= Power barrier for an exterior cone
{title2=$g=-r^\nu\cos(\beta\psi)$}
For a planar set outside a global solid cone of half-angle $\alpha$, choose $0<\alpha_0<\min(\alpha,\pi/2)$, $\beta=\pi/[2(\pi-\alpha_0)]$, and $0<\nu<\beta$. If $\psi$ is the angle from the opposite axis, the <Laplacian in polar coordinates> gives
$$
g=-r^\nu\cos(\beta\psi)<0,\qquad
\Delta g=(\beta^2-\nu^2)r^{\nu-2}\cos(\beta\psi)>0.
$$
The cosine has a positive uniform lower bound outside the original cone. On a bounded set, $\Delta g$ therefore has a positive uniform lower bound as well. The positive function $-g$ is a <barrier for the Dirichlet problem>. An unbounded set does not receive the same global lower bound from this construction.
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