Regularity for Fully Nonlinear Elliptic Equations with Neumann Boundary Data

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1 Communications in Patial Diffeential Equations, 31: , 2006 Copyight Taylo & Fancis Goup, LLC ISSN pint/ online DOI: / Regulaity fo Fully Nonlinea Elliptic Equations with Neumann Bounday Data EMMANOUIL MILAKIS AND LUIS E. SILVESTRE Depatment of Mathematics, Univesity of Texas at Austin, Austin, Texas, USA We obtain local C, C 1, and C 2 egulaity esults up to the bounday fo viscosity solutions of fully nonlinea unifomly elliptic second ode equations with Neumann bounday conditions. Keywods Fully nonlinea elliptic equations; Neumann bounday conditions; Viscosity solutions. Mathematics Subject Classification Pimay 35J65; Seconday 35B Intoduction The theoy of viscosity solutions gives a solid famewok to study fully nonlinea elliptic equations, and povides a poweful way to pove existence and uniqueness in a vey geneal setting. The question of egulaity of the coesponding solutions (that in pinciple ae meely continuous has been studied extensively in the last decade. Thee ae good esults fo inteio egulaity as well as fo egulaity up to the bounday in the case of the Diichlet poblem. Howeve, fo the Neumann poblem, thee ae still not many esults. We intend to wok in that diection. On the othe hand thee ae seveal aticles coesponding to uniqueness, compaison theoems, Hölde and Lipschitz continuity fo solutions of geneal fully nonlinea second ode elliptic equations with Neumann type bounday conditions. We efe the eade inteested in the viscosity solutions appoach to Ishii and Lions (1990 and Ishii (1991 whee the authos investigate uniqueness esults that yield existence via an adaptation of the classical Peon s method. A late aticle of Bales (1993 gives uniqueness and Lipschitz egulaity esults fo quite geneal bounday conditions in the case whee the bounday is assumed to be smooth enough and the diffeential opeatos ae basically Lipschitz. Canny (1996 concened with C egulaity of solutions fo less egula opeatos. He achieved these esults Received Mach 1, 2005; Accepted Januay 1, 2006 Addess coespondence to Luis E. Silveste, Depatment of Mathematics, Univesity of Texas at Austin, 1 Univesity Station C1200, Austin, TX , USA; lsilvest@math.utexas.edu 1227

2 1228 Milakis and Silveste unde some mild geometic conditions upon the domain. We also efe the eade inteested in the classical solutions appoach to the aticle of Lions and Tudinge (1986 whee the authos poved, using the continuity method, that a poblem with an oblique deivative condition at the bounday has a C 2 solution if the equation is convex. In the pesent pape we will conside the egulaity fo viscosity solutions of fully nonlinea unifomly elliptic second ode equations with Neumann bounday data. We will always conside the domain to be the uppe half ball, and the Neumann bounday data to be given on its base. We will use the following notation: B 1 = x n x < 1 x n > 0 = x n x < 1 x n = 0 The vecto = is the inne nomal to (the base of B 1. The aticle is oganized as follows: The fist two sections ae the intoduction and peliminaies. The thid section is devoted to obtaining an extension of the Alexandoff Backelman Pucci (ABP estimate to Neumann bounday conditions. In the fouth section we pove the C egulaity up to the bounday fo the solution of the homogeneous poblem { F D 2 u = 0 in B 1 (1.1 u = 0 in In section five we develop some popeties fo sup- and inf-convolutions and in section six we obtain the C 1 egulaity up to the bounday fo the solution u of poblem (1.1. In the seventh section we get the C 2 egulaity up to the bounday fo u assuming that F is convex (o concave. In sections eight and nine we extend ou esults to moe geneal poblems coesponding to x dependence on F o to inhomogeneous ight hand side. At the end, thee is an appendix with the poof of a egulaity esult fo Diichlet bounday conditions. We expect that ou esults can be extended to moe geneal (nonflat domains, but we have not woked in that diection yet. It is ou intention to use these esults fo the uppe half ball in a fothcoming aticle (Milakis and Silveste, In pepaation associated with Signoinilike obstacle poblems. We believe that it is insightful to think of the Neumann condition as pat of the equation and not as bounday data. Ou esults ae local in the sense that we only equie a Neumann condition in a piece of the bounday, then obtain egulaity thee egadless of how the function behaves fa fom those points. This could be thought as an inteio egulaity esult, if we think of the Neumann condition as pat of the equation. 2. Peliminaies Fist of all, we make some emaks about ou notation. When we say that a function touches anothe function u fom above (o esp. fom below at a point x, we mean that x = u x and y > u y (o esp. y < u y fo evey y in a neighbohood of x. Stictly speaking, it is not the functions themselves but thei gaphs that touch each othe at the point x u x = x x.

3 Regulaity fo Fully Nonlinea Elliptic Equations 1229 When we say that u solves F D 2 u = 0inB 1, we always mean in the viscosity sense, and F is always assumed to be unifomly elliptic with constants and. A constant is consideed univesal when it depends only on,, and n (the dimension. When we say u 0on, we also efe to the viscosity sense. By this we mean that fo any smooth function touching u fom above at a point x 0 in we have x 0 0. Similaly, when we say u 0in, we mean that if touches u fom below at a point x 0, then x 0 0. We also use the notation S f, S f, and S f as in Caffaelli and Cabé (1995. Fo Diichlet bounday data, the egulaity up to the bounday is faily well undestood. The following popositions ae moe o less well known fo the specialists. Howeve, since we could not find any efeence whee these popositions ae poven, we give the poofs in the appendix of this aticle. Poposition 2.1. Let u be a solution of F D 2 u = 0 in B 1 such that u C B 1 and the estiction of u to is C fo some <1, then u is C B 1/2 up to the bounday. Moeove, we have the estimate u C ( u C u B 1/2 C B 1 C F 0 (2.1 fo a constant C depending only on n,,, and (C as 0. Poposition 2.2. Let u be a solution of F D 2 u = 0 in B 1 such that u C B 1 and the estiction of u to is C 1 fo some >0, then u is C 1 B 1/2 up to the bounday, whee = min 0 fo a univesal 0. Moeove, we have the estimate u C ( u C u 1 B 1/2 C B 1 C 1 F 0 (2.2 fo a constant C depending only on n,,, and (C as 0. We ae going to develop coesponding esults fo Neumann bounday data. In many poofs we use that a C (o C 1,oC 2 estimate on plus an inteio estimate implies the estimate all the way up to the bottom. This is a standad pocedue in the egulaity theoy that we illustate in the following popositions. Poposition 2.3. Let u be a continuous function in B 1 that satisfies C inteio estimates. By this we mean that if x 0 B 1, u x u y C 1 osc u fo evey x y B x y /2 x 0 (2.3 x 0 Let us also suppose that u is C at the bottom bounday, i.e., Then u C B 1/2 and u x u y C 0 x y fo x and y B 1 (2.4 u x u y CC 0 x y (2.5 fo evey x B 1/2 and y B 1, whee C depends only on the constant of (2.3.

4 1230 Milakis and Silveste Simila esults fo C 1 and C 2 estimates ae also valid. The statement fo C 2 will be needed late in the aticle and is poven in the appendix fo completeness. Poposition 2.4. Let u C B 1 be a viscosity solution of F D 2 u = 0 in B 1 fo a unifomilly elliptic convex function F. We known fom Caffaelli and Cabé (1995 that solutions of such equations have inteio C 2 estimates fo a univesal constant. Let us also assume that u is C 2 on fo the same. By this we mean that fo evey x, thee is a second ode polynomial P x such that fo any y B 1, Then u C 2 B 1 and u y P x y C 0 x y 2 (2.6 whee C is a univesal constant. u C 2 C C 0 The poofs of Popositions 2.3 and 2.4 ae done in the appendix. 3. An Extension of the ABP Estimate We obtain an extension to the ABP estimate to Neumann bounday conditions whee by S class we mean the usual function space dealing with the Pucci s extemal opeatos (see Section 2.2 of Caffaelli and Cabé, Poposition 3.1. Let u C B 1 be a function that belongs to S f in B 1 such that it satisfies u = g in in the viscosity sense. Then inf x n >0 ( u x inf u C u= u whee u is the convex envelope of u and C is a univesal constant. 1/n f x dx n C sup g (3.1 Poof. To simplify the notation we can suppose inf B x n >0 u x = 0. Fom Caffaelli and Cabé (1995, Chapte 3 we know that u C 1 1 B. Let u x 0 = inf B u. We will follow the usual pocedue of finding a subset of u B. Let us define the following set (see Figue 1 { = A n A max g A inf } u 2 Take a vecto A such that A max g and A inf 2 u (3.2

5 Regulaity fo Fully Nonlinea Elliptic Equations 1231 Figue 1. The set. Theefoe A x x 0 u x 0 is a linea function that coincides with u at x 0 and is below u in x n > 0. Then thee is a tanslation A x b such that it touches u fom below in a point that is not in x n > 0. Since A max g, then A x b cannot touch u at the bottom. Theefoe it touches at an inteio point and A u B. Thus we have: u B The set is the uppe cap of a ball. Let R = inf. If max g>r, then is 2 empty. If max g<r/3, then CR n, whee C depends only on dimension. To summaize, one of the following two happens: 1. inf B u<4sup g; 2. CR n. In the second case, we follow the usual poof of the ABP-estimate to obtain f x n dx CR n u= u fo a univesal constant C. And theefoe, combining the two cases, (( 1/n inf u C max f x dx n sup g u= u u which is equivalent to what we wanted to pove. 4. Hölde Regulaity In the pesent section we intend to pove C egulaity fo the solution up to the bounday. We ae going to use the following eflection popety and the Hölde egulaity fo functions in S class whee by S class we mean the usual function space dealing with S and S. We point out that this egulaity has been poven in a much moe geneal situation (see Canny, In ou case we can povide a simple poof.

6 1232 Milakis and Silveste Poposition 4.1 (Reflection Popety. Let u B 1 be a function that belongs to S f in B 1 such that it satisfies u = 0 in in the viscosity sense. Then the eflected function { u u x when x = n 0 (4.1 u x x n when x n < 0 belongs to the class S f in B 1, whee f is eflected the same way as u. Poof. We will show that u belongs to S f in B 1, whee f is also eflected in the same way as u. Fo, let us conside the function v = u x n. It is clea that v S f in B 1 as well as v S f in B1, since the Pucci extemal opeatos depend only on the second deivatives and ae invaiant unde symmeties. When >0, then v cannot be touched by any smooth function fom above at the points in. Indeed, if was such a test function, then x x n would touch u in the Neumann bounday, and theefoe at the contact point. But x x n x n would also touch u fom above at the same point in the bounday, then, obtaining a contadiction. Theefoe, since v can neve be touched fom above at any point in, and in the est of B 1, v is in the S class, then v S f in B 1, when >0. Similaly, we obtain v S f in B 1, when <0. But v u unifomly as 0. Since the classes of S and S ae closed unde unifom limits, then u belongs to both, in othe wods u S f in B 1. Poposition 4.2. Let u B 1 be a function that belongs to S f in B 1 such that it satisfies u = 0 in the viscosity sense in. Then u C B 1/2 up to the bounday, fo a univesal >0. Moeove, we have the estimate ( u C C u B 1/2 L B 1 f L n B 1 Poof. Since the eflected function u of Poposition 4.1 is in the class S acoss the bounday. Then u is C in B 1/2 by inteio estimates (see Caffaelli and Cabé, 1995, Section 4.3. Thus u C B 1/2 up to the bounday. The estimate follows fom the C estimates fo the S class (see Caffaelli and Cabé, 1995, Poposition Coollay 4.3. Let u be a solution of a fully nonlinea unifomly elliptic equation F D 2 u = 0 in B 1 with Neumann data u = 0 in in the viscosity sense. Then u C B 1/2 up to the bounday, fo a univesal >0. Moeove, we have the estimate u C ( u C B 1/2 L B 1 F 0 5. Sup- and Inf-Convolutions In Jensen (1988, the autho intoduced the concept of sup- and inf-convolutions to pove compaison pinciples fo viscosity solutions of second ode patial

7 Regulaity fo Fully Nonlinea Elliptic Equations 1233 diffeential equations. We will see in this section, that this concept applies up to the bounday in ou situation. Let u B 1. We will conside the following definition of sup- and infconvolutions: u x = sup {u y 1 } x y 2 (5.1 y B 1 u x = inf y B 1 {u y 1 x y 2 } (5.2 The following popety is standad and its poof can be found in Caffaelli and Cabé (1995, Theoem 5.1. Poposition 5.1. The sup-convolution satisfies the following popeties: 1. u C B 1 ; 2. u u unifomly as 0; 3. Fo any point x 0 B 1, thee is a concave paaboloid of opening 2 fom below. Hence, u C 1 1 by below. that touches u Lemma 5.2. Suppose u satisfies u 0 in, in the viscosity sense. Fo any x B 1, the sup (esp. inf in(5.1 (esp. (5.2 is achieved fo a y 0 B 1 \. Poof. Since we conside u to be continuous and B 1 is a compact set, then the supemum in (5.1 is achieved. We have to check that fo the y 0 that achieves this supemum is not in. Suppose that y 0, u x = sup y B 1 {u y 1 x y 2 } Then = u y 0 1 x y 0 2 (5.3 u y 0 1 x y x y 2 u y (5.4 fo evey y B 1. Theefoe function v y = u y 0 1 x y x y 2 touches u fom above at the point y = y 0. By the Neumann bounday condition in the viscosity sense, we have v y 0 0. But v y 0 = 2 y 0 x <0 since x B 1 and y 0. Lemma 5.3. Let u be a subsolution of the equation F D 2 u 0 and u 0 in the viscosity sense. Then u is also a subsolution of the same equation (same conclusion holds with supesolutions if we conside u instead of u. Poof. Suppose that P x touches u fom above at a point x 0.

8 1234 Milakis and Silveste If x 0 B 1, then u x 0 = u y 0 1 x 0 y 0 2 fo some y 0 B 1. Now u x u x y 0 x 0 1 x 0 y 0 2 in a neighbohood of x 0. Theefoe Q x = P x x 0 y 0 1 x 0 y 0 2 touches u fom above at the point y 0. Since u is a subsolution we have F D 2 P x 0 = F D 2 Q y 0 0. If x 0 x < 1, then u x 0 = u y 0 1 x 0 y 0 2 fo some y 0 B 1. Now u x = u y 0 1 x y 0 2, theefoe P x 0 2 y 0 x 0 0. Poposition 5.4. Let u be a subsolution of the equation F D 2 u 0 and u 0. Let v be a supesolution of the equation F D 2 v 0 and v 0. Then u v S in n B 1 and u v 0 in. Poof. The poof uses sup- and inf-convolutions. The poof that u v S ( in n B 1 can be found in Caffaelli and Cabé (1995, Theoem 5.3. We will concentate hee in the bounday condition. We know that u and u satisfy also the same inequality fo the nomal deivatives in the bounday. Let x 0. Suppose that P x touches u v fom above at a point x 0. Let y 0 and y 1 be the point that ealize the supemum and infimum espectively: u x 0 = u y 0 1 x 0 y 0 2 (5.5 v x 0 = v y 1 1 x 0 y 1 2 (5.6 Then u x = u y 0 1 x y 0 2 and v x = v y 1 1 x y 1 2 fo any x. Theefoe, u x v x = u y 0 1 x y 0 2 v y 1 1 x y 1 2 = G x Then, P x also touches G x fom above at x 0, thus P x 0 2 y 0 x 0 2 y 1 x 0 0 Remak 5.5. As the efeee pointed out, it is also possible to pove Poposition 5.4 using a doubling vaiables type agument, as it is standad in viscosity solutions theoy. Fo a geneal desciption of the method see Candall et al. ( Hölde Estimates fo the Fist Deivatives The main esult of this section is the following theoem. Theoem 6.1. Let u be a solution of F D 2 u = 0 in B 1 and u = 0 in. Then u is C 1 B 1/2 up to the bounday, fo a univesal >0. Moeove, we have the estimate u C ( u C F 0 1 B 1/2 C B (6.1 1 fo a univesal constant C.

9 Regulaity fo Fully Nonlinea Elliptic Equations 1235 Ou poof of Theoem 6.1 is an adaptation of the poof of Coollay 5.7 in Caffaelli and Cabé (1995 (the inteio C 1 egulaity fo unifomly elliptic equations. We will use the following lemma, whose poof can be found in Caffaelli and Cabé (1995, Lemma 5.6. Lemma 6.2. Let 0 < <1, 0 < 1, and K>0 be constants. Let u L 1 1 satisfy u L 1 1 K. Define, fo h with 0 < h 1, v h x = u x h u x h x I h whee I h = 1 1 h if h>0 and I h = 1 h 1 if h<0. Assume that v h C I h and v h C I h K, fo any 0 < h 1. We then have: 1. If <1 then u C 1 1 and u C 1 1 CK, 2. If >1 then u C and u C 0 1 CK, whee the constants C in 1. and 2. depend only on. Poof of Theoem 6.1. Let T B be the space of functions that ae C in the hoizontal diections, T B = v C sup v x v y < x y x y x y =0 The nom in this space is given by v T B = v C B sup x y B x y =0 v x v y x y Let be any unit vecto paallel to (i.e., =0. Fo any h<1/8, fom Poposition 5.4, we have that v h x = 1 u x h u x S ( in B h n 7/8 and v h = 0in B 7/8. Hence, by Poposition 4.2 popely escaled v h C s v C h C s u C B (6.2 s /2 T B s whee 0 <<s 7 s,0<h<, is univesal and C s depends on n,,,, 8 2 and s. We can make slightly smalle if needed so that thee is an intege i such that i < 1 and i 1 > 1. Fom Coollay 4.3, we know that u T ( u u C F 0 B 7/8 C B 7/8 C B 1 Let K = ( u F 0 C B so that u 1 T CK. B 7/8 We can apply now (6.2 with = and = 1 <s= 7/8 toget v h T B 1 C 1 u T B 7/8 C 1 K

10 1236 Milakis and Silveste We apply Lemma 6.2 with =. Recall that we can do this fo any unit vecto paallel to. We obtain u T 2 B 1 C 1 K We epeat this pocess with = 2 to obtain u T CK. If we choose 3 2 i1 = 5/8, at the end we obtain u T CK 1 B 3/4 Then we apply (6.2 with = 1 and get v 1 h C u C CK B 5/8 T 1 B 3/4 Since v 1 h is a diffeence quotient of u fo h and is any vecto paallel to, we obtain u C 1 B 5/8 and u C 1 B 5/8 CK. Finally, we apply Poposition 2.2 popely escaled to obtain u C 1 1 B 1/2 and ( u C C 1 1 B u u 1/2 C B 5/8 C 1 B 5/8 F 0 CK 7. Hölde Estimates fo the Second Deivative The following lemma was fist obseved by Kylov (1983. Late a simple poof was given by Caffaelli (to appea in the context of viscosity solutions, but unfotunately he did not publish it. His poof can be found in new editions of the book of Gilbag and Tudinge (2001, Theoem 9.31 stated in a slightly diffeent but equivalent way. Lemma 7.1. Let u B 1 be such that u = 0 in, and u S in B 1. Then thee is a C function A such that fo evey x B 1/2, C x n 1 u x A x x n C x n 1 whee x = x x n, >0 is univesal, C and A C depend on n,,, and linealy on sup B u. The function A is then the nomal deivative of u at the bounday. 1 We can apply this lemma to the nomal deivative of a Neumann type poblem to find the following estimate. Coollay 7.2. Let u be a solution of F D 2 u = 0 in B 1 and u = 0 in, then thee is a C function A such that C x n 2 u x u x 0 A x 2 x2 n C x n 2 (7.1 In the pevious statements we think of A x to be u x 0.

11 Regulaity fo Fully Nonlinea Elliptic Equations 1237 Poof. The nomal deivative u is well defined since u C 1 by Theoem 6.1. We see that u satisfies the hypothesis of Lemma 7.1, then Theefoe C x n 1 u x A x x n C x n 1 u x u x 0 = xn 0 xn 0 u x y dy A x y C y 1 dy A x 2 x2 n C x n 2 Similaly, we show u x u x 0 A x 2 x 2 n C x n 2. Poposition 7.3. Let u be a solution of F D 2 u = 0 in B 1 and u = 0 in. Then if we take its estiction to, v x = u x 0, then v solves (in the viscosity sense the equation ( D F 2 v 0 0 A x = 0 whee A is a C function, fo a univesal >0. Poof. Let be a smooth function on touching v fom below in a point in that, fo simplicity, we will conside to be the oigin. We want to extend to B 1 and tanslate it to tun it into a test function which touches u fom below in the inteio of B, fo an abitaily small. Let A be the function of Coollay 7.2. Fo a small >0, let x = x A 0 2 x2 n x 2 Fom Coollay 7.2, we know that u x u x 0 A x 2 x2 n C x n 2 u x 0 A 0 2 x2 n C x 2 since A is C u x 0 A 0 2 x2 n 2 x 2 fo x small enough x 2 x 2 Let >0 be chosen so that the above computation is valid fo x <. We will conside two cases: whethe A 0 0 o not. If A 0 0, we tanslate in the inne nomal diection. h x = x x n h

12 1238 Milakis and Silveste We choose h such that ( A 0 h h 2. Theefoe u 0 h 0 u x h x when x =. The function u h cannot have a local minimum in because u = 0 and h > 0 in thee. Theefoe u h must have a local minimum at some point x 1 B. Since F D2 u = 0inB 1, then F D2 h x 1 0. Since we can do all this fo abitaily small, and then we can also take 0, we obtain ( D F A 0 In the case A 0 >0, the only diffeence is that we tanslate in the oute nomal diection h x = x x n h and then if we choose <A 0, the same easoning as above applies. We can do the same thing fo test functions touching v fom above, theefoe we obtain in the viscosity sense ( D F 2 v 0 0 A x = 0 Theoem 7.4. Assume F to be a convex function and let u be a solution of F D 2 u = 0 in B 1 and u = 0 in. Then u is C 2 B 1/2 up to the bounday, fo a univesal >0. Moeove, we have the estimate u C ( u C F 0 2 B 1/2 C B (7.2 1 fo a univesal constant C. Poof. By Poposition 7.3, the estiction v x = u x 0 satisfies equation ( D F 2 v 0 0 A x = 0 whee A is C. By the C 2 estimates fo elliptic equations in Caffaelli and Cabé (1995, we conclude that v C 2 B 2/3. By Coollay 7.2, then u is C 2 at the bounday B 2/3, and then fom Poposition 2.4 we obtain the desied estimate. 8. Inhomogeneous Equations In this section we study the egulaity in the case when we have a nonzeo ighthand side. The poofs ae based on a petubation of the homogeneous case. Fo simplicity, in this section we keep the left-hand side independent of x. The poofs of the coesponding esults with x dependent left-hand side follow the same spiit but ae moe complicated. We will outline the geneal case in the next section. Theoem 8.1. Let u be a solution of F D 2 u = f x in B 1 and u = g in, fo a bounded function g and f L n B 1. Then u is C B 1/2 up to the bounday. Fo a

13 Regulaity fo Fully Nonlinea Elliptic Equations 1239 univesal >0. Moeove, we have the estimate u C ( u C g B 1/2 C B 1 L f L F 0 n (8.1 fo a univesal constant C. Poof. If we can show that any function u, continuous in B such that F D 2 u = f x in B and u = g in satisfies osc B /2 u 1 osc u C F 0 2 C g L f L n (8.2 fo univesal constants >0 and C. Then applying (8.2 to tanslations of u we obtain a C modulus of continuity fo u at the bottom by a standad iteative agument. Then (8.1 follows by inteio egulaity (o by Poposition 2.1. So we ae going to show (8.2. Let u be as above. Let v be the solution of the poblem: F D 2 v = 0 in B v = 0 in v = u in x n > 0 The function v is in the class S /n F 0 in B. We can eflect it using Poposition 4.1 to get a function v S /n F 0 in. By simple compaison pinciple, Similaly, Theefoe, max B v = max v max min B osc B v C F 0 2 max u C F 0 2 v min u C F 0 2 v osc u C F 0 2 (8.3 Since v S /n F 0 in, we can apply Hanack inequality to obtain osc v = osc B /2 /2 v 1 osc Combining (8.3 with (8.4 we get osc B /2 fo some univesal constant C. v C F 0 2 = 1 osc v C F 0 2 (8.4 v 1 osc u C F 0 2 (8.5

14 1240 Milakis and Silveste Let w = u v. Then w satisfies the following elations: w S /n f in B w = g in w = 0 in x n > 0 Applying Poposition 3.1 to w and w, we obtain Theefoe, osc B /2 Adding (8.5 with (8.6, we finally get osc B /2 u osc B /2 max w C ( f L n g L w osc w C ( f L n g L (8.6 v osc w B /2 1 osc u C F 0 2 C f L n g L (8.7 Theoem 8.2. Conside g C B 1 and f Lp B 1 fo some 0 1 and p>n. Let u be a solution of F D 2 u = f in B 1 and u = g in. Then u is C 1 B 1/2 whee = min 0 1 n/p and 0 is a univesal constant. Moeove, we have the estimate u C ( u C g 1 B 1/2 C B 1 C f L p F 0 (8.8 whee C is a constant depending only on n,,, and. Poof. By inteio estimates (o by Poposition 2.2 it is enough to find a C 1 estimate fo the points in. Moeove, fo poving (8.8 it is enough to get a univesal estimate at the oigin, and then apply it to escaling and tanslations of u. Without loss of geneality we can assume u 0 = 0 and g 0 = 0. Let M = u g C B 1 C f L p F 0. We want to show that thee is a univesal 0, and a univesal constant <1, such that fo = min 0 1 n/p, thee is a constant C 1, depending only on n,, and and a sequence of vectos A k such that A k = 0 and osc u x A k x C 1 M k 1 (8.9 B k A k1 A k CM k (8.10 We will show this by induction. We choose a C 1 > 1 so that (8.9 holds fo k = 0 with A 0 = 0. To complete an induction poof, we assume that we aleady have a sequence of vectos A k so that (8.9 holds fo k = 0 1 K; we have to show that thee is a vecto A K1 such that (8.9 holds fo k = K 1.

15 Regulaity fo Fully Nonlinea Elliptic Equations 1241 Let = K and B = A K. Let v be the solution of the following poblem (like in the poof of Theoem 8.1: F D 2 v = 0 in B v = 0 in v = u B x in x n > 0 Fom maximum pinciple (Poposition 3.1 we know osc B v osc u B x C 2 F 0 2 (8.11 Now we apply the C 1 estimates to v. Theoem 6.1 tells us that v is well defined up to the bounday. Let A = v 0, by the bounday conditions A = 0. Rescaling of the C 1 estimate gives ( osc B v x A x 1 C ( oscb A C v F 0 osc v 1 1 F 0 (8.12 (8.13 fo any /2. Whee 1 is the of Theoem 6.1. We choose small enough so that C 0 1 = 1 < 1, fo some positive constant. Combining (8.12 fo = with (8.11, we get osc v x A x 1 osc u B x C 3 F 0 2 (8.14 B Let w = u B x v. Recall B = 0. Like in the poof of Theoem 8.1, we have w S /n f in B w = g in w = 0 in x n > 0 Then, by Poposition 3.1, we have ( sup w x C g L C Adding (8.14 with (8.15 we get osc u x A B x B 1/n f n dx C g C 1 C f L p 1 n/p ( osc u B x C F 0 2 C g C 1 C f L p 2 n/p (8.16

16 1242 Milakis and Silveste By the inductive hypothesis we have osc B u B x C 1 M K 1, theefoe, osc u x A B x M ( 1 C 1 K 1 C 2 ( 2K K 1 K 2 n/p B (8.17 Now we will choose the ight constants 0 and C 1. We choose 0 so that 0 = 1 /2, = min 0 1 n/p, and C 1 lage enough so that C 2 1 3C 2. Replacing in (8.17, we get osc u x A B x B M ( 1 /2 C 1 K 1 C 2 ( 2K K 1 K 2 n/p 2 C 1 K 1 M 1 /2 C 1 K 1 MC 1 K1 1 (8.18 Fom (8.13, (8.11, (8.9, and that = K,weget A CM K (8.19 Taking A K1 = A B, we finish the inductive poof of (8.9 and (8.10. Let A = lim k A k. We claim that Indeed, fom (8.9 and (8.10 we get u x A x CM x 1 osc B k u x A x osc B k u x A k x 2 k A k A (8.20 C 1 M k 1 CM k j=k j (8.21 C 1 M k 1 CM 1 k 1 1 (8.22 CM k 1 (8.23 This implies the C 1 estimate at the oigin, and the estimate follows by tanslation and inteio estimates. Theoem 8.3. Fo F a convex, let u be a solution of F D 2 u = f in B 1 and u = g in, fo a C 1 function g, and f C ( B ( 1 ( >0. Then u is C 2 B 1/2 up to the bounday, fo = min 0. Whee 0 > 0 is a univesal constant. Moeove, we have the estimate u C ( u C g 2 B 1/2 C B 1 C 1 f F 0 C (8.24 B 1 fo a constant C depending only on n,,, and.

17 Regulaity fo Fully Nonlinea Elliptic Equations 1243 Poof. The poof will follow the same ideas as Theoem 8.2, but instead of appoximating with planes, we must use paaboloids, and we have to use Theoem 7.4 instead of 6.1. Let M = u g C B 1 C 1 f C F 0. We want to show that thee is an >0, and a univesal constant <1, such that thee is a constant C 1, depending only on n,, and, and a sequence of paaboloids P k x = 1 2 xt Q k x A k x u 0 such that F Q k = f 0, A k = 0 and Q k jn = jn P k x = j g 0 fo evey j n, and osc u x P k x C 1 M k 2 B k (8.25 A k1 A k CM k 1 (8.26 Q k1 Q k CM k (8.27 As befoe, this will show that estimate (8.24 holds punctually at x = 0; and then this implies the full estimate (8.24 by tanslations and inteio estimates. We can subtact a suitable plane to u such that u 0 = 0 and g 0 = u 0 = 0. So we suppose u 0 = 0 and g 0 = 0. We will show (8.25 by induction. We choose C 1 > 1 so that (8.25 holds k = 0 with A 0 = 0 and Q 0 the symmetic matix such that j g 0 = Q jn fo evey j n, Q ij = 0 fo i j n and Q nn chosen so that F Q = f 0. Note that Q C F 0 f 0 CM fo a univesal constant C. This is the only pat whee the tem F 0 in the definition of M mattes. To complete an induction poof, we assume that we aleady have such a sequence of paaboloids P k = 1 2 xt Q k x A k x so that (8.25, (8.26, and (8.27 hold fo k = 0 1 K; we have to show that thee is anothe paaboloid P K1 such that (8.25 (8.27 hold fo k = K 1. Let = K. Let v be the solution of the following poblem (like in the poof of Theoem 8.2: F D 2 v Q K = 0 in B v = 0 in v = u P k x in x n > 0 Fom maximum pinciple (Poposition 3.1 and that F Q K = 0, we know that osc B v osc u P k x (8.28 Now we apply the C 2 estimates to v. Theoem 7.4 tells us that v and D 2 u ae well defined up to the bounday. Let B = v 0 and R = D 2 v 0, and let P be the paaboloid x T Rx B x. By the bounday conditions n P = 0 and by the equation and the fact that D 2 v is continuous up to the bounday F Q K R = 0. Since F Q K = 0, escaling of the C 2 estimate gives osc B v x P x 2 1 C osc v (8.29

18 1244 Milakis and Silveste B C osc v (8.30 R C osc v ( fo any /2, whee 1 is the of Theoem 7.4. We choose small enough so that C 0 1 = 1 < 1, fo some positive constant. Combining (8.29 fo = with (8.28, we get osc v x P x 1 2 osc u P K x (8.32 B Let w = u P K x v. Like in the poof of Theoem 8.2, we have w S /n f f 0 in B w = g x n P K x in w = 0 in x n > 0 Recall that j g = Q K jn = jn P K 0 and g 0 = 0, so g n P K aound x = 0. Then, by Poposition 3.1, we have is of ode x 1 ( sup w x C g n P K L C Adding (8.32 with (8.33 we get 1/n f f 0 n dx C g C 1 2 C f C 2 (8.33 osc u x P K x P x 1 2 osc u P K x C g C 2 C f B 1 C 2 (8.34 By the inductive hypothesis, osc B u P K x C 1 M K 2. Thus we get osc B u x P K x P x M ( 1 2 C 1 K 2 C 2 K 2 (8.35 Now we will choose the ight constants 0 and C 1. We choose 0 so that 0 = 1 /2, = min 0, and C 1 lage enough so that 2 C 2 1 C 2. Replacing in (8.35, we get osc u x P K x P x M ( 1 /2 2 C 1 K 2 C 2 K 2 2 B 2 C 1 K 2 M 1 /2 2 C 1 K 2 Fom (8.30, (8.28, and that = K,weget MC 1 K1 2 (8.36 B C 1 K (8.37

19 Regulaity fo Fully Nonlinea Elliptic Equations 1245 And fom (8.31 and (8.28, we get R C K (8.38 Taking P K1 = P K P, we finish the inductive poof of (8.25, (8.26, and (8.27. Like in the poof of Theoem 8.2, this implies a C 2 estimate punctually at the oigin. Thus the estimate follows. 9. Geneal Equations This section is concened with the Hölde egulaity fo the fist deivatives of solutions of F D 2 u x = f x (9.1 fo x B 1 and u = g on. In Caffaelli and Cabé (1995, Chapte 8, inteio egulaity esults ae obtained fo an equation like (9.1 by a petubation agument of the homogeneous case. With the esults we have so fa in this aticle, we can extend the poof in Caffaelli and Cabé (1995 fo C 1 egulaity up to the bounday in the Neumann poblem. It is a key idea to think of the Neumann condition as pat of the equation and not the bounday data. Afte all, if the Neumann condition is pat of the equation, then we ae actually talking about inteio egulaity. The esults so fa in this aticle povide us with good estimates fo the equation that we obtain when we feeze the value of x. We assume that the oscillation of function F M x in x is sufficiently small so that the Neumann condition u = g has a C ight-hand side, and that the coesponding homogeneous equation with constant coefficients F D 2 u x 0 = 0 u = g y 0 in B in has C 1 estimates fo any fixed x 0, whee y 0 is the pojection of x 0 into. We intend to follow poof in Caffaelli and Cabé (1995, Theoem 8.3 fo the inteio egulaity. We ealize eveything follows the same way as long as we modify two lemmas. We ae just going to outline the equied modifications. The poof of Lemma 9.1 (Poposition 4.14 in Caffaelli and Cabé, 1995 is based on the ABP estimate and C egulaity of the solution. Since we have a coesponding esult by Popositions 3.1 and 4.2, we can extend the lemma to ou case. Lemma 9.1. Let u be continuous in B 1 that belongs to S f in B 1 such that it satisfies u = g on and f is a continuous function. Denote by the estiction of u on B 1 x n > 0 and let x y be a modulus of continuity of ; that is is a nondeceasing function with lim 0 = 0 such that x y x y

20 1246 Milakis and Silveste fo all x y B 1 x n > 0. In addition let K be a positive constant such that L K, g L K and f L n B 1 K. Then thee exists a modulus of continuity of u in B 1, i.e., is nondeceasing, lim 0 = 0 and u x u y x y fo all x y in B 1. Whee depends on n,,, K, and. Next we adapt Lemma 8.2 in Caffaelli and Cabé (1995 (see also Theoem 2.1 in Swiech, 1997 to ou case. We conside the function F M x F M 0 x = F x = sup \ O 1 M which measues the oscillation of F in x nea the oigin (ecall that denotes the space of symmetic matices. Lemma 9.2. Suppose that F is continuous in x, F 0 x 0, and = F is Hölde continuous in B 1 (fo some exponent in 0 1. Let u 0 be a continuous function on B 1 x n > 0, has = s as modulus of continuity on B 1 x n > 0 and satisfy u 0 L B 1 x n >0 K, fo some positive constant K. Then, given >0, thee exists >0 depending only on n K such that if f is Hölde continuous in B 1, g is Hölde continuous on, L n B 1 f L n B 1 and g g 0 L fo g 0 K then any two viscosity solutions v and w of, espectively, F D 2 v x = f x in B 1 v = u 0 on B 1 x n > 0 v = g on and F D 2 w 0 = 0 in B 1 w = u 0 on B 1 x n > 0 w = g 0 on satisfy v w L B 1 Note that it is no estiction to assume F 0 x 0, since F D 2 u x = f x may be witten as F D 2 u x F 0 x = f x F 0 x. Fo the poof of the pevious lemma we efe the eade to Caffaelli and Cabé (1995, Lemma 8.2 (o Theoem 2.1 in Swiech, 1997 the poof of which is still valid in ou case due to Lemma 9.1. As we mentioned befoe, in ode to pove C 1 estimates fo ou solution, we would like to use the feezing agument as in Theoem 8.3 of Caffaelli and Cabé

21 Regulaity fo Fully Nonlinea Elliptic Equations 1247 (1995. In fact if we follow the lines of the poof, we ealize that all the aguments ae applied pefectly in ou case, except the fact that we use domains B, instead of balls, with the Neumann condition on the bottom. But this is not a poblem since we teat the Neumann conditions as a pat of ou equation and thus we also apply fo them the compactness method (see Lemma 9.2. To conclude we state the main theoem. Theoem 9.3. Assume that F 0 x 0, F is continuous in x B 1 and both = F and f ae Hölde continuous in B 1 and g is Hölde continuous on. Suppose that thee is a constant 0 < <1 such that fo any u 0 C B 1 x n > 0 and g 0 K, thee exists a w C B 1 C1 B 1/2 which is viscosity solution of F D 2 w 0 = 0 in B 1 w = u 0 on B 1 x n > 0 w = g 0 on Assume that 0 < <, 0 > 0, and C 1 > 0. Then thee exists >0 depending on n, and the C 1 nom such that if ( / B 0 n 1/n and ( / B 0 f n 1/n C 1 1 fo all 0, then any viscosity solution u of { F D 2 u x = f x in B 0 0 w = g on 0 is C 1 in the sense that thee is an affine function l such that u l L B 0 C l C 2 fo all 0 and ( C 2 C 1 0 u L 0 0 g C C 1 whee C>0 depends only on n, and the C 1 nom of w. 10. Appendix In this appendix, we will give a poof of Popositions 2.1, 2.2, 2.3, and 2.4. Fo functions vanishing in the bounday, the C 1 egulaity follows fom Lemma 7.1. Lemma Let u be a solution of F D 2 u = 0 in B 1 such that u C B 1 and u = 0 in, then u is C 1 B 1/2 up to the bounday, fo a univesal. Moeove, we have the

22 1248 Milakis and Silveste estimate u C 1 B 1/2 C ( u C B 1 F 0 (10.1 fo a univesal constant C. Poof. The function u belongs to S ( F 0 in B n 1, and it vanishes in, theefoe we can apply Lemma 7.1 to obtain u x A x x n C x n 1 fo all x B 1/2, whee C = C 0 u L B 1 and A C C 1 u L B 1, fo univesal constants C 0 and C 1. Theefoe u x A 0 x n u x A x x n x n A x A 0 C 0 u L B 1 x n 1 x n C 1 u L B 1 x C u L B 1 x 1 fo a univesal C. Then u is punctually C 1 at the oigin. In the same way, we can show it is punctually C 1 at evey point in B 1/2 with a unifom bound. Now the lemma follows using the inteio C 1 estimates fo the equation F D 2 u = 0. Popositions 2.1 and 2.2 follow fom Lemma 10.1 in the same way Theoem 8.2 follows fom Theoem 6.1. We ae going to give a detailed poof of Poposition 2.2, that is the one that we actually use in this aticle. The poof of Poposition 2.1 is vey simila. The poof is witten almost with the same wods as the poof of Theoem 8.2 to stess the similaity. Poof of Poposition 2.2. Set g x = u x 0. We ae going to use an iteation pocess simila to the poof of Theoem 6.1. By subtacting a suitable plane at the oigin, we can suppose that u 0 = g 0 = 0 and n 1 g 0 = 0. We ae going to pove a ight decay fo the function at the oigin, and fom thee the estimate follows. Let M = u L B 1 F 0 g C. 1 We want to show that thee is a univesal 0, and a univesal constant <1, such that fo = min 0, thee is a constant C 1, depending only on n,, and and a sequence of eal numbes a k such that osc u x a k x n C 1 M k 1 B k (10.2 a k1 a k CM k (10.3 We will show this by induction. We choose C 1 > 1 so that (10.2 holds fo k = 0 with a 0 = 0. To complete an induction poof, we assume that we aleady have a sequence a k so that (10.2 holds fo k = 0 1 K; we have to show that thee is a eal numbe a K1 such that (10.2 holds fo k = K 1.

23 Regulaity fo Fully Nonlinea Elliptic Equations 1249 Let = K and b = a K. Let v be the solution of the following poblem F D 2 v = 0 in B v = 0 in v = u b x n in x n > 0 Fom maximum pinciple, we know osc B v osc u b x C 2 F 0 2 (10.4 Now we apply the C 1 estimates to v. Lemma 10.1 tells us that v is well defined up to the bounday. Let a = n v 0, by the bounday conditions v 0 = 0 0 a. Rescaling of the C 1 estimate gives ( osc B v x a x n 1 C ( oscb a C v F 0 osc v 1 1 F 0 (10.5 (10.6 fo any /2. Whee 1 is the of Lemma We choose small enough so that C 0 1 = 1 < 1, fo some positive constant. Combining (10.5 fo = with (10.4, we get osc v x a x 1 osc u b x C 3 F 0 2 (10.7 B Let w = u b x n v. We have w S /n w = g w = 0 in B in in x n > 0 Then, by the maximum pinciple, sup w x C g L C g C 1 ( Adding (10.7 with (10.8 we get osc u x a b x n 1 osc u b x n C F 0 2 C g C 1 B (10.9 By the inductive hypothesis osc B u b x n C 1 M K 1. Replacing, ( ( osc u x a b x n M 1 C 1 K 1 C 2 2K K 1 (10.10 B

24 1250 Milakis and Silveste Now we will choose the ight constants 0 and C 1. We choose 0 so that 0 = 1 /2, = min 0, and C 1 lage enough so that C 2 1 3C 2. Replacing in (10.10, we get ( ( osc u x a b x n M 1 /2 C 1 K 1 C 2 2K K 1 B 2 C 1 K 1 M 1 /2 C 1 K 1 MC 1 K1 1 (10.11 Fom (10.6, (10.4, and that = K,weget a CM K (10.12 Taking a K1 = a b, we finish the inductive poof of (10.2 and (10.3. Let a = lim k a k. We claim that indeed, fom (10.2 and (10.3 we get u x a x n CM x 1 osc u x a x n osc u x a x n k a k a (10.13 B k B k C 1 M k 1 CM k j=k j (10.14 C 1 M k 1 CM 1 k 1 1 (10.15 CM k 1 (10.16 This implies the C 1 estimate at the oigin, and the estimate follows by tanslation and inteio estimates. Now let us pove Popositions 2.3 and 2.4. Poof of Poposition 2.3. Let x B 1/2 and y B 1. Let x = x x n, whee x n 1 and x n. We conside two cases whethe x y x n /2 o not. If x y x n /2, we apply (2.3 fo = x n and x 0 = x, then u x u y x y C 1 but osc B x u 2C 0 2x n by (2.4, since = x n osc u x u x u y x y C2 C 0 CC 0 fo a constant C depending only on the constant of (2.3.

25 Regulaity fo Fully Nonlinea Elliptic Equations 1251 If x y > x n /2, we apply (2.4 to obtain u x u y u x u x 0 u y u x 0 C 0 x n y x 0 C x y 5C 0 x y whee fo the last inequality we used that x n < 2 x y, and since y x 0 x y x n, then y n 3 x y. Putting the two cases togethe, we obtain (2.5. Poof of Poposition 2.4. We ae going to show that fo any x B 1/2 second ode polynomial P x such that fo any y B 1, thee is a u y P x y C C 0 x y 2 (10.17 The statement of the theoem clealy follows fom this. Let us wite x = x x n and x = x 0 be the pojection of x on the bounday. We know fom the assumptions that thee is a polynomial P x such that The function v = u P x solves u y P x y C 0 y x 2 (10.18 F D 2 v A = 0 in B 1 whee A is the constant matix A = D 2 P x. Fom Caffaelli and Cabé (1995, this equation has a C 2 inteio estimate that does not depend on A. Applying it in the ball B xn /2 x and ecalling (10.18 we obtain that thee is a polynomial R such that R x = v x C 0 x n 2 R x C C 0 x n 1 D 2 R x C C 0 x n and ( v y R y C sup B x n/2 x v 1 x n 2 y x 2 C C 0 y x 2 (10.19 Let us define Now, if y x <x n /2 we have P x = P x R u y P x y = v y R y C C 0 y x 2

26 1252 Milakis and Silveste We ae only left to pove (10.17 fo the case y x >x n /2. In that case, u y P x y u y P x y R y C 0 y x 2 R x R x y x y x t D 2 R y x C 0 y x 2 C C 0 x 2 n C C 0 y x 2 y x x 1 n y x 2 x n which finishes the poof. Remak The poof of a C 1 estimate up to the bounday woks the same way eplacing the second ode polynomials by fist ode ones. Acknowledgment We would like to thank Pof. Luis Caffaelli fo the fuitful discussion of ideas egading this wok. Refeences Bales, G. (1993. Fully nonlinea Neumann type bounday conditions fo second-ode elliptic and paabolic equations. J. Diffeential Equations 106(1: Caffaelli, L. A. Pesonal Communication. Caffaelli, L. A., Cabé, X. (1995. Fully Nonlinea Elliptic Equations. Ameican Mathematical Society Colloquium Publications. Vol. 43. Povidence, RI: Ameican Mathematical Society. Candall, M. G., Ishii, H., Lions, P.-L. (1992. Use s guide to viscosity solutions of second ode patial diffeential equations. Bull. Ame. Math. Soc. (N.S. 27(1:1 67. Canny, T. R. (1996. Regulaity of solutions fo the genealized inhomogeneous Neumann bounday value poblem. J. Diffeential Equations 126(2: Gilbag, D., Tudinge, N. S. (2001. Elliptic Patial Diffeential Equations of Second Ode. Classics in Mathematics. Repint of the 1998 edition. Belin: Spinge-Velag. Ishii, H. (1991. Fully nonlinea oblique deivative poblems fo nonlinea second-ode elliptic PDEs. Duke Math. J. 62(3: Ishii, H., Lions, P.-L. (1990. Viscosity solutions of fully nonlinea second-ode elliptic patial diffeential equations. J. Diffeential Equations 83(1: Jensen, R. (1988. The maximum pinciple fo viscosity solutions of fully nonlinea second ode patial diffeential equations. Ach. Rational Mech. Anal. 101(1:1 27. Kylov, N. V. (1983. Boundedly inhomogeneous elliptic and paabolic equations in a domain. Izv. Akad. Nauk SSSR Se. Mat. 47(1: Lions, P.-L., Tudinge, N. S. (1986. Linea oblique deivative poblems fo the unifomly elliptic Hamilton-Jacobi-Bellman equation. Math. Z. 191(1:1 15. Milakis, E., Silveste, L. Regulaity fo the nonlinea signoini poblem. In pepaation. Swiech, A. (1997. W 1 p -inteio estimates fo solutions of fully nonlinea, unifomly elliptic equations. Adv. Diff. Eq. 2(6: