On torsional rigidity and principal frequencies: an invitation to the Kohler−Jobin rearrangement technique

Laboratoire d’Analyse, Topologie, Probabilités, Aix-Marseille Université, 39 Rue Frédéric Joliot Curie, 13453 Marseille Cedex 13, France
lorenzo.brasco@univ-amu.fr

Received: 14 March 2013
Revised: 15 June 2013

Abstract

We generalize to the p-Laplacian Δ_p a spectral inequality proved by M.-T. Kohler−Jobin. As a particular case of such a generalization, we obtain a sharp lower bound on the first Dirichlet eigenvalue of Δ_p of a set in terms of its p-torsional rigidity. The result is valid in every space dimension, for every 1 < p < ∞ and for every open set with finite measure. Moreover, it holds by replacing the first eigenvalue with more general optimal Poincaré-Sobolev constants. The method of proof is based on a generalization of the rearrangement technique introduced by Kohler−Jobin.