14-69 Antonio Greco, Raffaella Servadei

Hopf's lemma and constrained radial symmetry for the fractional Laplacian (228K, pdf) Oct 1, 14

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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 $\Omega$}

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