Homogenization of Nonlinear Degenerate Non-monotone Elliptic Operators in Domains Perforated with Tiny Holes

Abstract

The paper deals with the homogenization problem beyond the periodic setting, for a degenerated nonlinear non-monotone elliptic type operator on a perforated domain ? ? in ? N with isolated holes. While the space variable in the coefficients a 0 and a is scaled with size ? (?>0 a small parameter), the system of holes is scaled with ? 2 size, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The homogenization problem is formulated in terms of the general, so-called deterministic homogenization theory combining real homogenization algebras with the Σ-convergence method. We present a new approach based on the Besicovitch type spaces to solve deterministic homogenization problems, and we obtain a very general abstract homogenization results. We then illustrate this abstract setting by providing some concrete applications of these results to, e.g., the periodic homogenization, the almost periodic homogenization, and others.