Pólya-type inequalities on spheres and hemispheres
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911024825761792 |
|---|---|
| author | Freitas, Pedro Mao, Jing Salavessa, Isabel |
| author_facet | Freitas, Pedro Mao, Jing Salavessa, Isabel |
| contents | Given an eigenvalue $λ$ of the Laplace-Beltrami operator on $n-$spheres or $-$hemispheres, with multiplicity $m$ such that $λ=λ_{k}=\dots = λ_{k+m-1}$, we characterise the lowest and highest orders in the set $\left\{k,\dots,k+m-1\right\}$ for which Pólya's conjecture holds and fails. In particular, we show that Pólya's conjecture holds for hemispheres in the Neumann case, but not in the Dirichlet case when $n$ is greater than two. We further derive Pólya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres. We also obtain direct and reversed Li-Yau inequalities for $\mathbb{S}^2$ and $\mathbb{S}^4$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_07277 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pólya-type inequalities on spheres and hemispheres Freitas, Pedro Mao, Jing Salavessa, Isabel Spectral Theory Differential Geometry 58C40, 35P15, 35J25, 53C40, 53C41, 53A07, 53A05, 53C35 Given an eigenvalue $λ$ of the Laplace-Beltrami operator on $n-$spheres or $-$hemispheres, with multiplicity $m$ such that $λ=λ_{k}=\dots = λ_{k+m-1}$, we characterise the lowest and highest orders in the set $\left\{k,\dots,k+m-1\right\}$ for which Pólya's conjecture holds and fails. In particular, we show that Pólya's conjecture holds for hemispheres in the Neumann case, but not in the Dirichlet case when $n$ is greater than two. We further derive Pólya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres. We also obtain direct and reversed Li-Yau inequalities for $\mathbb{S}^2$ and $\mathbb{S}^4$, respectively. |
| title | Pólya-type inequalities on spheres and hemispheres |
| topic | Spectral Theory Differential Geometry 58C40, 35P15, 35J25, 53C40, 53C41, 53A07, 53A05, 53C35 |
| url | https://arxiv.org/abs/2204.07277 |