Calculus of Variations and Geometric Measure Theory

A. Braides - G. Dal Maso - C. Le Bris

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

created by braidesa on 29 Feb 2024
modified on 01 Mar 2024


Submitted Paper

Inserted: 29 feb 2024
Last Updated: 1 mar 2024

Year: 2024

ArXiv: 2402.19031 PDF


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 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.

Keywords: Homogenization, Gamma-convergence, stability, perforated domains, H-convergence