*Accepted Paper*

**Inserted:** 7 oct 2016

**Last Updated:** 19 apr 2017

**Journal:** ESAIM: Control Optim. Calc. Var.

**Year:** 2017

**Doi:** 10.1051/cocv/2017027

**Abstract:**

We provide an approximation result for free-discontinuity functionals of the form \[\mathcal{F}(u)=\int_\Omega f(x,u,\nabla u)dx+\int_{S_u\cap\Omega}\theta(x,\nu_u)d\mathcal{H}^{n-1},\quad u\in SBV^2(\Omega),\] where $f$ is quadratic in the gradient-variable and $\theta$ is an arbitrary smooth Finsler metric. The approximating functionals are of Ambrosio-Tortorelli type and depend on the Hessian of the edge variable through a suitable nonhomogeneous metric $\phi$.

**Keywords:**
$\Gamma$-convergence, Finsler metrics, Ambrosio-Tortorelli approximation, anisotropic free-discontinuity functionals

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