Homogenization and correctors of a class of elliptic problems in perforated domains

Abstract

In this paper, we study the homogenization and the correctors for a class of linear elliptic problems in a periodically perforated domain when the oscillating matrix field also depend on a weakly converging sequence. We prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. Using the periodic unfolding method, we first derive the homogenized problem, then we study the convergence of the energy of the solution and the related corrector, which are the main results of this paper. As a particular case, we obtain a corrector result for the Laplacian with a linear nonhomogeneous Neumann condition on the hole, in the case of a nonzero Neumann data with a zero average. This remained an open problem since the corresponding homogenized results given in [Asymptotic Anal. 1 (1988), 115–138].