On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation

TL;DR

This paper investigates non-local functionals that approximate Sobolev norms and total variation, providing new methods to determine exact constants in their Gamma-convergence and constructing smooth recovery sequences.

Contribution

It introduces a novel approach to Gamma-convergence for these functionals, enabling the calculation of exact constants in certain cases and the construction of smooth recovery families.

Findings

Exact constants identified in special cases

New approach to Gamma-convergence developed

Existence of smooth recovery sequences demonstrated

Abstract

We consider the family of non-local and non-convex functionals introduced by H. Brezis and H.-M. Nguyen in a recent paper. These functionals Gamma-converge to a multiple of the Sobolev norm or the total variation, depending on a summability exponent, but the exact values of the constants are unknown in many cases. We describe a new approach to the Gamma-convergence result that leads in some special cases to the exact value of the constants, and to the existence of smooth recovery families.