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

TL;DR

This paper extends the understanding of singularly perturbed Perona-Malik functionals to higher dimensions, establishing $ ext{Gamma}$-convergence and equicoerciveness results using new density techniques in BV spaces.

Contribution

It generalizes one-dimensional results to arbitrary dimensions and introduces a novel density result in BV spaces for functionals with discontinuities.

Findings

Established $ ext{Gamma}$-convergence in arbitrary dimensions.

Proved a new density result in BV spaces with vanishing diffuse gradient.

Provided a framework for approximating functionals with cohesive energies.

Abstract

In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and $\Gamma$-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.