| Abstract: |
| Unlike in the local case, Hopf`s Lemma does not hold true, in general, for sign-changing supersolutions to equations driven by the fractional Laplacian, as proven in a recent paper by Dipierro, Soave and Valdinoci. We investigate the validity of Hopf`s Lemma for a (possibly sign-changing) function $u \in H^s_0(\Omega)$ satisfying
\[ (-\Delta)^s u(x) \geq c(x)u(x) \quad \text{in }\Omega,\]
where $\Omega \subset \mathbb{R}^N$ is an open set, $c \in L^\infty(\Omega)$ and $(-\Delta)^s u$ is the fractional Laplacian of $u$. We show that, under suitable assumptions, the validity of Hopf`s Lemma for $u$ at a point $x_0 \in \partial \Omega$ is essentially equivalent to the validity of Hopf`s Lemma for the Caffarelli-Silvestre extension of $u$ at the point $(x_0,0) \in \mathbb{R}^N \times \mathbb{R}^+$. The results have been obtained in collaboration with Azahara DelaTorre. |
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