Asymptotic behavior of nonlinear elliptic systems on varying domains
Abstract
We consider a monotone operator of the form Au = −div(a(x, Du)), with Ω ⊆ Rn and a : Ω×MM×N → MM×N , acting on W1,p 0 (Ω, RM). For every sequence (Ωh) of open subsets of Ω and for every f ∈ W−1,p0 (Ω, RM), 1/p+ 1/p0 = 1, we study the asymptotic behavior, as h → +∞, of the solutions uh ∈ W1 0 (Ωh, RM) of the systems Auh = f in W−1,p0 (Ωh, RM), and we determine the general form of the limit problem.
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ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS ON VARYING DOMAINS∗ JUAN CASADO DIAZ†AND ADRIANA GARRONI‡ SIAM J. MATH. ANAL.c °2000 Society for Industrial and Applied Mathematics Vol. 31, No. 3, pp. 581–624 Abstract. We consider a monotone operator of the form Au =−div(a(x, Du)), with Ω ⊆Rn and a:Ω×M M×N→M M×N, acting on W1,p 0(Ω,RM). For every sequence (Ωh) of open subsets of Ω and for every f∈W−1,p0(Ω,RM), 1/p+1/p0= 1, we study the asymptotic behavior, as h→+∞, of the solutions uh∈W1 0(Ωh,RM) of the systems Auh=fin W−1,p0(Ωh,RM), and we determine the general form of the limit problem. Key words. homogenization, perforated domains, Dirichlet systems AMS subject classifications. 35B27, 35B40 PII. S0036141097329627 1. Introduction. In this paper we study the asymptotic behavior of the solutions of elliptic nonlinear systems, of Mequations and Nvariables, on varying domains with Dirichlet boundary conditions. Namely, let Ω be a bounded open subset of RNand let 1 <p<+∞. We regard Aas a vector monotone operator defined from W1,p(Ω,RM)toW −1,p0(Ω,RM), mapping u∈W1,p(Ω,RM)in Au =−div¡a(x, Du)∈W−1,p0(Ω,RM). The function a:Ω×M M×N7→ MM×Nis a Carath´eodory function which satisfies the standard assumptions of strong monotonicity and H¨older continuity (see conditions (i)–(iv) in section 5). Given an arbitrary sequence of open subsets Ωnof Ω and given an arbitrary f∈W−1,p0(Ω,RM), we consider the solutions unof the following systems with Dirichlet boundary condition un∈W1,p 0(Ωn,RM),Au n =fin Ωn.(1.1) We set un= 0 in Ω\Ωnand regard the sequence (un) as a sequence in W1,p 0(Ω,RM). Our results describe the asymptotic behavior of (un)asn→∞and characterize the limit function as the solution of a variational “limit problem.” The main result of this paper is given by the following compactness theorem. Theorem 1.1. Let Ωnbe an arbitrary sequence of open subsets of Ω. Then there exist a subsequence of Ωn, still denoted by Ωn, a measure µin the class Mp 0(Ω) of positive Borel measures not charging set of p-capacity zero, and a vector function F:Ω×R M→R M , such that for every f∈W−1,p0(Ω,RM)the sequence (un)of solutions of problems (1.1) converges weakly in W1,p 0(Ω,RM)to the solution uof the variational problem u∈W1,p 0(Ω,RM)∩Lp µ(Ω,RM), ZΩ a(x, Du)Dv dx +ZΩ F(x, u)vdµ =hf,vi ∀v∈W1,p 0(Ω,RM)∩Lp µ(Ω,RM). (1.2) ∗Received by the editors November 5, 1997; accepted for publication (in revised form) September 9, 1998; published electronically February 24, 2000. http://www.siam.org/journals/sima/31-3/32962.html †Departamento de ecuaciones differenciales, C. Tarfia Sin, 41012 Sevilla, Spain (jcasado@ numer.us.es). ‡Dipartimento di Matematica, Universit`a “La Sapienza,” Piazzale A. Moro 2, 00185 Roma, Italy ([email protected]). 581 Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
582 JUAN CASADO DIAZ AND ADRIANA GARRONI By Lp µ(Ω,RM) we denote the standard Lpspaces with respect to the measure µ. Note that in this general case the usual “extra term” is given by RΩF(x, u)vdµ. The problem considered in the present paper has been studied, under various degree of generality, by many authors, with several approaches and in different frameworks. Most of the known results are given under assumptions involving the geometry or the capacity of the closed sets Ω \Ωn, which in general imply that the measure µ in the limit problem is a Radon measure (see, for instance, [20], [22], and [7] for the linear case, and [24], [25], [26], [27], [21], and [3], for monotone operators). The class Mp 0(Ω) described above also includes measures which take the value +∞on large families of sets; in this way, Dirichlet problems in subdomains of Ω can be written in the form (1.2) for a suitable choice of µ. Indeed, it is easy to see that, if Eis a closed subset of Ω and the measure µis defined by µ(B)=0ifC p (B∩E)=0 +∞otherwise, (1.3) for every Borel subset Bof Ω, where Cpdenotes the p-capacity, then problem (1.2) is equivalent to u∈W1,p 0(Ω \E,RM),Au=fin Ω \E. In view of the latter remark, the compactness result above will be proved in a more general formulation (see Theorem 6.4) for a sequence of problems of the type un∈W1,p 0(Ω,RM)∩Lp µn(Ω,RM), ZΩ a(x, Dun)Dv dx +ZΩ Fn(x, un)vdµ n=hf,vi ∀v∈W1,p 0(Ω,RM)∩Lp µn(Ω,RM), (1.4) which for a suitable choice of (µn)inM p 0 (Ω) reduce to (1.1), and includes also Schr¨odinger systems with positive oscillating potentials. A further motivation for the study of problem (1.4) is given by the recent applications to a relaxed formulation of some optimal design problems (see, for instance, [2]). The compactness result in the setting of (1.4) was first proved for the scalar case M= 1, using Γ-convergence techniques, in [13] and [14] when p= 2 and Ais a symmetric linear elliptic operator, and in [10] if Ais p−1 homogeneous, under the assumption that it is the subdifferential of a convex functional. These results were generalized using Tartar’s energy method in [11] for the general scalar linear case, and subsequently for the nonlinear case under an assumption of homogeneity of order p−1 for the operator A(see [15] and [16]). In these cases the extra term which appears in the limit problem is proved to be of the form RΩ|u|p−2uv , dµ. The case of systems is much less investigated. Previous results have been obtained only in the framework of linear symmetric elliptic operators in [18]. Further reference on this subject can be found in the book [9] and in the papers [11] and [16], which contain a wide bibliography. Our result provides a description of the limiting behavior of sequences of Dirichlet boundary value problems not only for monotone operators of Leray–Lions type, but also covering the case of systems related to linear possibly nonsymmetric operators or nonlinear homogeneous operators, which were not included in previous results. The proof follows the lines of [3], where Theorem 1.1 is obtained in the scalar case, under Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 583 some additional assumptions on the sequence (Ωn) which imply in particular that the measure µin the limit problem is bounded. The idea of the proof is essentially to compare our sequence of problems with a sequence of model problems for which the behavior is known (for instance scalar problems with the p-Laplace operator). In section 2 we recall some preliminary results and notation and in section 3 we state some known results in the study of the asymptotic behavior of scalar problems with the p-Laplace operator. Section 4 is dedicated to a careful study of the behavior of a sequence of “correctors” for the p-Laplace operator, as introduced in [16]. In section 5 we state the problem and we prove, following the line of [1] and [16], that a sequence of solutions of problems (1.4) which converges weakly in W1,p(Ω,RM) converges also strongly in W1,r(Ω,RM) for every r<p(see Proposition 5.4). In section 6 we prove the compactness result. In section 7 we prove a correctors result, in the general context of nonlinear monotone vector operators. Indeed, the sequence of gradients (Dun)ofsolutions of problems (1.1) converges to Du a priori only weakly in Lpby Theorem 1.1. Hence to obtain a strong convergence it is necessary to add a further sequence which depends only on the limit function u. The construction of such a sequence is provided by Theorem 7.1 and is new also in the case of linear systems. For previous correctors results, see, e.g., [7], [11], [3]. Section 8 is devoted to the analysis of some special cases. In particular we obtain a simpler form for the extra term and for the correctors in the linear case and in the homogeneous case, in agreement with the previous scalar results. The structure of the extra term is proved to depend only on the asymptotic behavior of the function a(x, ξ) for ξ→∞(see section 9). In the last section our result is applied also to the treatment of asymptotic problems in a class of pseudomonotone operators. The extension to the general pseudomonotone operators for the scalar case can be found in [5]. Throughout the paper we treat in detail only 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 operator A. The changes in the proofs can easily be performed using Proposition 3.2 of [17]. 2. Notation and preliminaries. Let Nand Mbe two positive integers, N≥2; by MM×Nwe denote the space of M×Nreal matrices. Let Ω be a bounded open subset of RN. We denote by W1,p 0(Ω,RM) and W1,p(Ω,RM), 1 <p<+∞, the usual Sobolev spaces (of RM-valued functions) and by W−1,p0(Ω,RM), 1/p0+1/p = 1, the dual of W1,p 0(Ω,RM). By W1,p c(Ω,RM) and W1,p loc (Ω,RM) we denote respectively the space of all functions in W1,p(Ω,RM) with compact support in Ω and the space of all functions which belong to W1,p(U, RM) for every open set U⊂⊂ Ω. When p= 2 we adopt the standard notation H1(Ω,RM), H1 0(Ω,RM), and H−1(Ω,RM). By Lp µ(Ω,RM), 1 ≤p≤+∞, we denote the usual Lebesgue space with respect to the measure µ.Ifµis the Lebesgue measure, we use the standard notation Lp(Ω,RM). When we consider space of scalar functions (M= 1), we omit RMin the notations above. Let u∈W1,p(Ω) and k∈R.ByT k uwe shall denote the truncation at the level kwhich is the function in W1,p(Ω) defined by Tku=(−k)∧u∨k. Let Abe an open set in RN,u:A→RMand a, b ∈R; we shall denote by {a≤|u|≤b} Athe set of all x∈Asuch that a≤|u(x)|≤b. When A= Ω we shall omit Ω in the notation above. We shall use om,n (respectively, on) to denote a sequence of real numbers such that limm→∞ limn→∞ om,n = 0 (respectively, limn→∞ on= 0). Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
584 JUAN CASADO DIAZ AND ADRIANA GARRONI If E⊆Ω, the (harmonic)p-capacity of Ein Ω, denoted by Cp(E), is defined as the infimum of ZΩ|Du|pdx over the set of all functions u∈W1,p 0(Ω) such that u≥1 almost everywhere (a.e.) in a neighborhood of E. We say that a property P(x) holds p-quasi everywhere (abbreviated as p-q.e.) in a set Eif it holds for all x∈Eexcept for a subset Nof Eof p-capacity zero. The expression µ-almost everywhere (abbreviated as µ-a.e.) refers, as usual, to the analogous property for a Borel measure µ. A function u:Ω →R Mis said to be p-quasi continuous if for every ε>0 there exists a set A⊆Ω, with Cp(A)<ε, such that the restriction of uto Ω\Ais continuous. It is well known that every u∈W1,p(Ω,RM)hasap-quasi continuous representative, which is uniquely defined up to a set of p-capacity zero. In the following we shall always identify uwith its p-quasi continuous representative, so that the pointwise values of a function u∈W1,p(Ω,RM) are defined p-q.e. in Ω. A subset Aof Ω is said to be p-quasi open in Ω if for every ε>0 there exists an open subset Aεof Ω, with Cp(Aε)<ε, such that A∪Aεis open. It is easy to see that if a function u:Ω→Ris p-quasi continuous, then the set {u>c}is p-quasi open for every c∈R. For all these properties of p-quasi continuous representatives of Sobolev functions we refer to [28, Chapter 3]. By a nonnegative Borel measure in Ω we mean a countably additive set function defined in the Borel σ-field of Ω and with values in [0,+∞]. By a nonnegative Radon measure in Ω we mean a nonnegative Borel measure which is finite on every compact subset of Ω. We shall always identify a nonnegative Borel measure with its completion. We say that a Radon measure νon Ω belongs to W−1,p0(Ω) if there exists f∈ W−1,p0(Ω) such that hf,ϕi=ZΩ ϕdν ∀ϕ∈C∞ 0(Ω) ,(2.1) where h·,·i denotes the duality pairing between W−1,p0(Ω) and W1,p 0(Ω). We shall always identify fand ν. Note that, by the Riesz theorem, for every positive functional f∈W−1,p0(Ω), there exists a Radon measure νsuch that (2.1) holds. We denote by Mp 0(Ω) the class of all Borel measures which vanish on the sets of p-capacity zero and satisfy the following condition: µ(B) = inf{µ(A): Ap-quasi open,B⊆A⊆Ω} for every Borel set B⊆Ω. It is well known that every Radon measure which belongs to W−1,p0(Ω) belongs also to Mp 0(Ω) (see [28, section 4.7]). 3. Preliminary results on the relaxed Dirichlet problem with the p- Laplace operator. Let Ω be a bounded open subset of RN,N≥2. Let 2 ≤p<+∞ and let µ∈M p 0 (Ω). In the following we shall consider the space W1,p 0(Ω) ∩Lp µ(Ω) of all functions u∈W1,p 0(Ω) such that RΩ|u|pdµ < +∞. With the norm kukW1,p 0(Ω)∩Lp µ(Ω) =ZΩ|Du|pdx +ZΩ|u|pdµ1 p Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 585 the space W1,p 0(Ω) ∩Lp µ(Ω) is a reflexive Banach space. Let fbe a functional belonging to (W1,p 0(Ω)∩Lp µ(Ω))0(the dual space of W1,p 0(Ω)∩ Lp µ(Ω)) and let us consider the following variational problem: u∈W1,p 0(Ω) ∩Lp µ(Ω), ZΩ|Du|p−2DuDv dx +ZΩ|u|p−2uv dµ =hf,vi ∀v∈W1,p 0(Ω) ∩Lp µ(Ω). (3.1) Since the operator from W1,p 0(Ω)∩Lp µ(Ω) to (W1,p 0(Ω)∩Lp µ(Ω))0mapping u∈W1,p 0(Ω)∩ Lp µ(Ω) to the functional defined by RΩ|Du|p−2DuDv dx +RΩ|u|p−2uv dµ for every v∈W1,p 0(Ω)∩Lp µ(Ω) is a maximal monotone operator and the space W1,p 0(Ω)∩Lp µ(Ω) is reflexive, we get that there exists a unique solution uof problem (3.1). Remark 3.1. It is easy to see that the dual of W1,p 0(Ω) ∩Lp µ(Ω), (W1,p 0(Ω) ∩ Lp µ(Ω))0, coincides with W−1,p0(Ω) + Lp0 µ(Ω), so that, in particular, an element of the space W−1,p0(Ω) can be seen as an element of (W1,p 0(Ω) ∩Lp µ(Ω))0. In what follows, with a slight abuse of notation, hf,viwill denote the duality pairing between (W1,p 0(Ω)∩Lp µ(Ω))0and W1,p 0(Ω)∩Lp µ(Ω), in the general case f∈(W1,p 0(Ω)∩Lp µ(Ω))0, and the duality pairing between W−1,p0(Ω) and W1,p 0(Ω), in the case f∈W−1,p0(Ω). Many results similar to those given in the linear case (comparison principle, compactness, etc.) have been proved by Dal Maso and Murat (see [16] and [15]) for nonlinear problems of the type (3.1) (in general for nonlinear homogeneous operators). Proposition 3.2. Let f1,f2∈W−1,p0(Ω) and let µ1,µ2∈M p 0 (Ω).Letu 1 , u 2 ∈W 1,p 0(Ω) be the solutions of problem (3.1) corresponding to f1,µ1and to f2,µ2. If 0≤f1≤f2and µ2≤µ1in Ω, then 0≤u1≤u2p-q.e. in Ω. Proof. See [15, Proposition 2.7]. In the space Mp 0(Ω) it is possible to introduce a notion of convergence relative to the p-Laplace operator, ∆pu= div(|Du|p−2Du). Definition 3.3. Let (µn)be a sequence of measures of Mp 0(Ω) and let µ∈ Mp 0(Ω). We say that (µn)γ−∆p-converges to the measure µif, for every f∈W−1,p0(Ω), the sequence (un)of solutions of problems un∈W1,p 0(Ω) ∩Lp µn(Ω), ZΩ|Dun|p−2DunDv dx +ZΩ|un|p−2unvdµ n=hf,vi ∀v∈W1,p 0(Ω) ∩Lp µn(Ω) (3.2) converges weakly in W1,p 0(Ω) to the solution uof problem (3.1). Theorem 3.4. Every sequence of measures in Mp 0(Ω) contains a γ−∆p-con- vergent subsequence. Proof. See [10, Theorem 2.1] or [15, Theorem 6.5]. Many properties of the measure µ∈M p 0 (Ω) can be studied by means of the Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
586 JUAN CASADO DIAZ AND ADRIANA GARRONI solution wof the problem w∈W1,p 0(Ω) ∩Lp µ(Ω), ZΩ|Dw|p−2DwDv dx +ZΩ|w|p−2wv dµ =ZΩ vdx ∀v∈W1,p 0(Ω) ∩Lp µ(Ω). (3.3) By the comparison principle (Proposition 3.2), the function wis bounded in L∞(Ω) by a constant which does not depend on µ(see [15, section 2]) and satisfies w≥0 p-q.e. in Ω. Theorem 3.5. Let µ∈M p 0 (Ω), let wbe the solution of problem (3.3) and let ν=1+∆ p win the sense of W−1,p0(Ω). Then νis a nonnegative Radon measure of W−1,p0(Ω) and ν(B∩{w>0})=ZB w p−1 dµ(3.4) for every Borel set B⊆Ω. Reciprocally, we have µ(B)= ZB 1 w p−1 dν if Cp(B∩{w=0})=0, +∞if Cp(B∩{w=0})>0 for any Borel set B⊆Ω. Proof. See [15, Theorem 5.1] and [11, Proposition 3.4] for the linear case. The next proposition gives two density results which will be useful in what follows. Proposition 3.6. Let µ∈M p 0 (Ω) and let wthe solution of problem (3.3). Then (a) the set {wψ :ψ∈C∞ 0(Ω)}is dense in W1,p 0(Ω) ∩Lp µ(Ω) and hence in W1,p loc (Ω) ∩Lp µ(Ω). (b) the set Λof all functions of the form wPl i=1 ai1Kiwhere ai∈Rand Kiare closed subsets of Ωsuch that w=0µ-a.e. on Ki∩Kj, with i6=j, is dense in Lp µ(Ω). Proof. The proof of part (a) is given in [15, Proposition 5.5]. In order to prove part (b), we consider the measure λ=wpµ. Since wbelongs to Lp µ(Ω), the measure λis a Borel bounded measure and therefore the set of all step functions of the form Pl i=1 ai1Ki, where ai∈Rand Kiare closed subsets of Ω such that, for i6=j, λ(Ki∩Kj) = 0, is dense in Lp λ(Ω). If ubelongs to Lp µ(Ω), then u/w belongs to Lp λ(Ω), and for every ζ∈Λwehave ZΩ ζ−u w pdλ =ZΩ|wζ −u|pdµ which gives part (b). Finally the solutions of problems (3.3) are useful to characterize the γ−∆p-con- vergence in Mp 0(Ω). Let (µn) be a sequence of measures in Mp 0(Ω), and let wnbe the solutions of the problems wn∈W1,p 0(Ω) ∩Lp µn(Ω), ZΩ|Dwn|p−2DwnDv dx +ZΩ|wn|p−2wnvdµ n=ZΩ vdx ∀v∈W1,p 0(Ω) ∩Lp µn(Ω). (3.5) Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 587 The following result characterizes the γ−∆p-convergence in terms of the convergence of the functions wn. Theorem 3.7. The following conditions are equivalent: (a) (wn)converges to wweakly in W1,p 0(Ω); (b) (µn)γ−∆p-converges to µ. Proof. See [15, Theorem 6.3] and [11, Theorem 4.3] for the linear case. Remark 3.8. If (µn)γ−∆p-converges to µ, then the sequence (wn) converges to w strongly in W1,r 0(Ω) for every 1 ≤r<pand hence, a subsequence of (Dwn) converges to Dw pointwise a.e. in Ω (see [15, Theorem 6.8]). 4. Sequences in the spaces W1,p∩Lp µn.In this section (µn) will be a sequence of Mp 0(Ω) which γ−∆p-converges to a measure µ∈M p 0 (Ω). We shall use the sequence (wn) of the solutions of problems (3.5) to investigate the behavior of an arbitrary sequence (un), with un∈W1,p(Ω) ∩Lp µn(Ω), which converges weakly in W1,p(Ω). By Remark 3.8 we may assume that (wn) and (Dwn), respectively, converge to wand Dw pointwise a.e. in Ω. Let us prove some technical lemmas that will be useful in the remainder of this paper. Lemma 4.1. Let Ω0be an open subset of Ω. For every ϕ∈W1,p 0(Ω0)∩L∞(Ω0) we have lim n→∞ZΩ0|Dwn|pϕdx+ZΩ 0|w n|pϕdµ n=ZΩ 0|Dw|pϕdx+ZΩ 0|w|pϕ dµ.(4.1) Proof. Let ϕ∈W1,p 0(Ω0)∩L∞(Ω0) and let us extend ϕto Ω by setting ϕ=0in Ω\Ω 0 .Thusw n ϕbelongs to W1,p 0(Ω), and we can take it as test function in (3.3). Therefore by Remark 3.8 we obtain lim n→∞ZΩ0|Dwn|pϕdx+ZΩ 0|w n|pϕdµ n = lim n→∞ZΩ0 wnϕdx−ZΩ 0|Dwn|p−2DwnDϕwndx =ZΩ0 wϕdx −ZΩ|Dw|p−2DwDϕw dx =ZΩ0|Dw|pϕdx+ZΩ 0|w|pϕ dµ, which concludes the proof. Lemma 4.2. Let Ω0be an open subset of Ω. For every ϕ, ψ ∈W1,p(Ω0)∩L∞(Ω0), with ϕor ψin W1,p 0(Ω0), we have lim n→∞ZΩ0|D(wnψ)|pϕdx+ZΩ 0|w nψ|pϕdµ n=ZΩ 0|D(wψ)|pϕdx+ZΩ 0|wψ|pϕ dµ. (4.2) Proof. Let ϕ, ψ ∈W1,p(Ω0)∩L∞(Ω0), with ϕor ψin W1,p 0(Ω0). Since for every ξ1,ξ 2∈R Nand for every p≥2 the following inequality holds |ξ1|p−|ξ 2 |p ≤p ¡|ξ 1 |+|ξ 2 |p−1 |ξ 1−ξ 2 |,(4.3) we have |ψDwn+wnDψ|p−|ψDwn|p≤p¡|ψDwn+wnDψ|+|ψDwn|p−1|wnDψ|, Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
588 JUAN CASADO DIAZ AND ADRIANA GARRONI where by Remark 3.8 the left-hand side converges pointwise to |ψDw +wDψ|p− |ψDw|pand the right-hand side is uniformly integrable. Then (|D(wnψ)|p−|ψDwn|p) converges to |D(wψ)|p−|ψDw|pstrongly in L1(Ω0). This implies that ZΩ0|D(wnψ)|pϕdx =ZΩ 0|D(wψ)|pϕdx−ZΩ 0|Dw|p|ψ|pϕdx+ ZΩ 0|Dwn|p|ψ|pϕdx+o n and therefore the conclusion follows from Lemma 4.1. Lemma 4.3. Let Ω0be an open subset of Ω, let u∈W1,p(Ω0)∩Lp µ(Ω0), and let (ψm)be a sequence of functions in C∞ 0(Ω0)such that (wψm)converges to ustrongly in W1,p loc (Ω0)∩Lp µ(Ω0). Then lim m→∞ lim n→∞ZΩ0|D(wnψm−u)|pϕdx+ZΩ 0|w nψ m|pϕdµ n=ZΩ 0|u|pϕdµ(4.4) for every ϕ∈W1,p c(Ω0)∩L∞(Ω0). Proof. As in the proof of Lemma 4.2, ¡|D(wnψm−u)|p−|D(wnψm)|pconverges to |D(wψm−u)|p−|D(wψm)|pstrongly in L1(Ω0)asn→∞. Let ϕ∈W1,p c(Ω0)∩L∞(Ω0); thus, by Lemma 4.2, we get ZΩ0|D(wnψm−u)|pϕdx+ ZΩ 0|w nψ m|pϕdµ n=ZΩ 0¡|D(w nψ m−u)|p−|D(wnψm)|pϕdx +ZΩ 0|D(w nψ m)|pϕdx+ZΩ 0|w nψ m|pϕdµ n=ZΩ 0¡|D(wψm−u)|p−|D(wψm)|pϕdx +ZΩ 0|D(wψm)|pϕdx+ZΩ 0|wψm|pϕdµ+o n=ZΩ 0|u|pϕdµ+o m,n. The conclusion follows by taking the limit first as n→∞and then as m→∞. Let Ω0be an open subset of Ω. The following theorem shows that if a sequence (un), with un∈W1,p(Ω0)∩Lp µn(Ω0), converges weakly in W1,p(Ω0) to a function u∈W1,p(Ω0), and there exists a constant C>0 such that ZΩ0|un|pdµn≤C(4.5) for every n∈N, then the function ubelongs to Lp µ(Ω0). Theorem 4.4. Let (un)be a sequence such that un∈W1,p(Ω0)∩Lp µn(Ω0). Suppose that (un)converges weakly in W1,p(Ω0)to some function u. Then lim inf n→∞ ZΩ0|Dun|pdx +ZΩ0|un|pdµn≥ZΩ0|Du|pdx +ZΩ0|u|pdµ.(4.6) In particular, if (4.5) holds, then u∈W1,p(Ω0)∩Lp µ(Ω0). The result of Theorem 4.4 can be obtained as a direct consequence of the Γ- convergence of the functionals RΩ0|Dun|pdx+RΩ0|un|pdµnto the functional RΩ0|Du|pdx+ RΩ0|u|pdµ proved in [10]. For the sake of completeness we shall give an alternative proof of Theorem 4.4 which does not involve Γ-convergence theory. Before proving Theorem 4.4, let us prove two preliminary lemmas. Lemma 4.5. Let (un)be a sequence such that un∈W1,p(Ω0)∩Lp µn(Ω0)and such that (4.5) holds. Suppose that (un)converges weakly in W1,p(Ω0)to some function u. Then {u=0} Ω 0⊇{w=0} Ω 0. Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 589 Proof. Taking into account the decomposition un=u+ n−u− n, where u+ nand u− n denote respectively the positive and the negative part of un, it is not restrictive to assume un≥0p-q.e. in Ω0. We shall prove first the result in the special case where the functions unand u belong to W1,p 0(Ω0), and we shall suppose, also, that there exists a constant K>0 such that un≤Kp-q.e. in Ω0and hence u≤Kp-q.e. in Ω0. For every m∈Nlet us consider the sequence (um n) of the solutions of the problems um n∈W1,p 0(Ω0)∩Lp µn(Ω0), ZΩ0|Dum n|p−2Dum nDv dx +ZΩ0|um n|p−2um nvdµ n =mZΩ 0¡|u n|p−2u n−|u m n|p−2 u m nvdx ∀v∈W1,p 0(Ω0)∩Lp µn(Ω0), (4.7) which, extended to Ω by setting um n= 0 in Ω\Ω0, are also the solutions of the following equivalent problems: um n∈W1,p 0(Ω) ∩Lp ˆµn(Ω), ZΩ|Dum n|p−2Dum nDv dx +ZΩ|um n|p−2um nvdˆµ n =mZΩ¡|u n|p−2u n−|u m n|p−2 u m nvdx ∀v∈W1,p 0(Ω) ∩Lp ˆµn(Ω), (4.8) where ˆµnis the measure defined by ˆµn(B)= µ n (B)ifC p (B∩(Ω \Ω0))=0, +∞if Cp(B∩(Ω \Ω0)) >0 for any Borel set B⊆Ω. By the comparison principle (Proposition 3.2) we have that 0≤um n≤m1 p−1Kwnp-q.e. in Ω.(4.9) By taking in (4.7) um n−unas a test function we get ZΩ0¡|Dum n|p−2Dum n−|Dun|p−2DunD(um n−un)dx +ZΩ0¡|um n|p−2um n−|u n |p−2 u n (u m n−u n )dµn +mZΩ0¡|um n|p−2um n−|u n |p−2 u n (u m n−u n )dx =−ZΩ0|Dun|p−2DunD(um n−un)dx −ZΩ0|un|p−2un(um n−un)dµn. (4.10) Since for every ξ1,ξ 2∈R Nand for every p≥2wehave (|ξ 1 | p−2 ξ 1−|ξ 2 |p−2 ξ 2 )(ξ1−ξ2)≥22−p|ξ1−ξ2|p,(4.11) Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
596 JUAN CASADO DIAZ AND ADRIANA GARRONI Proposition 5.4. Let (un), with un∈W1,p(Ω0,RM)∩Lp µn(Ω0,RM), be a sequence which converges to some function uweakly in W1,p(Ω0,RM). Suppose that there exists a sequence (fn), with fn∈(W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM))0, which converges to f∈(W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM))0in the sense of (HΩ0), such that un satisfies problem (5.2). Then (un)converges to ustrongly in W1,r(Ω0,RM)for every r<p, and hence a subsequence of (Dun)converges to Du pointwise a.e. in Ω0. Proof. The proof follows the lines of the one given in [16] (see also [1]). In the course this proof we shall denote by Ca positive constant independent on n. Let Ψ : R7→ RbeaC1function which satisfies the following properties: Ψ(t)=1 if |t|<1,Ψ(t)=0 if |t|≥2, |Ψ(t)|≤1∀t∈R,|Ψ 0 (t)|≤2∀t∈R, and let Φ(y)=Ψ(|y|)y. Let δ>0 and, for every n∈N, let δn≤δbe a positive real number that we shall fix later. For every such a δnwe define the function Φδn(y)= δ n Φ(y/δn). Given ϕ∈C1 0(Ω0), with ϕ≥0, we can take (Φδn(un)+w nΦ δ n(u n−u))ϕ as test function in problem (5.2), and we obtain ZΩ0 a(x, Dun)DΦδn(un)Dunϕdx +ZΩ 0¡a(x, Dun)−a(x, Du)DΦδn(un−u)D(un−u)wnϕdx +ZΩ 0 a(x, Dun)¡Φδn(un)⊗Dϕ +Φ δ n(u n−u)⊗D(w n ϕ) dx +ZΩ0 Fn(x, un)(wnΦδn(un−u)+Φ δ n(u n ))ϕdµ n =hf n,(w nΦ δ n(u n−u)+Φ δ n(u n ))ϕi −ZΩ0 a(x, Du)DΦδn(un−u)D(un−u)wnϕ dx. (5.4) Since (un) is bounded in W1,p(Ω0,RM), (wn) is bounded in W1,p(Ω), and |Φδn|≤ 2δ n≤2δ, by condition (iv), we have ZΩ0 a(x, Dun)¡Φδn(un)⊗Dϕ +Φ δ n(u n−u)⊗D(w n ϕ) dx≤Cδ.(5.5) From property (V), H¨older’s inequality, and Proposition 5.3 it follows that ZΩ0 Fn(x, un)Φδn(un−u)wnϕdµ n ≤Cδ,(5.6) while from property (VI) and the definition of the function Φ we get ZΩ0 Fn(x, un)Φδn(un)ϕdµ n≥0.(5.7) Since (Φδn(un−u)wnϕ) converges weakly to zero in W1,p 0(Ω,RM) and (fn) converges in the sense of (HΩ0), we have hfn,w nΦ δ n(u n−u)ϕi=o n.(5.8) Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 597 Moreover, as 0 <δ n≤δand the sequence (Φδn) is uniformly Lipschitz, it is easy to see that there exists a positive number ˜ δ≤δsuch that lim sup n→∞ hfn,Φδn(un)ϕi=hf,Φ˜ δ(u)ϕi.(5.9) Finally, since (DΦδn(un−u)D(un−u)) converges weakly to zero in Lp(Ω0,RM)we also obtain that ZΩ0 a(x, Du)DΦδn(un−u)D(un−u)wnϕdx =o n.(5.10) Thus, by assumptions (i)–(v), by (5.4)–(5.10), and by the definition of the function Φ, we get Z{|un|<δn}Ω0|Dun|pϕ dx, +Z{|un−u|<δn}Ω0|D(un−u)|pwnϕdx ≤CZ{δ n≤|un−u|<2δn}Ω0 (h+|Du|+|Dun|)p−2|D(un−u)|2dx +CZ{δn≤|un|<2δn}Ω0 (k+η|Dun|p−1)|Dun|dx +hf,Φ˜ δ(u)ϕi+Cδ +on, (5.11) where we also used the fact that the sequence (wn) is bounded in L∞(Ω). Now, since (un) is bounded in W1,p(Ω0,RM), there exists a positive constant Ksuch that ZΩ0 (h+|Du|+|Dun|)p−2|D(un−u)|2dx +ZΩ0 (k+η|Dun|p−1)|Dun|dx ≤K. In particular, if we fix J∈Nand γ>0, then we have J X j=1Z{2j−1γ≤|un−u|<2jγ}Ω0 (h+|Du|+|Dun|)p−2|D(un−u)|2dx +Z{2j−1γ≤|un|<2jγ}Ω0 (k+η|Dun|p−1)|Dun|dx≤K; so that, for every n∈N, there exists j(n)∈{1,...,J}such that Z{2j(n)−1γ≤|un−u|<2j(n)γ}Ω0 (h+|Du|+|Dun|)p−2|D(un−u)|2dx +Z{2j(n)−1γ≤|un|<2j(n)γ}Ω0 (k+η|Dun|p−1)|Dun|dx ≤K J. If in (5.11) we take δ=2 Jγand δn=2 j(n)−1 γ, then we get Z{|un|<γ}Ω0|Dun|pϕdx+Z{|un−u|<γ}Ω0|D(un−u)|pwnϕdx ≤C J+C2 Jγ+hf,Φ˜ δ(u)ϕi+on, (5.12) where we used the fact that δn≥γfor every n∈N. By Rellich’s theorem the sequence (un) converges to ustrongly in Lp loc(Ω0,RM), and hence, up to a subsequence, Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
598 JUAN CASADO DIAZ AND ADRIANA GARRONI pointwise a.e. in Ω0. Thus, by Egorov’s theorem, for every σ>0 there exists a subset Sof Ω0, with |S|<σ, such that (un) converges to uuniformly on Ω0\S. Now let ε>0. If we choose J∈Nand γ>0 such that 1/J < ε and δ=2 Jγ=ε, then by (5.12) we have lim sup n→∞ Z{|un|<γ}Ω0|Dun|pϕdx+Z{|un−u|<γ}Ω0|D(un−u)|pwnϕdx ≤Cε+hf,Φ˜ δ(u)ϕi. Moreover, for nlarge enough we have that Ω0\S⊆{|u n−u|<γ} Ω 0and {u= 0}Ω0\S⊆{|u n |<γ} Ω 0, so that we get lim sup n→∞ Z{u=0}Ω0\S|Dun|pϕdx +ZΩ 0\S|D(u n−u)|pw nϕdx (5.13) ≤Cε +hf,Φ˜ δ(u)ϕi, which, by using that 0 ≤˜ δ≤δ=εand Φ˜ δ(u)ϕconverges strongly to zero in W1,p 0(Ω,RM)∩Lp µ(Ω,RM)asεtends to zero, gives lim n→∞Z{u=0}Ω0\S|Dun|pϕdx +ZΩ 0\S|D(u n−u)|pw nϕdx =0.(5.14) By the arbitrariness of σ, we get, up to a subsequence, that (D(un−u)wn) and (Dun1{|u|=0}Ω0) converges to zero pointwise a.e. in Ω0. Moreover, since (wn) converges to wstrongly in Lp(Ω0,RM) and by Lemma 4.5 {w>0}⊇{|u|>0} Ω 0 , this implies that (Dun) converges pointwise to Du a.e. in {|u|>0}Ω0and hence, as |Du|= 0 a.e. in {|u|=0} Ω 0,(Dun) converges pointwise to Du a.e. in Ω0. Finally, since (un) is bounded in W1,p(Ω0,RM), we obtain that (un) converges to ustrongly in W1,r(Ω0,RM) for every r<p. Remark 5.5. Under the same assumptions of Proposition 5.4, by (v) and Proposition 5.4 we have that (a(x, Dun)) converges to a(x, Du) weakly in Lp0(Ω0,MM×N) and strongly in Lr(Ω0,MM×N) for every 1 ≤r<p 0 . Similarly we deduce that (a(x, D(un− u))) converges to zero weakly in Lp0(Ω0,MM×N) and strongly in Lr(Ω0,MM×N) for every 1 ≤r<p 0 . 6. The limit problem. In this section we shall prove the main result of this paper (Theorem 6.4). We shall consider a sequence (un) of solutions of problems (5.2), with Fn∈F(L) and µn∈M p 0 (Ω), which satisfies ZΩ0|Dun|pdx +ZΩ0|un|pdµn≤M,(6.1) where Mis a positive constant which depends on the sequence (un). We shall show that a cluster point uof such a sequence is a solution of a variational problem similar to (5.2). Namely we shall prove that the limit problem will be of the form u∈W1,p(Ω0,RM)∩Lp µ(Ω0,RM), ZΩ0 a(x, Du)Dv dx +ZΩ0 F(x, u)vdµ =hf,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM), (6.2) where µis a measure in Mp 0(Ω) and F(x, s) is a vector function in F(α, C, 1/(p−1)) for a constant Cwhich depends only on α,β,L,N, and p. Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 599 Remark 6.1. Let µ∈M p 0 (Ω) and let ˜ F,F ∈F(c 1 ,c 2,σ) be two functions such that for every s∈RMF(x, s)= ˜ F(x, s)µ-a.e. in {w>0}, where wis the solution of problem (3.3). If in problem (6.2) we change Fby ˜ Fwe obtain an equivalent problem. In particular the function F(x, s) can be defined arbitrarily in the set {w=0}. Let us introduce now a notion of convergence in the space Mp 0(Ω) ×F(c 1,c 2,σ), with c1>0, c2>0, and 0 <σ ≤1. Definition 6.2. Let (µn)be a sequence in Mp 0(Ω), let (Fn)be a sequence in F(c1,c 2,σ), let µ∈M p 0 (Ω) and F∈F(c 1 ,c 2,σ). We say that the pairs (µn,F n)γA- converge (in Ω) to the pair (µ, F)if the following property holds: for any open set Ω0⊆ Ω, for any sequence of functionals (fn)with fn∈(W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM))0, which converges to some f∈(W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM))0in the sense of (HΩ0) (according with Definition 5.1), and for any sequence (un)of solutions of problems (5.2) satisfying (6.1), all cluster points of the sequence (un)in the weak topology of W1,p(Ω0,RM)satisfy problem (6.2). The most important property of the γAconvergence is the following result. Proposition 6.3. Let ((µn,F n)) be a sequence in Mp 0(Ω) ×F(c 1 ,c 2,σ)which γA-converges to a pair (µ, F). Then for any open set Ω0⊆Ωand for any sequence (fn), with fn∈(W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM))0, which converges to some f∈(W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM))0in the sense of (HΩ0), the unique solution unof the problem un∈W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM), ZΩ0 a(x, Dun)Dv dx +ZΩ0 Fn(x, un)vdµ n=hf n,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM) (6.3) converges weakly in W1,p 0(Ω0,RM)to the unique solution uof the problem u∈W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM), ZΩ0 a(x, Du)Dv dx +ZΩ0 F(x, u)vdµ =hf,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM). (6.4) Proof. By using unas a test function in (6.3) and by taking into account Lemma 5.2, we deduce that the sequence (un) satisfies (6.1). This implies in particular that there exists a subsequence of (un) which converges weakly in W1,p 0(Ω0,RM)toa function u∈W1,p 0(Ω0,RM). By the definition of γA-convergence, the function u satisfies (6.4). Since this problem has a unique solution, the whole sequence (un) converges to u. The following theorem gives a compactness result for the γA-convergence. Theorem 6.4. Let (µn)be a sequence of measures in Mp 0(Ω) and let (Fn)be a sequence in F(L), with L>0. Then there exist an increasing sequence of integers (nj), a measure µ∈M p 0 (Ω), and a function F∈F(α, C, 1/(p−1)) such that the pairs (µnj,F n j)γA-converge to (µ, F)in Ω(according to Definition 6.2), where Cis a positive constant which depends only on α,β,L,N, and p. Remark 6.5. The compactness result stated in Theorem 6.4 can be proved under more general assumptions on (Fn). Namely, if the sequence (Fn) belongs to Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
600 JUAN CASADO DIAZ AND ADRIANA GARRONI F(c1,c 2,σ), for some constants c1>0, c2>0, and 0 <σ≤1, then there exist an increasing sequence of integers (nj), a measure µ∈M p 0 (Ω), and a function F∈F(c 0 1 ,c 0 2,σ0) such that the pairs (µnj,F n j)γA-converge to (µ, F)inΩ. The positive constants c1and c2depend only on α,β,c1,c2,N,p, and σ; while σ0= min{σ, 1/(p−σ)}. In order to simplify the exposition of the proof, we shall prove only the compactness result as stated in Theorem 6.4 (the proof of the general case stated in Remark 6.5 being analogous). Before proving Theorem 6.4 we need additional information on the behavior of the sequence (un) of solutions of problems (5.2). To this aim we shall compare (un) with the sequences (wnψm), ψm∈C∞ 0(Ω,RM), of correctors for the p-Laplacian, studied in section 4. In Lemma 6.6 and Propositions 6.7 and 6.8, we shall consider an open set Ω0⊆Ω, a sequence of measures (µn), a sequence of functions (Fn), two sequences of functionals (fn), (gn), two sequences of functions (un), (zn), a measure µ, two functionals f,g, and two functions uand zsuch that µn,µ∈M p 0 (Ω),F n ∈F(L), (µ n )γ −∆ p-converges to µ, (6.5) fn,g n∈(W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM))0, f,g ∈(W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM))0, fn→fin the sense of (HΩ0), gn→gin the sense of (HΩ0), (6.6) un,z n∈W1,p(Ω0,RM)∩Lp µn(Ω0,RM), u, z ∈W1,p(Ω0,RM)∩Lp µ(Ω0,RM), un*u in W1,p(Ω0,RM), zn*z in W1,p(Ω0,RM), (6.7) RΩ0a(x, Dun)Dv dx +RΩ0Fn(x, un)vdµ n=hf n,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM), RΩ0a(x, Dzn)Dv dx +RΩ0Fn(x, zn)vdµ n=hf n,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM). (6.8) Lemma 6.6. Assume that (6.5), (6.6), (6.7), and (6.8) hold. Then for every function ϕ∈C∞ c(Ω0), with ϕ≥0, we have the estimates lim sup n→∞ ¡ZΩ0|D(un−u)|pϕdx+ZΩ 0|u n|pϕdµ n≤CZΩ 0|u|pϕdµ(6.9) and lim sup n→∞ ZΩ0|D((un−zn)−(u−z))|pϕdx+ZΩ 0|u n−z n|pϕdµ n ≤CZΩ 0|u|pϕdµ+ZΩ 0|z|pϕdµ p−2 p−1ZΩ 0|u−z|pϕdµ 1 p−1, (6.10) where Cis a positive constant which depends only on α,β,L,N, and p. Proof. By Theorem 4.4 we have that uand zbelong to W1,p(Ω,RM)∩Lp µ(Ω,RM). Let ϕ∈C∞ c(Ω0), with ϕ≥0, let wnand wbe the solutions of problems (3.5) and (3.3). By Proposition 3.6 there exists a sequence (ψm)inC ∞ 0(Ω0,RM) such that (wψm) converges to u−zstrongly in W1,p loc (Ω0)∩Lp µ(Ω0). Thus, taking (un−zn−wnψm)ϕ Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 601 as a test function in the difference of the equations in (6.8), we get ZΩ0 [a(x, Dun)−a(x, Dzn)]D(un−zn−wnψm)ϕdx +ZΩ 0 [F n(x, un)−Fn(x, zn)](un−zn−wnψm)ϕdµ n =−ZΩ 0 [a(x, Dun)−a(x, Dzn)](un−zn−wnψm)⊗Dϕdx +hfn−gn,(un−zn−wnψm)ϕi=om,n. (6.11) Let us estimate the terms which appear in (6.11). By using assumption (ii) and Proposition 5.4, the sequences (|a(x, Dun)−a(x, D(un−u)|p0) and (|a(x, Dzn)− a(x, D(zn−z)|p0) are uniformly integrable and pointwise convergent respectively to |a(x, Du)|p0and |a(x, Dz)|p0. Therefore they converge strongly in L1(Ω0,MM×N) and hence, from (6.11), we deduce ZΩ0 [a(x, D(un−u)) −a(x, D(zn−z))]D(un−zn−(u−z))ϕdx +ZΩ 0 [F n(x, un)−Fn(x, zn)](un−zn)ϕdµ n =o m,n −ZΩ0 [a(x, Du)−a(x, Dz)]D(un−zn−wnψm)ϕdx +ZΩ 0 [a(x, D(un−u)) −a(x, D(zn−z))]D(wnψm−(u−z))ϕdx +ZΩ 0 [F n(x, un)−Fn(x, zn)]wnψmϕdµ n. Since ZΩ0 [a(x, Du)−a(x, Dz)]D(un−zn−wnψm)ϕdx =o m,n by properties (i) and (ii) of aand properties (II) and (III) of Fn,weget α ZΩ 0 |D(u n−z n−(u−z))|pϕdx+αZΩ 0 |u n−z n| pϕdµ n ≤βZΩ 0 (h+|D(u n−u)|+|D(z n−z)|) p−2|D(u n−z n−(u−z))||D(w nψ m−(u−z))|ϕdx +LZΩ 0 (|u n|+|z n|) p−2|u n−z n||w nψ m|ϕdµ n+o m,n =βZΩ0 (|D(un−u)|+|D(zn−z)|)p−2|D(un−zn−(u−z))||D(w nψ m−(u−z))|ϕdx +LZΩ 0 (|u n|+|z n|) p−2|u n−z n||w nψ m|ϕdµ n+o m,n. (6.12) Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
602 JUAN CASADO DIAZ AND ADRIANA GARRONI Using Young’s inequality and then H¨older’s inequality in (6.12), we obtain ZΩ0|D((un−zn)−(u−z))|pϕdx+ZΩ 0|u n−z n|pϕdµ n ≤CZΩ 0¡|D(u n−u)|+|D(z n−z)|) pϕdx p−2 p−1ZΩ 0|D(w nψ m−(u−z))|pϕdx 1 p−1 +CZΩ 0¡|u n|+|z n|pϕdµ np−2 p−1ZΩ 0|w nψ m|pϕdµ n1 p−1+o m,n. (6.13) Taking zn=z= 0 (and then gn= 0), in estimate (6.13), by Young’s inequality, we get ZΩ0|D(un−u)|pϕdx+ZΩ 0|u n|pϕdµ n ≤CZΩ 0|D(w nψ m−u)|pϕdx+CZΩ 0|w nψ m|pϕdµ n+o m,n, which by Lemma 4.3 implies (6.9). Finally, in order to get (6.10), it is enough to apply, in estimate (6.13), estimate (6.9) for unand zn, and Lemma 4.3. The following proposition gives a first version of the limit problem satisfied by u. Proposition 6.7. Let us assume (6.5), (6.6), (6.7), and (6.8). Then there exists a wµ-measurable vector function H, uniquely defined µ-a.e. in Ω0, such that the function usatisfies the problem u∈W1,p(Ω0,RM)∩Lp µ(Ω0,RM), ZΩ0 a(x, Du)Dv dx +ZΩ0 Hvdµ =hf,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM), (6.14) and |H|≤C|u| p−1µ-a.e. in Ω0.(6.15) Moreover, for every φ∈C∞ c(Ω0,RM)we have ZΩ0 Hwφdµ = lim n→∞hZΩ0 a(x, D(un−u))φ⊗D(wn−w)dx +ZΩ0 Fn(x, un)wnφdµ ni, (6.16) where wnand ware the solutions of problems (3.5) and (3.3). Proof. Given φ∈C∞ c(Ω0,RM), we take wnφas the test function in the equation satisfied by un(see (6.8)) and we get ZΩ0 a(x, Dun)φ⊗Dwndx +ZΩ0 a(x, Dun)Dφwndx +ZΩ0 Fn(x, un)wnφdµ n=hf n,w nφi. (6.17) Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 603 By Remark 5.5 we have lim n→∞hfn,w nφi−ZΩ 0 a(x, Dun)Dφwndx=hf,wφi−ZΩ 0 a(x, Du)Dφ wdx. Let us define a distribution Tin Ω0by hT,φi= lim n→∞hZΩ0 a(x, Dun)φ⊗Dwndx −ZΩ0 a(x, Du)φ⊗Dw dx +ZΩ0 Fn(x, un)wnφdµ ni for every φ∈C∞ c(Ω0,RM). Since the norm of wnin W1,p 0(Ω) ∩Lp µn(Ω) is bounded, by (6.1) and property (V) we have ZΩ|a(x, Dun)||Dwn|dx +ZΩ|Fn(x, un)||wn|dµn≤C, and hence Tis continuous with respect to the uniform convergence and it can be represented by a vector Radon measure (T1,...,T M) such that hT,φi= M X i=1 ZΩ0 φidTi∀φ∈C∞ c(Ω0,RM),(6.18) where φ1,...,φ Mare the components of the vector function φ. Thus taking the limit in (6.17) we obtain ZΩ0 a(x, Du)D(wφ)dx +hT,φi=hf,wφi.(6.19) Since by conditions (ii) and (iv) and Proposition 5.4, a(x, Dun)−a(x, D(un−u)) converges to a(x, Du) strongly in Lp0(Ω,MM×N), we can write hT,φi= lim n→∞hZΩ0 a(x, D(un−u))φ⊗D(wn−w)dx +ZΩ0 Fn(x, un)wnφdµ ni. (6.20) Let us prove (6.16). For every φ∈C∞ c(Ω0,RM), with φ≥0, by assumptions (V) and (v), Proposition 5.4, H¨older’s inequality, Lemma 4.3, and estimate (6.9), we have |hT,φi|≤Clim sup n→∞ hZΩ0 (k(x)+η|D(u n−u)|p−1 )|D(w n−w)||φ|dx +ZΩ0|Fn(x, un)|wn|φ|dµni ≤Clim sup n→∞ hZΩ|D(un−u)|p−1|D(wn−w)||φ|dx +ZΩ0|un|p−1wn|φ|dµni ≤Clim sup n→∞ hZΩ0|D(un−u)|p|φ|dxp−1 pZΩ|D(wn−w)|p|φ|dx1 p (6.21) +ZΩ0|un|p|φ|dµnp−1 pZΩ0|wn|p|φ|dµn1 pi ≤CZΩ0|u|p|φ|dµp−1 pZΩ0|w|p|φ|dµ1 p. Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
604 JUAN CASADO DIAZ AND ADRIANA GARRONI Let us denote by |Ti|the total variation of the measures Ti,i=1...,M. Taking into account that for every open subset Aof Ω0we have |Ti|(A) = sup {hTi,ϕi:ϕ∈C∞ 0(A),sup |ϕ|≤1}, by (6.21) we get |Ti|(A)≤CZA|u|pdµp−1 pZA|w|pdµ1 p (6.22) for every open subset Aof Ω0. Since |u|pµ,|w|pµ, and |Ti|are finite measures, (6.22) holds for every Borel subset of Ω0. This implies that the measures Tiare absolutely continuous with respect to the measure |w|pµ, and hence to the measure wµ. Since wµ isaσ-finite measure we can apply the Radon–Nikod´ym derivation theorem and we find a wµ-measurable vector function H=(H 1 ,...,H M) such that Ti(A)=ZA H i wdµ for every Borel subset Aof Ω0and i=1,...,M, so that, by (6.20) and (6.18), (6.16) holds. We can suppose that Hi(x)=0 forµ-a.e. xin {w=0}∀i=1,...,M.(6.23) Thus by (6.22) we get ZA|Hi|wdµ ≤CZA|u|pdµp−1 pZA|w|pdµ1 p for every Borel subset Aof Ω0. Using Young’s inequality, we obtain ZA|Hi|wdµ ≤C1 p0εp0ZA|u|pdµ +εp pZA|w|pdµ for every Borel subset Aof Ω0and for every ε>0. Thus (first reasoning for ε∈Q and then arguing by density) we get |Hi(x)|w(x)≤C1 p0εp0|u(x)|p+εp p|w(x)|p (6.24) for µ-a.e. xin Ω0and for every ε>0. If x∈Ω0satisfies w(x)>0 and (6.24) holds true for any ε, by choosing ε=|u(x)|p−1 p/|w(x)|p−1 pin (6.24) and taking into account (6.23), we get |Hi(x)|≤C|u(x)| p−1 ,µ-a.e. x∈Ω0, and hence (6.15) is proved. Condition (6.14) follows from (6.19), (6.18), and the density result given by Proposition 3.6. Finally the vector function His uniquely determined µ-a.e. in Ω0by (6.14) and (6.15). Indeed, by (6.14) His uniquely determined µ-a.e. in {w>0}, and by (6.15) we have H=0µ-a.e. in {|u|=0} Ω 0 . Then the conclusion follows by Lemma 4.5. In order to study the dependence of the function Hon the function u, let us consider a sequence of functionals (gn) and a sequence of functions (zn) which satisfy Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 605 (6.6), (6.7), and (6.8). By Proposition 6.7, applied to (zn), we get that there exists a wµ-measurable vector function H0, uniquely defined µ-a.e. in Ω0, such that z∈W1,p(Ω0,RM)∩Lp µ(Ω0,RM), ZΩ0 a(x, Dz)Dv dx +ZΩ0 H0vdµ =hg,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM), (6.25) |H0|≤C|z| p−1 ,µ-a.e. in Ω0,(6.26) and ZΩ0 H0wφdµ = lim n→∞hZΩ0 a(x, D(zn−z))φ⊗D(wn−w)dx +ZΩ0 Fn(x, zn)wnφdµ ni. (6.27) The following proposition compares the function Hwith the function H0. Proposition 6.8. The vector functions Hand H0satisfy |H−H0|≤C ¡|u|+|z| p p−2 p−1|u−z|1 p−1,µ-a.e. in Ω0 (6.28) and (H−H0)(u−z)≥α|u−z|p,µ-a.e. in Ω0.(6.29) Proof. Let us first prove (6.28). Consider φ∈C∞ c(Ω0,RM) and let wnand wbe the solutions of problems (3.5) and (3.3). By (6.16), (6.27), and by assumptions (ii) and (III), we have ZΩ0 (H−H0)wφdµ ≤ZΩ0 (a(x, D(un−u)) −a(x, D(zn−z)))φ⊗D(wn−w)dx +ZΩ0 (Fn(x, un)−Fn(x, zn))wnφdµ n +o n ≤CZΩ 0¡h+|D(u n−u)|+|D(z n−z)|p−2|D((un−zn)−(u−z))||D(w n−w)||φ|dx +CZΩ0 (|un|+|zn|)p−2|un−zn||φ|w ndµn+on ≤CZΩ0¡|D(un−u)|+|D(zn−z)|p−2|D((un−zn)−(u−z))||D(w n−w)||φ|dx +CZΩ0 (|un|+|zn|)p−2|un−zn||φ|w ndµn+on. (6.30) By using H¨older’s inequality, (6.9), applied to unand znand (6.10), we get ZΩ0 (H−H0)wφdµ ≤CZΩ0|u|p|φ|dµ +ZΩ|z|p|φ|dµp−2 p−1ZΩ0|u−z|p|φ|dµ1 p(p−1) ZΩ0|w|p|φ|dµ1 p. Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
612 JUAN CASADO DIAZ AND ADRIANA GARRONI But the choice ζ=u/w in (7.2) is not possible since we do not know, a priori, if Rm n(x, s) is a Carath´eodory function; so Rm n(x, u(x)) may not even be measurable. We avoid this problem using the function wζ to approach u. This approach is always possible by Proposition 3.6 part b. Remark 7.3. When we consider Rm n(x, ζ), the value of ζon Ki∩Kj,i6=j, is not relevant. Indeed by taking in (7.2) un=u= 0 (then fn= 0) and ζ=s1Ki∩Kjwe deduce lim sup m→∞ lim sup n→∞ ZKi∩Kj Rm n(x, s)dx =0 ∀s∈R M. Remark 7.4. If Kis a compact subset of Ω0such that µ(K) = 0, estimate (7.2) with ζ= 0 implies that Dunconverges strongly to Du in Lp(K, RM). Proof of Theorem 7.1.Let s∈RMand let Kbe a closed subset of Ω0. By Lemma 6.6, for every ϕ∈C∞ c(Ω0,RM), with ϕ≥1Kin Ω0,wehave lim sup n→∞ ZK|D((un−sm n)−(u−sm))|pdx ≤CZΩ0|u|pϕdµ+ZΩ 0|s m|pϕdµ p−2 p−1ZΩ 0|u−s m|pϕdµ 1 p−1. (7.3) If ϕnow decreases to 1K, by the fact that (sm) tends to sw strongly in W1,p 0(Ω,RM)∩ Lp µ(Ω,RM) and from (7.3) we deduce lim sup m→∞ lim sup n→∞ ZK|D((un−sm n)−(u−sm))|pdx ≤CZK|u|pdµ +ZK|sw|pdµp−2 p−1ZK|u−sw|pdµ1 p−1. (7.4) Moreover, by inequality |ξ1|p−|ξ 2 |p ≤p(|ξ 1 |p−1+|ξ 2 |p−1 )|ξ 1−ξ 2 |∀ξ 1 ,ξ 2∈M M×N, we get |D((un−sm n)−(u−sm))|p−|D((un−sm n)−(u−sw))|p ≤p|D((un−sm n)−(u−sm))|p−1+|D((un−sm n)−(u−sw))|p−1|D(sm−sw)|, and then by the strong convergence of (sm)inW 1,p 0(Ω,RM) we deduce lim m→∞ lim sup n→∞ ZK|D((un−sm n)−(u−sm))|pdx −|D((un−sm n)−(u−sw))|pdx =0. Thus by (7.4) and the definition of Rm n,weget lim sup m→∞ lim sup n→∞ ZK|D(un−u−Rm n(x, s))|pdx ≤CZK|u|pdµ +ZK|sw|pdµp−2 p−1ZK|u−sw|pdµ1 p−1. (7.5) Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 613 Consider now ζ=Pl i=1 si1Ki, with si∈RMand Kiclosed subsets of Ω0such that w=0µ-a.e. on Ki∩Kj, for i6=j. By Lemma 4.5, we also have |u|=0µ-a.e. on Ki∩Kj, for i6=j. Then, if K=Sl i=1 Ki, by using (7.5) and H¨older’s inequality we get lim sup m→∞ lim sup n→∞ ZK|Dun−Du −Rm n(x, ζ)|pdx ≤lim sup m→∞ lim sup n→∞ l X i=1 ZKi|D(un−u−Rm n(x, si))|pdx ≤C l X i=1ZKi|u|pdµ +ZKi|wsi|pdµp−2 p−1ZKi|u−wsi|pdµ1 p−1 =CZK|u|pdµ +ZK|wζ|pdµp−2 p−1ZK|u−wζ|pdµ1 p−1, which concludes the proof. 8. Particular cases. In this section, we shall prove that some assumptions on the function a, as homogeneity or linearity, are inherited by function F. In [6] we construct an example which shows that the function Fin general can be nonlinear and nonhomogeneous. Homogeneous case. Let abe a function which satisfies conditions (i)–(v), as at the beginning of section 5. Let us assume in addition that asatisfies the following homogeneity condition: (vi) for a.e. x∈Ω, for every t∈R, and for every ξ∈MM×N, a(x, tξ)=|t| p−2 ta(x, tξ). Moreover, let (µn) be a sequence in Mp 0(Ω), and let (Fn) be a sequence of functions in F(L) which satisfies the following condition: (VII) for every x∈Ω, for every t∈R, and for every s∈RM, Fn(x, ts)=|t| p−2 tFn(x, s). Under these assumptions we have the following result. Theorem 8.1. If the function asatisfies conditions (i)–(vi) and the sequence (Fn)satisfies conditions (I)–(VII), then in Theorem 6.4 the function Fcan be chosen satisfying F(x, ts)=|t| p−2 tF(x, s) for every x∈Ω, for every t∈R, and for every s∈RM. Proof. Assumptions (vi) and (VII) imply that for every t∈Rand for every q∈QM, the solution qm nof (6.42) satisfies (tq)m n=tqm n,µ-a.e. in Ω, where (tq)m nis the solution of problem (6.42) with qreplaced by tq, which converges, according with Theorem 6.9, to some function (tq)mweakly in W1,p 0(Ω,RM) for every m∈N. Then taking the limit as n→∞we have (tq)m=tqm,µ-a.e. in Ω. Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
614 JUAN CASADO DIAZ AND ADRIANA GARRONI Therefore, the functions Hm qand Hm tq defined by (6.43) satisfy Hm tq =tHm q,µ-a.e. in Ω for every t∈Rand for every q∈QM. Thus, using that for every q∈QM, the function G(x, q) in the proof of Theorem 6.4 is defined as the limit in mof Hm q, the continuity of G(x, s) with respect to the variable sand that the function F(x, s) satisfies (6.49), we conclude the proof. In this special case we have the following result for the correctors defined by (7.1). Theorem 8.2. Assume that the function aand the sequence (Fn)satisfy, respectively, properties (vi) and (VII). Then, the function Rm ndefined by (7.1) satisfies Rm n(x, ts)=tRm n(x, s)(8.1) for almost every x∈Ω, for every s∈RM, and for every t∈R. Proof. Assumptions (vi) and (VII) imply that, for every t∈Rand for every s∈RM,(ts)m n=tsm n, where sm nis the solution of (6.32) and (ts)m nis the solution of problem (6.32) with sreplaced by ts. Thus the conclusion follows by the definition of Rm n. Linear case. Let us consider now the linear case, i.e., let us assume, with slight abuse of notation, that the function a(x, ξ) is of the form a(x)ξ, where a(x)isa measurable function from Ω on the linear applications from MM×Nto MM×Nwhich satisfies these hypotheses: (il) there exists a constant α>0 such that for every ξ∈MM×Nand for a.e. x∈Ω, we have a(x)ξξ ≥α|ξ|2; (iil) there exists a constant β>0 such that for every ξ∈MM×Nand for a.e. x∈Ω, we have |a(x)ξ|≤β|ξ|. Remark 8.3. Hypotheses (il) and (iil) imply (i)–(v) at the beginning of section 5 for p=2. Let us denote by Fl(L), with L>0, the class of all vector functions from Ω×RM to RMwhich are linear in the second argument (i.e., of the form F(x)s) and which satisfy the following two conditions: (Il) for every s∈RMand for every x∈Ωwehave F(x)ss ≥α|s|2; (IIl) for every s∈RMand for every x∈Ωwehave |F(x)s|≤L|s|. Remark 8.4. It is easy to see that the class Fl(L) defined above is contained in the class F(L) defined in section 5. We are now in a position to state the following result. Theorem 8.5. Assume that in Theorem 6.4, Au =−div (a(x)Du), with a(x)satisfying (il)and (iil),and that the sequence (Fn)belongs to Fl(L). Then, the function Fwhich appears in the statement of Theorem 6.4 can be chosen in the class Fl(L0), with L0>0different, in general, from L. Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 615 Proof. We have already shown in Theorem 8.1 that Fis homogeneous in its second argument. The additivity of Fcan be proved essentially with the same argument. For the corrector result, as in section 7, let us assume that (µn)γ−∆-converges to µand that the pairs (µn,F n)γA-converge to (µ, F ) according with Definition 6.2 (where Au =−div (a(x)Du)). In this case the function Rm n:Ω×R M→M M×Nis given by Rm n(x, s)=Dsm n(x)−D(ws)(x), where for every s∈RM,sm nis the solution of the problem sm n∈H1 0(Ω,RM)∩L2 µn(Ω,RM), ZΩ a(x)Dsm nDv dx +ZΩ Fn(x)sm nvdµ n=mZΩ (w ns−s m n)vdx ∀v∈H1 0(Ω,RM)∩L2 µn(Ω,RM). (8.2) Clearly, the functions Rm nare linear in their second argument, and hence they are Carath´eodory functions. This allows us to improve Theorem 7.1. Theorem 8.6. Let Ω0be an open subset of Ω.Let(u n ), with un∈H1(Ω0,RM)∩ L2 µn(Ω0,RM), be a sequence which converges weakly in H1(Ω0,RM)to some function u and satisfies (6.1). Assume also that there exists a sequence (fn), with fnbelonging to (H1 0(Ω0,RM)∩L2 µn(Ω0,RM))0, converging to some functional f∈(H1 0(Ω,RM)∩ L2 µ(Ω,RM))0in the sense of (HΩ0), such that (un)satisfies the following problem: un∈H1(Ω0,RM)∩L2 µn(Ω0,RM), ZΩ a(x)DunDv dx +ZΩ Fn(x)unvdµ n=hf n,vi ∀v∈H1 0(Ω0,RM)∩L2 µn(Ω0,RM). (8.3) Then, for every function ψ∈H1(Ω0,RM)∩L∞(Ω0,RM)and for every closed set K⊂Ω, we have lim sup m→∞ lim sup n→∞ ZK|Dun−Du −Rm n(x)ψ|2dx ≤CZK|u−wψ|2dµ.(8.4) In particular, if u/w belongs to L∞(K, RM), then we have lim sup m→∞ lim sup n→∞ ZKDun−Du −Rm n(x)u w 2dx =0.(8.5) In order to prove Theorem 8.6, we need some preliminary lemmas. Lemma 8.7. Let W= sup{kwnkL∞(Ω)}. Then for every s∈RM, the solutions sm nof (8.2) satisfy lim sup m→∞ lim sup n→∞ Z{|sm n|≥2kW|s|} |Ds m n|2dx ≤C|s|2 k∀k∈N.(8.6) Proof. For any j∈N, let us consider the function Φj:RM7→ RMdefined by Φj(ζ)= 0if|ζ|≤2 j−1 W|s|, |ζ|−2 j−1 W|s| 2 j−1 W|s|ζif 2j−1W|s|<|ζ|<2jW|s|, ζif |ζ|≥2 jW|s|. (8.7) Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
616 JUAN CASADO DIAZ AND ADRIANA GARRONI Taking Φj(sm n) as a test function in (8.2), we get ZΩ a(x)Dsm nD[Φj(sm n)] dx +ZΩ Fn(x)sm nΦj(sm n)dµn+mZΩ (sm n−wns)Φj(sm n)dx =0, which implies αZ{|sm n|≥2jW|s|} |Ds m n|2dx ≤− Z{2 j−1 W|s|≤|sm n|<2jW|s|} a(x)Dsm nD[Φj(sm n)] dx +mZΩ|sm n−wns||Φj(sm n)|dx ≤CZ{2j−1W|s|≤|sm n|<2jW|s|} |Dsm n|2dx +mZΩ|sm n−wns||Φ1(sm n)|dx, (8.8) where we used that |Φj(sm n)|≤|Φ 1 (s m n)|for every j∈N, and the fact that, in the set {2j−1W|s|≤|s m n|<2 jW|s|}, we have D[Φj(sm n)] = Dsm n(2|sm n|−2j−1W|s|)/2j−1W|s|, and hence |D[Φj(sm n)]|≤3|Dsm n|. On the other hand, by (6.38) we can see that for every k∈Nwe have k X j=1 Z{2j−1W|s|≤|sm n|<2jW|s|} |Dsm n|2dx ≤ZΩ|Dsm n|2dx ≤C|s|2 and therefore, for every k∈N, there exists j(k), with 1 ≤j(k)≤k, such that Z{2j(k)−1W|s|≤|sm n|<2j(k)W|s|} |Dsm n|2dx ≤C|s|2 k. By (8.8), applied with j(k), we deduce that for any k∈Nwe have Z{|sm n|≥2kW|s|} |Ds m n|2dx ≤Z{|sm n|≥2j(k)W|s|} |Ds m n|2dx ≤C|s|2 k+mZΩ|sm n−wns||Φ1(sm n)|dx, thus taking the limit as n→∞and using that Φ1(ws) = 0, we obtain lim sup n→∞ Z{|sm n|≥2kW|s|} |Ds m n|2dx ≤C|s|2 k+mZΩ|sm−ws||Φ1(sm)−Φ1(ws)|dx ≤C|s|2 k+CmZΩ|sm−ws|2dx. Since, by Theorem 6.9, the second term on the right-hand side tends to zero when m tends to infinity, estimate (8.6) is proved. Lemma 8.8. For every function ϕ∈H1 0(Ω) ∩L∞(Ω), with ϕ≥0, we have lim sup m→∞ lim sup n→∞ ZΩ|Rm n(x)|2ϕdx ≤CZΩ w2ϕ dµ.(8.9) Proof. Let s∈RM, with |s|≤1. Let us define Ψk:RM7→ RMby Ψk(ζ)= ζ−Φ k (ζ), where Φkis the function defined by (8.7). Taking Ψk(sm n−wns)ϕ, with Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 617 ϕ∈H1 0(Ω) ∩L∞(Ω) and ϕ≥0, as a test function in (8.2), we have ZΩ a(x)Dsm nD[Ψk(sm n−wns)]ϕdx+ZΩ a(x)Dsm nΨk(sm n−wns)⊗Dϕ dx +mZΩ (sm n−wns)Ψk(sm n−wns)ϕdx+ZΩ F n(x)s m nΨ k(s m n−w ns)ϕdµ n=0, where the second term tends to zero when nand then mtend to infinity and where the third term in the left-hand side is positive. This permits us to write αZ{|sm n−wns|≤2k−1W|s|} |D(sm n−wns)|2ϕdx+ZΩ F n(x)(sm n−wns)Ψk(sm n−wns)ϕdµ n ≤CZ{2 k−1W|s|<|s m n−w ns|<2 kW|s|} |Dsm n||D(sm n−wns)|ϕdx +CZ{|sm n−wns|≤2k−1W|s|} |D(wns−ws)||D(sm n−wns)|ϕdx(8.10) +CZ{|sm n−wns|≤2k−1W|s|} |Dws||D(sm n−wns)|ϕdx +CZΩ|w ns||Ψk(sm n−wns)|ϕdµ n+o m,n. Since for k≥2 |sm n|≥|s m n−w n s|−|w n s|≥|s m n−w n s|−W|s|≥2 k−2 W|s|(8.11) in the set {|sm n−wns|≥2 k−1 W|s|},byH¨older’s inequality, (6.38), and Lemma 8.7, we obtain Z{2k−1W|s|<|sm n−wns|<2kW|s|} |Dsm n||D(sm n−wns)|ϕdx ≤CkϕkL ∞(Ω) √k−2|s|2+om,n. By the definition of Ψkand (Il), we have ZΩ Fn(x)(sm n−wns)Ψk(sm n−wns)ϕdµ n≥αZΩ (s m n−w ns)Ψk(sm n−wns)ϕdµ n ≥αZΩ|Ψ k(s m n−w ns)|2ϕdµ n. Therefore, using Young’s inequality in (8.10) and taking into account that |s|≤1 and that the third term of the right-hand side of (8.10) tends to zero when nand mtend to infinity, we get Z{|sm n−wns|≤2k−1W|s|} |D(sm n−wns)|2ϕdx+ZΩ|Ψ k(s m n−w ns)|2ϕdµ n ≤CkϕkL ∞(Ω) √k−2+ZΩ|D(wns−ws)|2ϕdx+ZΩ|w ns|2ϕdµ n+o m,n ≤CkϕkL∞(Ω) √k−2+CZΩ w2ϕdµ+o m,n, (8.12) Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
618 JUAN CASADO DIAZ AND ADRIANA GARRONI where in the last inequality we used Lemma 4.2. Thus by (8.12), (8.11), and Lemmas 8.7 and 4.2, we have ZΩ|Rm n(x)s|2ϕdx =ZΩ|D(s m n−ws)|2ϕdx ≤2Z{|sm n−wns|≤2k−1W|s|} |D(sm n−wns)|2ϕdx+2 Z{|sm n−wns|≤2k−1W|s|} |D(wns−ws)|2ϕdx +2Z{|sm n−wns|>2k−1W|s|} |Dsm n|2ϕdx+2Z{|sm n−wns|>2k−1W|s|} |D(ws)|2ϕdx ≤CkϕkL ∞(Ω)1 √k−2+1 k−2+CZΩ w2ϕdµ+o m,n, which by the arbitrariness of kimplies lim sup m→∞ lim sup n→∞ ZΩ|Rm n(x)s|2ϕdx ≤CZΩ w2ϕ dµ. Since |Rm n(x)|= max{|Rm n(x)s|:|s|≤1}≤ N X i=1 |Rm n(x)ei|, where {ei:i≤i≤N}is the canonical basis of RN, Lemma 8.8 is proved. Remark 8.9. If in Lemma 8.8, ϕbelongs to C∞ c(Ω), then estimate (8.9) may be easily deduced from estimate (6.9) in Lemma 6.6. Remark also that Lemmas 8.7 and 8.8 can be easily generalized to the nonlinear case. Proof of Theorem 8.6.By Lemma 8.8, for every closed K⊂Ω0and for every function ψ∈H1(Ω0,RM), with ψ≥0, we have lim sup m→∞ lim sup n→∞ ZK|Rn m(x)|2ψdx ≤CZK w 2ψ dµ.(8.13) Indeed it is enough in (8.9) to take ϕequals to ϕnψ, with ϕn∈H1 0(Ω0,RM)∩ L∞(Ω0,RM) decreasing to the characteristic function of K. Consider ψ∈H1(Ω0,RM)∩L∞(Ω0,RM) and let Kbe a closed subset of Ω0.By Theorem 7.1, for any function ζ=Pl i=1 si1Ki, with si∈Rand Kiclosed subsets of Ω0, such that K=Sl i=1 Kiand w=0µ-a.e. on Ki∩Kj, for i6=j,wehave ZK|Dun−Du −Rm n(x)ψ|2dx ≤2ZK|Dun−Du −Rm n(x)ζ|2dx +2ZK|R m n(x)|2 |ψ−ζ|2dx ≤CZK|u−wζ|2dµ +2 l X i=1 ZKi|Rm n(x)|2|ψ−si|2dx +om,n ≤CZK|u−wζ|2dµ +C l X i=1 ZKi|wψ −siw|2dµ +om,n =CZK|u−wζ|2dµ +CZK|wψ −wζ|2dµ +om,n, (8.14) Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 619 where we used (8.13). In order to obtain (8.4), it is enough to take ζ=ζk, where (ζk) is a sequence of step functions such that (wζk) converges strongly to wψ in L2 µ(Ω0,RM). Assume now that u/w belongs to L∞(K, RM) and take ε>0. By estimates (8.4) and (8.13), we get ZKDun−Du −Rm n(x)u w 2dx ≤ZK Dun−Du −Rm n(x)u w+ε 2 dx +ZK|Rm n(x)|2 u w+ε−u w 2 dx ≤ZK u−wu w+ε 2 dµ +Cε2° ° ° u w° ° ° 2 L∞(K,RM)ZK|Rm n(x)|21 (w+ε)2dµ +om,n ≤ZK u−wu w+ε 2 dµ +Cε2° ° ° u w° ° ° 2 L∞(K,RM)ZK w2 (w+ε)2dµ +om,n. (8.15) By using that ubelongs to L2 µ(Ω,RM) and the dominated convergence theorem, the first integral of the right-hand side of (8.15) tends to zero when εtends to zero. Since εw (w+ε)2=ε w+ε w w+ε≤1, and hence by the fact that ν=wp−1µis a Radon measure, εZK w2 (w+ε)2dµ ≤ZK wdµ<+∞, we get that the second integral on the right-hand side of (8.15) tends to zero when ε tends to zero. We deduce (8.5) taking the limit in n,m, and then in ε. 9. Asymptotically equivalent operators. We saw in the previous sections that the properties of the function Fwhich appears in the limit problem (6.2) are strictly related to the properties of the function awhich define the differential operator A. The next proposition shows, in some sense, how the function Fdepends on the behavior of a(x, ξ) when |ξ|is large. Let ˜a:Ω×M M×N→M M×Nbe a Carath´eodory function which satisfies conditions (i)–(v), and suppose that the following property lim |ξ|→∞ |a(x, ξ)−˜a(x, ξ)| |ξ|p−1=0(9.1) holds uniformly with respect to xin Ω. Let ˜ Abe the differential operator given by ˜ Au =−div (˜a(x, Du)). Proposition 9.1. Suppose that the pair (µn,F n), according to Definition 6.2, γA-converges to (µ, F). If the functions aand ˜asatisfy condition (9.1), then we also have that (µn,F n) γ˜ A-converges to (µ, F). Proof. According to the definition of the γ˜ A-convergence, we have to show that for any open subset Ω0of Ω, for any sequence of functionals (fn), with fn∈ (W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM))0, which converges to some f∈(W1,p 0(Ω0,RM)∩ Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
620 JUAN CASADO DIAZ AND ADRIANA GARRONI Lp µ(Ω0,RM))0in the sense of (HΩ0), and for any sequence (un) which satisfies (6.1) and un∈W1,p(Ω0,RM)∩Lp µn(Ω0,RM), ZΩ0 ˜a(x, Dun)Dv dx +ZΩ0 Fn(x, un)vdµ n=hf n,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM), (9.2) every cluster point of the sequence (un) in the weak topology of W1,p(Ω0,RM) satisfies problem u∈W1,p(Ω0,RM)∩Lp µ(Ω0,RM), ZΩ0 ˜a(x, Du)Dv dx +ZΩ0 F(x, u)vdµ =hf,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µ(Ω0,RM). (9.3) If unsatisfies (9.2), then it also satisfies un∈W1,p(Ω0,RM)∩Lp µn(Ω0,RM), ZΩ0 a(x, Dun)Dv dx +ZΩ0 Fn(x, un)vdµ n=hg n,vi ∀v∈W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM), where gn=fn−div [a(x, Dun)−˜a(x, Dun)]. Therefore, once we show that (div [a(x, Dun)− ˜a(x, Dun)]) converges in the sense of (HΩ0) to div [a(x, Du)−˜a(x, Du)], by the γA- convergence of (µn,F n)to(µ, F), we can deduce that usatisfies (9.3). In order to prove that (−div [˜a(x, Dun)−a(x, Dun)]) converges in the sense of (HΩ0), let us consider vn∈W1,p 0(Ω0,RM)∩Lp µn(Ω0,RM) such that (vn) converges weakly to some vin W1,p 0(Ω0,RM). Since by Proposition 5.4 the sequence (Dun) converges to Du pointwise a.e. in Ω0, by Egorov’s theorem, for every δ>0, there exists a set E⊆Ω0, with |E|<δ, such that (Dun) converges uniformly to Du in Ω0\E.Thusweget lim n→∞ ZΩ0 [˜a(x, Dun)−a(x, Dun)]Dvndx =ZΩ0\E [˜a(x, Du)−a(x, Du)]Dv dx + lim n→∞ ZE [˜a(x, Dun)−a(x, Dun)]Dvndx. (9.4) Let us estimate the last limit in (9.4). By (9.1), for every ε>0 there exists M>0 such that |˜a(x, ξ)−a(x, ξ)|≤ε|ξ| p−1 whenever |ξ|>M. Thus, since (vn) and (un) are bounded in W1,p(Ω0,RM), by Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
ASYMPTOTIC BEHAVIOR OF NONLINEAR ELLIPTIC SYSTEMS 621 H¨older’s inequality, we get lim n→∞ZE [˜a(x, Dun)−a(x, Dun)]Dvndx ≤lim n→∞ ZE∩{|Dun|>M}Ω0|˜a(x, Dun)−a(x, Dun)||Dvn|dx +ZE∩{|Dun|≤M}Ω0|˜a(x, Dun)−a(x, Dun)||Dvn|dx ≤C(ε+Mp−1δ(p−1)/p). (9.5) Now taking the limit as δgoes to zero and then the limit as Mgoes to infinity, by (9.4) and (9.5) we obtain lim n→∞ ZΩ0 [˜a(x, Dun)−a(x, Dun)]Dvndx =ZΩ0 [˜a(x, Du)−a(x, Du)]Dv dx, which concludes the proof. Corollary 9.2. Let (Fn)be a sequence in F(L)which satisfies condition (VI) and assume that the function asatisfies the following condition: There exists a Carath´eodory function ˜asuch that lim t→∞ a(x, tξ) |t|p−2t=˜a(x, ξ)(9.6) uniformly in x, for every ξ∈MM×N. Suppose that the pair (µn,F n), according to Definition 6.2, γA-converges to (µ, F). Then the function Falso satisfies condition (VI). Proof. It is easy to see that ˜asatisfies conditions (i)–(vi) and that condition (9.6) implies condition (9.1). Thus by the previous theorem the sequence of pairs (Fn,µ n)γ˜ A-converges to (µ, F) and by Theorem 8.1 the function Fsatisfies condition (VI). 10. General operators. In this section we shall prove that the results given in the previous sections hold for a class of more general operators. Actually, let 2≤p<+∞and let b:Ω×R M×M M×N7→ MM×Nbe a Carath´eodory function such that: (i0) there exists a constant α>0 such that (b(x, 0,ξ 1)−b(x, 0,ξ 2))(ξ1−ξ2)≥α|ξ1−ξ2|p for every s∈RM, for every ξ1,ξ 2∈M M×N, and for a.e. x∈Ω; (ii0) there exists a constant β>0 and a function h∈Lp p−2(Ω) (p/(p−2)=+∞ if p= 2) such that |b(x, 0,ξ 1)−b(x, 0,ξ 2)|≤β(h(x)+(|ξ 1 |+|ξ 2 |) p−2 )|ξ 1−ξ 2 | for every ξ1,ξ 2∈M M×Nand for a.e. x∈Ω; (iii0) there exists a constant γ>0 and a function k∈Lp0(Ω) such that |b(x, s1,ξ)−b(x, s2,ξ)|≤γ ¡k(x)+(|s 1 |+|s 2 |) q+|ξ|rmin{|s1−s2|,1} for every s1,s 2∈R M, for every ξ∈MM×Nand for a.e. x∈Ω, where qand rare constants which satisfy 0 ≤q<N(p−1)/(N−p)ifp<N,q≥0if p≥Nand 0 ≤r<p−1. Downloaded 09/30/16 to 150.214.182.15. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php