  In this work we examine the stability of some classes of integrals, and 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 been recently examined in the framework
of elliptic PDE. 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 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 on 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.
