Krahn's proof of the Rayleigh conjecture revisited

Abstract

The paper is a discussion of Krahn's proof of the Rayleigh conjecture in his 1925 paper in Mathematische Annalen. The result is that amongst all membranes of the same area and the same physical properties, the circular one has the lowest ground frequency. We show how Krahn's approach coincides with the modern techniques of geometric measure theory using the co-area formula. We furthermore discuss some issues and generalisations of his proof.