Calculus of Variations and Geometric Measure Theory

R. Marziani

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

created by marziani on 23 Jun 2022
modified on 19 Apr 2023


Accepted Paper

Inserted: 23 jun 2022
Last Updated: 19 apr 2023

Journal: ESAIM: Control Optim. Calc. Var.
Pages: 32
Year: 2022


In this paper we study the 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.

Keywords: Γ-convergence, deterministic and stochastic homogenisation., Singular perturbation, phase-field approximation, free discontinuity problems