Homogenization Structures and Applications I

Abstract

We lay the foundations of a mathematical theory of homogenization structures and show how the latter arises in the homogenization of partial differential equations. We find out that the concept of a homogenization structure turns out to be exactly the right tool that is needed to systematically extend homogenization theory beyond the classical periodic setting. This permits to work out various outstanding nonperiodic homogenization problems that were out of reach till then for lack of an appropriate mathematical framework. The classical Gelfand representation theory is one of our main tools and our basic approach is an adaptation of the two-scale convergence method.