Spherical caps do not always maximize Neumann eigenvalues on the sphere
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908288086441984 |
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| author | Bucur, Dorin Laugesen, Richard S. Martinet, Eloi Nahon, Mickaël |
| author_facet | Bucur, Dorin Laugesen, Richard S. Martinet, Eloi Nahon, Mickaël |
| contents | We prove the existence of an open set $Ω\subset\mathbb{S}^2$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_15385 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spherical caps do not always maximize Neumann eigenvalues on the sphere Bucur, Dorin Laugesen, Richard S. Martinet, Eloi Nahon, Mickaël Analysis of PDEs Spectral Theory 35P15, 58J50 We prove the existence of an open set $Ω\subset\mathbb{S}^2$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary. |
| title | Spherical caps do not always maximize Neumann eigenvalues on the sphere |
| topic | Analysis of PDEs Spectral Theory 35P15, 58J50 |
| url | https://arxiv.org/abs/2503.15385 |