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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.00455 |
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| _version_ | 1866915472220356608 |
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| author | Wheeler, Miles H. |
| author_facet | Wheeler, Miles H. |
| contents | In this note we construct smooth bounded domains $Ω\subset \mathbb R^2$, other than disks, for which the overdetermined problem $$
\left\{
\begin{alignedat}{2}
Δu + λu &= 0 &\qquad& \text{ in } Ω, \newline
u &= b &\qquad& \text{ on } \partial Ω, \newline
\frac{\partial u}{\partial n} &= c &\qquad& \text{ on } \partial Ω
\end{alignedat}
\right. $$ has a solution for some constants $λ,b,c \ne 0$. These appear to be the first counterexamples to a conjecture of Willms and Gladwell [WG94]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane Wheeler, Miles H. Analysis of PDEs In this note we construct smooth bounded domains $Ω\subset \mathbb R^2$, other than disks, for which the overdetermined problem $$ \left\{ \begin{alignedat}{2} Δu + λu &= 0 &\qquad& \text{ in } Ω, \newline u &= b &\qquad& \text{ on } \partial Ω, \newline \frac{\partial u}{\partial n} &= c &\qquad& \text{ on } \partial Ω \end{alignedat} \right. $$ has a solution for some constants $λ,b,c \ne 0$. These appear to be the first counterexamples to a conjecture of Willms and Gladwell [WG94]. |
| title | Non-symmetric solutions to an overdetermined problem for the Helmholtz equation in the plane |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2509.00455 |