A two scale Γ -convergence approach for random non-convex homogenization

Leonid Berlyand, Etienne Sandier, Sylvia Serfaty

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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.

Original languageEnglish (US)
Article number156
JournalCalculus of Variations and Partial Differential Equations
Volume56
Issue number6
DOIs
StatePublished - Dec 1 2017

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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