Resolutivity of boundary functions

Summary

A function on the boundary of a bounded open set is resolutive when the Perron method gives a well-defined harmonic function from it. This function is then the generalised solution of the Dirichlet problem with that boundary function as data. The Perron method squeezes the solution between an upper envelope built from superharmonic functions and a lower envelope built from subharmonic functions. Resolutivity says that the two envelopes meet and give a harmonic function. It is the right notion of "solvable data" for the generalised Dirichlet problem of the Perron method, and it holds for every continuous boundary function.