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

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 ${\mathcal{D}\mathcal{M}}^p(\mathbb{X})$ 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 ${\mathcal{D}\mathcal{M}}^{\infty }(\mathbb{X})$ 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 ${\mathcal{D}\mathcal{M}}^{\infty }(\mathbb{X})$ and ultimately to extend our discussion on the Gauss–Green formulas.