where Sènè stands for the set of n æ n real symmetric matrices, and æ is a bounded open set in IR n,typically with a suæciently regular boundary, mean

Transcription

1 GOOD AND VISCOSITY SOLUTIONS OF FULLY NONLINEAR ELLIPTIC EQUATIONS 1 ROBERT JENSEN 2 Department of Mathematical and Computer Sciences Loyola University Chicago, IL 60626, U.S.A. rrj@math.luc.edu MACIEJ KOCAN 3 Department of Mathemetics University of Cologne Cologne 50923, Germany ANDRZEJ SWIE, CH 4 School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, U.S.A. swiech@math.gatech.edu Abstract. We introduce the notion of a ëgood" solution of a fully nonlinear uniformly elliptic equation. It is proven that ëgood" solutions are equivalent to L p -viscosity solutions of such equations. The main contribution of the paper is an explicit construction of elliptic equations with strong solutions that approximate any given fully nonlinear uniformly elliptic equation and its L p -viscosity solution. The results of the paper also extend some results about ëgood" solutions of linear equations. Section 1. Introduction è1:1è We are interested in nonlinear elliptic partial diæerential equations of the form G x; uèxè;duèxè;d 2 uèxè =0 for x 2 æ: Here G: æ æ IR æ IR n æsènè!ir; Mathematics Subject Classiæcation 35J60, 35J65, 35J25, 49L25 2 Supported by NSF grants DMS , DMS and DMS Supported by an Alexander von Humboldt Fellowship. 4 Supported by NSF grant DMS Part of this work was completed while this author was visiting the University of Cologne, supported by the TMR Network ëviscosity Solutions and their Applications". 1

2 where Sènè stands for the set of n æ n real symmetric matrices, and æ is a bounded open set in IR n,typically with a suæciently regular boundary, meaning here the uniform exterior cone condition. We will always assume that Gèx; r;p;xèiscontinuous in èr;p;xè with modulus of continuity independent of x 2 æ, and is jointly measurable in all variables èx; r;p;xè. We will typically require that G of è1.1è satisfy the following structure conditions: è1:2è jgèx; r; p; Xè,Gèx; r; q; Xèjæjp,qj; è1:3è P, èx, Y è Gèx; r;p;xè,gèx; r;p;yèp + èx,yè; for all x 2 æ, r 2 IR, p; q 2 IR n and X; Y 2Sènè. Here P æ are the Pucci extremal operators deæned as P + èxè =,trèx + è+ætrèx, è; P, èxè=,ætrèx + è+trèx, è; where trèxè is the trace of X; æ; and æ are positive constants which are æxed for all time; and given X 2Sènè, X + and X, are its positive and negative part èx = X +, X, è. Thus è1.3è amounts to uniform ellipticity of G, with ellipticity constants and æ. Sometimes we will allow G to be just degenerate elliptic, meaning that è1:4è Gèx; r; p; Xè Gèx; r; p; Y è whenever X, Y is nonnegative deænite. As for the dependence on r, wewill assume that è1:5è r 7! Gèx; r; p; Xè is uniformly continuous, uniformly for x 2 æ and bounded èr;p;xè. Typically we will also request that G be proper, i.e. that è1:6è r 7! Gèx; r; p; Xè is nondecreasing. We will frequently write è1.1è as è1:7è F èx; u; Du; D 2 uè=fèxè bysetting fèxè =,Gèx; 0; 0; 0è, F èx; r;p;xè=gèx; r;p;xè+fèxè, so that è1:8è F èx; 0; 0; 0è 0: In what follows we will move freely between è1.7è and è1.1è. As explained above, f is just a measurable function on æ. Regarding its behavior, we will require that è1:9è f 2 L p èæè; pép 0 ; 2

3 where p 0 é n is a constant such that the generalized maximum principle holds for p é p 0 èsee ë2ë, ë6ë, ë10ëè. Clearly, F of è1.7è will satisfy versions of è1.2è í è1.6è whenever G of è1.1è does, and vice versa. We recall next the deænitions of strong and L p -viscosity solutions of è1.7è. We refer the reader to ë2ë, ë5ë and ë10ë for the theory of L p -viscosity solutions and to ë1ë for an overview of recent results on fully nonlinear elliptic equations. A function u 2 W 2;p G èæè is a strong solution of è1.1è if è1.1è holds pointwise a.e., i.e. x; uèxè;duèxè;d 2 uèxè =0 for a.e. x 2 æ: A function u 2 Cèæè is an L p -viscosity subsolution èrespectively, supersolutionè of è1.1è if for every ëtest function" ' 2 W 2;p èæè and al maximum èrespectively, minimumè point ^x 2 æ of u, ' èrespectively, ess lim inf G x; uèxè;d'èxè;d 2 'èxè 0 x!^x ess lim sup G x!^x x; uèxè;d'èxè;d 2 'èxè 0:è A function u 2 Cèæè is an L p -viscosity solution of è1.1è if it is both an L p -viscosity subsolution and an L p -viscosity supersolution of è1.1è. In case of linear equations è1:10è, nx i;j=1 a ij èxèu xi x j èxè+ nx j=1 b j èxèu xj èxè+cèxèuèxè,fèxè=0 anotion of so-called good solution has been proposed in ë3ë. Namely, u 2 Cèæè is a good solution of è1.10è if there is a sequence u m of strong solutions of approximate problems, nx i;j=1 a m ij èxèum x i x j èxè+ nx j=1 b m j èxèum x j èxè+c m èxèu m èxè,f m èxè=0 such that u m! u in Cèæè. It was proved in ë8ë èsee also ë5ëè that the notions of good and L p -viscosity solutions of è1.10è coincide, at least in the case with b j 0, c 0 and f 2 L 1 èæè. We will say that the functions G 1 ;G 2 ; :::G m ; ::: satisfy structure conditions uniformly in m if è1.2è, è1.3è, è1.5è are satisæed uniformly in m with the same æxed ; æ;æ, and if jg m èx; 0; 0; 0; èjgèxè for some g 2 L p èæè. It was hinted in ë5ë how to extend the notion of a good solution to fully nonlinear equations. Here we make this precise. Deænition 1.1. We say that u 2 Cèæè is a good solution of è1.1è if there exist G m satisfying structure conditions uniformly in m and strong solutions u m of G m = 0 in æ, such that u m! u in Cèæè and G m converge to G in the following sense è1:11è G m èx; t; p; Xè! Gèx; t; p; Xè for a.e. x 2 æ and all èt; p; Xè 2 IRæIR n æsènè: 3

4 The requirement that the constants ; æ; æ be æxed for all equations comes from the fact that the constant p 0 in è1.9è depends on them and on diam èæè. The main result of this paper shows that the notions of L p -viscosity solution and good solution of è1.1è coincide, generalizing the results of ë8ë and ë5ë for linear equations to the general case of f 2 L p èæè and nonzero b j and c. This paper provides another tool for the analysis nonlinear elliptic partial diæerential equations with measurable spatial dependence. For example, using our main theorem, the results in ë3ë on uniqueness of good solutions translate immediately into corresponding results on uniqueness of L p -viscosity solutions. Section 2. Constructing strong solutions We are going to construct a strong solution of the Dirichlet problem è2:1è,æu + Gèx; u; Du; D 2 uè=0 in æ; u = è when G is bounded. This will turn out to be an important construction in proving that an L p -viscosity solution of è1.1è is a good solution of è1.1è as well. Proposition 2.1. Let G: æ æ IR æ IR n æsènè!irbemeasurable, bounded and satisfy è1.2è, è1.4è and è1.5è, let è 2 Cè@æè, and let æ satisfy uniform exterior cone condition. Then the Dirichlet problem è2.1è has a strong solution u 2 CèæèëW 2;p èæè for every pé1. For existence we do not require G to be proper the only essential ingredients are the boundedness and ellipticity of G and in fact we need this greater generality later. However, if G satisæes è1.6è, then the constructed strong solution is unique. Proof. We will solve è2.1è by the æxed point method. To this end, for any given v 2 Cèæè we will consider the Dirichlet problem è2:2è,æu + Gèx; vèxè; Du; D 2 uè=0 in æ; u = è Since the equation in è2.2è is independent of u, it follows from the general theory èsee ë4ë, Theorem 4.1è that è2.2è has an L p -viscosity solution èfor any ænite pè incèæè, which we are going to denote by Tv. By ë2ë, Proposition 3.5, Tv is twice pointwise diæerentiable a.e., thus gèxè =Gèx; vèxè;dètvèèxè;d 2 ètvèèxèè is well-deæned and g 2 L 1 èæè since G is bounded. It follows that Tv is a pointwise a.e. í and therefore L p -viscosity, see ë10ë, Corollary 1.6 í solution of the Dirichlet problem è2:3è,æu =,gèxè in æ; u = è However, è2.3è clearly has a unique strong solution, which must coincide with Tv, and it follows that Tv 2 W 2;p èæè for every ænite p. In particular, Tv is a unique strong solution of è2.2è. To ænish the proof it is now enough to show that the map T : Cèæè! Cèæè has a æxed point. This, however, is fairly obvious. For R suæciently large T is a compact mapping 4

5 from the closed ball of radius R in Cèæè to itself. Hence T has a æxed point and the proof is complete. We close this section with a direct construction of good solutions of è2:4è F èx; u; Du; D 2 uè=fèxè in æ; u = è under the assumptions: è 2 Cè@æè, æ satisæes uniform exterior cone condition, F is measurable and satisæes è1.2è, è1.3è, è1.5è, è1.6è, è1.8è, and è1.9è. Under the same conditions we know that è2.4è also has an L p -viscosity solution see Theorem 4.1 in ë4ë. The construction uses Proposition 2.1, illustrating èin a simpler contextè how it is applied in the following section. We will rewrite the diæerential equation in è2.4è as,æu + Gèx; u; Du; D 2 uè=0; where Gèx; r; p; Xè = trèxè + Fèx; r; p; Xè,fèxè. Clearly G is degenerate elliptic, i.e. it satisæes è1.4è. Without loss of generality we may replace è2.4è by è2:5è,æu + Gèx; u; Du; D 2 uè=0 in æ; u = è For m =1;2;::: consider truncating functions m : IR! IR given by è2:6è m èrè = 8 é :,m for ré,m, r for r 2 ë,m; më, m for rém. For every m we will consider an approximating Dirichlet problem è2:7è,æu + m Gèx; u; Du; D 2 uè =0 in æ; u = è Since m ègèx; r;p;xèè satisæes the conditions of Proposition 2.1 and G is proper, è2.7è has a unique strong solution u m 2 CèæèëW 2;p èæè for every pé1. The family of equations satisæes structure conditions uniformly in m and so by Proposition 4.2 in ë4ë the u m are precompact in Cèæè, and therefore passing to a subsequence if necessary we can assume that u m! u in Cèæè. The function u 2 Cèæè is a desired good solution since the approximations in è2.7è obviously converge to G in the sense of è1.11è. Section 3. Viscosity solutions are good solutions Consider the Dirichlet problem è3:1è F èx; Du; D 2 uè=fèxè in æ; u = è 5

6 We will prove that then every L p -viscosity solution of è3.1è is a good solution. The fact that good solutions are L p -viscosity solutions is obvious from the deænition of good solutions and the general theory convergence and stability of L p -viscosity solutions èsee ë2ë, Theorem 3.8è. Theorem 3.1. Let F be measurable and satisfy è1.2è, è1.3è, è1.8è, let f satisfy è1.9è, let è 2 Cè@æè, and æ satisfy uniform exterior cone condition. Then every L p -viscosity solution of è3.1è is a good solution in the sense of Deænition 1.1, i.e. there is a sequence of operators F m, independent of u, satisfying è1.2è, è1.3è and è1.8è, a sequence f m 2 L p èæè and a sequence u m 2 Cèæè ë W 2;p èæè of strong solutions of è3:2è F m èx; Du m ;D 2 u m è=f m èxè in æ such that è3:3è u m! u in Cèæè; F m converge to F in the sense of è1.11è: è3:4è F m èx; t; p; Xè! F èx; t; p; Xè for a.e. x 2 æ and all èt; p; Xè 2 IR æ IR n æsènè; and è3:5è f m! f in L p èæè and a.e. in æ: Observe that F in è3.1è is independent of u.the result holds with u dependence as well, assuming that F is proper; the proof is the same in all essential features as it is without u dependence. However, the introduction of u dependence introduces additional terms which just clutter up the proof and further obscure the fundamental ideas behind the proof. For these reasons we present the result without u dependence. Recall that Jensen in ë8ë èsee also ë5ëè proved that L n -viscosity solutions of linear equations è1.10è with f 2 L 1 èæè are good solutions. Our result generalizes this to general èsubject to structure conditionsè fully nonlinear equations and f 2 L p èæè;pép 0. Proof. STEP 0. Fix a countable, dense in IR n æsènè sequence èp i ;X i è 2 IR n æsènè, i =1;2;:::. STEP 1. Choose a sequnce ~ f m 2 Cèæè ë L 1 èæè such that Consider the Dirichlet problem æ m = kf, ~ f m k L p èæè! 0: è3:6è P, èd 2 wè, æjdwj = f, ~ f m in æ; w =0 By Corollary 3.10 in ë2ë, è3.6è has a unique strong solution w, and by the maximum principle kwk L 1 èæè Cæ m. Let u m = u, w. It follows that u m is an L p -viscosity solution of è3:7è F èx; Du m ;D 2 u m è ~ f m èxè in æ 6

7 and è3:8è ku, u m k L 1 èæè = kwk L 1 èæè Cæ m : Similarly, solving P + èd 2 wè+æjdwj = f, ~ f m in æ; w =0 and setting u m = u, w we conclude that u m is an L p -viscosity solution of è3:9è F èx; Du m ;D 2 u m è ~ f m èxè in æ and è3:10è ku, u m k L 1 èæè Cæ m : Also u m = u = è = u m STEP 2. Let æ m æ be a subdomain of æ with smooth boundary and such that è3:11è x 2 æ n æ m è 1 m : Next we are going to regularize u m and u m on æ m by means of the by-now standard process of sup-inf convolution, see ë9ë. An equivalent approximation procedure was used in ë8ë without bringing up the connection to sup-inf convolutions. This connection has been pointed out in ë5ë, and here we will follow the approach of ë5ë. Recall that for a given continuous function w: æ! IR, for æé0its sup-convolution w æ and its inf-convolution w æ are deæned as w æ èxè = sup y2æ wèyè, 1 2æ jx, yj2 ; w æ èxè = inf wèyè+ 1 jx,yj2 y2æ 2æ ; x2æ: Now for æ; æ é 0we consider w æ;æ = w æ+æ. It is well known that if æ; æ are suæciently æ small, then w æ;æ is C 1;1 on æ m, and w æ;æ converge to w as æ; æ è 0, see ë9ë. Moreover, this approximation procedure respects viscosity subsolutions, see ë5ë, Section 4. First we will consider èu m è æ;æ. Since ~ f m are bounded it follows from the results in ë8ë and ë5ë èin particular see Proposition 4.6 in ë5ëè that for every suæciently small æ one can choose a suitable æ = æèæè so that u + m =èu m è æ;æ 2 W 2;1 èæ m è and is a strong solution of a perturbed version of è3.7è, namely è3:12è F m x; Du + mèxè;d 2 u + mèxè f m èxè for a.e. x 2 æ m : Here è3:13è F m èx; p; Xè =FèT + m x; p; Xè and f mèxè = ~ f m èt + m xè 7

8 with T + m x = x + ædu+ m. While the equations considered in ë8ë and ë5ë were purely second order, the computations carried out there show that ærst order terms can be accommodated as well. Similarly, setting w æ;æ =èw æ+æ è æ,weconclude that u, m =èu m è æ;æ 2 W 2;1 èæ m è and è3:14è F m x; Du, mèxè;d 2 u, mèxè f m èxè for a.e. x 2 æ m ; where now è3:15è F m èx; p; Xè =FèT, m x; p; Xè and f m èxè = ~ f m èt, m xè with T, m x = x, ædu, m. Derivations of è3.12è and è3.14è use the fact that there is æ m é 0, independent ofæ, such that è3:16è DT æ m æ m I a.e. in æ m : See ë8ë and Section 4 of ë5ë for details. It follows from è3.16è that èt æ mè,1 map null sets into null sets, and therefore F m and F m are measurable. Moreover, by è3.16è the composition with T æ m is an approximate identity inl p èæ m è and therefore decreasing æ if necessary we can achieve that è3:17è kf m, ~ f m k L p èæ mè ; kf m, ~ f m k L p èæ mè 1 m and è3:18è Z æm æ æf èt æ m x; p i;x i è,fèx; p i ;X i è æ æ æ dx 1 m for i =1;2;:::;m: Further, without loss of generality we may also assume that è3:19è ku m, u + mk L 1 èæ mè ; ku m, u, mk L 1 èæ mè Cæ m: Finally, redeæning u, m = u, m, 3Cæ m and u + m = u + m +3Cæ m we obtain that è3.12è and è3.14è still hold, while by è3.8è, è3.10è and è3.19è è3:20è u, 5Cæ m u, m u, Cæ m ; u + Cæ m u + m u +5Cæ m on æ m : STEP 3. Before going any further with the construction of approximating equations, here we will establish some limiting properties of the approximations constructed in Step 2, which will be needed later in Step 6. From è3.18è, for every èp i ;X i è and æ 00 æ Z æ00 æ æf èt æ m x; p i;x i è,fèx; p i ;X i è æ æ æ dx! 0 as m!1: 8

9 By a diagonal argument we construct a subsequence m k and a null set N æ such that x 2 æ nn èfèt æ m k x; p i ;X i è!fèx; p i ;X i è as k!1; for all i; and using structure conditions è1.2è and è1.3è we can generalize this to è3:21è x 2 æ nn èfèt æ m k x; p; Xè! F èx; p; Xè as k!1; for all èp; Xè 2 IR n æsènè: STEP 4. Next choose a constant M m m so that for a.e. x 2 æ m è3:22è æ æ æf m èx; Du + mèxè;d 2 u + mèxèè + æu + mèxè æ æ æ ; æ ææf m èx; Du, m ;D2 u, mè+æu, mèxè æ æ æmm : Writing è3:23è G m èx; p; Xè =,trèxè+ Mm F m èx; p; Xè+trèXè ; it follows from è3.12è and è3.22è that è3:24è G m x; Du + mèxè;d 2 u + mèxè f m èxè for a.e. x 2 æ m : Similarly, è3:25è G m x; Du, mèxè;d 2 u, mèxè f m èxè for a.e. x 2 æ m ; where now è3:26è G m èx; p; Xè =,trèxè+ Mm èf m èx; p; Xè+trèXèè : The next step of the proof is similar to the proof of Theorem 3.30 in ë8ë. Let : æ m æir! ë0; 1ë be a continuous function such that è3:27è èx; tè =0 if u + mèxè t; èx; tè =1 if u, mèxè t: This can be done due to è3.20è. Deæne è3:28è H m èx; r;p;xè=èx; règ m èx; p; Xè+è1,èx; rèè G m èx; p; Xè =,trèxè+èx; rè Mm F m èx; p; Xè+trèXè +è1,èx; rèè Mm èf m èx; p; Xè+trèXèè ; è3:29è h m èx; rè =èx; rèf m èxè+è1,èx; rèè f m èxè and consider the Dirichlet problem è3:30è H m èx; v; Dv; D 2 vè=h m èx; vè in æ m ; v = u m : 9

10 All assumptions of Proposition 2.1 are satisæed and thus è3.30è has a strong solution v m 2 Cèæ m è ë W 2;p èæ mè. Now we deæne è3:31è G m èx; p; Xè =H m èx; v m èxè;p;xè; g m èxè=h m èx; v m èxèè; so that v m solves è3:32è G m x; Dv m èxè;d 2 v m èxè =g m èxè for a.e. x 2 æ m ; v m = u m : Observe that by construction è3:33è kg m k L 1 èæ mè k~ f m k L 1 èæè ; jtrèxè+g m èx; p; Xèj M m : Moreover, from è3.17è and è3.29è è3:34è kg m, ~ f m k L p èæ mè kf m, ~ f m k L p èæ mè + kf m, ~ f m k L p èæ mè 2 m : Next we claim that è3:35è v m u + m +8Cæ m on æ m : n o To show è3.35è consider æ + m = x 2 æ m : v m èxè éu + mèxè. Then èx; v m èxèè 0 on æ + m and therefore g m èxè = f m èxè and G m èx; p; Xè = G m èx; p; Xè for x 2 æ + m. It follows that v m is a strong solution of G m = f m on æ + m, and since by è3.25è u, m is a supersolution of the same equation, by the minimum principle inf u, æ + m, v m inf u, + m, v m m min + mn@æm by è3.20è, and using è3.20è again yields è3.35è. A symmetric argument shows that è3:36è v m u, m, 8Cæ m on æ m : Putting è3.35è, è3.36è and è3.20è together gives u, m, u + m ; inf u, m, è3:37è u, 13Cæ m v m u +13Cæ m on æ m : STEP 5. We will extend g m and G m to the whole æ according to è3:38è gm èxè for x 2 æ f m èxè = m, Gm èx; p; Xè for x 2 æ F 0 for x 2 æ n æ m, m èx; p; Xè = m,,trèxè for x 2 æ n æ m. 10

11 Recalling è3.33è, by Proposition 2.1 the Dirichlet problem è3:39è F m èx; Dw; D 2 wè=f m in æ; u = è has a unique strong solution u m 2 Cèæè ë W 2;p èæè for every pé1. Moreover, by Remark 4.3 in ë4ë there exists a modulus of continuity determined only by ; æ;n;p;æ;kf m k L p èæè ; the modulus of continuity of è, diam èæè and the parameters of the cone condition for æ èand therefore independent of mè such that è3:40è ju m èxè, èèyèj èjx,yjè for x 2 æ; y STEP 6. Denoting by u the modulus of continuity of u on æ we conclude from è3.11è and è3.40è that è3:41è ju m èxè, uèxèj è 1 m è+ uè 1 m è for x 2 æ n æ m: Since both u m and v m solve the same equation G m = g m in æ m while v m = u m, by the maximum principle and è3.41è and this, together with è3.37è, yields sup ju m, v m j sup ju m, uj èm 1è+ uèm 1è; x2æm x2@æm è3:42è sup ju m, uj èm 1è+ uèm 1è+13Cæ m: x2æm Using è3.41è again we conclude that and therefore è3.3è follows. From è3.34è sup ju m, uj è 1 m è+ uè 1 m è+13cæ m; x2æ kf m, fk L p èæè kfk L p èænæmè + kg m, ~ f m k L p èæ mè + kf, ~ f m k L p èæè kfk L p èænæmè + 2 m + æ m; which, together with è3.11è, establishes è3.5è along a subsequence. We will ænish the proof by showing that the convergence in è3.4è holds along a subsequence m k constructed in Step 3 í recall è3.21è. To this end, we will show that F mk converge to F pointwise a.e. in the sense that è3:43è x 2 ænn è F mk èx; p; Xè! F èx; p; Xè as k!1; for all èp; Xè 2 IR n æsènè: However, F m is just a convex combination of truncations of F m and F m deæned in è3.13è and è3.15è èrecall è3.38è, è3.31è, è3.28è, è3.26è and è3.23èè, and hence è3.43è follows easily from è3.21è. 11

12 References ë1ë L.A. Caæarelli and X. Cabre, Fully nonlinear elliptic equations, American Mathematical Society, Providence, 1995 ë2ë L. Caæarelli, M.G. Crandall, M. Kocan and A. Swie, ch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 è1996è, 365í397 ë3ë M. C. Cerrutti, L. Escauriaza and E. B. Fabes, Uniqueness in the Dirichlet problem for some elliptic operators with discontinuous coeæcients, Ann. Mat. Pura Appl. 163 è1993è, 161í180 ë4ë M.G. Crandall, M. Kocan, P.L. Lions and A. Swie, ch, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electronic J. Diæerential Equations 24 è1999è, 1í20 ë5ë M.G. Crandall, M. Kocan, P. Soravia and A. Swie, ch, On the equivalence of various weak notions of solutions of elliptic PDE's with measurable ingredients, in ëprogress in elliptic and parabolic partial diæerential equations", èa. Alvino et al. eds.è, Pitman Research Notes in Math., vol. 50, 1996, 136í162 ë6ë L. Escauriaza, W 2;n a priori estimates for solutions to fully non-linear equations, Indiana Univ. Math. J. 42 è1993è, 413í423 ë7ë D. Gilbarg and N.S. Trudinger, Elliptic partial diæerential equations of second order, 2nd edition, Springer-Verlag, Berlin Heidelberg New York, 1983 ë8ë R. Jensen, Uniformly elliptic PDEs with bounded, measurable coeæcients, J. Fourier Anal. Appl. 2 è1996è, 237í259 ë9ë J.M. Lasry and P.L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math. 55 è1986è, 257í266 ë10ë A. Swie, ch, W 1;p -interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Diæerential Equat. 2 è1997è, 1005í