*Accepted Paper*

**Inserted:** 11 aug 2017

**Last Updated:** 21 feb 2019

**Journal:** Arch. Rational Mech. Anal.

**Year:** 2018

**Doi:** 10.1007/s00205-018-01344-7

**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 $C^1$ 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 $E_k\subset \Omega$ 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 \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
$L^1(\Omega;\mathbb{R}^n)$.