A Rough Divergence Theorem

TL;DR

This paper generalizes the divergence theorem to domains with inner boundaries by introducing a new boundary integral concept and characterizing the domain conditions through functions of bounded fluctuation.

Contribution

It establishes a divergence theorem for rough domains with inner boundaries using a novel boundary integral and characterizes domain conditions via functions of bounded fluctuation.

Findings

Boundary integral expressed via a surface functional

Necessary and sufficient conditions for divergence theorem validity

Introduction of the space of functions with bounded fluctuation

Abstract

A generalized divergence theorem is established allowing for domains with inner boundaries. The normal trace of a rough integrand is not a Radon measure; rather, the boundary integral is expressed via a surface functional continuous with respect to the uniform convergence of integrands. We provide necessary and sufficient analytic and geometric conditions on the domain for the validity of the theorem. Central to this characterization is the introduction of the space of functions having bounded fluctuation, whose norm is precisely defined so that the divergence theorem holds if and only if the characteristic function $\chi_U$ of the integration domain $U \subset \R^m$ has finite norm.