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.

arXiv commentIn the final version we have explained how to apply our proof of the divergence theorem to more general domains including polytopes. The article will be published in the Bulletin of the London Mathematical Society

References in corpus (5)

• Gamma-convergence of nonlocal perimeter functionals
• Fractional div-curl quantities and applications to nonlocal geometric equations
• A general framework for nonlocal Neumann problems
• A Remake of Bourgain-Brezis-Mironescu Characterization of Sobolev Spaces
• Leibniz rules and Gauss-Green formulas in distributional fractional spaces

Cited by in corpus (2)

• A general framework for nonlocal Neumann problems
• Convergence rates of the fractional to the local Dirichlet problem