A sharp lower bound for a class of non-local approximations of the total variation

TL;DR

This paper establishes a precise lower bound for a class of non-local functionals that approximate the total variation, enhancing understanding of their behavior in characterizing functions of bounded variation.

Contribution

It provides a new sharp lower bound for the liminf of non-local functionals related to total variation, with explicit coefficients for different parts.

Findings

Sharp lower bound for non-local functionals established

Explicit coefficients for total variation parts derived

Improved understanding of non-local approximations of total variation

Abstract

We study a class of non-local functionals that was introduced by Brezis-Seeger-Van Schaftingen-Yung (2022), and can be used to characterize functions of bounded variation. We give a new lower bound for the liminf of these functionals, involving the three different parts of the total variation, with sharp coefficients.