A density result in $GSB{D}^p$ with applications to the approximation of brittle fracture energies

We prove that any function in $GSB{D}^p(\mathrm{\Omega })$, with $\mathrm{\Omega }$ a $n$-dimensional open bounded set with finite perimeter, is approximated by functions $u_k\in SBV(\mathrm{\Omega };{\mathbb{R}}^n)\cap L^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^n)$ whose jump is a finite union of $C^1$ hypersurfaces. The approximation takes place in the sense of Griffith-type energies $\int_\Omega W(e(u)) ,$\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 $E_k\subset \mathrm{\Omega }$ whose measure tends to 0 and if $\mathrm{\mid }u{\mathrm{\mid }}^r\in L^1(\mathrm{\Omega })$ with $r\in (0,p]$, then $\mathrm{\mid }{u}_k-u{\mathrm{\mid }}^r\to 0$ in $L^1(\mathrm{\Omega })$. Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce $\mathrm{\Gamma }$-convergence approximation \emph{`a 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 $L^1(\mathrm{\Omega };{\mathbb{R}}^n)$.