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

Leonid Berlyand, Etienne Sandier, Sylvia Serfaty

Research output: Contribution to journalArticle

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

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Two-scale Convergence
Homogenization
Stochastic Homogenization
Young Measures
Non-convexity
Cell
Singular Perturbation
Term
Variational Problem
Paradigm
Lower bound
Energy
Profile
Framework

Keywords

  • 35B27
  • 35J20
  • 60H25

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

A two scale Γ -convergence approach for random non-convex homogenization. / Berlyand, Leonid; Sandier, Etienne; Serfaty, Sylvia.

In: Calculus of Variations and Partial Differential Equations, Vol. 56, No. 6, 156, 01.12.2017.

Research output: Contribution to journalArticle

Berlyand, Leonid ; Sandier, Etienne ; Serfaty, Sylvia. / A two scale Γ -convergence approach for random non-convex homogenization. In: Calculus of Variations and Partial Differential Equations. 2017 ; Vol. 56, No. 6.
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