Asymptotic behavior of nonlinear systems in varying domains with boundary conditions on varying sets

ESAIM: COCV 15 (2009) 49–67 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv:2008021 www.esaim-cocv.org ASYMPTOTIC BEHAVIOR OF NONLINEAR SYSTEMS IN VARYING DOMAINS WITH BOUNDARY CONDITIONS ON VARYING SETS Carmen Calvo-Jurado1, Juan Casado-D´ ıaz2and Manuel Luna-Laynez2 Abstract. For a fixed bounded open set Ω ⊂RN, a sequence of open sets Ωn⊂Ω and a sequence of sets Γn⊂∂Ω∩∂Ωn, we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on Ωn, satisfying Neumann boundary conditions on Γnand Dirichlet boundary conditions on ∂Ωn\Γn. We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on Ωnand Γnlocally. Mathematics Subject Classification. 35B40. Received March 15, 2007. Revised April 10, 2007 and June 26, 2007. Published online March 6, 2008. 1. Introduction For a given Lipschitz bounded open set Ω ⊂RN,N≥2, a sequence of open sets Ωn⊂Ω and a sequence of sets Γn⊂∂Ω∩∂Ωn, we study the asymptotic behavior of the solution unof the nonlinear elliptic system ⎧ ⎪ ⎨ ⎪ ⎩ −div (a(x, Dun)−Gn)=gnin Ωn un=0 on∂Ωn\Γn (a(x, Dun)−Gn)ν=0 onΓ n, (1.1) where a:Ω×RM×N→RM×N,M≥1, is a Carath´eodory function which satisfies standard assumptions so that the operator v∈W1,p 0(Ω)M→−div a(x, Dv)∈W−1,p(Ω)M,p≥2, defines a monotone operator in the sense of Leray and Lions [16] (see Sect. 2for the precise assumptions on a)andνdenotes the unitary outward normal to Ω. The sequences gnand Gnare assumed to converge in Lp(Ω)Mweakly and Lp(Ω)M×Nstrongly to some functions gand Grespectively. Assuming that unW1,p(Ωn)Mis bounded (this holds for example if there exists C>0 independent of nwith vW1,p(Ωn)≤C∇vLp(Ωn)N, for every v∈W1,p(Ωn), v=0on∂Ωn\Γn) and extending unby zero outside Ωn, we prove the existence of a nonnegative Borel measure μin Ω which does not charge sets of p-capacity zero, and a μ-Carath´eodory function F:Ω×RM→RMsatisfying monotonicity and continuity properties related Keywords and phrases. Homogenization, varying domains, nonlinear problems. 1Dpto. de Matem´aticas, Escuela Polit´ecnica, Avenida de la Universidad s/n, 10071 C´aceres, Spain. [email protected] 2Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Fac. de Matem´aticas, C. Tarfia s/n, 41012 Sevilla, Spain. [email protected]; [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2008 50 C. CALVO-JURADO, J. CASADO-D´ IAZ AND M. LUNA-LAYNEZ to those imposed to a(see (3.20), (3.21), (3.22)), such that unconverges weakly in W1,p(Ω)Mand strongly in W1,q(Ω)M,1≤q<p,tothesolutionuof the problem ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u∈W1,p(Ω)M∩Lp μ(Ω)M Ω a(x, Du):Dv dx+Ω F(x, u)vdμ=Ω gv dx+Ω G:Dv dx ∀v∈W1,p(Ω)M∩Lp μ(Ω)M, (1.2) which (if μis smooth) can be written as −div (a(x, Du)−G)+F(x, u)μ=gin Ω (a(x, Du)−G)ν+F(x, u)μ=0 on∂Ω.(1.3) The pair (F, μ) does not depend on gnor Gn, and it depends on Ωnand Γnlocally in the sense that if we consider a Lipschitz open set ω⊂Ω and we replace in (1.1)Ω nby Ωn∩ω,Γ nby Γn∩ω, then the previous result holds with (F, μ) replaced by (F|¯ω,μ |¯ω). The term F(x, u)μin (1.2) is similar to the strange term which appears in the homogenization of Dirichlet problems on varying domains (see [1,3–12,19,20]). In fact, if Γnis empty, our result follows from [5]. When Γnis not empty the main difference is that now μis defined on ¯ Ω and not only on Ω and then the term F(x, u)μdoes not only appears in the equation but also in the boundary conditions of (1.3). Taking Ω = Ωnfor every n∈N, the above result proves that the boundary condition corresponding to the limit of a sequence of nonlinear elliptic systems with Dirichlet an Neumann conditions on varying subsets of ∂Ω is a Fourier-Robin condition. Indeed, the proof of this fact was the origin of the present work. We have preferred to present here the more general case where the open sets Ωnare variable, in order to show that the homogenization of elliptic Dirichlet problems in varying domains (corresponding to Γn=∅) and the homogenization of elliptic problems with Neumann and Dirichlet conditions imposed on varying sets of the boundary admit a common formulation. As in the case of Dirichlet problems on varying domains [9], we observe that (1.1)canbewritteninsuchway that its structure is similar to (1.2). For this purpose, it is enough to define μnas (Cpstands for the p-capacity, see Sect. 2) μn(B)=+∞if Cp(B∩(Ω \(Ωn∪Γn))) >0 0ifCp(B∩(Ω \(Ωn∪Γn))) = 0,∀B⊂Ω Borel, (1.4) and Fn: Ω ×RM−→ RMas, for example, Fn(x, s)=|s|p−2s.Then(1.1)isequivalentto ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ un∈W1,p(Ω)M∩Lp μn(Ω)M Ω a(x, Dun):Dv dx+Ω Fn(x, un)vdμn=Ω gnvdx+Ω Gn:Dv dx ∀v∈W1,p(Ω)M∩Lp μn(Ω)M. (1.5) Hence, we can consider (1.1) as a particular case of (1.2). For this reason, better than the homogenization of (1.1), we will study the homogenization of (1.5) for a sequence μnof Borel measures in Ω (not necessarily defined from sequences Ωn,Γ nas above) which vanish on sets of p-capacity zero and a sequence Fn: Ω×RM−→ RMof monotone μn-measurable functions (see Sect. 2for the precise hypotheses on Fn). We prove that, in this more general form, the problem is stable for homogenization, i.e. for every sequences μnand Fnthere exist μ and Fsuch that, at least for a subsequence, the limit problem of (1.5) is still given by (1.2). Throughout the paper we just consider the case p≥2. The case 1 ≤p<2 can be treated in a similar way, after proper modification on the growth and coerciveness hypotheses for the functions aand Fn.Thecaseof linear equations and μnconcentrated on ∂Ω (which for problem (1.1)meansΩ n=Ωforeveryn∈N) has been studied in [2], see also [13] for related problems. SYSTEMS IN VARYING DOMAINS WITH VARYING BOUNDARY CONDITIONS 51 2. Notations and definitions The minimum and the maximum of two numbers a,bare respectively denoted by a∧b,a∨b. The scalar product of two matrices A, B ∈RM×Nwill be denoted by A:B. For a Borel set B⊂RNand a Borel measure μin B,wedenotebyLq μ(B), 1 ≤q≤+∞, the usual Lebesgue spaces with respect to the measure μ.Ifμis the Lebesgue measure, we use the standard notation Lq(B). For every Lipschitz open set O⊂RN,wedenotebyW1,q(O), 1 ≤q≤+∞, the usual Sobolev spaces. We recall that, since we are assuming OLipschitz, the elements of W1,q(O)haveatraceon∂O and then, they are defined in O.Moreover,C∞(O)isdenseinW1,q(O)ifq<+∞. For every subset Υ of ∂O, we define W1,q Υ(O) as the closure in W1,q(O) of the functions in C∞(O) which vanish in a neighborhood of Υ. In the case Υ = ∂O, we write W1,q 0(O) instead of W1,q Υ(O). Along the paper we denote by pa fixed number such that p≥2. Also we consider a bounded Lipschitz open set Ω ⊂RN,N≥2, and a bounded open set ˆ Ω, such that Ω⊂ˆ Ω. We denote by P:W1,p(Ω) −→ W1,p 0(ˆ Ω) a bounded linear operator such that P(u)=uin Ω,∀u∈W1,p(Ω).(2.6) This operator is also chosen bounded from W1,p(Ω)∩L∞(Ω) into W1,p 0(ˆ Ω)∩L∞(ˆ Ω) and such that it transforms nonnegative functions into nonnegative functions. The existence of this extension operator is guaranteed because Ω is Lipschitz (see e.g.[17]). For a Lipschitz open set ω⊂Ω,we denote Sω={ϕ:ϕ∈W1,∞(ω),ϕ= 0 in a neighborhood of ∂ω ∩Ω}. Also, we define the bounded linear operators Zω:W1,p ∂ω∩Ω(ω)→W1,p(Ω), Qω:W1,p ∂ω∩Ω(ω)→W1,p 0(ˆ Ω) as Zω(u)=uin ω 0in Ω\ω, Qω=P◦Zω. When u=(u1,...,u M) is vectorial, we denote P(u)=(P(u1),...,P(uM)),Z ω(u)=(Zω(u1),...,Z ω(uM)), Qω(u)=(Qω(u1),...,Q ω(uM)). For E⊂ˆ Ωand1<p<+∞,thep-capacity of Ein ˆ Ω, denoted by Cp(E), is defined by Cp(E)=infˆ Ω |∇u|pdx:u∈W1,p 0(ˆ Ω),u≥1 a.e. in a neighborhood of E. This definition depends on ˆ Ω, however the sets of p-capacity zero are independent of ˆ Ω. We say that a property P(x) holds quasi everywhere (abbreviated as q.e.) in a set B⊂ˆ Ωifitholdsfor all x∈B\N,withCp(N)=0. A function u:ˆ Ω−→ Ris said to be quasi continuous if for every ε>0thereexistsasetB⊂ˆ Ω, with Cp(B)<ε, such that the restriction of uto ˆ Ω\Bis continuous. It is well known (see e.g.[14,15,21]) that every u∈W1,p(ˆ Ω) has a quasi continuous representative. We shall always identify u∈W1,p(ˆ Ω) with this quasi continuous representative. A subset Oof ˆ Ω is said to be quasi open if for every ε>0thereexistsB⊂ˆ Ω, with Cp(B)<ε, such that O∪Bis open. 52 C. CALVO-JURADO, J. CASADO-D´ IAZ AND M. LUNA-LAYNEZ Following [8,9], for every Borel subset Bof ˆ Ω, we denote by Mp 0(B) the class of all non negative Borel measures μin Bwhich vanish on Borel sets of p-capacity zero and satisfy the following condition μ(E)=inf{μ(O∩B): Oquasi open, E⊂O⊂ˆ Ω},∀E⊂BBorel. (2.7) We will denote by a:Ω×RM×N−→ RM×Na Carath´eodory function such that there exist two positive constants α,γ,andr∈Lp p−2(Ω) satisfying a(x, 0) = 0 a.e. x∈Ω,(2.8) (a(x, ξ1)−a(x, ξ2)) : (ξ1−ξ2)≥α|ξ1−ξ2|p,∀ξ1,ξ 2∈RM×N,a.e. x∈Ω,(2.9) |a(x, ξ1)−a(x, ξ2)|≤r(x)+γ(|ξ1|+|ξ2|)p−2|ξ1−ξ2|,∀ξ1,ξ 2∈RM×N,a.e. x∈Ω.(2.10) Observe that these hypotheses imply in particular that there exist β>0andh∈Lp(Ω) such that a(x, ξ):ξ≥α|ξ|p,∀ξ∈RM×N,a.e. x∈Ω,(2.11) |a(x, ξ)|≤h(x)+β|ξ|p−1,∀ξ∈RM×N,a.e. x∈Ω.(2.12) For every n∈N, we will also consider μn∈M p 0(Ω) and Fn: Ω ×RM−→ RMsuch that Fn(·,s)μn-measurable, ∀s∈RM,(2.13) Fn(x, 0) = 0,μ n-a.e. x∈Ω,(2.14) (Fn(x, s1)−Fn(x, s2))(s1−s2)≥α|s1−s2|p,∀s1,s 2∈RM,μ n-a.e. x∈Ω,(2.15) |Fn(x, s1)−Fn(x, s2)|≤γ(|s1|+|s2|)p−2|s1−s2|,∀s1,s 2∈RM,μ n-a.e. x∈Ω.(2.16) Thus, for β>0asabove,wehave Fn(x, s)s≥α|s|p,∀s∈RM,μ n-a.e. x∈Ω,(2.17) |Fn(x, s)|≤β|s|p−1,∀s∈RM,μ n-a.e. x∈Ω.(2.18) Remark 2.1. To simplify the exposition we have considered p≥2. The case 1 <p<2 can be studied similarly after proper modification on the hypotheses on aand Fn. We can also consider hypotheses less restrictive than (2.10)and(2.16), assuming that a(x, ξ)andFn(x, s) are locally H¨older continuous with respect to ξand srespectively. We denote by Ca generic constant which does not depend on nand can change from line to line. We denote by Om,n and Ongeneric sequences of real numbers which can change from line to line and satisfy lim m→∞ lim n→∞ |Om,n|=0,lim n→∞ On=0. SYSTEMS IN VARYING DOMAINS WITH VARYING BOUNDARY CONDITIONS 53 3. Homogenization result In this section we state the main results of the paper, relative to the homogenization problem (1.5). Theorem 3.1. Let a,μnand Fnbe in the conditions of Section 2. Then, there exist a subsequence of n,still denoted by n,ameasureμ∈M p 0(Ω) andafunctionF: Ω ×RM−→ RM,with F(·,s)μ-measurable, ∀s∈RM,(3.19) F(x, 0) = 0,μ-a.e. x∈Ω,(3.20) |F(x, s2)−F(x, s1)|≤C1(|s1|+|s2|)p(p−2) p−1|s2−s1|1 p−1,∀s1,s 2∈RM,μ-a.e. x∈Ω,(3.21) (F(x, s2)−F(x, s1))(s2−s1)≥C2|s2−s1|p,∀s1,s 2∈RM,μ-a.e. x∈Ω,(3.22) such that the following homogenization result holds: Let ω⊂Ωbe a Lipschitz open set and consider a sequence gn∈Lp(ω)Mwhich converges weakly in Lp(ω)Mto a function g, a sequence Gn∈Lp(ω)M×Nwhich converges strongly in Lp(ω)M×NtoafunctionGandasequenceun∈W1,p(ω)M∩Lp μn(ω\∂ω ∩Ω) which satisfies unW1,p(ω)M∩Lp µn(ω\∂ω∩Ω)M≤C, (3.23) and ⎧ ⎨ ⎩ω a(x, Dun):Dv dx+ω Fn(x, un)vdμn=ω gnvdx+ω Gn:Dv dx ∀v∈W1,p ∂ω∩Ω(ω)M∩Lp μn(ω)M. (3.24) Then, every cluster point uof unintheweaktopologyofW1,p(ω)Msatisfies ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u∈W1,p(ω)M∩Lp μ(ω\∂ω ∩Ω)M ω a(x, Du):Dv dx+ω F(x, u)vdμ=ω gv dx+ω G:Dv dx ∀v∈W1,p ∂ω∩Ω(ω)M∩Lp μ(ω)M. (3.25) Moreover, the measure μcan be taken independently of a. The proof of this result is carried on in Section 6. Todoit,inSection5we consider a bounded open set ˆ ΩwithΩ⊂ˆ Ωandthen,forunand uas in the statement of Theorem 3.1, we estimate the difference between unand the corrector with limit urelative to the homogenization problem for the operator v→ −div |∇v|p−2∇v+|v|p−2vdμnin ˆ Ω with Dirichlet conditions. The properties of this corrector will be recalled in Section 4. As a consequence we obtain some estimates for Dun(Lems. 5.2 and 5.3) which allow us to prove (Prop. 5.4) the existence of μ∈M p 0(Ω) and T∈Lp μ(ω\∂ω ∩Ω)Msuch that ubelongs to Lp μ(ω\∂ω ∩Ω)Mand ω a(x, Du):Dv dx+ω Tvdμ=ω gv dx+ω G:Dv dx, for every v∈W1,p ∂ω∩Ω(ω)M∩Lp μ(ω)M(see [8,10]). The estimates obtained in Section 5prove that Tis of the form F(x, u(x)) (estimate (5.60)), but for a function Fonly defined on the set of pairs (x, v(x)) such that vis the limit of some sequence vnin the conditions of the sequence unwhich appears in the statement of Theorem 3.1. We will prove in Lemma 6.1 that the set of such functions vis large enough to allow us to define Fin the whole of Ω×RMand then to conclude Theorem 3.1. We will also prove in Section 6the following consequence of Theorem 3.1. 54 C. CALVO-JURADO, J. CASADO-D´ IAZ AND M. LUNA-LAYNEZ Theorem 3.2. Under the same assumptions that Theorem 3.1, the following results hold: (i) For every λ>0, the unique solution unof ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ un∈W1,p ∂ω∩Ω(ω)M∩Lp μn(ω)M ω a(x, Dun):Dv dx+λω |un|p−2unvdx+ω Fn(x, un)vdμn= ω gnvdx+ω Gn:Dv dx ∀v∈W1,p ∂ω∩Ω(ω)M∩Lp μn(ω)M, (3.26) converges weakly in W1,p ∂ω∩Ω(ω)Mand strongly in W1,q ∂ω∩Ω(ω)M,1≤q<p, to the unique solution uof ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u∈W1,p ∂ω∩Ω(ω)M∩Lp μ(ω)M ω a(x, Du):Dv dx+λω |u|p−2uv dx+ω F(x, u)vdμ=ω gv dx+ω G:Dv dx ∀v∈W1,p ∂ω∩Ω(ω)M∩Lp μ(ω)M. (3.27) (ii) Assume that there exists (a Poincar´e’s constant) CP>0such that υLp(ω)≤CP∇υp Lp(ω)N+υp Lp µn(ω)1 p,∀υ∈W1,p ∂ω∩Ω(ω)∩Lp μn(ω),∀n∈N.(3.28) Then, the unique solution unof ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ un∈W1,p ∂ω∩Ω(ω)M∩Lp μn(ω)M ω a(x, Dun):Dv dx+ω Fn(x, un)vdμn=ω gnvdx+ω Gn:Dv dx ∀v∈W1,p ∂ω∩Ω(ω)M∩Lp μn(ω)M, (3.29) converges weakly in W1,p ∂ω∩Ω(ω)Mand strongly in W1,q ∂ω∩Ω(ω)M,1≤q<p, to the unique solution uof ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u∈W1,p ∂ω∩Ω(ω)M∩Lp μ(ω)M ω a(x, Du):Dv dx+ω F(x, u)vdμ=ω gv dx+ω G:Dv dx ∀v∈W1,p ∂ω∩Ω(ω)M∩Lp μ(ω)M. (3.30) Remark 3.3. As we said in the Introduction, the homogenization of problem (3.29) gives in particular the homogenization of problem (1.1). For this, given a sequence of Lipschitz open sets Ωn⊂Ω and a sequence Γn⊂ ∂Ω∩Ωn, we define a sequence of measures μn∈M p 0(Ω) by (1.4). Then, problem (1.1), understood in the variational form ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ un∈W1,p(Ωn)M,u n=0q.e. on∂Ωn\Γn Ωn a(x, Dun):Dv dx=Ωn gnvdx+Ωn Gn:Dv dx ∀v∈W1,p(Ωn)M,v=0q.e. on∂Ωn\Γn, is equivalent to problem (3.29)withω=Ωand(forexample)Fn(x, s)=|s|p−2s. An interesting particular case is when Ωn=Ωforeveryn∈N,i.e. when we have a nonlinear homogenization problem in a fixed bounded open set Ω ⊂RN, where we impose Dirichlet and Neumann conditions in varying subsets of the boundary. SYSTEMS IN VARYING DOMAINS WITH VARYING BOUNDARY CONDITIONS 55 In this case, by Proposition 4.4 the measure μis supported on ∂Ω, and thus, if μis sufficiently smooth, the limit problem (3.30) is equivalent to the Fourier-Robin problem −div (a(x, Du)−G)=gin Ω (a(x, Du)−G)ν+F(x, u)μ=0 on∂Ω. Remark 3.4. Similarly to the homogenization of nonlinear Dirichlet problems in varying domains (see e.g.[5]), some properties on aand Fnare inherited by F.Namely,wehave: (i) If a(x, ξ) is linear with respect to ξand Fn(x, s) is linear with respect to s, for every n∈N(so, p=2), then, F(x, s) is linear with respect to s. (ii) If aand Fn,n∈N, satisfy the homogeneity assumption a(x, λξ)=|λ|p−2λa(x, ξ),∀λ∈R,∀ξ∈RM×N,a.e. x∈Ω, Fn(x, λs)=|λ|p−2λFn(x, s),∀λ∈R,∀s∈RM,μ n-a.e. x∈Ω, then Falso satisfies F(x, λs)=|λ|p−2λF (x, s),∀λ∈R,∀s∈RM,μ-a.e. x∈Ω. The proof of these results is analogous to the corresponding one of Theorems 8.1 and 8.5 in [5] and follows from the fact that the functions qm nof Lemma 6.1 satisfy (λq1+τq2)m n=λ(q1)m n+τ(q2)m n,∀q1,q 2∈QM,∀λ, τ ∈R, if we assume (i), and (λq)m n=λ(q)m n,∀q∈QM,∀λ∈R, if we assume (ii). Thus the functions Tqdefined by Lemma 6.1 satisfy Tλq1+τq2=λTq1+τTq2,∀q1,q 2∈QM,∀λ, τ ∈R, if we assume (i), and Tλq =|λ|p−2λTq,∀q∈QM,∀λ∈R, if we assume (ii). 4. Preliminaries In this section we recall some results related to the homogenization of the p-Laplace operator with Dirichlet boundary conditions in varying domains. From them we will obtain other results we will use later. Throughout this section, we consider a sequence ˆμn∈M p 0(ˆ Ω) and we denote by wnthe solution of the problem ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ wn∈W1,p 0(ˆ Ω) ∩Lp ˆμn(ˆ Ω) ˆ Ω |∇wn|p−2∇wn∇vdx+ˆ Ω |wn|p−2wnvdˆμn=ˆ Ω vdx ∀v∈W1,p 0(ˆ Ω) ∩Lp ˆμn(ˆ Ω). (4.31) The following result has been proved in [8,10]. 56 C. CALVO-JURADO, J. CASADO-D´ IAZ AND M. LUNA-LAYNEZ Proposition 4.1. Let wnbe the sequence defined by (4.31).Thenwnis nonnegative q.e. in ˆ Ωand its norm in W1,p 0(ˆ Ω) ∩L∞(ˆ Ω) ∩Lp ˆμn(ˆ Ω) is bounded. Up to a subsequence, there exists a nonnegative function w∈ W1,p 0(ˆ Ω) ∩L∞(ˆ Ω), such that wnconverges weakly to win W1,p 0(ˆ Ω),stronglyinW1,q 0(ˆ Ω),1≤q<p,and weakly-∗in L∞(ˆ Ω). Moreover, there exists a unique measure ˆμ∈M p 0(ˆ Ω) such that wis the solution of ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ w∈W1,p 0(ˆ Ω) ∩Lp ˆμ(ˆ Ω) ˆ Ω |∇w|p−2∇w∇vdx+ˆ Ω |w|p−2wv dˆμ=ˆ Ω vdx ∀v∈W1,p 0(ˆ Ω) ∩Lp ˆμ(ˆ Ω). (4.32) The interest of wnis that for a function ψsmooth enough, the sequence wnψprovides a corrector with limit wψ relative to the homogenization problem for the operator v→−div |∇v|p−2∇v+|v|p−2vdμnin ˆ Ω with Dirichlet conditions [8,10]. The following properties of wn,wand ˆμhave been proved in [8,10](seealso[5]). Proposition 4.2. The sequence of solutions wnof (4.31), the function wand the measure ˆμgiven by Proposition 4.1 satisfy (a) For every Borel set B⊂ˆ Ωwith Cp(B∩{w=0})>0, it holds ˆμ(B)=+∞. (b) The set {wϕ :ϕ∈C∞ c(ˆ Ω)}is dense in W1,p 0(ˆ Ω) ∩Lp ˆμ(ˆ Ω). (c) For every ψ, ϕ ∈W1,p(ˆ Ω) ∩L∞(ˆ Ω), we have lim n→∞ ˆ Ω |∇((wn−w)ψ)|pϕdx+ˆ Ω |wnψ|pϕdˆμn=ˆ Ω |wψ|pϕdˆμ. (4.33) (d) If vnis a sequence in W1,p(ˆ Ω) which converges weakly in W1,p(ˆ Ω) to a function v, then it holds lim inf n→∞ ˆ Ω |∇(vn−v)|pdx+ˆ Ω |vn|pdˆμn≥ˆ Ω |v|pdˆμ. (4.34) In particular, if vnLp ˆµn(ˆ Ω) is bounded, the function vis in Lp ˆμ(ˆ Ω). Remark 4.3. We will apply the previous results to the sequence ˆμndefined by (see Sect. 2) ˆμn(B)=μn(B∩Ω),∀B⊂ˆ ΩBorel.(4.35) From Proposition 4.2, every function v∈Lp ˆμn(ˆ Ω) vanishes q.e. on {wn=0}. So, although in Section 2we have considered Fndefined on ˆ Ω×RM, only its values in {wn>0}×RMare relevant. In the present paper, we are interested in a sequence of measures ˆμnhaving their supports contained in a fixed closed set (see (4.35)). We will use the following result. Proposition 4.4. If there exists a compact set K⊂ˆ Ωsuch that supp (ˆμn)⊂K, for every n∈N, then the measure ˆμgiven by Proposition 4.1 also satisfies supp (ˆμ)⊂K. Proof. Since −div(|∇wn|p−2∇wn)=1inˆ Ω\K,andwnconverges weakly to win W1,p 0(ˆ Ω) and strongly in W1,q 0(ˆ Ω), 1 ≤q<p, we deduce that −div(|∇w|p−2∇w)=1 in ˆ Ω\K. (4.36) Then, taking in (4.32)v=wϕ,withϕ∈C∞ c(ˆ Ω\K), we get ˆ Ω |w|pϕdˆμ=0,∀ϕ∈C∞ c(ˆ Ω\K).(4.37) SYSTEMS IN VARYING DOMAINS WITH VARYING BOUNDARY CONDITIONS 57 On the other hand, from w≥0q.e.inˆ Ω, (4.36) and the strong maximum principle for the p-Laplace operator (see e.g. [18]), we deduce w>0inˆ Ω\K. Together with (4.37), this implies that the support of ˆμis contained in K. Better than Proposition 4.2 (b) and (d), we will use the following results. Proposition 4.5. Assume that the support of the measure ˆμgiven by Proposition 4.1 is contained in Ωand let ωbe a Lipschitz open subset of Ω.Then,theset Dω={wϕ :ϕ∈S ω∩C∞(ω)},(4.38) is dense in W1,p ∂ω∩Ω(ω)∩Lp ˆμ(ω). Proof. First of all, we remark that for every uin W1,p ∂ω∩Ω(ω)∩Lp ˆμ(ω), there exist a sequence un∈W1,p ∂ω∩Ω(ω)∩ Lp ˆμ(ω)andOn⊂ˆ Ωopen,with∂ω ∩Ω⊂On, such that un=0q.e. inOn∩ωand unconverges to u in W1,p ∂ω∩Ω(ω)∩Lp ˆμ(ω). For this purpose, we use that by definition of W1,p ∂ω∩Ω(ω), there exists a sequence On⊂ˆ Ω open, with ∂ω ∩Ω⊂Onand ψn∈C∞(ω), with ψn=0inOn∩ω, such that ψnconverges to uin W1,p(ω). Then we take un=(ψn∧u+)+−((−ψn)∧u−)+. In order to prove Proposition 4.5, it is then enough to show that for every u∈W1,p ∂ω∩Ω(ω)∩Lp ˆμ(ω)such that there exits an open set O⊂ˆ Ωwith∂ω ∩Ω⊂O,u=0q.e. inO∩ω,andforeveryε>0, there exists ϕ∈C∞ c(ˆ Ω) which vanishes in a neighborhood of ∂ω ∩Ω such that u−wϕW1,p(ω)∩Lp ˆµ(ω)<ε. (4.39) Using a regularization by convolution, it is enough to prove (4.39)forϕin W1,p 0(ˆ Ω) ∩L∞(ˆ Ω) which vanishes on a neighborhood of ∂ω ∩Ω. Given u,Oand εas above, we observe that Zω(u)isinW1,p(Ω) ∩Lp ˆμ(Ω) and, since supp(ˆμ)⊂Ω, we have that Qω(u)∈W1,p 0(ˆ Ω) ∩Lp ˆμ(ˆ Ω) (P,Zωand Qωare defined in Sect. 2). Thus,takinganopensetOwith ∂ω ∩Ω⊂O,O⊂Oand a function ψ∈C∞(ˆ Ω), with ψ=1in ˆ Ω\O,ψ=0inO, the function u∗=Qω(u)ψ is in W1,p 0(ˆ Ω) ∩Lp ˆμ(ˆ Ω), vanishes in Oand is equal to uq.e. in ω. We define μ∗∈M p 0(ˆ Ω) by μ∗(B)=ˆμ(B)ifCp(B∩O)=0 +∞if Cp(B∩O)>0,∀B⊂ˆ ΩBorel, and we observe that since u∗=0inOthen u∗∈Lp μ∗(ˆ Ω). From Proposition 4.2(b) applied to μ∗we derive that taking w∗as the solution of ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ w∗∈W1,p 0(ˆ Ω) ∩Lp μ∗(ˆ Ω) ˆ Ω |∇w∗|p−2∇w∗∇vdx+ˆ Ω |w∗|p−2w∗vdμ∗=ˆ Ω vdx ∀v∈W1,p 0(ˆ Ω) ∩Lp μ∗(ˆ Ω), (4.40) there exists ϕ∗∈C∞ c(ˆ Ω) such that u∗−w∗ϕ∗W1,p 0(ˆ Ω)∩Lp µ∗(ˆ Ω) <ε. 64 C. CALVO-JURADO, J. CASADO-D´ IAZ AND M. LUNA-LAYNEZ the strong convergence of Gn−Hnin Lp(ω)M×N,(5.51)and(5.59), we have ω [a(x, Dun)−a(x, Dzn)] : D(un−zn)ϕpdx+ω [Fn(x, un)−Fn(x, vn)](un−zn)ϕpdμn= ω (g−h)(u−z)ϕpdxω (G−H):D((u−z)ϕp)dx+ −pω [a(x, Du)−a(x, Dz)] : [(u−z)⊗∇ϕ]ϕp−1dx+On =ω [a(x, Du)−a(x, Dz)] : D(u−z)ϕpdx+ω (T−T)(u−z)ϕpdμ+On,(5.62) which by (5.46) (applied to unand zn) implies ω (T−T)(u−z)ϕpdμ= lim n→∞ ω [a(x, D(un−u)) −a(x, D(zn−z))] : D(un−zn−u+z)ϕpdx +ω [Fn(x, un)−Fn(x, zn)](un−zn)ϕpdμn.(5.63) Using in the right-hand side of this inequality (2.9), (2.15)and(4.42), we deduce ω (T−T)(u−z)ϕpdμ≥αω |D(un−u−zn+z)|pϕpdx+ω |un−zn|pϕpdμn ≥α Pω |u−z|pϕpdμ+On,∀ϕ∈Sω,ϕ≥0.(5.64) From the measures derivation theorem, this proves (5.61).  6. Proof of the main results In this section we prove that there exists a μ-Carath´eodory function F:¯ Ω×RM→RMsuch that the function Tgiven by Proposition 5.4 satisfies T(x)=F(x, u(x)) μ-a.e. in ω\∂ω ∩Ω. We start with the following lemma. Its proof is completely similar to the one of Theorem 6.9 in [5], and thus we omit it. Lemma 6.1. We consider a subsequence of nsuch that there exists the measure μdefined in the beginning of Section 5. Then, up to another subsequence, we have that for every q∈QMand every m∈N, the solution qm nof ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ qm n∈W1,p(Ω)M∩Lp μn(Ω)M Ω a(x, Dqm n):Dv dx+Ω Fn(x, qm n)vdμn=mΩ [|wnq|p−2wnq−|qm n|p−2qm n]vdx ∀v∈W1,p(Ω)M∩Lp μn(Ω)M, (6.65) converges to a function qmweakly in W1,p(Ω)M. This function satisfies that there exists Tm q∈Lp(Ω) such that ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ qm∈W1,p(Ω)M∩Lp μ(Ω)M Ω a(x, Dqm):Dv dx+Ω Tm qvdμ=mΩ [|wq|p−2wq −|qm|p−2qm]vdx ∀v∈W1,p(Ω)M∩Lp μ(Ω)M. (6.66) SYSTEMS IN VARYING DOMAINS WITH VARYING BOUNDARY CONDITIONS 65 When mtends to infinity, the sequence qmconverges to wq strongly in W1,p(Ω)M∩Lp μn(Ω)Mand the sequence Tm qconverges strongly in Lp μ(Ω)MtoafunctionTq. Definition 6.2. We consider the subsequence of ngiven by Lemma 6.1. Then, we define F:Ω×QM→RMby F(x, q)=Tq(x),∀q∈QM,μ-a.e. x∈Ω. By Lemma 6.1,(5.53), (5.60)and(5.61), it is easy to show that for every q1,q 2∈QMand μ-a.e. x∈Ω, we have F(x, 0) = 0,(6.67) |F(x, q2)−F(x, q1)|≤C1(|q1|+|q2|)p(p−2) p−1|q2−q1|1 p−1w(x)p−1,(6.68) (F(x, q2)−F(x, q1))(q2−q1)≥C2|q2−q1|pw(x)p−1.(6.69) Using (6.68), we can extend by continuity Fto Ω×RM. Then, we define F: Ω ×RM→RMby F(x, s)=⎧ ⎨ ⎩ Fx, s w(x)if w(x)>0 |s|p−2sif w(x)=0. Thanks to Lemma 6.1,Proposition5.4 and estimate (5.60) we can now prove Theorem 3.1. ProofofTheorem3.1.We take unand uas in the statement of the theorem. By Proposition 5.4,thereexists T∈Lp μ(ω)Msuch that uis a solution of (5.51). Applying (5.60), with zreplaced by qm,wehave |T−Tm q|≤C(|u|+|qm|)p(p−2) p−1|u−qm|1 p−1μ-a.e. in ω\∂ω ∩Ω,(6.70) and therefore, taking the limit as mtends to infinity, we obtain |T−F(x, wq)|≤C(|u|+|wq|)p(p−2) p−1|u−wq|1 p−1μ-a.e. in ω\∂ω ∩Ω. Thus, for every simple function φ(x)=l i=1 siχBi(x), with si∈RM,BiBorel, we have |T−F(x, wφ)|≤C(|u|+|wφ|)p(p−2) p−1|u−wφ|1 p−1μ-a.e. in ω\∂ω ∩Ω. Finally, taking in this inequality φas a sequence φnsuch that wφnconverges μ-a.e. to uin ω\∂ω ∩Ω (the existence of such sequence is an easy consequence of Prop. 4.5) and passing to the limit in nthanks to the continuity of Fwith respect to its second variable, we get T=F(x, u)μ-a.e. in ω\∂ω ∩Ω. This proves (3.25) thanks to (5.51) and the fact that the functions of W1,p ∂ω∩Ω(ω) are zero q.e. on ∂ω ∩Ω.  Proof of Theorem 3.2. Let us just prove (ii). The proof of (i) is much simpler. Thanks to (3.28) and the assumptions on aand Fn,problem(3.29) has a unique solution. Moreover, taking unas test function in (3.29) and using (3.28), (2.11)and(2.17)wegetthatunsatisfies (3.23). Thus, from W1,p ∂ω∩Ω(ω)closed,Theorem3.1 and Proposition 5.1, we deduce that there exists a subsequence of unwhich converges weakly in W1,p ∂ω∩Ω(ω)Mand strongly in W1,q ∂ω∩Ω(ω)M,1≤q<p,toasolutionuof (3.30). If we prove that uis unique, then the whole sequence unwill converge to uand the proof of (ii) will be finished. For this purpose, it is enough to prove that the measure μalso satisfies (3.28)andthen,from(2.9)and(3.22), we will get the uniqueness of solution of problem (3.30). 66 C. CALVO-JURADO, J. CASADO-D´ IAZ AND M. LUNA-LAYNEZ Let υbe in W1,p ∂ω∩Ω(ω)∩Lp μ(ω). Using Proposition 4.5,weconsiderψm∈S ωsuch that wψmconverges to υ in W1,p ∂ω∩Ω(ω)∩Lp μ(ω). From (3.28), for every m∈N,wehave ω |wnψm|pdx≤Cp Pω |∇(wnψm)|pdx+ω |wnψm|pdμn ≤Cp Pω (|∇(wnψm)|p−|∇((wn−w)ψm)|p)dx +Cp Pˆ Ω∇(wn−w)Qω(ψm) pdx+ˆ Ω |wnQω(ψm)|pdˆμn. Since wnconverges strongly to win W1,q(ˆ Ω), 1 ≤q<p, reasoning similarly to the proof of (5.46), we have that |∇(wnψm)|p−|∇((wn−w)ψm)|pconverges strongly to |∇(wψm)|pin L1(ω) and thus ω (|∇(wnψm)|p−|∇((wn−w)ψm)|p)dx→ω |∇(wψm)|pdx, whereas from (4.33)andμ=ˆμin supp(ˆμ)=Ωwehave ˆ Ω |∇((wn−w)Qω(ψm))|pdx+ˆ Ω |wnQω(ψm)|pdˆμn→ω |wψm|pdμ. Thus, using also the semicontinuity of the norm in Lp(ω), we get ω |wψm|pdx≤Cp Pω |∇(wψm)|pdx+ω |wψm|pdμ. Taking then mtending to infinity we derive ω |υ|pdx≤Cp Pω |∇υ|pdx+ω |υ|pdμ,∀υ∈W1,p ∂ω∩Ω(ω)∩Lp μ(ω). This finishes the proof of (ii). Acknowledgements. This work has been partly supported by the projects MTM2005-04914 of the Ministerio de Eduaci´on y Ciencia of Spain and FQM-309 of the Junta de Andaluc´ıa. References [1] C. Calvo-Jurado and J. Casado-D´ıaz, The limit of Dirichlet systems for variable monotone operators in general perforated domains. J. Math. Pures Appl. 81 (2002) 471–493. [2] C. Calvo-Jurado, J. Casado-D´ıaz and M. Luna-Laynez, Homogenization of elliptic problems with the Dirichlet and Neumann conditions imposed on varying subsets. Math. Meth. Appl. Sci. 30 (2007) 1611–1625. [3] J. Casado-D´ıaz, Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains. J. Math. Pures Appl. 76 (1997) 431–476. [4] J. Casado-D´ıaz, Homogenization of Dirichlet problems for monotone operators in varying domains. Proc. Roy. Soc. Edinburgh A 127 (1997) 457–478. [5] J. Casado-D´ıaz and A. Garroni, Asymptotic behavior of nonlinear elliptic systems on varying domains. SIAM J. Math. Anal. 31 (2000) 581–624. SYSTEMS IN VARYING DOMAINS WITH VARYING BOUNDARY CONDITIONS 67 [6] D. Cionarescu and F. Murat, Un terme ´etrange venu d’ailleurs, in Nonlinear partial differential equations and their applications, Coll`egedeFranceseminar,Vols.IIandIII,H.Br´ezis and J.-L. Lions Eds., Research Notes in Math. 60 and 70,Pitman, London (1982) 98–138 and 154–78. [7] G. Dal Maso and A. Defranceschi, Limits of nonlinear Dirichlet problems in varying domains. Manuscripta Math. 61 (1988) 251–278. [8] G. Dal Maso and A. Garroni, New results on the asymptotic behaviour of Dirichlet problems in perforated domains. Math. Mod. Meth. Appl. Sci. 3(1994) 373–407. [9] G. Dal Maso and U. Mosco, Wiener-criterion and Γ-convergence. Appl. Math. Optim. 15 (1987) 15–63. [10] G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for the Dirichlet problems in perforated domains with homogeneous monotone operators. Ann. Sc. Norm. Sup. Pisa 7(1997) 765–803. [11] G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains. Ann. Inst. H. Poincar´eAnal.NonLin´eaire 21 (2004) 445–486. [12] G. Dal Maso, A. Garroni and I.V. Skrypnik, A capacitary method for the asymptotic analysis of Dirichlet problems for monotone operators. J. Anal. Math. 71 (1997) 263–313. [13] A. Damlamian and T. Li, Boundary homogenization for elliptic problems. J. Math. Pures Appl. 66 (1987) 351–361. [14] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992). [15] H. Federer and W.P. Ziemer, The Lebesgue set of a function whose distribution derivaties are p-th power sumable. Indiana Univ. Math. J. 22 (1972) 139–158. [16] J. Leray and J.-L. Lions, Quelques r´esultats de Visik sur les probl`emes elliptiques non lin´eaires par les m´ethodes de MintyBrowder. Bull. Soc. Math. France 93 (1965) 97–107. [17] J. Necas, Les m´ethodes directes en th´eorie des ´equations elliptiques. Masson, Paris (1967). [18] J. Serrin, Local behaviour of solutions of quasilinear equations. Acta Math. 111 (1964) 302–347. [19] I.V. Skrypnik, Asymptotic behaviour of solutions of nonlinear elliptic problems in perforated domains. Mat. Sb. 184 (1993) 67–90. [20] I.V. Skrypnik, Averaging of quasilinear parabolic problems in domains with fine-grained boundary. Diff. Equations 31 (1995) 327–339. [21] W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Berlin (1989).