Proposition 1.1. Let

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 finding a harmonic function in a, say, smoothly bounded domain Ω ⊂ Rn matching a given continuous function f on ∂Ω 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. [50] for a very nice historical account. Here, we want to focus on algebraic properties of solutions to the Dirichlet problem when Ω is an ellipsoid and the data f possess nice algebraic properties. Thus, we first present the following proposition