The Anzellotti-Gauss-Green formula and least gradient functions in metric measure spaces

TL;DR

This paper extends the Anzellotti-Gauss-Green formula to metric measure spaces and applies it to analyze least gradient functions, advancing the understanding of BV functions and divergence in non-smooth settings.

Contribution

It introduces a Gauss-Green formula in metric measure spaces and uses it to study least gradient functions, bridging a gap in analysis on non-smooth spaces.

Findings

Gauss-Green formula established for BV functions in metric spaces

Application of the formula to analyze least gradient functions

Enhanced understanding of divergence and BV functions in metric measure spaces

Abstract

In the framework of the first-order differential structure introduced by Gigli, we obtain a Gauss-Green formula on regular bounded open sets of metric measure spaces, valid for BV functions and vector fields with integrable divergence. Then, we study least gradient functions in metric measure spaces using this formula as the main tool.