2019/06/18 by Buffa, Vito, Comi, Giovanni Eugenio, Miranda, Michele
#26A45 #26B20 #30L99 #53C23 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1906.07432
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 (\mathbbX,d,μ) equipped with a non-negative Radon measure μ finite on bounded sets. Then, we extend the concept of divergence-measure vector fields DMp(\mathbbX) for any p∈[1,∞] 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 DM^∞(\mathbbX) vector field. This differential machinery is also the natural framework to specialize our analysis for RCD(K,∞) spaces, where we exploit the underlying geometry to determine the Leibniz rules for DM^∞(\mathbbX) and ultimately to extend our discussion on the Gauss-Green formulas.