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.