The $\mathrm{\Gamma }$-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension

In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and $\mathrm{\Gamma }$-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functionals with discontinuities penalized with a “cohesive” energy, that is, whose cost depends on the actual opening of the discontinuity.