Surfaces with Commuting Boundary Laplacian and Dirichlet-to-Neumann Map
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915177686892544 |
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| author | Speciel, Romain |
| author_facet | Speciel, Romain |
| contents | For $M\subset \mathbb{R}^{d\geq 3}$ a smooth, connected, compact $d$-dimensional submanifold with boundary, equipped with the standard metric, the Laplacian on $\partial M$ is known to commute with the corresponding Dirichlet-to-Neumann map if and only if $M$ is a ball. In this paper, we investigate the $d=2$ case and show that, surprisingly, there exists a one-parameter family of submanifolds of $\mathbb{R}^2$ as above for which the boundary Laplacian and the Dirichlet-to-Neumann map commute, thus answering an open problem posed by Girouard, Karpukhin, Levitin, and Polterovich. We then classify all such Riemannian surfaces of genus $0$ or whose boundary has $k\geq 3$ connected components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Surfaces with Commuting Boundary Laplacian and Dirichlet-to-Neumann Map Speciel, Romain Differential Geometry Analysis of PDEs Spectral Theory 58J50 (Primary) 35P05 (Secondary) For $M\subset \mathbb{R}^{d\geq 3}$ a smooth, connected, compact $d$-dimensional submanifold with boundary, equipped with the standard metric, the Laplacian on $\partial M$ is known to commute with the corresponding Dirichlet-to-Neumann map if and only if $M$ is a ball. In this paper, we investigate the $d=2$ case and show that, surprisingly, there exists a one-parameter family of submanifolds of $\mathbb{R}^2$ as above for which the boundary Laplacian and the Dirichlet-to-Neumann map commute, thus answering an open problem posed by Girouard, Karpukhin, Levitin, and Polterovich. We then classify all such Riemannian surfaces of genus $0$ or whose boundary has $k\geq 3$ connected components. |
| title | Surfaces with Commuting Boundary Laplacian and Dirichlet-to-Neumann Map |
| topic | Differential Geometry Analysis of PDEs Spectral Theory 58J50 (Primary) 35P05 (Secondary) |
| url | https://arxiv.org/abs/2503.00270 |