Pólya's conjecture for thin products
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908441174343680 |
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| author | He, Xiang Wang, Zuoqin |
| author_facet | He, Xiang Wang, Zuoqin |
| contents | Let $Ω\subset \mathbb R^d$ be a bounded Euclidean domain. According to the famous Weyl law, both its Dirichlet eigenvalue $λ_k(Ω)$ and its Neumann eigenvalue $μ_k(Ω)$ have the same leading asymptotics $w_k(Ω)=C(d,Ω)k^{2/d}$ as $k \to \infty$. G. Pólya conjectured in 1954 that each Dirichlet eigenvalue $λ_k(Ω)$ is greater than $w_k(Ω)$, while each Neumann eigenvalue $μ_k(Ω)$ is no more than $w_k(Ω)$. In this paper we prove Pólya's conjecture for thin products, i.e. domains of the form $(aΩ_1) \times Ω_2$, where $Ω_1, Ω_2$ are Euclidean domains, and $a$ is small enough. We also prove that the same inequalities hold if $Ω_2$ is replaced by a Riemannian manifold, and thus get Pólya's conjecture for a class of ``thin" Riemannian manifolds with boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12093 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pólya's conjecture for thin products He, Xiang Wang, Zuoqin Spectral Theory Differential Geometry 35P15, 35P20, 58J50 Let $Ω\subset \mathbb R^d$ be a bounded Euclidean domain. According to the famous Weyl law, both its Dirichlet eigenvalue $λ_k(Ω)$ and its Neumann eigenvalue $μ_k(Ω)$ have the same leading asymptotics $w_k(Ω)=C(d,Ω)k^{2/d}$ as $k \to \infty$. G. Pólya conjectured in 1954 that each Dirichlet eigenvalue $λ_k(Ω)$ is greater than $w_k(Ω)$, while each Neumann eigenvalue $μ_k(Ω)$ is no more than $w_k(Ω)$. In this paper we prove Pólya's conjecture for thin products, i.e. domains of the form $(aΩ_1) \times Ω_2$, where $Ω_1, Ω_2$ are Euclidean domains, and $a$ is small enough. We also prove that the same inequalities hold if $Ω_2$ is replaced by a Riemannian manifold, and thus get Pólya's conjecture for a class of ``thin" Riemannian manifolds with boundary. |
| title | Pólya's conjecture for thin products |
| topic | Spectral Theory Differential Geometry 35P15, 35P20, 58J50 |
| url | https://arxiv.org/abs/2402.12093 |