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Pólya's conjecture up to $ε$-loss and quantitative estimates for the remainder of Weyl's law

Renjin Jiang, Fanghua Lin

Abstract

Let $Ω\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For any $ε\in (0,1)$ we show that for any Dirichlet eigenvalue $λ_k(Ω)>Λ(ε,Ω)$, it holds \begin{align*} k&\le (1+ε)\frac{|Ω|ω(n)}{(2π)^n}λ_k(Ω)^{n/2}, \end{align*} where $Λ(ε,Ω)$ is given explicitly. This reduces the $ε$-loss version of Pólya's conjecture to a computational problem. This estimate is based on quantitative estimates on the remainder of the Weyl law with explicit constants, which we give a new proof without using Neumann eigenvalues. Our arguments in deriving such uniform estimates yield also, in all dimensions $n\ge 2$, classes of domains that may even have rather irregular shapes or boundaries but satisfy Pólya's conjecture. Another key observation is that on strip-tiling domains (and therefore any triangles for instance) one actually has better eigenvalue estimates than Pólya conjectured.

Pólya's conjecture up to $ε$-loss and quantitative estimates for the remainder of Weyl's law

Abstract

Let be a bounded Lipschitz domain. For any we show that for any Dirichlet eigenvalue , it holds \begin{align*} k&\le (1+ε)\frac{|Ω|ω(n)}{(2π)^n}λ_k(Ω)^{n/2}, \end{align*} where is given explicitly. This reduces the -loss version of Pólya's conjecture to a computational problem. This estimate is based on quantitative estimates on the remainder of the Weyl law with explicit constants, which we give a new proof without using Neumann eigenvalues. Our arguments in deriving such uniform estimates yield also, in all dimensions , classes of domains that may even have rather irregular shapes or boundaries but satisfy Pólya's conjecture. Another key observation is that on strip-tiling domains (and therefore any triangles for instance) one actually has better eigenvalue estimates than Pólya conjectured.

Paper Structure

This paper contains 13 sections, 21 theorems, 205 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Weyl's law and Pólya's conjecture
  3. Pólya's conjecture up to $\epsilon$-loss and quantitative remainder estimate
  4. An approach to Pólya's conjecture and concrete examples
  5. Our method and plan of the paper
  6. Refined eigenvalue estimates
  7. Product domains
  8. Strip-tiling domains
  9. Pólya's conjecture on a class of domains
  10. The case of two dimension
  11. Three and higher dimensions
  12. Some other class satisfying Pólya's conjecture
  13. Quantitative remainder estimate

Key Result

Theorem 1.3

Let $\Omega\subset{\mathbb R}^n$ be a bounded Lipschitz domain, $n\ge 2$. For any $0<\epsilon<1$, it holds for all $\lambda_k(\Omega)\ge \Lambda(\epsilon,\Omega)$ that where $\Lambda(\epsilon,\Omega)\ge \mathrm{width}(\Omega)^{-2}$ is such that Moreover, if $\Omega$ is convex, then $\Lambda(\epsilon,\Omega)$ can be taken smaller as

Figures (5)

  • Figure 1: Examples of strip-tiling domains in two dimension
  • Figure 2: Examples satisfying Pólya's conjecture
  • Figure 3: The 1/2 sum is controlled by the 1/4 ball
  • Figure 4: Examples of strip-tiling domains in two dimension
  • Figure 5: Whitney decomposition for the complementary of a convex set

Theorems & Definitions (48)

  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4: Minimal Admissible rectangle
  • Theorem 1.5
  • Corollary 1.6
  • Definition 1.7: Strip-tiling domains
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Corollary 1.11
  • ...and 38 more