Fractional Sobolev norms and BV functions on manifolds

TL;DR

This paper characterizes functions of bounded variation and Sobolev functions on compact manifolds using limits of fractional Sobolev seminorms, providing a new perspective on these function spaces.

Contribution

It introduces a novel representation of BV and Sobolev seminorms on manifolds as limits of fractional seminorms, extending classical Euclidean results.

Findings

BV and Sobolev seminorms are represented as limits of fractional seminorms on manifolds

Characterization of functions of bounded variation on compact manifolds

Application to sets of finite perimeter

Abstract

The bounded variation seminorm and the Sobolev seminorm on compact manifolds are represented as a limit of fractional Sobolev seminorms. This establishes a characterization of functions of bounded variation and of Sobolev functions on compact manifolds. As an application the special case of sets of finite perimeter is considered.