Boundary measures, generalized Gauss-Green formulas, and mean value property in metric measure spaces

Niko Marola, Michele Miranda, Nageswari Shanmugalingam

Research output: Contribution to journalArticleScientificpeer-review

5 Citations (Scopus)

Abstract

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1, 1)-Poincaré inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss-Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss-Green formula we introduce a suitable notion of the interior normal trace of a regular ball.

Original languageEnglish
Pages (from-to)497-530
Number of pages34
JournalRevista Matematica Iberoamericana
Volume31
Issue number2
DOIs
Publication statusPublished - 2015
MoE publication typeA1 Journal article-refereed

Keywords

  • Dirichlet form
  • Doubling measure
  • Functions of bounded variation
  • Gauss-Green theorem
  • Green function
  • Harmonic function
  • Inequality
  • Metric space
  • Minkowski content
  • Newtonian space
  • Perimeter measure
  • Poincaré
  • Singular function

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