Hopf’s lemma and constrained radial symmetry for the fractional Laplacian

In this paper we give a new proof of 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 $\delta_R(x) = \mathop{\rm dist}(x, \partial B_R)$ and $B_R \subset \Omega$ is a ball such that $z \in \partial B_R$. More precisely, we show that\[\liminf_{B \ni x \to z} \frac{u(x)}{\, (\delta_R(x))^s} \gt 0 \textrm{ .}\]Also we consider the overdetermined problem\begin{cases}(-\Delta)^s \, u = 1 & \textrm{in}\; \Omega \\u = 0 & \textrm{in} \; \mathbb{R}^N \setminus \Omega \\\lim \limits_{\Omega \ni x \to z} \frac{u(x)}{(\delta_\Omega(x))^s} = q(\lvert z \rvert) & \textrm{for every } z \in \partial \Omega \textrm{ .}\end{cases}Here $\Omega$ is a bounded open set in $\mathbb R^N$, $N\geq 1$, containing the origin and satisfying the interior ball condition, $\delta_\Omega(x)=\mathrm{dist}(x, \partial \Omega)$, and $(-\Delta)^s , s\in (0,1)$, is the fractional Laplace operator defined, up to normalization factors, as\[(-\Delta)^s \, u(x) = \textrm{P.V.} \int_{\mathbb{R}^N}\frac{\, u(x) - u(y) }{{\lvert x - y \rvert }^{N + 2s}} dy\]We show that if the function $q(r)$ grows fast enough with respect to $r$, then the problem admits a solution only in a suitable ball centered at the origin. The proof is based on a comparison principle proved along the paper, and on the boundary point lemma mentioned before.

Keywords

overdetermined problems, comparison principle, Hopf’s lemma

2010 Mathematics Subject Classification

35B51, 35N25, 35S15

Full Text (PDF format)

Accepted 4 June 2015

Published 8 July 2016