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 infinity p, where the absolutely continuous part of the symmetric gradient is in L-p, with p > 1, and SBDp, whose functions are in SBD infinity p and the jump set has finite Hn-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 infinity p, SBVp spaces, obtaining also more regularity for the absolutely continuous part of the approximating functions. As an application, the sharp version of two Gamma-convergence results for energies defined on SBD2 is derived.