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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2405.16354 |
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| _version_ | 1866909241383583744 |
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| author | Steinerberger, Stefan |
| author_facet | Steinerberger, Stefan |
| contents | Let $Ω\subset \mathbb{R}^d$ be a bounded domain and let $λ_1, λ_2, \dots$ denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider inequalities for $λ_n$ that are independent of the domain $Ω$. A well--known such inequality follows from the Berezin--Li--Yau approach. The purpose of this paper is to point out a certain degree of flexibility in the Li--Yau approach. We use it to prove a new type of two-point inequality which are strictly stronger than what is implied by Berezin-Li-Yau itself. For example, when $d=2$, one has $ 2 λ_n + λ_{2n} \geq 10 πn/|Ω|.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16354 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universal lower bounds for Dirichlet eigenvalues Steinerberger, Stefan Spectral Theory Let $Ω\subset \mathbb{R}^d$ be a bounded domain and let $λ_1, λ_2, \dots$ denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider inequalities for $λ_n$ that are independent of the domain $Ω$. A well--known such inequality follows from the Berezin--Li--Yau approach. The purpose of this paper is to point out a certain degree of flexibility in the Li--Yau approach. We use it to prove a new type of two-point inequality which are strictly stronger than what is implied by Berezin-Li-Yau itself. For example, when $d=2$, one has $ 2 λ_n + λ_{2n} \geq 10 πn/|Ω|.$ |
| title | Universal lower bounds for Dirichlet eigenvalues |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2405.16354 |