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

Abstract

We study the $\hat{I}“$-convergence of sequences of free discontinuity functionals with linear growth defined in the space $\mathrm{BD}$ of functions with bounded deformation. We prove a compactness result with respect to $\hat{I}“$-convergence and outline the main properties of the $\hat{I}“$-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 $\mathrm{BD}$.

38 pages