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

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.