On the approximation of SBD functions and some applications

Three density theorems for three suitable subspaces of SBD functions, in the strong BD topology, are proven. The spaces are SBD, SBD∞p , where the absolutely continuous part of the symmetric gradient is in Lp, with p > 1, and SBDp, whose functions are in SBD∞p and the jump set has finite scrH n -1-measure. This generalizes on the one hand the density result [J. Math. Pures Appl. (9), 83 (2004), pp. 929-954] by Chambolle and, on the other hand, extends in some sense the three approximation theorems in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), pp. 369-413] by de Philippis, Fusco, and Pratelli for SBV , SBV∞p , SBV p spaces, obtaining also more regularity for the absolutely continuous part of the approximating functions. As an application, the sharp version of two Γ-convergence results for energies defined on SBD2 is derived.