Skip to main content
SpringerLink
Log in

Spectral stability of the Robin Laplacian

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We consider the Robin Laplacian in two bounded regions Ω1 and Ω2 of ℝN with Lipschitz boundaries and such that Ω2 ⊂ Ω1, and we obtain two-sided estimates for the eigenvalues λ n,2 of the Robin Laplacian in Ω2 via the eigenvalues λ n, 1 of the Robin Laplacian in Ω1. Our estimates depend on the measure of the set difference Ω\Ω2 and on suitably defined characteristics of vicinity of the boundaries ∂Ω1 and ∂Ω2, and of the functions defined on ∂Ω1 and on ∂Ω2 that enter the Robin boundary conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. I. Burenkov, Sobolev Spaces on Domains (B.G. Teubner, Stuttgart, 1998).

    MATH  Google Scholar 

  2. V. I. Burenkov and E. B. Davies, “Spectral Stability of the Neumann Laplacian,” J. Diff. Eqns. 186(2), 485–508 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  3. V. I. Burenkov and P. D. Lamberti, “Spectral Stability of Nonnegative Self-adjoint Operators,” Dokl. Akad. Nauk. 403(2), 159–164 (2005) [Dokl. Math. 72 (1), 507–511 (2005)].

    MathSciNet  Google Scholar 

  4. V. I. Burenkov and P. D. Lamberti, “Spectral Stability of General Non-negative Self-adjoint Operators with Applications to Neumann-Type Operators,” J. Diff. Eqns. 233(2), 345–379 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  5. V. I. Burenkov, P. D. Lamberti, and M. Lanza de Cristoforis, “Spectral Stability of Nonnegative Self-adjoint Operators,” Sovr. Mat., Fundam. Napr. 15, 76–111 (2006).

    Google Scholar 

  6. E. B. Davies, Spectral Theory and Differential Operators (Cambridge Univ. Press, Cambridge, 1995), Cambridge Stud. Adv. Math. 42.

    Google Scholar 

  7. E. B. Davies, Heat Kernels and Spectral Theory (Cambridge Univ. Press, Cambridge, 1989), Cambridge Tracts Math. 92.

    MATH  Google Scholar 

  8. K. Deimling, Nonlinear Functional Analysis (Springer, Berlin, 1985).

    MATH  Google Scholar 

  9. 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. Diff. Eqns. 216(1), 109–133 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  10. E. H. Lieb and M. Loss, Analysis (Am. Math. Soc., Providence, RI, 2001).

    MATH  Google Scholar 

  11. M. Marcus and V. J. Mizel, “Absolute Continuity on Tracks and Mappings of Sobolev Spaces,” Arch. Ration. Mech. Anal. 45, 294–320 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Nečas, Les méthodes directes en théorie des équations elliptiques (Masson et Cie, Paris, 1967).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121, Padova, Italy

    V. I. Burenkov & M. Lanza de Cristoforis

Authors
  1. V. I. Burenkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Lanza de Cristoforis
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burenkov, V.I., Lanza de Cristoforis, M. Spectral stability of the Robin Laplacian. Proc. Steklov Inst. Math. 260, 68–89 (2008). https://doi.org/10.1134/S0081543808010069

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543808010069

Keywords

Search

Navigation