The Poisson's Problems on graphs

Abstract

In this paper we study the problem \[ \begin{cases} -Δ_d u = μ_0 &\text{ in } G\\ u = 0 &\text{ on } \partial G \end{cases} \] where, $\hat{I}{”}_d$ represent the discret Laplacian, and $\hat{I}{¼}_0$ 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.