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

Lorenzo Brasco

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 2, page 315-338
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Brasco, Lorenzo. "On torsional rigidity and principal frequencies: an invitation to the Kohler−Jobin rearrangement technique." ESAIM: Control, Optimisation and Calculus of Variations 20.2 (2014): 315-338. <http://eudml.org/doc/272938>.

@article{Brasco2014,
abstract = {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 &lt; p &lt; ∞ 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.},
author = {Brasco, Lorenzo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {torsional rigidity; nonlinear eigenvalue problems; spherical rearrangements; -Laplacian},
language = {eng},
number = {2},
pages = {315-338},
publisher = {EDP-Sciences},
title = {On torsional rigidity and principal frequencies: an invitation to the Kohler−Jobin rearrangement technique},
url = {http://eudml.org/doc/272938},
volume = {20},
year = {2014},
}

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
PB - EDP-Sciences
VL - 20
IS - 2
SP - 315
EP - 338
AB - 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 &lt; p &lt; ∞ 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.
LA - eng
KW - torsional rigidity; nonlinear eigenvalue problems; spherical rearrangements; -Laplacian
UR - http://eudml.org/doc/272938
ER -

References

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