Antonio Greco, Raffaella Servadei
Hopf's lemma and constrained radial symmetry for
the fractional Laplacian
(228K, pdf)
ABSTRACT. In this paper we prove Hopf's boundary point lemma for the fractional
Laplacian. With respect to the classical formulation, in the non-local
framework the normal derivative of the involved function~$u$ at~$z \in
\partial \Omega$ is replaced with the limit of the ratio
$u(x)/(\delta_R(x))^s$, where $\delta_R(x)=\mathop{
m dist}(x,
\partial B_R)$ and $B_R \subset \Omega$ is a ball such that $z \in \partial
B_R$. More precisely, we show that
$$
\liminf_{B
i x o z}
rac{u(x)}{\, (\delta_R(x))^s}>0\,.
$$
Also we consider the extit{overdetermined} problem
$$
egin{cases}
(-\Delta)^s \, u = 1 &\mbox{in $\Omega$}