Barrier

Summary

A barrier at a boundary point $\xi$ of a bounded open set is a superharmonic function that vanishes at $\xi$. It is positive at every other point of the closure. Barriers are the local test for whether the Perron method gives the right boundary value at $\xi$. A boundary point is regular for the Dirichlet problem if and only if it has a barrier. Simple geometric conditions, such as an exterior ball or an exterior cone at $\xi$, produce explicit barriers.