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The sharp estimate of nodal sets for Dirichlet Laplace eigenfunctions in polytopes

Yingying Cai, Jinping Zhuge

Abstract

Let $P$ be a bounded $n$-dimensional Lipschitz polytope, and let $\varphi_λ$ be a Dirichlet Laplace eigenfunction in $P$ corresponding to the eigenvalue $λ$. We show that the $(n-1)$-dimensional Hausdorff measure of the nodal set of $\varphi_λ$ does not exceed $C(P)\sqrtλ$. Our result extends the previous ones in quaisconvex domains (including $C^1$ and convex domains) to general polytopes that are not necessarily quasiconvex.

The sharp estimate of nodal sets for Dirichlet Laplace eigenfunctions in polytopes

Abstract

Let be a bounded -dimensional Lipschitz polytope, and let be a Dirichlet Laplace eigenfunction in corresponding to the eigenvalue . We show that the -dimensional Hausdorff measure of the nodal set of does not exceed . Our result extends the previous ones in quaisconvex domains (including and convex domains) to general polytopes that are not necessarily quasiconvex.
Paper Structure (10 sections, 11 theorems, 90 equations)

This paper contains 10 sections, 11 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Monotonicity of doubling index
  3. Polytopes
  4. Geometry of polytopes
  5. Maximum star-shape radius (MSR)
  6. Uniform bound of doubling index
  7. Estimates of nodal sets
  8. Nodal sets of harmonic functions
  9. Dirichlet eigenfunctions
  10. Outline of the proof of Lemma \ref{['lem.monotonicityDL']}

Key Result

Theorem 1.1

Let P be a bounded n-dimensional Lipschitz polytope. Let $\varphi_{\lambda}$ be a Dirichlet eigenfunction of the Laplace operator corresponding to the eigenvalue $\lambda$, i.e., Then, the $(n-1)$-dimensional Hausdorff measure of the nodal set of $\varphi_{\lambda}$, denoted by $\mathbb{H}^{n-1}(Z(\varphi_{\lambda}))$, satisfies where $C$ depends only on $P$ and $n$.

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 12 more