Homogenization of systems with equi-integrable coefficients

ESAIM: COCV 20 (2014) 1214–1223 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv/2014013 www.esaim-cocv.org HOMOGENIZATION OF SYSTEMS WITH EQUI-INTEGRABLE COEFFICIENTS Marc Briane1and Juan Casado-D ´ ıaz2 Abstract. In this paper we prove a H-convergence type result for the homogenization of systems the coefficients of which satisfy a functional ellipticity condition and a strong equi-integrability condition. The equi-integrability assumption allows us to control the fact that the coefficients are not equi-bounded. Since the truncation principle used for scalar equations does not hold for vector-valued systems, we present an alternative approach based on an approximation result by Lipschitz functions due to Acerbi and Fusco combined with a Meyers Lp-estimate adapted to the functional ellipticity condition. The present framework includes in particular the elasticity case and the reinforcement by stiff thin fibers. Mathematics Subject Classification. 35B27, 49K20. Received October 1st, 2013. Revised February 1st, 2014. Published online August 8, 2014. 1. Introduction This paper is devoted to the asymptotic behavior of vector-valued systems with Mequations, in a regular bounded open set Ωof RN, −Div (AnDun)=fin Ω un=0 on∂Ω, (1.1) where Anis a sequence of equi-coercive (in a functional sense, see Eq. (2.1)) but not necessarily equi-bounded tensor-valued functions. The equi-bounded case was studied by Spagnolo [30] by G-convergence, and by Murat, Tartar [28](seealso[31]) by H-convergence. Then, in the scalar case assuming the L1-boundedness and the equi-integrability of the sequence |An|, Carbone and Sbordone proved a compactness result for the equation (1.1) using De Giorgi’s Γ-convergence [20,21] (see, also [7,19]forapresentationofΓ-convergence). On the other hand, Fenchenko and Khruslov [23](seealso[24,25]) were the first to show the appearance of nonlocal effects in the homogenization of equation (1.1)when|An|is bounded in L1(Ω) but not equi-integrable. To this end they considered a medium reinforced by very thin fibers. Several works [3,8,9,12,13,15,16,27] have extended this seminal article on the limit closure of equations (1.1) in connection with the Beurling–Deny [6]representation of Dirichlet forms. All these contributions are strongly based on the truncation principle (which is also called Keywords and phrases. Homogenization, vector-valued systems, not equi-bounded coefficients, equi-integrable coefficients, H-convergence. 1Institut de Recherche Math´ematique de Rennes, INSA de Rennes, France. [email protected] 2Departemento. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Spain. [email protected] Article published by EDP Sciences c EDP Sciences, SMAI 2014 HOMOGENIZATION OF SYSTEMS WITH EQUI-INTEGRABLE COEFFICIENTS 1215 the Markov property in Dirichlet forms theory) and the maximum principle as a by-product, which are specific to the scalar case. The vector-valued case for which (1.1) is actually a system is much more delicate since the truncation principle does not hold. In the case of elasticity [4] the homogenization of fiber reinforced media can lead to nonlocal effects as in the scalar case. Under this particular geometry and the boundedness of |An|in L1(Ω), the appearance of nonlocal effects is induced by the loss of equi-integrability for |An|. However, contrary to the scalar case which is constrained by the Beurling-Deny representation, when the fibers are very stiff – i.e.,|An|is not equi-bounded in L1(Ω) – fourth-order derivatives may appear at the limit in dimension three [29]aswellasindimension two [10,11]. Actually, Camar-Eddine and Seppecher [17] proved that in dimension three the Γ-convergence closure for the L2-strong topology of the elastic energies associated with equations (1.1) agrees with the whole set of the lower semi-continuity quadratic functionals which are null for the rigid displacements. This closure set thus contains functionals with nonlocal terms and derivatives at any order. In the more general framework of systems (1.1) the closure set is far to be clear. A first step in the understanding of the homogenization of such a system, with an equi-coercive but not equi-bounded sequence of tensor-valued functions, would be to know if the sole equi-integrability of |An|in L1(Ω) implies a compactness result for the sequence (1.1) as in the scalar case [18]. Up to our knowledge there is no general result in this direction. In this paper we prove a H-convergence type result (see Thm. 2.3) for system (1.1) assuming that there exists a functionally equi-coercive and equi-bounded sequence Bnof tensor-valued functions such that lim n→∞ An−BnL1(Ω)(M×N)2=0.(1.2) This assumption includes the case where Antakes non-uniformly bounded values only in some set Fnwith |Fn|→0, as in the fiber reinforcement setting (see Rem. 2.2). Contrary to the truncation of the solution un of (1.1) used in the scalar case [18], our method is based on the approximation result by Lipschitz functions due to Acerbi and Fusco [1,2], which can be regarded as a truncation of the gradient Dun.Moreover,the Lp-Meyers estimate is an alternative key ingredient of our approach. At this level we give an extension of the proof of [5](seeProp.3.1), which takes into account that the strong ellipticity of Anis functional rather than pointwise. This functional ellipticity allows us to include the elasticity case in a general framework. Indeed, the pointwise ellipticity with respect to the symmetrized gradient combined with Korn’s inequality implies a functional ellipticity with respect to the whole gradient. In view of the compactness result [18] which is restricted to the scalar case, our conjecture is that a H-convergence result holds for system (1.1)ifAnis functionally equi-coercive but simply bounded and equiintegrable in L1(Ω)(M×N)2. This is of course a weaker condition than (1.2). Very recently we have proved the conjecture in [14] but only for dimension N= 2. The method based on a div-curl lemma also provides a compactness result in dimension N>2, under the assumption that |An|is bounded in Lρ(Ω)withρ>N−1 2.This condition is more restrictive than the equi-integrability in L1(Ω), and cannot be compared to condition (1.2). In particular, it is not sharp in the fiber reinforcement setting contrary to condition (1.2). To conclude, the present approach is quite different, does work in any dimension and is well adapted to the fiber reinforcement problem. Notations •Mand Nare two positive integers. •Adenotes a tensor-valued function taking its values in R(M×N)2. •: denotes the scalar product in RM×N,i.e. ξ :η=trξTηfor any ξ,η ∈RM×N. •∇udenotes the gradient of the scalar distribution u:RN→R. •Du denotes the Jacobian matrix of the vector-valued distribution u:RN→RM,i.e. Du := ∂ui ∂xj1≤i≤M, 1≤j≤N . 1216 M. BRIANE AND J. CASADO-DIAZ •div denotes the classical divergence operator acting on the vector-valued distributions. •Div denotes the vector-valued differential operator taking the divergence of each row of a matrix-valued distribution, Div U:= ⎡ ⎣ N  j=1 ∂Uij ∂xj⎤ ⎦1≤i≤M ,for U:RN→RM×N. •cdenotes a positive constant which may vary from line to line. 2. The main result Let Ωbe a regular bounded domain of RN,N≥2, and let Mbe a positive integer. Consider a sequence An, n∈N, of tensor-valued functions in L∞(Ω)(M×N)2which satisfies the following properties: •there exists a sequence Bnof tensor-valued functions in L∞(Ω)(M×N)2such that ∀u∈H1 0(Ω)M,⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ α  Ω |Du|2dx≤min   Ω AnDu :Du dx,  Ω BnDu :Du dx β−1  Ω |BnDu|2≤  Ω BnDu :Du dx, (2.1) for given constants α, β > 0, and lim n→∞ An−BnL1(Ω)(M×N)2= 0; (2.2) •there exists a constant C>0 such that the generalized Cauchy–Schwarz inequality holds (Anξ:η)2≤C(Anξ:ξ)(Anη:η),a.e. in Ω, ∀(ξ,η)∈RM×N×RM×N.(2.3) Remark 2.1. Assumption (2.1) implies the pointwise estimates α|ξ|2|η|2≤Bn(ξ⊗η):(ξ⊗η)≤β|ξ|2|η|2,a.e. in Ω, ∀(ξ,η)∈RM×RN.(2.4) We refer to Lemma 22.5 of [19] for a similar computation. For the reader’s convenience, let us check briefly (2.4): Putting in the first inequality of (2.1) the functions u(x):=ϕ(x)cos(kη·x)ξ,withk≥1, ϕ∈C1 c(Ω), (ξ,η)∈RM×RN, it follows that α  Ω |ξ|2|η|2ϕ2(x)sin 2(kη·x)dx≤  Ω Bn(x)(ξ⊗η):(ξ⊗η)ϕ2(x)sin 2(kη·x)dx+O(k−1). Then, passing to the limit as k→∞and using the arbitrariness of ϕwe get the first inequality of (2.4). Similarly, we deduce from the second inequality of (2.1)that Bn(x)(ξ⊗η) 2≤βBn(x)(ξ⊗η):(ξ⊗η)a.e.x∈Ω, which implies Bn(x)(ξ⊗η)≤β|ξ||η|a.e. x∈Ω, (2.5) and thus the second inequality of (2.4). Now, decomposing any matrix Q∈RM×Non the canonical basis, namely Q= M  i=1 N  j=1 Qij fi⊗ej, we easily obtain from inequality (2.5) the existence of a constant γ>0 depending on β,M,N, such that Bn(x):= max Q∈RM×N,|Q|=1 Bn(x)Q≤γa.e. x∈Ω. (2.6) HOMOGENIZATION OF SYSTEMS WITH EQUI-INTEGRABLE COEFFICIENTS 1217 Remark 2.2. Condition (2.2) is equivalent to the existence of a functionally equi-coercive, equi-bounded sequence ˜ Bnin L∞(Ω)(M×N)2such that for any ε>0, there exists a sequence of measurable sets Fε nof Ω satisfying |An−˜ Bn|≤εa.e. in Ω\Fε n,lim n→∞ |Fε n|= 0 and lim n→∞  Fε n |An|dx=0.(2.7) Indeed, condition (2.7) implies convergence (2.2)withBn:= ˜ Bn,since lim sup n→∞   Ω |An−Bn|dx≤lim sup n→∞ ε|Ω\Fε n|+  Fε n |An|dx+  Fε n |Bn|dx=ε|Ω|. Conversely, assume that convergence (2.2) holds. For ε>0, define the measurable set Fε nand the tensor-valued function ˜ Bnby Fε n:= |An−Bn|>ε and ˜ Bn:= Bn.(2.8) The first assertion of (2.7) is clearly satisfied. Then, by the strong convergence (2.2) Lebesgue’s measure of Fε n tends to 0, and  Fε n |An|dx≤  Fε n |An−Bn|dx+|Fε n|BnL∞(Ω)(M×N)2→0, which yields (2.7). Note that if the equi-coerciveness of Anand Bnare pointwise rather than functional, then we may take ε= 0 in condition (2.7)andchoose Fn:= |An−Bn|>1and ˜ Bn:= Anin Ω\Fn Bnin Fn. Condition (2.7) is relevant in the case of a fiber reinforced medium for which the set Fε nis composed of stiff thin fibers, while the surrounding medium has uniformly bounded coefficients. In this setting, the loss of equi-integrability with respect to condition (2.7) corresponds to lim sup n→∞  Fε n |An|dx>0.(2.9) In fact, condition (2.9) may lead to pathologies in the homogenization process of system (2.10): nonlocal effects in conductivity [3,9,16,23] and in elasticity [4], but also the appearance of second gradients in elasticity [10,11,29]. We have the following H-convergence type result: Theorem 2.3. Assume that the conditions (2.1)and (2.2)are fulfilled. Then, there exist a subsequence of n, still denoted by n, and a tensor-valued function Bin L∞(Ω)(M×N)2satisfying the estimates (2.1)such that for any distribution f∈H−1(Ω)M, the solution unin H1 0(Ω)Mof the equation −Div (AnDun)=fin Ω, (2.10) satisfies the convergences unu weakly in H1 0(Ω)Mand AnDunBDu weakly in L1(Ω)M×N,(2.11) where uis the solution in H1 0(Ω)Mof −Div (BDu)=fin Ω. (2.12) Moreover, the limit tensor Bonly depends on the sequence Bn. 1218 M. BRIANE AND J. CASADO-DIAZ 3. Proofs Proof of Theorem 2.3.First note that using a density argument we are led to the case where the right-hand side fof equation (2.10) belongs to W−1,p(Ω)wherepis the Meyers exponent obtained in Proposition 3.1 below. Due to the functional ellipticity (2.1) the sequence unis bounded in H1 0(Ω)M, thus up to a subsequence converges weakly to some function uin H1 0(Ω)M. The Murat–Tartar H-convergence [28,31] for linear scalar operators can be extended without restriction to the linear operators Div (BnD·). Hence, there exists a subsequence of n, still denoted by n, and a tensor-valued function Bsuch that for any distribution f∈H−1(Ω)M,thesolutionvn∈H1 0(Ω)Mof the equation −Div (BnDvn)=fin Ω, (3.1) satisfies the convergences vnv weakly in H1 0(Ω)Mand BnDvnBDv weakly in L2(Ω)M×N,(3.2) where vis the solution in H1 0(Ω)Mof (2.12). From the convergences (3.2) and the lower semi-continuity of the L2-norm we easily deduce that the homogenized tensor Bsatisfies the estimates (2.1)and(2.6). Therefore, up to extract a subsequence we can assume that the sequence BnH-converges to some tensor-valued function B satisfying the estimates (2.1)and(2.6). The proof is divided in two steps: First step: Let unbe the solution of equation (2.10), and let ¯unbe the solution in H1 0(Ω)Mof −Div (BnD¯un)=fin Ω. (3.3) First of all, by virtue of Proposition 3.1 the sequence ¯unis bounded in W1,p(Ω)M. Then, by the Lusin type approximation theorem due to Acerbi and Fusco [1](maintheoremp.1),[2](Lem.[II-6]),foranyk≥1, there exists a function ¯uk nin W1,∞(Ω)M∩H1 0(Ω)Msuch that ¯uk nW1,∞(Ω)M≤k, (3.4) {¯uk n=¯un}≤c kp¯unp W1,p(Ω)M.(3.5) Now, we will prove that lim k→∞ lim sup n→∞   Ω AnDun−D¯uk n:Dun−D¯uk ndx=0.(3.6) By equations (2.10)and(3.3)wehave  Ω AnDun−D¯uk n:Dun−D¯uk ndx =f,un−¯uk n−  Ω BnD¯uk n:Dun−D¯uk ndx−  Ω (An−Bn)D¯uk n:Dun−D¯uk ndx =  Ω BnD¯un−D¯uk n:Dun−D¯uk ndx−  Ω (An−Bn)D¯uk n:Dun−D¯uk ndx. (3.7) HOMOGENIZATION OF SYSTEMS WITH EQUI-INTEGRABLE COEFFICIENTS 1219 To estimate the first term on the right-hand side of (3.7), we use that (3.4)and(3.5)implythat¯uk nis bounded in W1,p(Ω)Mindependently of nand k. Therefore, using the bound (2.6)andH¨older’s inequality, we get   Ω BnD¯un−D¯uk n:Dun−D¯uk ndx ≤γ  {¯uk n=¯un}|D¯un|+|D¯uk n||Dun|+|D¯uk n|dx ≤γ¯unW1,p(Ω)M+¯uk nW1,p(Ω)MunH1(Ω)M+¯uk nH1(Ω)M{¯uk n=¯un} 1 2−1 p ≤ck1−p 2.(3.8) To estimate the second term on the right-hand side of (3.7), we take m>0 and we use the decomposition   Ω (An−Bn)D¯uk n:Dun−D¯uk ndx ≤  {|Dun|≤m} (An−Bn)D¯uk n:Dun−D¯uk ndx +  {|Dun|>m} AnD¯uk n:Dun−D¯uk ndx +  {|Dun|>m} BnD¯uk n:Dun−D¯uk ndx .(3.9) The first term on the right-hand side of this inequality can be estimated by   {|Dun|≤m} (An−Bn)D¯uk n:Dun−D¯uk ndx ≤k(m+k)An−BnL1(Ω)(M×N)2.(3.10) Taking into account (2.3), we can estimate the second term on the right-hand side of (3.9)by   {|Dun|>m} AnD¯uk n:Dun−D¯uk ndx ≤C  {|Dun|>m} AnD¯uk n:D¯uk ndx1 2  {|Dun|>m} AnDun:Dundx1 2 +  {|Dun|>m} AnD¯uk n:D¯uk ndx ≤ck  {|Dun|>m} |An|dx1 2 +k2  {|Dun|>m} |An|dx. (3.11) For the third term on the right-hand side of (3.9)weuse   {|Dun|>m} BnD¯uk n:Dun−D¯uk ndx ≤γkunH1(Ω)M{|Dun|>m} 1 2+γk 2{|Dun|>m}.(3.12) 1220 M. BRIANE AND J. CASADO-DIAZ The estimates (3.9)to(3.12) show that   Ω (An−Bn)D¯uk n:Dun−D¯uk ndx ≤k(m+k)An−BnL1(Ω)(M×N)2 +ck   {|Dun|>m} |An|dx1 2 +k2  {|Dun|>m} |An|dx +γkunH1(Ω)M{|Dun|>m} 1 2+γk 2{|Dun|>m}.(3.13) Also note that sup n∈N{|Dun|>m}≤1 msup n∈N  Ω |Dun|dx≤C m, hence supn∈N{|Dun|>m}→0asm→∞. Then, since by (2.2)Anis equi-integrable in L1(Ω)(M×N)2,we can pass to the limsup successively in nand in min inequality (3.13)toget lim n→∞  Ω (An−Bn)D¯uk n:Dun−D¯uk ndx=0.(3.14) Finally, (3.7) combined with (3.8)and(3.14) yields lim sup n→∞   Ω AnDun−D¯uk n:Dun−D¯uk ndx≤ck1−p 2,∀k≥1,(3.15) which implies the double limit (3.6)sincep>2. Second step: Determination of the limit of the flux AnDun. On the one hand, by (2.3), the Cauchy–Schwarz inequality and the estimates (2.2), (2.6)wehave  ΩAnDun−D¯uk ndx ≤c  Ω |An|1 2AnDun−D¯uk n:Dun−D¯uk n1 2dx ≤c  Ω |An|dx1 2  Ω AnDun−D¯uk n:Dun−D¯uk ndx1 2 ≤c  Ω AnDun−D¯uk n:Dun−D¯uk ndx1 2 .(3.16) This combined with (3.6) yields lim k→∞ lim sup n→∞  ΩAnDun−D¯uk ndx=0.(3.17) On the other hand, using the functional ellipticity (2.1)andtheH¨older inequality combined with estimate (3.5), we have  Ω |Dun−D¯un|2dx ≤2  ΩDun−D¯uk n 2dx+2  {¯uk n=¯un}D¯uk n−D¯un 2dx ≤c  Ω AnDun−D¯uk n:Dun−D¯uk ndx+ck2−p,(3.18) HOMOGENIZATION OF SYSTEMS WITH EQUI-INTEGRABLE COEFFICIENTS 1221 which by (3.6) implies that lim n→∞  Ω |Dun−D¯un|2dx=0.(3.19) Now, consider the decomposition of the flux AnDun=BnD¯un+AnDun−D¯uk n+(An−Bn)D¯uk n+BnD¯uk n−D¯un,(3.20) which by (2.6), (3.4), (3.5)gives  Ω |AnDun−BnD¯un|dx ≤  ΩAnDun−D¯uk ndx+k  Ω |An−Bn|dx+ck1−p, ∀k≥1.(3.21) This combined with the double limit (3.17)and(2.2) yields lim n→∞  Ω |AnDun−BnD¯un|dx=0.(3.22) Let us conclude. By limit (3.19)unand ¯unactually converge weakly in H1 0(Ω)Mto the same limit u.Moreover, thanks to the H-convergence of Bnto Band the associated convergence of the flux (3.2), the sequence BnD¯un converges to BDu weakly in L2(Ω)M×N. This combined with (3.22) implies that the sequence AnDunalso converges to BDu weakly in L1(Ω)M×N. Hence, we deduce the convergences (2.11) and the limit equation (2.12). Note that the convergences (2.11) only depend on the sequence nfor which BnH-converges. Moreover, the limit equation (2.12) only depends on the H-limit of Bn. Therefore, Theorem 2.3 is proved.  In the Proof of Theorem 2.3 we have used the following extension to systems of the celebrated Meyers Lpestimate [26]. The proof follows the scheme of [5] with an adaptation due to the functional ellipticity (2.1). We will give a sketch of the proof to illuminate this point. Proposition 3.1 (Meyers Lp-estimate).Let α, β > 0and let Ωbe a regular bounded open set of RN.There exist a number p>2and a constant C>0which only depend on α, β and Ω, such that for any Btensor-valued function satisfying the conditions (2.1),(2.6), and for any f∈W−1,p(Ω)M, the solution u∈H1 0(Ω)Mof −Div (BDu)=fin Ω. (3.23) satisfies the estimate uW1,p(Ω)M≤CfW−1,p(Ω)M.(3.24) Proof. Consider the decomposition of [5] in the proof of Theorem 4.3: 1 γ+cB=B1+B2with B1:= 1 γ+c(Bs+cI)andB2:= 1 γ+c(Ba−cI),(3.25) where Bsis the symmetric part of B,Bais the antisymmetric part of B,γ>0isgivenby(2.6)andc>0. We have ⎧ ⎨ ⎩  Ω B1Du :Du dx≥μ  Ω |Du|2dx, ∀v∈H1 0(Ω)M |B1|≤1,|B2|≤νa.e. in Ω, (3.26) where by choosing c>γ2−α2 2α, 0<ν:= γ2+c2 γ+c<μ:= α+c γ+c<1.(3.27) 1222 M. BRIANE AND J. CASADO-DIAZ Following [5]thesolutionuof (3.23)satisfies u+Ru=1 γ+c(−Δ)−1fwhere R:= (−Δ)−1(Δ+T1+T2),T i:= −Div (BiD·).(3.28) First, let us estimate the bound of the operator Rfrom H1 0(Ω)Minto H1 0(Ω)M.SinceΔis an isometry from H1 0(Ω)Monto H−1(Ω)M,wehaveby(3.26) RH1 0(Ω)M≤ (−Δ)−1H1 0(Ω)M,H−1(Ω)MΔ+T1H1 0(Ω)M,H−1(Ω)M+T2H1 0(Ω)M,H−1(Ω)M =Δ+T1H1 0(Ω)M,H−1(Ω)M+ν. (3.29) Moreover, noting that Δ+T1is a self-adjoint operator we have again by (3.26) Δ+T1H1 0(Ω)M,H−1(Ω)M=sup v∈H1 0(Ω)M\{0}  Ω(I−B1)Dv :Dv dx  Ω|Dv|2dx ≤1−μ. (3.30) Therefore, we obtain that RH1 0(Ω)M≤1−μ+ν<1.(3.31) Next, let us estimate the bound of the operator Rfrom W1,p 0(Ω)Minto itself, for p>2. At this level, the proof is different from the one of [5]. Let Sbe the operator defined by S:Lp(Ω)M×N→W1,p 0(Ω)M,Sh:= Δ−1Div(h).(3.32) Then, denoting by DthederivativeoperatorwehavethatD◦R◦Sis a linear operator which maps continuously Lp(Ω)M×Ninto itself. By the Riesz–Thorin theorem (see, e.g.,[22] Sect. 10.11) we have for p, q ∈(2,∞)with p<q, D◦R◦SLp(Ω)M×N≤D◦R◦Sθ L2(Ω)M×ND◦R◦S1−θ Lq(Ω)M×Nwith 1 p=θ 2+1−θ q,(3.33) which fixing q>2 and using (3.31) implies that lim sup p→2,p>2 D◦R◦SLp(Ω)M×N≤D◦R◦SL2(Ω)M×N =R◦SL2(Ω)M×N,H1 0(Ω)M≤RH1 0(Ω)M<1. (3.34) We also have for any v∈W1,p 0(Ω)M, RvW1,p 0(Ω)M=D(Rv)Lp(Ω)M×N =(D◦R◦S)DvLp(Ω)M×N≤D◦R◦SLp(Ω)M×NvW1,p 0(Ω)M.(3.35) Therefore, estimates (3.34)and(3.35)giveRW1,p 0(Ω)M<1forpclose to 2, which combined with equation (3.28) concludes the proof.  Acknowledgements. The authors wish to thank the unknown referees for their careful reading and relevant comments which have improved the presentation of the paper. They are grateful for support from the Spanish Ministerio de Econom´ıa y Competitividad through Project MTM2011-24457, and from the Institut de Recherche Math´ematique de Rennes. This work was partly carried out while J. Casado D´ıaz was visiting the INSA de Rennes in June 2013.