Dirichlet problems on graphs with ends

TL;DR

This paper extends classical potential theory to solve Dirichlet problems on graphs with both finite boundary points and multiple ends, combining previous approaches to address more complex graph structures.

Contribution

It introduces a unified method for solving Dirichlet problems on graphs with finitely many ends and boundary points, bridging previous discrete potential theory results.

Findings

Successfully solves Dirichlet problems on graphs with ends and boundary points.

Provides a unified framework combining previous approaches.

Enhances understanding of potential theory on complex graph structures.

Abstract

In classical potential theory, one can solve the Dirichlet problem on unbounded domains such as the upper half plane. These domains have two types of boundary points; the usual finite boundary points and another point at infinity. W. Woess has solved a discrete version of the Dirichlet problem on the ends of graphs analogous to having multiple points at infinity and no finite boundary. Whereas C. Kiselman has solved a similar version of the Dirichlet problem on graphs analogous to bounded domains. In this work, we combine the two ideas to solve a version of the Dirichlet problem on graphs with finitely many ends and boundary points of the Kiselman type.