# On $BV$ functions and essentially bounded divergence-measure fields in metric spaces

## TL;DR

This paper develops a theory of functions of bounded variation and divergence-measure fields in metric measure spaces using Gigli's differential structure, extending classical concepts and formulas like Gauss-Green to these spaces.

## Contribution

It introduces a new notion of BV functions and divergence-measure fields in metric spaces, extending classical analysis tools to non-smooth settings with a focus on RCD spaces.

## Key findings

- Defined BV functions via vector fields in metric spaces
- Extended divergence-measure fields to metric measure spaces
- Established Gauss-Green formulas in this generalized setting

## Abstract

By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation ($BV$) in terms of suitable vector fields on a complete and separable metric measure space $(\mathbb{X},d,\mu)$ equipped with a non-negative Radon measure $\mu$ finite on bounded sets. Then, we extend the concept of divergence-measure vector fields $\mathcal{DM}^p(\mathbb{X})$ for any $p\in[1,\infty]$ and, by simply requiring in addition that the metric space is locally compact, we determine an appropriate class of domains for which it is possible to obtain a Gauss-Green formula in terms of the normal trace of a $\mathcal{DM}^\infty(\mathbb{X})$ vector field. This differential machinery is also the natural framework to specialize our analysis for ${\mathsf{RCD}(K,\infty)}$ spaces, where we exploit the underlying geometry to determine the Leibniz rules for $\mathcal{DM}^\infty(\mathbb{X})$ and ultimately to extend our discussion on the Gauss-Green formulas.