Homogenization in perforated domains beyond the periodic setting

Abstract

We study the homogenization of a second order linear elliptic differential operator in an open set in R N with isolated holes of size ε>0. The classical periodicity hypothesis on the coefficients of the operator is here substituted by an abstract assumption covering a variety of concrete behaviours such as the periodicity, the almost periodicity, and many more besides. Furthermore, instead of the usual “periodic perforation” we have here an abstract hypothesis characterizing the manner in which the holes are distributed. This is illustrated by practical examples ranging from the classical equidistribution of the holes to the more complex case in which the holes are concentrated in a neighbourhood of the hyperplane { x N =0}. Our main tool is the recent theory of homogenization structures and our basic approach follows the direct line of two-scale convergence.