The convergence, as e $0, of the functional F()=aN u(x)(x,x/e) associated with a given L function u with support in a fixed compact set is studied. The test functions (x, y) are continuous on R rv x R N and periodic in y. A convergence theorem is proved under the weaker assumption that u remains in a bounded subset of L2. Finally, the use of multiple-scale expansions in homogenization is justified, and a new approach is proposed for the mathematical analysis of homogenization problems.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.