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 M_0) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices H-converge to a matrix A^0, we prove that there exist a subsequence and a measure mu^0 in M_0 such that the limit problem is the relaxed Dirichlet problem corresponding to A^0 and mu^0. We also prove a corrector result which provides an explicit approximation of the solutions in the H^1-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.