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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/2512.06740 |
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| _version_ | 1866915980358189056 |
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| author | Chen, Hang Wu, Bohan |
| author_facet | Chen, Hang Wu, Bohan |
| contents | We investigate the overdetermined problem given by
\begin{equation*}
Δu=0 \text{ in } Ω,\quad \frac{\partial u}{\partialν} =σ_1 u \text{ on } \partial Ω, \quad |\nabla u|=\text{constant on } \partial Ω,
\end{equation*} where $Ω$ is a connected compact Riemannian surface with smooth boundary $\partial Ω$, and $σ_1$ is the first nonzero Steklov eigenvalue of $Ω$. We prove that this overdetermined problem admits a nontrivial solution if and only if $Ω$ is $σ$-homothetic to either the flat unit disk or a flat cylinder $[-T,T]\times S^1$ for some $T\ge T_1$. This gives a complete answer to the question raised by Payne and Philippin in [Z. Angew. Math. Phys. 42(6), 864-873, 1991] for $σ=σ_1$ and arbitrary surfaces. In particular, we completely characterize compact domains in 2-dimensional space forms for which the overdetermined problem is solvable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06740 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Payne-Philippin's overdetermined problems on compact surfaces Chen, Hang Wu, Bohan Differential Geometry 35N25, 58J32, 53C18, 53C24 We investigate the overdetermined problem given by \begin{equation*} Δu=0 \text{ in } Ω,\quad \frac{\partial u}{\partialν} =σ_1 u \text{ on } \partial Ω, \quad |\nabla u|=\text{constant on } \partial Ω, \end{equation*} where $Ω$ is a connected compact Riemannian surface with smooth boundary $\partial Ω$, and $σ_1$ is the first nonzero Steklov eigenvalue of $Ω$. We prove that this overdetermined problem admits a nontrivial solution if and only if $Ω$ is $σ$-homothetic to either the flat unit disk or a flat cylinder $[-T,T]\times S^1$ for some $T\ge T_1$. This gives a complete answer to the question raised by Payne and Philippin in [Z. Angew. Math. Phys. 42(6), 864-873, 1991] for $σ=σ_1$ and arbitrary surfaces. In particular, we completely characterize compact domains in 2-dimensional space forms for which the overdetermined problem is solvable. |
| title | Payne-Philippin's overdetermined problems on compact surfaces |
| topic | Differential Geometry 35N25, 58J32, 53C18, 53C24 |
| url | https://arxiv.org/abs/2512.06740 |