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.

Mots clés : Shape optimization, Eigenvalues, Symmetrization, Euler equation, Shape derivative

[1] L. Barbosa, P. Bérard, Eigenvalue and “twisted” eigenvalue problems, applications to cmc surfaces, J. Math. Pures Appl. (9) 79 no. 5 (2000), 427-450 | MR | Zbl

[2] M. Belloni, B. Kawohl, A symmetry problem related to Wirtingerʼs and Poincaréʼs inequality, J. Differential Equations 156 no. 1 (1999), 211-218 | MR | Zbl

[3] F. Brock, A. Henrot, A symmetry result for an overdetermined elliptic problem using continuous rearrangement and domain derivative, Rend. Circ. Mat. Palermo (2) 51 no. 3 (2002), 375-390 | MR | Zbl

[4] A.P. Buslaev, V.A. Kondratév, A.I. Nazarov, On a family of extremal problems and related properties of an integral, Mat. Zametki (64) 51 no. 3 (1998), 830-838 | MR

[5] G. Croce, B. Dacorogna, On a generalized Wirtinger inequality, Discrete Contin. Dyn. Syst. 9 no. 5 (2003), 1329-1341 | MR | Zbl

[6] B. Dacorogna, W. Gangbo, N. Subía, Sur une généralisation de lʼinégalité de Wirtinger, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 29-50 | EuDML | Numdam | MR | Zbl

[7] L. Damascelli, B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differential Equations 206 no. 2 (2004), 483-515 | MR | Zbl

[8] G. Dinca, F. Isaia, Generalized Pohozaev identity and a non-existence result for the p-laplacian: weak solutions, Adv. Differential Equations 14 no. 5–6 (2009), 497-540 | MR | Zbl

[9] B. Emamizadeh, M. Zivari-Rezapour, Monotonicity of the principal eigenvalue of the p-laplacian in an annulus, Proc. Amer. Math. Soc. 136 no. 5 (2008), 1725-1731 | MR | Zbl

[10] L. Erbe, M. Tang, Uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball, J. Differential Equations 138 no. 2 (1997), 351-379 | MR | Zbl

[11] F. Farroni, R. Giova, T. Ricciardi, Best constant and extremals for a vector Poincaré inequality with weights, Sci. Math. Jpn. 71 no. 2 (2010), 111-126 | MR | Zbl

[12] B. Franchi, E. Lanconelli, J. Serrin, Existence and uniqueness of nonnegative solutions of quasilinear equations in ${ℝ}^n$, Adv. Math. 118 no. 2 (1996), 177-243 | MR | Zbl

[13] G. Franzina, P.D. Lamberti, Existence and uniqueness for a p-laplacian nonlinear eigenvalue problem, Electron. J. Differential Equations 26 (2010) | EuDML | MR | Zbl

[14] P. Freitas, A. Henrot, On the first twisted Dirichlet eigenvalue, Commun. Anal. Geom. 12 no. 5 (2004), 1083-1103 | MR | Zbl

[15] N. Fusco, F. Maggi, A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) (2006) | MR | Zbl

[16] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel (2006) | MR | Zbl

[17] A. Henrot, M. Pierre, Variation et optimisation de formes. Une analyse géométrique, Mathématiques & Applications (Berlin) vol. 48, Springer, Berlin (2005) | MR

[18] B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics vol. 1150, Springer-Verlag, Berlin (1985) | MR | Zbl

[19] B. Kawohl, Symmetry results for functions yielding best constants in Sobolev-type inequalities, Discrete Contin. Dyn. Syst. 6 no. 3 (2000), 683-690 | MR | Zbl

[20] B. Kawohl, V. Fridman, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolin. 44 no. 4 (2003), 659-667 | EuDML | MR | Zbl

[21] B. Kawohl, M. Lucia, S. Prashanth, Simplicity of the principal eigenvalue for indefinite quasilinear problems, Adv. Differential Equations 12 no. 4 (2007), 407-434 | MR | Zbl

[22] O.A. Ladyzhenskaya, N.N. Uralt´Seva, Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968) | MR | Zbl

[23] A.I. Nazarov, On the symmetry of extremals in the weight embedding theorem, J. Math. Sci. (New York) 107 no. 3 (2001), 3841-3859 | MR | Zbl

[24] P. Pucci, J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 no. 3 (1986), 681-703 | MR | Zbl

[25] J. Simon, Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim. 2 no. 7–8 (1980), 649-687 | MR | Zbl