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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.09272 |
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Table of 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 $ω$.