On formulae decoupling the total variation of BV functions

TL;DR

This paper develops formulae to represent the singular part of the derivative measure of BV functions as limits of non-local functionals, including fractional Laplacian rescalings that converge to this singular component.

Contribution

It introduces new formulae linking non-local functionals and the singular measure derivative of BV functions, including fractional Laplacian rescaling convergence.

Findings

Rescalings of fractional Laplacian converge to the singular derivative measure.

New formulae characterize the singular part of BV functions' derivatives.

Non-local functionals effectively capture the measure's singular component.

Abstract

In this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non-local functionals. One special case shows that rescalings of the fractional Laplacian of a function $u\in SBV$ converge strictly to the singular portion of $Du$.