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

In this paper we studythe asymptotics of singularly perturbed phase-transition functionals of the form $$F_k(u)=\frac{1}{{\epsilon }_k}{\int }_A{f}_k(x,u,{\epsilon }_k\mathrm{\nabla }u)dx,$$ where $u\in [0,1]$ is a phase-field variable, ${\epsilon }_k>0$ a singular-perturbation parameter, i.e., ${\epsilon }_k\to 0$, as $k\to +\mathrm{\infty }$, 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 $F_k$ $\mathrm{\Gamma }$-converge (up to subsequences) to a surface functional of the form $$F_{\mathrm{\infty }}(u)={\int }_{S_u\cap A}{f}_{\mathrm{\infty }}(x,{\nu }_u)d{\mathcal{H}}^{n-1},$$ where $u\in BV(A;\{0,1\})$ and $f_{\mathrm{\infty }}$ 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 $\mathrm{\Gamma }$-convergence result for stationary random integrands.