A proof of the Faber-Krahn inequality for the first eigenvalue of the p-Laplacian

Abstract

In this work we present an elementary proof of the Faber-Krahn inequality for the first eigenvalue of the p-Laplacian on bounded domains in Rn. Let λ1 be the first eigenvalue and λ1* be the first eigenvalue for the ball of the same volume. Then we show that λ1 = λ1* iff the domain is a ball. Our proof makes considerable use of the corresponding Talenti's inequality and some well known properties of the first eigenfunetion.