Exterior disc points and exterior cone points are regular

In the exterior-disc case put $\mathrm{\Phi }(z)=\frac{\zeta -c}{z-c},b(z)=\mathrm{Re}\mathrm{\Phi }(z)-1.$ The center $c$ is outside $\bar{\mathrm{\Omega }}$, so $\mathrm{\Phi }$ and $b$ are holomorphic and harmonic respectively on $\mathrm{\Omega }$. Moreover $\mathrm{\mid }\mathrm{\Phi }(z)\mathrm{\mid }=R\mathrm{/}\mathrm{\mid }z-c\mathrm{\mid }<1$ for $z\in \mathrm{\Omega }$, hence $b<0$, while $b(z)\to 0$ as $z\to \zeta$. On the compact set $\mathrm{\partial }\mathrm{\Omega }\setminus V$, for any neighbourhood $V$ of $\zeta$, the continuous function $\mathrm{Re}\mathrm{\Phi }-1$ has a strictly negative maximum: equality could hold only when $\mathrm{\Phi }=1$, namely at $z=\zeta$. Thus $b$ is a global barrier and [L1] makes $\zeta$ regular.

In the exterior-cone case, after translation and rotation take $\zeta =0$ and suppose that near $0$ the domain lies in $S=\{r{e}^{it}:0<r<r_0,\mathrm{\mid }t\mathrm{\mid }<\theta \},0<\theta <\pi \mathrm{/}2.$ Choose ${\theta }^{\mathrm{\prime }}$ with $\theta <{\theta }^{\mathrm{\prime }}<\pi \mathrm{/}2$, put $\lambda =\pi \mathrm{/}(2{\theta }^{\mathrm{\prime }})$, and use the branch of $z^{\lambda }$ on the larger sector $\mathrm{\mid }t\mathrm{\mid }<{\theta }^{\mathrm{\prime }}$. Then $q(z)=-\mathrm{Re}(z^{\lambda })$ is harmonic and negative on $\mathrm{\Omega }\cap S$, and tends to $0$ at the origin. On a sufficiently small circle $\mathrm{\mid }z\mathrm{\mid }=\rho$, the angular margin gives $q(z)\le -{\rho }^{\lambda }\cos (\lambda \theta )<0(z\in \mathrm{\Omega }\cap \mathrm{\partial }D(0,\rho ))\mathrm{.}$ Thus [L2] globalizes $q$ to a barrier, and [L1] gives regularity.

The two barrier constructions prove the two regularity criteria.