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Hausdorff measure bound for the nodal sets of Neumann Laplace eigenfunctions

Shaghayegh Fazliani

Abstract

We study the nodal sets of Neumann Laplace eigenfunctions in a bounded domain with $\mathcal{C}^{1,1}$ boundary. We show that for $u_λ$ such that $Δu_λ+ λu_λ= 0 $ with the Neumann boundary condition $\partial_νu_λ= 0$, we have $\mathcal{H}^{n-1}(\{u_λ= 0\}) \leq C \sqrtλ$.

Hausdorff measure bound for the nodal sets of Neumann Laplace eigenfunctions

Abstract

We study the nodal sets of Neumann Laplace eigenfunctions in a bounded domain with boundary. We show that for such that with the Neumann boundary condition , we have .
Paper Structure (7 sections, 19 theorems, 88 equations)

This paper contains 7 sections, 19 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Extension and Reflection Across the Boundary
  3. Frequency Function and Doubling Index
  4. Two Important Lemmas
  5. Proof of the Main Theorem
  6. Reflection Across the Boundary
  7. Examples of the Nodal Sets of Neumann Laplace Eigenfunctions

Key Result

Theorem 1

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $\mathcal{C}^{1,1}$ boundary, and let $u_\lambda$ be a solution of NeumannLaplaceEigenfunction in $\Omega$. Denote by $Z_{u_\lambda} := \{u_\lambda = 0\}$ the zero set of $u_\lambda$. Then, there exists a constant $C > 0$, depending only on the

Theorems & Definitions (33)

  • Theorem 1
  • Lemma 1
  • proof
  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 2
  • proof
  • ...and 23 more