THE DEPENDENCE OF THE FIRST EIGENVALUE OF THE INFINITY LAPLACIAN WITH RESPECT TO THE DOMAIN

NAVARRO, J. C. ; ROSSI, J. D. ; ANTOLIN, A. SAN ; SAINTIER, N.

Glasgow mathematical journal, Tome 56 (2014) no. 2, pp. 241-249

Résumé

In this paper we study the dependence of the first eigenvalue of the infinity Laplace with respect to the domain. We prove that this first eigenvalue is continuous under some weak convergence conditions which are fulfilled when a sequence of domains converges in Hausdorff distance. Moreover, it is Lipschitz continuous but not differentiable when we consider deformations obtained via a vector field. Our results are illustrated with simple examples.

Détail

DOI : 10.1017/S0017089513000219

Mots-clés : 35J60, 35P30

Bibliographie

[1] 1.Anane, A., Simplicité et isolation de la premiere valeur propre du p-Laplacien avec poinds, C. R. Acad. Sci. Paris Série I 305, (1987), 725–728. Google Scholar

[2] 2.Aronsson, G., Extensions of functions satisfying Lipschitz conditions, Ark. Math. 6 (1967), 551–561. Google Scholar

[3] 3.Aronsson, G., Crandall, M. G. and Juutinen, P., A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. 41 (2004), 439–505. Google Scholar

[4] 4.Belloni, M. and Kawohl, B., The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p → ∞, ESAIM Control Optim. Calc. Var. 10 (2004), 28–52. Google Scholar | DOI

[5] 5.Bhattacharya, T., Di Benedetto, E. and Manfredi, J., Limits as p → ∞ of Δu = f and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino (1991), 15–68. Google Scholar

[6] 6.Crandall, M. G., Ishii, H. and Lions, P. L., User's guide to viscosity solutions of second-order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1–67. Google Scholar | DOI

[7] 7.García-Azorero, J. and Peral, I., Existence and non-uniqueness for the p-Laplacian: Nonlinear eigenvalues, Comm. Partial Differ. Equ. 12 (1987), 1389–1430. Google Scholar

[8] 8.Garcia-Melian, J. and De Lis, J. Sabina, On the perturbation of eigenvalues for the p-Laplacian, C. R. Acad. Sci. Paris Séries I 332 (2001), 893–898. Google Scholar

[9] 9.Henrot, A., Minimization problems for eigenvalues of the Laplacian, J. Evol. Equ. 3 (2003), 443–461. Google Scholar

[10] 10.Henrot, A. and Pierre, M., Variation et optimization de forme, Mathématiques et Applications 48 (Springer, Berlin, Germany, 2005). Google Scholar

[11] 11.Jensen, R., Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), 51–74. Google Scholar | DOI

[12] 12.Juutinen, P. and Lindqvist, P., On the higher eigenvalues for the ∞-eigenvalue problem, Calc. Var. Partial Differ. Equ. 23 (2) (2005), 169–192. Google Scholar | DOI

[13] 13.Juutinen, P., Lindqvist, P. and Manfredi, J. J., The ∞-eigenvalue problem, Arch. Rational Mech. Anal. 148 (1999), 89–105. Google Scholar

[14] 14.Lindqvist, P., On the equation div(|∇ u|p-2 ∇ u) + λ |u|p-2u =0, Proc. Amer. Math. Soc. 109 (1990), 157–164. (Addendum to: On the equation div(|∇ u|p-2 ∇ u) +λ |u|p-2u =0, Proc. Amer. Math. Soc. (1992), 583–584). Google Scholar

[15] 15.Lindqvist, P., A nonlinear eigenvalue problem. Topics in Mathematical Analysis, 175203, Ser. Anal. Appl. Comput., 3 (World Scientific, Hackensack, NJ, 2008). Google Scholar | DOI

[16] 16.Munkres, J., Topology, 2nd ed (Prentice Hall, Upper Saddle River, NJ, 1999). Google Scholar

[17] 17.Simon, J., Optimal design for Neumann condition and for related boundary value conditions, in Boundary control and boundary variations, Lecture Notes in Control and Information Sciences, 100 (J. P. Zolezio, Editor) (Springer, New York, 1988). Google Scholar