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

TL;DR

This paper investigates the asymptotic behavior of singularly-perturbed elliptic functionals, proving their $ ext{Gamma}$-convergence to brittle energy functionals and establishing stochastic homogenisation results for random integrands.

Contribution

It provides explicit formulas for the limit integrands and demonstrates the decoupling of volume and surface terms in the $ ext{Gamma}$-limit, extending to stochastic homogenisation.

Findings

$ ext{Gamma}$-convergence to brittle energies under superlinear growth

Explicit asymptotic formulas for bulk and surface integrands

Decoupling of volume and surface terms in the limit

Abstract

We study the limit behaviour of singularly-perturbed elliptic functionals of the form \[ \mathcal F_k(u,v)=\int_A v^2\,f_k(x,\nabla u)\.dx+\frac{1}{\varepsilon_k}\int_A g_k(x,v,\varepsilon_k\nabla v)\.dx\,, \] 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., $\varepsilon_k \to 0$, as $k\to +\infty$. 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 $\mathcal F_k$ $\Gamma$-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 $\mathcal F_k$ decouple in the limit. The abstract $\Gamma$-convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.