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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2405.18079 |
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| _version_ | 1866918071506042880 |
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| author | Freitas, Pedro Gama, Miguel |
| author_facet | Freitas, Pedro Gama, Miguel |
| contents | We provide an answer to a question raised by Levine and Weinberger in their $1986$ paper concerning the difference between Dirichlet and Neumann eigenvalues of the Laplacian on bounded domains in $\mathbb{R}^{n}$. More precisely, we show that for a certain class of domains there exists a sequence $p(k)$ such that $λ_{k}\geq μ_{k+ p(k)}$ for sufficiently large $k$. This sequence, which is given explicitly and is independent of the domain, grows with $k^{1-1/n}$ as $k$ goes to infinity, which we conjecture to be optimal. We also prove the existence of a sequence, now not given explicitly and only of order $k^{1-3/n}$ but valid for bounded Lipschitz domains in $mathbb{R}^{n} (n\geq4)$, for which a similar inequality holds for all $k$. We then frame these general results with some specific planar Euclidean examples such as rectangles and disks, for which we provide bounds valid for all eigenvalue orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the (growing) gap between Dirichlet and Neumann eigenvalues Freitas, Pedro Gama, Miguel Spectral Theory We provide an answer to a question raised by Levine and Weinberger in their $1986$ paper concerning the difference between Dirichlet and Neumann eigenvalues of the Laplacian on bounded domains in $\mathbb{R}^{n}$. More precisely, we show that for a certain class of domains there exists a sequence $p(k)$ such that $λ_{k}\geq μ_{k+ p(k)}$ for sufficiently large $k$. This sequence, which is given explicitly and is independent of the domain, grows with $k^{1-1/n}$ as $k$ goes to infinity, which we conjecture to be optimal. We also prove the existence of a sequence, now not given explicitly and only of order $k^{1-3/n}$ but valid for bounded Lipschitz domains in $mathbb{R}^{n} (n\geq4)$, for which a similar inequality holds for all $k$. We then frame these general results with some specific planar Euclidean examples such as rectangles and disks, for which we provide bounds valid for all eigenvalue orders. |
| title | On the (growing) gap between Dirichlet and Neumann eigenvalues |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2405.18079 |