An isoperimetric inequality for a nonlinear eigenvalue problem

Croce, Gisella ; Henrot, Antoine ; Pisante, Giovanni

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 21-34 / Harvested from Numdam

Résumé

On montre une inégalité isopérimétrique du type Rayleigh–Faber–Krahn pour une généralisation non-linéaire de la première valeur propre de Dirichlet torsadée, définie par λ p,q (Ω)= inf {∥∇v∥ L p (Ω) ∥v∥ L q (Ω) ,v≠0,v∈W 0 1,p (Ω),∫ Ω|v| q-2 vdx=0}. Plus précisément, on montre que le minimum parmi les ensembles de volume donné est lʼunion de deux boules égales.

Abstract

We prove an isoperimetric inequality of the Rayleigh–Faber–Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue, defined by λ p,q (Ω)= inf {∥∇v∥ L p (Ω) ∥v∥ L q (Ω) ,v≠0,v∈W 0 1,p (Ω),∫ Ω|v| q-2 vdx=0}. More precisely, we show that the minimizer among sets of given volume is the union of two equal balls.

Détail

Publié le : 2012-01-01

MR 2876245 | Zbl 1243.49048 | 1 citation dans Numdam

DOI : https://doi.org/10.1016/j.anihpc.2011.08.001

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