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

Summary 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 λ ₁ ^* be the first eigenvalue for the ball of the same volume. Then we show that λ₁=λ ₁ ^* 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 eigenfunction. Entrata in Redazione il 20 aprile 1998.