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Asymptotic Analysis of Spectral Problems in Thick Multi-Level Junctions

  • T. A. Mel’nyk
Chapter

Abstract

Spectral boundary-value problems are considered in a new kind of perturbed domain, namely, thick multi-level junctions. Boundary-value problems in thick one-level junctions (thick junctions) have been intensively investigated recently (see, for instance, [BlGaGr07], [BlGaMe08], [Me08] and, the references there). In [MeNa97]–[Me(3)01], classification of thick one-level junctions was given and basic results were obtained both for boundary-value and spectral problems in thick junctions of different types. It was shown that qualitative properties of solutions essentially depend on the junction type and on the conditions given on the boundaries of the attached thin domains. It is known that the asymptotic behavior of the spectrum of a perturbed spectral problem is highly sensitive to perturbation, and it is unexpected. This was also observed for spectral problems in thick junctions with Neumann conditions ([MeNa97] and [Me00]), with Dirichlet conditions ([Me99] and [Me(3)01]), with Fourier conditions ([Me(2)01]) and with Steklov ones ([Me(1)01]).

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References

  1. [BlGaGr07]
    Blanchard, D., Gaudiello, A., Griso, G.: Junction of a periodic family of elastic rods with a 3d plate. Parts I, II. J. Math. Pures Appl., 88, 1-33, 149-190 (2007).MATHMathSciNetGoogle Scholar
  2. [BlGaMe08]
    Blanchard, D., Gaudiello, A., Mel'nyk, T.A.: Boundary homogenization and reduction of dimension in a Kirchhoff-Love plate. SIAM J. Math. Anal., 39, 1764-1787 (2008).MATHCrossRefMathSciNetGoogle Scholar
  3. [Me08]
    Mel'nyk, T.A.: Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1. Math. Methods Appl. Sci., 31, 1005-1027 (2008).MATHCrossRefMathSciNetGoogle Scholar
  4. [MeNa97]
    Mel'nyk, T.A., Nazarov S.A.: Asymptotics of the Neumann spectral problem solution in a domain of “thick comb”-type. J. Math. Sci., 85, 2326-2346 (1997).CrossRefMathSciNetGoogle Scholar
  5. [Me99]
    Mel'nyk, T.A.: On free vibrations of a thick periodic junction with concentrated masses on the fine rods. Nonlinear Oscillations, 2, 511-523 (1999).MATHMathSciNetGoogle Scholar
  6. [Me00]
    Mel'nyk, T.A.: Asymptotic analysis of a spectral problem in a periodic thick junction of type 3:2:1. Math. Methods Appl. Sci., 23, 321-346 (2000).MATHCrossRefMathSciNetGoogle Scholar
  7. [Me(1)01]
    Mel'nyk, T.A.: Asymptotic behavior of eigenvalues and eigenfunctions of the Steklov problem in a thick periodic junction. Nonlinear Oscillations, 4, 91-105 (2001).MATHMathSciNetGoogle Scholar
  8. [Me(2)01]
    Mel'nyk, T.A.: Asymptotic behaviour of eigenvalues and eigenfunctions of the Fourier problem in a thick junction of type 3:2:1, in Grouped and Analytical Methods in Mathematical Physics, Academy of Sciences of Ukraine, Kiev (2001), 187-196.Google Scholar
  9. [Me(3)01]
    Mel'nyk, T.A.: Vibrations of a thick periodic junction with concentrated masses. Math. Models Methods Appl. Sci., 11, 1001-1029 (2001).MATHCrossRefMathSciNetGoogle Scholar
  10. [Me(4)01]
    Mel'nyk, T.A.: Hausdorff convergence and asymptotic estimates of the spectrum of a perturbed operator. Z. Anal. Anwendungen, 20, 941-957 (2001).MATHMathSciNetGoogle Scholar
  11. [Me06]
    Mel'nyk, T.A.: Asymptotic behaviour of eigenvalues and eigenfuctions of the Fourier problem in a thick multilevel junction. Ukrainian Math. J., 58, 220-243 (2006).CrossRefMathSciNetGoogle Scholar
  12. [Me94]
    Mel'nyk, T.A. : Spectral properties of the discontinuous self-adjoint operator-functions. Reports Nat. Acad. Sci. Ukraine, 12, 33-36 (1994).MathSciNetGoogle Scholar
  13. [HrMe96]
    Hryniv, R.O., Mel'nyk, T.A.: On a singular Rayleigh functional. Math. Notes, 60, 97-101 (1996). (Russian edition: Matem. zametki. (1), 60 (1996): 130-134).CrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.National Taras Shevchenko University of KyivKyivUkraine

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