Overdetermined problems with fractional Laplacian

Abstract

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

Added a missing assumption (1.3) in Theorem 1.1 and Theorem 1.2, which is used in the proof of Lemma 4.3