Multiscale homogenization of convex functionals with discontinuous integrand

Abstract

This article is devoted to obtain the $\Gamma$-limit, as $\epsilon$ tends to zero, of the family of functionals $$F_{\epsilon }(u)={\int }_{\Omega }f(x,\frac{x}{\epsilon },...,\frac{x}{{\epsilon }^n},\nabla u(x))dx$$, where $f=f(x,y^1,...,y^n,z)$ is periodic in $y^1,...,y^n$, convex in $z$ and satisfies a very weak regularity assumption with respect to $x,y^1,...,y^n$. We approach the problem using the multiscale Young measures.

18 pages; a slight change in the title; to be published in J. Convex Anal. 14 (2007), No. 2