Potential Theory

Abstract

Liouville is not usually mentioned in historical accounts of potential theory despite the fact that he discussed the relation between harmonic and holomorphic functions in the plane and the uniqueness of an equilibrium distribution on given conductors in a more elegant way than his contemporaries and contributed substantially to the theory of ellipsoidal harmonics. It is the main purpose of this chapter to remedy this neglect and in particular to present a most remarkable but unfinished and unpublished contribution by Liouville to general potential theory. In this work, Liouville anticipated many of the central ideas and theorems concerning integral operators, in particular, their spectral theory, including the Rayleigh-Ritz method for determining eigenvalues. As background material, I shall first sketch the state of affairs in potential theory around 1840 and discuss Liouville’s published contributions.