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&= \lambda_{ 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\;\; $\partial\O$} \end{ aligned} \right. \end{ equation*} admits a nontrivial solution for $p\in [1, 2]$ and where $\lambda_{ s, p}= \lambda_{ s, p}(\O)$ 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}.
Categories: math.AP