Mathematical Research Letters

Volume 23 (2016)

Number 3

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

Pages: 863 – 885

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n3.a14

Authors

Antonio Greco (Department of Mathematics and Informatics, Università di Cagliari, Italy)

Raffaella Servadei (Dipartimento di Scienze di Base e Fondamenti, Università degli Studi di Urbino, Italy)

Abstract

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

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Published 8 July 2016