Remarks on the Works of O. Perron and L.A. Lusternik on the Dirichlet Problem *

Perron’s function u and Lusternik’s function z are the best solutions of the Dirichlet problem, in the sense that there is no other harmonic function which takes a given value at any other point than those at which u and z take given values. In spite of their general character, the three conditions presented in this chapter, Condition A' and Condition B,' suffer the lack of explicitness. Therefore, it is desirable to have another criterion for the regularity of a boundary point. The chapter explores whether a necessary and sufficient condition of regularity can be obtained if Lusternik’s condition be replaced by a stronger one.