Pseudo Harmonic Measures and the Dirichlet Problem

Abstract

For the case of plane regions bounded by finitely many disjoint Jordan curves, the solution of the Dirichlet problem can be expressed in terms of the classical harmonic measure of boundary arcs. At an appropriate stage in the development it is, in fact, useful to observe that the existence of such harmonic measures is equivalent to solvability of the Dirichlet problem (although one subsequently proves that all such regions are Dirichlet regions). We propose here to carry over this order of ideas to a quite general setting, in which arbitrary regions and ideal boundary structures are allowed. The counterparts of the classical harmonic measures of arcs are then harmonic functions with analogous boundary properties, but they no longer appear as measures in the boundary sets, in general. We shall refer to them as “pseudo harmonic measures”. Our main result shows how pseudo harmonic measures can be used to solve the Dirichlet problem.