is e d n o t t oask on ly tho se q ue st io ns whose answers can be computed (b u t t h e re may bes e v e r a l s c h o o ls o f t h o u g h t h e r e as w e l l ) .We t r y t o p r o v id e i n ge ne ra l a r i c h s e l e c t i o n o f m a t e r i a l and t o i n d i c a t eas w e l l c u r r e n t a rea s o f i n t e r e s t and d i f f e r e n t p o i n t s o f v ie w.p e c i a l l y i n t e re s t e d i n t h e i n t e r a c t i o n o f id ea s fr om a p p ar e nt ly d i f f e r e n ta r e a s a n d t h e i r s y n t h e s i s i n t h e d is c o ve r y process. I n t h i s d i r e c t i o n wea l s o f e e l t h a t t h e u s e o f l a n g u a g e i s enr i ched by know ledge o f o the r l an -guages.

I f t h i s s t u d y c a n b e d i r e c t e d o r g u i d e d a l s o b y

We a r e e s-

We t r y whenever p o s s i b l e t o e x h i b i t p a t t e r n s and s t r u c t u r e and w i l l


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v i ROBERT CARROLL

em phasize s t r u c t u r e a s p r o v i d i n g a c r a d l e f o r t h e n u t u r i n g o f t h e o r y .w i l l g i v e t o t a l l y e l e me n ta ry i n t r o d u c t i o n s t o many ar e as w i t h c o m pl et e de-t a i l s and w i l l t h e n c o n t i n u e t o d e v e lo p t h e t hemes i n v a r i o u s ways a t v a r i -o us p l a c e s i n t h e b oo k.r e s u l t t h a t may b e needed f o r i l l u m i n a t i o n ( w i t h r e f e r e n c e s ) an d no a po lo gyseems n e ce ss a ry f o r o m i t t i n g t h e p r o o f .t im e s b u t t h e n ec es sa ry d e t a i l s a r e u s u a l l y t h e r e i n t h e t e x t o r i n t h e a p-pend ices ,i n a way we ha ve f o u n d p e r s o n a l l y i n s t r u c t i v e i n l e a r n i n g a n d w h i c h we h av eused e f f e c t i v e l y i n teac h ing . Fo r examp le i n Chapter 2, 83-5, we develop anumber o f s t r u c t u r a l f o rm u l as a n d r e s u l t s , i n w o r k i n g o u t t h e n e c es s ar y t e c h -n i c a l m a ch ine ry a s we go a lo ng , som et im es i n a h e u r i s t i c m an ne r. I n f a c twe do n o t p r o v e t h e a b s t r a c t s p e c t r a l t he ore m i n H i l b e r t space f o r a s e l fa d j o i n t o p e r a t o r as s uc h (i t s s t a t e d h ow ev er i n 2. 2) ,a nd we do n o t g i v ean " a x io m a t i c " t r e a t m e n t o f s p e c t r a l meas ures, p r o j e c t i o n o p e r a t o r s , e t c .However we g i v e t h e n ec e s sa r y f o r m u l a s , d e t a i l s , a nd b ac k gr ou nd t o d e a l w i t ha l l t h e s e i d e a s a nd us e t h e m a t e r i a l i n a way wh i ch amoun ts t o p ro v i ng e .g .t h e s p e c t r a l th eo re m a f t e r a l l . I n f a c t i n t h i s way much more i s done,int h a t c o n n e c t i o ns b et we en v a r i o u s p o i n t s o f v i e w a r e d i s p l a y e d a s wel1,andone sees t h e r o l e o f t h e v a r io u s i n g r e d i e n t s i n p r a c t i c e .needed i s p r ov e d o r s k e t c h ed more o r l e s s c o m p l e t e l y so t h a t t h e d e t a i l scan be f i l l e d i n i n any case. The pr es en ta t i on th us may appear somewhatd i s j o i n t e d a t t i m es b u t we ha ve f o u nd i t p e d a g o g i c a l l y m o r e s a t i s f a c t o r ytha n a theo rem-p roo f fo r ma t and i t h as m o re m ea ni ng p e r s o n a l l y t o p r oc e edi n t h i s way. I n t h i s s p i r i t we h a ve o rg a n i z ed much m a t e r i a l t h r o u g h o u t t h ebook i n a r e ma rk f o r m a t ( i n s t e a d o f t h e or e m - pr o o f) w i t h t h e p r o o f s o f s t a t e -ments i n d i c a t e d o r c a r r i e d o u t i n t h e t e x t , a l o ng w i t h t h e g en er al d i sc u s -s i o n . E x er c is e s a r e t h e n i nt e r s p e r s e d t h r o ug h o u t t h e t e x t .We h av e e x t r a c t e d m a t e r i a l f r o m many s o ur c es w i t h a m p le re fe re n ce s .v a r i o u s i d e a s o f p r o o f o r p r e s e n t a t i o n , w h i ch we hav e f ou nd p a r t i c u l a r l yi l l u m i n a t i n g o r s t i m u l a t i n g , a r e h o p e f u l l y c onveyed t o t h e r e a d e r . I n o r -d er t o i n c l u d e enough m a t e r i a l t o j u s t i f y a t i t l e as p r e t e n t i o u s as "mathe-m a t i c a l p h y s i cs " we h av e r e s o r t e d t o c e r t a i n s pa ce s a v i n g d e vi c es ( t o m i n i -m i z e t h e num ber o f pa ge s an d t h e p r i c e ) .goes o n t h e r e a r e p r o g r e s s i v e l y fe w e r d i s p l a y e d f o r m u la s a nd we u se t h e f o l -l o w in g s u b s t i t u t e . T h e re a re 6 dark symbo ls , *, *, 0 , b y +, ., h i c h a r eused as d i s p l a y " i n d i c a t o r s " i n t h e t e x t i n t h e f o l l o w i n g o rd e r : * , A , 0 ,

We

O c c a s i on a l ly ( b u t r a r e l y ) we w i l l s i m p l y s t a t e a

The p ac e may ap pe ar t o b e f a s t a t

Once bey ond t h e f i r s t c h a p t e r some o f t h e m a t e r i a l i s p r e s en t ed

What i s a c t u a l l y

Thus

Thus i n p a r t i c u l a r a s t h e book


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PREFACE v i i

6 , 6 , .,*, *A, ..., *., A*, A A , ..., A m , ..., .*, ..., . m y ***, **A,...,**my *A*, . . . T h i s tends t o make th e t e x t r a t h e r de nse a t times bu t w i th al i t t l e pa ti e nc e and p r a c t i c e this n o t a t i o n is q u i t e e f f i c i e n t and useful.There is a g r e a t d e a l on f u n c t i o n a l a n a l y s i s i n the book, probably enoughf o r a semester c o u r s e i n f u n c t i o n a l a n a l y s i s , and most d e t a i l s a r e p ro v id ed .In p a r t i c u l a r t h e t h eo r y o f d i s t r i b u t i o n s o r g e n e r a l i z e d f u n c t i o n s i s d e v e l-oped i n several ways.c h a o s , b la c k h o l e s , i nd ex t h e o r y , s u p e r s t r i n g s , e t c . ) we do manage to touchupon many topics of current interest ( e . g . s u p e r c o n d u c t i v it y , g aug e f i e l dtheory , geomet ric qua n t iz a t io n , Feynman in te g r a l s , quantum f i e l d theory , in -ver se p rob lems, so l i to n theory , etc .) , some of i t i n c o n s id e r ab l e d e t a i l( e .g . inverse s c a t t e r i n g a n d s o l i t o n t h e o r y ) .many i n t er m s of o v e r a l l p e r s p e c t i v e ) s e c t i o n s ba se d on t he au thor ' s workand this s h o u l d n o t b e c o n s t r u e d e n t i r e l y a s v a n i t y ( i n p a r t i c u l a r i t a l l o w sus t o develop con s id era ble de t a i l in ar ea s which we know b e s t ) .ia l in e .g . 51 .6 , 1 .11, 2 .6 , 2 . 7 provide s a good model f o r d is cu ss i ng c er -tain areas of research and we have employed i t s u c c e s s f u l l y i n l e c t u r e s ; t h et h e o r y o f n e c e s s ar y i n g r e d i e n t s s uc h a s s p e c t r a l me as ur es e t c . i s developedas one goes along and this seems t o make f o r meaningful pedagogy. In a se ns eone of th e main co n tr ib u ti o n s of th e book may inv olv e Chapter 2 where ar a t h e r f u l l d i s c u s s i o n o f i n v e r s e s c a t t e r i n g a n d e l e m e nt a r y s o l i t o n t h e o r yi s g i v e n . T he re a r e a number o f new r e s u l t s and a l o t o f r e c e n t m a t e r i a l .We have not sp ent much t ime on phys ic al de r i va t i on s or the phi losophy ofphys ics . Th is i s a s e r i o u s gap b u t o ne n o t p o s s i b l e t o b r i d g e unde r t h e im-posed s p a c e l i m i t a t i o n s . I t i s v e r y p r o d u c t i v e t o l i n k mathematical devel-opment w i t h phys ica l r eason ing . For example a nice complex of ideas revol-ves a ro un d c a u s a l i t y , h yp er bo l i c PDE, F o u r i e r t r a n s f o r m s a n d Pal ey-Wieneri d e a s , s c a t t e r i n g , t r i a n g u l a r i t y o f o p e r a t o r s , e t c . S i m i la r l y one h as i de -a s o f cohom ology, g au ge t h e o r y , c u r r e n t s , c h a r g e s , e t c . i n f i e l d t h e o r y . Wf e e l t h e p r e s e n t e r a t o b e r e v o l u t i o n a r y i n s c i e n c e a n d mathematics andhave t r ied to develop enough machinery to help the r e a d e r s t o r m the b a r r i -cades . In the a r e a o f n o n l i n e a r P D E f o r example the methods of fun ct i on alan aly s is have reached a very hybr id ab s t ra c t form,and we have pre fer red t og i v e a p r e s e n t a t io n o f e a r l i e r v e rs i o ns o f the theory ,where there is morec o n t a c t w i t h th e o r ig in a l p rob lems,and mot iva t ion i s more v i s ib le . One cane mph asize h e r e t h a t i t i s w is e t o s t a y r e a so n a bl y c l o s e t o the source o fmathematical problems i n phys ics i n o r d e r t o r e t a i n n o u r i s h m e n t a n d v i t a l i t y .

Although there a r e many om iss ions (not hing abo ut

There are some (c le a r l y too

The mater-


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A b s tr a ct io n f o r i t s e l f i s o f te n a t t r a c t i v e b u t we pursue t h i s o n l y i n t h ei n t e r e s t o f n u t r i e n t s t r u c t u r e . One s ho u ld be f r e e t o use i n t u i t i o n , p i c -tu res , ana logy , e t c . t o develop the app rop r i a te l anguage f o r wha tever phy -s i c s i s un d er c o n s i d e r a t i o n . The r e l i g i o n o f e mb alm in g m ath em atic s i n a x i -o m a ti c sys te ms does n o t p r o v e t o o p r o f i t a b l e i n m a th e ma ti ca l p h y s i c s ( a l -though the reader w i l l d e t e c t v e s ti g es o f a fo rm er f l i r t a t i o n w i t h t he Museo f N. Bo ur ba ki) . The book makes ve ry modest cla im s. We hope i t can be use-f u l as a te x t , even a more or l e s s i n t r o d u c to r y t e x t , w h i l e s e r v i n g as ag u i d e t o some r e se a rc h ar e as o f c u r r e n t i n t e r e s t .s o p h i s t i c a t e d m a t e r i a l w i t h h o p e f u l l y enough r i g o r t o be b e l i e v a b l e andenough h e u r i s t i c c o n t en t t o s t i m u l a t e f u r t h e r s tu dy .The a u th o r wo u ld l i k e t o thank L. N ac hb in f o r a d d i n g t h i s book t o t h e N ota sde Matemat ica ser ies.va ri ou s pe op le who made i t p o s s i b l e t o t r a v e l t o con fe rences and g i ve sem i-n a r t a l k s i n t he p a st 3 yea rs w h i l e t he book was be ing w r i t t en ; we ment ioni n p a r t i c u l a r L . Bragg, J. Dettman, J. Donaldson, A. F a v i n i , T . G i l l , R.G i l b e r t , T. K a i l a t h , E. Magenes, P . McCoy, C. Pucci , L . Raphael, F. Santosa,and W . Zachary. I w ou ld a l s o l i k e t o a ckn ow le dg e r e l e v a n t c o n v e r s a t i o n d u r -i n g t h i s p e r i o d w i t h t h e above p e op le as w e l l as w i t h ( i n p a r t i c u l a r ) A .Aros io , C. Baiocchi , M. Berger, M. Berna rd i , A. Brucks te i n , M. Cheney, D.Colton, J. Cooper, S . D o l z y c k i , C. Foias , J . Go lds te in , D . Isaacson, H .Kaper, T. Kappeler , D. Kaup, M. Kon, I . Lasiecka, P . Lax, T. Mazumdar, J.Neuberger, P. Newton, R. Newton, A. Pazy, H. P o l l a k , J. Rose, T. Seidman,G. Strang, W. Strauss , W . Symes, P. Tondeur, G. Toth, and A . Yag le (w i tha p o lo g ie s f o r o m i s si o ns ) . F i n a l l y t he book i s d e d ic a t ed t o my wi fe Joan .

There i s a l o t o f f a i r l y

We w ou ld a l s o l i k e t o a c kn ow le dg e t h e s u p p o r t o f


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i x

TABLE OF CONTENTS

PREFACECHWCER 1, CCASrSlCAL IDEM A I D PR0BCW

1.2.3,4.5-6.7.8.9,

10.11-

IntraZluttiunSome prel iminary uariat ianal i d e a svarious d i f f e ren t i a l eq wt i an s and th e i r o r ig insLinear second arder PDEF u r t h e r t a p i o i n the calculus af var iat iansSpec t r a l t hea ry f a r a rd inaq d i f f e ren t i a l ope ra t a r s ,transmutatian, and inverse problemsIntruductian t o classical mechanicsIntraduct inn ta qwntwn mechanicsPeak problems i n PDESame nanlinear PDEI l l posed prablems and regular i ra t ian

1.2-3.4.5.6.7,8-9.

10,11.

IntroductionSca t t e r ing t hea rg I (apera tar theory)Scat ter ing theory 11 (3-D)Scat ter ing theorg 111 (a medley of themes)Scat ter ing thearg IV ( spec t r a l methads i n 3-D)Systems and half li n e prahlemsRefatians between pate nt ia l s and spe ct ra l h t aln t roduct ian ta sa l i t an theorySa l i t ans v ia Am sgstemsSalikan thearg (Hamiltanian structure)frame tapics i n integrable systems

CHI\PCER 3- WmE N0NCIIEAR ANAWZS: S6NE GEQHREERZC F0Rl MCI$I1. In t raduct ian2. Manlinear analysis

V

121016253 54957657486

991011081191371471681831922012 1 1

227227


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X ROBERT CARROLL

3.4,5,6,7,8,9,10

Nnnotane nperatarsenpalogical methodsConvex analysisNonlinear semigrnups and mnnatane setsUariak ional inequaI t iesQuuankwn field thenryGauge fields (phys i c s )Gauge f ields (makhematics) and geometricquantiaatian

APPENDIX A.APPENDIX 3.APPENDIX C.REFERENCESINDEX

I N C R 0 D U C Z 1 0 N CO C Z N E A R F U I C C I B N A C A N A C W I SR C E C C E D C @ PI C9 I N FUNCCMNAC ANACwl$INCFER0DUCCLQ)N E0 DIFFEFERENCIAI: GE0mECRy

2382522642722832862 4301311329351377393


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T

CHAPTER 1CLASSI CAL I DEAS AND PROBLEM S

1. ZbllR0DLIC&I0N. C l a s s i c a l l y i t was easy t o lo ok f o r "mean ing" o r perhaps" S tr u c tu r e " i n m ath em at ic al p h ys ic s i n t h e a re as i n v o l v i n g t h e c a l c u l u s o fv a r i a t i o n s (s ee e.g. [ L l ] where l i t e r a r y c i t a t i o n s appear as c h a pt e r i n t r o -d u c t i o n s a nd c f . a l s o [ Ca l; Co l, 2; Gl ;I l; Yl ]) . We s h a l l u se t h i s v a r i a t i o n a lthem e a s a v e h i c l e t o e n t e r t h e s u b j e c t o f m at h em a t ic a l phys i cs . I t w i l ll e ad t o d e r i v a t i o n s o f many i m p o r ta n t d i f f e r e n t i a l e q ua ti on s and p r o v i d e i n -s i g h t i n t o many p h y s ic a l p ro ble ms v i a a m i n i m i z a t i o n ( o r b e t t e r e x tr e m a l )d i r e c t i v e .v a r i o u s i m p o r t a n t m a t he m a ti ca l t e c h n i q u e s whose f u r t h e r s t u d y h as l e d t o t h edevelopmen t o f who le a reas o f ma themati cs as w e l l a s t o f r u i t f u l a p p l ic a t io ni n p h ys ic s .Thus 2,3 and 5 w i l l d e al w i t h v a r i a t i o n a l i d ea s and t h e o r i g i n o f someb a s i c d i f f e r e n t i a l e qu atio ns . 14 discusses some fundamental methods andr e s u l t s o f e x is te n ce , uniqueness, e t c. f o r c l a s s i c a l 1 n e a 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 ( P D E ) . 96 d e a ls w i t h some id e a s o f s p e c t r a l t h e o r y , t r a n s -m u ta t io n , and i n ve r s e t h e or y f o r t y p i c a l o r d i n a r y d i f f e r e n t i a l e q ua t io n s( O D E ) ; t h i s theme i s p i c k ed up a g a in l a t e r i n Ch ap te r 2 and developed exten-s i v e l y . 917 and 8 g i ve i n t r od uc t i o ns t o c l a ss i ca l and quantum mechan ics, p re -s e n t i n g v a ri ou s p o i n t s o f v ie w and n o t a t i o n s t o b e r e f e r r e d t o f r e q u e n t l y i no t h e r p a r t s of the book.t i o n s i n PDE ( v a r i a t i o n a l - o p e r a t i o n a l p ro ble m s) and i n d i c a t e s f i r s t someb a s i c l i n e a r t h e o r y .f o r t he Nav ie r-S tokes equa t i ons , abou t wh i ch fu r t h e r remarks and i nd i ca t i o nso f p r o o f s w i l l be g i v e n l a t e r a t v a r i o u s p l a c e s ( e. g. S s l. 10 , 3.7, e t c . ) . l o g i v e s some f u r t h e r n o n l i n e a r p ro ble ms a nd r e s u l t s . I n p a r t i c u l a r we a r eab le t o make con tac t w i t h some re ce nt work on th e G inzburg-Landau equa t ionsand i n t rodu ce ideas about so l i t o ns , vo r t i ce s , gauge i nva r i an ce , Yang -M i l l s -Higgs .equat ions, e tc . Such themes w i l l a l s o b e p i c k e d u p a g a i n l a t e r .

Furthermore we w i l l be a bl e t o d i s p l a y q u i c k l y a nd n a t u r a l l y

9 nt r odu ces t h e id ea o f weak prob lems and so l u -IThen we develop th e framework and s t a t e some re s u l t s


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2 ROBERT CARROLL

F i n a l l y , i n $11, we g i v e some t y p i c a l r e s u l t s on ill posed problems andT ik ho no v t y p e r e g u l a r i z a t i o n i n o r d e r t o i n d i c a t e an i m p or t an t d i r e c t i o n i nc u r r e n t r e s e a r c h .2. B0mE PRECImlNAR& UARZAt10NAL IDEA$.h i s t o r i c a l i e r e s t.EMAIRPLE 2.1 (ErNECt'S MU)).

L e t us s t a r t w i t h a few p ro ble ms o f

I ma gi ne t h a t we l o o k a t a f i s h as i n d i c a t e d

T hus t h e d i s t a n c e s b a nd d f r o m t h e a i r - w a t e r i n t e r f a c e a r e known a nd a t c =co n s ta n t i s known.b u t t h e v e l o c i t i e s o f l i g h t i n a i r (v,) and i n w a t e r (v,) a r e assumed t o beknown.v e r i f i e d ex pe r i me n t al ly ( i f t h e f i s h i s w i l l i n g - o r d ea d) .t h i s l a w howev er f r om F e r m a t 's p r i n c i p l e o f l e a s t t i m e w hic h s t a t e s t h a t t h et im e r e q u i re d f o r t h e l i g h t t o p ass f r om A t o B s h a l l b e a minimum (one as-sumes he re t h a t l i g h t t r a v e l s i n s t r a i g h t l i n e s ) .a = bTane; L = bSece; Lw = dSec$; t = L / v ; and tw Lw/vw. Thus t h et i m e o f p as sa ge i s T = ta + tw (b /va)Sece t ( d /L w )S e c$ w h i l e k = a t c =bTane t dTanQ. I f o n e s o l v e s t h e s e c o n d e q u a t i o n f o r Q = $ ( e ) a n d i n s e r t si t i n t h e f i r s t e q u a ti o n we w ou ld o b t a i n T = T ( e ) . T h e n s e t t i n g T ' ( e ) 0o ne w ou ld f i n d v a l u e s o f e f o r wh ic h T ( e ) i s ex tr em e (max o r min o r i n f l e c -t i o n ) . The c a l c u l a t i o n c an be s h or te n ed b y d i f f e r e n t i a t i n g b o t h e q u a ti o n sw i t h r e sp ec t t o e and e l i m i n a t i n g d $/de; t h e r e s u l t i s th e n (*) ( e x e r c i s e ) .EXAIIPCE 2.2 ( B a A C H l $ e 0 C W N E PR08tEm).n o u l l i b r o t h e r s a n d c an b e s o l v e d b y v a r i o u s m etho ds ( c f . [Col-3;Yl]) . Thei n g e n io u s t e c h n i q ue d e v e lo p ed b y E u l e r ( w h i c h we p r e s e n t h e r e f o r m a l l y ) c a nb e e x te nd e d a nd g e n e ra l i ze d an d i s a m a z ing l y p r o d u c t i ve . T hus on e im a g in e sa u n i f o rm b a l l w i t h a h o l e i n i t s l i d i n g under g r a v i t y w i t h o u t f r i c t i o n on aw i r e whose s ha pe i s t o b e d e t e r m i n ed so t h a t t h e t i m e o f d e sc e nt f ro m A t oB sha l l be a m in imum. A

The p o s i t i o n o f t h e o r i g i n o f r e f r a c t i o n i s not known

S n e l l ' s l a w s ay s t h a t ( * ) [S in e /va ] = [ Si n$ /v W ] and t h i s i s e a s i l yLe t us deduce

We w r i t e (A) c = dTan$;a a a a

T h i s p ro b le m g o e s b a ck to t h e B e r -

(2 .2)


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VARIATIONAL I D E A S 32Equa t i ng po ten t i a l and k i ne t i c ene rgy one has mgy = (1/2)mv where v = d s / d t

= i ( l + y ) ( s i n c e j = y ' i by t he cha in ru l e ) and hence i(l+y")' = 2gy sow i t h s d e n o t i n g a r c l e n g t h . One wri tes ( e = d/d t , ' % d/dx) 5 = ( i t 2 4

I 2 4(2 .3 ) T = T ( y ) = j o O ( d t / d x ) d x = ~ ~ o [ ( l + y ' 2 ) / 2 g y ] t d x =

We a d m i t i n t o c o m p e t i t i o n as ad m is s ab le f u n c t i o n s y t he c o l l e c t i o n A = t y EC (O,xo), y(0) = 0, y ( x o ) = yo) and ask for y E A su ch t h a t T ( y ) 5 T ( t ) f o ra l l z E A (he re Cn(O .xo) deno tes n t imes con t i nuo us l y d i f f e r e n t i a b le f unc -t i o n s on ( 0 , ~ ~ ) ) . A p r i o r i such a p ro bl em w i t h g e n e ra l F need not have anys o l u t i o n y E A and such a so lu t i on need n o t be un ique. However i n the pre-s en t s i t u a t i o n t h e r e i s a s o l u t i o n w hic h t u rn s o u t t o be t h e a r c o f a c y-c l o i d - n o t r e a l l y a s u r p r i s e t o Newton f o r example. We r e c a l l t h a t a c y -c l o i d i s t h e p a t h t r a c e d b y a p o i n t on t h e c i rc u m f er e n ce o f a c i r c l e whent h e c i r c l e r o l l s on a s t r a i g h t l i n e , and c y c l o id s were o f more i n t e r e s t i nNewton's t ime. Now l e t us f o l l o w E u l e r and assume f i r s t t h a t t h e r e i s am i n im i z in g f u n c t i o n y E A ( n o t n e c e s s a r i ly u n iq u e) an d f i x i t .tv E C (O,xo), v ( 0 ) = 0 = v ( x o ) ) a n d E E R ( R = real numbers) .f i xed and then z = y + 9 E A so t h a t T ( y ) 5 T(z ) .= T ( E ) and then ?(O) 5 ?(E) f o r a n y E (y and v a r e f i x e d ) .hypotheses on F now i n ( 2 . 3 ) s o t ha t one may d i f f e r e n t i a t e u n d e r t h e i n t e -g r a l s i g h w i t h r es p ec t t o E i n t h e f or mu la(2 .4 ) T ( E ) = fo oF(x,y+Eq,y'+Ev')dx( c f . any r e as o n a bl e book on advanced c a l c u l u s f o r d i f f e r e n t i a t i o n u nd er t h ei n t e g r a l s i g n ) . Thus f o r m a l l y

1

L e t 6 =P i c k v E CD

We w r i t e T (z ) = T ( y t E v )Make app rop r i a teA

X'v

rv N1Now by s tanda rd c r i t e r i a f o r ext reme va lues o f C f u n c t i o n s T we want T ' (0)= 0 so (2.5) = 0 f o r E = 0.(2 .6 ) loFyq + F , v ' ] dx = 0Yf o r a l l v E @ ( t he a rgumen t o f F and F We w i l lsom etim es r e f e r t o t h i s p ro c ed u re o f r e d u c i n g T ( y ) t o ? ( E ) and the subse-quen t ana l ys i s as E u l e r ' s t r i c k . Now f o r m a l l y 0 ) $OF , v l d x = -loodl ,dxY X Ys i n c e v vani shes a t 0 and xo. Iy"we must a l l ow th e fun c t i o n y t o h ave a no th e r d e r i v a t i v e i n g e ne ra l i f 0 ) i s

S ince v E 6 was a r b i t r a r y we have0X

i n (2 .6 ) i s ( x ,y , y' )) .Y Y 'U

However since D F = FYI, t Fy l yy ' t Fx y ' Y ' Y


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4 ROBERT CARROLL

t o be used.Reymond ind ica ted be low (wh ich ac tua l l y shows tha t y" does make sense when

9 0) .This p rocedure can be c ircumvented by a techn iq ue o f du Bo is

Thus h e u r i s t i c a l l y l e t us use 0 ) and ( 2 . 6) t o o b t a i nF Y ' Y '(2.7) c 0 [ F y - DxFyl]gdx = 0It w i l l f o l l o w by Lemna 2.3 b e l o w t h a t [ 1 = 0 i n (2 .7 ) so we w i l l have theEu ler equat ions(2.8) D x F y i (X,Y,Y ' = Fy(X.Y,Y'Note t h a t t h i s i s a second o r d e r n o n l in e a r d i f f e r e n t i a l e q ua ti on y " F +F Y l y y J 2 + F[ ( l t y ' )/2gy]", a f t e r a c l e v e r ch an ge o f v a r i a b l e s ( c f . [ C O ~ ] ) , ( 2 .8 ) r e -duces t o t h e e q u at io n f o r a c y c l o i d ( e x e r c i s e ) .examples o f E u l e r e q u at io n s su ch as ( 2 .8 ) ( w i t h e a s i e r c a l c u l a t i o n s ) i n t h et e x t .LEmmA 2.3. Assume fooG(x)v(x)dx = 0 f o r a l l 9 E d where G i s assumed con-tinuous on [O,xo].R o o d :G i s 0 i n some i n t e r v a l I as shown

Y l Y '= F ( i f Fylyl 4 0). F o r t h e b r a c h i s to c h r o n e p ro b le m w i t h F =We w i l l g i ve many imp or ta n t

Y'X, Y

To complete th e pr ese nt d is cu ss ion we need two lemnas.X

Then G z 0.Assume G # 0 so,fo r some x1 E [O,xo],G(xl) > 0 say. B y c o n t i n u i t y

and fo r 0 < n < G(x l) we can f i n d a su b i n t er va l J C I such tha t G (x ) 2 n i nJ . Now choose v E a s i n d i c a t e d so t h a t v = 0 o u t s i d e o f I and v 2 n i n J .(2 .10) 0 = laovGdx = jI vGdx :p G d x L n2 l e n g t h J > 0

Jthen im p l ie s a f a l l a c y i n reason ing somewhere and by con t r ad ic t i on we con-c l u d e t h a t t h e lemma i s t r u e . QEDCEIIIIRA 2.4. Assume say H E Co(O,xo) and fo H(x )n (x )dx = 0 f o r any n E C o ( O ,x o) s a t i s f y i n g $o n (x ) dx = 0. Then H(x) = c .R o o d :/do Hdx o r (*) #o(H - c )dx = 0.l2oHndx = 0 and we can choose rl = H - c by (*).

xO

I f H = c c e r t a i n l y $oHndx = 0 and we choose c now by t he r u l e cxo =It f o l l o w s t h a t f t o ( H - c ) dx

Then i n p a r t i c u l a r #o(H - c)ndx = 0 s i n c e2


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VARIATIONAL IDEAS 5

= 0 and hence H z c.Now t h e t e c h n i q u e o f du B o i s R eymond a l l u d e d t o a f t e r 0 ) go es a s f o l l o w s .I n s t e a d o f p a s s i n g f r o m (2 .6 ) t o 0 ) we w r i t e(2 .11 )H ence f ro m (2 .6 ) o n e o b ta in s 80 FYI - G l v ' d x = 0 f o r 9 E @ ( so 0 = 9' E Cow i t h $ o q ' dx = 0 ) a nd f r o m L e n a 2 .4 i t f o l l o w s t h a t(2.12) FYI - loxyd5 = 0

QED

X xOI Fy(S,y(C),y ' (C))dC = G(x) ; I v F y dx = -['Gcp'dx0 0

T h i s r e p l a c e s E u l e r ' s e q u a t i o n ( 2 . 8 ) .m e d i a t e l y ( fu nd am e nt al t he or em o f c a l c u l u s ) t h a t F E C and ( 2 . 8 ) i s v a l i di n a ny c a se .a nd be lo ng s t o Co ( e x e r c i s e - cf . [Co l ,3 ;Gl ] - s im p l y w o rk f r om A F ' /A x

However f rom (2 .12) i t f o l l o w s i m -1Y 'F u r t h e r i f F # 0 ( L e g e n d r e c o n d i t i o n ) t h e n y " makes senseY ' Y '

Yu s i n g ( 2 . 8 ) ) .T h i s t e c h n i q u e i l l u s t r a t e dt i o n s a n d P D E i n a b e a u t i f uEXAIUPLE 2.5, L e t R c Rn be

n Example 2.2 e x te n ds t o mu1 t i d i m e n s i o n a l s i t u a -way. Cons ide r f o r examp le

an open s e t w i t h a smoo th enough boundary r sot h a t t h e c l a s s i c a l G r ee n' s th eo rem s a p p l y i n t h e f o rm(2 .13 ) -1 Auvdx = J,I DjuDjv dx - unvdowhere Dsuch n r e g u l a r ) .t o m in i m i z in g t h e D i r i c h l e t f un c t i o n a l

(2 .14 ) D(u ) = 1 (D ju) dxf o r u E A .l ea ds t o d i f f i c u l t i e s w hic h we i l l u s t r a t e b elow.1m a l l y and s e t = { c p E C (n), cp = 0 on r l .f u n c t i o n u E A , f i x it, a n d f o r cp E @ f i x e d c o n s i d e r v = u + w E A .D ( v ) E(E) nd t h e s t i p u l a t i o n a ( 0 ) ~ t i ( c ) i a ( d / d E ) i j ( E ) I E = O = 0 l e a d s t o( 6 ) Jnly i e l d s ( + ) J R Aucpdx = 0 f r o m w h ic h Au = 0 i n R by an argument based onLemma 2.3 (e x e rc is e) .s o l v i n g t h e D i r i c h l e t p ro ble m Au = 0 i n R w i t h u = f on r.assumpt ion u E C2 i n p a s si n g f r o m ( 6 ) t o ( + ) t ak e t e s t f u n c t io n s cp E C,"(n)

52 ra /a x . a n d un d en ot es t h e e x t e r i o r n or ma l d e r i v a t i v e (we w i l l c a l lj J 1L e t A = I u E C ( a ) , u = f on r l a n d c o n s i d e r t h e q u e s t i o n

21One w o u ld l i k e t o a ssum e f E Co b u t even t h i s n a t u r a l h y p o t he s i sF i r s t l e t us p roc eed f o r -

Assume D(u ) has a m in im iz ingS e t

2DjuDjcp dx 0 . I f we assume u E C ( n ) t h e n an a p p l i c a t i o n o f ( 2. 1 3)Hence f o r m a l l y , m i n i m i z i n g D (u ) f o r u E A a mo un ts t o

To a v o i d t h e


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6 ROBERT CARROLL

f o r e xa mp le so t h a t ( 4 ) a nd ( 2 .1 3 ) g i v e

(2.15)where ( , ) d en ot es a d i s t r i b u t i o n b r a c k e t ( c f . A pp en dix B ) . V a r i o u s d i s -t r i b u t i o n a l t y p e a rg um e nt s ( Weyl I s l e m a , e t c . ) c an now be in v o ke d t o a s s e r tt h a t Au = 0 i n s2 i n a c la s s i c a l s e n s e ( c f . [ J l ] ) .h a v i o r i s s t i l l a p rob lem .f o l l o w i n g exam ple. Take a u n i t c i r c l e R c R so t h a t

I b u d x = ( u , b ) = ( A u , ~ ) = 0nHowever the boundary be-

To see t h i s l e t us e .9 . e x t r a c t f ro m [C o l] t h e2

L e t f ( e ) = (1 /2 )a0 t 1 anCosne t bnSinne and t r y u = ( 1 / 2 ) f o ( r ) t 1 f n ( r )Cosne t g n ( r )S i n n e a s i n [ C o l] .g i ve s f o r t h e m i n i m iz i ng f u n c t i o n

S t r a i g h t f o r w a r d c a l c u l a t i o n ( e x e r c is e )

(2 .17)w i t h m i n D (v ) = D ( u ) = 1 nn ( an + bn) w h ich o f co urse mus t make sense .r e c a l l he re ( c f . [ Co l;D l] ) t h a t f o r c o n t i n u ou s f on [-7r,n] o ne e x pe c ts t h a t1 (an t bn) < - w h i l e f o r f E C we have 1 n (an t bn) < m.1f E C t h e c a l c u l a t i o n s w i l l make sense but f E C o i s not enough.an exam pl e ( c f . [C o l ] ) t a ke f ( e ) = 1 ( l / n ) Co s( n e ) w hic h i s Co w i t h au n i f o r m l y co n ve r ge n t s e r i e s r e p r e s e n t a t i o n b u t 1 maf = 1 k / k = m.E U m P L E 2.6. A no th er i m p o r t a n t exam ple i n t h e same s p i r i t i n v o l v e s t h ee q u a t i o n f o r s u r f a c e s o f m i ni m al a r e a s p an n in g a g i v e n " fr a me ". Thus l e t nC R w i t h b ou n da ry r and l e t a f u n c t i o n z = f ( x , y ) b e p r e s c r i b e d on r .s i d e r a s u r f a c e u = u ( x , y ) ove r n w i t h u = z = f on r whose a r e a i s t h e n(2 .18)L e t A = { u E C1; u = z = f on r l and a sk f o r a m i n i m i z i n g o b j e c t u E A f o rS - i . e . S ( u ) 2 S (v ) f o r a l l v E A . T h i s p ro bl em , j u s t as Example 2.5, mu stbe s t u d i e d c a r e f u l l y b e f o r e a p r e c i s e t h e o r y c an be p r od u ce d . Many u ne x-p e c t e d t h i n g s c an h ap pe n ( f o r w h i c h we r e f e r e. g. t o [ Co l, 2; Y1 ] i n a g e n e r a ls en se a nd t o r e ma rk s b e l ow f o r some s p e c i f i c p a t h o l o g y ) . I f o n e p r oc e ed s t od e r i v e E u l e r e q u a t io n s f o r ( 2 .1 8 ) a s i n Example 2 . 5 t h e re r e s u l t s ( e x e r c i s e )

u = ( 1 / 2 ) a o + 1 rn[anCosne t bnSinne ]2 2 We

2 2 1 2 2 2 Thus e .g . f o rI ndeed as2

4

2 Con-

2 2 4S ( U ) = J [ I t ux + uy] dAn

2 2Y YY2 . 1 9 ) u X x [ l + u ] - ~ u ~ ~ u ~ u ~u [l + uX] = 0

H ere on e m u st u se i n t e g r a t i o n f o rm u la s o f t h e f o l l o w i n g t y p e ( m )


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V A R I A T I O N A L I D E A S 9

d i f f e r e n t i a l fo rm s b u t f o r now l e t u s make a fe w c l a s s i c a l comments. F i r s to ne t a k es t h e ma g ne ti c i n d u c t i o n B IJH (IJ = p e r m e a b i l i t y m a t r i x ) as t h er e a l m a g n e t ic f i e l d s t r e n g t h and t h e n somehow o ne has t o c ho os e u n i t s ( u n i t shave a lways been an i n p e n e t r a b l e m y s t e r y t o t h e a u t h o r a nd we w i l l say asl i t t l e as p o s s i b l e a bo ut them - s e e [ S 1 2 l f o r d e t a i l s ) . I n p a r t i c u l a r f o rM a x w e l l ' s e q u a ti o n s t h e r e i s a k i n d o f h o r r o r s t o r y c on ne cte d w i t h u n i t s ,f a c t o r s o f 471, e t c . ( s e e [ S l ] ) and we w i l l t h e r e f o r e a t v a r i ou s p l ac e s i nt h i s book ch oo se v a r i o u s e s s e n t i a l l y e q u i v a l e n t fo rm s o f ( 2 . 23 ) w i t h o u t a nya t t e m p t t o c o n n e c t t he m.

- L a

Thus cons ider e .g .A -L A

(2 .24) Cur l E + ( l / c ) a @ a t = 0; D i v B = 0; D i v E = P;AC u r l B - ( i / c ) a S j a t = ( l / c ) ?

2 2 2 2o r i n t r o d u c i ng a f a c t o r o f c o n ly i n ?i ( * ) C u r l E + Bt = 0; D i v B = 0; E t- c Cur l B = - J ; D i v E = P. We w i l l u s u a l l y r e f e r t o (2 .24) or (**) now asM a x w e l l ' s e q u a t i o n s . a r e t h e o b s er va b le s b u t i n s t u d y i n gt h e s e e q u a t i o n s i t i s i m p o r t a n t t o use gauge p o t e n t i a l s ( a bo u t w h ic h a g r e a td e a l w i l l be s a i d l a t e r ) . Thus, a t l e a s t l o c a l l y , one w r i t e s ( * @ ) = C u r l i iand E = - A t - Gradq. W i t h t h i s c h o i c e on e has o f c o ur s e a u t o m a t i c a l l y D i v B= 0 and Cur l (E + At) = 0 = C u r l E + Bt.(use (** h e r e )( 2 . 2 5 )

A2 - AT h e f i e l d s f and

A A

2 2 - L A I t r em ain s t h en o n l y t o s o l v e

22hp - ( l / c ) q t t = - P - Dt[Div A + ( l / c ) v t ] ;A i - ( l / c 2 ) i t t = - ( 1 / c 2 ) i + G r a d [ ( l / c 2 )qt .t D i v 21

.AI t i s e asy t o c heck t h a t i f o n e f i n d s a s o l u t i o n ( Eo ,Bo ) t o (**) v i a g a u g ep o t e n t i a l s (q o,A O) t h e nA

* a( 2 . 2 6 ) = q0 - x t ; A = A + AX0Ag i v e s t h e same ( Eo ,B o) a nd s a t i s f i e s t h e e q u a t i o n s ( 2 .2 5 ) a g a i n ( e x e r c i s e ) .

These t r a n s f o r m a t i o n s ( 2 . 2 6 ) a r e c a l l e d gauge t r a n s f o r m a t i o n s a nd h av e f a rr e a c h i n g i m p or ta n c e i n a g e n er a l f o r m u l a t i o n a s i n C ha pt er 3. I n p a r t i c u l a r2g iven such (qO,AO) one can choose (q,A) so t h a t (*A) D i v A + ( l / c ) v t = 0( e x e r c i s e - p u t ( 2 . 26 ) i n t h i s e qu a t io n t o o b t a i n Ax - ( l / c ) x t t =- [ D i v A + ( l / c )Dtvo] = f w i t h f known).so c a l l e d L o r e n t z g auge an d ( 2 . 2 5 ) d e co u pl es t o g i v e (**) & - ( l / c ) v t t =2 . 2 -- P w i t h AA - ( l / c ) A t t = - ( l / c ) J .

2 a A

2T h e c o n d i t i o n (*A) d e t e r m i n e s t h e2

A We w i l l show l a t e r how t o e xp re s s a l l


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10 ROBERT CARROLL

t h i s v i a v a r i a t i o n a l p r i n c i p l e s a nd s y m p l e c t i c g eo me try . The n o t a t i o n i sp u t i n t o c o n t ra v a r i a nt - c o v a r i a n t f or m and i n t o d i f f e r e n t i a l g eo m et ri c la n -guage i n Chapte r 3 .3. V A R I 0 W D I F F E R E N C I A L E Q11ACZQ)NS A ND C H E I R 0 R I G I W . We c o n t i n ue i n t h es p i r i t o f 2 t o d e r i v e v a r i o u s e q u at io n s a nd i n d i c a t e p ro ble ms .n iq ue s o f s o l u t i o n a r e d e ve lo pe d h e u r i s t i c a l l y a n d v a r io u s m at he m at ic alm ac hin ery ( t o b e e s t a b l is h e d r i g o r o u s l y l a t e r ) w i l l b e m o t i v a t e d .E M N P L E 3.1 . L e t u s use t h e v a r i a t i o n a l t e c h n i q u e o f 52 t o d e r i v e t h e equa-t i o n o f m o t i o n f o r a v i b r a t i n g s t r i n g .P ) i s s t r e t c h e d betw een 0 5 x 5 L w i t h e nd p o in t s f i x e d ( u ( 0 , t ) = u ( L , t ) = 0 )and a f t e r a n i n i t i a l ( s m a l l ) d is p l ac e me n t u(x,O) = f ( x ) t h e s t r i n g i s r e -l e a s e d t o v i b r a t e (we assum e u t (x y O ) = i n i t i a l v e l o c i t y = 0 f o r s i m p l i c i t y ) .The k i n e t i c e n er gy i s T = (1/2)10 putdx ( p = d e n s i t y ) an d f o r s ma ll d i s p l a c em en ts t h e p o t e n t i a l e n er gy U = ( 1 /2 ) J0 u ux dx a p p r o x i m a t e l y ( e x e r c i s e - c f .[ C ol ]) . The l e a s t a c t i o n p r i n c i p l e o f E x e rc is e 2.10 t h e n a sk s t h a t

Some tech-

T hus assum e a s t r i ng ( unde r t ens i on

L 2L 2

2L( 3 . 1 ) ( 1 / 2 ) \ '1 [PU; - uux ]dxd t = A(u)t n 0

s h o ul d be " s t a t i o n a r y " ( o r m in im a l h e r e ) r e l a t i v e t o t h e a dm is sa bl e c l a s s A= { u E Cu ( x ,t , ) p r e s c r i b e d o r d e t e r m i n e d } . A ss um in g P and P c on s t an t f o r s i m p l i c i t y

= 0 ( c = u / p ) . L e t us use t h i sn e o b t a i n s ( e x e r c i s e ) ( * ) ut te q u a t i o n now t o m o t i v a t e a num ber o f m a t h e m a ti c a l t e c h n i q u e s .s e rv e t h a t t h i s i s a h y p e r b o l i c e q u a t i o n ( t h e wave e q u a t io n ) w i t h " ch ar ac -t e r i s t i c " l i n e s x * c t = k ( t o b e d is cu ss ed l a t e r ) and i n f a c t t h e g e n er als o l u t i o n u = F ( x + c t ) t G ( x - c t ) f o r F,G E Ch e re t o g i v e a d ' A l e m b e r t s o l u t i o n (A) u ( x , t ) = ( 1 / 2 ) [ r ( x + c t ) t i ' ( x -c t ) ]where 7 i s t h e o dd p e r i o d i c e x te n s i on o f f ( o f p e r i o d 2L ).

1 i n ( x , t ) ( o r p i e c e w is e s mo oth ); u ( 0 , t ) = u ( L , t ) = 0; u ( x , t o ) a n d2- uxx F i r s t we o b-

2 a r b i t r a r y can be p a r t i c u l a r i z e d

(3 .2 ) -.> xN ote t h a t u i m m e d ia t el y s a t i s f i e s ( * ) w i t h u ( x , O ) = f ( x ) on [O,L] and ut (x ,0) 0 .odd ) and u ( L , t ) = ( 1 / 2 ) [ ? ( L t c t ) t F ( L - c t ) ] = 0 ( b y p e r i o d i c i t y ) .a r r i v e a t t h e same a nsw er b y s e p a r a t i o n o f v a r i a b l e s . We t r y t o b u i l d up as o l u t i on o f ( * ) i n t er ms o f e l e m en ta r y p r o d u c ts u = X ( x ) T ( t ) w h i c h l e a d s t o( a ) X " = - A X and T" = - A c T ( i . e . X " T c 2 = XT " w h ic h can o n l y h o l d f o rX " / X = T"/c T = k ( c o n s t a n t ) - t h a t k = - A

A t t h e end p o i n t s u ( 0 , t ) ( 1 / 2 ) [ f z ( c t ) t f u ( - c t ) ] = 0 ( s i n c e f" i sNow l e t us

2 2 22 2 du e t o b ou nd ar y c o n d i t i o n s i s


22/410

DIFFERENTIAL EQUATIONS 11

l e f t as an e x e r c is e ) . We b u i l d i n t h e bo un da ry c o n d i t i o n s an d t h e c o n d i-t i o n u t( x, O) = 0 v i a X ( 0 ) = X(L) = 0 w i t h T ' ( 0 ) = 0.nn/L w i t h X = Xn = Sin (nnx /L ) and Tn = C o s ( n n c t / L ) . T h e X e qu a t io n i n 0 )w i t h b o u n d a r y c o n d i t i o n s X ( 0 ) = X(L) = 0 i s a S t u r m - L i o u v i l l e p ro b le m w hic hi s s o l v a b l e o n l y f o r t h e e ig en va lu es ( t h e X n a r e c a l l e d e i g e n f u n c t i o n s ) .Now un = X n T n s a t i s f i e s (*) e x c e pt f o r t h e i n i t i a l c o n d i t i o n f ( x ) = u(x,O)a nd t o a c c o m p l i s h t h i s we t r y an i n f i n i t e sum

T h i s le a d s t o X = A n =

(3 .3 ) u ( x , t ) = lmn X n ( x ) T n ( t ) = 1 b n S i n ( n n x / L ) C o s ( n a c t / L )1E v i d e n t l y a f i n i t e sum w i l l g e n e r a l l y n o t g i v e f ( x ) = u(x,O) so we must t r yan i n f i n i t e sum; o n t h e o t h e r hand w h i l e a ny f i n i t e sum s a t i s f i e s (*) p l u su ( 0 , t ) = u ( L , t ) = 0 w i t h u (x,O) = 0 one may have convergen ce problems upond i f f e r e n t i a t i n g t h e i n f i n i t e sum. I n any e ve n t t h e r e i s no hope u n l es s wecan s a t i s f y

t

( 3 . 4 ) f ( x ) = 1 b n S i n ( n n x / L )w hich i s c a l l e d a F o u r i e r s e r i e s ( n o te t h a t t h e s e r i e s i s p e r i o d ic o f p e r i od2L a nd i s a n o dd f u n c t i o n so i t r e p r e s e n t s ? ( x) o n ( - - , m ) ) .i s v a l i d ( i n some s en se ) an d th en , f o r m a l l y , upon n o t i n g t h a t ( e x e r c i s e )Suppose (3.4)

( L / 2 ) f o r m = n( 3 . 5 ) f i n ( n n x / L ) S i n ( m n x / L ) d x = { for +0

i t f o l l o w s t h a t ( m u l t i p l y i n g ( 3 . 4) b y S i n ( m x / L ) a nd i n t e g r a t i n g t er m wi se )( 3 .6 ) bm = ( z / L ) J f ( x ) S i n ( m n x / L ) d xWe remark t h a t f o r ? a s i n d i c a t e d i n ( 3 . 2 ) b = O ( l / m ) i s e xpe cte d b u t f o rN rnf o n l y PC, bm = O ( l / m ) wou ld be norma l .f C os AS in B) ( 3 . 3 ) a nd ( 3 . 6 ) l e a d t o

L0 2

F u r t h e r ( r e c a l l S i n ( A + B ) = SinACosB

(3 .7 )w h ic h o f c o u rs e r e p r e s e n t s ( 1 / 2 ) [ F ( x + c t ) + ? ( x - c t ) ] an d on e a r r i v e s f o r m a l l ya t ( A ) a g a i n . G e n e r a l l y t h e m e t h o d o f s e p a ra t io n o f v a r i a b l e s w i l l a p p l y t omany p ro b le m s w he re o ne does n o t k now a p r i o r i a s o l u t i o n l i k e (A) so we w i l lw a nt t o e xa min e t h e m eth od an d l o o k a t t h e m a t h e m a t i c a l q u e s t i o n s i t posesf o r v a l i d i t y .a r e a g en e ra l c on se qu en ce o f t h e f a c t t h a t Xn s a t i s f i e s a S t u r m - L i o u v i l l ep r o b l e m a nd t h u s t h e ex p a n si o n ( 3 . 4 ) i s a g e n e r a l q u e s t i o n , n am ely , s t u d y

u ( x , t ) = 1 b n ( l / 2 ) [ S i n n n ( x + c t ) / L + S i n n n ( x - c t ) / L ]

I n pa s s in g we m en t io n t h a t t h e o r t h o g o n a l i t y c o n d i t i o n s ( 3 .5 )


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12 ROBERT CARROLL

t h e e xp an s io n o f ( s u i t a b l e ) f u n c t i o n s f i n an i n f i n i t e s e r ie s o f o r th og on a le i g e n fu n c t i o n s . T h is i s b e s t t r e a t e d i n t h e c o n t e xt o f H i l b e r t s paces ( o rr i g g e d H i l b e r t s p ac es ) a nd h e lp s e x p l a i n t h e n eed f o r H i l b e r t s pace t e c h -n iq u e s i n m a th e m a t i ca l p h ys i cs .REmARK 3.2, L e t us u se Exam ple 3.1 e ve n f u r t h e r t o m o t i v a t e c e r t a i n m et ho dsi n v o l v i n g g e n e ra l i z ed f u n c t i o n s o f d i s t r i b u t i o n s ( c f . A pp en di x B). F i r s t wed e f i n e t h e F o u r ie r t r a ns f o r m ( f o r n i c e f u n c t i o n s f ) ( + ) F f ( h ) = f ( h ) =/I ( x ) e x p ( i h x ) d x . The F o u r i e r t r a n s f o r m c an be e x te n de d t o a l a r g e c l as so f d i s t r i b u t i o n s f E 3 f o r e xa m ple (a nd be yo nd ) a nd th e i n ve rs i o n f o rm u lai s g i ve n by ( c f . Appe ndix B f o r a l l d e t a i l s h e re ) ( m ) f ( x ) = F-'fA(x) =(1/21r)/ : ?(h)exp(- ihx)dh.g(x-S)dg = 1: f (x -S )g (S )d S f o r s u i t a b l e f , g and t h e n F ( f * g ) = FfFg.t h e 6 f u n c t i o n ( wh ic h i s n o t a f u n c t i o n a t a l l b u t a measure) i s d e f i n e d byi t s a c t i o n on t e s t f u nc t io n s ~p E C E (C: = Cm f u n c t i o n s w i t h c o mpa ct s u p p o r t )by t h e r u l e ( 6 , ~ ) ~ ( 0 ) . Now t h i n k o f (*) (u = c u x x ) a s a n e q u a t i o n on(-a,-) i n x, t 1. 0, w i t h i n i t i a l d a t a u(x,O) = f ( x ) = F ( x ) o n ( - m , m ) andut(x,O) = 0 ( t h i s i s c a l l e d a Cauchy p r ob le m ).s i g h t has a F o u r i e r t r a n s f o r m i n x so t h a t F u(x ,y ) = i ? (h , t ) .

2" 2 2AX /Ii s f i e s ( n o t e F f " = ( - i x ) f ) (*A) Ctt - c A u = 0; ~ ( x , o ) = f ( h ) . Conse-q u e n t l y ( s i n c e ut( h, O) = 0 ) (*.) $ ( h , t ) = ?(h )Coshct = F R u x ( t ) where p x ( t )i s t h e m ea su re d e f i n e d b y ( * 6 ) p & t ) = ( 1 / 2 ) [ 6 ( x - c t ) + G (x+ c t ) ] . To seet h i s , s i m p l y com pute f o r ex am ple ( t r e a t i n g t h e 6 s y m b o l i c a l l y as a f u n c t i o nf o r p u r p o s e s o f i n t e g r a t i o n - w h ic h i s e x p l a i n e d i n A pp en di x B )

A

-O n e d e f i n e s a c o n v o l u t i o n ( f * g ) ( x ) = LI ( S )

A l s o

2tt,

S u p p o s e t h a t e v e r y t h i n g i ns a t -hen

A

mA ( x - c t ) = eihx6(x-ct)dx = (e ihx,6(x-ct ) ) = e i h c t

m(3.8)

I t f o l l o w s t h a t F p x ( t ) = Coshct and hence by t h e c o n v o l u t i o n t h e o r e n(3.9) U ( x , t ) = F * p x ( t ) = ( 1 / 2 ) [ F ( x + c t ) -t F ( x - c ~ ) ]w h ic h a gr ee s w i t h ( 3 . 7 ) o r (A). To check ( 3 . 9 ) compute f o r examp le F *G ( x - c t ) = /I G(S-c t )F (x-c )dS = ( s ( c - c t ) , F ( x - c ) ) = F ( x - c t ) . Thus we have de-v e lo p ed a n o t h e r way t o s o l v e (*) ( t h e F o u r i e r m et ho d) wh i ch i n v o l v e s t h e u seo f F o u r i e r t r an s f or m s a nd g e n e r a l i z e d f u n c t i o n s .a b l e o f g r e a t g e n e r a l i z a t i o n .L e t us c o n t i n u e o u r p r o gr am o f i n t r o d u c i n g v a r i o u s p ro bl em s a nd me th od s b yd e s c r i b i n g some c l a s s i c a l d i f f e r e n t i a l eq u at io n s o f e v o l u t i o n t yp e . H ereo ne t h i n k s o f some p h y s i c a l s ys tem e v o l v i n g i n t i m e fr om a g i v e n i n i t i a l

T h i s m e th od a l s o i s c ap -


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DIFFERENTIAL EQUATIONS 13

s t a t e .E xa mp le 2.11, a r e o f t h i s t y p e as a r e e .g . t h e p a r t i c l e e q u a t i o n s o f E xample2.10 ( i n i t i a l s t a t e s m ust be p r e s c r ib e d i n a s a t i s f a c t o r y m an ne r) .c o n s i d e r a g a i n t h e wave e q u a t i o n .REmARK 3.3. A d i f f e r e n t i a l p ro b l em i s s a i d t o be w e l l posed i f t h e s o l u t i o ndepends ( i n some m a nn er ) c o n t i n u o u s l y o n t h e b o un d ar y c o n d i t i o n s o r d a t a -w hi ch f o r a p u r e e v o l u t i o n p ro bl em means t h e i n i t i a l c o n d i t i o n s . Thus l e tu(x,O) = F(x ) and u t (x ,O) = G ( x) i n t h e C auchy p r o b l e m f o r ( * ) ( F n ee d n o tbe 7 anymore).

The C auchy p r ob l em f o r t h e wave e qu a t i o n , or t h e f i e l d eq ua tio ns o f

F i r s t

The ( u n iq u e ) s o l u t i o n , c a l l e d d ' A le m b e r t s o l u t i o n , i sx + t

( 3 . 1 0 ) u ( x , t ) = ( l / Z ) [ F ( x + c t ) + F ( x - c t ) ] + ( 1 / 2 ) ( G ( c ) d ch x - tThe p i c t u r e b e lo w show s how t h e s o l u t i o n a t ( x , t ) d ep en ds o n t h e d a t a a l o n g

The l i n e s x t c t = k a r e c a l l e d c h a r a c t e r i s t i c s h e r e a nd d e l i m i t t h e dom ainso f d ep en de nc e a nd o f i n f l u e n c e . L e t a c om p ac t s e t K b e g i v e n i n t h e u pp erh a l f p l a n e a nd t h e c om pa ct s e t ;on t h e x a x i s b e t h e r e b y d e t e r m i n e d a sshown.m easurem ent e r r o r e t c . ) b u t su pp ose a t l e a s t t h a t f o r any s u ch K we c a n f i n dF*,G* so t h a t s u p lF * ( x) - F ( x ) l 5 E and sup l G* ( x ) - G ( x ) l 2 E ( s u p o v e r K ) .Then f r o m ( 3 . 1 0 ) - ( 3 . 1 1 ) w i t h u * % (F*,G*) ( *= ) s u p [ u * ( x , t ) - u ( x , t ) l 5 E +( 1 / 2 ) 2 c T ~ = E ( l + c T ) ( su p f o r ( x , t ) E K ) . Thus u n i f o r m c o n t r o l o f t h e d a t aon compac t se ts on (--,-I i n s u r e s u n i f o r m c o n t r o l o f t h e s o l u t i o n o n com pacts e t s i n t h e up pe r h a l f p la n e.s u i t a b l y g e n e r a l i z e d - c an be u sed t o c h a r a c t e r i z e h y p e r b o l i c o p e r a t o r s( c f . [ C l ;Ga;]).EXACAmPLE 3 - 4 - We c o n s i d e r n e x t t h e s i m p l e s t p a r a b o l i c e q ua t i on , n am ely, t h eh e a t e q u a t i o n (A*) ut = u x x ( as su min g t h e ( c o n s t a n t ) t he rm a l c o n d u c t i v i t y i sn o r m a l iz e d b y a cha ng e o f v a r i a b l e s t o b e 1 ) .am ple t h e e v o l u t i o n o f t e m pe r at u re u i n a b a r s e t be tw een x = 0 and x = Lw i t h U ( 0, t) = u ( L , t ) = 0 and u(x ,O) = f ( x ) . A s o l u t i o n by s ep a r a ti o n o f 2v a r i a b l e s i s p o s s ib l e , f o l l o w i n g E x am ple 3 .1 , a nd o ne a r r i v e s a t X " = - A X2and T ' = - A T w i t h X (0 ) = X(L) = 0.

Suppose the da ta F and G a r e i m p e r f e c t l y known ( as i s n or ma l w i t hL v

As a m at t er o f f a c t t h i s k i n d o f p r o p e r t y -

T h i s c o u l d d e s c r ib e f o r e x-

C onsequen t l y X n = S i n ( n n x / L ) a s b e f o r e


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14 ROBERT CARROLL2( A n = nn/ L) and Tn = e x p ( - A n t ) w i t h

(3.12)2 2 2

u ( x , t ) = 1 bne'(n t/L ) S i n ( n a x / L ) ; f ( x ) = 1 b n S i n ( n n x / L )The same F o u r i e r t h e o r y a p p l i e s t o t h e e xp a ns io n o f f ( c f . ( 3 . 4 ) - ( 3 . 6 ) ) b u tf o r t > 0 t h e be ha v i o r o f t h e i n f i n i t e s e r i e s f o r u i n ( 3. 1 2) i s v a s t l y d i f -f e r e n t f r o m t h a t o f t h e s e r i e s , i n ( 3 . 3 ) .e ve n ha ve t r o u b l e c o n v e r g i n g ( i f f i s s ay o n l y c o n t i n u o u s w i t h n o " c om pa ta -b i l i t y " c o n d i t io n s a t 0 a n d L ) .u n i f o r m co nv erg en ce i s a ss u re d i n ( 3 .3 ) b u t a f t e r t w o te rm w is e d e r i v a t i v e sone expec t s t r oub l e . On t h e o t h e r hand i n (3 .1 2) f o r t > 0 one has a con-v e r g e n c e f a c t o r e x p [ - n 2 n 2 t / L 3 w hi ch e a t s up p o ly n o m i a l s i n n f o r b r e a k f a s t .One c an d i f f e r e n t i a t e t er m wi se i n x o r t a r b i t r a r i l y o f t e n i n ( 3 . 1 2 ) s i n c et h i s o n l y b r i n g s down p o l y n o m ia l s i n n .e l l i p t i c f o r t > 0 ( c f . [ C l ; M i l ; T r l ] ) a nd u Cm i n ( x , t ) a s t h e a bo ve a r g u -ment w i l l show.REmARK 3.5 .

Indeed a t f i r s t s i g h t ( 3 . 3 ) may2For f Q ? a s i n (3 .2 ) w i t h bn = O ( l / n )

2

I n f a c t t h e h e a t eq u a t i o n i s h yp o-

C o ns id e r a g a i n t h e h e a t e q u a t i o n (A*) wi th Cauchy da ta u (x ,O) =f ( x ) ( - m < x < mca l l ed a C auchya n d u ( L , t ) = h ( tv a l u e - boundaryI n t h e p r es e nt s

where e.g. f i s c o n t i n u o u s an d b ounde d. T h i s i s a g a i ni n i t i a l v a l u e ) p ro bl em a nd when c o n d i t i o n s u ( 0 , t ) = g ( t )a r e a l s o p r e s cr i be d t h e p ro ble m i s c a l l e d a m ix ed i n i t i a lv a l u e p r o b le m o r a C auchy p r o b le m w i t h b o u n da r y c o n d i t i o n s .t u a t i o n t h e s o l u t i o n can be w r i t t e n v i a t h e h ea t k e rn e l

Thus c o n s i d e r f o r s u i t a b l e f (e.g. f c o nt in uo u s a nd I f ( x ) l 5 c he re bu t somee x p o n e n t i a l g r o w t h o f f i s a l s o p e rm i t t ed ) (Am) u ( x , t ) = iz K(x-S , t ) f (S )dSOne che cks e a s i l y t h a t f o r t > O,Kxx = Kt ( e x e r c i s e ) an d by g ro w th s t i p u l a -t i o n s i t i s p er mi ss ab le t o d i f f e r e n t i a t e u nder t h e in t e g r a l s i g n i n ( A m ) t oo b t a i n u x x = u t. The p o i n t o f t h i s r em ar k i s t o show now t h a t as t -+ 0,K(x-S , t ) -, ( x - s ) a n d t h u s u ( x , t ) -f ( 6 ( x - S ) , f ( S ) ) = f ( x ) . We do t h i s byc l a s s i c a l m ethods ( c f . [ J l ] ) i n n o t i n g f i r s t t h a t K (x -S ,t ) > 0 w i t hr, K(x-S , t )dS = 1 f o r t > 0 ( e x e r c i s e ) and f o r any 6 > 0

K(x-S , t )dS = 0m(3 .14)u n i f o r m l y f o r x E R. To check (3.14 ) (and (A&) a s w e l l ) s e t x-6 = (4t) 'n sot h a t J K (x -S ,t )d S f o r I x - L I > 6 equa l s n-'/ e x p [ - l n ( l d r l f o r In( > 6 / 4 4 t .The c o n c l u s i o n i s i m m ed ia t e. Now, g i v e n E > 0 t h e r e e x i s t s 6 s u c h t h a t

2


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DIFFERENTIAL EQUATIONS 1 5

( f ( x ) - f ( S ) ( 5 E f o r ( x - F ( 5 2 6 . L e t M = su p ( f ( x ) ( a nd w r i t e ( I x - y I I )(3 .15)

m

l u ( x , t ) - f ( y ) l = I / K ( x - S , t ) [ f ( S ) - f ( y ) l d S ( 5mK ( x - S , t ) I f ( S ) - f ( Y ) I d s + IK ( X - { , t ) l f ( S ) - f ( y ) l d S 5 2M/

J K( X-, , )2MdS 5 2MI K ( X - S 9 t dS tI X - S I I 6 1 x 4 126 I x-s 126Is - Y I < 2 6j K(X-E,t)dS + E 1 K(x-S,t)dC + 0I x -5 126T h i s k i n d o f s i t u a t i o n ( an d ar gu me nt ) o c c u r s r e p e a t e d l y i n s o l v i n g PDE.

Some k i n d o f k e r n e l t e n d s t o a 6 f u n c t i o n as a p a ra m e te r ( o r v a r i a b l e ) goest o a g i v e n v a l u e ( s e e e. g. Exa mple 3 . 7 w he re t h e P o is s o n i n t e g r a l f o r m u l ai s d i sc u ss e d ).EWAmPLE 3.6. T h e S c h r o d i n g e r e q u a t i o n i s b e t t e r d i s cu ss ed l a t e r i n a quan-tum mechan ica l f ramework b u t we w i l l r e c o r d h e r e a p r e l i m i n a r y v e r s i o n ( c f .[ S c l ; T r l ] ) . Thus f o r a q u a n t i t y JI ( w a v e f u n c t i o n ) s u c h t h a t l J I l r e p r e s e n t sa p r o b a b i l i t y d e n s i t y on e has ( h = h /2n w he r e h i s Planck; co ns ta n t ) (.+)( l / i ) J I t = (h/2m)JIxx - q$ w he re q r e p r e s e n t s a p o t e n t i a l . One c an w r i t e t h i s2 2a l s o a s (Am) ihJIt = - (h /2m)JIxx + q$ where energy E Q p /2m t q, E Q i 7 i D t ,p % -ibDx and t h e r e i s a f o r c e F = - g r a d q ( t h u s ( A m ) says iW t = W whereH Q H a m i l t o n i a n o p e r a t o r w i t h q z p o t e n t i a l e n e rg y . T h i s e q u a t i o n has somed e f e c t s ( i t s n o t L o re n tz i n v a r i a n t f o r example i n t h e f o r m ( l / i ) J I t =(R /2m)A$ i n R X R ) b u t i t s t i l l has been ve ry u s e f u l .fo rms (A+) i n t ( D t + - i w ) we o b t a i n (**) jXxq$ = - A 2 $ ($ = rI J I (x , t )e x p ( i w t ) d t ; A 2 = 2 m w / h ) .quantum s c a t t e r i n g t h e o r y f o r e xa mple ( c f . [C2,3;Chl;Shl ;R1] and Ch apt er 2 ) .The a b s t r a c t t h e o r y o f (Am) i n t h e f o rm 0 . ) ihJIt = HJI; J I ( 0 ) = J Io w i t h J I ( t )t a k i n g v a lu e s i n a H i l b e r t spa ce f o r exa mp le has t h e s o l u t i o n (a*) $(t)e x p ( -i t H /f i )J I a nd o ne s t u d i e s t h e o p e r a t o r e x p ( - i t H / h ) i n C h ap te r 2.EXAmPLE 3.7.e t c . and i s t h e p r o t o t y p i c a l e l l i p t i c o pe r a to r . L e t us examine b r i e f l y t h ep ro b le m a r i s i n g f r o m E x am ple 2. 5 f r o m t h e p o i n t o f v i e w o f PDE.has a , say bounded r e g i on ( = open s e t t o w h i ch G r een ' s theor em s a pp l y ) R CRn and a p r e s c r i b e d c o n t i n u o u s f u n c t i o n f g i v e n o n t h e b o u n d a r y r o f R.One poses t h e D i r i c h l e t pr ob le m o f f i n d i n g u E C ( 0 ) n C o ($ ) s uc h t h a t A u =0 i n R and u = f on r . F o r " w i l d " s e t s R t h i s m ay be t h e w rong p r ob l em andon e i s f o rc e d t o l o o k e.g . a t L s o l u t i o n s t a k i n g b o u n d a r y v a l u e s w e a k l y o ro ne im po se s s t r o n g e r c o n d i t i o n s on f a n d a s k s sa y f o r u E C2+a(R) (Ho lderc o n t i o u o u s s ec on d d e r i v a t i v e s ) . We w i l l n o t d e a l w i t h t h e p a t ho l o gy h er e

2

3 I f o ne F o u r i e r t r a n s -

T h i s e q u a t i o n h as b ee n s t u d i e d e x t e n s i v e l y i n

0The Lap lac e op e ra to r has come up a l r e ad y i n Examples 2.5, 2.11,

Thus one

2

2


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1 6 ROBERT CARROLL

and r e f e r e.g. t o [ G i l ; L i ly 2 ;M a l] . F or r e l a t i v e l y n i c e n a s above t h e r e i sa s a t i s f a c t o r y t h e o r y (some o f wh ic h i s d e s c ri b e d l a t e r ) and we s i m p ly g i v ehere an example i n t h e pl an e i n or de r t o e x h i b i t an e x p l i c i t s o l u t io n v i at h e P o is s on i n t e g r a l f o rm u l a.t e r e d a t t h e o r i g i n a nd w r i t e t h e L a pl ac e o p e r at o r A i n p o l a r c o or d in a te sso t h a t (06) Dr + ( l / r ) D r + ( l / r )De]u = 0 w i t h u = f ( 8 ) on t h e c i r c l e r =a. We s e p a r a t e v a r i a b l e s a g a i n a nd t r y b u i l d i n g b l oc k s R ( r) @ ( e) s a t i s f y i n g(a+) [r R " + r R ' ] / R = -(8 /S) X .Sinne ( A n = n ) an d R= rn d et er mi ne s t h e n t h e s o l u t i o n s o f t h e R e q u a t i o nf i n i t e a t r 0. Thus we consider (om) u ( r , e ) = (1 /2 )ao + 1 rn[anCos neb S in n e ] w h ere t h e an a n d bn a r e d e te rm in e d v i a (6*) ( e ) = a0/2 + 1" a"[a Cos ne + b nS in n e] . One c h ec k s e a s i l y ( e x e r c i s e ) t h a t (U) an = ( l / n a1 f (e)Cos nede; bn = (l/nan)1:" f ( e ) S i n nede s o t h a t(3 .16)

Thus take n t o be a c i r c l e o f r a d i u s a cen-2 2 2

2 2 By p e r i o d i c i t y 8 has the form Cosne o

n ' n

2n 2au ( r , e ) = ( 1 / 2 n ) \ f ( e ) d e + ( l / n ) l y ( r / a I n [ f ( $ )

where P,(e-$) = ly ( r /a ) 'Cos n (e -$ ) .t h a t 1 zn = l / ( l - z ) f o r 121 < 1 so 1 zn = z / ( l - z ) .

Now t o sum th e s e r ie s i n Po w e r e c a l lT h e n , w r i t i n g 0 - 4 = 5

(3 .17 ) 1 ( r /a )nCosn5 = (1/2)17 (r/a)'[ein5 + I =( r /a ) [Cosc - ( r / a )1 / [ 1 - 2 ( r / a ) ~ o s c + ( r / a l 2 1

C om bining t h i s w i t h (3 .1 6 ) we h ave (e x e rc i se )(3 .1 8 ) u ( r , e ) = P ( r ,e ,$ ) f ( $ )d $ ; P = ( a - r ) / [a -2arCo s(e-$) + r 1211T h i s i s t h e P ois so n i n t e g r a l f o r m ul a.0 and as r -+ a an argu me nt s i m i l a r t o t h a t i n Exa mple 3 .5 shows t h a t P ( r , e ,4 ) + t i ( & ) ( e x e r c i r e - c f . [ J l ] ) .4 . L I N E A R 9 E C 0 N D 0 R D E R P D E . I n t h i s s e c t i o n we w i l l g i v e a s k e t c h o f someb a s i c i d ea s and the or em s i n l i n e a r PDE. Var ious i mp or t an t theo rems w i l l bep r ov e d by c l a s s i c a l m et ho ds ; l a t e r i n Ch a pt e r 3 we w i l l u s e f u n c t i o n a l a n a l -y t i c m ethods more h e a v i l y i n d e a l i n g w i t h a b s t r a c t PDE. We w i l l c o n c e n t r a t eo n se co n d o rd e r e q u a t i o n s s i n ce th e y a re o f m ost f r e q u e n t occu ra nce i n m ath-e m a ti c a l p h y s i c s ( t h e m a in e x c e p t i o n b e in g f o u r t h o r d e r e q u at i on s i n t h et h e o ry o f e l a s t i c i t y and some hi g h e r o r d e r n o n l i n e a r eq u at io ns i n s o l i t o nt h e o r y ) . P r i o r t o say 19 45 t h i s was r e a l l y a l l t h a t was known an d t h e t h e o r y

2 2 2 2jo2=For r < a on e c hec ks e a s i l y t h a t AP =


28/410

PARTIAL DIFFERENTIAL EQUATIONS 17

c o n s i s te d m a i n l y i n a l a r g e bag o f t r i c k s . A ro un d 1950, w i t h t h e a d ve nt o ft h e t h e o r y o f d i s t r i b u t i o n s , and th u s w i t h a la ng ua ge i n w h ic h t o t a l k a bo utd i f f e r e n t i a t i o n and l i n e a r PDE,not o n l y w er e g e n er a l a nd f a r r e a c h i n g t h eo -r i e s d ev elo pe d b u t h i g h e r or d e r e qu a ti on s c o u l d be n a t u r a l l y t r e a t e d ( c f .he r e e .g . [ C l ;Col ; F r l ;G2;H1 ;M11; J1 ;Pa l ;Sa l ;T r l 1) T h i s a 1 1 o c c ur e d i n c o n-j u n c t i o n w i t h t h e d ev elo pm en t o f t h e t h e o r y o f l o c a l l y c on ve x t o p o l o g i c a lv e c t o r s pa ce s ( TVS) i n f u n c t i o n a l a n a l y s i s an d t h e r e was a b e a u t i f u l a ndp r o f ou n d i n t e r a c t i o n (some o f t h e e n s u i ng ma c hi ne ry w i l l b e u se d a t t i m e sl a t e r i n t h i s b oo k) . S ub seq ue nt d ev el op me nt w i t h s t r o n g g e o m e t r i c a l i n t e r -a c t i o n i n v o l v e d p s e u d o d i f f e r e n t ia l o p er a to r s, o s c i l l a t o r y i n t e g r a l s andF o u r i e r i n t e g r a l o p e r a t o r s , m i c r o l o c a l i z a t i o n , e t c . However h e r e i n t h i ssec t i on w e w i l l p roc ee d c l a s s i c a l l y a nd t h i n k o f PDE more i n t h e c o n te x t o fa dva nce d c a l c u l u s ( t h e n a t u r a l h a b i t a t o f co u r s e ) . T he re a r e many e x c e l l e n ts ou rc es o f a d d i t i o n a l m a t e r i a l and we me nt i on o n l y [Col ;Gbl ;J l ;Mkl ;Pel ; Z1 ] .F i r s t we w i l l c l a s s i f y second o r d e r l i n e a r P D E i n t h e s t a n d a r d way.s i m p l i c i t y t a k e 2 i n de p e n de n t v a r i a b l e s x,y a nd c o n s i d e r

F o r

( 4 . 1 )where A,B, . . . a r e fu n c t i o n s o f ( x , y ) a lo n e ( n o t o f u ) .s t a r t a t some p o i n t Po = ( xo , yo ) and make a co o r d i n a t e change ( x , y ) + ( 6 , ~ )a r ound Po f Po.t h e r e d u c t i o n c an o n l y o c c u r a t Po i n g e ne r al ( n o t i n a n ei g hb o rh o o d o f Po)b u t f o r two v a r i a b l e s t h e r e i s more f l e x i b i l i t y . Thus w r i t e C = t ; (x ,y) andn = ~ ( x , y ) assumed C - nXsy # 0X Yi n a ne i ghbo r hood o f Po and then one has ux = u6Lx + u ~ , ; ~ , e t c . P u t t i n gsuch exp r ess i ons i n ( 4 . 1 ) we o b ta i n (*) x u + 2BuEn t C u n n + . . . + Fu = Gwhere i n p a r t i c u l a r ( A ) + B ( E x i i y + 5 n+ C S y ~ y ; C = An: + 2Bn n + Cny . One de f i nes now a d i s c r i m i n e n t ( a ) A =B - A C and th e n a s t r a i g h t f o r w a r d c a l c u l a t i o n g i v es ( e x e r c i s e )t H E 0 R E m 4 - 1 .D E F L N I t L 0 N 4.2, The e q ua t i on ( 4 . 1) i s s a id t o be h y p er b o l i c , e l l i p t i c , o rp a r a b o l i c a t Po ( o r i n a r e g i o n a r o u nd P o ) i f r e s p . A > 0, A < 0, o r A = 0.One ch ec ks e a s i l y t h a t t h e wave e q u a t i o n u t t = Au i s h y p e r b o l i c , t h e L a p la c ee q u a t i o n Au = 0 i s e l l i p t i c , dnd th e h e at e qu a t io n ut = Au i s p a r a b o l i c .EXAIRPLE 4.3. = 0 i s e l l i p t i c f o r y > 0,

Auxx t 2Buxy t Cuyy + Dux t Euy + Fu = GThe i dea i s now t o

i n o r d e r t o r ed uc e ( 4 . 1) t o a " c a n o n i c a l " f o r m i n a n ei gh bo rh oo dOne knows ( c f . [ P e l ] ) t h a t f o r m o re t h a n t w o i n d ep e n d en t v a r i a b l e s0

2 f u n c t i o n s ) w i t h J = a ( c , n ) / a ( x , y ) = 5 Q& , - 4

2 5 5 2.= AS, t 2B5 5X Y x xCCy, B = A 5 n y 12 .y

2 = z2 - i?AJ2 so t h a t s g n x = sgnA.

The T r i c o m i e q u a t i o n y u x x + uYY


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C l e a r l y e l l i p t i c e q u at io n s have no c h a r a c t e r i s t i c s (a nd t h e s o l u t i o n o f e.g .Au 0 i s Cm i n t h e i n t e r i o r o f n ) w hereas f o r h y p e r b o l i c e q u a t i o n s t h ec h a r a c t e r i s t i c s p l a y a n i m p o r t a nt r o l e i n t h e th e o r y. I n a g e ne ra l sensec h a r a c t e r i s t i c s r e pr es e n t cu rv es a l o ng w hich jump d i s c o n t i n u i t i e s o f f i r s td e r i v a t i v e s c an o c c u r o r a l o n g w h ic h s i n g u l a r i t i e s c an b e p ro pa ga te d. " I n -i t i a l " d a t a (u and un = norm al d e r i v a t i v e ) c an no t be pr e s cr ib e d a r b i t r a r i l ya l o n g a c h a r a c t e r i s t i c " i n i t i a l " cu rv e. Examples o f such b e h a v i o r w i l l bei l l u s t r a t e d f r o m t i m e t o t i m e a s we go a l on g . A no th er p r o f o u n d d i f f e r e n c ebetween h y p e r b o l i c an d e l l i p t i c e q u a t i o n s c on ce rn s t h e t y p e o f p r ob le m swhic h ar e w e l l posed. Fo r example we showed i n Remark 3.3 t h a t t h e C a u c h yp r ob l em f o r t h e wave equ a t i on i n one space d i m ens i on was w e l l posed . Look -i n g a t t h e P o is so n i n t e g r a l f o rm u l a ( 3 .1 8 ) we se e a l s o t h a t t h e D i r i c h l e tp ro b le m f o r t h e L a pl ac e eq u a t io n on a c i r c l e i s w e l l posed ( e x e r c i s e - showt h a t I f * - f l 5 E on r i m p l i e s I u * - u I 5 E i n n) . However a wel l known ex-a m p le o f H adam ard shows t h a t t h e Cauchy p r o b l e m f o r t h e L a p l a c e e q u a t i o n i sn o t w e l l p os ed . = 0 i n , t h e h a l f p la ne y L 0 l e t u(x,O)YY= 0 and u (x,O) = S i n (n x ) /n . The s o l u t i o n i s u = un = S i n h ( n y ) S i n ( n x ) / n 2 a n das n -f m y g n ( x ) = S i n ( n x ) / n -f 0 u n i fo r m l y on t h e l i n e b u t f o r say y > 0f i x e d a nd x # mn f i x e d ( s a y x = n /2 e ve n) u n(x ,y ) o s c i l l a t e s w i l d l y w i t h am-p l i t u d e g o in g t o m . On t h e o t h e r h and t h e D i r i c h l e t p r o bl e m f o r t h e w avee q u a t i o n i s n o t w e l l p os ed . C o n s i d er e .g . a s q u a r e 0 5 x 5 1, 0 5 y 5 1,w i t h u = 0 i n t h e i n t e r i o r an d b ou nd ar y c o n d i t i o n s u(x,O) = f o ( x ) , u ( x , l )= f l ( x ) , u ( 0 , y ) = go ( y ) , and u ( 1 , y ) = g l ( y ) ( w i t h some c o m p a t a b i l i t y a t t h ec o rn e rs f o r c o n t i n u i t y s ay ) . XYhence e.g. ux(x ,O) = f;(x ) we must have u x ( x , l ) = f i ( x ) = f i ( x ) .q u e n t l y a r b i t r a r y ( c o nt in u o u s) b ou nd ary f u n c t i o n s fT he re a r e a few t y p i c a l c l a s s i c a l a rg um en ts w h ic h a p p ly t o t h e t h r e e p r o t o -t y p i c a l e q u a t i o n s i n q u e s t i o n s o f e x i s t e n c e an d u n iq ue ne ss a nd we w i l ls k e t c h t h i s h e r e . F i r s t f o r t h e L a pl ac e e q u a t i o n Au = 0 i n a " r e g u l a r " R CRn f o r exam ple ( i . e . G r e en ' s t he or em a p p l i e s t o R) one speaks o f a fundamen-t a l ( o r e l em e n ta r y) s o l u t i o n E o f t he e qu at i o n i n t h e s p i r i t AE = 6 where 6i s t h e D i r a c m ea sure a t t h e o r i g i n . H ere ( B ) E ( r ) = r 2 - n /( 2 - n )w n f o r n > 2and E ( 1 / 2 n ) l o g r f o r n = 2 where tun i s t h e s u r f a c e ar ea o f t h e u n i t s phe re(un = 2 n n" /r (n /2 ) ). N o te t h a t f o r s uc h r a d i a l f u n c t i o n s t h e L a p l a ce e qua -t i o n AE = 0 becomes Err t [ ( n - l ) / r ] E r = 0. T h e r e s u l t AE = 6 o r more gen-e r a l l y , f o r r = Ix-SI, AE = ~ ( x - S ) , w i l l e n s u e f r o m t h e a n a l y s i s t o f o l l o w( e x e r c i s e ) .

In de ed f o r uxx f uY

XY

S i nce u = 0 i m p l i e s u x (x , y) = c o n s t a n t a n dc a nn o t b e p r e s c r i b e d .

Conse-

Now f o r R g i v e n we e x c i s e a sm a l l b a l l B(S,E) o f r a d i u s E


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20 ROBERT CARROLL

around 5 w i t h b ou nd ar y B = S ( S , E ) a s ph er e o f r a d i u s E

I n t h e r e g i o n nE = R-B, E ( r ) i s h ar mo ni c ( i . e . AE = 0 ) and i f Au E Co i n R( w i t h u E Co (E) ) t h e n o n e o b t a i n s( 4 . 4 )

t

( n o t e t h e e x t e r i o r n o r m a l n on S p o i n t s i n w a r d ) . Now f o r r = I x - c l = E onS , do 'L w n r n-1 , E ?, r2-n/ (2-n)wn, and En 'L - ( 2 - r 1 ) r l - ~ / ( 2 - n ) w ~ . H e n c e t h el a s t i n t e g r a l i n (4.4) -+ u(S) as E + 0 ( exe r c i se ) and consequen t l y( 4 . 5 ) u ( E ) = I EAudx - j [Eu, - Enu]doR rT h i s f o r m u la f o r Au = 0 i s s l i g h t l y m is le ad in g s i nc e i t seems t o sugges tt h a t u and un cou l d bo t h be p r e sc r i be d on r - however we w i l l s e e t h a t t h es o l u t i o n t o t h e D i r i c h l e t prob lem i s u ni qu e so u a lone on r d e te r m in e s u i nR.c e n t e r E w i t h aB = S = r i t f o l l o w s t h a tI f we a p p l y ( 4 . 5 ) t o a h ar mo ni c f u n c t i o n u w i t h R a b a l l o f r a d i u s R and( 4 . 6 ) u ( 5 ) = -Is [Eu, - Enu]do = ( l/un Rn- ' u do

x-E 1 =RHere Is Eun = E ( R ) l S undo = E(R) lB Audx = 0 by Green ' I s theorem app l ied to uand v = 1.CKE0REill 4.5. If Au = 0 i n a r e g i o n c o n t a i n i n g B ( E , R ) t hen ( 4 . 6 ) h o lds -i . e . t h e v a l u e a t t h e c e n te r 5 o f t h e s ph ere S equa ls t he sphe r i ca l meanva lue .

We have proved t h e mean val ue theorem

rHE0REm 4.6 (mAXImUJlI PRINCIPLE), L e t R be a bounded po l y go na l l y connec tedopen set and Au = 0 i n R ( u E C O ( 6 ) ) . Then u a t t a i n s i t s maximum and m in i -mum va lues on a R = r . I f u # c t h e n i n f a c t i t a t t a i n s i t s m aximum a nd m i n i -mum only on r .Phood:t e r i o r p o i n t .u ( x ) = M i n n .l a r g e s t b a l l c en te re d a t 5 and l y i n g i n R.b a l l an d u se ( 4 . 6 ) .

L e t M = max u f o r x E 5 and suppose u(xo) = M where xo E n i s an i n -We w i l l show u(x) = M f o r a n y x n a n d h e n c e b y c o n t i n u i t y

F i r s t we n o t e t h a t i f u ( 5 ) = M f o r E E R t hen u z M o n t h eThus M = s p h e r i c a l a ve ra ge o f u o v e r S ( c , R ) and hence

Indeed, l e t B(S,R) be such a


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b y c o n t i n u i t y u z M o n S (5,R ) (e x e rc i se ) .B(S,r ) C B ( S , R ) so u : i n B (S ,R). B y p o l y -gonal ly connected we mean xo and y c an be j o i n e d b y a p o l y g o n a l a r c a (= af i n i t e number o f s t r a i g h t l i n e s egm ents) .p o i n t s xo, xl, ..., x = y on a w hi ch a r e c e n t e r s o f b a l l s B (x ., R. ) h a v i ngn J Jt h e p r o p e r t i e s t h a t B (x .,R .) C R a n d x . E B ( X ~ - ~ , R ~ - ~ )e x e r c i s e ) .J J J

T h is h o l ds a l s o f o r a ny b a l lNow l e t y E R be a r b i t r a r y .

One ca n f i n d a f i n i t e s e qu en ce o f

(4 .7 )

S ince u = M i n Bo = B(x ,R ) one has u(x, ) = M and hence u = M i n B1 = B(xl,R 1 ) , .... I t f o l l o w s t h a t u ( y ) = M.CHEbREll 4 .7 ,0 i n n, u = f on r(u E Co(?t) R C (a)) as a t mo st one s o l u t i o n ( i t may n o tha ve a n y ) . F u r t h e r g i v e n t wo s o l u t i o n s u * and u c o r r e s p o n d i ng t o d a t a f *and f on r , i f I f * - f ) 5 E on r t h e n I u * - u l 5 E i n R .Phoofi:i s f i e s Au = 0 i n R w i t h u = 0 on r . By t h e ma x-m in p r i n c i p l e u = @ i n ;.I n t h e s econd s i t u a t i o n i f 7 = u* - u th e n A = 0 i n R w i t h lcl 5 E on r .

0 0 The same argument app l ies t o m in im a .U nde r t h e h y po t he s es o f T heorem 4 .6 t h e D i r i c h l e t p r o b l e m Au =2

I f o ne h ad tw o s o l u t i o n s u1 a nd u2 f o r d a t a f then u1 - u 2 = u s a t -It f o l l o w s t h a t 5 E i n by Theorem 4 .6. 4 EDWe w i l l d i s c u s s t h e D i r i c h l e t p ro bl em v i a H i l b e r t s pa ce m etho ds l a t e r . Nowf o r h y p e r b o l i c e q u a t i o n s we go t o t h e p r o t o t y p i c a l w a ve e q u a t i o n a n d co n s i -d e r t h e m et ho d o f s p h e r i c a l means ( c f . [C 1 ,4 ; Co l ;D i l; J l] ) . T h i s a l s o g i v e su s a g o o d co n te x t i n w h i c h t o ex am in e some d i s t r i b u t i o n f o r m u l a s an d t o il-l u s t r a t e t h e u se fu ln es s o f d i s t r i b u t i o n t ec hn iq ue s.Appendix B f o r n o ta t i o n e t c . ) u x ( t ) E E; an d A x ( t ) E E; by

T h u s o n e d e f i n e s ( c f .

. .( A x ( t ) , 9 ( x ) ) = [ n / u n t n l ] l 9 ( x ) d x

I x I f t( t h e s u b s c r i p t x i n p x and A x i s n o t a p a r t i a l d e r i v a t i v e ) .( r es p . A x ( t ) ) r e p r es e n t s p h e r i c a l ( r e s p . s o l i d ) mean v a l u e o p e r a t o r s ( c f .( 4 . 6 ) ) . One c he ck s t h a t { u x ( t ) , p ( x ) ) = ( u ( 1 ) , 9 ( t y ) ) ( e x e r c i s e ) an d (*)( l / w n t n-1 ) f 4 d x = ( t / n ) ( A x ( t ) , 4 ) = ( t / n ) ( M x ( t ) , d w h e r e don = t

T h e n v x ( t )

YD ( p ( t ) , 9 ) = ( 1 1 ~ ~ ) [I Yia~ iax i ]dnn = (i/ontn-l) a ~ / a v ] d o n =dQn


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22 ROBERT CARROLL

ag/aw d en ot es t h e e x t e r i o r no rm al d e r i v a t i v e , and t h e i n t e g r a t i o n s i n (**)a r e r e s p e c t i v e l y o v e r Iy I = 1, 1x1 = t, and 1x1 5 t. S i m i l a r l y (*A) D $ A x( t ) , g ) - [n /unti n t e g r a t i o n s o v e r 1x1 2 and 1x1 = t r e s p e c t iv e l y . We t h i n k o f p x ( t ) andA x ( t ) as d i s t r i b u t i o n v a lu ed f u n c t io n s o f t a nd h ave shown t h a t i n t h e se n seo f weak v e c t o r v a lu ed d i f f e r e n t i a t i o n

2 n t l]Id x + [n/untn]/ vdon = ( n /t ) ( u x ( t ) - Ax ( t ) , v ) , w i t h

(4 .9)I n f a c t o n e c an show t h a t t + v x ( t ) E C (El) i n t h e sense o f s t r o n g d i f f e r -e n t i a t i o n ( c f . [C4,5]) b u t we d o n ' t n ee d t h i s h e re . One c an d i f f e r e n t i a t ea g a i n i n t h e same s p i r i t and fr o m ( 4 . 9 ) o ne o b t a i n s ( *e ) { D t + [ ( n - 1 ) / t l D t lp x ( t ) = A u x ( t ) . N ot e a l s o f r o m ( 4 .8 ) a nd ( 4 . 9) t h a t p,(t) * 6 ( x ) as t + 0w h i l e D t p x ( t ) + 0.and i n f a c t D [U ( t ) * T] = D t p x ( t ) * T e t c . i n a s t ro n g o r wehk sen se .CHE0REm 4.8,[ ( n - l ) / t ] u t = Au w i th u (x ,O) = T and ut(x,O) = 0.Thus u s a t i s f i e s a Cauchy p ro bl em f o r a s p e c i a l c as e o f t h e EPD e q u a t i o n( * & ) w t t t [ ( 2 m + l ) / t ]w t Aw where 2mt1 = n - 1 o r m = ( n /2 ) - 1 . I n f a c tt h i s s o l u t i o n i s u n iq u e i n D; ( c f . [C 4,5]) b u t we o m i t t h e p r o o f .us use the mean va lue opera to r p x ( t ) and Theorem 4.8 t o s o l ve th e wave equa-t i o n wh ic h c o rr es p on ds t o m = -1 /2 i n ( *& ) .s t e i n ( c f . [ C4,5;Dil;W3]) f o r t h e s o l u t i o n o f ( *& ) w h ic h was v e r i f i e d i n t h ed i s t r i b u t i o n c o n t e x t by t h e a u t h or .( *&) w i t h w m(0 ) = T and wT(0) = 0 one has f o r any in te ge r p such m+p 1. -1 /2

(4 .10 ) wm( t = [r m t l ) - 2 m / 2 p r ( m + p+ l )] [ 1/ t ) Dt]Pk2(m+p)wmtp() ]T he fo rm u la can be ch e cke d d i r e c t l y b u t [C4,5] p ro v id e s a m ore e le g a n t p ro o f ,I t i s a ssumed h e r e t h a t wm+' i s k nown and t h i s i s as su re d by t h e f o rm u la f o rs > q 1. -1 /2 (cf . [C4,5 ] - n o t e wq i s known f o r q = ( n / 2 ) - 1 by Theorem4.8)

D tu x ( t ) = ( t / n ) U X ( t ) ; D t A x ( t ) = ( n / t ) h , ( t ) - A x ( t ) lt x

2

Now f o r T E D; a r b i t r a r y u,(t) * T E D; i s w e l l d e f in edt x

For T E D; a r b i t r a r y u ( x , t ) = p x ( t ) * T s a t i s f i e s u t t +

Now l e t

We ca n use a f o rm u la o f W e in-

T h u s w r i t i n g wm f o r t he s o l u t i o n o f

( 4 . 1 1 ) w S ( t ) = [~r(s+i)t-2~/r(qtl)r(s-q)ll n qtl ( t -n ) s-q-lwq(q)dn0

( ag a in ( 4. 11 ) can a l s o b e v e r i f i e d d i r e c t l y and we r e f e r t o [C4,5] f o r mean-i n g ) . Now f o r m = -1 /2 (wave equ a t i on ) and d imens ion n = 3 f o r s i m p l i c i t ywe c an o b t a i n t h e c l a s s i c a l P oi ss on s o l u t i o n by t a k i n g p = 1 so m+p = 1 / 2 =(n /2 )- 1 an d (w' = p ( t ) * T ) (*+) w-'(t) = [r(l/2)t/2r(3/2)](1/t)Ot(tw4) =Dt(tw') = w + t D ii = p x ( t ) * T t ( t 2 / 3 ) A x ( t ) * AT. One can also checkt


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d i r e c t l y t h a t i f G = v s a t i s f i e s v " + ( 2 / t ) v ' = A v t h e n u = ( t v ) ' s a t i s f i e su " = A u . T h e r e f o r eCHEBREIII 4.9.D; and wT(0) = 0 i s g i v e n by ( 4 .1 0 ) - (4 . 1 1 ) w he re w'I2-'s known from Theo-rem 4.8. I n p a r t i c u l a r f o r m = -1 /2 o ne o b t a i n s s o l u t i o n s o f t h e wave e qua-t i o n .Now n o t e t h a t f o r d = v a g a in t h e f u n c t i o n 9 = t v a l s o s a t i s f i e s 9 = LLPw i t h g ( 0 ) = 0 a nd g t ( 0 ) = v ( 0 ) .R w i t h i n i t i a l v al ue s W(0) = T E D; and Wt(0) = S E D; i s ( * m ) W ( t ) =D t [ t v x ( t ) * T I + t v x ( t ) * S.f u n c t i o n s now t h e s o l u t i o n W ( t ) = W ( x, t) a t a p o i n t ( x , t ) d ep end s o n l y o nt h e d a t a S,T o n t h e s u r f a c e r o f t h e i n t e r s e c t i o n o f t h e r e t r o g r a d e l i g h tcone t h r u ( x , t ) w i t h t h e i n i t i a l h yp er pl an e t = 0

The ( u n i q u e ) s o l u t i o n o f ( * 6 ) f o r m > - 1 /2 w i t h w m( 0) = T E

Hence t h e s o l u t i o n o f t h e wave e q u a t i o n i n

I t i s i n t e r e s t i n g t o n o t e t h a t f o r S and T3

For T a f u n c t i o n x e n o t e h e re f o r m a l l y

( c f . ( 4 . 6 ) ) . T h is f a c t i s a v e r s i o n o f w ha t i s c a l l e d H uy ge n's p r i n c i p l e .The p i c t u r e a l s o a l l o w s us t o f o r m u l a t e an e ne rg y p r i n c i p l e .f i x e d l e t Q~ b e t h e t r u n c a t e d c o ne i n ( 4 . 12 ) b ou nd ed b y B ( x , t ) = { S ; t ( S - x l- t 1 , B ( x , t - T ) = 1 5 ; I c - x lt h e e n e r g y o f u i n B C R3 a t t im e t by

Thus f o r ( x , t )2D e f i n e2 25 ( t - T ) I , and th e l a t e r a l s u r f a c e A T .

2CHEbREm 4-10.ene r gy s a t i s f i e s E( u ,B ( x, t - T ,T ) 5 E ( u, B (x , t ) O ) .Pkoo6:a x i ) ] - [ut t 1 (au /ax i ) 3n o t i n g t h a t t h e r i g h t s i d e i s a d i ve r ge n c e, t o o b t a i n (u) 0 =2 2(au/ax i ) - ( u t t 1 ( a u/ a xi ) )vt]da ( i n t e g r a l o v e r anT) where v = ( v 1 y v 2 y v 3 yv ) i s t h e e x t e r i o r u n i t no rm al t o anT.

Suppose utt = Au ( u E C i n t h e c one o f ( 4 . 1 2 ) ). Then the

T he re i s an o bv io u s i d e n t i t y (A*) 2ut(Au - u t t ) = 2 1 a/ax i ) [Ut (au/2 2Given Au = u t t n o w , i n t e g r a t e (.*) o v e r QT,[ 12utvi

On th e to p v = ( O , O , O , l ) , o n t h et


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24 ROBERT CARROLL

2b o t t o m w = O , O , O , - l ) , and on A T , 1 w = w t so w t = 1/J2. Hence(4 .15) 0 = - zE(u ,B(x , t -T ) ,T ) + ZE(U,B(x,t),O) +

2 2The l a s t t e k n ca n be w r i t t e n as (A*) 42 1 [I 2ututvi(au/axi) - 1 v t (au /ax i )- u t I vi]do = -42 1 [ l ( v t (au /ax i ) - utvi) ]do 5 0. Consequent ly E(u,B(x,t H E 0 R E N 4.11.t he cone i n ( 4 . 12 ) , t hen u E 0 i n t h i s cone.P4006:H e n c e t h e i n t e g r a n d I ( a u / a x . ) t u t = 0 a t any v a l u e o f T 5 t so u = con-

2 2 2t -T,T) 5 E(u,B(x,t) ,O). QEO

2If u = utt, u E C , and u(x,O) = ut(x ,O) = 0 on t h e ba se o fEv i den t l y E ( u ,B ( x , t ) ,O ) = 0 so E(u,B(x,t-T,T) = 0 f o r any T 5 .2 2

1s t a n t a nd t h e c o n s t a n t m u st b e 0 b y c o n t i n u i t y . QEDT h i s shows t h a t s o l u t i o n s o f Au = u tt a t ( x , t ) a r e d et er mi ne d by t h e i r i n i -t i a l d a ta u and u t o n t h e base o f t h e r e t r o g r a d e l i g h t cone a t ( x , t ) andt h a t s o l u t i o n s w i t h t h e same i n i t i a l d a ta a r e i d e n t i c a l ( i . e . u niq ue ne ssho l ds ) . One can e a s i l y show a l s o f r o m T heorem 4 .10 t h a t i f t h e i n i t i a l d a t au (x ,O ) and u t( x,O) van i sh ou t s i de o f some compact se t t hen con se r va t i on o f3 3ene r gy ho l d s i n t h e f o r m E(u,R ,T) = E(u,R ,0 ) ( e x e r c i s e ) .F o r t h e h e a t e q u a t i o n ( a nd wave e q u a t i o n ) t h e r e a r e a l s o maximum p r i n c i p l e so f s p e c i a l f or m s w h ic h l e a d t o u n iq ue n es s a nd w e l l p ose dne ss r e s u l t s ( c f .[ J l ; S p l ; P r l ; Z l ] ) .t h e h ea t e q u a t i o n t o i l l u s t r a t e t h e m a t t e r . Thus

We m e n t i o n h e r e o n l y a s i m p l e on e d i m e ns i o na l t h eo re m f o r

CHEBREN 4.12.t h e c l os e d r e c t a n g l e R : 0 5 x 5 L, 0 5 t 5 T .and min imum values on the base t = 0 o r on th e l a t e r a l s i de s x = 0, x = L .Phood: L e t M = max u i n R and suppose th e max o f u on t h e b as e a nd l a t e r a ls id es i s M- E ( E > 0 ) . L e t (

0444704434 Mathematical Physics

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