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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.15029 |
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| _version_ | 1866909465707544576 |
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| author | Kim, Hanna N. Laugesen, Richard S. |
| author_facet | Kim, Hanna N. Laugesen, Richard S. |
| contents | The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the corresponding eigenvalue of a disjoint union of two equal disks, for Robin parameters in $[-4π,4π]$. This sharp inequality was known previously only for negative parameters in $[-4π,0]$, by Girouard and Laugesen. Their proof fails for positive Robin parameters because the second eigenfunction on a disk has non-monotonic radial part. This difficulty is overcome for parameters in $(0,4π]$ by means of a degree-theoretic approach suggested by Karpukhin and Stern that yields suitably orthogonal trial functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two disks maximize the third Robin eigenvalue: positive parameters Kim, Hanna N. Laugesen, Richard S. Spectral Theory The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the corresponding eigenvalue of a disjoint union of two equal disks, for Robin parameters in $[-4π,4π]$. This sharp inequality was known previously only for negative parameters in $[-4π,0]$, by Girouard and Laugesen. Their proof fails for positive Robin parameters because the second eigenfunction on a disk has non-monotonic radial part. This difficulty is overcome for parameters in $(0,4π]$ by means of a degree-theoretic approach suggested by Karpukhin and Stern that yields suitably orthogonal trial functions. |
| title | Two disks maximize the third Robin eigenvalue: positive parameters |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2501.15029 |