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
Revised: 15 June 2013
We generalize to the p-Laplacian Δp 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 Δp 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.
Mathematics Subject Classification: 35P30 / 47A75 / 49Q10
Key words: Torsional rigidity / nonlinear eigenvalue problems / spherical rearrangements
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