An isoperimetric inequality for a nonlinear eigenvalue problem
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

λ p,q (Ω)= inf {v L p (Ω) v L q (Ω) ,v0,vW 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.

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) ,v0,vW 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.

@article{AIHPC_2012__29_1_21_0,
     author = {Croce, Gisella and Henrot, Antoine and Pisante, Giovanni},
     title = {An isoperimetric inequality for a nonlinear eigenvalue problem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {21--34},
     publisher = {Elsevier},
     volume = {29},
     number = {1},
     year = {2012},
     doi = {10.1016/j.anihpc.2011.08.001},
     zbl = {1243.49048},
     mrnumber = {2876245},
     language = {en},
     url = {archive.numdam.org/item/AIHPC_2012__29_1_21_0/}
}
Croce, Gisella; Henrot, Antoine; Pisante, Giovanni. An isoperimetric inequality for a nonlinear eigenvalue problem. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 1, pp. 21-34. doi : 10.1016/j.anihpc.2011.08.001. http://archive.numdam.org/item/AIHPC_2012__29_1_21_0/

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