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A note on the critical set of harmonic functions near the boundary

Carlos Kenig, Zihui Zhao

Abstract

Let $u$ be a harmonic function in a $C^1$ domain $D\subset \mathbb{R}^d$, which vanishes on an open subset of the boundary. In this note we study its critical set $\{x \in \overline{D}: \nabla u(x) = 0 \}$. When $D$ is a $C^{1,α}$ domain for some $α\in (0,1]$, we give an upper bound on the $(d-2)$-dimensional Hausdorff measure of the critical set by the frequency function. We also discuss possible ways to extend such estimate to all $C^1$-Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set $\{x \in \overline{D}: u(x) = 0 = |\nabla u(x)| \}$ (see [KZ1, KZ2]).

A note on the critical set of harmonic functions near the boundary

Abstract

Let be a harmonic function in a domain , which vanishes on an open subset of the boundary. In this note we study its critical set . When is a domain for some , we give an upper bound on the -dimensional Hausdorff measure of the critical set by the frequency function. We also discuss possible ways to extend such estimate to all -Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set (see [KZ1, KZ2]).
Paper Structure (8 sections, 12 theorems, 161 equations)

This paper contains 8 sections, 12 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Two-dimensional case
  4. Domain transformation
  5. Normalized frequency function and the doubling property
  6. Normalized doubling property and the monotonicity formula
  7. Normalized frequency function beyond the critical scale
  8. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $D$ be a $C^{1,\alpha}$ domain in ${\mathbb{R}}^d$ with constant $C_\alpha$ ($\alpha\in (0,1)$) such that $0\in \partial D$. Let $u$ be a harmonic function in $D \cap B_{2R}(0)$ such that $u=0$ in $\partial D \cap B_{2R}(0)$. Then the critical set satisfies that where the constant $C$ depends on the upper bound of the frequency function $N(0,2R)$ of $u$ in $B_{2R}(0)$ (see the definition of

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Theorem 1.11 in HJ
  • Remark 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.3
  • Definition 2.5
  • Lemma 2.8
  • Remark 2.10
  • ...and 22 more