Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains

Rendiconti di Matematica, Serie VII Volume 16, Roma (1996), 387-413 Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains J. CASADO-D´ IAZ Riassunto: In un lavoro del 1982, D. Cioranescu e F. Murat hanno considerato il problema soddisfatto dal limite udi una successione undi soluzioni di 0−∆un=fin Ωn, un=0 su ∂Ωn, dove Ωn`e una successione di insiemi aperti che sono contenuti in un fissato insieme aperto limitato Ω. Tale studio richiede di imporre numerose ipotesi sulla successione Ωn. I risultati di D. Cioranescu e F. Murat sono stati estesi in seguito da N. Labani e C. Picard al caso del p-Laplaciano. Nel presente lavoro, noi dimostriamo che le ipotesi su Ωnpossono essere ridotte a un’unica ipotesi, la seguente: esiste una successione zn∈W1,p(Ω)che vale zero su Ω\Ωne che converge a 1 debolmente in W1,p(Ω). Il problema di omogeneizzazione nel caso generale in cui non si fa alcuna ipotesi sulla successione Ωn`e stato risolto da G. Dal Maso e U. Mosco con metodi di Γ-convergenza e recentemente G. Dal Maso e A. Garroni hanno risolto il problema generale con metodi prossimi a quelli usati da D. Cioranescu e F. Murat. Abstract: In a paper of 1982, D. Cioranescu and F. Murat considered the problem satisfied by the limit uof the sequence unsolution of 0−∆un=fin Ωn, un=0 on ∂Ωn, where Ωnis a sequence of open sets which are contained in a fixed bounded open set Ω. In order to make this, they imposed several hypotheses about the sequence Ωn. Their results were later extended to the p-Laplacian operator by N. Labani and C. Picard. In 388 J. CASADO-D´ IAZ [2] the present paper, we prove that these hypotheses may be reduced to the following one: There exists a sequence zn∈W1,p(Ω)which is zero in Ω\Ωnand which converges weakly to 1 in W1,p(Ω). Indeed, G. Dal Maso and U. Mosco have solved the above homogenization problem in the general case in which we do not make any hypothesis about Ωnusing Γconvergence methods and recently, G. Dal Maso and A. Garroni have also solved this general problem by a method close to the one used by D. Cioranescu and F. Murat. – Introduction The contents of this paper is concerned with the study of the homogenization problem (0.1) &−∆pun=fin D#(Ωn), un∈W1,p 0(Ωn), where Ωndenotes a sequence of open sets contained in a fixed bounded open set Ω⊂IR N,pis a given number with 1 <p<+∞,fis an element of W−1,p"(Ω) and ∆pis the p-lapacian operator defined by −∆pu=−div |∇u|p−2∇u. The solutions unof (0.1) are bounded in W1,p 0(Ω) (we identify unwith its extension by zero to Ω\Ωn) and so, there exists a subsequence which converges weakly to a function uin W1,p 0(Ω). The homogenization problem is to find the equation satisfied by the function uand also, to give an approximate representation of the gradient of unin the strong topology of Lp(Ω) using the function uand some explicit auxiliary functions (corrector problem). In the case p= 2, this homogenization problem has been solved by D. Cioranescu and F. Murat in [6] (see also [15]) assuming the following hypotheses about the sequence Ωn: Key Words and Phrases: Homogenization – Perforated domains A.M.S. Classification: 35B40 This work has been partially supported by the Project EURHomogenization, Contrat SC1-CT91-0732 of the Program SCIENCE of the Commission of the European Communities and by the Project PB92-0696 of the DGICYT of Spain. [3] Existence of a sequence satisfying Cioranescu-Murat etc. 389 There exists a sequence of functions wnand a distribution µsatisfying wn∈H1(Ω),(H1) wn= 0 in Ω\Ωn,(H2) wn31 in H1(Ω),(H3) µ∈W−1,∞(Ω),(H4)                    for any sequence vnand any vsatisfying vn3vin H1(Ω),v n= 0 in Ω\Ωn, and for any ϕ∈D(Ω),we have !Ω∇wn∇(ϕvn)→1µ, ϕv2. (H5) It was then proved in [15] that hypothesis (H4) can be weakened in µ∈ H−1(Ω). These hypotheses are justified in [6] by several examples, the most typical case being when Ωnis obtained by removing from Ωthe union of closed balls of radius ε N N−2 ncentered at the centers of cubes of size εnwhich cover IRNperiodically. With these assumptions, D. Cioranescu and F. Murat prove that the limit uof the sequence unsatisfies (0.2) &−∆u+µu =fin D#(Ω), u∈H1 0(Ω). Moreover, they prove that when ubelongs to W1,∞(Ω), un−wnuconverges strongly to zero in H1 0(Ω). Their method has been generalized by N. Labani and C. Picard to the case of the p-Laplacian in [16]. The goal of the present paper is to prove the existence of a sequence wnand of a distribution µsatisfying properties similar to (H1),..., (H5) for the p-Laplacian, starting from the only assumption that the wnsatisfy (H1), (H2), (H3) (with H1(Ω) replaced by W1,p(Ω), see Theorem 2.1). In this case, µis no more in W−1,∞(Ω) but only in the set of bounded nonnegative measures vanishing on the sets of p-capacity zero. We will also generalize the results obtained in [6] to this new context. In particular we obtain an improvement of the corrector result given in 390 J. CASADO-D´ IAZ [4] [6], proving that it is enough to have u∈W1,p 0(Ω)∩L∞(Ω) in order to have the strong convergence of un−wnuin W1,p 0(Ω) (see Theorem 4.1). The method we use here can be extended to the case of general quasilinear problems, but for our purpose it is enough to study the p-laplacian, because in the general case we will obtain better results reasoning by comparison. This will be carried out in [3] and [4], where we will use the results obtained in the present paper to solve on the other hand general monotone problems, and on the other one quasi-linear problems with a perturbation term, which is quadratic with respect to the gradient, respectively. Hypothesis (H1), (H2), (H3) mean that Ω\Ωnare small enough. In the limit, Ωnfills the whole of Ω. Indeed, the general problem in which we do not assume any hypothesis about the sequence Ωnhas been solved by G. Dal Maso and U. Mosco ([9], [10]) in the linear case and by G. Dal Maso and A. Defranceschi [7] in the monotone case, using Γ-convergence methods. To use Γ−convergence, the problem has to be written as a minimization problem, which is not always possible for a general quasi-linear problem. Also there is no corrector result in these papers, while this is essential for us, in order to apply the comparison method which allows us to study more general equations. On the other hand, G. Dal Maso and A. Garroni [8] have recently used a different argument to study the linear case without any hypotheses about Ωn. This argument, which is close to the one used in [6] (the main difference lies in the definition of the function wn), does not need symmetry assumptions and gives a corrector result. The method used in [8] has been extended by G. Dal Maso and F. Murat [11], [12] to the case of monotone operators assuming a homogenity hypothesis for the operator. Using the corrector result which appears in [8] or [11], [12] and the comparison method we are able to solve in [5] the case of general monotone systems without homogeneity hypothesis, and without any hypothesis on the open sets. 1 – Preliminaries and notation Throughout the present paper: -Ωdenotes a bounded open set contained in IRN. -Lp(Ω, dµ), 1 ≤p<+∞, denotes the space of functions with power p integrable in Ωwith respect to the measure µ. [5] Existence of a sequence satisfying Cioranescu-Murat etc. 391 -L∞(Ω, dµ) denotes the space of functions essentialy bounded in Ωwith respect to the measure µ. - If the measure µis the Lebesgue measure we abbreviate the notation using Lp(Ω) or L∞(Ω). - For 1 ≤p≤+∞we denote p#the conjugate exponent of pdefined by 1 p+1 p"= 1. -D(Ω) denotes the space of infinitely derivable functions with compact support contained in Ω. The dual space of D(Ω) is the space of distributions which will be denoted by D#(Ω). -W1,p(Ω) denotes the usual Sobolev space of functions of Lp(Ω) with distributional derivatives in Lp(Ω). -W1,p 0(Ω) denotes the closure of D(Ω) in W1,p(Ω). For 1 ≤p<+∞, the dual space of W1,p 0(Ω) will be denoted by W−1,p"(Ω) -∇denotes the gradient operator. - div denotes the divergence operator. -∆ pdenotes the p-Laplacian operator, i.e. ∆pu= div |∇u|p−2∇u. -χSdenotes the characteristic function of the set S, i.e. χS(x)=1if x∈S,χS(x) = 0 if x-∈ S. -Mb(Ω) denotes the space of bounded Borel measures in Ω. - cap(S) denotes the p-capacity of the set S⊂Ωwith respect to Ω(pwill be specified by the context), which is defined in the following way: If Sis a compact set, the capacity of Sis defined by cap(S) = inf1!Ω|∇ϕ|p:ϕ∈D(Ω),ϕ≥χS2. If Sis an open set, the capacity of Sis defined by cap(S) = sup1cap(K): K⊂S, K compact2. If Sis an arbitrary set, the capacity of Sis defined by cap(S) = inf1cap(G): S⊂G⊂Ω,Gopen2. It is well known (see e.g. [14], [13], [19]) that a function of W1,p(Ω) has a representative which is defined quasi-everywhere, i.e. except on a set of zero p-capacity. In the whole of the present paper we will select this representative for any function u∈W1,p(Ω). 392 J. CASADO-D´ IAZ [6] -Mp b(Ω) denotes the set of nonnegative bounded Borel measures vanishing on the sets of zero capacity. By the above mentioned result, the functions of W1,p(Ω) are µ-measurable for µ∈M p b(Ω). We have (1.1) W1,p(Ω)∩L∞(Ω)5→L∞(Ω, dµ)5→Lq(Ω, dµ) for any 1 ≤q<+∞, where the last inclusion holds since µbelongs to Mb(Ω). - For r≥0, Tr:IR4→ IR is the truncation function definded by Tr(s)=     rif s≥r sif −r≤s≤r −rif s≤−r, while Rr:IR4→ IR is the function defined by Rr(s)=           0 if |s|≤r 2 2 r|s|−1 if r 2≤|s|≤r 1 if |s|≥r. The following properties of the function ξ∈IR N4→|ξ|p−2ξ∈IR N will be often used: For any ξ,η∈IR N, we have for p≥2 (|ξ|p−2ξ−|η|p−2η)(ξ−η)≥22−p|ξ−η|p,(1.2) --|ξ|p−2ξ−|η|p−2η--≤(p−1)3|ξ|p−2+|η|p−24|ξ−η|(1.3) and for 1 ≤p≤2 (|ξ|p−2ξ−|η|p−2η)(ξ−η)≥(p−1) |ξ−η|2 |ξ|2−p+|η|2−p,(1.4) --|ξ|p−2ξ−|η|p−2η--≤22−p|ξ−η|p−1.(1.5) [7] Existence of a sequence satisfying Cioranescu-Murat etc. 393 Inequality (1.4) will be used in the following form: given u, v ∈ W1,p(Ω), 1 <p<2, then (1.6) !Ω|∇(u−v)|p≤ ≤1 (p−1)p 2!Ω5(|∇u|p−2∇u−|∇v|p−2∇v)(∇u−∇v)6p 2· ·5|∇u|2−p+|∇v|2−p6p 2≤ ≤2p−1 (p−1)p 27!Ω (|∇u|p−2∇u−|∇v|p−2∇v)(∇u−∇v)8p 2· ·7!Ω3|∇u|p+|∇v|p482−p 2. 2 – The main result and its proof Let us consider a fixed bounded open set Ω⊂IR Nand a sequence of open sets Ωncontained in Ω. In the whole of the present paper, the functions of W1,p 0(Ωn) will be always extended by zero outside of Ωnand therefore considered defined as in the whole of Ω. Theorem 2.1 establishes the existence of a sequence satisfying properties analogous to those of the sequence wndefined in [6]. Theorem 2.1. Assume that there exists a sequence zn∈W1,p(Ω), with zn=0in Ω\Ωn, which converges weakly in W1,p(Ω) (1 <p<∞)to a function z. Assume also that there exists a constant ρ>0with z≥ρ quasi-everywhere in Ω. Then, there exists a subsequence of n(which will still denoted by nto simplify the notation), a sequence of functions wn and a measure µsatisfying wn∈W1,p(Ω),(P1) wn=0in Ω\Ωn,(P2) 0≤wn≤1,(P3) wn31weakly in W1,p(Ω)and strongly in W1,q(Ω),1≤q < p,(P4) 394 J. CASADO-D´ IAZ [8] µ∈M p b(Ω),(P5) &for any ϕ∈W1,p 0(Ω)∩L∞(Ω)we have 9Ω|∇wn|pϕ→9Ωϕdµ, (P6)            for any vn∈W1,p 0(Ωn)and for any v∈W1,p 0(Ω)such that vn3vin W1,p 0(Ω),we have v∈L1(Ω, dµ)and !Ω|∇wn|p−2∇wn∇vn→!Ω v dµ. (P7)            for any vn∈W1,p(Ω),such that vn=0in Ω\Ωnand vn30in W1,p(Ω),we have !Ω|∇wn|p−2∇wn∇vn→0. (P8) Moreover, if the properties (P1), (P2), ...,(P8) hold true for the same subsequence nand for some ˆwnand ˆµ, then we have &ˆµ=µ ˆwn−wn→0in W1,p(Ω)strongly. Remark 2.1. It will be proved below that in Property (P7), vactually belongs to Lp(Ω,dµ), (see Theorem 3.1). Remark 2.2. The sequence wnwill provide us a corrector for the homogenization problem &−∆pun=fin D#(Ωn), un∈W1,p 0(Ωn). Remark that the behaviour of wnis similar to of a sequence ˜wnwhich satisfies (µdoes not belong in general to W−1,p"(Ω) and therefore such that ˜wndoes not exist in general)      ˜wn∈W1,p(Ωn),˜wn= 0 in Ω\Ωn, −∆p˜wn=µin D#(Ωn), ˜wn31 in W1,p(Ω). [9] Existence of a sequence satisfying Cioranescu-Murat etc. 395 Compare this one with the homogenization problem &−div (An∇un)=fin D#(Ω), un∈H1 0(Ω) where An∈L∞(Ω)N×Nare such that there exist α,β>0 with αI≤ An≤βIin the sense of the matrices. The idea of L. Tartar (see [18]) to construct a corrector for this problem is to consider for any iwith 1≤i≤Na sequence wi nsuch that      wi n∈H1(Ω), −div (An∇wi n)=−div (A0∇xi)=−div (A0ei) in D#(Ω), wi n3x iin H1(Ω), where A0will be the H-limit of An. Proof of Theorem 2.1. The proof of Theorem 2.1 will be divided in nine steps. Step 1: Definition of the subsequence n, of the sequence wnand of µ; the subsequence wnsatisfies (P1), (P2) and (P3). Proof. Define A=:{vn}:vn∈W1,p(Ω) : vn= 0 in Ω\Ωn,v n31 in W1,p(Ω);, α= inf 0lim inf n→∞ !Ω|∇vn|p:{vn}∈A <. The set Ais not empty since the sequence vn=T+ ρ(zn) ρbelongs to A. For any k∈IN, consider a sequence {vk n}∈Asuch that lim inf n→∞ !Ω|∇vk n|p<α+1 k. Defining ˜vk nas ˜vk n=T+ 1(vk n), we have that {˜vk n}∈A,0≤˜vk n≤1, and lim inf n→∞ !Ω|∇˜vk n|p≤lim inf n→∞ !Ω|∇vk n|p<α+1 k. Rellich-Kondrachov’s compactness and Lebesgue’s dominated convergence theorems imply that the embedding W1,p(Ω)∩L∞(Ω)5→Lq(Ω) 402 J. CASADO-D´ IAZ [16] By Step 7, the second term of the right-hand side tends to zero as ntends to infinity for kfixed, while by Step 3 the first term tends to zero when nand ktend to infinity. This proves that (2.6) lim k→∞ lim sup n→∞ ---!Ω|∇wn|p−2∇wn∇vn−!Ω Tk(v)dµ---=0, which implies that 9ΩTk(v)dµ is bounded independently of k. Hence, from the Beppo Levi’s monotone convergence theorem, v∈L1(Ω, dµ) and Tk(v) converges strongly to vin L1(Ω, dµ). To prove (2.5) it is now enough to write lim sup n→∞ ---!Ω|∇wn|p−2∇wn∇vn−!Ω v dµ---≤lim k→∞ !Ω|v−Tk(v)|dµ+ + lim k→∞ lim sup n→∞ ---!Ω|∇wn|p−2∇wn∇vn−!Ω Tk(v)dµ--- which is zero. Step 9: Uniqueness. Proof. Let ˆwnbe another sequence which together with some ˆµ satisfies properties (P1), (P2),..., (P8). Using Property (P8) with vn= wn−ˆwn, we have !Ω (|∇wn|p−2∇wn−|∇ˆwn|p−2∇ˆwn)(∇wn−∇ˆwn)→0. Inequality (1.2) or (1.6) then gives the strong convergence to zero of wn− ˆwnin W1,p(Ω). On the other hand by (P6) and this strong convergence, for any function ϕ∈D(Ω), we have !Ω ϕdˆµ= lim n→∞ !Ω|∇ˆwn|pϕ= lim n→∞ !Ω|∇wn|pϕ=!Ω ϕdµ, i.e. µ=ˆµ. [17] Existence of a sequence satisfying Cioranescu-Murat etc. 403 3 – Semicontinuity We will now improve the result already obtained in (P7) and to prove that every function v∈W1,p 0(Ω) which is the weak limit in W1,p 0(Ω) of a sequence vn∈W1,p 0(Ωn) belongs to Lp(Ω,dµ). We will also obtain a semicontinuity result for the energy. Theorem 3.1. Consider a sequence vn∈W1,p 0(Ωn)which converges weakly in W1,p 0(Ω)to a function v. Then (3.1) v∈Lp(Ω, dµ)and lim inf n→∞ !Ω|∇vn|p≥!Ω|∇v|p+!Ω|v|pdµ. Remark 3.1 Theorem 3.1 can be deduced in a straightforward way from the Γ-convergence result given in [7], where the result is actually stronger because no hypothesis on the sequence Ωnis imposed there. This general result can also be obtained by a method close to the present one (see [5]) which also allows one to obtain the corrector result of Theorem 4.1 in a framework where no hypothesis is imposed on the sequence Ωn. Remark 3.2 A consequence of Theorem 3.1 is that for a given v∈ W1,p 0(Ω) which is not zero µ-almost everywhere there does not exist any sequence vn∈W1,p 0(Ωn) which converges strongly in W1,p 0(Ω) to v. Proof. As for Theorem 2.1, the proof of Theorem 3.1 will be divided in several steps which are interesting in themselves and which establish in particular that for z∈W1,p(Ω)∩L∞(Ω), the sequence wnzsatisfies properties similar to those of wn. Step 1: If z∈W1,p 0(Ω)∩L∞(Ω), then we have (3.2) !Ω|∇(wnz)|p→!Ω|∇z|p+!Ω|z|pdµ. Proof. Write (3.3) !Ω|∇(wnz)|p=!Ω|z∇wn+wn∇z|p= =!Ω5|z∇wn+wn∇z|p−|z∇wn|p6+!Ω|z∇wn|p. 404 J. CASADO-D´ IAZ [18] In the first integral of the right-hand side of (3.3) by Lagrange’s theorem we have --|z∇wn+wn∇z|p−|z∇wn|p--≤ ≤p5|z∇wn+wn∇z|p−1+|z∇wn|p−16|wn∇z|. Note that the right-hand side is equi-integrable. Therefore the left hand side, which converges almost everywhere, converges strongly in L1(Ω) to |∇z|p. For the second integral of the right-hand side of (3.3), we use (P6), obtaining that !Ω|z|p|∇wn|p→!Ω|z|pdµ. This completes the proof of (3.2). Step 2: Consider a sequence vn∈W1,p 0(Ωn) which converges weakly in W1,p 0(Ω) to a function v. Then for any function ϕ∈W1,p(Ω)∩L∞(Ω), we have (3.4) !Ω ϕ|∇wn|p−2∇wn∇vn→!Ω vϕdµ. Proof. Suppose first that vnis also bounded in L∞(Ω). Then, using (P7), Rellich-Kondrachov’s compactness theorem and the pointwise convergence of ∇wn, we have (3.5)        lim n→∞ !Ω ϕ|∇wn|p−2∇wn∇vn= lim n→∞ !Ω|∇wn|p−2∇wn∇(vnϕ) −lim n→∞ !Ω vn|∇wn|p−2∇wn∇ϕ=!Ω vϕdµ. In the general case, (vnis not in L∞(Ω)), by the above proved, we have that !Ω ϕ|∇wn|p−2∇wn∇Tk(vn)→!Ω Tk(v)ϕdµ, ∀k∈IN and then, by the Lebesgue’s dominated convergence theorem, we get lim k→∞ lim n→∞ !Ω ϕ|∇wn|p−2∇wn∇Tk(vn)=!Ω vϕdµ. [19] Existence of a sequence satisfying Cioranescu-Murat etc. 405 To finish the proof of (3.4) it is then enough to prove that lim k→∞ lim n→∞ !Ω ϕ|∇wn|p−2∇wn∇(vn−Tk(vn)) = 0 or, using H¨older’s inequality and that ϕ∈L∞(Ω), that (3.6) lim k→∞ lim n→∞ !|vn|≥k|∇wn|p=0. Applying (2.2) with vk n=wnRk(vn), we obtain (3.7) lim n,k→∞ "!Ω|∇wn|pRk(vn)+!Ω wnR# k(vn)|∇wn|p−2∇wn∇vn#=0. But ---!Ω wnR# k(vn)|∇wn|p−2∇wn∇vn---≤2 k---!k≥|vn|≥k/2|∇wn|p−2∇wn∇vn---, which, using that wnand vnare bounded in W1,p 0(Ω), implies lim n,k→∞ !Ω R# k(vn)|∇wn|p−2∇wn∇vn=0 and therefore, by (3.7) we have lim k→∞ lim sup n→∞ !Ω|∇wn|pRk(vn)=0. which gives (3.6), by Rk(vn)≥χ{|vn|≥k}. Step 3: Let z∈W1,p(Ω)∩L∞(Ω). Then for any sequence vn∈ W1,p 0(Ωn) which converges weakly in W1,p 0(Ω) to a function v, we have (3.8) !Ω|∇(wnz)|p−2∇(wnz)∇vn→!Ω|∇z|p−2∇z∇v+!Ω|z|p−2zvdµ. Proof. As in Step 1, it is easy to see, using (1.3) or (1.5), that |∇(wnz)|p−2∇(wnz)−|z∇wn|p−2z∇wn 406 J. CASADO-D´ IAZ [20] is equi-integrable in Lp"(Ω) and converges pointwise to |∇z|p−2∇z, and thus converges strongly in Lp"(Ω)Nto |∇z|p−2∇z. Therefore lim n→∞ !Ω|∇(wnz)|p−2∇(wnz)∇vn= =!Ω|∇z|p−2∇z∇v+ lim n→∞ !Ω|z|p−2z|∇wn|p−2∇wn∇vn. To obtain (3.8) it is enough to use Step 2 with ϕ=|z|p−2z. Step 4: Proof of (3.1). Using the convexity inequality |ξ|p≥|η|p+p|η|p−2η(ξ−η),∀ξ,η∈IR N, we have for k∈IN !Ω|∇vn|p≥!Ω|∇(wnTk(v))|p+ +p!Ω|∇(wnTk(v))|p−2∇3wnTk(v)43∇vn−∇(wnTk(v))4. Using (3.2) and (3.8) (with z=Tk(v)) and then &|∇Tk(v)|p−2∇Tk(v)(∇v−∇Tk(v)) = 0 a.e. in Ω |Tk(v)|p−2Tk(v)(v−Tk(v)) ≥0µ-a.e. in Ω we obtain lim inf n→∞ !Ω|∇vn|p≥!Ω|∇Tk(v)|p+!Ω|Tk(v)|pdµ+ +p!Ω|∇Tk(v)|p−2∇Tk(v)(∇v−∇Tk(v))+ +p!Ω|Tk(v)|p−2Tk(v)(v−Tk(v)) dµ ≥ ≥!Ω|∇Tk(v)|p+!Ω|Tk(v)|pdµ. The Beppo Levi’s monotone convergence theorem then implies that v belongs to Lp(Ω,dµ) and that Tk(v) converges in Lp(Ω,dµ) to v. Using also the convergence of Tk(v) to vin W1,p 0(Ω) we obtain (3.1). [21] Existence of a sequence satisfying Cioranescu-Murat etc. 407 4 – Corrector In the case where the lim inf in (3.1) is actually a limit, and where the inequality is actually an equality, we have the following corrector result, which provides an approximate representation of the gradient of vnin the strong topology of Lp(Ω)N. Theorem 4.1. Consider a sequence vn∈W1,p 0(Ωn)which converges weakly in W1,p 0(Ω)to a function v. Assume that (4.1) lim n→∞ !Ω|∇vn|p=!Ω|∇v|p+!Ω|v|pdµ. Then we have (4.2) lim k→∞ lim sup n→∞ !Ω|∇(vn−wnTk(v))|p=0. Remark 4.1 In particular, when v∈W1,p(Ω)∩L∞(Ω), (4.2) implies that vn−wnv→0 in W1,p 0(Ω). Proof. Step 1. In this step we do not use hypothesis (4.1). We will prove that (4.3) lim sup k→∞ lim sup n→∞ !Ω|∇vn|p−2∇vn3∇vn−∇(wnTk(v))4=0 implies that (4.2) holds true. Indeed, If p≥2, by (1.2), (3.8) (see Step 4 in the proof of Theorem 3.1) and using that Tk(v) converges strongly to vin W1,p(Ω)∩Lp(Ω,dµ), we have (4.4)                              lim sup k→∞ lim sup n→∞ !Ω|∇vn−∇(wnTk(v))|p≤ ≤2p−2$lim sup k→∞ lim sup n→∞ !Ω|∇vn|p−2∇vn3∇vn−∇(wnTk(v))4− −lim k→∞ lim n→∞!Ω|∇(wnTk(v))|p−2∇(wnTk(v))3∇vn−∇ (wnTk(v))4%= =2 p−2lim sup k→∞ lim sup n→∞ !Ω|∇vn|p−2∇vn(∇vn−∇(wnTk(v))) 408 J. CASADO-D´ IAZ [22] If 1 <p<2, we use (1.6) which gives lim sup k→∞ lim sup n→∞ !Ω|∇(vn−wnTk(v))|p≤2p−1 (p−1)p 2$lim sup k→∞ lim sup n→∞ !Ω 3|∇vn|p−2∇vn−|∇(wnTk(v))|p−2∇(wnTk(v))43∇vn−∇(wnTk(v))4%p 2· ·$lim sup k→∞ lim sup n→∞ !Ω3|∇vn|p+|∇(wnTk(v))|p4%2−p 2. By (3.2) and the strong convergence of Tk(v) to vin W1,p 0(Ω)∩Lp(Ω, dµ) we have lim sup k→∞ lim sup n→∞ !Ω|∇(wnTk(v))|p=!Ω|∇v|p+!Ω|v|pdµ < +∞. Applying then (3.8) as in (4.4), we get (4.5)          lim sup k→∞ lim sup n→∞ !Ω|∇(vn−wnTk(u))|p≤ ≤C7lim sup k→∞ lim sup n→∞ !Ω|∇vn|p−2∇vn3∇vn−∇(wnTk(v))48p 2. In both case 2 ≤p<+∞and 1 <p<2, we have proved that (4.3) implies (4.2). Step 2. Define B=:{zn}:zn∈W1,p 0(Ωn): zn3vin W1,p 0(Ω);, and note that (4.1) and Theorem 3.1 imply that the sequence vnsatisfies lim n→∞ !Ω|∇vn|p= min 1lim inf n→∞ !Ω|∇zn|p:{zn}∈B2. Then (see Steps 2 and 3 in the proof of Theorem 2.1) vnsatisfies the following property: (4.6)          ∀zk n∈W1,p 0(Ωn) such that zk n30 in W1,p 0(Ω) when n, k→∞, we have lim n,k→∞ !Ω|∇vn|p−2∇vn∇zk n=0. [23] Existence of a sequence satisfying Cioranescu-Murat etc. 409 Step 3. In order to prove (4.3), we cannot apply directly (4.6) since vn−wnTk(v) is not in general bounded in W1,p 0(Ω) independently of n and k. To bypass this difficulty, we write lim sup k→∞ lim sup n→∞ !Ω|∇vn|p−2∇vn3∇vn−∇(wnTk(v))4≤ ≤lim sup k→∞ lim sup n→∞ !Ω|∇vn|p−2∇vn(∇vn−∇Tk(vn))+ + lim sup k→∞ lim sup n→∞ !Ω|∇vn|p−2∇vn3∇Tk(vn)−∇(wnTk(v))4. Now, by (4.6), lim sup k→∞ lim sup n→∞ !Ω|∇vn|p−2∇vn(∇vn−∇Tk(vn)) ≤ ≤lim sup k,n→∞ !Ω|∇vn|p−2∇vn(∇vn−∇Tk(vn)) = 0 while for kfixed (4.6) with zk nindependent of kimplies lim sup n→∞ !Ω|∇vn|p−2∇vn3∇Tk(vn)−∇(wnTk(v))4=0. This proves (4.3). 5 – Homogenization As an application of the results established in the previous sections, let us make a brief study of the homogenization problem for the pLaplacian in perforated domains. (This result will be used in [3] and [4] to obtain similar homogenization results for more general quasi-linear problems). For a general result without any hypothesis about the sequence Ω\Ωn, see [7], [12]. Theorem 5.1. Consider a sequence Ωn(whose existence is given in Theorem 2.1) for which there exist wnand µsatisfying (P1), (P2),... , 410 J. CASADO-D´ IAZ [24] (P8). Then the following homogenization result holds: For any f∈ W−1,p"(Ω), the solution unof the problem (5.1) &−∆pun=fin D#(Ωn), un∈W1,p 0(Ωn), converges weakly in W1,p 0(Ω)to the solution uof the problem (5.2) &−∆pu+|u|p−2uµ =fin D#(Ω), u∈W1,p 0(Ω)∩Lp(Ω, dµ), which is equivalent to the variational formulation (5.3)              u∈W1,p 0(Ω)∩Lp(Ω, dµ), !Ω|∇u|p−2∇u∇v+!Ω|u|p−2uv dµ =1f,v2, ∀v∈W1,p 0(Ω)∩Lp(Ω, dµ). The sequence unalso satisfies (5.4) lim n→∞ !Ω|∇un|p=!Ω|∇u|p+!Ω|u|pdµ and so the corrector result of Theorem 4.1 applies (5.5) lim k→∞ lim sup n→∞ !Ω|∇(un−wnTk(u))|p=0. Proof. Using unas test function in (5.1) we prove that the sequence unis bounded in W1,p 0(Ω). We thus can extract a subsequence of unwhich converges weakly in W1,p 0(Ω) to some u. By Theorem 3.1 ubelongs to W1,p 0(Ω)∩Lp(Ω, dµ). For the sake of simplicity, let us still denote this subsequence by un. (Indeed, we will prove that usatisfies (5.3), which has a unique solution, and thus uniqueness will imply the convergence of the whole sequence un). [25] Existence of a sequence satisfying Cioranescu-Murat etc. 411 Let us first prove the corrector result (5.5). For this, we use un− wnTk(u) as test function in (5.1). This gives lim k→∞ lim sup n→∞ !Ω|∇un|p−2∇un3∇un−∇(wnTk(u))4= = lim k→∞ lim sup n→∞ 1f,un−wnTk(u)2=0. This is analogous to (4.3), and by the proof of Step 1 of Theorem 4.1 this implies (5.5). Now, for ϕ∈D(Ω) we take wnϕ∈W1,p 0(Ωn) as test function in (5.1). We obtain !Ω|∇un|p−2∇un∇(wnϕ)=1f,wnϕ2. The right hand side satisfies 1f,wnϕ2→1f,ϕ2. On the other hand, using the corrector result (5.5), then (3.8) and then that Tk(u) converges strongly to uin W1,p 0(Ω)∩Lp(Ω, dµ), we have lim n→∞ !Ω|∇un|p−2∇un∇(wnϕ)= = lim k→∞ lim n→∞ !Ω|∇(wnTk(u))|p−2∇(wnTk(u)) ∇(wnϕ)= =!Ω|∇u|p−2∇u∇ϕ+!Ω|u|p−2uϕ. Therefore (5.6) !Ω|∇u|p∇u∇ϕ+!Ω|u|puϕdµ =1f,ϕ2,∀ϕ∈D(Ω). By density, (5.6) holds for any ϕ∈W1,p 0(Ω)∩Lp(Ω, dµ) and so uis the (unique) solution of (5.3). To obtain (5.4), use (5.5) or more directly, take unas test function in (5.1).