293 33 15MB
English Pages 104 [113] Year 1995
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
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)
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
This page intentionally left blank
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
http://dx.doi.org/10.1090/coll/043
AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME 43
Fully Nonlinear Elliptic Equations Luis A. Caffarelli Xavier Cab re
American Mathematical Society Providence, Rhode Island
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
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
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
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
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
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
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
103
http://dx.doi.org/10.1090/coll/043/01
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
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
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) .
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
This page intentionally left blank
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
http://dx.doi.org/10.1090/coll/043/02
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
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
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.
Licensed to AMS. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms
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,/-