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The divergence theorem and nonlocal counterparts

Solveig Hepp, Moritz Kassmann

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.

The divergence theorem and nonlocal counterparts

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.
Paper Structure (4 sections, 9 theorems, 83 equations)

This paper contains 4 sections, 9 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. The nonlocal setting
  3. The local divergence theorem
  4. Fractional operators, perimeters, and the mean curvature revisited

Key Result

Proposition 1.1

Let $\Omega \subset {\mathbb{R}^d}$ be measurable, $f: \mathbb R^d \times \mathbb R^d \to \mathbb{R}$ antisymmetric and sufficiently regular and $\mu(h)\mathrm{d} h$ a symmetric measure. Then

Theorems & Definitions (18)

  • Proposition 1.1: nonlocal divergence theorem
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • ...and 8 more