Gamma -convergence and stochastic homogenisation of singularly-perturbed elliptic functionals

Abstract

We study the limit behaviour of singularly-perturbed elliptic functionals of the form Fk(u,v)=∫Av2fk(x,∇u)dx+1εk∫Agk(x,v,εk∇v)dx,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathscr {F}}_k(u,v)=\\int _A v^2\\,f_k(x,\ abla u)\\, \ extrm{d}x+\\frac{1}{\\varepsilon _k}\\int _A g_k(x,v,\\varepsilon _k\ abla v)\\, \ extrm{d}x, \\end{aligned}$$\\end{document}where u is a vector-valued Sobolev function, v in [0,1] a phase-field variable, and varepsilon _k>0 a singular-perturbation parameter; i.e., varepsilon _k rightarrow 0, as krightarrow +infty . Under mild assumptions on the integrands f_k and g_k, we show that if f_k grows superlinearly in the gradient-variable, then the functionals {mathscr {F}}_kGamma -converge (up to subsequences) to a brittle energy-functional; i.e., to a free-discontinuity functional whose surface integrand does not depend on the jump-amplitude of u. This result is achieved by providing explicit asymptotic formulas for the bulk and surface integrands which show, in particular, that volume and surface term in {mathscr {F}}_kdecouple in the limit. The abstract Gamma -convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.