Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains

Abstract

We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in M0+(Ω)) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices H-converge to a matrix A0, we prove that there exist a subsequence and a measure μ0 in M0+(Ω) such that the limit problem is the relaxed Dirichlet problem corresponding to A0 and μ0. We also prove a corrector result which provides an explicit approximation of the solutions in the H1-norm, and which is obtained by multiplying the corrector for the H-converging matrices by some special test function which depends both on the varying matrices and on the varying domains.