Fully Nonlinear Elliptic Equations 0821804375, 9780821804377

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Titles in This Series 43 Lui s A. Caffarelli an d Xavier Cabr£, Full y nonlinea r elliptic equations, 1 99 5 42 Victo r Guillemin and Shlomo Sternberg, Variation s on a theme by Kepler, 1 99 0 41 Alfre d Tarski and Steven Givant, A formalization o f set theory withou t variables , 1 98 7 40 R . H. Bing, Th e geometric topology of 3-manifolds , 1 98 3 39 N . Jacobson, Structur e an d representation s o f Jordan algebras , 1 96 8 38 O . Ore, Theor y of graphs, 1 96 2 37 N . Jacobson, Structur e of rings , 1 95 6 36 W . H. Gottschalk and G. A. Hedlund, Topologica l dynamics , 1 95 5 35 A . C. Schaeffer and D. C. Spencer, Coefficien t region s for Schlich t functions , 1 95 0 34 J . L. Walsh, Th e location of critica l points of analytic and harmonic functions, 1 95 0 33 J . F. Ritt, Differentia l algebra , 1 95 0 32 R . L. Wilder, Topolog y o f manifolds, 1 94 9 31 E . Hille and R. S* Phillips, Functiona l analysi s and semigroups , 1 95 7 30 T . Radd, Lengt h and area, 1 94 8 29 A . Weil, Foundation s of algebraic geometry, 1 94 6 28 G . T. Whyburn, Analyti c topology, 1 94 2 27 S . Lefschetz, Algebrai c topology, 1 94 2 26 N . Levinson, Ga p an d density theorems, 1 94 0 25 Garret t BirkhofT, Lattic e theory, 1 94 0 24 A . A. Albert, Structur e o f algebras, 1 93 9 23 G . Szego, Orthogona l polynomials , 1 93 9 22 C . N. Moore, Summabl e series and convergenc e factors , 1 93 8 21 J . 1VL Thomas, Differentia l systems , 1 93 7 20 J . L . Walsh, Interpolatio n an d approximatio n b y rationa l function s i n th e comple x domain, 1 93 5 19 R . E. A. C. Paley and N. Wiener, Fourie r transform s i n the complex domain , 1 93 4 18 M L Morse, Th e calculus of variations i n the large, 1 93 4 17 J . M. Wedderburn , Lecture s on matrices , 1 93 4 16 G . A. Bliss, Algebrai c functions, 1 93 3 15 M . H. Stone, Linea r transformations i n Hilber t spac e and thei r application s to analysis, 1932 14 J . F. Ritt, Differentia l equation s from th e algebraic standpoint, 1 93 2 13 R . L. Moore, Foundation s of poin t se t theory, 1 93 2 12 S . Lefschetz, Topology , 1 93 0 11 D . Jackson, Th e theor y of approximation, 1 93 0 10 A . B. Coble, Algebrai c geometry an d thet a functions , 1 92 9 9 G . D. BirkhorT, Dynamica l systems , 1 92 7 8 L . P. Eisenhart, Non-Riemannia n geometry , 1 92 2 7 E . T. Bell, Algebrai c arithmetic, 1 92 7 6 G . C. Evans, Th e logarithmic potential, discontinuous Dirichle t and Neuman n problems , 1927 5.1 G . C . Evans , Functional s an d thei r applications ; selecte d topics , includin g integra l equations, 1 91 8 5.2 O . Veblen, Analysi s situs, 1 92 2 4 L . E. Dickson, O n invariant s and th e theory o f number s W. F. Osgood, Topic s in th e theory o f function s o f several complex variables, 1 91 4 {Continued in the back of this publication)

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AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME 43

Fully Nonlinear Elliptic Equations Luis A. Caffarelli Xavier Cab re

American Mathematical Society Providence, Rhode Island

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2000 Mathematics Subject Classification. Primar y 35J60 ; Secondar y 35B45 , 35B65. ABSTRACT. Thi s boo k provide s a self-containe d developmen t o f th e regularit y theor y o f solution s of fully nonlinea r ellipti c equations . Thes e partia l differentia l equation s aris e i n contro l theor y an d optimization. Th e goa l o f thi s wor k i s t o exten d th e classica l Schaude r an d Calderon-Zygmun d regularity theorie s fo r linea r ellipti c equation s t o th e full y nonlinea r context . Th e boo k contain s a detaile d presentatio n o f al l th e technique s needed . W e d o no t trea t the m i n thei r greates t generality; rathe r w e presen t th e ke y idea s an d prov e al l th e result s neede d fo r th e subsequen t theory. We develo p th e theor y o f viscosit y solution s o f nonlinea r equations , th e Alexandrof f estimat e and Krylov-Safono v Harnac k inequalit y fo r viscosit y solutions , Jensen' s uniquenes s theor y fo r viscosity solutions , Evan s an d Krylo v regularit y theor y fo r conve x full y nonlinea r equations , an d finally th e regularit y theor y fo r full y nonlinea r equation s wit h variabl e coefficients .

Library o f Congres s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Caffarelli, Lui s A . Fully nonlinea r ellipti c equation s / Lui s A . Caffarelli , Xavie r Cabre . p. cm . — (Colloquiu m publication s / America n Mathematica l Society , ISS N 0065-9258 ; v. 43 ) Includes bibliographica l references . ISBN 0-821 8-0437- 5 (alk . paper ) 1. Differential equations , Elliptic . 2 . Differential equations , Nonlinear . I . Cabre, Xavier , 1 966 II. Title . III . Series : Colloquiu m publication s (America n Mathematica l Society) ; v . 43 . QA377.C241 99 5 515' .353—dc20 95- 502

4 CIP

C o p y i n g an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r 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 customar y acknowledgmen t o f th e sourc e i s given . Republication, systemati c copying , o r multipl e reproductio n o f an y materia l i n thi s publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addresse d t o th e Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-229 4 USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 1 99 5 b y th e America n Mathematica l Society . Al l right s reserved . Reprinted b y th e America n Mathematica l Society , 1 997 , 2011 . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This publicatio n wa s printe d fro m file s prepare d b y th e author s usin g A^S-T^, the America n Mathematica l Society' s TJjj X macr o system . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 41 3

61 51 41 31 21 1

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Contents Introduction 1. Preliminarie s 5 1.1. Basi c Notatio n an d Terminolog y 5 1.2. Tangen t Paraboloid s an d Secon d Orde r Differentiabilit y 6 2. Viscosit 1 y Solution s o f Elliptic Equation s 1 2.1. Viscosit y Solution s 2 2.2. Th e Clas s S o f Solution s o f Uniforml y Ellipti c1 Equation s 4 2.3. Example s o f Fully Nonlinea1 r Ellipti c Equation s 7 Notes 9 3. Alexandrof f Estimat e an d Maximu m Principl e 2 1 3.1. Alexandroff-Bakelman-Pucc i Estimat e 2 1 Notes 2 8 4. Harnac k Inequalit y 2 9 4.1. Tw o Importan t Tool s 2 9 4.2. Harnac k Inequalit y 3 1 4.3. C a Regularit y 3 7 5. Uniquenes s o f Solution s 4 5.1. Jensen' s Approximat e Solution s 4 5.2. Uniquenes s fo r F(D 2u) = 0 4 5.3. C 1 '" Regularit y fo r F(D 2u) = 0 4 5.4. Application s t o Concav e Equation s 4 Notes 4

3 3 5 6 8 9

6. Concav e Equation s 5 1 6.1. Evans-Krylo v Theore m 5 1 6.2. C 2 ' a Regularit y fo r F(D 2u) = 0 5 4 7. W 2* Regularit y 5 7.1. W 2* Estimate s 5

9 9

8. Holde r Regularit y 7 8.1. C 2 '* Estimate s 7 8.2. C 1 '" Estimate s 7

3 3 8

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vi CONTENT

S

9. Th e Dirichle t Proble m fo r Concav e Equation s 8 9.1. Bernstein' s Techniqu e 8 9.2. C 2 ' Q Estimat e u p t o th e Boundar y fo r F(D 2u) = 0 8 9.3. Th e Dirichle t Proble m 9

5 5 8 5

Bibliography 9

9

Index

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Introduction A In 1 97 9 Krylo v an d Safono v [KS1 ,2 ] prove d th e Harnac k inequalit y fo r solu tions o f secon d orde r ellipti c equation s i n nondivergenc e for m wit h measurabl e coefficients. This opened the way to the development o f a regularity theory for fully nonlin ear equations. Firs t wa s Evans-Krylov C 2 , a interio r estimat e [E3] , [K3,4] and the n up t o th e boundar y regularit y tha t allowe d fo r th e developmen t o f th e continuit y method (see , fo r instance, [GT]) . Simultaneously, Crandall-Lion s [CL ] an d Evan s [El,2 ] develope d th e concep t of weak solution s o f linear or nonlinear equation s in nondivergence for m (th e 'Vis cosity" method) , whic h took th e place tha t th e Dirichle t principl e an d the concep t of variational solutio n enjoy s i n the divergenc e for m theory . This idea , togethe r wit h Jensen' s uniquenes s theore m [J] , provided a powerfu l compactness proo f which allowed one of the authors to develop a perturbation the ory, showin g tha t regularit y o f solutions i s preserve d unde r suitable perturbation s of th e equation s (se e [CI,2,3]) . Relate d result s wer e obtaine d b y differen t mean s by Safonov [Sl,2 ] an d Trudinge r [Tl,3] . These note s ar e base d o n a serie s o f lecture s a t th e Couran t Institute , Ne w York University , i n th e sprin g o f 1 993 . W e than k D . Phare s an d C . Warfiel d fo r a very efficient typin g o f this manuscript .

B In thes e note s w e develop th e regularit y theor y o f solution s o f full y nonlinea r equations, tha t is , elliptic equation s o f the for m (0.1) F(D

2

u,x) =

f(x);

here D 2u denote s th e Hessia n o f th e functio n u. W e describ e th e perturbatio n theory introduce d i n [Cl,2 ] fo r equations o f th e for m (0.2) F(D

2

u) = 0

(which correspon d t o "homogeneou s equation s wit h constan t coefficients " i n th e linear case). B y mean s of this theory , w e obtain C a , C 1 , a , C 2 a an d W 2>p interior a priori estimate s fo r solution s o f (0.1 ) . Throughou t thes e notes , w e wil l alway s assume tha t (0.1 ) i s uniformly elliptic . The simples t cas e i s the on e of linea r equations, i n which we ma y assum e tha t (0.2) i s Au = 0 . Recal l tha t on e ma y estimat e th e derivative s o f a harmoni c l

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2

INTRODUCTION

function (i.e. , a solutio n o f Au = 0 ) i n th e interio r o f a domai n b y th e oscillatio n of th e functio n itself . Th e genera l ide a i s tha t th e sam e propert y remain s tru e fo r linear smal l perturbation s o f th e Laplacian . Mor e precisely , suppos e tha t u i s a solution o f a linea r uniforml y ellipti c equatio n i n nondivergenc e form : (0.3) a

ij(x)diju

=

f(x)

(we use summation conventio n ove r repeated indices) . W e then have , for a bounde d solution u o f (0.3 ) i n th e uni t bal l B 1 o f R n : (a) (Cordes-Nirenber g typ e estimates) . Le t 0 < a < 1 an d assum e tha t llaij ~ a(B1 /2) an d (c) (Calderon-Zygmund) . I f a^ ar e continuou s i n B\ an d / € L V{B\), fo r som e 1 < p < oo , the n u e W 2*{Bl/2) an d \\u\\ W2lP{Bl/2) < C(||tt|| L oo (Bl) + ||/||LP(BI))-

Here w e exten d thes e result s t o solution s o f (0.1 ) . Eve n i n th e linea r case , the techniqu e provide s ne w result s sinc e th e closednes s o f a ^ t o 6ij i s measure d i n L n -norm instea d o f L°°-nor m (recal l tha t n i s Euclidean dimension) . The main tool in the new approach i s the Alexandroff-Bakelman-Pucci estimat e and maximu m principle . W e describ e ho w t o us e thi s estimat e an d th e Calderon Zygmund cub e decompositio n t o (1) contro l th e distributio n functio n o f a solution ; thi s wil l lea d t o th e Harnac k inequality, an d henc e t o C a regularity ; (2) approximat e i n L°° a solution by affine function s (respectively , paraboloids) ; this will lea d t o C 1 ,a (respectively , C 2,a) estimates ; (3) contro l th e curvatur e o f tangent paraboloid s t o th e grap h o f a solution ; thi s will lead t o W 2,p estimates . Therefore, a ke y poin t i s t o understan d th e partia l derivative s o f a functio n b y its polynomia l approximations . Th e metho d roughl y describe d abov e i s basicall y nonlinear, i n th e sens e tha t i t doe s no t rel y o n differentiatin g equatio n (0.1 ) . Thi s implies tha t th e result s i n thi s wa y obtaine d appl y t o genera l (i.e. , no t necessaril y smooth) full y nonlinea r equations , suc h a s Pucci , Bellma n an d Isaac s equations . See [E3] , [CNS] , [CKNS ] an d [Kl ] wher e regularit y i s attacke d b y differentiatin g equation (0.1 ) . In Chapte r 1 we introduc e som e terminolog y an d describ e th e relatio n (whic h we hav e alread y mentione d above ) amon g differentiabilit y propertie s o f a functio n u an d th e tangen t paraboloid s t o th e grap h o f u. Throughout thes e notes , we will conside r viscosit y solution s o f (0.1 ) ; we defin e them an d giv e thei r basi c propertie s i n Chapte r 2 . Thi s ver y wea k concep t o f solution will le t u s defin e a clas s o f function s containin g al l classica l solution s o f linear an d nonlinea r ellipti c equation s wit h fixe d ellipticit y constant s an d bounde d measurable coefficient s (se e Section 2.2). I n Section 2.3 we give important example s of full y nonlinea r equations . Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms

INTRODUCTION

3

Chapter 3 is devoted t o prove th e Alexandrofl-Bakelman-Pucci estimat e and maximum principl e fo r viscosity solutions , whic h will pla y a ke y rol e i n all th e future regularit y theory . Harnack inequalit y fo r viscosity solution s i s proved i n Chapte r 4 usin g the Alexandroff estimat e an d th e Calderon-Zygmun d technique ; th e proo f i s essentiall y the sam e a s the first on e discovere d b y Krylov an d Safono v i n [KS1,2]. A s a consequence o f the Harnac k inequality , w e get interior C a regularit y fo r solutions of (0.1). I n Section 4.3 , we also prov e globa l C Q regularity . In Chapte r 5 we describ e Jensen' s approximat e solution s o f (0.2), first intro duced i n [J] . We use them to prove uniqueness for the Dirichlet proble m correspond ing to (0.2) . Section s 5.3 and 5. 4 contain other applications of Jensen's approximat e solutions; w e obtain basi c properties o f the first an d secon d derivative s o f solution s of (0.2) . Fo r instance , w e prove C 1 , a interio r regularit y fo r solution s of (0.2). Chapter 6 is devoted t o concave (o r convex) equation s o f the for m (0.2) . We prove that viscosit y solutions of concave equations F(D 2u) = 0 are C 2,oc and satisf y an interio r C 2 , a a priori estimat e in terms of ||tt||£oo. Thi s fac t will imply tha t th e regularity theor y o f chapters 7 and 8 for equation (0.1 ) applie s t o any functiona l F(M, x) which is a concave function o f M. W e obtain this interior estimate combin ing Evans-Krylo v theore m (se e Sectio n 6.1 ) an d a new proo f (describe d i n Section 6.2) o f the C 1 ' 1 regularit y o f viscosity solution s of concave equation s F(D 2u) = 0. In Chapte r 7 w e prov e W 2,v interio r a priori estimate s fo r solution s o f 2 F(D u,x) = f(x). A basic assumptio n neede d i s that th e "homogeneous equa tions wit h constant s coefficients " 2

(0.4) F(D

u(x),x0)=0

1 ,1

have (fo r an y fixed XQ) C interio r a priori estimates ; thi s assumptio n i s fulfille d when F is concave in D2u, as we prove in Chapter 6. C 2 ' a an d C1,a interio r a priori estimate s fo r solutions o f (0.1 ) ar e proved in Chapter 8 , assumin g tha t th e sam e typ e o f estimates hol d fo r solution s of (0.4). In the cas e of C2,a estimates , this assumptio n i s fulfilled b y any concav e functiona l F\ for th e C 1 ,a estimates , th e assumptio n i s fulfilled b y an y functiona l F , as we show in Chapte r 5. In Chapte r 9 we continu e th e study o f solutions o f concave equation s o f the form (0.2) . Sectio n 9. 1 describes Bernstein' s technique , which i s an alternative wa y of proving, onl y fo r smoot h solutions , th e C 1 , 1 interio r a priori estimat e of Section 6.2. I n Section 9. 2 w e prove a C 2>a a priori estimat e u p to the boundar y fo r such solutions, which was independently discovere d b y Krylov, an d Caffarelli , Nirenber g and Spruck . Thi s C 2 , a estimat e u p to the boundar y wil l let us apply th e metho d of continuit y an d solv e th e Dirichle t proble m fo r F{D 2u) = 0 (see Sectio n 9.3) .

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CHAPTER 1

Preliminaries 1.1. Basi c Notatio n an d Terminolog y K n wil l denote th e n-dimensiona l Euclidea n spac e wit h norm s

|x|oo = m a x { | x i | , . . . , | x n | } . If B T = B r(xo) = {x € R n : \x — XQ\ < r} i s a bal l (b y bal l w e alway s mea n open ball) , w e denote B ar(x0) b y B ar. We consider th e ope n cub e Qr(x0) = {x G Mn : |a ? - ar 0|oo < ^ } with cente r x 0 an d side-lengt h r . fi wil l always b e a bounde d domai n (i.e. , bounded ope n connecte d set ) o f R n . Throughout thes e notes , A and A are tw o fixed constant s suc h tha t 0 < A < A ; they wil l b e calle d ellipticit y constants . A constant i s called universal i f it depend s only o n n , A and A (recal l tha t n i s dimension) . C will denot e a positiv e constant ; i t ma y b e differen t i n eac h inequalit y o r formula. diam(fi) an d \Cl\ will denot e th e diamete r an d th e n-dimensiona l Lebesgu e measure o f f2 , respectively . Given a functio n u , w e denote b y u + an d W th e positiv e an d negativ e parts of u, respectively , s o tha t u = u^ — u~. Th e suppor t o f u wil l b e denote d b y suppw . We wil l als o writ e du dxi d2u dxidxj

diU — Ui,

D2u wil l denote th e Hessia n o f tx , i.e. th e symmetri c matri x wit h entrie s Uij. A functio n L i n E n wil l b e calle d affine i f L(x) = l 0 + l{x), where i 0 i s a rea l numbe r an d / i s a linea r function . 5

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6

. PRELIMINARIE S

A paraboloid is a polynomial i n ( # i , . . . , xn) o f degree 2 . An y paraboloi d P ca n be writte n a s P(x) = L(x) + - x l Ax, where L i s an affin e functio n an d A = D 2P is a, symmetric matrix . Throughout thes e notes , smoot h mean s o f C° ° class . Wk'p(Q) denote s th e Sobole v spac e o f function s whic h belong , togethe r wit h their derivative s u p t o orde r fc, to L P (Q). C*' a (ft) an d C k^(U) denot e Holde r ( 0 < a < 1 ) an d Lipschit z (a = 1 ) spaces; here k i s a nonnegativ e integer . Thei r norm s ar e

IMIc*.«(R) = Nlc*(fi) + I ^ k n where (1-1) M a . O

= SU

P

_-n——^-.

1,2. Tangen t Paraboloid s an d Secon d Orde r Differentiabilit y In thi s sectio n w e deduc e som e secon d orde r differentiabilit y propertie s o f a function u fro m knowledg e abou t th e tangen t paraboloid s t o the grap h o f u. Thes e results wil l b e use d i n th e regularit y theor y o f futur e sections . Se e [G] , [Cm ] an d [W] for th e definitio n o f Campanat o norm s an d mor e genera l results . Let u s sa y tha t P i s a paraboloi d o f opening M wheneve r (1.2) P(x)

2

= lo + l(x)±™\x\

,

where M i s a positiv e constant , 1 $ is a constan t an d I i s a linea r function . P i s convex whe n w e have - f i n (1 .2 ) an d concav e whe n w e have — i n (1 .2) . Given tw o continuou s function s u an d v define d i n a n ope n se t A an d a poin t XQ E A) w e say tha t v touches u b y abov e a t XQ in A wheneve r u(x) < v(x) u(x0) -

Vx€i,

V(XQ).

We also hav e th e analogou s definitio n o f touchin g b y below . Let u b e a continuou s functio n define d i n fi an d A b e an ope n subse t o f ft. Fo r xo € A , w e defin e

(1.3) e(%A)(x

0)

to be the infimum o f all positive constants M fo r which there is a convex paraboloi d of openin g M tha t touche s u b y abov e a t XQ in A. W e defin e (1 .3 ) t o b e o o i f n o such constan t M exists . I t i s eas y t o se e tha t 0(tt , A) i s a measurabl e functio n in A. Using concav e paraboloid s tha t touc h u b y below , w e similarly defin e e{u,A){x0)e [0,oo]

,

We finally conside r 9(U,A)(XQ)

=

su p {©(iz, A)(x0),Q(u, A)(x

Q)}

< oo.

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1.2. TANGEN T PARABOLOID S AN D SECON D ORDE R DIFFERENTIABILIT Y 7

Given XQ 6 ft, w e sa y tha t u i s C 1 , 1 b y abov e a t # 0 [resp . C 1 ' 1 b y belo w a t xo; C 1 , 1 at*o?o ] i f @(u,A)(xo) < o o [resp . Q(U^A)(XQ) < oo ; 0(w , ^4)(x0) < oo] , for som e neighborhoo d A o f XQ. Propositio n 1 . 2 below wil l justify ou r terminolog y (note tha t th e spac e C 1 ' 1 wa s alread y define d i n Sectio n 1 .1 ) . If u i s C 1 , 1 a t # 0 then t t is differentiable a t #0 , since w lies between tw o tangen t paraboloids i n a neighborhoo d o f XQ. Let u s conside r th e secon d differentia l quotient s o f u a t xo 2

(LA) l±

u{x

hU(X0)

Q

—

+ h) + ujxp -h)- 2u(x

—

0)

,

here h e R n an d we assume that rc 0 +/i and x 0 —h belong to ft. Not e that A^P = M [resp. A ^ P = —M] when P i s a conve x [resp . concave ] paraboloi d o f openin g M . It follow s that , fo r an y XQ £ ft, (1.5) - e ( u , B

w(x0))(x0)

^

A u x

l ( o)
2u € L p (ft) and 2

(1.7) | p PROOF.

^||LP(n) 2t*l|L~(B) < 2\\9(u,e)\\ Loo{By Since Ui = diU € W 1 ,O0(B) an d i? is convex, we have that ui is continuous and I

-£t*i(te + ( 1 - t)y)d t o i

J0

for an y x,y € B . Usin g that ||I? 2ti||z,«>(B) ^ 2||9( ,u,e)||x/oo(B), w e finally conclude (1.10). • Proposition 1 . 1 will be used in Chapter 7 to prove W 2,p estimates , Propositio n 1.2 will be needed, together with the following result, in the proof of the Alexandroff estimate for viscosity solutions (se e Chapter 3), THEOREM 1 .3 . Let H:T$ d c l ^ E H ea Lipschitz map. Then H is differentiable almost everywhere in B d, Let Ac Bd be such that \Bd\A\ = 0 and H is differeniiable at every point in A, Then (1.11) \H{B

d)\
0. (5) M+{M) + M~(N) < M +{M + N) < M+{M) + M+(N). (6) A < - ( M ) + J M - ( J V ) < A < " ( ^ + ^ ) < A ^ - ( M ) + Af +(AT). (7)A T > 0 => A||AT|| < M'(N) < M +(N) < nA\\N\\. (8) Mr and M+ are uniformly elliptic with ellipticity constants A,nA . P R O O F . (1 ) , (2) , (3 ) an d (4 ) ar e clear . (5 ) an d (6 ) follo w immediatel y fro m (2.4) an d (2.5) . (7 ) i s clear an d (8 ) follow s fro m (5) , (6 ) an d (7) . •

We now defin e th e clas s 5 . DEFINITION 2.1 1 . Le t / b e a continuous functio n i n Q and A < A two positiv e constants. W e denot e b y 5(A , A, /) th e spac e o f continuou s function s u i n ft suc h that M+{D 2u, A , A) > f(x) i n th e viscosit y sens e i n ft. Similarly, 5(A , A, /) denote s the spac e of continuous functions u i n Q such tha t M~(D2U) A , A) < f(x) i n th e viscosit y sens e i n fi. We als o defin e 5(A, A, /) = 5(A , A, /) n S(A , A, / ), S*(A, A, /) = £(A , A, - l / l) O 5(A, A, | / | ).

Clearly, w e hav e tha t 5(A,A,/ ) C S*(\,AJ) an d 5(A,A,0 ) = 5*(A,A,0) . We wil l denot e 5,_5,5,5*(A , A , /) b y SL,_S,S,S*(f) whe n th e choice_o f A, A i s un derstood, an d 5 , 5 , 5 , 5 * ^ ^ , 0 ) b y 5,5,5,5*(A,A ) (o r simpl y 5 , 5 , 5 , 5 * ) . W e will call the function s i n 5,5,5(A , A , /) subsolutions , supersolution s an d solutions , respectively. W e now stat e som e propertie s o f these classe s o f functions . LEMMA 2.1 2 . _

(1) A 7 < A < A < A ' = » ^(A^A, /) C 5(A' , A7, / ); the same holds for 5 , 5 , 5 * . (2) t i e S ( A , A , / ) = » - u G S ( A , A > - / ) . (3) a > 0 , r > 0 , u e 5(A , A, / ), v(y ) = au(y/r) for y € rf t = > ^5(A,A,a/(y/r)/r2). (4) w G 5(A,A,/),< £ G C 2 (Q) an d X + ( D 2 ^ ( x ) ) < s(z ) /o r an y x € ft = » u-0£5(A,A,/-