[29] (with P. R. Garabedian) On existence theorems of potential theory and conformal mapping

Abstract

This is the second Garabedian–Schiffer paper, following (Schiffer and Garabedian, Trans. Amer. Math. Soc., 65, 187–238, 1949) by just a year in date of publication. The setting is again multiply connected plane domains, and the questions addressed are some of the most fundamental existence theorems in conformal mapping and potential theory. Specifically, in this one paper, the authors present new proofs of the existence of the mapping onto a parallel slit plane, of Green’s function for Laplace’s equation, and of the extremal function for a generalization of Schwarz’s lemma. The arguments make thorough use of the Bergman kernel and its relatives and provide a common approach to the different problems. Moreover, the existence of the kernel itself, also independently shown here as the solution of an extremal problem, is fairly considered to be less deep than previous proofs of the existence results that the paper aims for, a point made by the authors. If the kernel played a supporting role in Schiffer and Garabedian (Trans. Amer. Math. Soc., 65, 187–238, 1949), here it takes center stage.