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Geometric Versus Spectral Convergence for the Neumann Laplacian under Exterior Perturbations of the Domain

In: Integral Methods in Science and Engineering, Volume 1

Author

Listed:
  • J. M. Arrieta

    (Universidad Complutense de Madrid)

  • D. Krejĉiřík

    (ASCR, Nuclear Physics Institute)

Abstract

This chapter is concerned with the behavior of the eigenvalues and eigenfunctions of the Laplace operator in bounded domains when the domain undergoes a perturbation. It is well known that if the boundary condition that we are imposing is of Dirichlet type, the kind of perturbations that we may allow in order to obtain the continuity of the spectra is much broader than in the case of a Neumann boundary condition. This is explicitly stated in the pioneer work of Courant and Hilbert [CoHi53], and it has been subsequently clarified in many works, see [BaVy65, Ar97, Da03] and the references therein among others. See also [HeA06] for a general text on different properties of eigenvalues and [HeD05] for a study on the behavior of eigenvalues and in general partial differential equations when the domain is perturbed.

Suggested Citation

  • J. M. Arrieta & D. Krejĉiřík, 2010. "Geometric Versus Spectral Convergence for the Neumann Laplacian under Exterior Perturbations of the Domain," Springer Books, in: Christian Constanda & M.E. Pérez (ed.), Integral Methods in Science and Engineering, Volume 1, chapter 2, pages 9-19, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-4899-2_2
    DOI: 10.1007/978-0-8176-4899-2_2
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