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A two scale \(\Gamma \)-convergence approach for random non-convex homogenization

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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 \(\Gamma \)-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 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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Acknowledgements

The authors wish to thank the referee for his/her careful reading of the manuscript and comments. They also wish to thank V. Bergelson and E. Lesigne for pointing them to references [27] and [18]. The work of L.B. and E.S. was partially supported by NSF Grants DMS-1405769 and DMS-1106666.

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Authors and Affiliations

  1. Pennsylvania State University, University Park, PA, 16802, USA

    Leonid Berlyand

  2. LAMA – CNRS UMR 8050, Université Paris-Est, 61, Avenue du Général de Gaulle, 94010, Créteil, France

    Etienne Sandier

  3. UPMC Univ. Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, Sorbonne Universités, 4, place Jussieu, 75005, Paris, France

    Sylvia Serfaty

  4. Institut Universitaire de France, Paris, France

    Sylvia Serfaty

  5. Courant Institute, New York University, 251 Mercer st, New York, NY, 10012, USA

    Sylvia Serfaty

Authors
  1. Leonid Berlyand
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  2. Etienne Sandier
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  3. Sylvia Serfaty
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Correspondence to Etienne Sandier.

Additional information

Communicated by L. Ambrosio.

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Berlyand, L., Sandier, E. & Serfaty, S. A two scale \(\Gamma \)-convergence approach for random non-convex homogenization. Calc. Var. 56, 156 (2017). https://doi.org/10.1007/s00526-017-1249-y

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