Journal of Mathematical Sciences

, Volume 198, Issue 1, pp 13–30 | Cite as

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

  • V. M. Hut

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.


Asymptotic Expansion English Translation Elliptic Operator Spectral Problem Concentrate Masse 
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  1. 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).MATHMathSciNetGoogle Scholar
  2. 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).CrossRefMathSciNetGoogle Scholar
  3. 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. 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).MathSciNetGoogle Scholar
  5. 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).Google Scholar
  6. 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).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators [in Russian], Fizmatlit, Moscow (1993).Google Scholar
  8. 8.
    R. Courant and D. Hilbert, Methods of Mathematical Physics, Wiley (1953).Google Scholar
  9. 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).Google Scholar
  10. 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. 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. 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).MathSciNetGoogle Scholar
  13. 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. 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).CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    A. L. Pyatnitskii, G. A. Chechkin, and A. S. Shamaev, Homogenization. Methods and Applications [in Russian], Vol. 3, Tamara Rozhkovskaya, Novosibirsk (2007).Google Scholar
  16. 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).CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    E. Sánchez-Palencia, Non-Homogeneous Media and Vibration Theory, Vol. 127, Lecture Notes in Physics, Springer, Berlin (1980).Google Scholar
  18. 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).CrossRefMathSciNetGoogle Scholar
  19. 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).CrossRefMATHMathSciNetGoogle Scholar
  20. 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).CrossRefMATHMathSciNetGoogle Scholar
  21. 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).MATHMathSciNetGoogle Scholar
  22. 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).CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    A. Bourgeat, G. A. Chechkin, and A. L. Piatnitski, “Singular double porosity model,” Appl. Anal., 82, No. 2, 103–116 (2003).CrossRefMATHMathSciNetGoogle Scholar
  24. 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).CrossRefMathSciNetGoogle Scholar
  25. 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).CrossRefMATHMathSciNetGoogle Scholar
  26. 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).CrossRefMATHGoogle Scholar
  27. 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).CrossRefMATHGoogle Scholar
  28. 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).CrossRefMATHGoogle Scholar
  29. 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).CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    M. Lobo and E. Pérez, “Local problems for vibrating systems with concentrated masses: a review,” C. R. Mécanique, 331, 303–317 (2003).CrossRefMATHGoogle Scholar
  31. 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).CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    V. Rybalko, “Vibrations of elastic systems with a large number of tiny heavy inclusions,” Asymptotic Anal., 32, 27–62 (2002).MATHMathSciNetGoogle Scholar
  33. 33.
    J. Sánchez Hubert and E. Sánchez Palencia, Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer, Berlin (1989).CrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  • V. M. Hut
    • 1
  1. 1.Franko Lviv National UniversityLvivUkraine

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