Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients

J. Differential Equations 245 (2008) 2038–2054 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients Marc Briane a,∗, Juan Casado-Díaz b aCentre de Mathématiques, INSA de Rennes & IRMAR, France bDepartamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain article info abstract Article history: Received 22 November 2006 Revised 10 July 2008 Availableonline6August2008 MSC: 35J70 35B40 35B50 35B65 35B27 Keywords: Linear degenerate elliptic equations Continuity of solutions Uniform convergence of solutions Maximum principle Homogenization This paper deals with the behavior of two-dimensional linear elliptic equations with unbounded (and possibly infinite) coefficients. We prove the uniform convergence of the solutions by truncating the coefficients and using a pointwise estimate of the solutions combined with a two-dimensional capacitary estimate. We give two applications of this result: the continuity of the solutions of two-dimensional linear elliptic equations by a constructive approach, and the density of the continuous functions in the domain of the Γ-limit of equicoercive diffusion energies in dimension two. We also build two counter-examples which show that the previous results cannot be extended to dimension three. ©2008 Elsevier Inc. All rights reserved. 1. Introduction In this paper we study the behavior of degenerate linear elliptic equations posed in a bounded open subset Ωof RN, especially in the case N=2, of type −div(A∇u)=fin Ω, (1.1) *Corresponding author. E-mail addresses: [email protected] (M. Briane), [email protected] (J. Casado-Díaz). 0022-0396/$ – see front matter ©2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jde.2008.07.027 M. Briane, J. Casado-Díaz / J. Differential Equations 245 (2008) 2038–2054 2039 where the right-hand side fbelongs to H−1(Ω), and Ais a coercive but not necessarily bounded matrix-valued function. Indeed, the quadratic form relating to Acan even take infinite values (see Definition 2.1). Therefore, the solutions of (1.1)will be understood in the sense ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  Ω A∇u·∇udx<∞,  Ω A∇u·∇vdx=f,v,∀v∈H1 0(Ω) with  Ω A∇v·∇vdx<∞. (1.2) We will prove (see Proposition 2.3) that the solutions of this problem can be obtained as the limits of solutions of coercive problems with bounded coefficients, using some truncation of the matrixvalued A. In particular, this allows us to extend some classical properties of the solutions of linear elliptic equations with coercive and bounded operators, such as the maximum principle (see Theorem 2.4). The main results of the paper refer to the compactness of the solutions of (1.1)for the uniform convergence in dimension two. Recall that for any bounded open subset Ωof RN,N⩾1, any coercive matrix-valued function A∈L∞(Ω)N×N, and any fin W−1,q(Ω) with q>2, the solutions of (1.1)are Hölder-continuous in Ω(see e.g. [14,18,22]). As a consequence, if unis a bounded sequence in H1(Ω) of solutions of equations −div(An∇un)=fnin Ω, (1.3) where fnis bounded in W−1,q(Ω) for some q>2, and Anis uniformly coercive and bounded in L∞(Ω)N×N,thenunis compact in C0(Ω), i.e. uniformly convergent in any compact set of Ω. In the two-dimensional case the solutions of (1.3)are still continuous even if the diffusion matrix An is not bounded from above (see e.g. [15,17]). In general they are no longer Hölder-continuous (see Example 4.6), but we show in the present paper that the former uniform convergence result does subsist. More precisely, we prove (see Theorems 2.5 and 2.7) that the compactness of unin C0(Ω) still holds true without assuming any bound from above on the equicoercive sequence An.Moreover, if Ωis Lipschitz and unis compact in C0(∂Ω), then the sequence unis compact in C0(¯ Ω). The previous uniform convergence results are applied in two directions. On the one hand, we give an alternative proof (see Theorem 4.1) of the continuity of the solutions of (1.3)using the approximation by truncation of the matrix-valued Ancombined with the uniform convergence of the solutions of the equations with truncated coefficients (which are known to be continuous). On the other hand, the asymptotic behavior of sequences of solutions of (1.3)is strongly connected to the homogenization theory. The homogenization of sequences of elliptic equations has formed the subject of several works since the end of sixties. Assuming the uniformly boundedness of the coefficients Spagnolo [24] and Murat and Tartar [21,25] proved that the limit problem of (1.3) has the same structure. These results were extended in [10,12] and [20] under the weaker assumption of the L1-equiintegrability, as well as in the periodic case [5] under the control of a weighted Poincaré–Wirtinger inequality. Moreover, Fenchenko and Khruslov [16] proved that only L1-bounded coefficients may induce nonlocal effects in dimension three through jumping measures which modify the nature of the limit problem (also see the recent book [19] on the topic and another approaches in [1,4,9,11,20]). Mosco [20] showed the nonlocal terms arise naturally in the limit process using the Beurling–Deny [2] representation of the Dirichlet forms. Completing the previous work Camar-Eddine and Seppecher [11] proved that in dimension three any jumping measure can be obtained by the homogenization of a suitable equicoercive sequence of conductivity matrices An. Recently, we showed first in the periodic case [6], then in the general case [7,8], that, contrary to the dimension three, the dimension two prevents from the appearance of nonlocal effects. In [7] we proved the compactness of Eqs. (1.3)assuming the L1-boundedness of Anand using two-dimensional div–curl lemmas. In [8] we extended the previous result without assuming any bound from above on An. 2040 M. Briane, J. Casado-Díaz / J. Differential Equations 245 (2008) 2038–2054 The uniform convergence results of the present paper allow us to complete and improve the results of [8]. More precisely, the approach of [8] is based on a (exclusively) two-dimensional capacitary estimate (see Lemma 3.2). This tool is now combined with two extra ingredients: the truncation principle mentioned above and a pointwise estimate satisfied by the solutions of linear elliptic equations (see Theorem 2.4) which extends a classical one (see e.g. Theorem 8.16 of [18]). The combination of these three ingredients yields the uniform convergence of the sequence unof solutions of (1.3) without assuming a priori the continuity of its limit as in [8]. As a consequence of this approach, we prove (see Theorem 4.3) that the continuous functions are dense in the domain (endowed with the intrinsic norm) of the Γ-limit of any equicoercive sequence of linear diffusion energies in dimension two, which is a new contribution in the topic up to our knowledge. We conclude the paper with two counter-examples illustrating the gap between the dimension two and the higher one, concerning the former uniform convergence results. First, we give an example (see Proposition 4.7) of a sequence of solutions of three-dimensional Dirichlet problems, which does not converges uniformly. Then, we construct (see Proposition 4.9) a discontinuous solution of a three-dimensional linear elliptic equation with unbounded coefficients. The two counter-examples are based on fibers reinforced structures which were first used in [16] to derive nonlocal effects in homogenization. 2. Statement of the results 2.1. A pointwise estimate of solutions of linear elliptic equations We start the paper by giving a consistent definition of unbounded non-symmetric matrix-valued functions, with possibly infinite values: Definition 2.1. Let α>0 and let Ωbe a bounded open subset of RN.LetA(α,Ω) be the set of the matrix-valued functions Asuch that there exist P∈L∞(Ω)N×Nwith values on the set of orthogonal matrices, Nmeasurable functions d1,...,dN:Ω→[α,∞], a constant β>0, and a measurable skewsymmetric matrix-valued function B:Ω→RN×N,with |B|∞:= max 1⩽i,j⩽N|Bij|⩽βmin{d1,...,dN},a.e. in Ω, which satisfy for a.e. x∈Ω, A(x)ξ =P(x)tD(x)P(x)ξ +B(x)ξ, ∀ξ∈Vx:=ξ∈RN:DP(x)ξ<∞,(2.1) where Dis the diagonal matrix-valued function diag(d1,...,dN). Remark 2.2. Definition 2.1 allows us to give a sense to A(x)ξ ·ξ,fora.e.x∈Ωand any ξ∈RN,by A(x)ξ ·ξ:=D(x)P(x)ξ ·P(x)ξ if ξ∈Vx, ∞if ξ∈R2\Vx.(2.2) In the particular case of symmetric matrices this leads us to the following extensions of classical definitions: •For any A,B∈A(α,Ω),A⩽Bmeans that A(x)ξ ·ξ⩽B(x)ξ ·ξ⩽∞,for a.e. x∈Ω, ∀ξ∈R2.(2.3) •AsequenceAn∈A(α,Ω) is said to converge a.e. to A∈A(α,Ω) if A(x)ξ ·ξ=lim n→∞ An(x)ξ ·ξ∈[0,∞],for a.e. x∈Ω, ∀ξ∈R2.(2.4) M. Briane, J. Casado-Díaz / J. Differential Equations 245 (2008) 2038–2054 2041 Let A∈A(α,Ω) and denote by Asits symmetric part. The conditions d1,...,dN:Ω→[α,∞] and |B|∞⩽βmin{d1,...,dN}a.e. in Ω, read as, according to (2.3), αI⩽Asand |B|∞I⩽βAs,a.e. in Ω, (2.5) where Iis the unit matrix of R2×2. Denote by Tκ,forκ⩾0, the truncation at size κ,i.e.Tκ(t):=min(κ,max(−κ,t)) for any t∈R. Then, we have the following truncation result: Proposition 2.3. Let A be an element of A(α,Ω), and consider P, D, a, βas in Definition 2.1. For n ⩾α, let Dnbe the diagonal matrix-valued function with entries Tn(d1),...,Tn(dN),letB nbe the skew-symmetric matrix-valued function with entries (Bn)ij := Tnβ(Bij),andletA nbe the element of A(α,Ω)defined by (2.1) with P , Dnand Bn.Then,forany f ∈H−1(Ω) and any u ∈H1(Ω) which satisfies (1.2), the solution unof −div(An∇un)=finΩ, un−u∈H1 0(Ω), (2.6) converges strongly to u in H1(Ω). Moreover, we have lim n→∞ Ω An∇(un−u)·∇(un−u)dx =0.(2.7) A consequence of this truncation result is the following pointwise estimate satisfied by the solutions of (1.2): Theorem 2.4. For any α>0and any q >2, there exists a constant C >0with the following property: For any bounded open subset Ωof RN,A∈A(α,Ω),f∈W−1,q(Ω),q>2,m,M∈Rwith m <M, and any u ∈H1(Ω) satisfying (1.2)with (m−u)+,(u−M)+∈H1 0(Ω),wehave m−C|Ω| Nq−N−q Nq fW−1,q(Ω) ⩽u⩽M+C|Ω| Nq−N−q Nq fW−1,q(Ω),a.e. in Ω. (2.8) 2.2. Uniform convergence results The main results of the paper are the following compactness result and its refinement in Theorem 2.7: Theorem 2.5. Let α>0,letΩbe a bounded open subset of R2, with a Lipschitz boundary, and let Γbe a relatively open subset of ∂Ω. Consider a sequence Anin A(α,Ω),asequenceu nin H1(Ω) ∩C0(Ω ∪Γ) which converges weakly in W 1,p(Ω),withp∈(1,2], and uniformly in the closed subsets of Γtoafunction u∈W1,p(Ω) ∩C0(Ω ∪Γ), and consider a sequence fnin W −1,q(Ω),withq>2. Assume that An,u nand fn satisfy the following conditions: ∀n∈N,⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩  Ω An∇un·∇undx <∞,  Ω An∇un·∇vdx=fn,v,∀v∈H1 0(Ω) with  Ω An∇v·∇vdx<∞. (2.9) Then, the sequence unconverges to u in C0(Ω ∪Γ), i.e. uniformly in any compact set of Ω∪Γ. 2042 M. Briane, J. Casado-Díaz / J. Differential Equations 245 (2008) 2038–2054 We introduce the following notations: •We denote by B(x,r)the open ball of center x∈R2and of radius r>0, by ¯ B(x,r)its closure, and the corresponding half-balls by B(x,r)+:= B(x,r)∩y∈R2:y2>x2and ¯ B(x,r)+:= ¯ B(x,r)∩y∈R2:y2⩾x2. •For any bounded continuous function uinasubsetEof R2,weset osc Eu:=sup E u−inf Eu. As a consequence of Theorem 2.7, we have the following local estimates for the solutions of (1.2): Corollary 2.6. Let α,ε>0,p>1and q >2. Then, there exists a constant C >0such that the two following assertions hold: (i) For any x0∈R2,δ>0,A∈A(α,B(x0,2δ)),f∈W−1,q(B(x0,2δ)), and any function u in H1(B(x0,2δ)) ∩C0(B(x0,2δ)) satisfying (1.2)with Ω=B(x0,2δ),wehave osc ¯ B(x0,δ) u⩽εδ q−2 qfW−1,q(B(x0,2δ)) +δ p−2 p∇uLp(B(x0,2δ))2 +C δ2    u−1 4πδ2 B(x0,2δ) udx    L1(B(x0,2δ)) .(2.10) (ii) For any x0∈R2,δ>0,A∈A(α,B(x0,2δ)+),f∈W−1,q(B(x0,2δ)+), and any function u in H1(B(x0,2δ)+)∩C0(¯ B(x0,2δ)+)satisfying (1.2)with Ω=B(x0,2δ)+,wehave osc ¯ B(x0,δ)+ u⩽εδ q−2 qfW−1,q(B(x0,2δ)+)+δ p−2 p∇uLp(B(x0,2δ)+)2 +C1 δ2    u−1 2δ δ  −δ u(t,0)dt   L1(B(x0,2δ)+)+osc [−2δ,2δ]×{0}u.(2.11) Using Corollary 2.6 the following result is a refinement of Theorem 2.5 in the case p=2, without assuming the continuity of the limit: Theorem 2.7. Let α>0,letΩbe a bounded open subset of R2, with a Lipschitz boundary, and let Γ be a relatively open subset of ∂Ω. Consider a sequence Anin A(α,Ω),asequenceu nin H1(Ω) ∩ C0(Ω ∪Γ)converging weakly in H1(Ω) and uniformly in the closed subsets of Γto a function u, and consider a bounded sequence fnin W −1,q(Ω),withq>2. Assume that An,u nand fnsatisfy (2.9).Then,the sequence unconverges to u in C0(Ω ∪Γ). Note that in Theorem 2.5 the continuity of the limit cannot be removed when p<2, as shown in Example 4.5 below. M. Briane, J. Casado-Díaz / J. Differential Equations 245 (2008) 2038–2054 2043 3. Proof of the results Proof of Proposition 2.3. First of all, note that the symmetric part As nof Anis a nondecreasing sequence according to (2.3), which satisfies As n⩾αIby (2.5), and As nconverges to the symmetric part Asof Aaccording to (2.4).Takingun−uas test function in (2.6)we get that there exists C>0 with  Ω An∇un·∇undx ⩽C,∀n∈N.(3.1) Due to the α-coerciveness of An,thesequenceun−uis bounded in H1 0(Ω). Hence, up to extracting asubsequence,thereexistsu∗∈H1(Ω),withu∗−u∈H1 0(Ω), such that unconverges weakly to u∗ in H1(Ω). By semicontinuity and the non-decrease of As nwe have for any k⩾α,  Ω Ak∇u∗·∇u∗dx ⩽liminf n→∞  Ω Ak∇un·∇undx ⩽liminf n→∞  Ω An∇un·∇undx ⩽C. Therefore, since As kconverges to Asin a nondecreasing way, we deduce from the monotone convergence theorem that  Ω A∇u∗·∇u∗dx =lim k→∞ Ω Ak∇u∗·∇u∗dx ⩽C.(3.2) Now, consider v∈H1 0(Ω) such that  Ω A∇v·∇vdx<∞.(3.3) Taking vas test function in (2.6)we obtain for any nand k, f,v= Ω An∇un·∇vdx= Ω (An−Ak)∇un·∇vdx+ Ω Ak∇un·∇vdx.(3.4) We will pass to the limit in the right-hand side of this equality first in n⩾kthen in k. Consider the second term of the right-hand side of (3.4).Sinceby(3.2)A∇u∗·∇u∗<∞a.e. in Ω,by(2.2)we have |DP∇u∗|<∞a.e. in Ω,henceAk∇u∗·∇vconverges to A∇u∗·∇va.e. in Ω.Moreover,bythe Cauchy–Schwarz inequality combined with the inequalities As k⩽As,|Bk|⩽|B|and (2.5),wehave 2|Ak∇u∗·∇v|⩽2|DkP∇u∗·P∇v|+2|Bk∇u∗·∇v| ⩽Ak∇u∗·∇u∗+Ak∇v·∇v+N|Bk|∞|∇u∗|2+N|Bk|∞|∇v|2 ⩽A∇u∗·∇u∗+A∇v·∇v+N|B|∞|∇u∗|2+N|B|∞|∇v|2 ⩽(1+Nβ)(A∇u∗·∇u∗+A∇v·∇v)∈L1(Ω). Therefore, by the weak convergence of unto u∗in H1(Ω), and the Lebesgue convergence theorem we get lim k→∞ lim n→∞ Ω Ak∇un·∇vdx=lim k→∞ Ω Ak∇u∗·∇vdx= Ω A∇u∗·∇vdx. 2044 M. Briane, J. Casado-Díaz / J. Differential Equations 245 (2008) 2038–2054 For the first term of the right-hand side of (3.4), the decomposition (2.1)for An,theCauchy–Schwarz inequality, the inequalities As k⩽As n⩽Asand |Bk|∞I⩽|Bn|∞I⩽βAs n(as a consequence of (2.5)), and estimate (3.1)yield  Ω (An−Ak)∇un·∇vdx  ⩽ Ω (An−Ak)∇un·∇undx1 2 Ω (An−Ak)∇v·∇vdx 1 2 + Ω N|Bn−Bk|∞|∇un|2dx1 2 Ω N|Bn−Bk|∞|∇v|2dx1 2 ⩽ Ω An∇un·∇undx1 2 Ω (A−Ak)∇v·∇vdx 1 2 + Ω 2NβAn∇un·∇undx1 2 Ω N|Bn−Bk|∞|∇v|2dx1 2 ⩽c Ω (A−Ak)∇v·∇vdx 1 2 +c Ω |Bn−Bk|∞|∇v|2dx1 2 . Since As−As k⩽Asand |Bn−Bk|∞I⩽βAs, we deduce from the Lebesgue convergence theorem that the two last terms of the previous inequality converge to zero as n⩾kthen ktend to ∞.So,weget lim k→∞lim sup n→∞  Ω (An−Ak)∇un·∇vdx =0, and thus, by (3.4)we obtain f,v= Ω A∇u∗·∇vdx,(3.5) for any v∈H1 0(Ω) which satisfies (3.3).Takingu−u∗as test function in the difference of (1.2) and (3.5)we deduce that u∗=u. Then, the whole sequence unconverges weakly to uin H1 0(Ω). Now, taking un−uas test function in (2.6)and passing to the limit in nwe obtain lim n→∞ Ω An∇un·∇(un−u)dx =lim n→∞f,un−u=0.(3.6) By (3.2)and reasoning similarly as in (3.4)we have lim n→∞ Ω An∇u·∇(un−u)dx =0, which combined with (3.6)yields the desired limit (2.7).2 Proof of Theorem 2.4. Set δ:=|Ω|1 N, and define Ωδ:=δ−1Ω. For a measurable function hin Ω,we denote by hδthe measurable function defined by hδ(y):=h(δ y), a.e. y∈Ωδ. M. Briane, J. Casado-Díaz / J. Differential Equations 245 (2008) 2038–2054 2045 Then, taking α,q,A,fand uas in the statement of Theorem 2.4, we define Aδ,uδby the previous rule, and fδ∈W−1,q(Ωδ)by fδ,ϕδW−1,q(Ωδ),W1,q 0(Ωδ):=f,ϕW−1,q(Ω),W1,q 0(Ω),∀ϕ∈W1,q 0(Ω). We have that the measure of Ωδis equal to 1. Moreover, making the former change of variables in (1.2), the function uδsatisfies ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ m−uδ+,uδ−M+∈H1 0Ωδ, Ωδ Aδ∇uδ·∇uδdy <∞,  Ωδ Aδ∇uδ·∇vδdy =δ2−Nfδ,vδ,∀vδ∈H1 0Ωδwith  Ωδ Aδ∇vδ·∇vδdy <∞. On the other hand, let Aδ nbe the truncated of Aδdefined as in Proposition 2.3, and let uδ nbe the solution of (2.6)with An,f,ureplaced respectively by Aδ n,fδ,uδ.Since(m−uδ)+,(uδ−M)+ and uδ n−uδbelong to H1 0(Ωδ),sodo(m−uδ n)+and (uδ n−M)+. Then, taking into account |Ωδ|=1, Theorem 8.16 of [18] implies the existence of a constant C>0 only depending on αand q, such that m−C fδ W−1,q(Ωδ)⩽uδ n⩽M+C fδ W−1,q(Ωδ),a.e. in Ωδ,∀n∈N. However, by Proposition 2.3thesequenceuδ nconverges strongly to uδin H1(Ω), thus in particular a.e. in Ω(up to a subsequence). Therefore, passing to the pointwise limit in the previous inequalities we get m−C fδ W−1,q(Ωδ)⩽uδ⩽M+C fδ W−1,q(Ωδ),a.e. in Ωδ.(3.7) Finally, noting that  fδ W−1,q(Ωδ)=inf ϕδ=0 fδ,ϕδW−1,q(Ωδ),W1,q 0(Ωδ) ϕδW1,q 0(Ωδ) =inf ϕ=0 f,ϕW−1,q(Ω),W1,q 0(Ω) δ N q−N+1ϕW1,q 0(Ω) =δN−1−N qfW−1,q(Ω), we easily deduce (2.8).2 Let us now prove the compactness results stated in Section 2.2. First of all recall the definition of the r-capacity: Definition 3.1. Let r∈(1,2).Ther-capacity of a subset Eof R2is defined by Cr(E):=inf R2 |∇u|rdx:u∈D1,rR2,u⩾1 a.e. in a neighborhood of E, where D1,r(R2)denotes the space of the functions uin L2r 2−r(R2),with∇u∈Lr(R2)2. We need the following capacitary estimate which is proved in [8]: 2046 M. Briane, J. Casado-Díaz / J. Differential Equations 245 (2008) 2038–2054 Lemma 3.2. For any r ∈(1,2), there exists a constant Rr>0such that for any a,b∈R2, and any curve L of extremities a,b, we have Cr(L)⩾Rr|a−b|2−r.(3.8) Proof of Theorem 2.5. For δ>0, we define Ωδby Ωδ:={x∈Ω∪Γ:dist(x,∂Ω \Γ)⩾δ}if Γ=∂Ω, ¯ Ωif Γ=∂Ω. Then, for any l∈N,wetakeδl>0 which converges to zero such that u(x)−u(y)<1 2l,∀x,y∈¯ Ωδlwith |x−y|⩽δl.(3.9) Now, since unconverges weakly in W1,p(Ω) and Ωis regular, we can take r∈(1,p)and a subsequence of un,stilldenotedbyun, which converges Cr-quasi-uniformly to u. Also using that un converges to uin C0(Γ ), we can choose this sequence in such a way that there exists an open subset Glof Ω, which satisfies (Rrdefined in Lemma 3.2) Cr(Gl)<Rrδ2−r l,(3.10) un(x)−u(x)<1 2l,∀x∈(Ω \Gl)∪(Γ ∩Ωδl), ∀n⩾l.(3.11) Let us prove that this subsequence unconverges to uin C0(Ω ∪Γ). This will prove that the whole sequence unconverges to uin C0(Ω ∪Γ).Wefixδ>0 and we take lsuch that 2δl<δ. Consider a connected component Oof Glsuch that O∩Ωδ=∅.SinceOis connected by curves, for any y1,y2∈O,thereexistsacurveL⊂O, which contains y1,y2. By Lemma 3.2 and (3.10)we have Rr|y1−y2|2−r⩽Cr(L)⩽Cr(O)⩽Cr(Gl)⩽Rrδ2−r l, hence diam(O)⩽δl. Therefore, since 2δl<δ,wehave ¯ O⊂Ωδl, which implies in particular that ∂O⊂ (Ω \Gl)∪(Γ ∩Ωδl). By (2.9), the fact that un∈C0(Ω ∩Γ),byTheorem2.4 and |O|⩽πδ2 l/4, we have (for another constant C) min ∂Oun−Cδ q−2 q lfnW−1,q(Ω) ⩽un⩽max ∂Oun+Cδ q−2 q lfnW−1,q(Ω),in O,∀n⩾l.(3.12) Moreover, thanks to (3.11)and (3.9)we have min ∂Oun⩾min ∂Ou−1 2l⩾u(x)−1 l,max ∂Oun⩽max ∂Ou+1 2l⩽u(x)+1 l,∀x∈O,∀n⩾l. Hence, inequality (3.12)implies that |un−u|⩽1 l+Cδ q−2 q lfnW−1,q(Ω),in O,∀n⩾l. M. Briane, J. Casado-Díaz / J. Differential Equations 245 (2008) 2038–2054 2053 Taking u−:= max(−u,0)as test function in (4.9), we deduce that u−=0 and so, u⩾0a.e.inΩ. Moreover, by Theorem 2.4 the function ubelongs to L∞(Ω).Takinguw2χˆ ωn k,n∈N∗,1⩽k⩽2n−1, as test function in (4.9),wealsoget  ˆ ωn k a|∇u|2w2dx +2 ˆ ωn k a∇u·∇wwdx=0, which by Young’s inequality implies  ˆ ωn k a|∇u|2w2dx ⩽4 ˆ ωn k a|∇w|2dx. This combined with the definitions (4.8)of aand (4.7)of w,yields 1 3n−  ωk n |∇u|2dx ⩽ ˆ ωn k a|∇u|2w2dx ⩽4 ˆ ωn k a|∇w|2dx ⩽c 4n. Then, using that u=0on∂Ω, we have by the Cauchy–Schwarz inequality −  ωk n udx=−  ωk n x3  0 ∂u ∂x3 (x,t)dt dx ⩽−  ωk n |∇u|2dx1 2 ⩽c√3 2n ,∀n∈N∗,1⩽k⩽2n−1. Now, assume that uis continuous in a neighborhood of the boundary Γ. Then, the previous estimate and u⩾0a.e.inΩimply that u=0onΓ.Sinceby(4.9)and ψ=0inΩ+,uis a harmonic function in H1 0(Ω+),wethushaveu=0a.e.inΩ+. On the other hand, since a∈L1(Ω),foranyϕ∈C1 c(Ω), the function ϕ(1−w)is a suitable test function for problem (4.9), which is equal to ϕin Ω−.Hence, taking into account that u=0a.e.inΩ+,weobtain  Ω− ∇u·∇ϕdx = Ω a∇u·∇ϕ(1−w)dx = Ω− ψϕdx. Therefore, uis solution of the Dirichlet problem −u=ψ=0inD(Ω−), u=0on∂Ω−,and ∂u ∂ν=0onΓ, which contradicts the Hopf maximum principle, i.e. the negativity of the normal derivative of uon Γ. 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