Stochastic Homogenization of Nonconvex Unbounded Integral Functionals with Convex Growth

Abstract

We consider the well-travelled problem of 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 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. ContentsDate: August 4, 2018. 1 2 M. DUERINCKX AND A. GLORIA A.3. Ergodic Weyl decomposition 53 A.4. Measurability results 54 A.5. Approximation results 59 References 62 Proposition 3.2 (Γ-lim inf inequality). For all ω ∈ Ω 1 , all bounded domains O ⊂ R d , and all sequences (u εProof. Let O, (u ε ) ε , and u be as in the statement. Then, for all ω ∈ Ω 1 and all k ∈ N, using the Γ-lim inf result for J k ε (·, ω; O) towards J k (·; O) (see Section 3.2), and thatso that, by monotone convergence, lim inf ε↓0 J ε (u ε , ω; O) ≥ lim k↑∞ J k (u; O) = J(u; O).From this Γ-lim inf result, we deduce the locality of recovery sequences, if they exist.

The additive structure of elliptic homogenization

“…Results partially overlapping with the argument presented here have recently appeared in [13]. There, the randomness of the environment is assumed to have a product structure, so that the Chatterjee-Stein method of normal approximation becomes available.…”

“…A version of the statement of joint convergence in law of (11.3)-(11.5)-(11.6) simultaneously is then obtained. As far as we understand, the notion of "path-wise theory" introduced in [13] refers to this joint convergence. Note that an analogue ofb r (z) is called "the homogenization commutator" in [13].…”

Manifold Constrained Non-uniformly Elliptic Problems

“…These are known in the literature as (p, q)-growth conditions or non-standard growth conditions. They have pioneered by Uraltseva & Urdaletova [64] and Zhikov [65,66,67] in the context of Homogenization (see also the recent paper [21]). In the setting of the Calculus of Variations they have been systematically studied by Marcellini [46,47].…”

From Statistical Polymer Physics to Nonlinear Elasticity

“…We now comment on the main two analytical simplifications of this work, namely that f ij and W have p-growth from above. We believe that at least parts of the results survive if we let f ij blow up at finite deformation, following the approach developed by Duerinckx and the second author in [17] in the continuum setting. In contrast, the growth condition on W is crucial for our arguments to work.…”