# Dirichlet-to-Neumann maps on Trees

## TL;DR

This paper explores the Dirichlet-to-Neumann map on trees for solutions to mean value formulas, revealing local and nonlocal operator behaviors and providing existence and uniqueness results for associated boundary value problems.

## Contribution

It introduces two definitions of the Dirichlet-to-Neumann map on trees and characterizes them as local or nonlocal operators, extending classical concepts to discrete structures.

## Key findings

- Dirichlet-to-Neumann map is a local operator of order one for directed trees.
- For undirected mean value formulas, similar local results are obtained.
- An alternative nonlocal Dirichlet-to-Neumann map is characterized for certain parameters.

## Abstract

In this paper we study the Dirichlet-to-Neumann map for solutions to mean value formulas on trees. We give two alternative definition of the Dirichlet-to-Neumann map. For the first definition (that involves the product of a "gradient" with a "normal vector" and for a linear mean value formula on the directed tree (taking into account only the successors of a given node) we obtain that the Dirichlet-to-Neumann map is given by $g\mapsto cg'$ (here $c$ is an explicit constant). Notice that this is a local operator of order one. We also consider linear undirected mean value formulas (taking into account not only the successors but the ancestor and the successors of a given node) and prove a similar result. For this kind of mean value formula we include some existence and uniqueness results for the associated Dirichlet problem. Finally, we give an alternative definition of the Dirichlet-to-Neumann map (taking into account differences along a given branch of the tree). With this alternative definition, for a certain range of parameters, we obtain that the Dirichlet-to-Neumann map is given by a nonlocal operator (as happens for the classical Laplacian in the Euclidean space).