A Two scale Γ-convergence Approach for Random Non-Convex Homogenization

2016/01/17 by Berlyand, Leonid, Sandier, Etienne, Serfaty, Sylvia
#35B27 #35J20 #60H25 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1601.04344

Abstract

We propose an abstract framework for the homogenization of random functionals which may contain non-convex terms, based on a two-scale Γ-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 Γ-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 Müller, 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.

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