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

TL;DR

This paper investigates harmonic functions in metric measure spaces with doubling measures and Poincaré inequalities, establishing mean value properties, Green function analysis, and Gauss-Green formulas for sets of finite perimeter.

Contribution

It introduces a new characterization of harmonic functions via mean value properties and develops Gauss-Green formulas in the context of metric measure spaces.

Findings

Characterization of harmonic functions through mean value properties.

Detailed description of Poisson kernels in metric measure spaces.

Gauss-Green formulas for sets of finite perimeter with Minkowski content.

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)- Poincar\'e 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.