Overdetermined problems with fractional Laplacian

Abstract

Let $N\geq 1$ and $s\in (0,1)$. In the present work we characterize bounded open sets $\Omega$ with $ C^2$ boundary (\textit{not necessarily connected}) for which the following overdetermined problem \begin{equation*} ( -\Delta)^s u = f(u) \text{ in $\Omega$,} \qquad u=0 \text{ in $\mathbb{R}^N\setminus \Omega$,} \qquad(\partial_{\eta})_s u=Const. \text{ on $\partial \Omega$} \end{equation*} has a nonnegative and nontrivial solution, where $\eta $ is the outer unit normal vectorfield along $\partial\Omega$ and for $x_0\in\partial\Omega$ \[ \left(\partial_{\eta}\right)_{s}u(x_{0})=-\lim_{t\to 0}\frac{u(x_{0}-t\eta(x_0))}{t^s}. \] Under mild assumptions on $f$, we prove that $\Omega$ must be a ball. In the special case $f\equiv 1$, we obtain an extension of Serrin's result in 1971. The fact that $\Omega$ 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.