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

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.

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.

Keywords: Shape optimization, Eigenvalues, Symmetrization, Euler equation, Shape derivative

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