On the approximation of $SBD$ functions and some applications

TL;DR

This paper proves new density theorems for subspaces of $SBD$ functions, extending previous approximation results and applying them to improve $ ext{Gamma}$-convergence results for energies on $SBD^2$ functions.

Contribution

It introduces three density theorems for $SBD$ subspaces, generalizing prior results and enhancing regularity of approximations, with applications to $ ext{Gamma}$-convergence.

Findings

Established density theorems for $SBD$, $SBD^p_ty$, and $SBD^p$ spaces.

Extended approximation theorems from $SBV$ to $SBD$ spaces.

Derived sharper $ ext{Gamma}$-convergence results for $SBD^2$ energies.

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.