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

Abstract

By employing the differential structure recently developed by N. Gigli, we can extend the notion of divergence-measure vector fields $\\mathcal{DM}^p(\\mathbb{X})$, $1\\le p \\le \\infty$, to the very general context of a (locally compact) metric measure space $(\\mathbb{X},d,\\mu)$ satisfying no further structural assumptions. Here we determine the 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 give first a notion of functions of bounded variation ($BV$) in terms of suitable vector fields, to determine the Leibniz rules for $\\mathcal{DM}^{\\infty}(\\mathbb{X})$ and ultimately to extend our discussion on the Gauss-Green formulas.