On the Faber–Krahn inequality for the Dirichlet p-Laplacian

Abstract

A famous conjecture made by Lord Rayleigh is the following: “The first eigenvalue of the Laplacian on an open domain of given measure with Dirichlet boundary conditions is minimum when the domain is a ball and only when it is a ball”. This conjecture was proved simultaneously and independently by Faber [G. Faber, Beweiss dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförfegige den leifsten Grundton gibt. Sitz. bayer Acad. Wiss. (1923) 169–172] and Krahn [E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaftdes Kreises. Math. Ann. 94 (1924) 97–100.]. We shall deal with the p-Laplacian version of this theorem.