$\Gamma$-convergence and stochastic homogenisation of phase-transition functionals

Abstract

In this paper we studythe asymptotics of singularly perturbed phase-transition functionals of the form \[ F_k(u)=\frac{1}{ε_k}\int_A f_k(x,u,ε_k\nabla u)\,dx\,, \] where $u\in [0,1]$ is a phase-field variable, ${\epsilon }_k>0$ a singular-perturbation parameter, i.e., ${\epsilon }_k\to 0$, as $k\to +\infty$, and the integrands $f_k$ are such that, for every $x$ and every $k$, $f_k(x,\cdot ,0)$ is a double well potential with zeros at 0 and 1. We prove that the functionals $F_k$ $\Gamma$-converge (up to subsequences) to a surface functional of the form \[ F_\infty(u)=\int_{S_u\cap A}f_\infty(x,ν_u)\,d\mathcal H^{n-1}\,,\] where $u\in BV(A;\{0,1\})$ and $f_{\infty }$ is characterised by the double limit of suitably scaled minimisation problems. Afterwards we extend our analysis to the setting of stochastic homogenisation and prove a $\Gamma$-convergence result for stationary random integrands.