Calculus of Variations and Geometric Measure Theory

A. Bach - R. Marziani - C. I. Zeppieri

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

created by zeppieri1 on 10 Feb 2021
modified on 18 Feb 2021



Inserted: 10 feb 2021
Last Updated: 18 feb 2021

Year: 2021


We study the limit behaviour of singularly-perturbed elliptic functionals of the form \[ F_k(u,v)=\int_A v^2\,f_k(x,\nabla u)dx+\frac{1}{\varepsilon_k}\int_A g_k(x,v,\varepsilon_k\nabla v)dx\,, \] 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 $F_k$ $\Gamma$-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 bulk and surface integrands which show, in particular, that the volume and surface term in $F_k$ decouple in the limit.

The abstract $\Gamma$-convergence analysis is complemented by a stochastic homogenisation result for stationary random integrands.

Keywords: $\Gamma$-convergence, singular perturbation, stochastic homogenisation, elliptic approximation, free-discontinuity functionals