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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2013
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1311.7549 |
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Table of Contents:
- Let $N\geq 1$ and $s\in (0,1)$. In the present work we characterize bounded open sets $Ω$ with $ C^2$ boundary (\textit{not necessarily connected}) for which the following overdetermined problem \begin{equation*} ( -Δ)^s u = f(u) \text{ in $Ω$,} \qquad u=0 \text{ in $\mathbb{R}^N\setminus Ω$,} \qquad(\partial_η)_s u=Const. \text{ on $\partial Ω$} \end{equation*} has a nonnegative and nontrivial solution, where $η$ is the outer unit normal vectorfield along $\partialΩ$ and for $x_0\in\partialΩ$ \[ \left(\partial_η\right)_{s}u(x_{0})=-\lim_{t\to 0}\frac{u(x_{0}-tη(x_0))}{t^s}. \] Under mild assumptions on $f$, we prove that $Ω$ must be a ball. In the special case $f\equiv 1$, we obtain an extension of Serrin's result in 1971. The fact that $Ω$ is not assumed to be connected is related to the nonlocal property of the fractional Laplacian. The main ingredients in our proof are maximum principles and the method of moving planes.