Homogenization and boundary layers

Abstract

This paper deals with the homogenization of elliptic systems with a Dirichlet boundary condition, when the coefficients of both the system and the boundary data are ε-periodic. We show that, as ε → 0, the solutions converge in L 2 with a power rate in ε, and identify the homogenized limit system. Due to a boundary layer phenomenon, this homogenized system depends in a non-trivial way on the boundary. Our analysis answers a longstanding open problem, raised for instance in [6]. It substantially extends previous results obtained for polygonal domains with sides of rational slopes as well as our previous paper [14], where the case of irrational slopes was considered. © 2012 by Institut Mittag-Leffler.