The Poisson's Problems on graphs

TL;DR

This paper investigates the Poisson problem on graphs involving the discrete Laplacian and measure data, proving existence and uniqueness of solutions using adapted Perron's method and Balayage techniques.

Contribution

It introduces a novel approach to solving Poisson's problems on graphs by adapting Perron's method and Balayage, establishing existence and uniqueness results.

Findings

Proved the existence and uniqueness of solutions to the Poisson problem on graphs.

Developed an adaptation of Perron's method for graph-based PDEs.

Applied Balayage techniques to the graph setting for the first time.

Abstract

In this paper we study the problem \[ \begin{cases} -\Delta_d u = \mu_0 &\text{ in } G\\ u = 0 &\text{ on } \partial G \end{cases} \] where, $\Delta_d$ represent the discret Laplacian, and $\mu_0$ it is a measure defined in the vertex of the graph $G=(V,E)$. Here $V$ defined the vertex of the graph, $E$ its edges and $\partial G$ its boundary. We prove that this problem has an unique solution by using an adaption of the Perron's method for the graphs by using an idea known as Balayage.