Sharp inequalities and asymptotics for polyharmonic eigenvalues
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912431688646656 |
|---|---|
| author | Buoso, Davide Freitas, Pedro |
| author_facet | Buoso, Davide Freitas, Pedro |
| contents | We study eigenvalues of general scalar Dirichlet polyharmonic problems in domains in $\mathbb R^{d}$. We first prove a number of inequalities satisfied by the eigenvalues on general domains, depending on the relations between the orders of the operators involved. We then obtain several estimates for these eigenvalues, yielding their growth as a function of these orders. For the problem in the ball we derive the general form of eigenfunctions together with the equations satisfied by the corresponding eigenvalues, and obtain several bounds for the first eigenvalue. In the case of the polyharmonic operator of order $2m$ we derive precise bounds yielding the first two terms in the asymptotic expansion for the first normalised eigenvalue as $m$ grows to infinity. These results allow us to obtain the order of growth for the $k^{\rm th}$ polyharmonic eigenvalue on general domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp inequalities and asymptotics for polyharmonic eigenvalues Buoso, Davide Freitas, Pedro Analysis of PDEs Spectral Theory 35J30, 35P15, 49R05, 74K20 We study eigenvalues of general scalar Dirichlet polyharmonic problems in domains in $\mathbb R^{d}$. We first prove a number of inequalities satisfied by the eigenvalues on general domains, depending on the relations between the orders of the operators involved. We then obtain several estimates for these eigenvalues, yielding their growth as a function of these orders. For the problem in the ball we derive the general form of eigenfunctions together with the equations satisfied by the corresponding eigenvalues, and obtain several bounds for the first eigenvalue. In the case of the polyharmonic operator of order $2m$ we derive precise bounds yielding the first two terms in the asymptotic expansion for the first normalised eigenvalue as $m$ grows to infinity. These results allow us to obtain the order of growth for the $k^{\rm th}$ polyharmonic eigenvalue on general domains. |
| title | Sharp inequalities and asymptotics for polyharmonic eigenvalues |
| topic | Analysis of PDEs Spectral Theory 35J30, 35P15, 49R05, 74K20 |
| url | https://arxiv.org/abs/2506.12791 |