${\Gamma}$-convergence and homogenisation for free discontinuity functionals with linear growth in the space of functions with bounded deformation

TL;DR

This paper investigates the $ ext{Gamma}$-convergence of free discontinuity functionals with linear growth in the space of functions with bounded deformation, establishing compactness, integral representation, and applications to homogenisation.

Contribution

It provides new compactness and integral representation results for $ ext{Gamma}$-limits of free discontinuity functionals in ${ m BD}$, advancing homogenisation theory.

Findings

Established $ ext{Gamma}$-convergence compactness in ${ m BD}$

Derived integral representation of $ ext{Gamma}$-limits

Applied results to deterministic and stochastic homogenisation

Abstract

We study the $\Gamma$-convergence of sequences of free discontinuity functionals with linear growth defined in the space ${\rm BD}$ of functions with bounded deformation. We prove a compactness result with respect to $\Gamma$-convergence and outline the main properties of the $\Gamma$-limits, which lead to an integral representation result. The corresponding integrands are obtained by taking limits of suitable minimisation problems on small cubes. These results are then used to study the deterministic and stochastic homogenisation problem for a large class of free discontinuity functionals defined in ${\rm BD}$.