JournalsrmiVol. 38, No. 3pp. 883–946

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

  • Vito Buffa

    Smiling International School, Ferrara, Italy
  • Giovanni E. Comi

    Universität Hamburg, Germany
  • Michele Miranda Jr.

    Università degli Studi di Ferrara, Italy
On BV functions and essentially bounded divergence-measure fields in metric spaces cover
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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 (X,d,μ)(\mathbb{X},d,\mu) equipped with a non-negative Radon measure μ\mu finite on bounded sets. Then, we extend the concept of divergence-measure vector fields DMp(X)\mathcal{DM}^p(\mathbb{X}) for any p[1,]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 DM(X)\mathcal{DM}^\infty(\mathbb{X}) vector field. This differential machinery is also the natural framework to specialize our analysis for RCD(K,)\mathsf{RCD}(K,\infty) spaces, where we exploit the underlying geometry to determine the Leibniz rules for DM(X)\mathcal{DM}^\infty(\mathbb{X}) and ultimately to extend our discussion on the Gauss–Green formulas.

Cite this article

Vito Buffa, Giovanni E. Comi, Michele Miranda Jr., On BV functions and essentially bounded divergence-measure fields in metric spaces. Rev. Mat. Iberoam. 38 (2022), no. 3, pp. 883–946

DOI 10.4171/RMI/1291