A note on Neumann problems on graphs

TL;DR

This paper investigates Neumann boundary value problems for Laplacians on graphs, establishing conditions for unique solutions and providing analytic and probabilistic representations under various boundary conditions.

Contribution

It introduces new solvability criteria for Neumann problems on infinite and compactifiable graphs, linking heat semigroup properties to boundary value problem solutions.

Findings

Unique solvability under ultracontractivity assumptions

Analytic and probabilistic solution representations

Conditions for Neumann problem solutions on different graph types

Abstract

We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations.