Asymptotic behaviour of equicoercive diffusion energies in dimension two

Asymptotic behaviour of equicoercive diffusion energies in dimension two Marc BRIANE Juan CASADO-D´ IAZ Centre de Math´ematiques Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico I.N.S.A. de Rennes & I.R.M.A.R. Universidad de Sevilla m[email protected] jcasado[email protected] November 3, 2006 Abstract In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies Fn,n∈N, defined in L2(Ω), for a bounded open subset Ω of R2. We prove that, contrary to the three dimension (or greater), the Γ-limit of any convergent subsequence of Fnis still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence Fn, which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity. 1 Introduction This paper deals with the asymptotic behaviour of sequences of diffusion energies in a bounded open subset Ω of R2. The prototype of the diffusion energy is given by the following quadratic functional defined in L2(Ω): Fn(u):=     ZΩ An∇u·∇u dx if u∈H1 0(Ω). +∞if u∈L2(Ω) \H1 0(Ω), for n∈N,(1.1) where Anis a symmetric positive definite matrix-valued function in L∞(Ω)2×2. The knowledge of the limit behaviour of Fnis crucial in the homogenization theory applied to conduction problems (see e.g. [1] for an introduction), since Anthen represents the conductivity matrix of a given heterogeneous medium. In this context, Spagnolo [23], with the G-convergence theory, and Murat & Tartar [25], [20], with the H-convergence theory, proved the compactness of the sequence Fn, when Anis assumed to be both equicoercive and equibounded. A few times later, Buttazzo & Dal Maso [10] and Carbone & Sbordone [12] extended the result of compactness by only assuming that the sequence Anis bounded and equiintegrable in L1(Ω)2×2. At the same period, Fenchenko & Khruslov [15] showed that the equiintegrability condition cannot be relaxed since high conductivity regions in three dimension may induce nonlocal effects which correspond to a lack of compactness in the homogenization process (see also [2], [9], [4] for different approaches). 1 Nonlocal effects naturally appear in the limit behaviour of the diffusion energy. Indeed, using the Beurling-Deny [3] theory Mosco [18] proved in particular that any sequence Fn Γ-converges, up to a subsequence, for the strong topology of L2(Ω) (see Definition 3.1) to a Dirichlet form (see Definition 3.3). According to the Beurling-Deny formula any Dirichlet form can always be split up into three terms: a strongly local form (the diffusion part), a local form and a nonlocal one. Inversely, Camar Eddine & Seppecher [11] proved that any Dirichlet form in L2 loc(R3) can be obtained as the Γ-limit of a sequence of diffusion energies of type (1.1) with a suitable isotropic conductivity An. The nonlocal effects obtained in the previous works are based on three-dimensional microstructures whose model example is a medium reinforced by a periodic lattice of high conductivity thin fibers. Then, it is natural to ask if the appearance of nonlocal effects is specific to the three dimension (or greater). Recently, in [6] for periodic microstructures and more generally in [7], we showed that the answer is positive. Assuming that the sequence Anis both equicoercive and bounded in L1(Ω)2×2, we proved that the Γ-limit of any sequence of type Fnis a strongly local Dirichlet form. Therefore, the dimension two preserves the compactness in the homogenization process. The proof in [6], [7] is based on two-dimensional div-curl type lemmas which extend the one of Murat & Tartar [26], [21]. Note that the equicoerciveness assumption is essential to obtain strongly convergent sequences in L2(Ω). However, the use of div-curl lemmas is strictly limited to conductivity sequences which are bounded in L1(Ω)2×2. In this paper, we study the asymptotic behaviour of the sequence of diffusion energies (1.1), without assuming any boundedness assumption on An. Our approach is completely different of the one used in [6] or [7] for Anbounded in L1(Ω)2×2. The keyingredient of the method is a uniform convergence result satisfied by some energy minimizers (see Section 2). More precisely, we prove (see Theorem 2.1) that for any bounded energy (with respect to Fn) sequence in H1(Ω) which strongly converges in L2(Ω) to a continuous function, there exists a smaller energy subsequence which strongly converges to the same limit in L∞ loc(Ω). The proof of this result uses that the p-capicity, for p∈(1,2), of a continuous curve is positive (see Lemma 2.8), which is specific to the dimension two. This combined with the continuity of the limit and the maximum principle allows us to construct a uniformly convergent sequence. Up to our knowledge, the previous result provides new uniform estimates on solutions of uniformly elliptic partial differential equations without any control from above on the coefficients. Here, we give an example (see Corollary 2.5) of such an estimate for A-harmonic functions, where Ais any uniformly elliptic (but not necessarily uniformly bounded) matrix-valued in L∞(O), for a bounded open subset Oof R2. This uniform estimate is used in the last section of the paper. More general cases are the subject of a work in progress [8]. On the other hand, thanks to the uniform convergence result of Section 2 and under the only assumption of equicoerciveness for An, we prove (see Section 3 and Theorem 3.6) that the diffusion energy FnΓ-converges (up to a subsequence) for the strong topology of L2(Ω) to a strongly local Dirichlet form F. Moreover, if the domain of the Γ-limit F contains the space C1 c(Ω) of the C1-regular functions with compact support in Ω, we obtain (see Theorem 3.4) the following representation formula F(u) = ZΩ A∇u·∇u dµ, ∀u∈C1 c(Ω),(1.2) where µis a Radon measure on Ω and Aa matrix-valued function in L∞ µ(Ω)2×2. In other words, the sequence of the diffusion energies Fnis relatively compact for the L2(Ω)- strong Γ-convergence topology in the set of the uniformly coercive diffusion energies. In particular, the compactness result implies that the limit of the energy density Andx is still a density of type A dµ. 2 The compactness result of Section 3 has a remarkable application in the periodic homogenization framework. In this context, the conductivity Anis a highly oscillating sequence defined by An(x) := Bn(x εn), where Bnis an equicoercive sequence of (0,1)2- periodic matrix-valued functions in L∞(R2)2×2and εnis a positive sequence converging to zero. Associated with Bnthe constant matrix A∗ n(see formula (4.4)) obtained, for a fixed n, from the periodic homogenization of Bn(x ε) as ε→0 (see e.g. [1]), plays a fundamental role in the homogenization process. Indeed, extending [6] we prove (see Theorem 4.1) that the asymptotic behaviour of the diffusion energies (1.1) is completely determined by the limit behaviour of the matrix A∗ n, according to the following alternative: •if the spectral radius ρ(A∗ n) of A∗ nis bounded, A∗ nconverges, up to a subsequence, to a matrix A∗and the Γ-limit Fof Fnsatisfies (1.2) with the constant density A∗dx, in the whole space H1 0(Ω); •if ρ(A∗ n) tends to +∞, the domain of the Γ-limit reduces to {0}. As an immediate consequence, the question on the asymptotic behaviour of the twodimensional periodic conduction problem (−div Bn(x εn)∇un=fin Ω un= 0 on ∂Ω,for f∈H−1(Ω),(1.3) is now definitively solved under the only assumption of equicoerciveness for Bn: •if ρ(A∗ n) is bounded, (1.3) converges, up to a subsequence, to the conduction problem with the constant conductivity lim n→+∞A∗ n; •if ρ(A∗ n) tends to +∞, the potential unof (1.3) strongly converges to zero in H1 0(Ω). The paper is organized as follows. Section 2 is devoted to the uniform convergence results and Section 3 to the Γ-convergence of sequences of diffusion energies of type (1.1). In Section 4 we apply the results of Section 3 to the periodic framework. Notations •N∗:= N\{0}denotes the set of the positive integers; •a∨b, resp. a∧b, denotes the maximum, resp. the minimum, of a, b ∈R; •B(x0, δ) denotes the disk of center x0∈R2and of radius δ > 0; •χEdenotes the characteristic of the set E; • ∃ lim means that the limit does exist; •Ω denotes an open subset of R2and ¯ Ω the closure of Ω in R2; •Hloc(Ω) means locally in the space H(Ω); •C(Ω) denotes the space of the continuous functions in Ω, C0(Ω) the subspace of C(Ω) composed of the functions which are zero on the boundary of Ω, Cc(Ω) the subspace of C0(Ω) composed of the functions with compact support in Ω, and Ck c(Ω), for k∈ N∩{+∞}, the subspace of Cc(Ω) composed of the k-th continuously differentiable functions in Ω; •D0(Ω) denotes the set of the distributions on Ω; •M(Ω) denotes the set of the Radon measures on Ω; 3 •a sequence µnin M(Ω) converges to µ∈M(Ω) in the weak ∗sense of the measures in Ω if lim n→+∞ZΩ ϕ dµn=ZΩ ϕ dµ, ∀ϕ∈C0(Ω), and the convergence is denoted by µn* µ in M(Ω) ∗; •q.e. means quasi-everywhere in the sense of the 2-capacity in R2, and a.e. means everywhere in the sense of the Lebesgue measure in R2; •for any p∈(1,2) and for any subset Eof R2,Cp(E) denotes the p-capacity of E with respect to R2, which is defined by Cp(E) := inf ZR2|∇u|pdx :u∈D1,p(R2), u ≥1 a.e. in a neighbourhood of E, where D1,p(R2) is the space of the functions uin L 2p 2−p(R2) such that ∇u∈L2(R2)2. 2 Uniform convergence results 2.1 Statement of the results Let Ω be an open subset of R2. In this section, we consider a given sequence of symmetric matrix-valued functions An∈L∞(Ω)2×2,n∈N, which satisfies the following equicoerciveness property in Ω ∃α > 0 such that ∀n∈N,∀ξ∈R2, Anξ·ξ≥α|ξ|2a.e. in Ω.(2.1) For any function u∈H1(Ω)∩C(Ω), we will study some questions related to the existence of sequences unin H1(Ω) which both converge uniformly to uin Ω and satisfy the following minimization property ∃lim n→+∞ZΩ An∇un·∇undx ≤lim inf n→+∞ZΩ An∇vn·∇vndx, for any sequence vnin H1(Ω) (some boundary conditions can be added), which strongly converges to uin L2(Ω). Our main result in this way is the following theorem: Theorem 2.1. Let ube a function in H1(Ω) ∩C(Ω) and let ˆunbe a sequence H1(Ω) which strongly converges to uin L2(Ω) and satisfies ∃lim n→+∞ZΩ An∇ˆun·∇ˆundx < +∞. Then, up to a subsequence of n, still denoted by n, there exists un∈H1(Ω) which satisfies lim sup n→+∞ZΩ An∇un·∇undx ≤lim n→+∞ZΩ An∇ˆun·∇ˆundx, (2.2) and un−→ ustrongly in L∞ loc(Ω).(2.3) Moreover, if the support of uis contained in a compact set Kof Ω, then we can take un such that un= 0 q.e. in Ω\K, for any n∈N. Remark 2.2. If in Theorem 2.1 the sequence ˆunis in H1 0(Ω) and uin H1 0(Ω) ∩C0(Ω), then we can choose unin H1 0(Ω), which strongly converges to uin L∞(Ω). To this end, it is enough to consider a bounded open set ˜ Ω such that ¯ Ω⊂˜ Ω and to apply the second part of Theorem 2.1 to the sequences ˜unand ˜ Andefined by ˜un(x):=un(x) if x∈Ω 0 if x∈˜ Ω\Ω,and ˜ An(x):=An(x) if x∈Ω I2if x∈˜ Ω\Ω. 4 Corollary 2.3. Consider ˆun∈H1(Ω) and u∈H1(Ω) ∩C(Ω) such that ˆun* u weakly in H1 loc(Ω) and div (An∇ˆun) = 0 in D0(Ω).(2.4) Then, we have the following uniform convergence ˆun−→ ustrongly in C(Ω).(2.5) Remark 2.4. Note that in Corollary 2.3 each function unis continuous in Ω by the De Giorgi-Stampacchia theorem (see e.g. [16] Chapter 8). Corollary 2.5. For any open subset Ωof R2and any compact subset Kof Ω, there exists a constant C > 0which only depends on Ωand Ksuch that, for any matrix-valued function A∈L∞(Ω)2×2satisfying the uniform coerciveness (2.1) and any function u∈H1(Ω) solution of div (A∇u) = 0 in D0(Ω), the following estimate holds true kukC(K)≤CkukH1(Ω). Remark 2.6. Corollaries 2.3 and 2.5 can be deduced from [8] where more general results, for non-necessarily homogeneous equations, are proved. 2.2 Proof of the results Let us now give the proof of the uniform convergence results stated in the previous section. We will need the two following lemmas: Lemma 2.7. Let Obe a bounded open subset of R2and let u∈H1(O)∩C(¯ O). Denote M:= max ∂O uand m:= min ∂O u. Then, for any function vsuch that v−ubelongs to H1 0(O), the functions (v−M)+and (m−v)+belong to H1 0(O). Proof. Consider ϕε∈C∞ c(O), for ε > 0, which strongly converges to v−uin H1 0(Ω) as ε→0. Since uis continuous in ¯ O, the functions Uε:= (u+ϕε−M−ε)+and uε:= (m−ε−u−ϕε)+have compact support in O. The functions Uεand uεbelong to H1(O), hence they also belong to H1 0(O). Therefore, using that Uεand uεstrongly converge respectively to (v−M)+and (m−v)+in H1(O), yields the result. Lemma 2.8. For any p∈(1,2) and any continuous curve Lof extremities a, b, we have Cp(L)≥Rp|a−b|2−p,(2.6) where Rp>0is the p-capacity of a unit segment in R2. Proof. Using a translation, a rotation and a homothety, we can reduce the proof to the case where a= (0,0), b= (1,0). Then, consider a curve Lof extremities (0,0), (1,0) and take a function ϕ∈C∞ c(R2) such that ϕ>χL. By the P´olya-Szeg¨o inequality [22] extended to any power p≥1 (see e.g. [24] and Chapter I.4 of [19]), it is known that the Steiner symmetrization ϕ∗of ϕwith respect to {x2= 0}, defined by its level sets (x1, x2)∈R2:ϕ∗(x1, x2)> c=(x1, x2)∈R2:|x2|<1 2{y∈R:ϕ(x1, y)> c}, 5 belongs to W1,p c(R2) and satisfies ZR2|∇ϕ∗|pdx ≤ZR2|∇ϕ|pdx. Moreover, since Land ϕare continuous, it is clear that ϕ∗>1 in [0,1] ×{0}, hence Cp([0,1] ×{0})≤ZR2|∇ϕ∗|pdx ≤ZR2|∇ϕ|pdx. Taking the infimum in ϕ, we get the desired estimate (2.6). Proof of Theorem 2.1. Using the density of H1(Ω) ∩C∞(Ω) in H1(Ω) (see e.g. [16]), we can also assume that ˆunis continuous in Ω. For any δ > 0, we define Ωδby Ωδ:= {x∈Ω : d(x, ∂Ω) > δ}. Since uis continuous in ¯ Ωδ, for any l∈N∗, there exists δl>0, with lim l→+∞δl= 0, such that |u(x)−u(y)|<1 2l,∀x, y ∈¯ Ωδl,with |x−y| ≤ δl.(2.7) Let p∈(1,2). Since ˆunweakly converges in H1(Ω), there exists (see e.g. [14]) a subsequence of ˆun, still denoted by ˆun, which converges to u Cp-quasi uniformly in every open set ω⊂Ω, with ¯ω⊂Ω (ωcan be chosen as Ω if Ω is smooth). Thus, we can choose this sequence in such a way that for any l∈N∗, there exists a relatively closed subset Klof Ω satisfying Cp(Ω \Kl)< Rpδ2−p l,(2.8) |ˆun(x)−u(x)|<1 2l,∀x∈Ωδl∩Kl,∀n≥l. (2.9) Then, we define unby      un:= ˆunin Kl, un−ˆun∈H1 0(Ω \Kl), ZΩ\Kl An∇un·∇undx ≤ZΩ\Kl An∇v·∇v dx, ∀v, v −ˆun∈H1 0(Ω \Kl).(2.10) Clearly, unsatisfies (2.2). Let us prove that unstrongly converges to uin L∞ loc(Ω). To this end, we fix δ > 0. We have |un(x)−u(x)|=|ˆun(x)−u(x)|<1 2l,∀x∈Ωδ∩Kl,∀n≥l, with δl< δ. (2.11) Consider a connected component Oof Ω \Klsuch that O∩Ωδ6= Ø. Since Ois opened and connected, it is connected by curves. Thus, for any y1, y2∈O, there exists a curve L⊂Owhich connects y1, y2. By Lemma 2.8 and (2.8), we have Rp|y1−y2|2−p≤Cp(L)≤Cp(O)≤Cp(Ω \Kl)≤Rpδ2−p l, hence diam (O)≤δl. Then, taking llarge enough such that 2δl< δ, we get that ¯ O⊂Ωδl, and in particular, ∂O ⊂Ωδl∩Kl. Denote mn:= min ∂O ˆunand Mn:= max ∂O ˆun. By Lemma 2.7 (un−Mn)+and (mn−un)−belong to H1 0(O), and by definition (2.10) un is An-harmonic in O. Then, the maximum principle yields mn≤un≤Mn,q.e. in O, ∀n∈N.(2.12) 6 On the other hand, since ∂O ⊂Ωδl∩Kl, we have by (2.9). mn≥min ∂O u−1 2land Mn≤max ∂O u+1 2l,∀n≥l. Moreover, ¯ O⊂Ωδl, diam (O)≤δland (2.7) imply that min ∂O u≥u(x)−1 2land max ∂O u≤u(x) + 1 2l,∀x∈O, ∀n≥l. Therefore, (2.12) combined with the two previous estimates yields |un−u| ≤ 1 2l,q.e. in O, hence the sequence unsatisfies the uniform convergence (2.3). Now, assume that the support of uis contained in a compact subset Kof Ω, and consider an open set ˜ Ω which contains Kand is strictly contained in Ω. For any ε > 0, let Sεbe the function defined by Sε(s) := (s−εsgn(s)))χ{|s|>ε}, for s∈R. Since un strongly converges to uin L∞(˜ Ω), the sequence εn:= kun−ukL∞(˜ Ω) tends to zero. Therefore, the sequence ˜un:= χ˜ ΩSεn(un) satisfies conditions (2.2), (2.3) and vanish q.e. in Ω \K. Proof of Corollary 2.3. Since ˆunsatisfies div (An∇ˆun) = 0 in D0(Ω), it is H¨older continuous in Ω by the De Giorgi-Stampacchia theorem. Then, the argument used in the proof of Theorem 2.1 proves that the sequence undefined by (2.10) strongly converges to uin L∞ loc(Ω). However, we have un= ˆunby construction, which yields the desired result.  Proof of Corollary 2.5. We reason by contradiction. If the result does not hold true, then, for any n∈N, there exist un∈H1(Ω), An∈L∞(Ω)2×2and γn>0 such that Anξ·ξ≥γn|ξ|2,∀ξ∈R2,a.e. x∈Ω, and kunkC(K)> nkunkH1(Ω).(2.13) Up to replace Anby An/γnand unby un/kunkC(K), we can assume that γn= 1 and kunkC(K)= 1. Then, Anis equicoercive and by (2.13) unstrongly converges to zero in H1(Ω). Therefore, by Corollary 2.3 unconverges uniformly to zero in K, in contradiction with kunkC(K)= 1.  3Γ-limit of equicoercive diffusion energies 3.1 Γ-convergence and Dirichlet forms In this section we first recall the definition of the De Giorgi Γ-convergence and some of its properties which will be used in the sequel. We refer to [13] for an exhaustive presentation of the Γ-convergence. Definition 3.1. A sequence of functionals Fn:L2(Ω) −→ [0,+∞] is said to Γ-converge to F:L2(Ω) −→ [0,+∞] for the strong topology of L2(Ω) if, for any uin L2(Ω), (i) the Γ-liminf inequality holds ∀un−→ ustrongly in L2(Ω), F(u)≤lim inf n→+∞Fn(un),(3.1) 7 (ii) the Γ-limsup inequality holds ∃¯un−→ ustrongly in L2(Ω), F(u) = lim n→+∞Fn(¯un).(3.2) Any sequence satisfing (3.2) will be called a recovery sequence for Fn, of limit u. In the sequel, we will always consider the Γ-convergence with respect to the strong topology of L2. Consequently, this topology will be not necessarily mentionned. Properties 3.2. a)Since L2(Ω) is separable, any sequence of functionals Fn:L2(Ω) −→ [0,+∞]has a subsequence which Γ-converges with respect to the strong topology of L2(Ω). b)Let Fn:L2(Ω) −→ [0,+∞]be a sequence of quadratic forms which Γ-converges to F. Then, Fis a quadratic form on L2(Ω) which is semi-lower continuous with respect to the strong topology of L2(Ω). c)Let Fn:L2(Ω) −→ [0,+∞]be a sequence of quadratic forms which Γ-converges to F. Let Φn,Φbe the polar forms respectively associated with Fn, F on their domains. Then, for any u∈L2(Ω), with F(u)<+∞, a sequence unin L2(Ω) is a recovery sequence (3.2) for Fn, of limit u, if and only if ∀vn−→ vstrongly in L2(Ω),with Fn(vn)≤c, lim n→+∞Φn(un, vn) = Φ(u, v),(3.3) or equivalently, (3.3) with v= 0. Now, we recall some notions about Dirichlet forms, which will be used in the statement of Theorem 3.8. We refer to [18] for more details in connection with the Γ-convergence. Definition 3.3. Let Xbe a Hausdorff, separable, locally compact space, and let mbe aσ-finite nonnegative Radon measure on X. Let Hbe the space L2 m(X) endowed with its Hilbert norm k·kH. Let F:H−→ [0,+∞] be a quadratic form of domain D(F) := {u∈H:F(u)<+∞}, whose polar form Φ is a bilinear form defined in D(F)×D(F). (i) The form Fis said to be closed if it is semi-lower continuous with respect to the norm k·kH. The form Fis said to be closable if there exists an extension ˜ Fof F in Hsuch that D(F)⊂D(˜ F). The closure of a closable form is its smallest closed extension in H. (ii) The form Fis said to be Markovian if ∀u∈D(F), v := (u∨0) ∧1∈D(F) and F(v)≤F(u). (iii) A Dirichlet form on His a closed Markovian quadratic form defined in H. (iv) The form Fis said to be regular if there exists a subset of D(F)∩C0(X), which is dense both in C0(X) with the uniform norm and in D(F) with the norm (F+k·kH)1/2. (v) The form Fis said to be local if Φ(u, v) = 0,∀u, v ∈D(F),with supp (u)∩supp (v) = Ø. The form Fis said to be strongly local if Φ(u, v) = 0,∀u, v ∈D(F),with u= cst in supp (v).(3.4) Thanks to the Beurling-Deny theory [3] any regular Dirichlet form Fon L2 m(X) can be split up on its domain into three specific forms: a strongly local form Fd, a local form and a nonlocal one. More precisely, the following representation formula holds for any u∈D(F), F(u) = Fd(u) + ZX u2(x)k(dx) + ZZX×X\diag u(x)−u(y)2j(dx, dy),(3.5) where Fdis called the diffusion part of F,kthe killing measure and jthe jumping measure. 8 3.2 Statement of the results As in Section 2, let us consider a bounded open subset Ω of R2, and a sequence of symmetric matrix-valued functions An∈L∞(Ω)2×2which satisfy (2.1). For any n∈N and any open subset ωof Ω, we define the quadratic form Fn(·, ω) in L2(ω) by Fn(u, ω):=     Zω An∇u·∇u dx if u∈H1 0(ω). +∞if u∈L2(ω)\H1 0(ω). (3.6) The form Fn(·,Ω) is simply denoted by Fn. Assume that FnΓ-converges to some quadratic form Ffor the topology of L2(Ω), which holds true for a subsequence in virtue of Properties 3.2 a). Since Fnis clearly Markovian, the properties (3.1), (3.2) of the Γ-convergence imply that Fis also Markovian. Moreover, thanks to Properties 3.2 b)Fis closed. Therefore, Fis a Dirichlet form in the sense of Definition 3.3 (iii). The following result gives a necessary and sufficient condition to have Fregular with C1 c(Ω) ⊂D(F). When this condition is satisfied, Fis a strongly local (3.4) Dirichlet form whose integral representation for regular functions is independent of the domain. Theorem 3.4. The domain D(F)of Fcontains C1 c(Ω) if and only if, for any x0∈ Ω, there exists δ > 0, two functions w1, w2in C1(B(x0, δ)) and two sequences w1 n, w2 n in H1(B(x0, δ)),n∈N, such that              B(x0, δ)⊂Ω, ∇w1(x0),∇w2(x0)are linearly independent, wi n−→ wistrongly in L2(B(x0, δ)) ,for i= 1,2, An∇wi n·∇wi nis bounded in L1(B(x0, δ)) ,for i= 1,2. (3.7) Assume that C1 c(Ω) is contained in D(F). Then, there exist a nonnegative Radon measure µon Ωand a nonnegative matrix-valued function Ain L∞ µ(Ω)2×2such that the regular part Arof Adµ with respect to the Lebesgue measure satisfies Arξ·ξ≥α|ξ|2,∀ξ∈R2,a.e. in Ω,(3.8) and such that, for any open set ωof Ω, the Γ-limit F(·, ω)of Fn(., ω)with respect to the strong topology of L2(ω)does exist on C1 c(ω)and reads as F(u, ω) = Zω A∇u·∇u dµ, ∀u∈C1 c(ω).(3.9) Moreover, for any u∈C1 c(ω)and any un∈H1 0(ω)which strongly converges to uin L2(ω) and such that Fn(un, ω)tends to F(u, ω), the sequence An∇un·∇unconverges to A∇u· ∇u dµ in the weak ∗sense of the measures in ω. Remark 3.5. In the second part of Theorem 3.4 the Γ-convergence of Fn(·, ω) to F(·, ω) holds true up to a subsequence which does depend on the open set ω. However, the integral representation (3.9) of F(u, ω), which is valid on C1 c(ω), is independent of ω. In fact, the integral expression (3.9) holds true for any u∈C1 0(ω), with A∇u·∇u∈ L1 µ(ω). Indeed, it is easy to check that these functions can be approximated by functions in C1 c(ω) in the strong topology of D(F(., ω)). Theorem 3.4 provides an integral representation of F, assuming that D(F) contains C1 c(Ω). The following result gives a corrector result: 9 the closed sets. Using the compactness of ¯ Ω, the characterization of the open sets implies that ¯ Ω/Ris compact and thus, the set Ω∗:= ¯ Ω/R\{[∂]}is locally compact. Using that the Γ-limit Fof Fn(3.6) is closed for the strong topology of L2(Ω) and the definition of the measure m, the restriction of Fto D(F)∩C0(Ω) is a closable form in L2 m(Ω∗), whose closure is denoted by F∗. Let us prove that F∗is a regular Dirichlet form according to Definition 3.3. Set H(s) := (s∨0) ∧1, for s∈R. Then, for any u∈D(F) and any recovery sequence un∈H1 0(Ω) associated with uand Fnby (3.2), we have F(H(u)) ≤lim inf n→+∞Fn(H(un)) ≤lim Fn(un) = F(u). In particular, this holds for any u∈D(F)∩C0(Ω), hence the restriction of Fto D(F)∩ C0(Ω) is Markovian. Since D(F)∩C0(Ω) is an algebra which separates points, the Stone-Weierstrass theorem shows that the functions of the form u+c, with u∈D(F)∩C0(Ω) and c∈R, are dense in C(¯ Ω/R). Now, consider v∈Cc(Ω∗), vn∈D(F)∩C0(Ω) and cn∈R, such that vn+cnconverges to vin C(¯ Ω/R). Since vand vnvanish in [∂], the sequence cnconverges to zero and thus vnconverges to vin C(¯ Ω/R). This proves that D(F)∩C0(Ω) is dense in C0(Ω∗), which implies that F∗is regular. Therefore, F∗is a regular Dirichlet form. It remains to prove that F∗is strongly local, i.e., the polar form Φ of Fsatisfies (3.4). Let u, v ∈D(F)∩C0(Ω) and c∈R, such that u=cconstant in supp (v). First, we assume that c≥0. Taking into account Remark 2.2, we consider two recovery sequences un, vn which strongly converge respectively to u, v in L∞(Ω) and such that Fn(un), Fn(vn) tend respectively to F(u), F(v). We also choose vnsuch that supp (vn)⊂supp (v). Let Hε, for ε > 0, be the function defined in Rby Hε(s):=   c c−εsif s < c −ε cif c−ε≤s≤c+ε s+εif s > c +ε, if ε < c, (note that Hε(0) = 0), and Hε(s) := (s−εsgn(s)) χ{|s|>ε}if c= 0. The sequence εn:= kun−ukL∞(Ω) converges to zero. Then, the sequence Hεn(un) satisfies the same properties than un, but we also have Hεn(un) = cin supp (v)⊃supp (vn), for εnsmall enough. Therefore, the Properties 3.2 c) of the recovery sequence vnyields Φ(u, v) = lim n→+∞ZΩ An∇(Hεn(un)) ·∇vndx = 0. In the case c < 0, we simply use the equality Φ(u, v) = −Φ(−u, v) = 0.  4 Application to the periodic case 4.1 Statement of the results In this section we consider a sequence Bnof symmetric matrix-valued functions in L∞(R2)2×2, which satisfies the following assumptions: Bnis Y-periodic, where Y:= (0,1)2, i.e., ∀n∈N,∀κ∈Z2, Bn(·+κ) = Bn(·) a.e. in R2,(4.1) Bnis equicoercive in R2, i.e., ∃α > 0 such that ∀n∈N,∀ξ∈R2, Bnξ·ξ≥α|ξ|2a.e. in R2.(4.2) 16 Let εnbe a sequence of positive numbers which tends to 0. From the sequences Bnand εn we define the highly oscillating sequence of matrix-valued functions Anby An(x):=Bnx εn,a.e. x∈R2.(4.3) In virtue of (4.1) and (4.2) Anis an equicoercive sequence of εn-periodic matrix-valued functions in L∞(R2)2×2. Let A∗ nbe the constant matrix defined by A∗ nλ·λ:= min ZY Bn(y)(λ+∇ϕ(y)) ·(λ+∇ϕ(y)) dy :ϕ∈H1 #(Y), λ ∈R2,(4.4) where H1 #(Y) denotes the set of Y-periodic functions in H1 loc(R2). The matrix A∗ nis symmetric and positive definite with A∗ n≥α I2. By the classical result of periodic homogenization (see e.g. [1]) A∗ n, for fixed n, is the homogenized matrix associated with the oscillating sequence An(x ε) as εtends to zero. Note that in the definition (4.3) of Anthe oscillations period εndepends on the sequence n. In this periodic framework we are interested in the asymptotic behaviour of the diffusion energy Fndefined by Fn(u):=     ZΩ An∇u·∇u dx if u∈H1 0(Ω) +∞if u∈L2(Ω) \H1 0(Ω), (4.5) as well as the conduction problem (−div (An∇un) = fin Ω un= 0 on ∂Ω,(4.6) for a given fin H−1(Ω). The following result shows that the asymptotic behaviour of the diffusion energy Fn(4.5) only depends on the limit of the spectral radius ρ(A∗ n) of the matrix A∗ n(4.4). Theorem 4.1. Let Ωbe a bounded open set of R2. Consider a highly oscillating sequence of matrix-valued functions Ansatisfying (4.1),(4.2) and (4.3). Then, we have the following alternative: If ρ(A∗ n)is bounded, there exists a subsequence, still denoted by n, and a positive definite matrix A∗such that A∗ n(4.4) converges to A∗in R2×2and Fn(4.5) Γ-converges for the strong topology of L2(Ω) to the quadratic form Fassociated with A∗by F(u):=     ZΩ A∗∇u·∇u dx if u∈H1 0(Ω) +∞if u∈L2(Ω) \H1 0(Ω). (4.7) If ρ(A∗ n)tends to +∞, the sequence FnΓ-converges for the strong topology of L2(Ω) to the quadratic form Fwhose domain is D(F) = {0}.(4.8) In term of the conduction problem (4.6) Theorem 4.1 implies the following result: Corollary 4.2. Let Ωbe a bounded open set of R2. Consider a highly oscillating sequence of matrix-valued functions Ansatisfying (4.1),(4.2) and (4.3). Then, we have the following alternative: 17 If ρ(A∗ n)is bounded, there exists a subsequence, still denoted by n, and a positive definite matrix A∗such that A∗ n(4.4) converges to A∗in R2×2and, for any fin H−1(Ω), the solution unof (4.6) weakly converges in H1 0(Ω) to the solution uof the conduction problem (−div (A∗∇u) = fin Ω u= 0 on ∂Ω.(4.9) If ρ(A∗ n)tends to +∞, the sequence unstrongly converges to 0in H1 0(Ω). Proof. Corollary 4.2 is a immediate consequence of Theorem 4.1 using the fact that the solution unof (4.6) is the minimizer of the functional u∈L2(Ω) 7−→ 1 2Fn(u)−ZΩ f u dx, and the minimizers convergence property of the Γ-convergence (see e.g. Corollory 7.24 p. 84 of [13]). Remark 4.3. Corollary 4.2 is an extension of the similar homogenization result obtained in [4] (by a complete different approach) under the assumption that the sequence of periodic matrix-valued Bn(4.3) is bounded in L1(Y)2×2. This condition is more restrictive since it is easy to check that the boundedness of Bnin L1(Y)2×2implies the boundedness of ρ(A∗ n). We can also build a periodic two-dimensional microstructure such that ρ(A∗ n) is bounded while kBnkL1(Y)2×2is not. Therefore, Corollary 4.2 provides a complete answer to the periodic homogenization of the conduction problems with equicoercive sequences of symmetric conductivities. 4.2 Proof of Theorem 4.1 The case where ρ(A∗ n)is bounded Let Xi n,i= 1,2, be the unique function in H1 #(Y), with zero Y-average value, solution of ∀ϕ∈H1 #(Y),ZY Bn∇Wi n·∇ϕ dy = 0,where Wi n(y):=yi+Xi n(y),(4.10) or equivalently, div Bnei+∇Xi n= 0 in D0(R2),(4.11) where (e1, e2) denotes the canonic basis of R2. Let wi nbe the highly oscillating sequence defined by wi n(x):=εnWi nx εn=xi+εnXi nx εn,for x∈Ω.(4.12) By (4.11) and the definition (4.3) of Anthe function wi nis clearly An-harmonic. Moreover, by the Y-periodicity of Bn∇Wi n·∇Wi n, by (4.10) and the definition (4.4) of A∗ n, we have for any bounded open subset ωof R2, Zω An∇wi n·∇wi ndx ≤cωZY Bn∇Wi n·∇Wi ndy =cωA∗ nei·ei≤c < +∞. This combined with the equicoerciveness of Animplies that the sequence wi nis bounded in H1 loc(R2) and thus weakly converges to xiin H1 loc(R2). Then, thanks to Corollary 2.3 the sequence wi nstrongly converges to xiin L∞(Ω). On the other hand, by the boundedness assumption on ρ(A∗ n) the sequence A∗ nconverges, up to a subsequence, to some constant 18 matrix A∗in the space R2×2. Moreover, the εn-periodicity of ∇wi nimplies that, for any i, j = 1,2, An∇wi n·∇wj n=Bn∇Wi n·∇Wj nx εn* A∗ei·ejweakly in M(R2)∗. So, as the gradients of the functions xi,i∈ {1,2}, are independent at each point of Ω, the sequences wi nsatisfy (3.12) for any open disk contained in Ω. Therefore, since wi n are An-harmonic and converge uniformly in Ω, the construction of Lemma 3.7 yields the measure µand the matrix-valued Aby dµ = (A∗e1·e1+A∗e2·e2)dx and Aei·ejdµ =A∗ei·ejdx, hence A∗= (A∗e1·e1+A∗e2·e2)A. Then, in virtue of Theorem 3.4 the Γ-limit Fof the sequence Fn(4.5) satisfies C1 c(Ω) ⊂D(F) and F(u) = ZΩ A∗∇u·∇u dx, ∀u∈C1 c(Ω).(4.13) Let us conclude. On the one side, the equicoerciveness of Anand the lower semicontinuity of the H1 0(Ω)-norm give D(F)⊂H1 0(Ω). On the other side, the density of C1 c(Ω) in H1 0(Ω), combined with the fact that D(F) is a Hilbert space and that from (4.13) a sequence of C1 c(Ω) which strongly converges in H1 0(Ω) also strongly converges in D(F), we get D(F) = H1 0(Ω) and equality (4.13) holds true in H1 0(Ω). Note that FnΓ-converges to Ffor the whole sequence such that A∗ nconverges to A∗in R2×2. The case where ρ(A∗ n)tends to +∞ We proceed by contradiction. We assume that the domain D(F) of the Γ-limit Fdoes not reduce to {0}. Then, we prove that ρ(A∗ n) is necessarily bounded. To this end, we proceed in two steps. In the first step, we prove that there exists a continuous function in D(F)\{0}. The second step is devoted to the proof of the boundedness of ρ(A∗ n). First step : D(F)∩C(Ω) 6={0}. Up to an extraction of a subsequence we can assume that the sequence Fndefined by (4.5) Γ-converges to some quadratic functional F:L2(Ω) −→ [0,+∞]. The starting assumption is that D(F)6={0}. Let u∈D(F)\{0}. By the equicoerciveness of Anthe function u belongs to H1 0(Ω). There exists a sequence unin H1 0(Ω) which strongly converges to uin L2(Ω) and such that Fn(un) tends to F(u). Up to enlarge the domain Ω and to extend the functions of H1 0(Ω) by 0 outside Ω, we may assume that the supports of u, unare contained in a fixed compact Kof Ω. Firstly, let us prove that, for any τ∈R2of small enough norm, the translated function u(·+τ) belongs to D(F) and F(u(·+τ)) = F(u). We follow the procedure given in the proof of Theorem 24.1 of [13]. Let τ∈R2and let κnbe a sequence in Z2such that τn:= εnκntends to τ. If τhas a small enough norm, then we have K−τn⊂Ω for any n∈N. Then, using successively the fact that un(·+τn) is equal to 0 in Ω \(K−τn), the change of variable y=x+τnand the εn-periodicity of An, we obtain F(un(·+τn)) = ZK−τn An(x)∇un(x+τn)·∇un(x+τn)dx =ZK An(y)∇un(y)·∇un(y)dy =Fn(un). Moreover, the sequence un(·+τn) strongly converges to u(·+τ). Therefore, the Γ-liminf inequality implies that F(u(·+τ)) ≤lim inf n→+∞Fn(un(·+τn)) = lim inf n→+∞Fn(un) = F(u), 19 which also yields F(u)≤F(u(·+τ−τ)) ≤F(u(·+τ)), and thus F(u(·+τ)) = F(u). Secondly, let δbe a small enough positive number and let vδbe the function defined on Ω by vδ(x) := 1 δ2ZδY u(x+y)dy, for x∈Ω, which is continuous on Ω. Let (Qj k)1≤j≤k, for k∈N∗, be a covering of the set δY by ksquares of side δ √kand let yj kbe the center of Qj k. Then, the sequence of convex combinations of translated of udefined by vk δ:= 1 δ2 k X j=1 |Qj k|u(·+yj k) strongly converges to vδin L2(Ω) as k→+∞, for fixed δ. Then, the lower semi-continuity and the convexity of Fyield F(vδ)≤lim inf k→+∞F(vk δ)≤lim inf k→+∞ 1 δ2 k X j=1 |Qj k|Fu(·+yj k)=F(u)<+∞. Therefore, vδbelongs to D(F)∩C(Ω). Since vδstrongly converges to u6= 0 in L2(Ω), vδ is a non-zero function in D(F)∩C(Ω) for δsmall enough, which concludes the first step. Second step : Boundedness of ρ(A∗ n). Let vbe a non-zero function in D(F)∩C(Ω) and let vnbe a sequence in H1 0(Ω) which strongly converges to vin L2(Ω) and such that Fn(vn) tends to F(v). By Theorem 2.1 the sequence vnuniformly converges to vin Ω. Since vis a non-zero continuous function on Ω, the uniform convergence of vnto vimplies that there exists a non-empty open subset ω0 of Ω and a constant c0>0 such that |vn(x)| ≥ c0a.e. x∈ω0.(4.14) Let λnbe a unit norm vector in R2such that A∗ nλn·λn=ρ(A∗ n). Let wnbe the highly oscillating sequence defined by wn(x):=λn·x+εnXnx εnx∈R2,where Xn:= (λn·e1)X1 n+(λn·e2)X2 n(4.15) and Xi n,i= 1,2, are the Y-periodic solutions of (4.11). Set ˜ Y:= (−1 2,3 2)2. Since the function Wn(y) := λn·y+Xn(y) is Bn-harmonic in R2, by Corollary 2.5 there exists a constant C > 0 such that for any n∈N, kWnkL∞(Y)≤CkWnkH1(˜ Y). Then, using the periodicity and the zero Y-average value of Xn, as well as the Poincar´e- Wirtinger inequality yields kXnkL∞(Y)≤1 + kWnkL∞(Y) ≤1 + kλn·ykL∞(˜ Y)+CkXnkH1(˜ Y)≤1 + 3 √2+ 4 CkXnkH1(Y) ≤C0+C0k∇XnkL2(Y)≤2C0+C0k∇WnkL2(Y). Moreover, the coerciveness of Bnand the definition (4.4) of A∗ nimply that αk∇Wnk2 L2(Y)2≤ZY Bn∇Wn·∇Wndy =A∗ nλn·λn=ρ(A∗ n), 20 which combined with the previous estimates gives kXnkL∞(Y)≤C0+C0 √αpρ(A∗ n). Therefore, by the Y-periodicity of Xnand the definition (4.15) of wnthere exists a constant c > 0 such that for any n∈N, kwnk2 L∞(Ω) ≤c+c ε2 nρ(A∗ n).(4.16) On the other hand, using the An-harmonicity of wn(4.15) and the Cauchy-Schwarz inequality yields ZΩ An∇wn·∇wnv2 ndx =−2ZΩ An∇wn·∇vnvnwndx ≤2ZΩ An∇wn·∇wnv2 ndx1 2ZΩ An∇vn·∇vnw2 ndx1 2 , hence the inequality ZΩ An∇wn·∇wnv2 ndx ≤4ZΩ An∇vn·∇vnw2 ndx. (4.17) Let us conclude. On the one side, thanks to the uniform estimate (4.14) and the εn-periodicity of An∇wn· ∇wnthe left hand-side of (4.17) is bounded from below by a positive constant times Zω0 An∇wn·∇wndx ≥cω0ZY Bn∇Wn·∇Wndy =cω0ρ(A∗ n), where cω0is a positive constant only depending on ω0. On the other side, thanks to the uniform estimate (4.16) combined with the boundedness of Fn(vn) the right hand-side of (4.17) is bounded from above by c+c ε2 nρ(A∗ n). 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