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) no. 1, pp. 21-34.

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

$${\lambda }^{p,q}\left(\Omega \right)=\inf \{\frac{\parallel \nabla v{\parallel }_{L^p\left(\Omega \right)}}{\parallel v{\parallel }_{L^q\left(\Omega \right)}},v\ne 0,v\in W_0^{1,p}\left(\Omega \right),\underset{\Omega }{\int }{\left|v\right|}^{q-2}vdx=0\}.$$

Plus précisément, on montre que le minimum parmi les ensembles de volume donné est lʼunion de deux boules égales.

We prove an isoperimetric inequality of the Rayleigh–Faber–Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue, defined by

$${\lambda }^{p,q}\left(\Omega \right)=\inf \{\frac{\parallel \nabla v{\parallel }_{L^p\left(\Omega \right)}}{\parallel v{\parallel }_{L^q\left(\Omega \right)}},v\ne 0,v\in W_0^{1,p}\left(\Omega \right),\underset{\Omega }{\int }{\left|v\right|}^{q-2}vdx=0\}.$$

More precisely, we show that the minimizer among sets of given volume is the union of two equal balls.

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