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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2404.09272 |
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| _version_ | 1866916205253623808 |
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| author | Pacella, Filomena Ruiz, David Sicbaldi, Pieralberto |
| author_facet | Pacella, Filomena Ruiz, David Sicbaldi, Pieralberto |
| contents | Given a bounded regular domain $ω\subset \mathbb{R}^{N-1}$ and the half-cylinder $Σ= ω\times (0,+\infty)$, we consider the relative overdetermined torsion problem in $Σ$, i.e.
\[\left\{
\begin{array}{ll}
Δ{u}+1=0 &\mbox{in $Ω$},\newline
\partial_ηu = 0 &\mbox{on $\widetilde Γ_Ω$},\newline
u=0 &\mbox{on $Γ_Ω$},\newline
\partial_νu =c &\mbox{on $Γ_Ω$}.
\end{array} \right.
\] where $Ω\subset Σ$, $Γ_Ω= \partial Ω\cap Σ$, $\widetilde Γ_Ω= \partial Ω\setminus Γ_Ω$, $ν$ is the outer unit normal vector on $Γ_Ω$ and $η$ is the outer unit normal vector on $\widetilde Γ_Ω$. We build nontrivial solutions to this problem in domains $Ω$ that are the hypograph of certain nonconstant functions $v : \overlineω \to (0, + \infty)$. Such solutions can be reflected with respect to $ω$, giving nontrivial solutions to the relative overdetermined torsion problem in a cylinder. The proof uses a local bifurcation argument which, quite remarkably, works for any generic base $ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09272 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nontrivial solutions to the relative overdetermined torsion problem in a cylinder Pacella, Filomena Ruiz, David Sicbaldi, Pieralberto Analysis of PDEs Given a bounded regular domain $ω\subset \mathbb{R}^{N-1}$ and the half-cylinder $Σ= ω\times (0,+\infty)$, we consider the relative overdetermined torsion problem in $Σ$, i.e. \[\left\{ \begin{array}{ll} Δ{u}+1=0 &\mbox{in $Ω$},\newline \partial_ηu = 0 &\mbox{on $\widetilde Γ_Ω$},\newline u=0 &\mbox{on $Γ_Ω$},\newline \partial_νu =c &\mbox{on $Γ_Ω$}. \end{array} \right. \] where $Ω\subset Σ$, $Γ_Ω= \partial Ω\cap Σ$, $\widetilde Γ_Ω= \partial Ω\setminus Γ_Ω$, $ν$ is the outer unit normal vector on $Γ_Ω$ and $η$ is the outer unit normal vector on $\widetilde Γ_Ω$. We build nontrivial solutions to this problem in domains $Ω$ that are the hypograph of certain nonconstant functions $v : \overlineω \to (0, + \infty)$. Such solutions can be reflected with respect to $ω$, giving nontrivial solutions to the relative overdetermined torsion problem in a cylinder. The proof uses a local bifurcation argument which, quite remarkably, works for any generic base $ω$. |
| title | Nontrivial solutions to the relative overdetermined torsion problem in a cylinder |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.09272 |