Representation formulas for \(L^\infty\) norms of weakly convergent sequences of gradient fields in homogenization (Q2838579)

The authors consider a domain \(\Omega\) of \(\mathbb R^d\), \(d=2,3\), and the elliptic problem \(-\text{div}(A^n\nabla u_n)=f\) posed in \(\Omega\), the coefficient matrices \(A^n(x)\) being given through \(A^n(x)=\sum_{i=1}^N\chi_n^iA_i\) where the \(\chi_n^i\) are the indicator functions of sets \(\omega_n^i\) satisfying \(\sum_{i=1}^N\chi_n^i=1\) in \(\Omega\). The Dirichlet boundary conditions \(u_n=g\) on \(\partial\Omega\) are imposed. The authors assume that \((A^n)_n\) \(G\)-converges to some \(A^H\), which implies that \((u_n)_n\) converges in the weak limit of \(H^1(\Omega)\) to the solution \(u^H\) of \(-\text{div}(A^H\nabla u^H)=f\) in \(\Omega\). The purpose of the paper is to prove upper and lower bounds for \(\lim\sup_n\|\chi_n^i\nabla u_n\|_{L^{\infty}(S)}\) and for \(\lim\inf_n\|\chi_n^i\nabla u_n\|_{L^{\infty}(S)}\) for every open subset \(S\Subset\Omega \). The authors indeed exhibit such bounds in terms of \(\|\mathcal M^i(\nabla u^H)\| _{L^{\infty}(S)}\), where the operator \(\mathcal M^i\) involves the solution of local problems. In the rest of the paper, the authors give some examples of these general results and concerning laminated microstructures, periodic microstructures and what they call graded microstructures. They also consider some optimal design problems.