Notes On Classical Potential Theory

NOTES ON CLASSICAL POTENTIAL THEORY M. Papadimitrakis Department of Mathematics University of Crete January 2004 2 Forword During the fall semester of the academic year 1990-1991 I gave a course on Classical Potential Theory attended by an excellent class of graduate students of the Department of Mathematics of Washington University. That was my first time to teach such a course and, I have to say, besides sporadic knowledge of a few facts directly related to complex analysis, I had no serious knowledge of the subject. The result was: many sleepless nights reading books, trying to choose the material to be presented and preparing hand-written notes for the students. The books I found very useful and which determined the choice of material ´ ements de la Th´eorie Classique du Potentiel” by M. Brelot were the superb “El´ and the “Selected Problems on Exceptional Sets” by L. Carleson. Other sources were: “Some Topics in the Theory of Functions of One Complex Variable” by W. Fuchs, unpublished notes on “Harmonic Measures” by J. Garnett, “Subharmonic Functions” by W. Hayman and P. Kennedy, “Introduction to Potential Theory” by L. Helms, “Foundations of Modern Potential Theory” by N. Landkof, “Subharmonic Functions” by T. Rado and “Potential Theory in Modern Function Theory” by M. Tsuji. This is a slightly expanded version of the original notes with very few changes. The principle has remained the same, namely to present an overview of the classical theory at the level of a graduate course. The part called “Preliminaries” is new and its contents were silently taken for granted during the original course. The main material is the Divergence Theorem and Green’s Formula, a short course on holomorphic functions (, since their real parts are the main examples of harmonic functions in the plane and, also, since one of the central results is the proof of the Riemann Mapping