Regularity up to the boundary for singularly perturbed fully nonlinear elliptic equations
aa r X i v : . [ m a t h . A P ] O c t Regularity up to the boundary for singularly perturbed fullynonlinear elliptic equations by G. C. Ricarte ∗ & J. V. da Silva † Abstract
In this article we are interested in studying regularity up to the boundary for one-phase singularlyperturbed fully nonlinear elliptic problems, associated to high energy activation potentials, namely F ( X , (cid:209) u e , D u e ) = z e ( u e ) in W ⊂ R n where z e behaves asymptotically as the Dirac measure d as e goes to zero. We shall establish globalgradient bounds independent of the parameter e . Keywords:
Fully nonlinear elliptic operators, one-phase problems, regularity up to the boundary, singu-larly perturbed equations, global gradient bounds.
AMS Subject Classifications 2010: 35B25, 35B65, 35D40, 35J15, 35J60, 35J75, 35R35.
Contents ∗ G LEYDSON C HAVES R ICARTE . Universidade Federal Cear´a - UFC. Department of Mathematics. Fortaleza - CE, Brazil - 60455-760.
E-mail address: [email protected] † J O ˜ AO V´ ITOR DA S ILVA . Universidad de Buenos Aires. FCEyN, Department of Mathematics. Buenos Aires, Argentina.
E-mailaddress: [email protected] LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS Throughout the last three decades or so, variational problems involving singular PDEs has received awarm attention as they often come from the theory of critical points of non-differentiable functionals. Thepioneering work of Alt-Caffarelli [1] marks the beginning of such a theory by carrying out the variationalanalysis of the minimization problem(
Minimum ) min ˆ W (cid:0) | (cid:209) v | + c { v > } (cid:1) dX , among competing functions with the same non-negative Dirichlet boundary condition.Since the very beginning it has been well established that such discontinuous minimization problemscould be treated by penalization methods. Indeed, Lewy-Stampacchia, Kinderlehrer-Nirenberg, Caffarelliamong others were the precursors of such an approach to the study of problem D u e = z e ( u e ) over of 70sand 80s. Linear problems in non-divergence form was firstly considered by Berestycki et al in [2]. Teixeirain [7] started the journey of investigation into fully nonlinear elliptic equations via singular perturbationmethods:(1.1) F ( X , D u e ) = z e ( u e ) in W , where z e ∼ e − c ( , e ) . The problem appears in nonlinear formulations of high energy activation models, see[6] and [7]. It can also be employed in the analysis of overdetermined problems as follows. Given W ⊂ R n a domain and a non-negative function j : W → R , it plays a crucial role in Geometry and MathematicalPhysics the question of finding a compact hyper-surface ¶ W ′ ⊂ W such that the following elliptic boundaryvalue problem(1.2) F ( X , (cid:209) u , D u ) = W \ W ′ u = j on ¶ W u = W ′ u n = y in ¶ W ′ , can be solved. Problems as (1.1) became known over the years in the Literature as cavity type problems . u ≡ W ′ F ( X , (cid:209) u , D u ) = { u > } u = j on ¶ W R n u u n = y on ¶ { u > } W Figure 1: Configuration for Free Boundary ProblemHereafter in this paper, F : W × R n × Sym ( n ) → R is a fully nonlinear uniformly elliptic operator, i.e,there exist constants L ≥ l > Unif. Ellip. ) l k N k ≤ F ( X , −→ p , M + N ) − F ( X , −→ p , M ) ≤ L k N k , LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS M , N ∈ Sym ( n ) , N ≥ , −→ p ∈ R n and X ∈ W . As usual Sym ( n ) denotes the set of all n × n symmetricmatrices. Moreover, we must to observe the mapping M F ( X , −→ p , M ) is monotone increasing in thenatural order on Sym ( n ) and Lipschitz continuous. Under such a structural condition, the theory of viscositysolutions provides a suitable notion for weak solutions. Definition 1.1 ( Viscosity solution).
For an operator F : W × R n × Sym ( n ) → R , we say a function u ∈ C ( W ) is a viscosity supersolution (resp. subsolution) to F ( X , (cid:209) u , D u ) = f ( X ) in W , if whenever we touch the graph of u by below (resp. by above) at a point Y ∈ W by a smooth function f ,there holds F ( Y , (cid:209) f ( Y ) , D f ( Y )) ≤ f ( Y ) ( resp. ≥ f ( Y )) . Finally, we say u is a viscosity solution if it is simultaneously a viscosity supersolution and subsolution. Remark . All functions considered in the paper will be assumed continuous in W , namely C -viscositysolutions, see Caffarellli-Cabr´e [3] and Teixeira [7]. However, we also can to consider L p -viscosity notionfor such a solutions, see for example Winter [8].In [6], several analytical and geometrical properties of such a fully nonlinear singular problem (1.1)were established. Notwithstanding, regularity up to the boundary for approximating solutions has not beenproven in the literature yet. This is the key goal of the present article. More precisely, we shall prove auniform gradient estimate up to the boundary for viscosity solutions of the singular perturbation problem( E e ) (cid:26) F ( X , (cid:209) u e , D u e ) = z e ( u e ) in W u e = j on ¶ W , where we have: the singular reaction term z e ( s ) = e z (cid:0) s e (cid:1) for some non-negative z ∈ C ¥ ([ , ]) , a param-eter e >
0, a non-negative j ∈ C , g ( W ) , with 0 < g <
1, and, a bounded C , domain W (or ¶ W for short).Throughout this paper we will assume the following bounds: k j k C , g ( W ) ≤ A and k z k L ¥ ([ , ]) ≤ B . Theorem 1.3 ( Global uniform Lipschitz estimate).
Let u e be a viscosity solution to the singular pertur-bation problem ( E e ) . Then under the assumptions ( F1 ) − ( F3 ) there exists a constant C ( n , l , L , b , A , B , W ) > independent of e , such that k (cid:209) u e k L ¥ ( W ) ≤ C . Our new estimate allows us to obtain existence for corresponding free boundary problem (1.2), keepingthe prescribed boundary value data, see Theorem 4.8. Finally, we should emphasize our estimate general-izes the local gradient bound proven in [7], see also [6] for a rather complete local analysis of such a freeboundary problem.Although we have chosen to carry out the global analysis for the homogeneous case, the results pre-sented in this paper can be adapted, under some natural adjustments, for the non-homogeneous case, (cid:26) F ( X , (cid:209) u e , D u e ) = z e ( u e ) + f e ( X ) in W u e ( X ) = j ( X ) on ¶ W , with 0 ≤ c ≤ f e ≤ c .Our approach follows the pioneering work of Gurevich [4], where it is introduced a new strategy toinvestigate uniform estimate up to boundary of two-phase singular perturbation problems involving linearelliptic operators of type L u = ¶ i ( a i j ¶ j u ) . This method has been successfully applied by Karakhanyan in[5] for the one-phase problem in the case involving nonlinear singular/degenerate elliptic operators of the p -Laplacian type D p u e = z e ( u e ) . LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS The paper is organized of following way: In Section 2 we shall introduce the notation which will beused throughout of the paper, as well as we set up the structural assumptions for fully nonlinear ellipticoperators. In Section 3 we discuss about the existence of appropriated notion of weak solutions to problem( E e ), namely Perron’s type solutions , see Theorem 3.2. The Section 4 is devoted to prove the main Theorem1.3, for this reason it contains several keys Lemmas which are standard in the global regularity theory forelliptic operators in accordance with Gurevich [4] and Karakhanyan [5], as well as Teixeira [7] and Ricarte-Teixeira [6] for the corresponding local fully nonlinear singular perturbation theory. The free boundaryproblem, namely Theorem 4.8 is obtained as consequence of global Lipschitz regularity. Finally, the lastSection 5 is an Appendix where we prove two technical Lemmas (respectively Lemmas 5.1 and 5.2) thatplay an important role in order to prove the main Theorem 1.3 in Section 4.
We shall introduce some notations and structural assumptions which we will use throughout this paper. X n indicates the dimension of the Euclidean space. X H + is the half-space { X n > } . X H : = { X = ( X , . . . , X n ) ∈ R n : X n = } indicates the hyperplane. X ˆ X is the vertical projection of X on H . X C X : = (cid:8) Y ∈ H + : | Y − ˆ Y | ≥ | Y − X | (cid:9) is the cone with vertex at point X ∈ H . X B r ( X ) is the ball with center at X and radius r , and, B r the ball B r ( ) . X B + r ( X ) : = B r ( X ) ∩ H + . X B ′ r ( X ) is the ball with center at X and radius r in H . Remark . Throughout this article
Universal constants are the ones depending only on the dimension,ellipticity and structural properties of F , i. e., n , l , L and b .Also, following classical notation, for constants L ≥ l > P + l , L ( M ) : = l · (cid:229) e i < e i + L · (cid:229) e i > e i and P − l , L ( M ) : = l · (cid:229) e i > e i + L · (cid:229) e i < e i the Pucci’s extremal operators , where e i = e i ( M ) are the eigenvalues of M ∈ Sym ( n ) .We shall introduce structural conditions that will be frequently used throughout of this paper: (F1) ( Ellipticity and Lipschitz regularity condition ) For all M , N ∈ Sym ( n ) , −→ p , −→ q ∈ R n , X ∈ W P − l , L ( M − N ) − b |−→ p − −→ q | ≤ F ( X , −→ p , M ) − F ( X , −→ q , N ) ≤ P + l , L ( M − N ) + b |−→ p − −→ q | (F2) ( Normalization condition ) We shall suppose that, F ( X , , ) = (F3) ( Small oscillation condition ) We must to assumesup X ∈ W Q F ( X , X ) ≪ Q F ( X , X ) : = sup M ∈ Sym ( n ) \{ } | F ( X , , M ) − F ( X , , M ) |k M k LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS Remark . Assumption ( F1 ) is equivalent to notion of uniform ellipticity Unif. Ellip. when −→ p = −→ q .The assumption ( F2 ) is not restrictive, since we can always redefine the operator in order to check it. Thesmallest regime on oscillation of F , namely condition ( F3 ) , depends only on universal parameters, see [8]. Example 2.3 ( Isaacs type operators).
An example which we must have in mind are the Isaacs’ operatorsfrom stochastic game theory(2.1) F ( X , −→ p , M ) : = sup a ∈ A inf b ∈ B (cid:16) Tr h A a , b ( X ) · M i + D B a , b ( X ) , −→ p E(cid:17) (cid:18) resp. inf A sup B ( · · · ) (cid:19) , where A a , b is a family of measurable n × n real symmetric matrices with small oscillation satisfying l k x k ≤ x T A a , b ( X ) x ≤ L k x k , ∀ x ∈ R n and k B a , b k L ¥ ( W ) ≤ b . In this Section we shall comment on the existence of appropriated viscosity solutions to the singularlyperturbed problem ( E e ). Such a solutions are labeled by Perron’s type solutions . Theorem 3.1 ( Perron’s type method , [6]) . Let f ∈ C , ([ , ¥ )) be a bounded function. Suppose thatthere exist a viscosity sub-solution u ∈ C ( W ) ∩ C , ( W ) (respectively super-solution u ∈ C ( W ) ∩ C , ( W ) ) toF ( X , (cid:209) w , D w ) = f ( w ) satisfying u = u = g ∈ C ( ¶ W ) . Define the set of functions S : = (cid:26) v ∈ C ( W ) (cid:12)(cid:12)(cid:12)(cid:12) v is a viscosity super-solution toF ( X , (cid:209) w , D w ) = f ( w ) such that u ≤ v ≤ u (cid:27) . Then, (3.1) u ( X ) : = inf v ∈ S v ( X ) , for x ∈ W is a continuous viscosity solution to F ( X , (cid:209) w , D w ) = f ( w ) in W with u = g continuously on ¶ W . Existence of Perron’s type solution to ( E e ) will follow by choosing u : = u e and u : = u e as solutions tothe boundary value problems: ( F ( X , (cid:209) u e , D u e ) = sup [ , ¥ ) z e ( u e ( X )) in W u e ( X ) = j ( X ) on ¶ W and (cid:26) F ( X , (cid:209) u e , D u e ) = W u e ( X ) = j ( X ) on ¶ W , We must note that for each e > u e and u e follows as consequence of standardmethods of sub and super solutions. Moreover, we have that u ∈ C ( W ) ∩ C , ( W ) and u ∈ C ( W ) ∩ C , ( W ) are viscosity subsolution and supersolution to ( E e ) respectively. Finally, as consequence of the Theorem3.1 we have the following existence Theorem: Theorem 3.2 ( Existence of Perron’s type solutions, [6]).
Given W ⊂ R n be a bounded Lipschitz domainand g ∈ C ( ¶ W ) be a nonnegative boundary datum. There exists for each e > fixed, a nonnegative Perron’stype viscosity solution u e ∈ C ( W ) to ( E e ) . In this section, we shall present the proof of Theorem 1.3. Thus let us assume the assumptions ofproblem ( E e ).We make a pause as to discuss some remarks which will be important throughout this work. Firstly itis important to highlight that is always possible to perform a change of variables to flatten the boundary. LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS ¶ W is a C , set, the part of W near ¶ W can be covered with a finite collection of regions that canbe mapped onto half-balls by diffeomorphisms (with portions of ¶ W being mapped onto the “flat” parts ofthe boundaries of the half-balls). Hence, we can use a smooth mapping, reducing this way the general caseto that one on B + , and, the boundary data would be given on B ′ .Previously we start the proof of the global Lipschitz estimative, we need to assure the non-negativityand boundedness of solutions to ( E e ). This statement is a consequence of the Alexandroff-Bekelman-PucciMaximum Principle, see [3] for more details. Lemma 4.1 ( Nonnegativity and boundedness, [6] and [7]).
Let u e be a viscosity solution to ( E e ) . Thenthere exists a universal constant C > such that ≤ u e ( X ) ≤ C k j k L ¥ ( W ) in W . We will now establish a universal bound for the Lipschitz norm of u e up to the boundary. The proofwill be divided in two cases. Case 1: Lipschitz regularity up to the boundary in the region { ≤ u e ≤ e } . Theorem 4.2.
Let u e be a viscosity solution to ( E e ) . For X ∈ { ≤ u e ≤ e } ∩ B + there exists a universalconstant C > independent of e such that | (cid:209) u e ( X ) | ≤ C . Proof.
We denote by d ( X ) : = dist ( X , H ) the vertical distance. If d ( X ) ≥ e , then B e ( X ) ⊂ B + for e ≪
1. Therefore, from local gradient bounds [6, 7], there exists a universal constant C > e , such that | (cid:209) u e ( X ) | ≤ C . On the other hand, if d ( X ) < e , then it is sufficient to prove that there exists a universal constant C > e , such that(4.1) u e ( ˆ X ) ≤ C e . Indeed, suppose that (4.1) holds. Consider h : B + → R to be the viscosity solution to the Dirichlet problem (cid:26) F ( Y , (cid:209) h , D h ) = B + h = u e on ¶ B + . From C , a regularity estimates up to the boundary (see for instance [8, Theorem 3.1]), we know that h ∈ C , a (cid:16) B + (cid:17) with the following estimate | (cid:209) h ( Y ) | ≤ c (cid:16) k h k L ¥ ( B + ) + k j k C , g ( B ′ ) (cid:17) ≤ C in B + and by Comparison Principle we have u e ≤ h in B + . Hence, it follows from assumption (4.1) that u e ( Y ) ≤ h ( Y ) ≤ h ( ˆ X ) + C | Y − ˆ X | ≤ C e if Y ∈ B + e ( ˆ X ) LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS C , a regularity estimates from [8], we obtain | (cid:209) u e ( X ) | ≤ C ( n , l , L , b , B ) . In order to prove (4.1) suppose, by purpose of contradiction, there exists e > u e ( ˆ X ) ≥ k e for k ≫ . We shall denote r : = dist ( ˆ X , { ≤ u e ≤ e } ) . Consider X ∈ { ≤ u e ≤ e } ∩ ¶ B + r ( ˆ X ) a point to which the distance is achieved, i.e., r = | X − ˆ X | . Thereafter, let C ˆ X be the cone with vertex at ˆ X ∈ H . Suppose initially that X ∈ C ˆ X then B r ( X ) ⊂ B + . X ˆ X b b r B r ( X ) B + r ( ˆ X ) C ˆ X { X n = } b { ≤ u e ≤ e } Figure 2: Geometric argument for the case X ∈ C ˆ X .Now, let us define, v e : B → R by v e ( Y ) : = u e ( X + ( r / ) Y ) e . Therefore, v e satisfies in the viscosity sense F e ( Y , (cid:209) v e , D v e ) = e (cid:16) r (cid:17) z ( v e ) : = g ( Y ) , where(4.2) F e ( Y , −→ p , M ) : = e (cid:16) r (cid:17) F X + r Y , e r · p , e (cid:18) r (cid:19) M ! . Now note that g ∈ L ¥ ( B ) , since r < e and F e satisfies ( F1 ) − ( F3 ) with constant ˜ b = r · b . Moreover,since v e ( ) ≤ v e ( Y ) ≤ c for Y ∈ B , i.e., u e ( X ) ≤ c e , X ∈ B r ( X ) . LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS Z ∈ B ′ r ( ˆ X ) . It follows that j ( Z ) ≥ j ( ˆ X ) − A · | Z − ˆ X | ≥ k e − r · A ≥ ( k − A ) e , since r < e . Define the scaled function w e : B + → R , w e ( Y ) : = u e ( ˆ X + r Y ) e . It readily follows that (cid:26) F e ( Y , (cid:209) w e , D w e ) = B + w e ( Y ) ≥ k − A on B ′ , where F e is as in (4.2). Therefore according to Lemma 5.1, w e ( Y ) ≥ c ( k − A ) in B + . In other words, we have reached that u e ( X ) ≥ c e ( k − A ) in B + r ( ˆ X ) . Hence c e ( k − A ) ≤ u e ( Z ) ≤ c e , ∀ Z ∈ ¶ B + r ( ˆ X ) ∩ ¶ B r ( X ) , which leads to a contradiction for k ≫ X C ˆ X , choose X ∈ { ≤ u e ≤ e } such that r : = dist ( ˆ X , { ≤ u e ≤ e } ) = | ˆ X − X | . From triangular inequality and the fact that r ≤ r we have | X − ˆ X | ≤ | X − ˆ X | + | ˆ X − ˆ X | ≤ r + r ≤ r + r . If X ∈ C ˆ X the result follows from previous analysis. Otherwise, let X be such that r : = dist ( ˆ X , { ≤ u e ≤ e } ) = | ˆ X − X | . As before we have | X − ˆ X | ≤ | ˆ X − X | + | ˆ X − ˆ X | ≤ r + r + r , since r ≤ r ≤ r . Observe that this process must finish up within a finite number of steps. Indeed, supposethat we have a sequence of points X j ∈ ¶ { ≤ u e ≤ e } , X j + C ˆ X j ( j = , , . . . ) satisfying, r j + : = dist ( ˆ X j , { ≤ u e ≤ e } ) = | X j + − ˆ X j | and(4.3) r j + ≤ r j ≤ r j + . Thus, it follows from (4.3) that | X j − ˆ X | ≤ r + r j (cid:229) i = i ≤ r . Therefore, up to a subsequence, X j → X ¥ ∈ B ′ r ( ˆ X ) with j ( X ¥ ) = e . However, j ( X ¥ ) ≥ j ( ˆ X ) − A · | ˆ X − X ¥ | ≥ e ( k − A ) ≫ e for k ≫ LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS ˆ X X ˆ X X r r ˆ X C ˆ X { ≤ u e ≤ e } C ˆ X { X n = } b b b b Figure 3: Geometric argument for the inductive process.
Case 2: Lipschitz regularity in the region B + / \ { ≤ u e ≤ e } . Theorem 4.3.
Let u e be a viscosity solution to ( E e ) . If X ∈ B + ∩ { u e > e } , then there exists a constantC = C ( n , l , L , b , A ) > such that | (cid:209) u e ( X ) | ≤ C . The proof of the theorem consists in analysing three possible cases (Lemmas 4.5, 4.6, 4.7 below).Henceforth we shall use the following notation d e ( X ) : = dist ( X , { ≤ u e ≤ e } ) and d ( X ) : = dist ( X , H ) . The next result is decisive in our approach.
Lemma 4.4.
Let u e be a viscosity solution to ( E e ) with j ∈ C , g ( B ′ ) . Then, for all X ∈ B ′ ∩ { u e > e } ,there exists a constant c = c ( n , l , L , b ) > such that j ( X ) ≤ e + c · d e ( X ) . Proof.
Let us suppose for sake of contradiction that there exists an e > X ∈ B ′ \ { ≤ u e ≤ e } suchthat j ( X ) ≥ e + k · d e ( X ) holds for k ≫
1, large enough. Let Z = Z e ∈ ¶ { ≤ u e ≤ e } be a point to which the distance is achieved,i.e. d e : = d e ( X ) = | X − Z | . We have two cases to analyse: If Z ∈ C X , then the normalized function v e : B + → R given by v e ( Y ) : = u e ( X + d e Y ) − ed e satisfies F e ( Y , (cid:209) v e , D v e ) = B + in the viscosity sense, where F e ( Y , −→ p , M ) : = d e F (cid:18) X + d e Y , −→ p , d e M (cid:19) . As in Theorem 4.2, F e satisfies ( F1 ) − ( F3 ) with constant ˜ b = d e b . Moreover, v e ( Y ) ≥ B + . LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS X ∈ B ′ d e ( X ) we should have for k ≫ j ( X ) ≥ j ( X ) − A d e ≥ e + k d e − A d e ≥ e + k d e , i.e, j ( X + d e Y ) − ed e ≥ k B ′ . In other words, v e ( Y ) ≥ ck ∀ Y ∈ B ′ . Hence, from Lemma 5.1 we have that v e ≥ ck in B + . In a more precise manner,(4.4) u e ( X ) ≥ e + Ck d e , X ∈ B + d e ( X ) . From now on, let us consider ˜ B : = B d e ( P ) , where P = P e : = Z + X − Z . If we define w e : = u e − e , thensince Z ∈ ¶ ˜ B , it follows that F e ( X , (cid:209) w e , D w e ) = B , (4.5) w e ( Z ) = u e ( Z ) − e = , (4.6) ¶w e ¶n ( Z ) ≤ | (cid:209) w e ( Z ) | ≤ C . (4.7)Therefore, from (4.5)-(4.7) we can apply Lemma 5.2, which gives w e ( P ) ≤ C · d e , i.e.,(4.8) u e ( P ) ≤ e + C d e . At a point P on ¶ B + d e ( X ) we have (according to (4.4) and (4.8)) e + kc d e ≤ u e ( P ) ≤ e + C d e which gives a contradiction if k has been chosen large enough.The second case, namely Z C X , it is treated similarly as in Theorem 4.2 and for this reason we omitthe details here. Lemma 4.5.
Let u e be a viscosity solution to ( E e ) and X ∈ B + ∩ { u e > e } such that d e ( X ) ≤ d ( X ) . Thenthere exists a universal constant C > , such that | (cid:209) u e ( X ) | ≤ C . Proof.
We may assume with no loss of generality that d e ( X ) ≤ . Otherwise, if we suppose that d e ( X ) > ,then the result would follow from [6, 7]. From now on, we select X e ∈ ¶ { ≤ u e ≤ e } a point which achievesthe distance, i.e., d e : = d e ( X ) = | X − X e | . Since | X e | ≤ | X | + d e ≤ , LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS X e ∈ B + ∩ { ≤ u e ≤ e } . This way, by applying Theorem 4.2, there exists a constant C = C ( n , l , L , b , A , B ) > | (cid:209) u e ( X e ) | ≤ C . By defining the re-normalized function v e : B → R as v e ( Y ) : = u e ( X + d e Y ) − ed e . Then, as before v e satisfies F e ( Y , (cid:209) v e , D v e ) = B , (4.9) v e ( Y e ) = , (4.10) | (cid:209) v e ( Y e ) | ≤ C , (4.11) v e ( Y ) ≥ Y ∈ B , (4.12)where F e ( Y , −→ p , M ) : = d e F (cid:18) X + d e Y , −→ p , d e M (cid:19) and Y e : = X e − X d e ∈ ¶ B . From (4.9)-(4.12) we are able to apply Lemma 5.2 and conclude that there exists a universal constant c > v e ( ) ≤ c . Moreover, from Harnack inequality v e ≤ C in B / . Therefore, by C , a regularity estimates (see for example [3]) we must have that | (cid:209) u e ( X ) | = | (cid:209) v e ( ) | ≤ d e k u e − e k ≤ C , and the Lemma is proved. Lemma 4.6.
For X ∈ B + ∩ { u e > e } such that d ( X ) < d e ( X ) ≤ d ( X ) , we have | (cid:209) u e ( X ) | ≤ C for some constant C = C ( n , l , L , b , A , B ) > .Proof. Similar to Lemma 4.5, we may assume that d e ≤ , otherwise, as in Lemma 4.5 the gradient bound-edness follows from local estimates [6, 7]. Define the scaled function v e : B → R by v e ( Y ) : = u e ( X + d Y ) − ed , where d = d ( X ) . Clearly F d ( Y , (cid:209) v e , D v e ) = B in the viscosity sense, and, from Harnack inequality v e ≤ Cv e ( ) ∼ d in B . By applying once more C , a regularity estimates, we obtain(4.13) | (cid:209) u e ( X ) | = | (cid:209) v e ( ) | ≤ C d . LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS u e − e in terms of the vertical distance d ( X ) . To this end,consider h the viscosity solution to the Dirichlet problem(4.14) (cid:26) F ( X , (cid:209) h , D h ) = B + h = u e on ¶ B + . Since 0 ≤ u e ≤ C ( n , l , L , b , B ) , it follows from C , a estimate up to boundary [8] that h ∈ C , a (cid:16) B + (cid:17) .Moreover | (cid:209) h ( X ) | ≤ C (cid:16) k h k L ¥ ( B + ) + k j k C , g ( B ′ ) (cid:17) ≤ C ( C + A ) : = C ∗ . From Comparison Principle, we have that u e ≤ h in B + . Hence,(4.15) u e ( X ) ≤ h ( X ) ≤ h ( ˆ X ) + C ∗ | X − ˆ X | ≤ j ( ˆ X ) + C ∗ d . Now, we have that | ˆ X | ≤ | X | + d ≤ , and, consequently we are able to apply Lemma 4.4 which gives(4.16) j ( ˆ X ) ≤ e + c · dist ( ˆ X , { ≤ u e ≤ e } ) ≤ e + c ( d e + d ) ≤ e + c d . Thus, it follows from (4.15) and (4.16) that u e ( X ) − e ≤ C d , where C : = C ( c + C ∗ ) . Finally, if we apply C , a estimate, Harnack inequality and estimate (4.13),respectively, we end up with | (cid:209) u e ( X ) | = | (cid:209) v e ( ) | ≤ d k u e − e k L ¥ (cid:18) B (cid:19) ≤ C which concludes the proof. Lemma 4.7.
If X ∈ B + ∩{ u e > e } and d ( X ) < d e ( X ) , then there exists a constant C = C ( n , l , L , b , A , B ) > such that | (cid:209) u e ( X ) | ≤ C . Proof.
Initially we will consider the case when d e ≤ . The following inclusion holds true: B + d e ( ˆ X ) ⊂ B + \ { ≤ u e ≤ e } . In fact, if Y ∈ B + d e ( ˆ X ) then | Y | ≤ | Y − X | + | X | ≤ d e + | X | ≤ . Now, using the same argument as in Lemma 4.6 (see (4.14)) we are able to estimate u e in B + d e ( ˆ X ) as follows u e ( Y ) ≤ u e ( ˆ Y ) + C ∗ d e ≤ e + c · dist ( ˆ Y , { ≤ u e ≤ e } ) + C ∗ d e . Since the distance function is Lipschitz continuous with Lipschitz constant 1, we havedist ( ˆ Y , { ≤ u e ≤ e } ) ≤ d e + | ˆ Y − X | ≤ d e . Therefore, u e ( Y ) ≤ e + (cid:18) c + C ∗ (cid:19) d e = e + c d e . LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS v e ( Y ) = u e ( Y ) − e in B + d e ( ˆ X ) , we have that F ( Y , (cid:209) v e , D v e ) = B + d e ( ˆ X ) in the viscosity sense. From C , a estimate up to boundary and Lemma 4.1, we have | (cid:209) u e ( X ) | = | (cid:209) v e ( X ) | ≤ C ( c + A ) . On the other hand, for the case d e ≥ we have the following inclusion B + ( ˆ X ) ⊂ B \ { ≤ u e ≤ e } . Inthis situation, since supp ( z e ) = [ , e ] , ( F ( X , (cid:209) u e , D u e ) = B + ( ˆ X ) ≤ u e = j ≤ C on B ′ ( ˆ X ) and, consequently, the estimate will follow from C , a estimates up to the boundary.An immediate consequence of Theorem 1.3 and Lemma 4.1is the existence of solutions via compact-ness in the Lip-Topology for any family ( u e ) e > of viscosity solutions to singular perturbation problem( E e ). We consequently obtain Theorem 4.8 ( Limiting free boundary problem).
Let ( u e ) e > be a family of solutions to ( E e ) . For every e k → + there exist a subsequence e k j → + and u ∈ C , ( W ) such that (1) u e kj → u uniformly in W . (2) F ( X , (cid:209) u , D u ) = in W ∩ { u > } in the viscosity sense. In this final section we are going to give the proof of some technical results, which were temporarilyomitted.
Lemma 5.1 ( Boundary’s estimates propagation Lemma).
Suppose that u ≥ is a viscosity solution to (cid:26) F ( X , (cid:209) u , D u ) = in B + u ≥ s > on B ′ . Then there exists a universal constant C = C ( n , l , L , b ) > such thatu ( X ) ≥ C s , X ∈ B + . Proof.
First of all consider the following Dirichlet problem(5.1) F ( X , (cid:209) w , D w ) = B + w = s on B ′ w = ¶ B ∩ { X n > } . From C , a regularity estimate, [8, Theorem 3.1] we have w ∈ C , a (cid:16) B + (cid:17) , and, by the Comparison Principle(5.2) 0 ≤ w ≤ s in B + . From now on, it is appropriate we define the following reflection U : B → R ,(5.3) U ( X ) : = (cid:26) w ( X ) if X ∈ B + ∪ B ′ s − w ( X , . . . , X n − − X n ) if X ∈ B ∩ { X n < } . LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS U is a viscosity solution to G ( X , (cid:209) U , D U ) = B , where G ( X , −→ p , M ) : = (cid:26) F ( X , −→ p , M ) if X n ≥ − F ( e X , −→ e p , e M ) if X n < , with e X : = ( X , . . . , X n − , − X n ) , e p : = ( − p , . . . , − p n − , p n ) , e M : = (cid:26) − M i j if 1 ≤ i , j ≤ n − i = j = nM i j otherwise.Thus, from (5.2), s ≤ U ≤ s in B − Hence, 0 ≤ U ≤ s in B . Moreover, from Harnack inequality we have thatsup B / U ≤ c inf B / U . Particularly, w ( X ) ≥ c − s in B + . Therefore, the proof follows through the previous inequality combined with the Comparison Principle.
Lemma 5.2 ( Hopf’s type boundary principle).
Let u be a viscosity solution to (cid:26) F ( X , (cid:209) u , D u ) = in B r ( Z ) u ≥ in B r ( Z ) . with r ≤ . Assume that for some X ∈ ¶ B r ( Z ) ,u ( X ) = and ¶ u ¶n ( X ) ≤ q , where n is the inward normal direction at X . Then there exists a universal constant C > such thatu ( Z ) ≤ C q r . Proof.
By using a scaling argument, we may assume r =
1. Indeed, it is sufficient to consider the scaledfunction v : B → R v r ( Y ) = u ( Z + rY ) r . As before, v r is a viscosity solution of F r ( Y , (cid:209) v r , D v r ) = B , with F r ( Y , −→ p , M ) : = rF (cid:18) Z + rY , −→ p , r M (cid:19) Let A : = B \ B be an annular region and define w : A → R by w ( Y ) : = m (cid:16) e − d | Y | − e − d (cid:17) LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS m and d will be chosen a posteriori . One can computer the gradient andHessian of w in A as follows ¶ i w ( Y ) = − md Y i e − d | Y | , ¶ i j w ( Y ) = md Y i Y j e − d | Y | − md e − d | Y | d i j , | (cid:209) w ( Y ) | = md e − d | Y | | Y | . In particular, for every M ∈ A l , L : = ( A ∈ Sym ( n ) (cid:12)(cid:12) l k x k ≤ n (cid:229) i , j = A i j x i x j ≤ L k x k , ∀ x ∈ R n ) we haveTr (cid:0) M · D w (cid:1) − b | (cid:209) w | = n (cid:229) i , j = m i j ¶ i j w − b · s n (cid:229) i = ( ¶ i w ) = md e − d | Y | Tr ( M · Y ⊗ Y ) − dm Tr ( M ) e − d | Y | − md b | Y | e − d | Y | ≥ md l | Y | e − m | Y | − dm n L e − d | Y | − md b | Y | e − d | Y | = md ( dl | Y | − b | Y | − n L ) e − d | Y | ≥ md (cid:18) dl − b − n L (cid:19) e − d | Y | in A , where x ⊗ x = ( x i x j ) i , j . Choose and fix d ≥ l ( b + n L ) . Then, it follows readily that P − l , L ( D w ) − b | (cid:209) w | ≥ A . Therefore, since r ≤
1, if d ∈ (cid:2) l ( ˜ b + n L ) , + ¥ (cid:1) , with ˜ b = rb , we have F r ( Y , (cid:209) w ( Y ) , D w ( Y )) ≥ A . Now by Harnack inequality v r ( ) ≤ sup B / v r ≤ c inf B / v r , Hence v r ( Y ) ≥ c − o v r ( ) in B . By choosing m = v r ( ) c (cid:18) e − d − e − d (cid:19) we have w ≤ v r on ¶ A and Comparison Principle gives that w ≤ v r in A Thus, if we label Y : = X − Zr then md e − d ≤ ¶w¶n ( Y ) ≤ ¶ v r ¶n ( Y ) ≤ q . Therefore, v r ( ) ≤ qd − c (cid:16) e d − (cid:17) , and by returning to the original sentence we can conclude that u ( Z ) ≤ c q r . LOBAL REGULARITY FOR CAVITY TYPE PROBLEMS Acknowledgements
The authors would like to thank Eduardo V. Teixeira for insightful comments and suggestions thatbenefited a lot the final outcome of this article. We would like to thank anonymous referee by the carefulreading and suggestions throughout this paper.This article is part of the second author’s PhD thesis which would like to thank to Department of Math-ematics at Universidade Federal do Cear´a-UFC-Brazil for fostering a pleasant and productive scientificatmosphere during the period of his PhD program. This work has received financial support from CAPES-Brazil, CNPq-Brazil and FUNCAP-Cear´a.
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