The divergence theorem and nonlocal counterparts

TL;DR

This paper introduces a novel proof of the classical divergence theorem using a nonlocal analog and rescaling, connecting local and nonlocal divergence concepts and defining nonlocal notions like fractional perimeter.

Contribution

It provides a new proof of the divergence theorem based on nonlocal analogs, linking classical and fractional geometric concepts.

Findings

Established a nonlocal divergence theorem framework

Connected nonlocal divergence with classical divergence

Defined nonlocal concepts such as fractional perimeter

Abstract

We present a new proof of the classical divergence theorem in bounded domains. Our proof is based on a nonlocal analog of the divergence theorem and a rescaling argument. Main ingredients in the proof are nonlocal versions of the divergence and the normal derivative. We employ these to provide definitions of well-known nonlocal concepts such as the fractional perimeter.