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

Abstract

AbstractWe study the limit behaviour of singularly-perturbed elliptic functionals of the form$$\begin{aligned} {\mathscr {F}}_k(u,v)=\int _A v^2\,f_k(x,\nabla u)\, \textrm{d}x+\frac{1}{\varepsilon _k}\int _A g_k(x,v,\varepsilon _k\nabla v)\, \textrm{d}x, \end{aligned}$$Fk(u,v)=∫Av2fk(x,∇u)dx+1εk∫Agk(x,v,εk∇v)dx,whereuis a vector-valued Sobolev function,$$v \in [0,1]$$v∈[0,1]a phase-field variable, and$$\varepsilon _k>0$$εk>0a singular-perturbation parameter; i.e.,$$\varepsilon _k \rightarrow 0$$εk→0, as$$k\rightarrow +\infty $$k→+∞. Under mild assumptions on the integrands$$f_k$$fkand$$g_k$$gk, we show that if$$f_k$$fkgrows superlinearly in the gradient-variable, then the functionals$${\mathscr {F}}_k$$Fk$$\Gamma $$Γ-converge (up to subsequences) to abrittleenergy-functional; i.e., to a free-discontinuity functional whose surface integrand doesnotdepend on the jump-amplitude ofu. 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$${\mathscr {F}}_k$$Fkdecouplein the limit. The abstract$$\Gamma $$Γ-convergence analysis is complemented by a stochastic homogenisation result forstationary randomintegrands.