| Contents |
| We characterize bounded open sets $\Omega$ with $ C^2$ boundary (not necessarily connected)
for which the following overdetermined problem
\[
( -\Delta)^s u = f(u) text{ in $\Omega$; } u=0 \text{ in $\mathbb{R}^N\setminus \Omega$; } (\partial_{\eta})_s u=Const. \text{ on $\partial \Omega$}
\]
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 main ingredients in our proof are maximum principles and the method of moving planes. |
|