Abstract
The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N 1,1-spaces) and the theory of heat semigroups (a concept related to N 1,2-spaces) in the setting of metric measure spaces whose measure is doubling and supports a 1-Poincaré inequality. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the heat semigroup in such metric spaces. We also give a new characterization of BV functions in terms of a near-diagonal energy in this general setting.
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Marola, N., Miranda, M. & Shanmugalingam, N. Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces. Potential Anal 45, 609–633 (2016). https://doi.org/10.1007/s11118-016-9560-3
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DOI: https://doi.org/10.1007/s11118-016-9560-3
Keywords
- Bakry–Émery condition
- Bounded variation
- Dirichlet form
- Doubling measure
- Heat kernel
- Heat semigroup
- Isoperimetric inequality
- Metric space
- Perimeter
- Poincaré inequality
- Sets of finite perimeter
- Total variation