The Poisson's Problems on graphs
Diego Alexander Castro Guevara
TL;DR
The paper addresses the Poisson problem on finite graphs with a measure data $\mu_0$, proving existence and uniqueness of a solution to $-\Delta_d u = \mu_0$ in $G$ with $u=0$ on $\partial G$ by adapting Perron's method to graphs and employing balayage. It develops a graph-based potential theory toolkit, including a divergence theorem, a discrete maximum principle, and harmonic lifting arguments, to establish the Perron construction and the convergence of a balayage-driven iterative scheme. The main contributions are the adaptation of classical continuous potential-theory techniques to the discrete graph setting and the demonstration that the Poisson problem admits a well-defined, unique solution on finite graphs. This work provides a rigorous framework for discrete Poisson equations and lays groundwork for applications in network analysis, numerical methods, and discrete potential theory 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, $Δ_d$ represent the discret Laplacian, and $μ_0$ 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.