Geometric inequalities between Dirichlet and Neumann eigenvalues
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915259092041728 |
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| author | Hatcher, Lawford |
| author_facet | Hatcher, Lawford |
| contents | Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $Ω\subseteq\Rbb^n$ is a topic that goes back at least to the work of Pólya \cite{polya}. We study the effect of the isoperimetric ratio of $Ω$ on the number $N(Ω)$ of Neumann eigenvalues that do not exceed the first Dirichlet eigenvalue, proving that $N(Ω)$ is bounded above and below by a constant multiple of the isoperimetric ratio in the case of convex domains. We also show that these estimates do not hold in the non-convex setting, addressing questions of Cox-MacLachlan-Steeves \cite{coxetal} and Freitas \cite{freitas}. Despite these counterexamples, we find similar estimates for polygonal domains in $\Rbb^2$ as well as certain families of fiber bundles that asymptotically collapse onto their base spaces, the motivating examples being tubular neighborhoods of submanifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_18517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric inequalities between Dirichlet and Neumann eigenvalues Hatcher, Lawford Spectral Theory Analysis of PDEs 35P05, 35B38, 35J05, 35J25, 58J50 Comparing Neumann and Dirichlet eigenvalues of the Laplacian on a bounded domain $Ω\subseteq\Rbb^n$ is a topic that goes back at least to the work of Pólya \cite{polya}. We study the effect of the isoperimetric ratio of $Ω$ on the number $N(Ω)$ of Neumann eigenvalues that do not exceed the first Dirichlet eigenvalue, proving that $N(Ω)$ is bounded above and below by a constant multiple of the isoperimetric ratio in the case of convex domains. We also show that these estimates do not hold in the non-convex setting, addressing questions of Cox-MacLachlan-Steeves \cite{coxetal} and Freitas \cite{freitas}. Despite these counterexamples, we find similar estimates for polygonal domains in $\Rbb^2$ as well as certain families of fiber bundles that asymptotically collapse onto their base spaces, the motivating examples being tubular neighborhoods of submanifolds. |
| title | Geometric inequalities between Dirichlet and Neumann eigenvalues |
| topic | Spectral Theory Analysis of PDEs 35P05, 35B38, 35J05, 35J25, 58J50 |
| url | https://arxiv.org/abs/2504.18517 |