Topics in Potential Theory: Quadrature Domains, Balayage and Harmonic Measure

Abstract: In this thesis, which consists of five papers (A,B,C,D,E), we are interested in questions related to quadrature domains. Among the problems studied are the possibility of changing the type of measure in a quadrature identity (from complex to real and from real signed to positive), properties of partial balayage, which in a sense can be used to generate quadrature domains, and mother bodies which are closely related to inversion of partial balayage.These three questions are discussed in papers A,D respectively B.The first of these questions (when trying to go from real signed to positive measures) leads to the study of approximation in the cone of positive harmonic functions. These questions are closely related to properties of the harmonic measure on the Martin boundary, and this relationship leads to the study of harmonic measures on ideal boundaries in paper E. Some other approaches to the same problem also lead to some extent to the study of properties of classical balayage in paper C.