Harmonic functions on metric measure spaces

TL;DR

This paper explores harmonic functions on metric measure spaces, establishing properties like Harnack estimates, maximum principles, and differentiability, and solving the Dirichlet problem using probabilistic and Perron methods.

Contribution

It introduces a comprehensive framework for harmonic functions on metric measure spaces, including new properties, solution methods, and measure relations.

Findings

Established Harnack estimates for harmonic functions.

Solved the Dirichlet problem via dynamical programming and Perron methods.

Proved Liouville type theorems for these functions.

Abstract

We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the H\"older and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method related to stochastic games. We employ the Perron method to construct a harmonic function with continuous boundary data. Finally, we discuss and prove the Liouville type theorems. Our results are obtained for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition. Relations between such measures are presented as well. The presentation is illustrated by examples.