Harmonic functions on metric spaces

Abstract

This paper explores a Dirichlet type problem on metric measure spaces. The problem is to find a Sobolev-type function that minimizes the energy integral within a class of "Sobolev" functions that agree with the boundary function outside the domain of the problem. This is the analogue of the Euler-Lagrange formulation in the classical Dirichlet problem. It is shown that, under certain geometric constraints on the measure imposed on the metric space, such a solution exists. Under the condition that the space has many rectifiable curves, the solution is unique and satisfies the weak maximum principle.