Anzellotti's pairing theory and the Gauss--Green theorem

TL;DR

This paper extends the Gauss-Green theorem to weakly differentiable functions and sets of finite perimeter by revisiting Anzellotti's pairing theory, providing a general measure-theoretic framework.

Contribution

It generalizes the Gauss-Green formula using Anzellotti's pairing theory for divergence measure fields and BV functions, broadening its applicability.

Findings

Established a general Gauss-Green formula for weakly differentiable functions.

Characterized the measure pairing for divergence measure vector fields and BV functions.

Extended the theoretical framework for divergence and boundary measure interactions.

Abstract

In this paper we obtain a very general Gauss-Green formula for weakly differentiable functions and sets of finite perimeter. This result is obtained by revisiting Anzellotti's pairing theory and by characterizing the measure pairing $(\boldsymbol{A}, Du)$ when $\boldsymbol{A}$ is a bounded divergence measure vector field and $u$ is a bounded function of bounded variation.