Homogenization of systems with equi-integrable coefficients (Q2926574)

The authors describe a H-convergence result for systems of \(M\) equations written as \(-\operatorname{func}\operatorname{div}(\mathbb{A}_{n}Du_{n})=f\) and posed in a smooth, bounded and open subset \(\Omega \) of \(\mathbb{R}^{N}\), \(N\geq 2\). The homogeneous Dirichlet boundary conditions \(u_{n}=0\) are imposed on \(\partial \Omega \). Here the matrix \(\mathbb{A}_{n}\) of tensor-valued functions is supposed to be equi-coercive but not necessarily equi-bounded. The authors impose different hypotheses on this sequence which involve another tensor-valued matrix \(\mathbb{B}_{n}\) and which imply the existence of a tensor-valued function \(\mathbb{B}\in L^{\infty }(\Omega )^{(M\times N)^{2}}\) such that for every \(f\in H^{-1}(\Omega )^{M}\) the solution \(u_{n}\in H_{0}^{1}(\Omega )^{M}\) is such that \((u_{n})_{n}\) converges to \(u\) in the weak topology of \(H_{0}^{1}(\Omega )^{M}\) and \((\mathbb{A}_{n}Du_{n})_{n}\) converges to \(\mathbb{B}Du\) in the weak topology of \(L^{1}(\Omega )^{M\times N}\), where \(u\) is the solution of the limit problem \(-\operatorname{func}\operatorname{div}(\mathbb{B} Du)=f\) in \(\Omega \) with homogeneous Dirichlet boundary conditions. The sequence \((\mathbb{B}_{n})_{n}\) is supposed to H-converge. For the proof, the authors use a Lusin approximation result obtained by \textit{E. Acerbi} and \textit{N. Fusco} [in: Material instabilities in continuum mechanics, Proc. Symp. Edinburgh/Scotl. 1985/86, 1--5 (1988; Zbl 0644.46026)] and a generalization of Meyers' \(L^{p}\) -inequality to systems.