Γ-convergence and stochastic homogenisation of phase-transition functionals

Abstract

In this paper, we study the asymptotics of singularly perturbed phase-transition functionals of the formℱk(u) = 1/εk∫Afk(𝑥,u,εk∇u)d𝑥,whereu∈ [0, 1] is a phase-field variable, εk> 0 a singular-perturbation parameteri.e., εk→ 0, ask→ +∞, and the integrandsfkare such that, for everyxand everyk,fk(x, ·, 0) is a double well potential with zeros at 0 and 1. We prove that the functionalsFkΓ-converge (up to subsequences) to a surface functional of the formℱ∞(u) = ∫Su∩Af∞(𝑥,𝜈u)dHn-1,whereu∈BV(A; {0, 1}) andf∞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 forstationary randomintegrands.