On $BV$ functions and essentially bounded divergence-measure fields in\n metric spaces Article Swipe
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· 2019
· Open Access
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· DOI: https://doi.org/10.48550/arxiv.1906.07432
· OA: W4288321641
By employing the differential structure recently developed by N. Gigli, we\nfirst give a notion of functions of bounded variation ($BV$) in terms of\nsuitable vector fields on a complete and separable metric measure space\n$(\\mathbb{X},d,\\mu)$ equipped with a non-negative Radon measure $\\mu$ finite on\nbounded sets. Then, we extend the concept of divergence-measure vector fields\n$\\mathcal{DM}^p(\\mathbb{X})$ for any $p\\in[1,\\infty]$ and, by simply requiring\nin addition that the metric space is locally compact, we determine an\nappropriate class of domains for which it is possible to obtain a Gauss-Green\nformula in terms of the normal trace of a $\\mathcal{DM}^\\infty(\\mathbb{X})$\nvector field. This differential machinery is also the natural framework to\nspecialize our analysis for ${\\mathsf{RCD}(K,\\infty)}$ spaces, where we exploit\nthe underlying geometry to determine the Leibniz rules for\n$\\mathcal{DM}^\\infty(\\mathbb{X})$ and ultimately to extend our discussion on\nthe Gauss-Green formulas.\n
