Γ-convergence and stochastic homogenisation of phase-transition functionals

In this paper, we study the asymptotics of singularly perturbed phase-transition functionals of the form ℱk(u) = 1/εk∫Afk(𝑥,u,εk∇u)d𝑥, where u ∈ [0, 1] is a phase-field variable, εk > 0 a singular-perturbation parameter i.e., εk → 0, as k → +∞, and the integrands fk are such that, for every x and every k, fk(x, ·, 0) is a double well potential with zeros at 0 and 1. We prove that the functionals Fk Γ-converge (up to subsequences) to a surface functional of the form ℱ∞(u) = ∫Su∩Af∞(𝑥,𝜈u)dHn-1, where u ∈ BV(A; {0, 1}) and f∞ 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.