Skip to main content
SpringerLink
Log in

Asymptotic Expansions of Eigenvalues and Eigenfunctions of a Vibrating System With Stiff Light-Weight Inclusions

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We study the spectral properties of a boundary-value problem for an elliptic second-order operator with singularly perturbed coefficients. The problem simulates resonance vibrations of an elastic system with finitely many stiff and, at the same time, light-weight inclusions of any shape. The ratio of the stiffness coefficients of the inclusions and the main body has an order of ε−1 as ε → 0, whereas the ratio of their mass densities has an order of εκ, where κ > 0. The complete asymptotic expansions of the eigenvalues and eigenfunctions of this problem are constructed and justified.

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. M. I. Vishik and L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with small parameter,” Usp. Mat. Nauk, 12, No. 5, 3–122 (1957).

    MATH  MathSciNet  Google Scholar 

  2. R. R. Gadyl’shin, “Ramification of a multiple eigenvalue of the Dirichlet problem for the Laplacian under singular perturbation of the boundary condition,” Mat. Zametki, 52, No. 4, 42–55 (1992); English translation : Math. Notes, 52, No. 4, 1020–1029 (1992).

    Article  MathSciNet  Google Scholar 

  3. Yu. D. Golovatyi and H. E. Hrabchak, “On the Sturm–Liouville problem on starlike graphs with “heavy” nodes,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 72, 63–78 (2010).

    Google Scholar 

  4. Yu. D. Holovatyi and V. M. Hut, “Vibrating systems with rigid light-weight inclusions: asymptotics of the spectrum and eigenspaces,” Ukr. Mat. Zh., 64, No. 10, 1315–1330 (2012); English translation: Ukr. Math. J., 64, No. 10, 1314–1329 (2012).

    MathSciNet  Google Scholar 

  5. Yu. D. Golovatyi, “Spectral properties of oscillatory systems with adjoined masses,” Trudy Mosk. Mat. Obshch., 54, 29–72 (1992); English translation: Mosc. Math. Soc., 23–59 (1993).

  6. V. V. Zhikov, “On gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients,” Algebra Anal., 16, Issue 5, 773–790 (2004); English translation: St. Petersburg Math. J., 16, No. 5, 773–790 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  7. V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators [in Russian], Fizmatlit, Moscow (1993).

    Google Scholar 

  8. R. Courant and D. Hilbert, Methods of Mathematical Physics, Wiley (1953).

  9. V. F. Lazutkin, “Quasiclassical asymptotics of eigenfunctions,” in: Outcomes of Science and Engineering: Current Problems of Mathematics. Fundamental Trends [in Russian], Vol. 34, All-Union Institute of Scientific and Technical Information (VINITI), Academy of Sciences of the USSR, Moscow (1988), pp. 135–174; English translation: KAM Theory and Semiclassical Approximations to Eigenfunctions, 24, 225–241 (1993).

  10. V. A. Marchenko and E. Ya. Khruslov, Boundary-Value Problems in Domains with Fine-Grained Boundaries [in Russian], Naukova Dumka, Kiev (1974).

    Google Scholar 

  11. T. A. Mel’nyk and S. A. Nazarov, “Asymptotic structure of the spectrum in the problem of harmonic vibrations of a hub with heavy arms,” Dokl. Ross. Akad. Nauk, 333, No. 1, 13–15 (1993).

    Google Scholar 

  12. T. A. Mel’nyk and S. A. Nazarov, “Asymptotic analysis of the Neumann problem on the junction of a body with thin heavy rods,” Algebra Anal., 12, No. 2, 188–238 (2000); English translation: St. Petersburg Math. J., 12, No. 2, 317–351 (2001).

    MathSciNet  Google Scholar 

  13. O. A. Oleinik, G. A. Iosifian, and A. S. Shamaev, Mathematical Problems in Elasticity and Homogenization [in Russian], Moscow State University, Moscow (1990); English translation: Springer North-Holland, London (1992).

    Google Scholar 

  14. E. Pérez, G. A. Chechkin, and E. I. Yablokova, “On eigenvibrations of a body with «light» concentrated masses on the surface,” Usp. Mat. Nauk, 57, No. 6, 195–196 (2002); English translation : Russian Math. Surveys, 57, No. 6, 1240–1242 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  15. A. L. Pyatnitskii, G. A. Chechkin, and A. S. Shamaev, Homogenization. Methods and Applications [in Russian], Vol. 3, Tamara Rozhkovskaya, Novosibirsk (2007).

  16. G. V. Sandrakov, “Homogenization of elasticity equations with contrasting coefficients,” Mat. Sb., 190, No. 12, 37–92 (1999); English translation : Math. Sb., 190, No. 12, 1749–1806 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  17. E. Sánchez-Palencia, Non-Homogeneous Media and Vibration Theory, Vol. 127, Lecture Notes in Physics, Springer, Berlin (1980).

    Google Scholar 

  18. G. A. Chechkin, “Asymptotic expansions of eigenvalues and eigenfunctions of an elliptic operator in the region containing a large number of “light” concentrated masses closely located on the boundary. Two-dimensional case,” Izv. Ross. Akad. Nauk, Ser. Matematika, 69, No. 4, 161–204 (2005).

    Article  MathSciNet  Google Scholar 

  19. G. A. Chechkin, “Splitting of a multiple eigenvalue in a problem of concentrated masses,” Usp. Mat. Nauk, 59, No. 4, 205–206 (2004); English translation : Russian Math. Surveys, 59, No. 4, 790–791 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  20. Y. Amirat, G. A. Chechkin, R. R. Gadyl’shin, “Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues,” Appl. Anal., 86, No. 7, 873–897 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  21. N. Babych and Yu. Golovaty, “Asymptotic analysis of vibrating system containing stiff-heavy and flexible-light parts,” Nelin. Granichn. Zad., 18, 194–207 (2008).

    MATH  MathSciNet  Google Scholar 

  22. N. Babych and Yu. D. Golovaty, “Low and high frequency approximations to eigenvibrations in a medium with double contrasts,” J. Comput. Appl. Math., 234, No. 6, 1860–1867 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Bourgeat, G. A. Chechkin, and A. L. Piatnitski, “Singular double porosity model,” Appl. Anal., 82, No. 2, 103–116 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  24. G. A. Chechkin, “Homogenization of a model spectral problem for the Laplace operator in a domain with many closely located «heavy» and «intermediate heavy» concentrated masses,” Probl. Mat. Anal., 32, 45–76 (2006); English translation : J. Math. Sci., 135, No. 6, 3485–3521 (2006).

    Article  MathSciNet  Google Scholar 

  25. G. A. Chechkin and T. A. Mel’nyk, “Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses,” Appl. Anal., 91, No. 6, 1055–1095 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  26. G. Geymonat, M. Lobo-Hidalgo, E. Sanchez-Palencia, and G. F. Roach, “Spectral properties of certain stiff problems in elasticity and acoustics,” Math. Meth. Appl. Sci., 4, 291–306 (1982).

    Article  MATH  Google Scholar 

  27. Yu. D. Golovaty, M. Lobo and E. Pérez, “On vibrating membranes with very heavy thin inclusions,” Math. Mod. Meth. Appl. Sci., 14, No. 7, 987–1034. (2004).

    Article  MATH  Google Scholar 

  28. D. Gómez, M. Lobo, S. A. Nazarov, and E. Pérez, “Asymptotics for the spectrum of the Wentzell problem with a small parameter and other related stiff problems,” J. Math. Pure Appl., 86, No. 5, 369–402 (2006).

    Article  MATH  Google Scholar 

  29. M. Lobo, S. A. Nazarov, and E. Pérez, “Eigenoscillations of contrasting non-homogeneous elastic bodies: asymptotic and uniform estimates for eigenvalues,” IMA J. Appl. Math., 70, No. 3, 419–458 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Lobo and E. Pérez, “Local problems for vibrating systems with concentrated masses: a review,” C. R. Mécanique, 331, 303–317 (2003).

    Article  MATH  Google Scholar 

  31. T. A. Mel’nyk, “Vibrations of a thick periodic junction with concentrated masses,” Math. Mod. Meth. Appl. Sci., 11, No. 6, 1001–1029 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  32. V. Rybalko, “Vibrations of elastic systems with a large number of tiny heavy inclusions,” Asymptotic Anal., 32, 27–62 (2002).

    MATH  MathSciNet  Google Scholar 

  33. J. Sánchez Hubert and E. Sánchez Palencia, Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer, Berlin (1989).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Franko Lviv National University, Lviv, Ukraine

    V. M. Hut

Authors
  1. V. M. Hut
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 55, No. 4, pp. 16–29, October–December, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hut, V.M. Asymptotic Expansions of Eigenvalues and Eigenfunctions of a Vibrating System With Stiff Light-Weight Inclusions. J Math Sci 198, 13–30 (2014). https://doi.org/10.1007/s10958-014-1769-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-014-1769-3

Keywords

Search

Navigation