$$\varvec{\Gamma }$$-Convergence and Stochastic Homogenization of Second-Order Singular Perturbation Models for Phase Transitions

Abstract

AbstractWe 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 $$\varepsilon $$ ε 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 $$\Gamma $$ Γ -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.