A two phase elliptic singular perturbation problem with a forcing term

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1 J. Math. Pures Appl. 86 (2006) A two phase elliptic singular perturbation problem with a forcing term Claudia Lederman, Noemi Wolanski Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina Received 12 April 2006 Abstract We study the following two phase elliptic singular perturbation problem: u ε = β ε ( u ε ) + f ε, in Ω R N,whereε>0, β ε (s) = 1 ε β(s ε ), with β a Lipschitz function satisfying β>0in(0, 1), β 0 outside (0, 1) and β(s)ds = M. The functions u ε and f ε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit (ε 0) and we show that limit functions are solutions to the two phase free boundary problem: u = fχ {u 0} in Ω \ {u>0}, u + 2 u 2 = 2M on Ω {u>0}, where f = lim f ε, in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case f ε 0. The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary Elsevier Masson SAS. All rights reserved. Résumé Nous étudions le problème de perturbations singulières elliptiques à deux phases suivant : u ε = β ε ( u ε ) + f ε, dans Ω R N,oùε>0, β ε (s) = 1 β(s ε ), β fonction lipschitzienne qui satisfait β>0sur(0, 1), β 0 hors de (0, 1) et β(s)ds = M. Les fonctions u ε et f ε sont uniformément bornées. The research of the authors was partially supported by Fundación Antorchas Project , UBACYT X052, ANPCyT PICT and CONICET PIP The authors are members of CONICET. * Corresponding author. addresses: clederma@dm.uba.ar (C. Lederman), wolanski@dm.uba.ar (N. Wolanski) /$ see front matter 2006 Elsevier Masson SAS. All rights reserved. doi: /j.matpur

2 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Une des motivations pour l étude de ce problème est qu on le trouve dans l analyse de la propagation des flammes à la limite des hautes énergies d activation, en présence de sources. Nous obtenons des estimations uniformes qui nous permettent de passer à la limite lorsque (ε 0) : nous montrons que les fonctions limites sont solution du problème de frontière libre, u = fχ {u 0} dans Ω \ {u>0}, u + 2 u 2 = 2M sur Ω {u>0}, où f = lim f ε, au sens de la viscosité et au sens ponctuel aux points réguliers de la frontière libre. De plus, nous montrons la régularité de la frontière libre, d où les fonctions limites sont solutions classiques de notre problème à frontière libre, sous certaines hypothèses. Une partie des résultats obtenus est originale, même dans le cas f ε 0. Les résultats obtenus s appliquent à d autres modèles de combustion. Par exemple aux modèles avec diffusion non locale et/ou avec transport. D autres applications sont considerées ici et nous obtenons, dans certains cas, la régularité globale de la frontière libre Elsevier Masson SAS. All rights reserved. MSC: 35R35; 35J65; 80A25; 35B65 Keywords: Free boundary problem; Two phase; Viscosity solutions; Regularity; Combustion 1. Introduction In [24] the following singular perturbation problem for a nonlocal evolution operator was considered: Study the uniform properties, and the limit as ε 0, of nonnegative solutions u ε of the problem: θ u ε + (1 θ) ( J u ε u ε) u ε t = β ε( u ε ) in R N (0, + ), u ε (x, 0) = u ε 0 (x) in RN, (1.1) where 0 <θ 1, ε>0, β ε (s) = 1 ε β(s ε ), with β a Lipschitz continuous function satisfying β>0in(0, 1), β 0 outside (0, 1) and β(s)ds = M. The symbol denotes spatial convolution and J = J(x) is an even nonnegative kernel with unit integral. Problem (1.1) arises in the analysis of the propagation of flames in the high activation energy limit, when admitting nonlocal effects (for the model, see [24] and the references therein). In [24] it was shown that the understanding of the nonlocal problem (1.1) reduces to the understanding of the local problem: u ε u ε t = β ε( u ε ) + f ε. (P ε (f ε )) It is worth noticing that problem P ε (f ε ) appears in other situations as well. For instance, in the study of the combustion model with transport, u ε + a ε (x, t) u ε + c ε (x, t)u ε u ε t = β ε( u ε ), (1.2) when a ε, u ε, c ε and u ε are uniformly bounded. Moreover, the elliptic version of P ε (f ε ), namely: u ε = β ε ( u ε ) + f ε, (E ε (f ε )) also appears in the analysis of the travelling wave solutions to a combustion model studied in [3]. In [24] a family of nonnegative solutions u ε (x, t) of equations P ε (f ε ) in a domain D R N+1 is considered. It is assumed that both families u ε and f ε are uniformly bounded in L norm in D. Uniform estimates are obtained for the family u ε that allow the passage to the limit, as ε 0. It is also shown that the limit function u is a solution of the free boundary problem: u u t = f u = 2M in D {u>0}, on D {u>0},

3 554 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) in a parabolic viscosity sense and in a pointwise sense at regular free boundary points. Here f = lim f ε, M is as above and the free boundary is defined as D {u>0}. In order to go further in the understanding of problem P ε (f ε ), we deal in the present paper with the elliptic version of it, i.e., with E ε (f ε ). We here consider a family u ε of solutions to E ε (f ε ) in a domain Ω R N, such that both families u ε and f ε are uniformly bounded in L norm in Ω, and we study the passage to the limit, as ε 0. Our aim is twofold: we are interested, on one hand, in discussing the problem when there is no sign restriction on u ε and, on the other hand, in studying the regularity of the free boundary for the limit functions topics that remained unexplored in [24]. We point out that there is a vast literature on problem E ε (f ε ) (and on the parabolic version of it, P ε (f ε ))inthe particular case that f ε 0. A well studied free boundary problem is obtained in the limit; see, for instance, [3,7,12, 13,16,19,23,24,27]. However, the extension of the results holding for E ε (f ε ) when f ε 0 to the case f ε 0 is not immediate, in particular when dealing with two phase functions. On one hand, new tools are required to obtain uniform estimates that allow the passage to the limit. We achieve here this purpose with the aid of the recent monotonicity formula of [8]. On the other hand, the presence of a forcing term in E ε (f ε ) which does not have a sign, introduces a new difficulty due to the occurrence of a free boundary Γ := Ω ( {u <0}\ {u >0}), that did not appear in the two phase homogeneous case (see [12,13,23]). In fact, we prove that the limit problem has two free boundaries: Γ + := Ω {u >0} (i.e., the one already appearing in the homogeneous problem) and Γ = Ω ( {u<0}\ {u>0}). We show that on Γ limit functions are solutions of an obstacle type problem and that on Γ + limit functions behave as those in the case f ε 0. More precisely, we first prove that any limit function u satisfies: u fχ {u 0} = Λ in Ω, with Λ a Radon measure supported on Ω {u>0} and f = lim f ε. This implies, in particular, that there is no jump of u on Γ. We then show that, under suitable assumptions, the limit function u is a solution of the free boundary problem: u = fχ {u 0} in Ω \ {u>0}, u + 2 u 2 (E(f)) = 2M on Ω {u>0}, in a pointwise sense at regular free boundary points, and in a viscosity sense. Here M and f are as above, u + = max(u, 0) and u = max( u, 0). The key tools here are: the monotonicity formula of [8] in the case of the pointwise sense result and some asymptotic development results proven in [24] for nonnegative functions with bounded heat (or Laplacian) at boundary points with a tangent ball in the case of the viscosity sense result. We also prove that, under certain conditions, the free boundary Ω {u>0} is locally a C 1,α surface and therefore, the free boundary condition, u + 2 u 2 = 2M on Ω {u>0}, (1.3) is satisfied in the classical sense. We obtain two different type of results. One of them, holding for one phase limits, in the lines of the regularity theory developed in [1] (and its extension to inhomogeneous problems in [20] and [22]) and other results in the lines of the regularity theory developed in [5,6] (and its recent extension to inhomogeneous problems in [9]). We remark that there are limit functions u which do not satisfy the free boundary condition (1.3) in the classical sense on any portion of Ω {u>0} (see examples in [24], Section 3). The hypotheses we assume here are necessary to rule out those examples. In particular, we need to assume some kind of nondegeneracy for u +, and we thus devote a complete section to the discussion of conditions implying this nondegeneracy. We point out that most of the regularity results we prove in this paper are new even when f ε 0 (see discussion in Remark 9.7). This is the case, in particular, of Theorems 9.5, 9.6 and 9.7 which are obtained by applying a local monotonicity formula recently proved by the authors, as well as its consequences (see [25]). We finally present applications of our results to the study of the regularity of the free boundary for the limit of different singular perturbation problems. Namely, for the limit of stationary solutions to the nonlocal combustion model studied in [24], for the limit of stationary solutions to (1.2), for the limit of the travelling wave solutions to

4 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) a combustion model first studied in [3] and for the limit of minimizers of an energy functional that we construct in Proposition 2.2. In particular, in the last two examples we prove that there is an open and dense subset R of the free boundary that is a C 1,α surface and the reminder of the free boundary has (N 1)-dimensional Hausdorff measure zero. In dimension 2 we prove that, in both cases, the whole free boundary is C 1,α and we get the same result in dimension 3 in the case of minimizers (Theorems 10.1 and 10.2). An outline of the paper is as follows. In Section 2 we obtain uniform estimates for our problem and also the first results on the passage to the limit ε 0. Section 3 contains some basic examples and Section 4 results on the behavior of limit functions near the free boundary. In Section 5 we prove nondegeneracy results for u +. Next, in Section 6 we obtain results on the asymptotic development at regular free boundary points. In Section 7 we obtain other asymptotic development results and we deal with the concept of viscosity solution to problem E(f). In Section 8 we analyze the behavior of limit functions which satisfy an additional uniform nondegeneracy assumption on u +. In Section 9 we study the regularity of the free boundary and finally, in Section 10 we discuss applications of our results. Notation and assumptions. Throughout the paper N will denote the spatial dimension. The set Ω {u>0} will be referred to as the free boundary. We will assume that the functions β ε are defined by scaling of a single function β : R R satisfying: (i) β is a Lipschitz continuous function, (ii) β>0in(0, 1) and β 0 otherwise, (iii) 1 0 β(s)ds = M. And then β ε (s) := 1 ε β(s ε ). In addition, the following notation will be used: S N-dimensional Lebesgue measure of the set S, H N 1 (N 1)-dimensional Hausdorff measure, B r (x 0 ) open ball of radius r and center x 0, B r (x 0 ) u = B 1 r (x 0 ) B r (x 0 ) u dx, B r (x 0 ) u = 1 H N 1 ( B r (x 0 )) B r (x 0 ) u dhn 1, χ S characteristic function of the set S, u + = max(u, 0), u = max( u, 0),, scalar product in R N, B ε (s) = s 0 β ε(τ) dτ. 2. Uniform estimates and passage to the limit In this section we consider a given family of solutions u ε (x) of the equations E ε (f ε ): u ε = β ε (u ε ) + f ε, in a domain Ω R N. We assume that both families u ε and f ε are uniformly bounded in L norm in Ω, and we obtain further uniform estimates on the family u ε that allow the passage to the limit, as ε 0. We then pass to the limit, and we show that the limit problem has two free boundaries: Γ + := Ω {u >0} (i.e., the free boundary that already appeared in the case f ε 0) and Γ := Ω ( {u<0}\ {u>0}) (a new free boundary, which was not present in the case f ε 0). We here show that on Γ limit functions are solutions of an obstacle type problem and we also draw our first conclusions on the behavior of limit functions on Γ +. More precisely, we prove that any limit function u satisfies: u fχ {u 0} = Λ in Ω, (2.1) with Λ a Radon measure supported on Ω {u>0} and f = lim f ε. This implies, in particular, that there is no jump of u on Γ.

5 Proof. Let η C c (Ω) be nonnegative and let: φ k = η ( 1 h(kv) ), 556 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Finally, we conclude the section by presenting an example of a family of uniformly bounded solutions of E ε (f ε ) in general settings, which is obtained by minimization of energy functionals. We start the section by proving a lemma that will be used throughout the paper (for the homogeneous version see [1], Remark 4.2). Lemma 2.1. Let v be a continuous nonnegative function in a domain Ω R N, v H 1 (Ω), such that v = g in {v>0} with g L (Ω). Then λ v := v gχ {v>0} is a nonnegative Radon measure with support on Ω {v >0}. where h(s) = max(min(2 s,1), 0). Then, gχ {v>0} φ k = Then, letting k, we obtain: which gives the desired result. Ω Ω Ω v φ k gχ {v>0} η Ω Ω v η ( 1 h(kv) ). v η, We will next obtain uniform Lipschitz estimates for our family. Before doing so we state the following monotonicity result from [8] that will allow us to obtain these estimates and that will also be used at other stages of our work: Theorem 2.1. Let u i, i = 1, 2, be nonnegative continuous functions in B 1 (0), which verify: (i) u i 1 in the sense of distributions in B 1 (0), (ii) u 1 (x).u 2 (x) = 0 for x B 1 (0), (iii) u 1 (0) = u 2 (0) = 0. Set Then, Suppose, in addition, that ( 1 Φ(r) = r 2 B r (0) u 1 (x) 2 x N 2 )( 1 dx r 2 B r (0) u 2 (x) 2 x N 2 ) dx. Φ(r) C ( 1 + u 1 2 L 2 (B 1/2 (0)) + u 2 2 L 2 (B 1/2 (0))) 2, 0 <r<1/4, C= C(N). (iv) u i (x) C x σ in B 1 (0), forsomec>0, σ>0. Then, the limit lim r 0 + Φ(r) exists. Proof. It follows from Theorems 1.3 and 1.4 and Remark 2.2 of [8]. As a consequence we obtain: Theorem 2.2. Let u ε be a family of solutions to E ε (f ε ) in a domain Ω R N such that u ε L (Ω) A 1 and f ε L (Ω) A 2 for some A 1 > 0, A 2 > 0. Let K Ω be compact and let τ>0 be such that B τ (x 0 ) Ω, for every x 0 K. There exists a constant L = L(N, τ, A 1, A 2, β ) such that u ε (x) L for x K. (2.2)

6 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Proof. We will follow arguments similar to those in Theorem 3 in [7]. In fact, let us first obtain estimate (2.2) for x 0 K {u ε < 0}. For that purpose we define, for x B 1 (0), u 1 (x) = 1 ( u ε τ 2 (τx + x 0 ) λ ) +, u2 (x) = 1 ( u ε A 2 τ 2 (τx + x 0 ) λ ), A 2 with λ = u ε (x 0 ). Then, using Lemma 2.1 we see that u i are under the assumptions of Theorem 2.1. This implies that, for 0 <r< 1 4, Φ(r) C 1, and thus, u 1 (0) 2 u 2 (0) 2 C 2, which gives (2.2) at x 0. Let us now consider x 0 K {0 u ε 2ε}. Without loss of generality we can assume that ε<1. For x B τ/2 (0) we define: v ε (x) = 1 ε uε (εx + x 0 ). The estimate obtained in {u ε < 0} implies that v ε C 3 in B τ/2 (0). By using Harnack inequality we get: v ε C 4, v ε C 5, in B τ/4 (0) and thus (2.2) holds at x 0. Let us finally consider x 0 K {u ε >ε}.wetakev ε satisfying: v ε = f ε in B τ/2 (x 0 ), v ε = 0 on B τ/2 (x 0 ), and let w ε = u ε v ε. Since β ε (u ε ) = 0in{u ε >ε},wehave: w ε = 0, w ε C6 in B τ/2 (x 0 ) { u ε >ε }, w ε C 7 on B τ/2 (x 0 ) { u ε >ε } (we have used the estimate obtained in {0 u ε 2ε}). We now fix ϕ C 0 (B τ/2(x 0 )) such that 0 ϕ 1 and ϕ 1inB τ/4 (x 0 ). Then the function, ϕ 2 w ε 2 + λ ( w ε) 2, is subharmonic in B τ/2 (x 0 ) {u ε >ε} if we choose a constant λ large enough (depending only on ϕ). Therefore, w ε C 8 in B τ/4 (x 0 ) {u ε >ε}, which gives (2.2) at x 0. The proof is complete. With the uniform estimate obtained in the previous result we can now pass to the limit as ε 0. Lemma 2.2. Let u ε be a family of solutions to E ε (f ε ) in a domain Ω R N. Let us assume that u ε L (Ω) A 1 and f ε L (Ω) A 2 for some A 1 > 0, A 2 > 0. For every ε n 0 there exist a subsequence ε n 0, a function u which is locally Lipschitz continuous in Ω and a function f L (Ω), such that (i) u ε n u uniformly on compact subsets of Ω, (ii) u ε n u in L 2 loc (Ω), (iii) f ε n f -weakly in L (Ω), (iv) u f in the distributional sense in Ω. (v) u = f in {u>0} {u<0}. Proof. The result follows arguing as in Lemma 3.1 in [12]. The previous result shows that the limit problem has two free boundaries: Γ + = Ω {u >0} and Γ = Ω ( {u <0} \ {u >0}). The next result will allow us to draw our first conclusions on the behavior of limit functions on these free boundaries.

7 558 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Proposition 2.1. Let u ε j be a family of solutions to E εj (f ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Then, u + fχ {u>0} = λ + u in Ω, u + fχ {u<0} = λ u in Ω, with λ + u and λ u nonnegative Radon measures supported on the free boundary Γ + = Ω {u>0}. It follows that u fχ {u 0} = Λ in Ω, with Λ a Radon measure supported on the free boundary Γ + = Ω {u>0}. In particular, u W 2,p loc in {u 0}, 1 <p<. Proof. From Lemma 2.1 we deduce that u + fχ {u>0} = λ + u in Ω, u + fχ {u<0} = λ u in Ω, with λ + u and λ u nonnegative Radon measures, λ+ u supported on Ω {u>0} and λ u supported on Ω {u<0}. Let us see that λ u is actually supported on Ω {u >0}. In fact, let x 0 Ω ( {u <0}\ {u >0}), and let B r (x 0 ) {u 0}. On one hand there holds that u f f L in B r (x 0 ). On the other hand, u + fχ {u<0} = λ u 0 so that u = u f L in B r (x 0 ). It follows that u W 2,p loc (B r(x 0 )), 1<p<, and thus, u + fχ {u<0} = 0 inb r (x 0 ). Therefore support λ u Ω {u>0}. Remark 2.1. By different arguments from those in Proposition 2.1 we can deduce that u f = μ in Ω (2.3) with μ a nonnegative Radon measure. In fact, reasoning in a similar way as in [12], Proposition 3.1, we can deduce that ( ) β εj u ε j CK, for every K Ω. (2.4) K Therefore there exists a nonnegative Radon measure μ such that β εj (u ε j ) μ weakly in Ω and such that (2.3) holds. Notice that, as in [24], (2.3) implies that f 0in{u 0}. Remark 2.2. When u ε j 0 we deduce the nonnegativity of the Radon measure Λ appearing in Proposition 2.1 from the fact that Λ = λ + u in Ω. Remark 2.3. Let us point out that when f ε j 0 there holds that Γ := Ω ( {u<0}\ {u>0}) =.Iff ε j 0 the boundary Γ may appear but, as we showed in Proposition 2.1, there holds that u W 2,p across it. On the other hand, we know that f 0in{u 0},soiff is continuous necessarily f 0inΓ. If x 0 Γ and f< c<0inb δ (x 0 ) {u<0} then we have the well known obstacle problem in a smaller ball B δ (x 0 ). Examples with Γ can be easily constructed in one dimension.

8 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Now we state two results that follow from the convergence result (Lemma 2.2) exactly as Lemmas 3.2 and 3.3 in [12]. Lemma 2.3. Let u ε j be a family of solutions to E εj (f ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Let x 0 Ω {u>0}, and let x n Ω {u>0} be such that x n x 0 as n. Let λ n 0, u λn (x) = 1 λ n u(x n + λ n x), and (u ε j ) λn (x) = 1 λ n u ε j (x n + λ n x). Assume that u λn U as n uniformly on compact sets of R N. Then, there exists j(n) + such that for every j n j(n)there holds that ε jn λ n 0, and (1) (u ε jn )λn U uniformly on compact sets of R N, (2) (u ε jn )λn U in L 2 loc (RN ). Also, we deduce that (3) u λn Uin L 2 loc (RN ). Lemma 2.4. Let u ε j be a solution to E εj (f ε j ) in a domain Ω j R N with Ω j Ω j+1 and j Ω j = R N such that u ε j U uniformly on compact sets of R N, f ε j 0 -weakly in L loc (RN ) and ε j 0. Let us assume that for some choice of positive numbers d n and points x n {U >0}, the sequence U dn (x) = d 1 n U(x n + d n x) converges uniformly on compact sets of R N to a function U 0. Let (u ε j ) dn (x) = d 1 n u ε j (x n + d n x). Then, there exists j(n) such that for every j n j(n), there holds that ε jn d n 0 and (1) (u ε jn )dn U 0 uniformly on compact sets of R N, (2) (u ε jn )dn U 0 in L 2 loc (RN ). We conclude the section by presenting an example of a family of uniformly bounded solutions of E ε (f ε ) in general settings. This family is obtained by minimization of energy functionals. We will come back to this family in forthcoming sections. Proposition 2.2. Let Ω R N be a bounded domain and let φ ε H 1 (Ω) be such that φ ε H 1 (Ω) A 1. Let f ε L (Ω) such that f ε L (Ω) A 2. There exists u ε H 1 (Ω) that minimizes the energy, 1 J ε (v) = 2 v 2 + B ε (v) + f ε v, among functions v H 1 (Ω) such that v = φ ε on Ω.HereB ε (s) = s 0 β ε(τ) dτ. Then, the functions u ε satisfy: Ω u ε = β ε ( u ε ) + f ε in Ω, and for every Ω Ω there exists C = C(Ω, A 1, A 2 ) such that u ε L (Ω C. ) Proof. The proof of the existence of a minimizer of J ε is standard and we omit it here. It is also standard the proof that a minimizer u ε is a solution to E ε (f ε ). It is easy to see that u ε H 1 (Ω) C with C independent of ε. Let us show that for every Ω Ω there exists C = C(Ω, A 1, A 2 ) such that u ε L (Ω C. ) In fact, since u ε is a solution to E ε (f ε ) in Ω there holds that u ε C 1,α (Ω). In particular, {u ε < 1} is open and (u ε + 1) is a nonnegative solution to

9 560 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) u = f ε in Ω { u ε < 1 }, u = 0 onω { u ε < 1 }, with uniformly bounded H 1 (Ω) norm. Thus, ( sup u ε + 1 ) C. Ω In particular, u ε is uniformly bounded from below. Now, (u ε + C + 1) is a nonnegative solution to, u f ε in Ω, with uniformly bounded H 1 (Ω) norm. We deduce that ( sup u ε + C + 1 ) C. Ω So that the uniform boundedness of u ε in Ω follows. 3. Basic limits In this section we analyze some particular limits that are crucial in the understanding of general limits. We need to prove first the following lemma: Lemma 3.1. Let u ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Then there exists χ L 1 loc (Ω) such that, for a subsequence, B εj (u ε j ) χ in L 1 loc (Ω), with χ M in {u>0}, χ 0 in {u<0}, χ(x) {0,M} a.e. in Ω. If, in addition, f ε j 0 in {u 0}, there holds that χ M or χ 0 on every connected component of {u 0}. Proof. We follow the same ideas as in Step IV in the proof of Theorem 3.1 in [23], where we had f ε 0. If f ε 0 we have, for every K Ω, ( ) ( ) ( ) Bεj u ε j = β εj u ε j u ε j CK β εj u ε j, (3.1) K K where the last term is bounded by a constant C K due to estimate (2.4). Since 0 B εj (u ε j ) M, then, there exists χ L 1 loc (Ω) such that, for a subsequence, B ε j (u ε j ) χ in L 1 loc (Ω). In order to see that necessarily χ = 0orχ = M, we modify the argument in [23] as follows. Let ρ 1,ρ 2 > 0 and K Ω. There exist 0 <η<1 and β η > 0 such that { { ( ) } x K ρ 1 <B εj u ε j <M ρ2 x K η< uε j < 1 η} ε j { } ( ) x K β εj u ε j β η ε j ( ) β εj u ε j 0. ε j β η This implies that { } x K ρ 1 <χ<m ρ 2 = 0, for every ρ 1,ρ 2 > 0 and K Ω,soχ(x) {0,M} a.e. in Ω. We now deduce that χ M in {u>0} and χ 0in{u<0} as in [13], Theorem 3.1. Finally, in case f ε j 0in{u 0},wetakeK {u 0} in (3.1), we observe that (as in [23]) the last term there goes to zero and the result follows. Proposition 3.1. Let u ε j ) in a domain Ω R N. Let x 0 Ω and suppose u ε j converge to u = α(x x 0 ) + 1 γ(x x 0) 1 uniformly on compact subsets of Ω, with α 0,γ >0, f ε j 0 -weakly in L (Ω) and ε j 0. Then α 2 γ 2 = 2M. K K

10 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Proof. The proof follows as that of Proposition 5.1 in [12]. Proposition 3.2. Let u ε j ) in a domain Ω R N. Let x 0 Ω and suppose u ε j converge to u = α(x x 0 ) + 1 uniformly on compact subsets of Ω, with α R, f ε j 0 -weakly in L (Ω) and ε j 0. Then α = 0 or α = 2M. Proof. First we see that necessarily α 0. In fact, if not we would have u 0inΩ, u(x 0 ) = 0 and u subharmonic in Ω and thus u 0, which is a contradiction. If α>0 we deduce that α = 2M proceeding as in the proof of Proposition 5.1 in [24], but using in the present case Lemma 3.1 above. Proposition 3.3. Let u ε j ) in a domain Ω R N. Let x 0 Ω and suppose u ε j converge to u = α(x x 0 ) + 1 +ᾱ(x x 0) 1 uniformly on compact subsets of Ω, with α>0, ᾱ>0, f ε j 0 uniformly on compact subsets of Ω and ε j 0. Then α =ᾱ 2M. Proof. The result was proven for the parabolic version of this problem in Proposition 5.3 in [12], for f ε 0, and it was extended to the case f ε 0 in Proposition 5.2 in [24], under the assumption that u ε 0. But the same proof in [24] is valid in the present case. Remark 3.1. We point out that all the situations present in Propositions 3.1, 3.2 and 3.3 can occur. We refer to Section 3 in [24] for examples of those situations. In particular, the analysis in [24] shows that for any given α [0, 2M ] there are examples of families u ε j of solutions to E εj (f ε j ) in R N, with f ε j 0 uniformly on compact sets of R N such that u ε j u = αx αx 1, uniformly on compact sets of RN. 4. Behavior of limit functions near the free boundary In this section we analyze the behavior of limit functions u = lim u ε, with u ε a family of solutions to problems E ε (f ε ). The following result says that a limit function is, in a sense, a supersolution to the free boundary problem E(f) this holding for any limit function, without imposing any additional hypothesis. Theorem 4.1. Let u ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Let x 0 Ω {u>0} and let γ 0 be such that lim sup u (x) γ. x x 0 Then, lim sup u + (x) 2M + γ 2. x x 0 Proof. The proof follows as that of Theorem 6.1 in [12]. In fact, we define: α = lim sup u(x) x x 0 u(x)>0 and, proceeding as in [12], we assume that α>0 and let x n x 0 be such that u(x n )>0 and u(x n ) α. Then we let z n Ω {u>0} be such that d n := x n z n =dist(x n, {u>0}). As in [12] we choose εn 0 := ε jn d n 0, and consider the sequences: u dn (x) = 1 d n u(z n + d n x), u ε0 n(x) = 1 d n u ε jn (z n + d n x).

11 562 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) There holds that the functions u dn satisfy (iv) and (v) in Lemma 2.2 with right-hand side f n and that u ε0 n are solutions to E ε 0 n (f ε0 n), where the sequences, f n (x) = d n f(z n + d n x), f ε0 n(x) = d n f ε jn (z n + d n x), converge uniformly to 0 as n on compact sets, since f ε j C. It follows that the blow-up limit u 0 = lim u dn = lim u ε0 n is harmonic in the set {u 0 > 0} {u 0 0}. Then we can argue as in [12]. In the present case we obtain (after a second blow up) a sequence εn 00 0 and solutions uε00 n ε00 to E ε 00(f n ) in B 1 (0) such that u ε00 n u 00 = αx μx 1 uniformly on compact subsets of B 1 (0), with f ε00 n 0 uniformly on compact sets. Therefore, Propositions 3.1, 3.2 and 3.3 apply and we arrive at the conclusion as in [12]. Theorem 4.2. Let u ε j be a solution to E εj (f ε j ) in a domain Ω j R N such that Ω j Ω j+1 and j Ω j = R N. Let us assume that u ε j converge to a function U uniformly on compact sets of R N, f ε j 0 -weakly in L loc (RN ) and ε j 0. Assume, in addition, that U is Lipschitz continuous in R N and {U >0}.Ifγ 0 is such that U γ in R N then, U + 2M + γ 2 in R N. n Proof. The proof follows as that of Theorem 6.2 in [12], since U is harmonic in the set {U >0} {U 0}. 5. Nondegeneracy results At different stages of our work we will prove results for u = lim u ε, with u ε solutions to E ε (f ε ), which hold under the assumption that u + satisfies some nondegeneracy condition at the free boundary (see Definition 5.1). The purpose of this section is to present results that imply some kind of nondegeneracy on u +. In particular we define the concept of minimal solution to problem E ε (f ε ) and we prove the uniform nondegeneracy of u + on the free boundary when u is the limit of any family of minimal solutions. We also prove the uniform nondegeneracy of u + on the free boundary when u is the limit of the minimizers to the energy functional constructed in Proposition 2.2. We point out that, from Section 3 in [24], we know that there are examples where u + degenerates at the free boundary. Therefore, some additional assumption is required if one wants to guarantee the nondegeneracy of u + at a free boundary point. Definition 5.1. Let v 0 be a continuous function in a domain Ω R N. We say that v is nondegenerate at a point x 0 Ω {v = 0} if there exist c>0 and r 0 > 0 such that one of the following conditions holds: v cr for 0 <r r 0, (5.1) B r (x 0 ) B r (x 0 ) v cr for 0 <r r 0, (5.2) sup v cr for 0 <r r 0, (5.3) B r (x 0 ) sup v cr for 0 <r r 0. (5.4) B r (x 0 ) Otherwise, we say that v degenerates at x 0. We say that v is uniformly nondegenerate on Γ Ω {v = 0} in the sense of (5.1) (resp. (5.2), (5.3) or (5.4)), if there exist c>0 and r 0 > 0 such that (5.1) (resp. (5.2), (5.3) or (5.4)) holds for every x 0 Γ.

12 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Remark 5.1. If v 0 is locally Lipschitz continuous in a domain Ω R N and v C in Ω (which will be our case), the four concepts of nondegeneracy of Definition 5.1 are equivalent. In fact, this can be seen by arguing in a similar way as in Remark 3.1 in [23]. There holds the following result which will be applied to our limit functions Proposition 5.1. Let u be a locally Lipschitz continuous function in a domain Ω R N satisfying that u C in Ω. Assume that u is nondegenerate at x 0 Ω {u>0} in the sense of (5.2). Then u + is nondegenerate at x 0 in the same sense. Proof. The result follows as Lemma 5.2 of [23], if we observe that in the present case u λn (x) = λ 1 n u(x 0 + λ n x) converges to u 0 with u 0 0. Our first result implying that u + is nondegenerate at a free boundary point is the following: Theorem 5.1. Let u ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Let x 0 Ω {u>0} and assume that there exists ν R N, with ν =1 such that {u>0} { x x 0,ν > 0} B r (x 0 ) lim inf = α 1, (5.5) r 0 + B r (x 0 ) and {u<0} { x x 0,ν < 0} B r (x 0 ) lim inf = α 2 (5.6) r 0 + B r (x 0 ) with α 1 + α 2 > 1/2. Then, there exists a constant C>0such that, for every r>0small, sup u Cr. B r (x 0 ) The constant C depends only on α 1 + α 2, N and the function β. If, instead of (5.6), we have we obtain the same conclusion. {u 0} { x x 0,ν < 0} B r (x 0 ) lim inf = α 2, r 0 + B r (x 0 ) u ε j 0 a.e. in ( {u 0} {u<0} ) { x x 0,ν < 0 } B r0 (x 0 ), ε j Proof. Case f ε 0. The proof was done in [12], Theorem 6.3 under assumption (5.6). Under assumption (5.7), the proof was done in [19], Proposition 4.1 and Remark 4.1, when u ε 0. It is not hard to see that the proof in [19] applies also under assumption (5.7) when there is no sign restriction on u ε. Case f ε 0. The proof was done in [24], Theorem 6.2, under assumption (5.7), when u ε 0. The result in the statement, both for (5.6) or (5.7), follows as in the case f ε 0 but treating the term f ε as shown in [24]. Remark 5.2. If in Theorem 5.1, instead of (5.7), we have the alternative condition: {u 0} { x x 0,ν < 0} B r (x 0 ) lim inf = α 2, r 0 + B r (x 0 ) u ε j 0 a.e. in{u 0} { x x 0,ν < 0 } B r0 (x 0 ), ε j we obtain the same conclusion. (5.7)

13 564 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Corollary 5.1. Let u ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Let x 0 Ω {u>0} be such that there exists an inward unit normal ν to {u >0} at x 0 in the measure theoretic sense (see Definition 6.1), and assume that one of the following conditions holds: (1) lim inf r 0 + {u<0} B r (x 0 ) B r (x 0 ) > 0, (2) lim inf r 0 + {u<0} B r (x 0 ) B r (x 0 ) = 0 and uε j ε j 0 a.e. in {u 0} { x x 0,ν < 0} B r0 (x 0 ). Then, the same conclusion of Theorem 5.1 holds. Proof. We first notice that there holds (5.5), with α 1 = 1/2, and {u 0} { x x 0,ν < 0} B r (x 0 ) lim inf = 1 r 0 + B r (x 0 ) 2. Then, in case (1) holds the result is an immediate consequence of Theorem 5.1. In case (2) holds the result follows from Remark 5.2. Corollary 5.2. Let u ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Let x 0 Ω {u>0} be such that there exists a ball B {u>0}, with x 0 B, and assume that one of the following conditions holds: (1) lim inf r 0 + {u<0} B r (x 0 ) B r (x 0 ) > 0, (2) lim inf r 0 + {u 0} B r (x 0 ) B r (x 0 ) > 0 and uε j ε j 0 a.e. in ({u 0} {u<0}) B c B r0 (x 0 ). Then, the same conclusion of Theorem 5.1 holds. Proof. The result follows from Theorem 5.1, since (5.5) is satisfied with ν the inward unit normal to B at x 0 and α 1 = 1/2. Corollary 5.3. Let u ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Let x 0 Ω {u>0} be such that there exists a ball B {u 0}, with x 0 B. Assume that lim inf r 0 + {u>0} B r (x 0 ) B r (x 0 ) > 0, and that one of the following conditions holds: (1) u<0 in B, (2) uε j ε j 0 a.e. in {u 0} B. Then, the same conclusion of Theorem 5.1 holds. Proof. The result follows from Theorem 5.1, since either (5.6) or a condition equivalent to (5.7) are satisfied, with ν the outward unit normal to B at x 0 and α 2 = 1/2. The nondegeneracy of u + at a point x 0 {u >0} can also be derived from Hopf s Principle under suitable assumptions on the smoothness of {u>0} at x 0 and on the sign of f. In fact, we have: Proposition 5.2. Let u ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Let x 0 Ω {u>0}, and assume that one of the following conditions holds: (1) {u>0} satisfies a Dini interior condition at x 0 and f 0 in B r0 (x 0 ) {u>0}, (2) there exists a ball B {u>0}, with x 0 B and f 0 in B, (3) {u>0} satisfies a Dini exterior condition at x 0, {u 0} B r0 (x 0 ) = and f 0 in B r0 (x 0 ) {u<0},

14 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) (4) there exists a ball B {u<0}, with x 0 B and f 0 in B. Then u + is nondegenerate at x 0. Proof. In case (1) or (2) hold, the result follows from the application of Hopf s Principle to u +. In case (3) or (4) hold, it follows from the application of Hopf s Principle to u and from Proposition 5.1. We next define the concept of minimal solution to problem E ε (f ε ) and prove a nondegeneracy result for this kind of solutions. We will follow the lines of [3], Section 4. Definition 5.2. Let u ε be a solution to E ε (f ε ) in a domain Ω R N with f ε L (Ω). We say that u ε is a minimal solution to E ε (f ε ) in Ω if whenever we have h ε a strong supersolution to E ε (f ε ) in a bounded subdomain Ω Ω, i.e., h ε W 2,p (Ω ) C ( Ω ), h ε ( β ε h ε ) + f ε in Ω, (5.8) which satisfies, in addition, h ε u ε on Ω, then h ε u ε in Ω. Proposition 5.3. Let u ε be a minimal solution to E ε (f ε ) in a domain Ω R N such that f ε L (Ω) < A. For every Ω Ω, there exist positive constants c 0, ρ and ε 0 depending only on N, A, dist(ω, Ω) and the function β, such that if ε ε 0 and x Ω, then ( u ε ) + (x) c0 dist ( x, { u ε ε }) if dist ( x, { u ε ε }) ρ. (5.9) Proof. Our proof is a modification of Theorem 4.1 in [3]. In fact, let us fix 0 <a<b<1 and κ>0 such that β(s) > κ for s [a,b]. Letx 0 Ω such that u ε (x 0 )>εand such that 2δ = dist(x 0, {u ε ε}) dist(ω, Ω) and δ<1. Without loss of generality we will assume that x 0 = 0. In B 2δ (0) there holds that u ε = f ε. By the Harnack inequality there holds that u ε (x) Cu ε (0) + Cδ 2 A in B δ (0), with C = C(N) > 0. We will exhibit a C 1 supersolution h ε satisfying (5.8) in B δ (0). In addition h ε = h ε (r) will depend only on r = x and will satisfy: h ε (0) = aε < u ε (0), and also h ε (δ) δd 1 for some D>0 depending only on N,a,b,κ,A. By our hypothesis that u ε is a minimal solution it follows that we cannot have h ε u ε everywhere on B δ (0). Hence δ D hε (δ) Cu ε (0) + Cδ 2 A which gives: u ε (0) c 0 δ, if δ δ 0, for constants c 0 and δ 0 depending only on N,a,b,κ,A. This is, (5.9) holds. We will take as h ε the function constructed in [3], i.e., εa for 0 r r 0, h ε (r) = εa + k 2 (r r 0) 2 for r 0 r λ, H A 2 (r δ)2 for λ r δ, and we will show that we can choose the numbers r 0,λ,k,H and A so that h ε has the desired properties for our problem, provided ε ε 0 = ε 0 (N,a,b,κ,A).

15 566 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) As done in [3], we first ask that h ε be C 1 and h ε (λ) = εb,thisis, We now take εb = H A 2 (λ δ)2, (5.10) εb = εa + k 2 (λ r 0) 2, (5.11) k(λ r 0 ) = A(δ λ). (5.12) λ = (1 μ 0 )δ, (5.13) r 0 = λ Cε(b a), (5.14) for some 0 <μ 0 < 1 and C >0 to be fixed later (notice that in order to have r 0 > 0 we need ε<cδ). We now obtain k,a and H from (5.11), (5.12) and (5.10), resp. Let us verify that in B δ (0), h ε ( β ε h ε ) + f ε. (5.15) In fact, in 0 r r 0, so (5.15) holds provided ε ε 0 (κ, A). Next, in r 0 r λ, also (5.16) holds, so we have: ( β ε h ε ) + f ε κ A, (5.16) ε β ε ( h ε ) + f ε κ 2ε, if we take ε 0 (κ, A) smaller. Now ( h ε k 1 + (N 1) λ r ) 0, r 0 and we can make λ r 0 r 0 1 provided ε Cδ,forsomeC depending on C,μ 0,b and a. Then h ε 2N kn = C 2 ε(b a) κ 2ε, if choose C big depending on N,b,a,κ. It remains to verify (5.15) in λ r δ.here ( β ε h ε ) + f ε A, and ( h ε = A 1 (N 1) δ r ) ( A 1 (N 1) r if we take μ 0 small depending on N. Replacing A gives, h ε 1 A, Cμ 0 δ for appropriate μ 0 = μ 0 (N,a,b,κ,A). This shows that (5.15) holds in B δ (0). We have to see now that h ε (δ) D δ. In fact, h ε (δ) = H A 2 (λ δ)2 = μ 0 C δ, μ 0 1 μ 0 ) A 2, and thus, a constant D = D(N,a,b,κ,A) with the desired property exists. We finally notice that the construction above fails when ε Cδ,forC = C(N,a,b,κ,A), but the result is immediate in this case since u ε (0)>ε. The proof is now complete. As a consequence we obtain:

16 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Corollary 5.4. Let u ε j be a family of minimal solutions to E εj (f ε j u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) with f ε j L (Ω) < A and ε j 0. For every Ω Ω, there exist positive constants c 0 and ρ depending only on N, A, dist(ω, Ω)and the function β, such that if x Ω, then u + (x) c 0 dist ( x,{u 0} ) if dist ( x,{u 0} ) ρ. (5.17) Proof. Let x 0 Ω such that u(x 0 )>0 and such that 2δ = dist(x 0, {u 0}) dist(ω, Ω). Then u>0inb 2δ (x 0 ). Moreover, if 0 < 2σ <2δ, there holds that if ε is small enough (we have dropped the subscript j). From the proof of Proposition 5.3, we know that (5.18) implies that u ε >ε in B 2σ (x 0 ), (5.18) u ε (x 0 )>c 0 σ, (5.19) if σ ρ and ε ε 0, for some constants c 0, ρ and ε 0 depending only on N, A, dist(ω, Ω)and the function β. Then, letting ε 0 in (5.19) first and then σ δ, we get: u(x 0 ) c 0 δ, which gives the desired result. Next, we prove the nondegeneracy of the limit of the minimizers constructed in Proposition 2.2. First, we follow closely the proof of Theorem 1.6 in [15] and we obtain: Proposition 5.4. Let u ε be a minimizer to J ε in the set of functions in H 1 (Ω) that are equal to φ ε on Ω where φ ε H 1 (Ω) C and f ε L (Ω) A with C,A independent of ε. Then, for every Ω Ω, there exist positive constants c 0, ρ and ε 0 depending only on N, A, dist(ω, Ω)and the function β, such that if ε ε 0 and x Ω, then ( u ε ) + (x) c0 dist ( x, { u ε ε }) if dist ( x, { u ε ε }) ρ. Proof. Let x 0 Ω such that u ε (x 0 )>εand let us call d 0 = dist{x 0, {u ε ε}} and w(x) = 1 d 0 u ε (x 0 + d 0 x). Then, in B 1 (0), w = d 0 f ε (x 0 + d 0 x), w(x) > ε d 0. Let ψ C (B 1 ) such that ψ 0inB 1/4, ψ 1inB 1 \ B 1/2.LetΩ Ω Ω, L u ε L (Ω ) and assume that B d0 (x 0 ) Ω. By Harnack inequality there exists a constant c>0 such that w(x) cw(0) + C 0 d 0 in B 1/2 for a certain constant C 0 depending on A.Letα>0 be such that u ε (x 0 ) = αd 0. With this notation me have α = w(0). We want to prove that there exist c,ρ > 0 such that α c if d 0 ρ. Let { min(w(x), (cα + C0 d 0 )ψ) in B 1/2, z(x) = w(x) outside B 1/2. Then, z H 1 (B 1 ) and z coincides with w on B 1 so that, since w is a local minimizer of the functional, [ ] J(v)= 1 2 v 2 + B ε/d0 (v) + d 0 f ε (x 0 + d 0 x)v dx, there holds that J(z) J(w). B 1

17 568 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Let D = B 1/2 {w>(cα + C 0 d 0 )ψ}. Observe that B 1/4 D and B ε/d0 (w) = M in B 1/4 whereas z = 0inB 1/4. Therefore, { Bε/d0 (w) B ε/d0 (z) } dx M B 1/4. Thus, So that, M B 1/4 Ad 0 D D [ w (cα + C0 d 0 )ψ ] dx D ψ 2 (cα + C 0 d 0 ) 2 C(cα + C 0 d 0 ) 2. M B 1/4 Ad 0 B 1/2 (cα + C 0 d 0 ) C(cα + C 0 d 0 ) 2. (5.20) Now, since u ε (x 0 )>εthere holds that α> d ε 0. Therefore, if d ε 0 1 there is nothing to prove. Thus, we may assume ε that d 0 1. Thus, since there is a point x on B d0 (x 0 ) such that u ε ( x) = ε, α = uε (x 0 ) ε + Ld 0 = ε + L 1 + L. d 0 d 0 d 0 Going back to (5.20) we have for d 0 ρ 1, 0 <k C(cα + C 0 d 0 ) Ccα + k 2, if ρ is small enough. Therefore, α c>0 and the proposition is proved. Then, proceeding as in Corollary 5.4 we get, Corollary 5.5. Let u = lim u ε j with ε j 0, where u ε j are minimizers of J εj in the set of functions in H 1 (Ω) that coincide with φ εj on Ω where φ εj H 1 (Ω) C and f ε j L (Ω) A with C,A independent of ε j. Then, for every Ω Ω, there exist positive constants c 0 and ρ depending only on N, A, dist(ω, Ω) and the function β, such that if x Ω, then u + (x) c 0 dist ( x,{u 0} ) if dist ( x,{u 0} ) ρ. Finally we prove a result, which will be applied to our limit functions, that relates the nondegeneracy in the sense of (5.17) with the four concepts of nondegeneracy of Definition 5.1 (recall Remark 5.1). Proposition 5.5. Let u be a locally Lipschitz continuous function in a domain Ω R N satisfying that u C in Ω. Assume that u + is locally uniformly nondegenerate in the sense that (5.17) holds on every compact subset of Ω. Then u + is locally uniformly nondegenerate on Ω {u>0} in the sense of (5.4) and consequently in the sense of (5.1), (5.2) and (5.3). Proof. The proof was done in Lemma 2.15 in [22] for the case in which C = 2. For arbitrary C we proceed exactly as in [22], considering in the proof the auxiliary subharmonic function v(x) = u(x) + C x x 1 2 2N. 6. Asymptotic development at regular free boundary points In this section we consider u = lim u ε, with u ε solutions to problems E ε (f ε ), and we prove that the free boundary condition, u + 2 u 2 = 2M, (6.1) is satisfied in a pointwise sense at any point x 0 {u>0} that has an inward unit normal in the measure theoretic sense (see Definition 6.1). The result holds if u + satisfies a nondegeneracy condition at the point (see Definition 5.1).

18 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) We remark that, as shown by the examples in Section 3 in [24], an assumption that guarantees nondegeneracy of u + is essential in order to get the free boundary condition (6.1). We refer to Section 5 for a discussion on conditions under which u + is nondegenerate at a free boundary point x 0. A key tool in this section is the monotonicity formula of [8] (see Theorem 2.1). Definition 6.1. We say that ν is the inward unit normal to the free boundary {u>0} at a point x 0 {u>0} in the measure theoretic sense, if ν R N, ν =1 and 1 lim r 0 r N χ {u>0} χ {x/ x x0,ν >0} dx = 0. (6.2) B r (x 0 ) Definition 6.2. We say that a point x 0 {u>0} is regular if there exists an inward unit normal to {u>0} at x 0 in the measure theoretic sense. We will need the following lemma: Lemma 6.1. Let u ε j converge to u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Let x 0 Ω {u>0} and let u λ (x) = λ 1 u(x 0 + λx) for λ>0. There exists δ 0 such that if, for a sequence λ n 0, u λn U uniformly on compact sets of R N, then there holds: for every r>0. ( 1 Φ U (r) := r 2 B r (0) U + (x) 2 x N 2 )( 1 dx r 2 B r (0) U (x) 2 x N 2 ) dx δ, Proof. We will assume without loss of generality that x 0 = 0 and that B 1 (0) Ω. Since u + f L and u f L (recall Proposition 2.1), we can apply Theorem 2.1 with u 1 = u + and u 2 = u. For r>0, let ( 1 u + (x) 2 )( 1 u (x) 2 ) Φ u (r) := r 2 x N 2 dx r 2 x N 2 dx. B r (0) Since u + and u are locally Lipschitz continuous, Theorem 2.1 implies, in particular, that there exists Noticing that there holds: δ := lim r 0 Φ u (r). Φ uλ (r) = Φ u (λr), we deduce that there exists lim λ 0 Φ uλ (r) and it coincides with δ, for every r>0. Let now λ n 0 be such that u λn U uniformly on compact sets of R N, and let r>0be fixed. By Lemma 2.3 we know that u λn U in L 2 loc (RN ). So that for a subsequence, that we still call λ n,wehave u λn U a.e. in R N.Also u λn (x) L for x < r 0 λ n, where L is the bound of u in some B r0 (0). Consequently, we may pass to the limit in the expression of Φ uλn (r) to conclude that ( 1 Φ uλn (r) r 2 B r (0) U + (x) 2 dx x N 2 )( 1 r 2 B r (0) B r (0) U (x) 2 ) dx. x N 2 So that the lemma is proved with δ = lim r 0 Φ u (r) independent of the sequence λ n. The main result in the section is

19 570 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Theorem 6.1. Let u ε j converge to u uniformly on compact subsets of Ω, f ε j f -weakly in L (Ω) and ε j 0. Let x 0 Ω {u>0} be a regular point. Assume that u + is nondegenerate at x 0. Then, there exist α>0 and γ 0 such that with u(x) = α x x 0,ν + γ x x 0,ν + o ( x x 0 ), α 2 γ 2 = 2M, where ν is the inward unit normal to {u>0} at x 0 in the measure theoretic sense. Proof. We will assume, without loss of generality, that x 0 = 0 and ν = e 1.Let u λ (x) = 1 λ u(λx), and let r>0 be such that B r (0) Ω. We have that u λ is Lipschitz continuous in B r/λ (0) uniformly in λ, and u λ (0) = 0. Therefore, for every λ n 0, there exists a subsequence, that we still call λ n, and a function U, Lipschitz continuous in R N, such that u λn U uniformly on compact sets of R N. By (6.2), it follows that for every k>0, {u λ > 0} {x 1 < 0} B k (0) 0 as λ 0, and {u λ 0} {x 1 > 0} B k (0) 0 as λ 0. It follows that U is nonnegative in {x 1 > 0} and harmonic in {U >0} and that U is nonpositive in {x 1 < 0} and harmonic in {U <0} (recall Lemma 2.2(v)). So that U is superharmonic in {x 1 < 0}. On the other hand, from Lemma 2.2(iv) we deduce that U is subharmonic in R N. Thus, U is harmonic in {x 1 < 0} and necessarily U(x)= γx 1 in {x 1 < 0}, for some γ 0. On the other hand, since {U >0} {x 1 > 0},byLemmaA.1in[6],thereexistsα 0 such that The nondegeneracy assumption of u + at x 0 implies that necessarily α>0. Let us now show that By Lemma 2.3 there exists a subsequence ε jn such that δ n := ε jn λ n U(x)= αx o( x ) in {x 1 > 0}. (6.3) α 2 γ 2 = 2M. (6.4) 0 and u δ n (x) := 1 λ n u ε jn (λ n x), u δ n U uniformly on compact sets of R N. Let f δ n(x) := λ n f ε jn (λn x). Then, f δ n 0 uniformly on compact sets of R N and u δ n is a solution to E δn (f δ n). Now let U λ (x) = 1 λ U(λx). Then for a sequence λ k 0, U λk αx + 1 γx 1, uniformly on compact subsets. As before, there exists a subsequence δ nk such that δ k := δ n k λ k uniformly on compact subsets. u δ k αx + 1 γx+ 1, 0 and that u δ k (x) := 1 λ k u δ n k (λ k x) satisfies that

20 C. Lederman, N. Wolanski / J. Math. Pures Appl. 86 (2006) Since u δ k are solutions to E δ k ( f δ k ) with f δ k 0 (they are rescalings of the functions f δ n k ) uniformly on compact sets of R N, we may apply Proposition 3.1, if γ>0, or Proposition 3.2, if γ = 0, and deduce that α 2 γ 2 = 2M. Let us show that we actually have: U(x)= αx + 1 γx 1. (6.5) In fact, {U >0}, U γ and thus by Theorem 4.2 we have U + 2M + γ 2 = α. Using that U 0in {x 1 = 0} we deduce that U αx 1 in {x 1 > 0}. Since U is globally subharmonic and satisfies (6.3) the application of Hopf s Principle yields U = αx 1 in {x 1 > 0}, which gives (6.5). Finally we observe that, by Lemma 6.1, there exists δ 0 independent of the sequence λ n such that δ Φ U (r) C N α 2 γ 2. (6.6) So that (6.5) holds with α>0, γ 0 satisfying (6.4) and (6.6). In particular, α and γ are independent of the sequence λ n. The theorem is proved. Remark 6.1. We point out that, from Section 3 in [24], we know that there are examples where u + degenerates at x 0, and such that the conclusion in Theorem 6.1 does not hold. We recall that in Section 5 we gave conditions under which u + is nondegenerate at a free boundary point x Viscosity solutions to the free boundary problem In this section we consider u = lim u ε, with u ε solutions to problems E ε (f ε ), and we prove that, under suitable assumptions, u is a viscosity solution of the free boundary problem E(f) (Corollaries 7.1 and 7.2). First, we prove results on asymptotic developments at free boundary points in which there is a tangent ball contained either in {u>0} or in {u 0} (Theorems 7.1 and 7.2). The corollaries follow as an immediate consequence. Some of these results hold if u + satisfies a suitable nondegeneracy condition (we refer to Section 5 for conditions implying the nondegeneracy of u + ). Definition 7.1. Let Ω be a domain in R N. For any function u on Ω we define: and Ω + (u) := Ω {u>0}, (7.1) Ω (u) := Ω {u 0}, (7.2) F(u)= Ω {u>0}. (7.3) Definition 7.2. Let u be a continuous function in a domain Ω R N. We say that a point x 0 F(u) is a regular point from the right if there is a tangent ball at x 0 from Ω + (u) for (i.e., there is a ball B {u>0}, with x 0 B). Analogously, we say that a point x 0 F(u) is a regular point from the left if there is a tangent ball at x 0 from Ω (u) (i.e., there is a ball B {u 0}, with x 0 B). Definition 7.3. Let u be a continuous function in a domain Ω R N.Letf L (Ω). Then u is called a viscosity supersolution of E(f) in Ω if: (i) u fχ {u 0} in Ω + (u). (ii) u fχ {u 0} in Ω (u).