A closure theorem for $\Gamma$-convergence and $H$-convergence with applications to non-periodic homogenization

Abstract

In this work we examine the stability of some classes of integrals, in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coefficients perturbed by a term vanishing at infinity, which has recently been examined in the framework of elliptic PDEs. We use localization techniques and higher-integrability Meyers-type results to provide a closure theorem by $\Gamma$-convergence within a large class of integral functionals. From such a result we derive stability theorems in homogenization which comprise the case of perturbations with zero average on the whole space. The results are also extended to the stochastic case, and specialized to the $G$-convergence of operators corresponding to quadratic forms. A corresponding analysis is also carried out for non-symmetric operators using the localization properties of $H$-convergence. Finally, we treat the case of perforated domains with Neumann boundary condition, and their stability.