Γ-convergence and stochastic homogenisation of phase-transition functionals

2022/06/27 by Roberta Marziani, Marziani, Roberta · 2 citations
Computer Science · Mathematics · #49J45 #49Q20 #74Q05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2206.13131

openalex publication_date 2022/06/27 · openalex created_date 2022/07/01 · openalex updated_date 2026/07/28

Abstract

In this paper we studythe asymptotics of singularly perturbed phase-transition functionals of the form F_k(u)=(1)/(ε_k)∫_A f_k(x,u,ε_k∇ u) dx , where u ∈ [0,1] is a phase-field variable, ε_k>0 a singular-perturbation parameter, i.e., ε_k → 0, as k→ +∞, and the integrands f_k are such that, for every x and every k, f_k(x,⋅ ,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_∞(u)=∫_{Su∩ A}f_∞(x,ν_u) d\mathcal H^n-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.

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