On the gap between Gamma-limit and pointwise limit for a non-local approximation of the total variation

TL;DR

This paper investigates how the Gamma-limit of non-local functionals approximating total variation depends on the interaction law, revealing it depends on the entire law shape and identifying cases where it matches the pointwise limit.

Contribution

It demonstrates the Gamma-limit's dependence on the full interaction law shape and identifies conditions where it coincides with the pointwise limit for smooth functions.

Findings

Gamma-limit depends on the full interaction law shape

Existence of interaction laws where Gamma-limit equals pointwise limit

Reduction of Gamma-limit computation to multi-variable minimum problems

Abstract

We consider the approximation of the total variation of a function by the family of non-local and non-convex functionals introduced by H. Brezis and H.-M. Nguyen in a recent paper. The approximating functionals are defined through double integrals in which every pair of points contributes according to some interaction law. In this paper we answer two open questions concerning the dependence of the Gamma-limit on the interaction law. In the first result, we show that the Gamma-limit depends on the full shape of the interaction law, and not only on the values in a neighborhood of the origin. In the second result, we show that there do exist interaction laws for which the Gamma-limit coincides with the pointwise limit on smooth functions. The key argument is that for some special classes of interaction laws the computation of the Gamma-limit can be reduced to studying the asymptotic behavior of suitable multi-variable minimum problems.