Regular boundary point

Summary

A boundary point of a bounded open set is regular if the generalized solution of the Dirichlet problem tends to the prescribed value there. This must hold for every continuous boundary function. The Dirichlet problem is solvable for all continuous data exactly when every boundary point is regular. Regularity is a local property: it holds at $\xi$ if and only if there is a barrier at $\xi$. It is characterized by Wiener's criterion in terms of capacity, and the irregular points always form a polar set.