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Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth

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Abstract

We consider the well-trodden ground of the problem of the homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has p-growth from below (with p > d, the dimension), then it admits homogenization regardless of growth conditions from above. This result, that crucially relies on the existence and sublinearity at infinity of correctors, is also new in the periodic case. In the case of nonconvex integrands, we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two-sided estimate by a convex integrand (the domain of which may depend on the space variable) that itself admits homogenization. This result is of interest to the rigorous derivation of rubber elasticity from polymer physics, which involves the stochastic homogenization of such unbounded functionals.

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Authors and Affiliations

  1. Université Libre de Bruxelles (ULB), Brussels, Belgium

    Mitia Duerinckx & Antoine Gloria

  2. Inria, Lille, France

    Mitia Duerinckx & Antoine Gloria

Authors
  1. Mitia Duerinckx
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  2. Antoine Gloria
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Correspondence to Antoine Gloria.

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Communicated by A. Braides

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Duerinckx, M., Gloria, A. Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth. Arch Rational Mech Anal 221, 1511–1584 (2016). https://doi.org/10.1007/s00205-016-0992-0

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