Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2506.00268 |
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Inhaltsangabe:
- We use shape derivative approach to prove that balls are the only convex and $C^{1,1}$ regular domains in which the fractional overdetermined problem \begin{equation*} \left\{\begin{aligned} \Ds u&= λ_{s, p} u^{p-1}\quad\text{in}\quad\Om \\ u &= 0\quad \text{in}\quad\R^N\setminus \Om\\ u/d^s&=C_0\quad\text{on\;\; $\partialØ$} \end{aligned} \right. \end{equation*} admits a nontrivial solution for $p\in [1, 2]$ and where $λ_{s, p}= λ_{s, p}(Ø)$ is the best constant in the family of Subcritical Sobolev inequalities. In the cases $p=1$ and $p=2$, we recover the classical symmetry results of Serrin, corresponding to the torsion problem and the first Dirichlet eigenvalue problem, respectively (see \cite{FS-15}). We note that for $p\in (1,2)$, the above problem lies outside the framework of \cite{FS-15}, and the methods developed therein do not apply. Our approach extends to the fractional setting a method initially developed by A. Henrot and T. Chatelain in \cite{CH-99}, and relies on the use of domain derivatives combined with the continuous Steiner symmetrization introduced by Brock in \cite{Brock-00}.