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$Γ$-convergence and stochastic homogenisation of phase-transition functionals

In this paper we studythe asymptotics of singularly perturbed phase-transition functionals of the form \[ F_k(u)=\frac{1}{ε_k}\int_A f_k(x,u,ε_k\nabla u)\,dx\,, \] where $u \in [0,1]$ is a phase-field variable, $ε_k>0$ a singular-perturbation parameter, i.e., $ε_k \to 0$, 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 $F_k$ $Γ$-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 $Γ$-convergence result for stationary random integrands.