Skip to main content
×
Home
    • Aa
    • Aa

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

  • Lorenzo Brasco (a1)
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 < 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.

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.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

A. Alvino , V. Ferone , P.-L. Lions and G. Trombetti , Convex symmetrization and applications. Ann. Institut Henri Poincaré Anal. Non Linéaire 14 (1997) 275293.

M. Belloni and B. Kawohl , The pseudo p-Laplace eigenvalue problem and viscosity solution as p → ∞. ESAIM: COCV 10 (2004) 2852.

T. Carroll and J. Ratzkin , Interpolating between torsional rigidity and principal frequency. J. Math. Anal. Appl. 379 (2011) 818826.

E. DiBenedetto , C1 + α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (1983) 827850.

L. Esposito and C. Trombetti , Convex symmetrization and Pólya-Szegő inequality. Nonlinear Anal. 56 (2004) 4362.

M. Flucher , Extremal functions for the Moser-Trudinger inequality in two dimensions. Comment. Math. Helv. 67 (1992) 471497.

M.-T. Kohler-Jobin , Symmetrization with equal Dirichlet integrals. SIAM J. Math. Anal. 13 (1982), 153161.

M.-T. Kohler-Jobin , Une méthode de comparaison isopérimétrique de fonctionnelles de domaines de la physique mathématique. I. Une démonstration de la conjecture isopérimétrique \hbox{$P\,\lambda^2\ge \pi \, j^4_0/2$}P λ2π j04/2 de Pólya et Szegő, Z. Angew. Math. Phys. 29 (1978) 757766.

G.M. Lieberman , Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis. Theory, Methods & Appl. 12 (1988) 12031219.

J. Moser , A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20 (1970/71) 10771092.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ESAIM: Control, Optimisation and Calculus of Variations
  • ISSN: 1292-8119
  • EISSN: 1262-3377
  • URL: /core/journals/esaim-control-optimisation-and-calculus-of-variations
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 1 *
Loading metrics...

Abstract views

Total abstract views: 23 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th September 2017. This data will be updated every 24 hours.