Exterior cone condition

An open set satisfies an exterior cone condition at $z$ if a solid cone with vertex $z$ lies outside the set in some neighbourhood of $z$. The stronger global version requires the closures of the cone and the set to meet only at $z$. In two dimensions, a power barrier for an exterior cone makes such a point a regular boundary point for the Dirichlet problem. Local cone conditions suffice for local boundary regularity; a global cone is an additional geometric assumption.

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 givesThe 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.