Non-isoparametric Serrin domains of $\mathbb{S}^3$ with connected toric boundary
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arXiv
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| Formato: | Preprint |
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2025
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| author | Bisterzo, Andrea Sakaguchi, Shigeru |
| author_facet | Bisterzo, Andrea Sakaguchi, Shigeru |
| contents | We investigate the overdetermined torsion problem
$\begin{cases} -Δu = 1 & \text{in}\ Ω\\ u=0 & \text{on}\ \partial Ω\\ \frac{\partial u}{\partial ν}=\text{const.} & \text{on}\ \partial Ω, \end{cases}$
where $Ω$ is a smooth Riemannian domain. Domains admitting a solution to this problem are called \textit{Serrin domains}, after the celebrated work of Serrin \cite{Se71}, where is proved that in $\mathbb{R}^n$ such domains are geodesic balls. In the present paper we establish the existence of two distinct types of Serrin domains of $\mathbb{S}^3$, respectively of small and large volume, each of whose boundary is connected and is neither isometric to a geodesic sphere nor to a Clifford torus. These domains arise as nontrivial perturbations of some classical symmetric solutions to the same problem. Our approach relies on an implicit construction based on the Crandall-Rabinowitz bifurcation theorem, which allows us to detect branches of non-radial solutions bifurcating from a family of radial ones. The resulting examples highlight new geometric configurations of the torsion problem in the three-dimensional sphere, providing another proof of the fact that the rigidity of Serrin-type results can fail in the presence of curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16531 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-isoparametric Serrin domains of $\mathbb{S}^3$ with connected toric boundary Bisterzo, Andrea Sakaguchi, Shigeru Analysis of PDEs 35N25, 35B32, 35B20, 35B06 We investigate the overdetermined torsion problem $\begin{cases} -Δu = 1 & \text{in}\ Ω\\ u=0 & \text{on}\ \partial Ω\\ \frac{\partial u}{\partial ν}=\text{const.} & \text{on}\ \partial Ω, \end{cases}$ where $Ω$ is a smooth Riemannian domain. Domains admitting a solution to this problem are called \textit{Serrin domains}, after the celebrated work of Serrin \cite{Se71}, where is proved that in $\mathbb{R}^n$ such domains are geodesic balls. In the present paper we establish the existence of two distinct types of Serrin domains of $\mathbb{S}^3$, respectively of small and large volume, each of whose boundary is connected and is neither isometric to a geodesic sphere nor to a Clifford torus. These domains arise as nontrivial perturbations of some classical symmetric solutions to the same problem. Our approach relies on an implicit construction based on the Crandall-Rabinowitz bifurcation theorem, which allows us to detect branches of non-radial solutions bifurcating from a family of radial ones. The resulting examples highlight new geometric configurations of the torsion problem in the three-dimensional sphere, providing another proof of the fact that the rigidity of Serrin-type results can fail in the presence of curvature. |
| title | Non-isoparametric Serrin domains of $\mathbb{S}^3$ with connected toric boundary |
| topic | Analysis of PDEs 35N25, 35B32, 35B20, 35B06 |
| url | https://arxiv.org/abs/2511.16531 |