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.
Mots clés : torsional rigidity, nonlinear eigenvalue problems, spherical rearrangements
@article{COCV_2014__20_2_315_0,
author = {Brasco, Lorenzo},
title = {On torsional rigidity and principal frequencies: an invitation to the {Kohler-Jobin} rearrangement technique},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {315--338},
publisher = {EDP-Sciences},
volume = {20},
number = {2},
year = {2014},
doi = {10.1051/cocv/2013065},
mrnumber = {3264206},
zbl = {1290.35160},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2013065/}
}
TY - JOUR AU - Brasco, Lorenzo TI - On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 315 EP - 338 VL - 20 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013065/ DO - 10.1051/cocv/2013065 LA - en ID - COCV_2014__20_2_315_0 ER -
%0 Journal Article %A Brasco, Lorenzo %T On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 315-338 %V 20 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013065/ %R 10.1051/cocv/2013065 %G en %F COCV_2014__20_2_315_0
Brasco, Lorenzo. On torsional rigidity and principal frequencies: an invitation to the Kohler-Jobin rearrangement technique. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 2, pp. 315-338. doi : 10.1051/cocv/2013065. http://www.numdam.org/articles/10.1051/cocv/2013065/
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