In Caffarelli and Silvestre's extension problem paper link of paper Lemma 4.2, in Lemma 4.2, it is given that the viscosity solution is $C^2$ away from $z=0$. How do we have a classical solution on $B_R \cup \{z \neq 0\}$? As $z=0$ is already contained in the Ball $B_R$. Please help me out with this doubt.
2026-02-23 02:32:11.1771813931
How is the viscosity solution is $C^2$ away from $z=0$. The domain $B_R \cup {z \neq 0}$ already contains the point $z=0$.
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Looking to the Lemma, I guess that $ \{z \ne 0\}$ is short for $\{ (x,z) \in \mathbb R^2 \mid z \ne 0\}$.