*Published Paper*

**Inserted:** 8 jun 2018

**Last Updated:** 13 dec 2019

**Journal:** SIAM J. Math. Anal.

**Volume:** 51

**Number:** 6

**Pages:** 5011-5048

**Year:** 2019

**Doi:** 10.1137/18M119522X

**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.

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