A Density Result in GSBD^p with Applications to the Approximation of Brittle Fracture Energies

Abstract

We prove that any function in ${GSBD^p(\Omega)}$, with ${\Omega}$ a n-dimensional open bounded set with finite perimeter, is approximated by functions ${u_k\in SBV(\Omega; \mathbb{R}^{n})\cap L^\infty(\Omega; \mathbb{R}^{n})}$ whose jump is a finite union of C¹ hypersurfaces. The approximation takes place in the sense of Griffith-type energies ${\int_\Omega W(e(u)) {\rm dx} +\mathcal{H}^{n-1}(J_u)}$, e(u) and J_u being the approximate symmetric gradient and the jump set of u, and W a nonnegative function with p-growth, p > 1. The difference between u_k and u is small in L^p outside a sequence of sets ${E_k\subset \Omega}$ whose measure tends to 0 and if ${|u|^r \in L^1(\Omega)}$ with ${r\in (0,p]}$, then ${|u_k-u|^r \to 0}$ in ${L^1(\Omega)}$. Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce ${\Gamma}$-convergence approximation à la Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are a priori not even in ${L^1(\Omega; \mathbb{R}^{n})}$.