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 $z \in \partial B_R$. Also we consider an overdetermined problem and we prove the that it 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.

### Hopf's lemma and constrained radial symmetry for the fractional Laplacian

#### 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 $z \in \partial B_R$. Also we consider an overdetermined problem and we prove the that it 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.
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2016
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11576/2626828
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