Bifurcating domains for an overdetermined eigenvalue problem in cylinders
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915684431167488 |
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| author | Lian, Yuanyuan Pacella, Filomena Sicbaldi, Pieralberto |
| author_facet | Lian, Yuanyuan Pacella, Filomena Sicbaldi, Pieralberto |
| contents | We study an overdetermined eigenvalue problem for domains $Ω$ contained in the half-cylinder $Σ=ω\times (0, +\infty)$, based on a bounded regular domain $ω\subset \mathbb{R}^{N-1}$. It is easy to see that in any bounded cylinder $Ω_{t}=ω\times (0, t)$, $t > 0$, the eigenvalue problem admits a one-dimensional positive eigenfunction which satisfies the overdetermined boundary conditions. The aim of the paper is to construct other domains $Ω\subset Σ$ for which there exists a positive eigenfunction that is a solution of the overdetermined problem. This is achieved by showing that branches of such domains bifurcate from the ``trivial'' domains $Ω_{t_j}$ at the values $t_{j} = \fracπ{2\sqrt{σ_j}}$ where $σ_j$ ($j\geq 1$) is a simple Neumann eigenvalue of the Laplace operator on $ω\subset \mathbb{R}^{N-1}$. The solutions can be reflected with respect to $ω$ to generate nontrivial solutions in a cylinder. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bifurcating domains for an overdetermined eigenvalue problem in cylinders Lian, Yuanyuan Pacella, Filomena Sicbaldi, Pieralberto Analysis of PDEs 35B32, 35G15, 35N25 We study an overdetermined eigenvalue problem for domains $Ω$ contained in the half-cylinder $Σ=ω\times (0, +\infty)$, based on a bounded regular domain $ω\subset \mathbb{R}^{N-1}$. It is easy to see that in any bounded cylinder $Ω_{t}=ω\times (0, t)$, $t > 0$, the eigenvalue problem admits a one-dimensional positive eigenfunction which satisfies the overdetermined boundary conditions. The aim of the paper is to construct other domains $Ω\subset Σ$ for which there exists a positive eigenfunction that is a solution of the overdetermined problem. This is achieved by showing that branches of such domains bifurcate from the ``trivial'' domains $Ω_{t_j}$ at the values $t_{j} = \fracπ{2\sqrt{σ_j}}$ where $σ_j$ ($j\geq 1$) is a simple Neumann eigenvalue of the Laplace operator on $ω\subset \mathbb{R}^{N-1}$. The solutions can be reflected with respect to $ω$ to generate nontrivial solutions in a cylinder. |
| title | Bifurcating domains for an overdetermined eigenvalue problem in cylinders |
| topic | Analysis of PDEs 35B32, 35G15, 35N25 |
| url | https://arxiv.org/abs/2512.16319 |