$\Gamma$-convergence and stochastic homogenization of second order singular perturbation models for phase transitions

TL;DR

This paper investigates the asymptotic behavior of complex random phase transition energies in heterogeneous materials, proving their convergence to a deterministic surface energy under certain conditions.

Contribution

It establishes $ ext{Gamma}$-convergence of random anisotropic functionals to a deterministic surface energy, incorporating second order perturbations and stochastic homogenization.

Findings

Almost sure $ ext{Gamma}$-convergence to a surface energy

Surface energy density is independent of space variable

Deterministic limit described by a cell formula

Abstract

We study the effective behavior of random, heterogeneous, anisotropic, second order phase transitions energies that arise in the study of pattern formations in physical-chemical systems. Specifically, we study the asymptotic behavior, as $\epsilon$ goes to zero, of random heterogeneous anisotropic functionals in which the second order perturbation competes not only with a double well potential but also with a possibly negative contribution given by the first order term. We prove that, under suitable growth conditions and under a stationarity assumption, the functionals $\Gamma$-converge almost surely to a surface energy whose density is independent of the space variable. Furthermore, we show that the limit surface density can be described via a suitable cell formula and is deterministic when ergodicity is assumed.