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
SERVADEI, RAFFAELLA
2016
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
