Second Order Equations of Elliptic and Parabolic Type

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Selected Title s i n Thi s Serie s 171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 1 99 8 170 V i k t o r Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c integrals , 1 99 7 169 S . K . G o d u n o v , Ordinar y differentia l equation s wit h constan t coefficient , 1 99 7 168 Junjir o N o g u c h i , Introductio n t o comple x analysis , 1 99 7 167 M a s a y a Y a m a g u t i , Masayosh i Hata , an d J u n Kigami , Mathematic s o f fractals , 1 99 7 166 Kenj i U e n o , A n introductio n t o algebrai c geometry , 1 99 7 165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g proble m i n Galois theory , 1 99 7 164 E . I . G o r d o n , Nonstandar d method s i n commutativ e harmoni c analysis , 1 99 7 163 A . Ya . D o r o g o v t s e v , D . S . Silvestrov , A . V . Skorokhod , an d M . I . Yadrenko , Probability theory : Collectio n o f problems , 1 99 7 162 M . V . B o l d i n , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d method s i n linea r statistical models , 1 99 7 161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c dynamics , 1 99 7 160 V . G . Osmolovskii , Linea r an d nonlinea r perturbation s o f th e operato r div , 1 99 7 159 S . Ya . K h a v i n s o n , Bes t approximatio n b y linea r superposition s (approximat e nomography), 1 99 7 158 Hidek i Omori , Infinite-dimensiona l Li e groups , 1 99 7 157 V . B . Kolmanovski i an d L . E . Shatkhet , Contro l o f system s wit h aftereffect , 1 99 6 156 V . N . Shevchenko , Qualitativ e topic s i n intege r linea r programming , 1 99 7 155 Yu . Safaro v an d D . Vassiliev , Th e asymptoti c distributio n o f eigenvalue s o f partia l differential operators , 1 99 7 154 V . V . Prasolo v an d A . B . Sossinsky , Knots , links , braid s an d 3-manifolds . A n introduction t o th e ne w invariant s i n low-dimensiona l topology , 1 99 7 153 S . K h . A r a n s o n , G . R . B e l i t s k y , an d E . V . Zhuzhoma , Introductio n t o th e qualitative theor y o f dynamica l system s o n surfaces , 1 99 6 152 R . S . Ismagilov , Representation s o f infinite-dimensiona l groups , 1 99 6 151 S . Yu . Slavyanov , Asymptoti c solution s o f th e one-dimensiona l Schrodinge r equation , 1996 150 B . Ya . Levin , Lecture s o n entir e functions , 1 99 6 149 Takash i Sakai , Riemannia n geometry , 1 99 6 148 Vladimi r I . Piterbarg , Asymptoti c method s i n th e theor y o f Gaussia n processe s an d fields, 1 99 6 147 S . G . Gindiki n an d L . R . Volevich , Mixe d proble m fo r partia l differentia l equation s with quasihomogeneou s principa l part , 1 99 6 146 L . Ya . Adrianova , Introductio n t o linea r system s o f differentia l equations , 1 99 5 145 A . N . A n d r i a n o v an d V . G . Zhuravlev , Modula r form s an d Heck e operators , 1 99 5 144 O . V . Troshkin , Nontraditiona l method s i n mathematica l hydrodynamics , 1 99 5 143 V . A . M a l y s h e v an d R . A . Minlos , Linea r infinite-particl e operators , 1 99 5 142 N . V . Krylov , Introductio n t o th e theor y o f diffusio n processes , 1 99 5 141 A . A . D a v y d o v , Qualitativ e theor y o f contro l systems , 1 99 4 140 Aizi k I . Volpert , V i t a l y A . Volpert , an d V l a d i m i r A . Volpert , Travelin g wav e solutions o f paraboli c systems , 1 99 4 139 I . V . Skrypnik , Method s fo r analysi s o f nonlinea r ellipti c boundar y valu e problems , 1 99 4 138 Yu . P . R a z m y s l o v , Identitie s o f algebra s an d thei r representations , 1 99 4 137 F . I . K a r p e l e v i c h an d A . Ya . Kreinin , Heav y traffi c limit s fo r multiphas e queues , 1 99 4 136 Masayosh i Miyanishi , Algebrai c geometry , 1 99 4 135 M a s a r u Takeuchi , Moder n spherica l functions , 1 99 4 134 V . V . Prasolov , Problem s an d theorem s i n linea r algebra , 1 994 133 P . I . N a u m k i n an d I . A . Shishmarev , Nonlinea r nonloca l equation s i n th e theor y o f waves, 1 99 4 (See th e AM S catalo g fo r earlie r titles ) Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

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Second Orde r Equations o f Ellipti c and Paraboli c Typ e

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10.1090/mmono/171

Translations o f

MATHEMATICAL MONOGRAPHS Volume 1 7 1

Second Orde r Equations o f Ellipti c and Paraboli c Typ e E. M . Landi s

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American Mathematical Societ y Providence, Rhode Island Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

EDITORIAL COMMITTE E AMS Subcommitte e Robert D . MacPherso n Grigorii A . Marguli s J a m e s D . Stashef f (Chair ) A S L S u b c o m m i t t e e Steffe n Lemp p (Chair ) I M S S u b c o m m i t t e e Mar k I . Freidli n (Chair ) E. M . J I A H U H C yPABHEHMH B T O P O r O nOPtfUK 3JIJmnTHHECKOrOH

A

nAPABOJIMMECKOr O TMnO

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«HAYKA», MOCKBA , 1 97 1 Translated fro m th e Russia n b y Tamar a Rozhkovskay a with th e participatio n o f Scientifi c Book s (RIMIB E NSU) , Novosibirsk , Russi a 1991 Mathematics Subject

Classification. Primar y 35J1 5 , 35K1 0 .

ABSTRACT. Th e book introduce s th e reader t o the theory o f linear ellipti c an d parabolic equation s of th e secon d order . I n additio n t o th e discussio n o f classica l result s fo r equation s wit h smoot h coefficients (Schaude r estimate s an d the solvability o f the Dirichlet proble m fo r elliptic equations ; the Dirichle t proble m fo r the heat equation) , th e book describe s propertie s o f solutions t o secon d order ellipti c an d paraboli c equation s wit h measurabl e coefficient s nea r th e boundar y an d a t infinity. The boo k present s a fine elementar y introductio n t o th e theory o f elliptic an d parabolic equa tions o f the secon d order . Th e precise an d clea r expositio n o f the materia l make s i t suitabl e fo r graduate student s a s wel l a s fo r researc h mathematician s wh o want t o ge t acquainte d wit h thi s area o f the theory o f partial differentia l equations .

Library o f Congres s Cataloging-in-Publicatio n D a t a Landis, E . M. (Evgen h MikhaTlovich ) [Uravneniia vtorog o poriadk a ellipticheskog o i parabolicheskogo tipov . English ] Second orde r equation s o f elliptic an d parabolic typ e / E . M . Landi s ; [translate d b y Tamar a Rozhkovskaya]. p. cm . — (Translation s o f mathematical monographs , ISS N 0065-928 2 ; v. 171) Includes bibliographica l reference s an d index . ISBN 0-821 8-0857- 5 (acid-fre e paper ) 1. Differentia l equations , Elliptic . 2 . Differential equations , Parabolic . I . Title . II . Series . QA377.L35131 99 7 97-3467 0 CIP

C o p y i n g an d reprinting . Individua l reader s o f this publication , an d nonprofit librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given . Republication, systemati c copying , o r multiple reproductio n o f any materia l i n this publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248, Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mail t o [email protected] . © 1 99 8 by the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the United State s Government . Printed i n the United State s o f America . @ Th e paper use d i n thi s boo k i s acid-fre e an d falls withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AMS home pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0

3 02 01 0 0 99 9 8

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Contents Preface t o th e Englis h Editio n i

x

Preface x

i

Chapter 1 . Ellipti c Equation1 s i n Nondivergenc e For m Introduction §1. Th e maximu m principl e 2 §2. 5-capacit y 9 §3. Th e lemm a o n th e norma l derivativ e an d th e stron g maximu m principle §4. Th e growt h lemm a §5. Th e behavio r o f solution s i n a neighborhoo d o f a boundar y poin 1t §6. Th e behavio r o f solution s a t infinit y 3 §7. Equation s o f Corde s type . A prior i estimate s fo r Holde r norm s 3 §8. Th e existenc e o f a solutio n t o th e Dirichle t proble m fo r a linea r equation 4 §9. Th e existenc e o f a solutio n t o th e Dirichle t proble m fo r a quasilin ear equatio n 5 §10. Th e Harnac k inequalit y an d th e Liouvill e theore m fo r equation s o f Cordes typ e 5

4 5 8 0 7 4 0 5

Chapter 2 . Ellipti c Equation s i n Divergenc e For m 6 9 §1. Th e existenc e an d uniquenes s o f a wea k solutio n t o th e Dirichle t problem 6 9 §2. Som e fact s abou t function s o f severa l rea l variable s 8 1 §3. A prior i estimate s fo r th e Holde r norm s o f solution s t o equation s in divergenc e for m 9 1 §4. A priori estimate s fo r th e Holde r norm s (continued ) 9 9 1 1 Chapter 3 . Paraboli c Equation s 1 1 §1. Definition s an d notatio n 1 1 §2. Th e maximu m principl e §3. Superparaboli c an d subparaboli c potential-typ e1 function s 2 §4. Th e uniquenes s o f a solutio n t o th e Cauch y problem , an d th e sta bilization o f a solutio n t o th e Cauch y proble m a1 s t— > o o 2 1 §5. Paraboli c s, /^-capacity 2 §6. Th e growt h lemm a 2 §7. Th e behavio r o f a solutio n i n a neighborhoo d o f a boundar y poin1 t 3 §8. Equation s o f Corde s type . Th e oscillatio n theore m an d corollarie1 s 4 §9. Equation s o f Corde s type . Th e Harnac k inequalit y an d corollarie1 s 4

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3 3 5 2 4 7 9 3 0 4

viii C O N T E N T

S

§10. Hea t potential s 5 §11. Th e existenc e o f a solutio n t o th e firs t boundar y valu e proble m i n a cylindrica l domain . Estimate s fo r th e derivative s o f a solutio n and th e compactnes 1 s theore m fo r solution s 5 §12. Constructio n o f the generalized solutio n t o the first boundar y valu e problem i n a bounde d domai n i n IR n+1 . Th e behavio r o f th e gen eralized 1 solutio n a t boundar y point s 6 Appendix 7 §1. Th e proo f o f Lemm 1 a 5. 1 fro m Chapte r 1 7 §2. Th e proo f o f th e Schaude r1 fixe d poin t theore m 8 1 §3. Isoperimetri c inequalit y 8 1 §4. Th e Schaude r estimate s 8 Bibliography 9 Index

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3

9

5 5 5 0 2 3 9 203

Preface t o th e Englis h Editio n The boo k introduce s th e reade r t o th e theor y o f linea r ellipti c an d paraboli c equations o f secon d order . Althoug h it s mai n goa l i s t o stud y equation s wit h bounded measurabl e coefficients, th e book also contains a number o f classical result s concerning equations with smooth coefficients. Fo r example, the Schauder estimate s and th e solvabilit y o f th e Dirichle t proble m fo r ellipti c equation s i n Holde r spaces , the Wiener criterion for the regularity of a boundary poin t i n the case of the Laplac e equation, th e solvailit y o f th e Dirichle t proble m fo r th e hea t equatio n b y mean s o f heat potentials , an d som e othe r result s ar e discusse d i n detail . For equation s i n nondivergenc e for m wit h discontinuou s coefficient s th e proof s of result s presente d i n th e boo k ar e base d entirel y o n th e classica l maximu m prin ciple. To construc t th e correspondin g subellipti c an d superellipti c functions , th e au thor use s th e Ries z potentials , whic h lead s t o a natural characteristi c o f set s unde r consideration i n term s o f s-capacity , wher e s depend s o n th e ellipticit y constan t e. Th e essenc e o f th e approac h ca n b e observe d i n th e growt h lemm a (Chapte r 1 , Lemma 4.1 ) , whic h i s permanently use d throughou t Chapte r 1 . In particular , wit h th e hel p o f the growt h lemma , th e autho r deduce s sufficien t conditions fo r th e regularit y o f a boundar y poin t i n th e cas e o f ellipti c operator s with measurabl e coefficient s (a n analo g o f the Wiene r criterion ) an d prove s a serie s of theorem s o f Phragmen-Lindelo f typ e o n th e behavio r o f solution s a t infinity . A simila r approac h i s applie d t o paraboli c equation s wit h mesuarabl e coeffi cients i n Chapte r 3 . Th e autho r introduce s specia l familie s o f subparaboli c func tions and , usin g th e notio n o f paraboli c capacity , establishe s a paraboli c analo g o f the growt h lemm a an d othe r result s simila r t o thos e i n th e ellipti c case . The abov e approac h i s als o use d t o obtai n a n estimat e fo r th e Holde r nor m o f a solutio n an d t o prove th e Harnac k inequality . However , i n this cas e the us e of th e growth lemm a require s a lowe r estimat e o f th e s-capacit y o f a se t i n term s o f it s measure, whic h lead s t o a stron g restrictio n o n th e ellipticit y constan t e . Namely , it mus t satisf y th e inequalitie s n < e < n + 2 , wher e n i s th e dimensio n o f th e space. Suc h restriction s o n th e sprea d o f eigenvalue s o f th e matri x o f coefficient s are usuall y calle d th e Corde s condition , an d th e correspondin g equation s ar e calle d the equation s o f Corde s type . In th e earl y seventies , whe n th e Russia n editio n o f th e boo k wa s published , i t seemed unlikel y tha t th e Corde s conditio n coul d b e avoided . I n th e lat e seventies , the Harnack inequality an d the Holder estimates for solutions were proved by Krylov and Safonov . Thei r result s ar e o f significan t importanc e i n th e theor y o f nonlinea r elliptic an d paraboli c equations . Th e proof s o f thes e result s requir e mor e powefu l analytic method s tha n th e method s presente d i n thi s book . I n particular , th e crucial ste p consist s i n usin g the estimate s foun d b y A. D. Aleksandro v an d Krylo v ix

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x PREFAC

E T O T H E ENGLIS H E D I T I O N

for th e maximu m valu e o f a solution i n terms o f the integra l nor m o f the right-han d side o f th e equation . Th e correspondin g bibliographi c remark s an d reference s t o Krylov-Safonov's work s are added b y the autho r t o the Englis h editio n o f the book . For detaile d proof s i n th e ellipti c case , w e als o mentio n th e secon d editio n o f th e book b y D . Gilbar g an d N . S . Trudinger , "Ellipti c partia l differentia l equation s o f second order " (Springer-Verlag , 1 983) . Chapter 2 i s devote d t o ellipti c equation s i n diveregenc e form . T o stud y suc h equations, i t i s necessar y t o appl y essentiall y differen t methods . I n th e cas e o f simplest ellipti c equation s i n divergenc e form , th e estimat e fo r th e Holde r nor m o f a solutio n wa s first establishe d b y D e Giorg i i n 1 957 . Anothe r proo f wa s suggeste d by Moser i n 1 960 . I n Chapte r 2 , the estimates fo r th e Holde r norm s o f solutions ar e derived wit h th e hel p o f som e technica l tool s (cf . Chapte r 2 , Theore m 2.2) , whic h are differen t fro m thos e use d b y D e Giorg i an d Moser . Although th e Russia n editio n o f thi s boo k wa s publishe d 2 5 year s ago , th e book remain s attractiv e fo r th e reader . I t i s a fine elementar y introductio n t o th e theory o f ellipti c an d paraboli c equation s o f secon d order . Th e precis e an d clea r exposition o f the materia l make s the boo k understandabl e fo r graduat e student s a s well a s mathematician s wh o wan t t o ge t acquainte d wit h thi s are a o f the theor y o f partial differentia l equations . To illustrat e mai n idea s an d methods , som e result s ar e describe d i n th e boo k for th e simples t equation s only . Fo r mor e complet e an d detaile d description , th e reader ca n b e referre d t o th e above-mentione d boo k b y Gilbar g an d Trudinge r an d the recen t boo k b y N . V . Krylov , "Lecture s o n ellipti c an d paraboli c equation s in Holde r spaces " (America n Mathematica l Society , 1 996) . However , significan t part o f th e boo k b y Landi s (i n particular , th e constructio n o f barrie r function s o f potential type , an d th e us e o f capacit y i n th e stud y o f equation s i n nondivergenc e form wit h discontinuou s coefficients ) present s a nontraditiona l approac h whic h i s of interes t fo r specialist s a t presen t an d canno t b e foun d i n othe r monograph s an d textbooks. Nina Ural'tsev a July 1 99 7

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Preface This boo k i s based o n lecture s give n b y th e autho r a t Mosco w Stat e Universit y in 1 967-1 968 . The reade r mus t b e familia r onl y wit h basi c notion s an d fact s fro m th e theor y of partia l differentia l equation s (see , e.g. , [Pel]) . Th e goa l o f th e boo k i s t o intro duce th e reade r t o som e recen t problem s i n th e theor y o f secon d orde r ellipti c an d parabolic equations . The autho r tried , o n on e hand , t o mak e th e expositio n self-containe d and , o n the othe r hand , t o kee p th e siz e o f th e boo k i n reasonabl e limits . Therefore , a limited numbe r o f topic s (a s ca n b e see n fro m th e tabl e o f contents ) hav e bee n chosen t o b e presente d systematicall y an d i n detail . The mai n too l i n the stud y o f solutions i n Chapte r 1 (nonselfadjoint equations ) and i n Chapte r 3 (paraboli c equations ) i s th e us e o f subfundamenta l an d super fundamental solutions , whic h ar e constructe d wit h th e hel p o f th e Ries z potential . Such a n approac h facilitate s th e qualitativ e stud y o f the behavio r o f solutions nea r boundary point s (theorem s of Wiener type ) an d a t infinit y (theorem s of Phragmen Lindelof typ e an d Liouville' s theorems) . Th e sam e metho d allow s u s t o obtai n a priori estimate s fo r th e Holde r norm s o f solution s provide d tha t th e root s o f th e characteristic equatio n ar e no t sprea d to o fa r apart . Thes e estimate s ar e use d i n the proo f o f the existenc e o f a solution t o a boundary valu e proble m fo r quasilinea r equations. Chapter 2 (selfadjoint ellipti c operators ) present s ne w and , i n author's opinion , promising method s base d o n theorem s fro m th e theor y o f function s o f severa l rea l variables. These main topics of the book are not discusse d i n other monographs . However , our intentio n t o presen t informatio n withou t referrin g th e reade r t o othe r book s made i t necessar y t o includ e muc h o f traditiona l materia l (th e Leray-Schaude r method, variationa l method s o f solutio n o f Dirichle t problem s i n energ y spaces , heat potentials , etc.) . Almost ever y sectio n conclude s wit h bibliographi c remark s an d comments. 1 Material tha t i s not difficul t i n principle, bu t involve s cumbersome calculations , which woul d interrup t th e exposition , i s delegate d t o th e Appendix . I n particular , the fixed poin t theore m w e us e i s proved i n § 2 of the Appendix . The autho r thank s Yu . S . Il'yashenk o wh o helpe d a t al l stage s o f preparin g the lecture s an d th e book . Hi s assistanc e allowe d u s t o avoi d certai n error s an d t o improve som e proofs .

1 Translator's note. Som e section s ar e provide d wit h additiona l bibliographi c remark s writte n by th e autho r fo r th e Englis h editio n i n 1 996 . I n th e lis t o f references , th e ne w reference s ar e marked b y th e asterisk .

xi

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xii P R E F A C

E

The autho r i s grateful t o M. I. Vishik who read th e manuscrip t an d mad e man y useful remarks . E. M . Landi s 1970

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10.1090/mmono/171/01

CHAPTER 1

Elliptic Equation s i n Nondivergenc e For m Introduction A linea r secon d orde r differentia l operato r o f th e for m

=£aaix)^+£bi{x)£+c{x),

a) L i,k=l i—\

X = \X\ , . . . , X

n),

is said t o b e elliptic i f th e quadrati c for m n

/ , dikiiik is positive definite . W e alway s assum e tha t a ^ = dkiThe operato r (1 ) i s sai d t o b e uniformly elliptic i n a domai n D i f ther e exis t two positiv e constant s C\ an d C 0 and M > 0 are constants calle d the ellipticity constants. Hereinafter, w e use th e followin g notation : R n i s th e n-dimensiona l Euclidea n space , QXR i s a n ope n bal l i n W 1 o f radiu s R wit h cente r x° , Sft i s the spher e \x — x°\ — R, Q^ R i s the spherica l laye r define d b y th e inequalitie s R\ ^ \x — x°\ < it^ , O i s th e origin . In th e abov e notation , Q^ i s the bal l o f radiu s R an d cente r a t th e origin . E i s th e closur e o f a se t E an d dE i s the boundar y o f E. In Chapte r 1 , we will usuall y conside r operator s L withou t lowe r terms :

i,fc=l

Throughout Chapte r 1 , unles s otherwis e stated , b y L w e mea n th e operato r {!'). For suc h operator s onl y th e rati o o f th e constant s a an d M i s important , bu t no t the constant s themselves . It i s mor e convenien t t o introduc e int o consideratio n th e quantit y Tii=laii(x) , n.

e — su p „

which i s calle d th e ellipticit y constan t o f th e operato r L. In othe r words , fo r an y x G D w e tak e th e rati o o f th e su m o f al l eigenvalue s of th e matri x ||a;/c(:r)| | t o th e smalles t eigenvalue , an d th e ellipticit y constan t e i s the supremu m o f thi s rati o ove r al l x £ D. We not e tha t e ^ n , an d i f e = n , the n th e operato r L become s th e Laplac e operator afte r th e multiplicatio n b y som e positiv e function . §1. Th e maximu m principl e 1 .1 . Let D C R n be a bounded domain, L the elliptic operator (V) in D, and u a superelliptic (subelliptic) continuous function in D. Then THEOREM

u(x) ^ mini / (u(x) vJ

dD

v

v

J

^ maxw) , x J

dD

G D.

It suffice s t o conside r th e case o f a superellipti c functio n u. W e nee d th e following lemma . LEMMA 1 .1 . /

(6) Lu

/

< 0 in

D,

then u does not attain the minimum value at any interior point of D.

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3

§1. TH E MAXIMU M PRINCIPL E

P R O O F . W e assum e th e contrary . Le t th e minimu m valu e o f u b e attaine d a t an interio r poin t x° o f D. W e mak e th e chang e o f variables x y unde r whic h th e operator L take s th e canonica l for m a t x°. W e obtai n

_ \ ^/

Q\

u

u

I

2u

_ST^d

>0>

which contradict s (6) . D P R O O F O F T H E O R E M 1 .1 . W e note tha t th e ellipticit y o f the operato r implie s the inequalit y an > 0 . Conside r a n auxiliar y functio n

> 0.

ve — u — sx\, e Since

Lv£ = Lu — ean < 0, we hav e vAx) ^ mirivJx) 3D

in vie w o f Lemm a 1 .1 . Sinc e

u(x) ^ v £(x) and v£(x) — • u(x) a

s e —> 0 ,

we hav e u(x) ^ minu(x) . D V

' 3D

K

J

EXERCISE 1 .1 . Prov e th e followin g version s o f th e maximu m principl e fo r th e operator (1 ) . (a) I f c(x) ^ 0 , the n a superellipti c (subelliptic ) functio n attain s th e negativ e minimum valu e (th e positiv e maximu m value ) o n th e boundar y o f a domain . (b) Le t c(x) ^ 0 and le t u(x) b e a solution t o th e equatio n Lu — 0 in a domai n D, wher e L i s the operato r (1 ) . The n u(x) doe s no t attai n th e zer o extrema l valu e inside th e domai n D (i.e. , u^0). (c) Le t L b e a uniforml y ellipti c operato r (1 ) , wit h th e coefficient s bi bounde d and th e coefficien t c(x) bounde d fro m above . The n ther e exist s a constan t do > 0 such tha t do depend s o n th e constant s a , M fro m th e inequalit y (2 ;) an d o n th e bounds o f the coefficient s bi and c . Fo r an y domai n D o f diameter les s than do an d any functio n u tha t i s a solutio n t o th e equatio n Lu = 0 in D an d i s continuou s i n D, th e followin g inequalit y holds :

max \u\ ^ 2 max \u\. D dD

We tur n t o a n operato r L o f th e for m (l 7 ). Fo r n = 2 we ca n mak e som e con clusions abou t th e globa l behavio r o f solution s usin g onl y th e maximu m principle . Let n = 2 and le t u(x,y) b e a solutio n t o th e equatio n (60 Lu

= 0

in a plane domai n D. W e note that an y linea r functio n ax + by + c satisfies equatio n (&). W e conside r th e se t o f point s (x,y) a t whic h u(x, y) > ax + by + c.

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1 4

. ELLIPTI C EQUATION S I N NONDIVERGENC E FOR M

FIGURE 1

Any connecte d componen t D f o f this set must reac h th e boundary o f D; otherwise , applying th e maximum principl e t o the functio n v(x, y) = u(x, y) — {ax + by + c) in th e domain D 1 ', we arrive a t a contradiction . The sam e assertio n i s true fo r the set of points a t whic h u(x, y) < ax + by + c. This fac t ca n be geometrically expresse d a s follows: I t is impossible t o cut a "cap " off th e grap h o f u by any plane (Figur e 1). We sa y that th e graph o f a continuou s functio n f(x,y) i s of generalized nonpositive curvatur e i f it is impossible to cut a "cap " of f the grap h o f / b y any plane. Thus, the graph of a solution to equation (6 ) is of generalized nonpositiv e curvature . Let a functio n f(x,y) b e continuou s i n th e entir e plan e an d le t th e grap h of / ( # , y) b e of generalize d nonpositiv e curvature . The n th e followin g Liouvill e theorem i s valid. LIOUVILLE THEOREM . Let

M(r)= ma

x \u(x,y)\.

Then either the inequality lim — ^ > 0 1—>oo T

holds or the graph of f is a cylinder whose generator is parallel to the plane xy. If th e graph o f a solutio n t o equation (6 ) is such a cylinde r an d the solutio n increases mor e slowly than a linear function , the n the solution is a constant (why?) . This theore m i s due to Adel'son-Vel'skii wh o gav e a rather complicate d proof . We do not prove this theorem, but establish another relate d property of a continuous function wit h th e graph o f generalized nonpositiv e curvature . W e mean a theore m of Phragmen-Lindelo f type . W e first recal l th e Phragmen-Lindelo f theore m t o clarify connection s betwee n thi s classica l theore m an d the theorem w e will prove . Furthermore, w e will conside r variou s generalization s o f the classical Phragmen Lindelof theorem .

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§1. T H E M A X I M U M P R I N C I P L E 5

P H R A G M E N - L I N D E L O F THEOREM . Let an analytic function f(z) be defined in the upper complex half-plane z and let the following inequality hold for any point x on the real axis:

IH5|/(z)|x

Then either \f(z)\ ^ 1 in the entire upper half-plane or (7) li

— lnM(r ) m^ > i—>oo r

0,

where M(r) = su p \f(z)\. \=r \z\=r

We not e tha t th e uppe r limi t ca n b e replace d b y th e lowe r limi t i n (7) , but w e will generaliz e th e weake r inequalit y (7 ) here . The classica l Phragmen-Lindelo f theore m ca n b e reformulate d a s a theore m about harmoni c functions . P H R A G M E N - L I N D E L O F THEORE M (fo r a harmoni c function) . Let u(x,y) be harmonic function in the upper half-plane y > 0 and let

lim u(x,y)

a

< 0

(x,2/)—>(x 0 ,0)

for any x$. Then either u(x,y) ^ 0 everywhere in the upper half-plane or M r *— lim —( )—„ > 0 ,

r—>oo r

where M(r) = su

p u(x,y).

x2-\-y2=r2

PROOF. Le t th e harmoni c functio n v(x, y) b e conjugate t o the function u(x, y). We se t f(z) = e u+lv and appl y th e classica l Phragmen-Lindelo f theore m t o th e functio n / . • This theore m (wit h som e modifications ) remain s vali d fo r a n arbitrar y contin uous functio n wit h th e grap h o f generalize d nonpositiv e curvature . W e formulat e the correspondin g assertion . T H E O R E M 1 .2 . Let f(x,y) be a continuous function in the half-plane y ^ 0 with the graph of generalized nonpositive curvature. If

(8) / ( x , 0 X

0

for all x, then one of the following assertions holds: (a) f(x,y)^0fory>0; (b) -,— M(r) n lim — ^ > 0 , i—>oo

r

where M(r)= su

p |/(x,2/)|

;

x2_|_y2_r2

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6

1. E L L I P T I C E Q U A T I O N S I N N O N D I V E R G E N C E F O R M

FIGURE 2

(c) there exists Y such that for y ^ Y the graph of f(x,y) is generator is parallel to the x-axis.

a cylinder whose

P R O O F . Instea d o f (8), we can write f(x, 0 ) < 0 since we can replace / b y / — e with arbitrar y e > 0 . Assuming tha t assertion s (a ) and (b) fail, w e show the validity o f assertion (c). Let f{x0,y0) =a > 0, y 0 > 0. We conside r th e functio n

0 , 77(0:3) > 0 and the point (3:2,77(0:2) ) ^ e s o n ^ n e plan e (x , z) abov e the segment joinin g th e points (xi , 77(0:1)) an d (#3,77(23)) . The n thi s segmen t cut s a "cap" of f the graph o f 77(0:) so that th e "cap " i s directed upwar d an d is located in the uppe r half-plan e (x,z) (Figur e 3). But i n this case , the plane i n the space (x , y, z) tha t contain s thi s segmen t and is paralle l t o the y-axis cut s a "cap " of f the graph o f Lp(x,y), whic h i s impossible. Consequently, rj(x) = cons t = A > 0 . We consider th e level set E = {(x, y) \ ip(x,y) = A}. Th e set E, bein g the level set o f a continuous function , i s closed. I t intersects eac h lin e x = const . O n each of these lines , w e take th e lowest poin t o f E an d denote th e obtained se t by EQ. We prove tha t EQ i s a line paralle l t o the x-axis.

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7

§1. T H E M A X I M U M P R I N C I P L E

Z

k

(X3,7/(X3)>

(x2,J7(x2))

(*i,?7(Xi))

FIGURE 3 (x 3 ,y 3 )

^ —

&\.y[)

^^^^^^VlW ^^^"""^

I

*2

*3

FIGURE 4

We assum e th e contrary . Le t ther e exis t tw o point s (#1 ,2/1 ) an d (#2,2/2 ) ° f ^ 0 such tha t 2/ 1 7 ^ 2/2- Fo r defmiteness , le t X2 < x\ an d 2/ 2 < 2/i - Ther e i s a poin t (X3,2/3) £ £? o that i s located t o th e lef t o f (#2,2/2 ) an d satisfie s th e relatio n 2/i 2/2 2/ 1 #1 xi - # 2

2/3 - #

3

We conside r th e trapezoi d wit h vertice s (#1 ,2/1 ) , (#i,0) , (#3,0) , an d (#3,2/3) . On th e lowe r bas e an d latera l sides , th e inequalit y ip(x,y) < A hold s everywher e except th e point s (#1 ,2/1 ) an d (#3,2/3) . O n the plan e (#,2/) , we draw a line I between the poin t (#2,2/2 ) an d th e segmen t [(#1 ,2/1 ) , (#3,2/3)] s o tha t th e lin e i s paralle l t o the segment an d intersect s bot h latera l side s of the trapezoid . Le t thi s lin e be give n by th e equatio n 2/ = kx + b and le t (xi,y[) an d (#3,2/3 ) be - the intersectio n point s o f th e lin e wit h th e latera l sides o f th e trapezoi d (Figur e 4) . On the segments [(#1 , 0), (#1,2/i)], [(#3, 0), (#3,2/3)], and [(#3 , 0), (#1, 0)] we have ip(x,y) < A — 6, where 6 is a positiv e number . W e choos e e > 0 so smal l tha t e\y - kx - b\ < 6

on th e sam e segments . I n th e spac e (#,2/ , 2), th e plan e z — A + e{y — kx — b)

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1 8

. ELLIPTI C EQUATION S I N NONDIVERGENC E FOR M

cuts a "cap " of f the graph o f ip(x,y). Indeed , 0 and (p(x, y) - [A + e(y - kx - b)] < 0 on al l sides of the trapezoid. Thus, th e line E$ is parallel t o the x-axis. Denotin g b y Y it s ordinate, w e se e that th e graph o f / ( x, y) contain s th e line y = Y, z — A\ — A-\- if^-Y. Settin g

fi(x,y) = f(x,y-Y)-A

1

and applyin g the obtained resul t t o f\, w e conclude that i f the grap h of the functio n f(x,y) contain s a line paralle l t o the x-axis, the n i t als o contain s anothe r suc h lin e with large r ordinate . T o complete th e proof, i t remain s t o note tha t i f the grap h of / contain s tw o lines paralle l t o th e x-axis , the n ther e i s on e more suc h lin e between them . T o prove thi s assertion , w e consider a plane containin g thes e line s and subtrac t fro m / th e linea r functio n whos e grap h i s thi s plan e (perhaps , w e must chang e th e sign o f the difference). The n th e projections o f these line s ont o the plan e (x , y) can be regarded a s the boundary o f the strip II , which mean s tha t the require d lin e exists . • T E S T QUESTION . I n which par t o f the proo f di d we use the fact tha t M(r) i s an uppe r boun d o f the module o f u (bu t not of u itself ) fo r x 2 + y 2 — r2l Wh y i s the conditio n o n the module necessar y here ?

Bibliographic remarks . Th e main goa l of this sectio n is to present th e properties o f a functio n tha t ca n be derived fro m th e maximum principl e regarde d a s a geometrica l propert y independentl y o f an equation satisfie d b y the function. I n this section , no t only th e initial function , bu t any function tha t differ s b y a linea r function i s assumed t o satisfy th e maximum principle . For a function define d i n the entire plan e an d possessing suc h propertie s Bern shtein [Bel ] proved th e following so-calle d "two-sided " Liouvill e theorem : a func tion i s a constan t i f i t i s bounde d fro m below an d fro m above . H e als o prove d that th e ordinary "one-sided " Liouvill e theore m fail s i n this case . I n the proof of Bernshtein, th e function i s assume d t o b e of certai n smoothness . Gelfan d mad e the proposa l tha t th e smoothness i s not essential; i t suffice s t o assum e continuity . This hypothesi s turn s ou t to be true. Th e corresponding resul t wa s established by Adel'son-Vel'skii [Ad]. The principa l ide a o f the proo f o f the Phragmen-Lindelof theore m fo r surface s of generalize d nonpositiv e curvatur e (cf . §1 ) is due to Adel'son-Vel'skii. Th e proof itself wa s obtaine d b y Gerasimov [Ge]. For n > 2 the Bernshtein theore m (henc e als o th e Adel'son-Vel'skii theorem ) fails. Th e corresponding exampl e wa s constructed b y Hopf [Hoi] . A s was noted by Kosmodem'yanskii , a sligh t modificatio n o f the Hopf exampl e lead s t o a coun terexample t o the theorem o f Phragmen-Lindelof typ e fo r surfaces o f generalize d nonpositive curvature . I n this section , thi s theore m i s formulated fo r n — 2 . However, in the case n > 2 it is also possible to make some conclusions about the global behavio r o f a function wit h th e graph o f generalized nonpositiv e curvature .

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§2. s-CAPACIT Y

9

The followin g theore m i s due to Kosmodem'yanskii . Le t f(x) b e a functio n define d in th e cylinde r n i=2

and havin g th e grap h o f generalize d nonpositiv e curvature . Suppos e tha t f(x) i s nonpositive o n th e latera l surfac e o f th e cylinder . Introduc e th e notatio n M(t) = m a x / ( x ) . X\=t

One o f th e followin g case s take s place : Case 1 . f(x) i s nonpositiv e everywher e i n th e cylinder . Case 2 . lirn| rri|_).00 M(x 1 1 )/\x \>0. Case 3 . M(x\) = cons t > 0 . In Cas e 3, the function f(x) attain s the maximum valu e on a line that i s parallel to th e xi-axis . Not e tha t th e rat e o f growt h i n Cas e 2 canno t b e improved . Th e fact tha t th e grap h ha s generalize d nonpositiv e curvatur e mean s tha t i t i s a saddl e in a sense . W e see that som e conclusions, especiall y i n the cas e n — 2 , can b e mad e from th e saddl e propert y alone . Bu t thi s propert y itsel f i s no t sufficien t fo r mor e serious assertions . W e nee d a quantitativ e estimat e fo r th e "saddle-property" . I n the followin g sections , w e will obtai n i t i n term s o f th e s-capacity . §2. s-capacit y Let s b e a positiv e number . Le t E an d T b e B-sets i n W 1 . W e conside r al l possible measure s / x on E, i.e. , completel y additiv e nonnegativ e function s tha t ar e defined o n a cr-algebra consistin g o f subsets o f E an d containin g B-sets (cf . Halmo s [Ha]). A measure \i i s sai d t o b e admissible i f th e followin g conditio n holds :

We se t sup/xE = Cj(E), where th e supremu m i s take n ove r al l admissibl e measures . The numbe r Cj(E) i s calle d th e relative s-capacity o f th e se t E wit h respec t to th e se t T . The relativ e capacit y o f a se t E wit h respec t t o it s complemen t wil l b e calle d simply th e s-capacity an d denote d b y C S(E). Thus , Cf\E(E) =

C

B(E).

We nee d som e propertie s o f th e s-capacity . THEOREM

2.1 . LetE l \, let Q p^ be the cylinder defined by the conditions n-l

2=1

and let p , h — p] o f th e x naxis. W e suppose tha t p i s distributed uniforml y o n [/) , h — p] wit h th e linea r densit y v suc h tha t i f x doe s no t belon g t o th e cylinder , the n

(11) v

t~P-t—;— w^ E ^ * ? + (*„-0 1

L s

In thi s case , w e hav e (12) C

vh

s(Qp,h)^v(h-2p)^

We estimat e th e integra l o n th e left-han d sid e o f the inequalit y (1 1 ) . Le t a poin t x lie o n th e latera l surfac e o f the cylinder . W e eliminat e th e segmen t [x n — p, xn + p] from th e segmen t [p, h — p] of the x n -axis. Th e remainin g se t (i t ma y coincid e wit h the entir e segmen t [/? , h — p]) will b e denote d b y H. W e hav e rp ~pdAt r /r i Jp h

ir


s

Y.^kiak[(s +

2)-e] + ]T>*7 i + c_

Therefore, fo r s > e — 2 and sufficientl y smal l r (r < ro, where r o depend s o n s and the operator , L) the functio n l/\x — x°\ s is subelliptic i n (D D Qf0) \ i x°), wher e D i s the domai n o f th e operator . Bibliographic remarks . Th e excellen t surve y concernin g th e notion s o f a capacity an d a potential a s well a s a detailed bibliograph y ca n b e foun d i n [Land]. We briefly discus s the difference betwee n the approac h i n this book an d th e tra ditional approac h t o th e us e o f capacities an d potentials . Th e traditiona l approac h consists i n considering th e potentia l (generate d b y th e Ries z potential 1 ) y

'J

E\x-y\

s

in th e entir e space , whic h require s th e conditio n s < n on s. I f we ar e intereste d in th e behavio r o f U(x) o n a set T a t a positive distanc e fro m £" , th e valu e o f s is no t essential . Fo r an y linea r uniforml y ellipti c operato r an d sufficientl y larg e 5 (perhaps, greate r tha n n) the functio n U(x) i s subelliptic. I f we ar e intereste d i n a solution t o a linear equatio n withou t th e right-han d side , then it is useful t o regar d U(x) a s a barrier; moreover , E is placed outsid e th e domai n wher e th e equatio n is considered, wherea s U(x) i s studied insid e thi s domain . For the classica l Wiener capacit y w e can give an equivalen t definitio n b y mean s of the Dirichlet integra l for a n external problem for the set E with the unit boundar y condition. A number o f authors prefe r t o work i n terms o f this definitio n an d its generalizations. Fo r equation s wit h selfadjoin t principa l par t thi s approac h i s mor e suitable becaus e i t allow s u s to deal wit h th e principa l par t o f the fundamenta l solution o r eve n the fundamenta l solutio n itsel f instea d o f its roug h approximation . Some work s usin g thes e method s wil l b e discusse d later . However, i n the author' s opinion , th e elementar y approac h presente d i n this book ha s som e advantages . First , i t is simple. Secon d (mor e serious ) i s that in considering nonselfadjoin t equatio n wit h discontinuou s coefficients , th e bes t wa y t o find charasteristi c feature s o f a solution i s to study th e spreadin g o f roots o f the characteristic equatio n a t a point. W e not e tha t th e spreadin g canno t b e remove d since the coefficients ar e discontinuous. Therefore , a constant appears , referre d t o as the ellipticity constant an d denoted b y e, i.e., the ratio of the trace of a matrix to th e smallest eigenvalu e (also , i t should b e minimize d ove r al l linea r transformations) . The behavio r o f a solution depend s o n this constant . I n a sense, it cannot b e bette r than i t is possible t o describe i n terms o f the potentia l wit h kerne l l / r e _ 2 . W e note tha t th e numbe r e plays th e rol e o f dimension i n a series o f exact theorem s (see, fo r example , Chapte r 1 , Theore m 5.3 , simila r t o the Uryso n theore m [Ur] , i n which th e exponen t l/( e — 3), similar t o th e Uryso n exponen t l / ( n — 3), cannot b e replaced b y -^K$ + e). In thi s section , w e also introduce d th e notion s o f s-capacity an d relativ e scapacity. However , onl y the s-capacit y i s essentially use d i n this book . Th e relativ e 1 For th e Riesz potentia l w e recommend [Land] , wher e al l necessary reference s t o origina l works ar e als o presented .

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14

1. E L L I P T I C E Q U A T I O N S I N N O N D I V E R G E N C E F O R M

s-capacity, introduce d i n order t o obtai n semi-additivity , wa s used elsewher e (we refer th e reader t o [Lai]) . §3. Th e lemma o n the normal derivativ e and th e strong maximu m principl e LEMMA 3. 1 (o n the normal derivative) . Let L be the operator (1 ) in the ball QQR and let u(x) be a function subelliptic (superelliptic) in Q° R and continuous in

QR. Suppose that x° G 5^ and > u(x 0))

u(x) < u(x°) (u(x)

for all x G Q°R. If there exists the inner normal derivative f ^ at the point x°, then

0

)•

P R O O F . Fo r definiteness , w e assum e tha t u(x) i s subelliptic . Le t e b e the ellipticity constan t o f the operato r L. W e set u(x°) = m. The n

max u(x) = m — a. a

> 0.

\x\ = R/2

We conside r th e auxiliary functio n v[x) = m — |-— —^ + e|e-2 x

Re

-2>

where e > 0 is so small tha t v(x) > m — a for \x\ = R/2, an d the functions u(x) and v(x) ar e defined i n the spherical laye r

By Lemm a 2.1 , Lv < 0, v\

^

mo UJ L o

* R/2,R U

^

u\\80,0

*LR/2)R

By Theore m 1.1, v(x) ^

u(x) fo

rx

G Q°R/2,R-

Since v(x°) — u(x°), w e have du dn

dv ^ dn X = XQ

M l-l 0 (althoug h w e hav e v > Cs{H) ^P > 0 in D) becaus e w e d o no t kno w i f ther e exist s a limi t o f U(x) a s x G D tend s t o I \ However, i n an y case , w e can asser t tha t lim v(x) xeD,x->r

> 0,

which i s sufficient fo r application s o f th e maximu m principle . Thus, n < v. W e conclud e tha t sup u(x)

^ su

+

^ > )

v^Qi

i

which complete s th e proo f o f th e lemm a sinc e e i s arbitrary .

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•

§4. T H E G R O W T H LEMM A

17

If s < n an d R < 1 in the assumption s o f Lemma 4.1 , then , usin g the inequalit y

cm > ^ which i s valid b y Theore m 2.6 , w e obtai n /x / (16) supw(x

x f £ measiJ \ , ) ^1 + — — ^ — su

x

p u(x)

.

However, th e conditio n J R < 1 is undesirable. Furthermore , th e inequalit y (1 6 ) ca n be improve d b y replacin g R s wit h R n o n th e righ t hand-sid e o f (1 6) . Now w e prov e th e followin g lemma . LEMMA

4.2 . If s [ 1 + r] ——— ] su p u{x), xeD \ R j T ,DnOx0

where rj > 0 is a constant depending on s and n. PROOF. W e first mak e th e similarit y transformatio n whic h transform s Q\ R t o Q% . A s wa s mentioned , th e operato r i s only multiplie d b y a positiv e number . Le t H g o to H' unde r thi s transformation . The n __, mea s H meas H = — . n R Consequently, meas# which complete s th e proo f o f th e lemma . • REMARK 4.1 . Remar k 2. 1 and Exercis e 1 .1 (a ) imply that i f the coefficien t c(x) of th e operato r (1 ) satisfie s th e conditio n c(x) ^ 0 and M/a < n + 2 , then Lemm a 4.2 remain s vali d fo r R < i?o , wher e Ro depend s o n M , c , an d th e bound s o f th e coefficients bi and c ^ 0 . EXERCISE

4.1 . Prov e th e sam e assertio n withou t th e assumptio n c ^ 0 .

Bibliographic remarks . W e consider a solution t o the Dirichle t proble m i n a domain D. Le t th e zer o boundary conditio n b e impose d o n a part o f the boundary . Suppose tha t th e solutio n i s bounded , say , b y 1 . I t i s clear tha t th e zer o boundar y condition o n a par t o f th e boundar y o f D affect s th e solutio n i n som e way ; th e solution i s clos e t o zer o nea r thi s par t o f th e boundary , bu t als o i t canno t b e clos e to 1 a t som e distanc e fro m thi s part . Fo r applications , i t i s importan t t o hav e a quantitative estimat e fo r thi s behavio r dependin g o n th e structur e o f th e par t o f the boundar y wit h zer o boundar y condition . Th e growt h lemm a turne d ou t t o b e a convenien t too l fo r this . O n on e hand , it s proo f i s elementary , an d o n th e othe r hand, i t i s convenien t fo r application s (cf . §§5-7 , 1 0) . Perhaps, th e autho r treat s th e lemm a wit h partiality , becaus e severa l theorem s in the qualitativ e theor y o f second orde r ellipti c equation s hav e bee n prove d b y th e author b y usin g version s o f thi s lemm a (cf. , fo r example , [La2]) . In this lemma , th e "capacity " o f the par t o f th e boundar y wit h zer o boundar y condition i s estimate d i n term s o f th e siz e o f a portio n o f th e complemen t o f th e

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1. E L L I P T I C E Q U A T I O N S I N N O N D I V E R G E N C E F O R M

18

domain i n which the solutio n i s considered. Thi s portio n i s located i n some ball. I n [La2] an d othe r work s b y th e author , th e siz e o f th e portio n wa s characterize d b y its measure . I t i s clear tha t measur e provide s onl y a rough sufficien t characteristic , which does not relat e to the essenc e of the matter . Th e capacity , o n the othe r hand , yields a n adequat e characteristic . I n term s o f capacity , th e lemm a wa s first prove d by Blokhin a [Bll ] (se e detail s i n [B1 2]) . §5. Th e behavio r o f solution s i n a neighborhoo d o f a boundar y poin t We consider a domain D' C D wit h boundar y Y. Le t e be a positive number. A point x° G T i s calle d a n e-regular boundary point i f th e followin g condition s hold . For ever y pai r e\ > 0 an d £ 2 > 0 there i s 6 > 0 suc h tha t fo r an y domai n D' c D with boundar y T' , a uniforml y ellipti c operato r L define d i n D' wit h th e ellipticit y constant e' ^ e , an d a functio n u(x) < 1 , subellipti c fo r L an d continuou s i n D , the inequalit y u\ n ^ 0 0 Ir'nQ^ ^ implies th e inequalit y u\ 0

THEOREM 5.1 . Let x° eY and

< £2 -

\D'nQf e ^ n . We set s = e - 2 and denote

Ca{Q{-m\D)=7m. The point x° is e-regular if the series 00

V 4™ 7r> 771=1

is divergent. PROOF. Le t e\ > 0 an d 82 > 0 b e given . W e conside r a subdomai n D' o f D with boundar y r x , a uniforml y ellipti c operato r L define d i n D' wit h th e ellipticit y constant e' suc h tha t e' ^ e , an d a subellipti c functio n u(x) tha t i s continuou s i n D an d satisfie s th e inequalitie s u(x) < 1 , u(x)\

n

< 0.

It i s require d t o sho w tha t ther e exist s 6 > 0 depending o n e , £1 , and £ 2 such tha t u(x) < 82 at point s x e D' insid e th e ^-neighborhoo d o f th e poin t x° (Figur e 7) . Denote b y m o th e smalles t natura l numbe r suc h tha t 4-mo rao le t ther e exis t a poin t x' G D' suc h tha t \x' -x*\ ^ 4 ~

m

, u{x')

^e

2.

Our goa l i s to sho w tha t th e numbe r m i s less tha n a constan t dependin g o n e , £1, and £2 . Fo r eac h i = m o , . . . , m w e introduc e th e notatio n M; = su

p i^(x)

.

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§5. TH E BEHAVIO R O F SOLUTION S NEA R A BOUNDAR Y POIN T

19

FIGURE 7

For eac h i — mo + l , . . . , r a w e conside r th e ball s Q\- x an d Q*_ (i_iy W e als o consider th e se t o f point s x G D' D Q^_(i_1) a t whic h u(x) > 0 . Denot e b y Di the componen t o f thi s se t tha t contain s a poin t o n th e spher e S%-i a t whic h th e function u take s th e valu e Mi. W e hav e

Cs{Qf-,\Di)

^ Cs{Qf-,\D) = lt.

Applying Lemm a 4. 1 t o th e ball s Q^-z an d Q^_ (i_1 )5 th e domai n Di, an d th e function u, w e obtai n M,_1^(l+ ^-4^)M, . Consequently,

hence

m 1

n

(i+€-4"7i)

V

m)'

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§5. T H E BEHAVIO R O F S O L U T I O N S N E A R A BOUNDAR Y P O I N T

23

Therefore, th e sequenc e {vm}, ra =

fc,fc + 1, . .. ({v

m},

rn

= k, k + 1 ,... )

is monotone i n every D^ ; moreover, i t i s bounded i n view of the maximum principl e and, consequently , i t converges . We not e tha t fo r solution s u m t o th e Dirichle t problem s wit h th e boundar y conditions Um\ =

F

we hav e m a x | u m -Vm\ < Dm

£ ,

where Vm =V

m+Vm.

Hence {u m} converges . 3. Part s 1 and 2 mean tha t fo r a monoton e sequenc e o f domain s th e sequenc e {um} converge s t o a functio n uj whic h i s independen t o f th e metho d o f extensio n of / . Henc e i t converge s fo r an y sequenc e o f domain s {Dm}. Indeed , i f fo r som e sequence o f domain s {D m} th e sequenc e {u m} doe s no t converge , the n w e ca n choose a subsequenc e {D mk} s o tha t D mk C D mk+1 an d th e sequenc e {u m} doe s not converg e either . 4. Th e limi t functio n Uf i s harmonic . Thi s fac t follows , e.g. , fro m th e com pactness theore m whic h assert s tha t an y sequenc e o f function s tha t ar e harmoni c and uniforml y bounde d i n D , i s compact i n an y subdomai n lyin g strictl y insid e D. REMARK 5.1 . Th e argument s i n 1 - 3 ca n b e applie d t o an y uniforml y ellipti c operator L provide d tha t th e Dirichle t proble m fo r L i s solvable i n an y subdomai n that lie s strictl y insid e D an d ha s sufficientl y smoot h boundary . Thus, th e abov e notio n o f a generalize d solutio n t o th e Dirichle t proble m fo r the Laplac e equatio n i s well defined . DEFINITION 5.1 . A poin t x° o n th e boundar y dD o f a domai n D i s sai d t o b e regular if fo r an y continuou s functio n / define d o n dD th e generalize d solutio n Uf to th e Dirichle t proble m wit h th e boundar y conditio n \dD

J

satisfies th e conditio n lim u f{x) = f(x°). xeD As above , w e se t -ym = C

x n-2{Q 4-m\D).

THEOREM 5. 4 (Wiene r criterion) . A point x° is regular if and only if the series oo

(21) £

A

m{n 2

~ hm

is divergent.

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24

1. ELLIPTI C EQUATION S I N NONDIVERGENC E FOR M

PROOF. Sufficiency. W e extend th e functio n b y continuit y t o th e domai n D and denot e th e extended functio n b y F. Fo r e > 0 we find £\ > 0 such tha t

\F(x)-f(x°)\ 0 such tha t

Vm(x) < ^ xeD

mnQf.

Hence ttmW-/(a50) n Q f °

.

Therefore,

uf(x)-f(x°)^e, xeDHQf. The invers e inequalit y i s obtained i n a simila r way . Consequently , u(x) converge s to f(x°) a s x— > x°. REMARK 5.2 . Instea d o f the Laplace operator , w e consider a uniformly ellipti c operator L with th e ellipticity constan t e' ^ e. Le t izm be a solution t o the problem Lum=0, w

mLn =

^

Let Um converge t o a solutio n u(x) o f the equation Lu — 0 . Pro m th e above proo f it i s clear tha t i f a point x° G 3D i s e-regular, the n Uf(x)— > /(x° ) a s x —> x°, x G D . Necessity. T o prov e th e necessit y w e nee d som e propertie s o f th e s-capacit y and especiall y o f the Wiener capacity . Thes e propertie s ar e established i n Lemma s 5.1 and 5.2 below . LEMMA 5.1 . Let E be a bounded closed set, G D E an open set, and e > 0 an arbitrary number. There exists an open set G, E C G C G f, with twice continuously differentiable boundary and a measure ji on G so that for

JG\v-An-2 we have U\ = 1 \dG

and (22) j[_d/i7m < «,

771=7710 — 1

where a wil l b e chosen later . Le t the boundary functio n / b e such tha t /(s°) = l , / ( * ) < !

, f(x)=0,

\x

- x°\ > 4~ m°.

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26

1. E L L I P T I C E Q U A T I O N S I N N O N D I V E R G E N C E F O R M

We exten d / b y continuit y t o th e entir e spac e s o tha t th e abov e condition s ar e preserved fo r th e extende d functio n F(x), i.e. , F(x) ^ 1 , F(x)

= 0 , \x-

x°\ ^ 4

_m

°.

We sho w tha t fo r 6 > 0 there i s a poin t x e D , \X — x°\ < oo

Let Uk\

Dk ~

Uf = li i

'

k—

Introducing th e notatio n Em =

--Qf-m\D,

we ca n writ e C n - 2 y^m) 7m

*

Hm = Em \ E

m+l

Let Then Hm C E

m-\.

:

n-2{Hrn)

In th e notatio n im

C

we hav e oo

m—rriQ

By Lemm a 5.1 , every Hm ca n b e embedde d i n a n ope n se t G m wit h twic e contin uously differentiabl e boundar y suc h tha t On Gmi w e introduce a measur e \im suc h tha t

tem \ x - y\ implies Um\ = and (23) /

1

l 0 , ! ^ i < q, 0 < q < 1 , we se t A measur e /i m o n £? m i s sai d t o b e admissibl e i f

^ ( | L < X fo

rX

£Em

J'EE,m \*-y We se t

where th e supremu m i s take n ove r al l admissibl e measure s o n E m. Prov e tha t th e point x° i s regula r i f an d onl y i f J2 Um(x°) = CXD . ra=l

Bibliographic remarks . W e hav e see n tha t sufficien t an d necessar y condi tions fo r th e regularit y o f a boundar y poin t i n th e cas e o f th e Dirichle t proble m for th e Laplac e equatio n ar e give n b y th e Wiene r criterio n (anothe r proo f ca n b e found i n [Ke ] an d [Lai]) . In 1 949 , Oleini k [01 2 ] prove d tha t th e criterio n fo r th e regularit y o f a bound ary poin t fo r a linea r secon d orde r equatio n wit h sufficientl y smoot h coefficient s coincides wit h a similar criterio n fo r th e Laplac e equation . Later , th e condition s o n coefficients wer e weakened . I n 1 962 , Herv e [He ] prove d tha t i t suffice s t o assum e that th e coefficient s satisf y th e Holde r condition . Recently , Krylo v [Kr ] prove d that instea d o f th e smoothnes s o f coefficient s on e ca n impos e a weake r condition ; namely, th e continuit y o f coefficient s provide d tha t th e modul e o f continuit y uni formly satisfie s th e Din i condition . However, i t turn s ou t tha t a simila r conditio n (perhaps , th e Din i conditio n itself) represent s a border case . I t is possible to construct a n example of an equatio n that ha s discontinuous coefficient s an d i s as close to the Laplace equation a s desired, but a poin t tha t i s regula r fo r th e Laplac e equatio n canno t b e regula r fo r th e constructed equatio n (se e [Lai , M]) . Moreover , a s wa s show n b y Zogra f [Zo] , coefficients ca n eve n b e mad e continuou s i n thi s example . The abov e i s tru e fo r a n equatio n i n nondivergenc e form . I n [LSW ] i t wa s shown tha t th e regularit y conditio n fo r a uniforml y ellipti c equatio n i n divergenc e form wit h arbitrar y measurabl e an d bounde d coefficient s coincide s wit h tha t fo r the Laplac e equation . If coefficient s ar e goo d (i n som e sense ) u p t o a neighborhoo d o f a boundar y point (fo r example , satisf y th e Holde r conditio n i n a neighborhoo d o f a boundar y point), the n th e Wiener conditio n is necessary an d sufficien t fo r the regularity o f the boundary point . Suppos e tha t th e Wiene r conditio n holds , i.e. , th e correspondin g series i s divergent , an d th e poin t i s regular , whic h mean s tha t th e generalize d solution i s continuous a t thi s point . Studyin g th e rat e o f divergenc e o f th e Wiene r series, w e ca n sa y mor e abou t th e behavio r o f th e solutio n a t thi s point . W e ca n indicate th e exac t boun d fo r th e modul e o f continuit y o f th e solutio n a t thi s point .

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30

1. E L L I P T I C E Q U A T I O N S I N N O N D I V E R G E N C E F O R M

This stud y wa s firs t develope d b y Maz'ya [Mai ] wh o use d a different approac h t o the definitio n o f capacity. The majo r par t o f thi s sectio n i s devote d t o th e proo f o f th e necessit y an d sufficiency o f the Wiener criterio n fo r the regularity o f a boundar y poin t fo r the Laplace equation . Th e sufficiency i s established i n a simple way , wherea s th e proof of the necessity occupie s th e rest o f the section (an d Appendix, §1 ) . It seem s tha t the situatio n shoul d b e th e opposite , i.e. , th e necessit y o f th e Wiene r criterio n should b e "almos t obvious " wherea s th e sufficiency "mus t b e proved." Th e matter is tha t introducin g th e notio n o f capacity , w e did no t prov e tha t th e extrema l measure exists . Indeed , w e had no need o f this fac t anywher e els e in the book. Bu t here, w e had to construct (cf . Appendix , §1 ) a measure tha t realize s the extremum. The constructio n itsel f i s not too complicated, bu t looks heavy in contrast wit h the simplicity o f other construction s i n this chapter . Additional bibliographi c remarks . Th e proble m o f th e regularit y o f a boundary poin t i n the case o f a nonsmoot h boundar y an d a secon d orde r linea r equation i n nondivergence for m ha s not been studie d lately . Perhaps , th e most interesting result s wer e obtaine d b y Yu. A . Alkhutov wh o suggeste d som e regularit y conditions i n the case o f a discontinuous boundary . §6. Th e behavior o f solutions a t infinit y We retur n t o an arbitrary uniforml y ellipti c equatio n o f the form (1 /) . I n this section, w e use the growth lemm a t o prove theorem s o f Phragmen-Lindelof type . By this , w e mean theorem s tha t dea l wit h subellipti c function s tha t ar e define d in a n unbounded domai n an d are nonpositive o n the boundary. Dependin g o n th e shape o f the domain , fo r such function s th e minimum rat e of growth a t infinity can be foun d i f there i s at leas t on e point o f the domain a t whic h th e function unde r consideration i s positive. THEOREM

6.1 . Let an unbounded domain D lie in the layer \xn\ < h, —o c < x % < oo , i

= 1 , . . . ,n — 1 .

Let L be a uniformly elliptic operator in D and let u be a subsolution that is continuous in the closure of D and nonpositive on the boundary of D. Then one of the following assertions holds: (a) u ^ 0 in D; (b) ,. M(r) lim — ^ n> 0 , r—>oo e

h

where M(r) = ma

x u(x)

and c > 0 is a constant depending on n and the ellipticity constant e of the operator L. P R O O F . B y using a similarit y transformation , i t i s possible t o reduc e th e re quired assertio n t o the case of some fixed h. I t is convenient t o take h = 1/2. Suppose tha t assertio n (a ) i s false , i.e. , u(x°) = a > 0 fo r a poin t x° = (x°1,x^...,xl) e D. W e set r 0 = \x°\ and r k = r 0 + 4/c , k = 0 , 1 , 2 , . . . . Le t k y b e a point o n the sphere S® wher e u(y k) = M(rk). W e consider th e balls Q\

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§6. T H E BEHAVIO R O F S O L U T I O N S A T I N F I N I T Y

31

and Q\ . Th e differenc e Q\ \D contain s a bal l o f radiu s 1 /4 . Settin g s — e — 2 , from Lemm a 4. 1 w e obtain th e relatio n sup u(x)

> ( l + £,C s(Q°1 /4)) su

xeDnQlk

p u(x)

= ( 1 + 7 ) su

xeDnQ\k

p u(x),

xeDnQ\k

where 7 > 0 depends o n e and n. B y th e maximu m principle , M(r f e + i) ^ su

p u(x).

xeDnQf

Therefore, M(rfc+i)>(l+7)Af(rfc). Hence M(r0 + 4k ) > a( l + 7) * = ae

kln1 +

< ^

and M(r 0 + 4 f c ) > a e

M

^[''0+4('=+1 )]

for sufficientl y larg e /c , which prove s th e theore m i n vie w o f th e maximu m princi ple. • Let Bh denot e th e exterio r o f th e doubl e circula r con e

i=l

We conside r a n unbounde d domai n suc h tha t a poin t x o f th e domai n i s locate d inside Bh i f \x\ i s sufficientl y large . W e wil l se e (cf . Theore m 6.2 ) ho w a func tion subellipti c i n thi s domai n an d nonpositiv e o n it s boundar y increase s whe n h increases, provide d tha t th e functio n i s positive somewher e insid e th e domain . Moreover, w e will assum e tha t h i s smal l enough : h < 1 (instead o f 1 , anothe r constant ca n b e take n here) . I f h i s large , the n th e fac t tha t th e con e i s doubl e i s not essential . Thi s cas e wil l b e discusse d i n Theore m 6.3 . T H E O R E M 6.2 . Let 0 < h < 1 and let D be an unbounded domain. Suppose that there exists R > 0 such that D \ Q° R C Bh- Let L be a uniformly elliptic operator in D and let a subelliptic function u(x) be continuous in the closure of D and nonpositive on the boundary of D. Then one of the following assertions holds: (a) u(x) ^ 0 everywhere in D; (b)

(25) li

m M(r)/r c/h >

0,

r—->oo

where M{r) = max u \x\=r

and c> 0 is a constant depending on e.

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32

1. E L L I P T I C E Q U A T I O N S I N N O N D I V E R G E N C E F O R M x

nl

\

^

\

k

^ y * ] ^

^ 0

c^

A

^

^

^

F I G U R E1 PROOF.

0

W e assum e tha t assertio n (a ) is false, i.e. , ther e exist s a point x° £ D

such tha t u(x°) = a > 0 . Set (26) |x°

| = r 0, r

k

= r 0 ( l + 8ft) fc, k

= 1 ,2,.. . .

Let i t attain th e maximum valu e M{rk) a t a point 7/ fc £ 5 ^. W e consider th e balls QV2hr an< ^ Qthr • Sinc e ft < 1 , the set Q\ hr \ D contain s a bal l o f radius hr (Figure 10). By Lemm a 4.1 , for s = e — 2 we obtain sup u(x)

> I 1 + rj ,^*Z

|

M(r

{2hrky

x£DnQy8hrh

k).

Setting C

V

'{Q\hrk) _ (2hrky

=

C a{Ql) 77-

= 7,

we find tha t 7 depend s onl y o n e, and since M(r f e + i) ^ su

p i6(x)

,

zeDnQ!"rfc

the inequalit y M(rfc+1)^(l +

7

)M(r f c )

holds. Henc e (27) M(r

k)^a(l

+

j) k.

From (26 ) and (27) it follow s tha t lnr fc - l n r 0 ln(l + 8ft) *

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k/4

§6. T H E B E H A V I O R O F S O L U T I O N S A T I N F I N I T Y

33

For sufficientl y larg e / c we hav e M{rk) > a e l n ( 1 + 7 )

N W ^ ar*

l n (1 + 8 h )

Consequently, fo r sufficientl y larg e r k w e obtai n

where c i s a constan t dependin g o n e , whic h yield s (25 ) i n vie w o f th e maximu m principle. • THEOREM

6.3 . Denote by K h the cone 2 n-l

i=l

Let h > 1 . Le £ D be an unbounded domain and let K^ belong to the complement of D. Let L be a uniformly elliptic operator in D and let a subelliptic function u(x) be continuous in the closure of D and be nonpositive on the boundary of D. Then one of the following assertions holds: (a) u(x) < 0 everywhere in D; (b) lim M(r)/r ch ' > 0, r—• o o

where M{r) — maxii(x) |x|=r

and c is a constant depending on e and s = e — 2 . P R O O F . W e assum e tha t assertio n (a ) i s false . Se t H — l/h. Ther e i s a poin t x° e D suc h tha t u(x°) = a > 0 . W e se t |rrr° | = r 0 an d r k = r 0 • 4 fc, k = 1 , 2 , . . ., and conside r th e ball s Q® k and Q® k i . Th e differenc e Q® k \ D contain s a bal l o f radius \r kH (assum e tha t it s cente r i s place d a t th e poin t x'). Settin g s — e — 2 and applyin g Lemm a 4. 1 an d th e maximu m principle , w e obtai n

/C

s(Qi

) \

= (l + ^-C s{Cfx)\ M{r

k.x)

=

( 1 + 7 tf 8 )M(r f c _ 1 ),

fc = l , 2 , . . . , where 7 > 0 is a constan t dependin g o n e. Henc e (28) M(r

k)>a{l

+

iH

8 k

).

Since lnr f c + i = lnr 0 + (f c + l ) l n 4 , we can expres s k a s follows : lnr f c + i - l n r 0 - I n 4 k ~I n4 2

The con e Kh differ s fro m E n \ B^ b y onl y th e propert y tha t i t consist s o f a singl e cavity .

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34

1. ELLIPTI C EQUATION S I N NONDIVERGENC E FOR M

Substituting thi s expressio n i n (28) , we obtai n M(rfc)>a(l + 7i7s)(lnrfc+1-lnro-ln4)/ln4 and, fo r sufficientl y larg e k, M(rk) >

a(l +

7

t f ' ) ( > n r f c +1 ) / 2 1 n 4 =

^ ( 1 + ^ ) 1 / 2I n4^ ^

The constan t c is taken s o as to satisf y th e inequalit y ln( l + ^/Hs) > 2cln4H s (th e choice i s possibl e becaus e H < 1 ) . Applyin g th e maximu m principle , w e arriv e a t the require d inequality . • 6.1 . I n Theore m 6.3 , we derive d th e inequalit y M(r) > ar ch * (fo r sufficiently larg e r) . Prov e tha t M(r) > ar Clh fo r s > 1 and sufficientl y larg e r . EXERCISE

In Theore m 6. 3 w e assume d tha t h > 1 . Th e choic e o f th e constan t 1 is no t essential. W e ca n tak e an y othe r constant . Th e poin t i s that i n th e estimat e

r—>oo ^

the sam e constan t c ca n b e take n onl y fo r thos e h tha t ar e greate r tha n som e hoThe constan t c decreases a s ho decreases. The cas e wher e K^ become s th e half-spac e (h = 0 ) i s o f specia l interest . I n this case , w e obtain a n exac t estimate ; namely , th e increas e i s not slowe r tha n tha t of a linea r function. 3 Considerin g th e Phragmen-Lindelo f theore m fo r a harmoni c function i n a half-plane , w e se e tha t i f th e harmoni c functio n i s define d i n th e upper half-plane , i s nonpositiv e o n th e x-axis , an d i s positiv e a t som e point , the n it necessaril y increase s a t infinit y no t mor e slowl y tha n a linea r function . W e wil l see tha t thi s fac t i s immanen t fo r subellipti c function s fo r operator s (1 /) . THEOREM 6.4 . Let an unbounded domain D lie in the half-space x n > 0 . Let L be a uniformly elliptic operator in D and let a subelliptic function u(x) be continuous in the closure of D and nonpositive on the boundary of D. Then one of the following assertions holds: (a) u(x) ^ 0 everywhere in D; (b)

(29) li

m — ^> 0, i—>oo T

where M(r) — max| r r | = r u{x). P R O O F . W e assum e tha t assertio n (a ) i s false , i.e. , ther e exist s a poin t x° = ( x ? , . . . , x°) £ D suc h tha t u(x°) = a > 0. W e se t

v(x)=u{x)-—. Denote b y D' th e se t o f point s x G D a t whic h v{x) > 0 and b y D" th e componen t of D' containin g th e poin t x°. However, i n thi s estimate , th e lowe r limi t wil l b e replace d wit h th e uppe r limit .

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§6. T H E B E H A V I O R O F S O L U T I O N S A T I N F I N I T Y

35

By th e maximu m principle , D" i s an unbounde d domain . Le t c be the constan t from Theore m 6. 2 correspondin g t o th e ellipticit y constan t o f th e operato r L. W e choose hj 0 < h < 1 , so a s t o satisf y th e conditio n

Take Bh tha t ha s th e sam e meanin g a s i n Theore m 6.2 . Tw o case s ar e possible . Case 1 . Ther e i s R suc h tha t

D"\Q°RcBh. Case 2 . Ther e i s a sequenc e {x m} suc h tha t x m G D" \ Bh an d \x rn\— » o c a s m— » oo. In Cas e 1 , we appl y Theore m 6.2 . In Cas e 2 , we hav e M (

"

m ) =

2^

+ V ( x m ) >

2^|

> 6 | a ; m |

'

where b is a constant . Thus, (29 ) hold s i n bot h cases . • T E S T QUESTION .

Wh y canno t li m b e replace d b y li m i n ou r proof ?

EXERCISE 6.2 . Sho w tha t fo r a harmoni c functio n define d o n th e plan e th e exact valu e o f c is TT/2 i n Theore m 6. 1 an d ir in Theore m 6.2 .

Unsolved problems . PROBLEM 1 . Le t Kh an d D hav e th e sam e meanin g a s i n Theore m 6.3 . I s i t true tha t fo r an y e > 0 there i s a smal l numbe r h > 0 such tha t

— Mir) lim —^ >

0

for a functio n u(x) tha t i s subharmoni c i n D, i s continuou s i n D, vanishe s o n dD, and i s positive i n D? 2 . Le t K' h b e th e con e YA=I X 1 < ^ ^ L x n > 0 . Le t D b e a n unbounded domai n i n K' h, le t L b e a uniforml y ellipti c operato r i n D, an d le t u(x) be subellipti c an d continuou s i n D , positiv e i n D an d vanis h o n dD. (a) I s i t tru e tha t T ^ M{r) lim = oc : PROBLEM

r—>oo T

(b) I s i t tru e tha t — M(r) lim — r ^ > 0 r—>oo T

e

for som e positiv e el The answe r i s unknow n eve n i n th e plan e case . PROBLEM 3 . Le t a functio n u(x,y) satisf y th e ellipti c equatio n (*) a(x,

y)u xx + 26(rr , y)uxy + c(x , y)uyy = 0

on the half-plane y > 0 and be continuous up to the x-axis. Suppos e that u(x,y) > 0 for y > 0 and u = 0 for y = 0 . Is i t tru e tha t ^(x , 2/) is a linea r function ?

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36

1. ELLIPTI C EQUATION S I N NONDIVERGENC E FOR M

FIGURE1

1

PROBLEM 4 . Equatio n (* ) ca n b e considere d o n th e entir e plane . Th e solutio n u(x,y) i s define d o n th e entir e plan e an d

M(r) lim