Stochastic homogenization of nonconvex unbounded integral functionals with generalized Orlicz growth

TL;DR

This paper develops a stochastic homogenization theory for nonconvex, unbounded integral functionals with generalized Orlicz growth, extending the understanding of their asymptotic behavior in random media.

Contribution

It provides a complete homogenization framework for nonconvex unbounded functionals with generalized Orlicz growth, including approximation results in Musielak-Orlicz spaces.

Findings

Established homogenization results under standard assumptions including coercivity and growth conditions.

Defined the limit energy in anisotropic Musielak-Orlicz spaces with approximation properties.

Utilized localization and truncation techniques in the proof of homogenization.

Abstract

We consider the homogenization of random integral functionals which are possibly unbounded, that is, the domain of the integrand is not the whole space and may depend on the space-variable. In the vectorial case, we develop a complete stochastic homogenization theory for nonconvex unbounded functionals with convex growth of generalized Orlicz-type, under a standard set of assumptions in the field, in particular a coercivity condition of order $p^->1$, and an upper bound of order $p^+<\infty$. The limit energy is defined in a possibly anisotropic Musielak-Orlicz space, for which approximation results with smooth functions are provided. The proof is based on the localization method of $\Gamma$-convergence and a careful use of truncation arguments.