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

• Niko Marola ^[1] ; Michele Miranda Jr. ^[3] ; Nageswari Shanmugalingam ^[2]
  1. [1] University of Helsinki

     University of Helsinki

     Helsinki, Finlandia

     
  2. [2] University of Cincinnati

     University of Cincinnati

     City of Cincinnati, Estados Unidos

     
  3. [3] Università di Ferrara

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 31, Nº 2, 2015, págs. 497-530
• Idioma: inglés
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• Resumen

  • 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.