Calculus of Variations and Geometric Measure Theory

V. Buffa - G. E. Comi - M. Miranda Jr

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

created by buffa on 17 Jun 2019
modified on 04 Jan 2022


Published Paper

Inserted: 17 jun 2019
Last Updated: 4 jan 2022

Journal: Rev. Mat. Iberoamericana
Pages: 65
Year: 2021
Doi: 10.4171/RMI/1291

ArXiv: 1906.07432 PDF


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.

Keywords: functions of bounded variation, metric measure spaces, divergence-measure fields, Gauss--Green formula, curvature dimension condition