Asymptotics and regularity of flat solutions to fully nonlinear elliptic problems
aa r X i v : . [ m a t h . A P ] O c t ASYMPTOTICS AND REGULARITY OF FLAT SOLUTIONS TOFULLY NONLINEAR ELLIPTIC PROBLEMS
DISSON DOS PRAZERES AND EDUARDO V. TEIXEIRAA
BSTRACT . In this work we establish local C , a regularity estimates for flat so-lutions to non-convex fully nonlinear elliptic equations provided the coefficientsand the source function are of class C , a . For problems with merely continuousdata, we prove that flat solutions are locally C , Log-Lip . Keywords:
Smoothness properties of solutions, optimal estimates, fully nonlin-ear elliptic PDEs
AMS Subject Classifications: C ONTENTS
1. Introduction 12. Hypotheses and main results 33. Geometric tangential analysis 54. C , a estimates in C , a media 75. Applications 106. Log-Lipschitz estimates in continuous media 11References 131. I NTRODUCTION
The goal of this paper is to obtain optimal estimates for flat solutions to a classof non-convex fully nonlinear elliptic equations of the form(1.1) F ( X , D u ) = G ( X , u , (cid:209) u ) . Under continuous differentiability with respect to the matrix variable and appro-priate continuity assumptions on the coefficients and on the source function, wepresent a Schauder type regularity result for flat solutions, namely for solutionswith small enough norm, | u | ≪ F : B × Sym ( n ) → R is assumed to be uniformly ellip-tic, namely, there exist constants 0 < l ≤ L such that for any M , P ∈ Sym ( n ) , with P ≥ X ∈ B ⊂ R n there holds(1.2) l k P k ≤ F ( X , M + P ) − F ( X , M ) ≤ L k P k . Under such condition it follows as a consequence of Krylov-Safonov Harnack in-equality that solutions to the homogeneous, constant coefficient equation(1.3) F ( D h ) = C , a , for some 0 < a <
1. Under appropriate hypotheses on G : B × R × R n → R , the same conclusion is obtained, i.e., viscosity solutionsare of class C , a . Thus, insofar as the regularity theory for equation of the form(1.1) is concerned, one can regard the right hand side G ( X , u , (cid:209) u ) as an ˜ a − Höldercontinuous source, f ( X ) . Therefore, within this present work, we choose to lookat the RHS G ( X , u , (cid:209) u ) simply as a source term f ( X ) , and equation (1.1) will bewritten as(1.4) F ( X , D u ) = f ( X ) . Regularity theory for heterogeneous equations (1.4) has been a central target ofresearch for the past three decades. While a celebrated result due to Evans andKrylov assures that solutions to convex equations are classical, i.e., C , a for some a >
0, the problem of establishing continuity of the Hessian of solutions to generalequations of the form (1.3) challenged the community for over twenty years. Theproblem has been settled in the negative by Nadirashvili and Vladut, [6, 7], whoexhibit solutions to uniform elliptic equations whose Hessian blows-up.In view of the impossibility of a general existence theory for classical solutionsto all fully nonlinear equations (1.3), it becomes a central topic of research thestudy of reasonable conditions on F and on u as to assure the Hessian of the so-lution is continuous. In such perspective the works [5] and [2] on interior C , a estimates for a particular class of non-convex equations are highlights. Anotherwork towards Hessian estimates of solutions to fully nonlinear elliptic equations is[8], where it is proven that if the operator is of class C in all of its arguments, thensmall solutions are classical.Inspired by problems of the form (1.1), in the present work, we obtain regularityestimates for flat solutions to heterogeneous equation (1.4), under continuity con-ditions on the media. We show that if X ( F ( X , · ) , f ( X )) is a -Hölder continuous,then flat solutions are locally C , a . In the case a =
0, namely when the coefficientsand the source are known to be just continuous, we show that flat solutions arelocally C , Log-Lip .The proofs of both results mentioned above, to be properly stated in Theorem2.2 and Theorem 2.3 respectively, are based on a combination of geometric tan-gential analysis and perturbation arguments inspired by compactness methods inthe theory of elliptic PDEs.We conclude this introduction explaining the heuristics of the geometric tangen-tial analysis behind our proofs. Given a fully nonlinear elliptic operator F , we lookat the family of elliptic scalings F m ( M ) : = m F ( m M ) , m > . EGULARITY FOR FLAT SOLUTIONS 3
This is a continuous family of operators preserving the ellipticity constants of theoriginal equation. If F is differentiable at the origin (recall, by normalization F ( ) = F m ( M ) → ¶ M ij F ( ) M i j , as m → . In other words, the linear operator M ¶ M ij F ( ) M i j is the tangential equation of F m as m →
0. Now, if u solves an equation involving the original operator F , then u m : = m u is a solution to a related equation for F m . However, if in addition it isknown that the norm of u is at most m , then it accounts into saying that u m is anormalized solution to the m -related equation, and hence we can access the univer-sal regularity theory available for the (linear) tangential equation by compactnessmethods. In the sequel we transport such good limiting estimates towards u m , prop-erly adjusted by the geometric tangential path used to access the tangential linearelliptic regularity theory.The paper is organized as follows. In Section 2 we state all the hypotheses,mathematical set-up and notions to be used throughout the whole paper. In thatSection we also state properly the two main Theorems proven in the work. InSection 3 we rigorously develop the heuristics of the geometric tangential analysisexplained in the previous paragraph. The proof of C , a estimates, Theorem 2.2,will be delivered in Section 4. Two applications of such a result will be discussedin Section 5. Theorem 2.3 will be proven in Section 6.2. H YPOTHESES AND MAIN RESULTS
Let us start off by discussing the hypotheses, set-up and main notations used inthis article. For B we denote the open unit ball in the Euclidean space R n . Thespace of n × n symmetric matrices will be denoted by Sym ( n ) . By modulus ofcontinuity we mean an increasing function v : [ , + ¥ ) → [ , + ¥ ) , with v ( ) = F : B × Sym ( n ) → R and f : B → R :(H1) There exist constants 0 < l ≤ L such that for any M , P ∈ Sym ( n ) , with P ≥ X ∈ B , there holds(2.1) l k P k ≤ F ( X , M + P ) − F ( X , M ) ≤ L k P k . (H2) F ( X , M ) is differentiable with respect to M and for a modulus of continuity w , there holds(2.2) k D M F ( X , M ) − D M F ( X , M ) k ≤ w ( k M − M k ) , for all ( X , M i ) ∈ B × Sym ( n ) .(H3) For another modulus of continuity t , there holds | F ( X , M ) − F ( Y , M ) | ≤ t ( | X − Y | ) · k M k , (2.3) | f ( X ) − f ( Y ) | ≤ t ( | X − Y | ) , (2.4) DISSON DOS PRAZERES AND EDUARDO V. TEIXEIRA for all X , Y ∈ B and M ∈ Sym ( n ) . It will also be enforced hereafter in this paperthe following normalization conditions:(2.5) F ( , n × n ) = f ( ) = Definition 2.1.
A continuous function u ∈ C ( B ) is said to be a viscosity subso-lution to (1.4) in B if whenever one touches the graph of u by above by a smoothfunction j at X ∈ B , there holds F ( X , D j ( X )) ≥ f ( X ) . Similarly, u is a viscosity supersolution to (1.4) if whenever one touches the graphof u by below by a smooth function f at Y ∈ B , there holds F ( Y , D f ( Y )) ≤ f ( Y ) . We say u is a viscosity solution to (1.4) if it is a subsolution and a supersolution of(1.4).Condition (H2) fixes a modulus of continuity w to the derivative of F . Theregularity estimates proven in this paper depends upon w . Condition (H3) setsthe continuity of the media. When t ( t ) ≈ t a , 0 < a <
1, the coefficients and thesource function are said to be a -Hölder continuous. In such scenario we provethat flat solutions are locally of class C , a – a sharp Schauder type of estimate fornon-convex fully nonlinear equations. Theorem 2.2 ( C , a regularity) . Let u ∈ C ( B ) be a viscosity solution toF ( X , D u ) = f ( X ) in B , where F and f satisfy (H1)–(H3) with t ( t ) = Ct a for some < a < . There exista d > , depending only upon n , l , L , w , a , and t ( ) , such that if sup B | u | ≤ d then u ∈ C , a ( B / ) and k u k C , a ( B / ) ≤ M · d , where M depends only upon n , l , L , w , and ( − a ) . If f is merely continuous, then even for the classical Poisson equation D u = f ( X ) , solutions may fail to be of class C . In connection to Theorem 5.1 in [9], in thispaper we show that flat solutions in continuous media are locally of class C , Log-Lip ,which corresponds to the optimal regularity estimate under such weaker conditions.
EGULARITY FOR FLAT SOLUTIONS 5
Theorem 2.3 ( C , Log-Lip estimates) . Let u ∈ C ( B ) be a viscosity solution toF ( X , D u ) = f ( X ) in B . Assume (H1)–(H3). Then there exist a d = d ( n , l , L , w , t ) such that if sup B k u k ≤ d , then u ∈ C , Log − Lip ( B ) and | u ( X ) − [ u ( Y ) + (cid:209) u ( Y ) · ( X − Y )] | ≤ − M d · | X − Y | log ( | X − Y | ) , for a constant M that depends only upon n , l , L , w , and ( − a ) .
3. G
EOMETRIC TANGENTIAL ANALYSIS
In this Section we provide a rigorous treatment of the heuristics involved in thegeometric tangential analysis explained at the end of the Introduction. The nextLemmas are central for the proof of both Theorem 2.2 and Theorem 2.3.
Lemma 3.1.
Let F : B × Sym ( n ) → R satisfy conditions (H1) and (H2). Given ≤ g < , there exists h > , depending only on n , l , L , w , and g , such that if usatisfies | u | ≤ in B and solves m − F ( X , m D u ) = f ( X ) in B , for < m ≤ h , sup M ∈ Sym ( n ) | F ( X , M ) − F ( , M ) |k M k ≤ h and k f k L ¥ ( B ) ≤ h , then one can find a number < s < , depending only on n , l and L , and aquadratic polynomial P satisfying m − F ( , m D P ) = , with k P k L ¥ ( B ) ≤ C ( n , l , L ) , for a universal constant C ( n , l , L ) > , such that sup B s | u − P | ≤ s + g . Proof.
Let us suppose, for the purpose of contradiction, that the Lemma fails tohold. If so, there would exist a sequence of elliptic operators, F k ( X , M ) , satisfyinghypotheses (H1) and (H2), a sequence 0 < m k = o ( ) and sequences of functions u k ∈ C ( B ) and f k ∈ L ¥ ( B ) , all linked through the equation(3.1) 1 m k F k ( X , m k D u k ) = f k ( X ) in B , in the viscosity sense, such that(3.2) k u k k ¥ ≤ , m k ≤ k , sup M ∈ Sym ( n ) | F k ( X , M ) − F k ( , M ) |k M k ≤ k and k f k k ¥ ≤ k ;however for some 0 < s < B s | u k − P | > s + g , DISSON DOS PRAZERES AND EDUARDO V. TEIXEIRA that for all quadratic polynomials P that satisfies1 m k F k ( , m k D P ) = . Passing to a subsequence if necessary, we can assume F k ( X , M ) → F ¥ ( X , M ) lo-cally uniform in Sym ( n ) . From uniform C estimate on F k and the coefficientoscillation hypothesis in (3.2), we deduce(3.4) 1 m k F k ( X , m k M ) → D M F ¥ ( , ) · M , locally uniform in Sym ( n ) . Also, by C , a a priori estimates for equation (3.1), upto a subsequence, u k → u ¥ locally uniform in B . Thus, by stability of viscositysolutions, we conclude(3.5) D M F ¥ ( , ) · D u ¥ = , in B . As u ¥ solves a linear, constant coefficient elliptic equation, u ¥ is smooth. Define P : = u ¥ ( ) + Du ¥ ( ) · X + X . D u ¥ ( ) X . Since k u ¥ k ≤
1, it follows from C estimates on u ¥ thatsup B r | u ¥ − P | ≤ Cr , for a constant C that depends only upon dimension n and ellipticity constants, l and L . Thus, if we select s : = − g r C , a choice that depends only on n , l , L and g , we readily havesup B s | u ¥ − P | ≤ s + g , Also, from equation (3.5), we obtain D M F ¥ ( , ) · D P = | m − k F k ( , m k D P | = o ( ) . Now, since F k is uniformly elliptic in B × Sym ( n ) and F k ( , ) =
0, it is possible tofind a sequence of real numbers ( a k ) ⊂ R with | a k | = o ( ) , for which the quadraticpolynomial P k : = P + a k | X | do satisfy m − k F k ( , m k D P k ) = . EGULARITY FOR FLAT SOLUTIONS 7
Finally we have, for any point in B s and k large enough,sup B s | u k − P k | ≤ | u k − u ¥ | + | u ¥ − P | + | P − P k |≤ s + g + s + g + | a k | s < s + g , which contradicts (3.3). Lemma 3.1 is proven. (cid:3) In the sequel, we transfer the geometric tangential access towards a smallnesscondition of the L ¥ of the solution. Lemma 3.2.
Let F satisfy (H1) and (H2) and ≤ a < be given. There exist smalla positive constant d > depending on n , l , L ,and a , and a constant < s < depending only on n , l , L and ( − a ) such that if u is a solution to (1.4) and k u k L ¥ ( B ) ≤ d , sup M ∈ Sym ( n ) | F ( X , M ) − F ( , M ) |k M k ≤ d / and k f k L ¥ ( B ) ≤ d / , then one can find a quadratic polynomial P satisfying (3.6) F ( , D P ) = , with k P k L ¥ ( B ) ≤ d C ( n , l , L ) for a universal constant C ( n , l , L ) > , and sup B s | u − P | ≤ d · s + a Proof.
Define the normalized function v = d − u . We immediate check that d − F ( X , d D v ) = f ( X ) d . If h is the number from Lemma 3.1, we choose d = h and the Lemma follows. (cid:3) C , a ESTIMATES IN C , a MEDIA
In this Section we show that if the coefficients and the source are a -Höldercontinuous, then flat solutions are locally of class C , a , i.e. That is, herein weassume(4.1) t ( t ) . Ct a , for some 0 < a < C >
0, where t is the modulus of continuity of the coeffi-cients and the source function appearing in (2.3) and (2.4). Under such condition,we aim to show that flat solutions are locally of class C , a .The idea of the proof is to employ Lemma 3.2 in an inductive process as toestablish the aimed C , a estimate for flat solutions under an appropriate smallnessregime for the oscillation of the coefficients and the source function. DISSON DOS PRAZERES AND EDUARDO V. TEIXEIRA
Lemma 4.1.
Let F , f and u be under the hypotheses of Lemma 3.2. Then thereexists a d = d ( n , l , L , w ) > , such that if sup B | u | ≤ d and t ( ) ≤ d / , then u ∈ C , a at the origin and | u − ( u ( ) + (cid:209) u ( ) · X + X t D u ( ) X ) | ≤ C · d | X | + a , where C > depends only upon n , l , L , w and ( − a ) .Proof. The proof consists in iterating Lemma 3.2 as to produce a sequence of qua-dratic polynomials(4.2) P k = X t A k X + b k · X + c k with F ( , D P k ) = , that approximates u in a C , a fashion, i.e.,(4.3) sup B s k | u ( X ) − P k ( X ) | ≤ ds ( + a ) k . Furthermore, we aim to control the oscillation of the coefficients of P k as(4.4) | A k − A k − | ≤ C ds a ( k − ) | b k − b k − | ≤ C ds ( + a )( k − ) | c k − c k − | ≤ C ds ( + a )( k − ) where C > s and d are the parameters from Lemma 3.2. Theproof of existence of polynomials P k verifying (4.2), (4.3) and (4.4) will be deliv-ered by induction. The case k = k th step of induction, i.e., by there exists a quadraticpolynomial P k satisfying (4.2), (4.3) and (4.4). We define˜ u ( X ) : = s ( + a ) k ( u ( s k X ) − P k ( s k X )) ;(4.5) ˜ F ( X , M ) : = s k a F ( s k X , s k a · M + D P k ) . (4.6)Notice that (cid:12)(cid:12) D M ˜ F ( X , M ) − D M ˜ F ( X , N ) (cid:12)(cid:12) ≤ w ( s k a k M − N k ) ≤ w ( k M − N k ) , that is, ˜ F fulfills (H2). It readily follows from (4.3) that ˜ u satisfies | ˜ u | L ¥ ( B ) ≤ d . Moreover, ˜ u solve ˜ F ( X , D ˜ u ) = s k a f ( s k X ) = : ˜ f ( X ) in the viscosity sense. From t -continuity of f and the coefficients of F , togetherwith the smallness condition t ( ) ≤ d / , we verify k ˜ f k ¥ ≤ d / , EGULARITY FOR FLAT SOLUTIONS 9 and likewise, sup M ∈ Sym ( n ) | ˜ F ( X , M ) − ˜ F ( , M ) |k M k ≤ d / . Applying Lemma 3.2 to ˜ u gives a quadratic polynomial ˜ P satisfying ˜ F ( , D ˜ P ) = | ˜ u ( X ) − ˜ P ( X ) | ≤ ds + a , for | X | ≤ s . The ( k + ) th step of induction is verified if we define P k + ( X ) : = P k ( X ) + s ( + a ) k ˜ P ( s − k X ) . To conclude the proof of current Lemma, notice that (4.4) implies that { A k } ⊂ Sym ( n ) , { b k } ⊂ R n , and { c k } ⊂ R are Cauchy sequences. Let us label the limiting quadratic polynomial P ¥ ( X ) : = X t A ¥ X + b ¥ X + c ¥ , where A k → A ¥ , b k → b ¥ and c k → c ¥ . It further follows from (4.4)(4.7) | P k ( X ) − P ¥ ( X ) | ≤ C d ( s a k | X | + s ( + a ) k | X | + s ( + a ) k ) , whenever | X | ≤ s k . Finally, fixed X ∈ B s , take k ∈ N such that s k + < | X | ≤ s k and conclude, by means of (4.3) and (4.7), that | u ( X ) − P ¥ ( X ) | ≤ C ds ( + a ) k ≤ C ds + a | X | + a , as desired. (cid:3) We conclude the proof of Theorem 2.2 by verifying that if t ( t ) = t ( ) t a , thesmallness condition of Lemma 4.1, namely t ( ) ≤ d / , is not restrictive. In fact, if u ∈ C ( B ) is a viscosity solution to(4.8) F ( X , D u ) = f ( X ) in B , the auxiliary function v ( X ) : = u ( m X ) m solves F m ( X , D v ) = f m ( X ) , where F m ( X , M ) : = F ( m X , M ) and f m ( X ) : = f ( m X ) . Clearly the new operator F m satisfies the same assumptions (H1)–(H3) as F , withthe same universal parameters l , L and w . Note however thatmax (cid:26) | f m ( X ) − f m ( Y ) | , | F m ( X , M ) − F m ( Y , M ) |k M k (cid:27) ≤ t ( ) m a | X − Y | a , for M ∈ Sym ( n ) . Thus if t m is the modulus of continuity for f m and F m , t m ( ) = t ( ) m a . Finally, we take m : = min ( , a √ d a p t ( ) ) , where d is the universal number from Lemma 3.2. In conclusion, if u solves (4.8)and satisfies the flatness condition k u k L ¥ ( B ) ≤ d : = dm , then Lemma 4.1 applied to v gives C , a estimates for v , which is transported to u accordantly. (cid:3)
5. A
PPLICATIONS
Probably an erudite way to comprehend Theorem 2.2 is by saying that if u solvesa fully nonlinear elliptic equation with C a coefficients and source, then if it is closeenough to a C , a function, then indeed u is C , a . This is particularly meaningful inproblems involving some a priori set data.In this intermediary Section, we comment on two applications of Theorem 2.2.The first one concerns an improvement of regularity for classical solutions in Höldercontinuous media. Corollary 5.1 ( C implies C , a ) . Let u ∈ C ( B ) be a classical, poitwise solutionto F ( X , D u ) = f ( X ) where F ( X , · ) ∈ C ( Sym ( n )) satisfy (H1)–(H2). Assume further that condition (H3)holds with t ( t ) = Ct a for some < a < . Then, u ∈ C , a ( B / ) , and k u k C , a ( B / ) ≤ C ( n , l , L , a , w , t ( ) , k u k C ( B ) ) . Proof.
We shall proof that u is C , a at the origin. To this end, define, for an r > v : B → R , by v ( X ) : = r u ( rX ) − (cid:20) r u ( ) + r (cid:209) u ( ) · X + X t D u ( ) X (cid:21) . We clearly have(5.1) v ( ) = | (cid:209) v ( ) | = | D v ( ) | ≤ V ( r ) , where V is the modulus of continuity for D u . Now, we choose 0 < r ≪ V ( r ) ≤ c n d , where c n is a dimensional constant and d is the number appearing in Theorem2.2. With such choice, v is under the condition of Theorem 2.2, for ˜ F ( X , M ) : = F ( rX , M + D u ( )) and ˜ f ( X ) = f ( rX ) . (cid:3) Remark . We remark that in the proof of Corollary 5.1, we can estimate theabsolute value of v using integral remainders of the Taylor expansion. Thus, thevery same conclusion of that Corollary holds true if we start up only with VMOcondition on D u . It is also interesting to highlight that Corollary 5.1 implies that EGULARITY FOR FLAT SOLUTIONS 11 if u is a viscosity solution in B of a non-convex, fully nonlinear equation underhypotheses (H1)–(H3). Then if u is C at a point p ∈ B , then indeed u is C , a in aneighborhood of p .The second application we explore here regards a mild extension of a recentresult due to Armstrong, Silvestre, and Smart [1], on partial regularity for solutionsto uniform elliptic PDEs. Corollary 5.3 (Partial regularity) . Let u ∈ C ( B ) be a viscosity solution to F ( D u ) = f ( X ) where F ∈ C ( Sym ( n )) satisfy c ≤ D u i u j F ( M ) ≤ c − for some constant c > and the source function f is Lipschitz continuous. Then, u ∈ C , − ( B \ S ) for aclosed set S ⊂ B , with Hausdorff dimension at most ( n − e ) for an e > univer-sal.Proof. The proof is obtained by similar the reasoning employed in [1]. Indeed, thesame conclusion of Lemma 5.2 from [1] follows by noticing that if f ∈ C , , then M − l , L ( D ( u e )) ≤ C and M + l , L ( D ( u e )) ≥ − C where M − l , L ( M ) : = inf l I n ≤ A ≤ L I n tr ( AM ) , M + l , L ( M ) : = sup l I n ≤ A ≤ L I n tr ( AM ) are the Pucci extremal operators. Lemma 7.8 of [4] can still be employed. Thevery same conclusion of Lemma 5.3 from [1] also holds true for equations withLipschitz sources. Indeed, using the same notation from that Lemma, if Y ∈ B issuch that there exist M ∈ Sym ( n ) , p ∈ R n and Z ∈ B ( Y , r ) such that | u ( X ) − u ( Z ) + p . ( Z − X ) + ( Z − X ) . M ( Z − X ) | ≤ r − d | Z − X | , X ∈ B , we define v ( X ) = r ( u ( Z + rX ) − u ( Z ) + r p . X + r X . MX ) and e F ( N ) = F ( N − M ) − F ( − M ) . Notice that e F ( X , D v ) = f ( Z + rX ) − F ( − M ) = e f ( X ) ∈ C , . Thus, applying Theorem 2.2 to v gives u is C , − in B ( Y , r ) . The proof of Theorem5.3 follows now exactly as in [1]. (cid:3)
6. L OG -L IPSCHITZ ESTIMATES IN CONTINUOUS MEDIA
In this Section we proof Theorem 2.3. Initially we show that under continuityassumption on the coefficients of F and on the source f , after a proper scaling,solutions are under the smallness regime requested by Lemma 3.2, with a =
0. Forthat define v ( X ) = u ( m X ) m , F m ( X , M ) : = F ( m X , M ) and f m ( X ) : = f ( m X ) , for a parameter m to be determined. Equation F m ( X , D v ) = f m ( X ) , is satisfied in the viscosity sense. Now we choose m so small that t ( m ) ≤ d / , where t is the modulus of continuity of the media and d > a =
0. In the sequel, define t m ( t ) : = t ( m t ) and note that max (cid:26) | f m ( X | , | F m ( X , M ) − F m ( , M ) |k M k (cid:27) ≤ t m ( | X − Y | ) . Thus, sup M ∈ Sym ( n ) | F m ( X , M ) − F m ( , M ) |k M k ≤ d / and k f m k L ¥ ( B ) ≤ d / . Now if we take k u k L ¥ ( B ) ≤ d : = dm then k v k L ¥ ( B ) ≤ d , Estimates proven for v gives the desired ones for u .The conclusion of the above reasoning is that we can start off the proof of The-orem 2.3 out from Lemma 3.2. That is, the proof of the current Theorem beginswith the existence of a quadratic polynomial P satisfying F ( , D P ) = s > B s | u − P | ≤ s d , holds, provided d is small enough, depending only on universal parameters. Asin Lemma 4.1, we shall prove by induction process the existence of a sequence ofpolynomials P k ( X ) = X t A k X + b k X + c k satisfying F ( , D P k ) = | u ( X ) − P k ( X ) | ≤ ds k for | X | ≤ s k . Moreover, we have the following estimates on the coefficients(6.3) | A k − A k − | ≤ C d | b k − b k − | ≤ C ds ( k − ) | c k − c k − | ≤ C ds ( k − ) . The case k = k th step of induction. Define the scaled function and the scaled operator˜ u ( X ) : = s k ( u ( s k X ) − P k ( s k X )) and ˜ F ( X , M ) : = F ( s k X , M + D P k ) . EGULARITY FOR FLAT SOLUTIONS 13
Easily one verifies that ˜ u is a viscosity solution to˜ F ( X , D ˜ u ) = f ( s k X ) : = ˜ f ( X ) . From the induction hypothesis, (6.2), ˜ u is flat, i.e., | ˜ u | L ¥ ( B ) ≤ d . Also, clearlysup M ∈ Sym ( n ) | ˜ F ( X , M ) − ˜ F ( , M ) |k M k ≤ d / and k ˜ f k L ¥ ( B ) ≤ d / . That is, ˜ u is entitled to the conclusion (6.1), thus there exists a quadratic polynomial˜ P with ˜ F ( , D ˜ P ) = | ˜ u ( X ) − ˜ P ( X ) | ≤ ds k for | X | ≤ s . The ( k + ) th step of induction follows by defining P k + ( X ) : = P k ( X ) + s k ˜ P ( s − k X ) . In view of the coefficient oscillation control (6.3), we conclude b k converges in R n to a vector b ¥ and c k converges in R to a real number c ¥ . Also | c k − c ¥ | ≤ C ds k , (6.4) | b k − b ¥ | ≤ C ds k . (6.5)The sequence of matrices A k may diverge, however, we can at least estimate(6.6) k A k k Sym(n) ≤ kC d . In the sequel, we define the tangential affine function ℓ ¥ ( X ) : = c ¥ + b ¥ · X and estimate, in view of (6.4), (6.5) and (6.6), for | X | ≤ s k ,(6.7) | u ( X ) − ℓ ¥ ( X ) | ≤ | u ( X ) − P k ( X ) | + | c k − c ¥ | + | ( b k − b ¥ ) || X | + | A k || X | ≤ ds k + C ds k + kC ds k ≤ C d ( k s k ) . Finally, fixed X ∈ B s , take k ∈ N such that s k + < | X | ≤ s k . From (6.7), we find | u ( X ) − ℓ ¥ ( X ) | ≤ − ( C d ) · | X | log | X | , as desired. The proof of Theorem 2.3 is concluded. (cid:3) R EFERENCES [1] Armstrong S., Silvestre, L. and Smart, C.
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To appear in Arch. Ration. Mech. Anal.U
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ORTALEZA , CE - B
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NIVERSIDADE F EDERAL DO C EARÁ , D
EPARTAMENTO DE M ATEMÁTICA , C
AMPUS DO P ICI - B
LOCO
914 , F
ORTALEZA , CE - B
RAZIL
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