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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2309.07977 |
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| _version_ | 1866914910540136448 |
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| author | Enciso, Alberto Fernández, Antonio J. Ruiz, David Sicbaldi, Pieralberto |
| author_facet | Enciso, Alberto Fernández, Antonio J. Ruiz, David Sicbaldi, Pieralberto |
| contents | If on a smooth bounded domain $Ω\subset\mathbb{R}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $Ω$ be a disk or an annulus? This question can be understood as a weaker analog of the well known Schiffer conjecture, in that the function $u$ is allowed to take a different constant value on each connected component of $\partial Ω$ yet many of the known rigidity properties of the original problem are essentially preserved. Our main result provides a negative answer by constructing a family of nontrivial doubly connected domains $Ω$ with the above property. As a consequence, a certain linear combination of the indicator functions of the domains $Ω$ and of the bounded component of the complement $\mathbb{R}^2\backslash\overlineΩ$ fails to have the Pompeiu property. Furthermore, our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07977 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Schiffer-type problem for annuli with applications to stationary planar Euler flows Enciso, Alberto Fernández, Antonio J. Ruiz, David Sicbaldi, Pieralberto Analysis of PDEs 35N25, 35Q31, 35B32 If on a smooth bounded domain $Ω\subset\mathbb{R}^2$ there is a nonconstant Neumann eigenfunction $u$ that is locally constant on the boundary, must $Ω$ be a disk or an annulus? This question can be understood as a weaker analog of the well known Schiffer conjecture, in that the function $u$ is allowed to take a different constant value on each connected component of $\partial Ω$ yet many of the known rigidity properties of the original problem are essentially preserved. Our main result provides a negative answer by constructing a family of nontrivial doubly connected domains $Ω$ with the above property. As a consequence, a certain linear combination of the indicator functions of the domains $Ω$ and of the bounded component of the complement $\mathbb{R}^2\backslash\overlineΩ$ fails to have the Pompeiu property. Furthermore, our construction implies the existence of continuous, compactly supported stationary weak solutions to the 2D incompressible Euler equations which are not locally radial. |
| title | A Schiffer-type problem for annuli with applications to stationary planar Euler flows |
| topic | Analysis of PDEs 35N25, 35Q31, 35B32 |
| url | https://arxiv.org/abs/2309.07977 |