Inequalities between Dirichlet and Neumann Eigenvalues on Surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912410177110016 |
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| author | Hua, Bobo Münch, Florentin Zhang, Haohang |
| author_facet | Hua, Bobo Münch, Florentin Zhang, Haohang |
| contents | For a bounded Lipschitz domain $Σ$ in a Riemannian surface $M$ satisfying certain curvature condition, we prove that $$μ_{3-β_1} \leq λ_{1},$$ where $μ_k$ ($λ_k$ resp.) is the $k$-th Neumann (Dirichlet resp.) Laplacian eigenvalue on $Σ$ and $β_1$ is the first Betti number of $Σ.$ If $Σ$ is smooth and simply connected, we can further derive the strict inequality $
μ_{3}< λ_{1}. $
This extends previous results on the Euclidean space to various curved surfaces, including the flat cylinder, the hyperbolic plane, hyperbolic cusp, collar, funnel, and minimal surfaces such as catenoid and helicoid. The novelty of the paper lies in comparing Dirichlet and Neumann Laplacian eigenvalues via the variational principle of the Hodge Laplacian on $1$-forms on a surface, extending the variational principle on vector fields in the Euclidean plane as developed by Rohleder. The comparison is reduced to the existence of a distance function with appropriate curvature conditions on its level sets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inequalities between Dirichlet and Neumann Eigenvalues on Surfaces Hua, Bobo Münch, Florentin Zhang, Haohang Differential Geometry Analysis of PDEs Spectral Theory 35J05, 35P15 For a bounded Lipschitz domain $Σ$ in a Riemannian surface $M$ satisfying certain curvature condition, we prove that $$μ_{3-β_1} \leq λ_{1},$$ where $μ_k$ ($λ_k$ resp.) is the $k$-th Neumann (Dirichlet resp.) Laplacian eigenvalue on $Σ$ and $β_1$ is the first Betti number of $Σ.$ If $Σ$ is smooth and simply connected, we can further derive the strict inequality $ μ_{3}< λ_{1}. $ This extends previous results on the Euclidean space to various curved surfaces, including the flat cylinder, the hyperbolic plane, hyperbolic cusp, collar, funnel, and minimal surfaces such as catenoid and helicoid. The novelty of the paper lies in comparing Dirichlet and Neumann Laplacian eigenvalues via the variational principle of the Hodge Laplacian on $1$-forms on a surface, extending the variational principle on vector fields in the Euclidean plane as developed by Rohleder. The comparison is reduced to the existence of a distance function with appropriate curvature conditions on its level sets. |
| title | Inequalities between Dirichlet and Neumann Eigenvalues on Surfaces |
| topic | Differential Geometry Analysis of PDEs Spectral Theory 35J05, 35P15 |
| url | https://arxiv.org/abs/2412.19480 |