A Two scale $\Gamma$-convergence Approach for Random Non-Convex Homogenization

TL;DR

This paper introduces a novel two-scale onvergence framework for homogenizing random, non-convex functionals, extending existing methods to handle complex oscillations and mesoscale phenomena.

Contribution

It develops an abstract onvergence approach using Young measures on micropatterns, enabling homogenization of non-convex random functionals and recovering known convex results.

Findings

Established a lower bound via a cell problem on expanding cells.

Extended homogenization results to non-convex stochastic functionals.

Applied the method to a non-convex variational problem with mesoscale features.

Abstract

We propose an abstract framework for the homogenization of random functionals which may contain non-convex terms, based on a two-scale $\Gamma$-convergence approach and a definition of Young measures on micropatterns which encodes the profiles of the oscillating functions and of functionals. Our abstract result is a lower bound for such energies in terms of a cell problem (on large expanding cells) and the $\Gamma$-limits of the functionals at the microscale. We show that our method allows to retrieve the results of Dal Maso and Modica in the well-known case of the stochastic homogenization of convex Lagrangians. As an application, we also show how our method allows to stochastically homogenize a variational problem introduced and studied by Alberti and M\"uller, which is a paradigm of a problem where an additional mesoscale arises naturally due to the non-convexity of the singular perturbation (lower order) terms in the functional.