Dirichlet-to-Neumann maps on Trees

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 $g\mapsto c{g}^{\prime }$ (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).

arXiv comment26 pages. Added references and referee's suggestions. Corrected typos. Accepted for publication in Potential Analysis

References in corpus (6)

• An extension problem related to the fractional Laplacian
• Geometric analysis on Cantor sets and trees
• Asymptotic statistical characterizations of p-harmonic functions of two variables
• The unique continuation property for a nonlinear equation on trees
• Estimates for Nonlinear Harmonic Measures on Trees
• Existence, uniqueness and decay rates for evolution equations on trees

Cited by in corpus (1)

• The two membranes problem in a regular tree