Pólya's conjecture up to $ε$-loss and quantitative estimates for the remainder of Weyl's law
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916054279651328 |
|---|---|
| author | Jiang, Renjin Lin, Fanghua |
| author_facet | Jiang, Renjin Lin, Fanghua |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pólya's conjecture up to $ε$-loss and quantitative estimates for the remainder of Weyl's law Jiang, Renjin Lin, Fanghua Spectral Theory Mathematical Physics Analysis of PDEs Classical Analysis and ODEs 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. |
| title | Pólya's conjecture up to $ε$-loss and quantitative estimates for the remainder of Weyl's law |
| topic | Spectral Theory Mathematical Physics Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2507.04307 |