A Tale of Ellipsoids in Potential Theory

Abstract

Ellipsoids possess several beautiful properties associ- ated with classical potential theory. Some of them are well known, and some have been forgotten. In this article we hope to bring a few of the \lost pieces of classical mathematics back to the lime- light. 1. Dirichlet's problem Let us start our story with the Dirichlet problem. This problem of nding a harmonic function in a, say, smoothly bounded domain R n matching a given continuous function f on @ gained huge attention in the second half of the nineteenth century due to its cen- tral role in Riemann's proof of the existence of a conformal map of any simply connected domain onto the disk. Later on Riemann's proof was criticized by Weierstrass, and, after a considerable turmoil, cor- rected and completed by Hilbert and Fredholm - cf. (49) for a very nice historical account. Here, we want to focus on algebraic properties of solutions to the Dirichlet problem when is an ellipsoid and the data f possess nice algebraic properties. Thus, we