Calculus of Variations and Geometric Measure Theory

L. Brasco

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

created by brasco on 01 Feb 2013
modified on 20 Jun 2013


Accepted Paper

Inserted: 1 feb 2013
Last Updated: 20 jun 2013

Journal: ESAIM: COCV
Pages: 27
Year: 2013


We generalize to the $p-$Laplacian $\Delta_p$ 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 $\Delta_p$ of a set in terms of its $p-$torsional rigidity. The result is valid in every space dimension, for every $1<p<\infty$ and for every open set having finite measure. Moreover, it holds by replacing the first eigenvalue with more general optimal Poincar\'e-Sobolev constants. The method of proof is based on a generalization of the rearrangement technique introduced by Kohler-Jobin.

Keywords: Nonlinear eigenvalue problems, Torsional rigidity, spherical rearrangements