$\Gamma$-convergence of convolution-type functionals for free discontinuity problems

TL;DR

This paper establishes $ ext{Gamma}$-convergence compactness and integral representation for a broad class of non-local energies related to free discontinuity problems, extending previous models and characterizing limit energies.

Contribution

It introduces a general $ ext{Gamma}$-convergence framework for non-local energies and characterizes the limit functionals in terms of minimization problems.

Findings

Proves compactness of non-local energies under $ ext{Gamma}$-convergence.

Provides integral representation of limit functionals on special functions of bounded variation.

Characterizes bulk and surface energy densities via minimization problems.

Abstract

We prove compactness with respect to $\Gamma$-convergence for a general class of non-local energies modelled after the ones considered in [Gobbino, CPAM (1998)]. We give an integral representation result for the limits, which are free discontinuity functionals defined on the space of generalised special functions of bounded variation. We then characterise the bulk and surface energy densities of the obtained limits by means of minimisation problems on small cubes for the approximating energies.