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

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 with 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.