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 $SBD$, $SB{D}_{\infty }^p$, where the absolutely continuous part of the symmetric gradient is in $L^p$, with $p>1$, and $SB{D}^p$, whose functions are in $SB{D}_{\infty }^p$ 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 $SBV$, $SB{V}_{\infty }^p$, $SB{V}^p$ 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 $SB{D}^2$ is derived.

References in corpus (4)

• Approximation of a Brittle Fracture Energy with a Constraint of Non-Interpenetration
• A density result in $GSB{D}^p$ with applications to the approximation of brittle fracture energies
• Integral representation for functionals defined on $SB{D}^p$ in dimension two
• Approximation of fracture energies with $p$-growth via piecewise affine finite elements

Cited by in corpus (2)

• Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity
• Phase-field approximation of functionals defined on piecewise-rigid maps