Potential Analysis

, Volume 45, Issue 4, pp 609–633 | Cite as

Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces

  • Niko Marola
  • Michele MirandaJr.
  • Nageswari Shanmugalingam


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.


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 

Mathematics Subject Classification (2010)

31C25 26B30 46E35 35K05 


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Niko Marola
    • 1
  • Michele MirandaJr.
    • 2
  • Nageswari Shanmugalingam
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  2. 2.Department of Mathematics and Computer ScienceUniversity of FerraraFerraraItaly
  3. 3.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

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