Journal of the Mathematical Society of Japan

Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems

Pier Domenico LAMBERTI and Massimo LANZA DE CRISTOFORIS

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Abstract

We consider the Dirichlet and the Neumann eigenvalue problem for the Laplace operator on a variable nonsmooth domain, and we prove that the elementary symmetric functions of the eigenvalues splitting from a given eigenvalue upon domain deformation have a critical point at a domain with the shape of a ball. Correspondingly, we formulate overdetermined boundary value problems of the type of the Schiffer conjecture.

Article information

Source
J. Math. Soc. Japan, Volume 58, Number 1 (2006), 231-245.

Dates
First available in Project Euclid: 17 April 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1145287100

Digital Object Identifier
doi:10.2969/jmsj/1145287100

Mathematical Reviews number (MathSciNet)
MR2204572

Zentralblatt MATH identifier
1099.35070

Subjects
Primary: 35P15: Estimation of eigenvalues, upper and lower bounds 35N05: Overdetermined systems with constant coefficients 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) [See also 45Gxx, 45P05]

Keywords
Dirichlet and Neumann eigenvalues and eigenfunctions Laplace operator overdetermined problems domain perturbation special nonlinear operators

Citation

LAMBERTI, Pier Domenico; LANZA DE CRISTOFORIS, Massimo. Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems. J. Math. Soc. Japan 58 (2006), no. 1, 231--245. doi:10.2969/jmsj/1145287100. https://projecteuclid.org/euclid.jmsj/1145287100


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References

  • M. S. Ashbaugh, Open problems on eigenvalues of the Laplacian, Analytic and geometric inequalities and applications, Math. Appl., 478, Kluwer Acad. Publ., Dordrecht, 1999, pp.,13–28.
  • T. Chatelain, A new approach to two overdetermined eigenvalue problems of Pompeiu type, Élasticité, viscoélasticité et contrôle optimal, Lyon, 1995, 235–242: (electronic), ESAIM Proc., 2, Soc. Math. Appl. Indust., Paris, 1997.
  • L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1998.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 1983.
  • D. Henry, Topics in nonlinear analysis, Trabalho de Matemática, 192, Univ. Brasilia, Março 1982.
  • P. D. Lamberti and M. Lanza de Cristoforis, A global Lipschitz continuity result for a domain dependent Dirichlet eigenvalue problem for the Laplace operator, Zeitschrift für Analysis und ihre Anwendungen, 24 (2005), pp.,277–304.
  • P. D. Lamberti and M. Lanza de Cristoforis, A global Lipschitz continuity result for a domain dependent Neumann eigenvalue problem for the Laplace operator, J. Differential Equations, 216 (2005), pp.,109–133.
  • P. D. Lamberti and M. Lanza de Cristoforis, An analyticity result for the dependence of multiple eigenvalues and eigenspaces of the Laplace operator upon perturbation of the domain, Glasg. Math. J., 44 (2002), pp.,29–43.
  • P. D. Lamberti and M. Lanza de Cristoforis, A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator, J. Nonlinear Convex Anal., 5 (2004), pp.,19–42.
  • P. D. Lamberti and M. Lanza de Cristoforis, A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Neumann problem for the Laplace operator, submitted, 2003.
  • P. D. Lamberti and M. Lanza de Cristoforis, Persistence of eigenvalues and multiplicity in the Dirichlet problem for the Laplace operator on nonsmooth domains, to appear on Math. Phys. Anal. Geom., 2003.
  • M. Lanza de Cristoforis, Properties and pathologies of the composition and inversion operator in Schauder spaces, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 15 (1991), pp.,93–109.
  • G. Prodi, Dipendenza dal dominio degli autovalori dell'operatore di Laplace, Istituto Lombardo, Rend. Sc., 128 (1994), pp.,3–18.
  • L. Tartar, An introduction to Sobolev spaces and Interpolation spaces, Carnegie-Mellon Univ., Pittsburgh, 2000.
  • G. M. Troianiello, Elliptic differential equations and obstacle problems, Plenum Press, New York and London, 1987.