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

TL;DR

This paper investigates the asymptotic behavior of phase-transition functionals using $ ext{Gamma}$-convergence, establishing their limit as surface functionals, and extends results to stochastic homogenisation with random integrands.

Contribution

It provides a rigorous $ ext{Gamma}$-convergence analysis of phase-transition functionals and extends the framework to stochastic homogenisation for stationary random integrands.

Findings

$ ext{Gamma}$-convergence of phase-transition functionals to surface functionals.

Characterization of the limit integrand $f_ ext{infty}$ via scaled minimization.

Extension of $ ext{Gamma}$-convergence results to stochastic homogenisation.

Abstract

In this paper we studythe asymptotics of singularly perturbed phase-transition functionals of the form \[ F_k(u)=\frac{1}{\epsilon_k}\int_A f_k(x,u,\epsilon_k\nabla u)\,dx\,, \] 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 \[ F_\infty(u)=\int_{S_u\cap A}f_\infty(x,\nu_u)\,d\mathcal H^{n-1}\,,\] 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.