Stochastic homogenization of nonconvex unbounded integral functionals with convex growth

TL;DR

This paper develops a comprehensive stochastic homogenization theory for nonconvex unbounded integral functionals with convex growth, addressing a gap in the understanding of such problems especially in the nonconvex case.

Contribution

It introduces new homogenization results for unbounded functionals with convex growth, including nonconvex cases via two-sided estimates, extending the theory to more complex models.

Findings

Homogenization holds for convex integrands with p-growth when p>d.

Homogenization extends to nonconvex integrands under two-sided convex estimates.

Results are applicable to the derivation of rubber elasticity from polymer physics.

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.