On the approximation of $SBD$ functions and some applications

Abstract: Three density theorems for three suitable subspaces of $SBD$ functions, in the strong $BD$ topology, are proven. The spaces are $SBD$, $SBD^p_\infty$, where the absolutely continuous part of the symmetric gradient is in $L^p$, with $p>1$, and $SBD^p$, whose functions are in $SBD^p_\infty$ and the jump set has finite $\mathcal{H}^{n-1}$-measure. This generalises on the one hand the density result by [Chambolle, 2004-2005] and, on the other hand, extends in some sense the three approximation theorems in by [De Philippis, Fusco, Pratelli, 2017] for $SBV$, $SBV^p_\infty$, $SBV^p$ spaces, obtaining also more regularity for the absolutely continuous part of the approximating functions. As application, the sharp version of two $\Gamma$-convergence results for energies defined on $SBD^2$ is derived.