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

Published online by Cambridge University Press:  02 September 2013

J. C. NAVARRO ,
J. D. ROSSI ,
A. SAN ANTOLIN  and
N. SAINTIER
J. C. NAVARRO
Affiliation:
Departamento de Análisis Matemático, Universidad de Alicante, Ap. Correos 99, 03080 AlicanteSpain e-mails: [email protected], [email protected], [email protected]
J. D. ROSSI
Affiliation:
Departamento de Análisis Matemático, Universidad de Alicante, Ap. Correos 99, 03080 AlicanteSpain e-mails: [email protected], [email protected], [email protected]
A. SAN ANTOLIN
Affiliation:
Departamento de Análisis Matemático, Universidad de Alicante, Ap. Correos 99, 03080 AlicanteSpain e-mails: [email protected], [email protected], [email protected]
N. SAINTIER
Affiliation:
Departamento de Matemática, FCEyN Universidad de Buenos Aires (1428) Buenos Aires, Argentina, and Instituto de Ciencias – Universidad Nacional de General Sarmiento, J. M. Gutierrez 1150, C.P. 1613 Los Polvorines, Pcia de Bs. As., Argentina e-mails: [email protected]; [email protected]
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Abstract

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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.

Keywords

35J6035P30
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 2 , May 2014 , pp. 241 - 249
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

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.Aronsson, G., Extensions of functions satisfying Lipschitz conditions, Ark. Math. 6 (1967), 551–561.Google Scholar
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.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.CrossRefGoogle Scholar
5.Bhattacharya, T., Di Benedetto, E. and Manfredi, J., Limits as p → ∞ of Δpu p = f and related extremal problems, Rend. Sem. Mat. Univ. Politec. Torino Special issue (1991), 15–68.Google Scholar
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.CrossRefGoogle Scholar
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.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.Henrot, A., Minimization problems for eigenvalues of the Laplacian, J. Evol. Equ. 3 (2003), 443–461.Google Scholar
10.Henrot, A. and Pierre, M., Variation et optimization de forme, Mathématiques et Applications 48 (Springer, Berlin, Germany, 2005).Google Scholar
11.Jensen, R., Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123 (1993), 51–74.CrossRefGoogle Scholar
12.Juutinen, P. and Lindqvist, P., On the higher eigenvalues for the ∞-eigenvalue problem, Calc. Var. Partial Differ. Equ. 23 (2) (2005), 169–192.CrossRefGoogle Scholar
13.Juutinen, P., Lindqvist, P. and Manfredi, J. J., The ∞-eigenvalue problem, Arch. Rational Mech. Anal. 148 (1999), 89–105.Google Scholar
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. 116 (1992), 583–584).Google Scholar
15.Lindqvist, P., A nonlinear eigenvalue problem. Topics in Mathematical Analysis, 175203, Ser. Anal. Appl. Comput., 3 (World Scientific, Hackensack, NJ, 2008).CrossRefGoogle Scholar
16.Munkres, J., Topology, 2nd ed (Prentice Hall, Upper Saddle River, NJ, 1999).Google Scholar
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