A density result in $GSBD^p$ with applications to the approximation of brittle fracture energies

Abstract

We prove that any function in $GSBD^p(Ω)$, with $Ω$ a $n$-dimensional open bounded set with finite perimeter, is approximated by functions $u_k\in SBV(Ω;\mathbb{R}^n)\cap L^\infty(Ω;\mathbb{R}^n)$ whose jump is a finite union of $C^1$ hypersurfaces. The approximation takes place in the sense of Griffith-type energies $\int_ΩW(e(u)) \,\mathrm{d}x +\mathcal{H}^{n-1}(J_u)$, $e(u)$ 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 $E_k\subset Ω$ whose measure tends to 0 and if $|u|^r \in L^1(Ω)$ with $r\in (0,p]$, then $|u_k-u|^r \to 0$ in $L^1(Ω)$. Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce $Γ$-convergence approximation \emph{à la} Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are \emph{a priori} not even in $L^1(Ω;\mathbb{R}^n)$.