Monotonicity formulas for harmonic functions on the infinite regular tree

TL;DR

This paper extends monotonicity formulas for harmonic functions to infinite regular trees, providing new theoretical tools and explicit examples for 2- and 3-regular cases.

Contribution

It introduces generalized monotonicity formulas for harmonic functions on infinite regular trees, expanding the theoretical framework beyond previous work.

Findings

Proved monotonicity of weighted Dirichlet energy on infinite regular trees

Established a Weiss-type monotonicity formula for these trees

Computed explicit examples for 2- and 3-regular trees

Abstract

We continue the program initiated in \cite{SVGS}. In this paper, we focus on the infinite $d-$regular tree, and prove the monotonicity of a weighted Dirichlet energy, a Weiss-type monotonicity formula, and a generalization of the Almgren monotonicity formula of \cite{SVGS} for $p\ge 1$. We also compute examples in the infinite $2-$ and $3-$regular trees.