Properties of a Hilbertian Norm for Perimeter

TL;DR

This paper extends the relationship between fractional Sobolev norms and perimeter to functions of bounded variation and provides an exact formula for sets of finite perimeter, advancing understanding of geometric measure theory.

Contribution

It generalizes previous results to BV functions and derives an exact formula for finite perimeter sets, deepening the link between Sobolev norms and geometric properties.

Findings

Extended the relationship to BV functions

Derived an exact formula for finite perimeter sets

Identified open questions for future research

Abstract

A recent paper of Jerison and Figalli proved a relationship between the $H^{1/2}$ norms of smoothed out indicator functions of sets and their perimeter. We continue this line of investigation and extend it in two ways. First, we describe a description of the situation with general functions of bounded variation, and show that a related quantity controls the size of the jump set. Second, we provide an exact formula in the case of a set of finite perimeter. Several questions remain and are presented here.