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Monotonicity formulas for harmonic functions on the infinite regular tree

Kathryn Atwood, Mariana Smit Vega Garcia, Richard Wang

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.

Monotonicity formulas for harmonic functions on the infinite regular tree

Abstract

We continue the program initiated in \cite{SVGS}. In this paper, we focus on the infinite 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 . We also compute examples in the infinite and regular trees.
Paper Structure (14 sections, 7 theorems, 118 equations, 4 figures)

This paper contains 14 sections, 7 theorems, 118 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Monotonicity of a weighted Dirichlet energy
  4. Weiss monotonicity formula
  5. Weiss monotonicity formula for the infinite $d-$regular tree with $d\ge 3$
  6. Weiss for the infinite 2-regular tree
  7. Monotonicity of an Almgren frequency function with $p\ge 1$
  8. Examples
  9. For the infinite $3$-regular tree
  10. Bounded Harmonic Function
  11. Unbounded harmonic functions
  12. For the $2$-regular tree
  13. Bounded harmonic function
  14. Unbounded harmonic function

Key Result

Theorem 1

Let $x_0$ be a vertex in the infinite $d-$regular tree $T_d=(V,E)$ and let $u$ be a harmonic function on $T_d$. Then, for any $1 \leq p <\infty$ is monotonically non-decreasing in $k$.

Figures (4)

  • Figure 1: Harmonic function on the infinite 3-regular tree described in Example \ref{['needweight']} (locally).
  • Figure 2: A sketch for the argument when $d=3$.
  • Figure 3: Sketch of the bounded harmonic function described in § \ref{['E:bounded']} (locally).
  • Figure 4: Sketch of the unbounded harmonic function described in § \ref{['SS:unbounded']}: each vertex $x$ has 2 neighbors with value $u(x)/2$, and one with value $2u(x)$.

Theorems & Definitions (14)

  • Example 1
  • Theorem 1: Discrete weighted Dirichlet energy
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • ...and 4 more