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Izv. RAN. Ser. Mat., 2005, Volume 69, Issue 4, Pages 161–204 (Mi izv652)  

This article is cited in 20 scientific papers (total in 20 papers)

Asymptotic expansions of eigenvalues and eigenfunctions of an elliptic operator in a domain with many “light” concentrated masses situated on the boundary. Two-dimensional case

G. A. Chechkin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider vibrations of a membrane which contains many “light” concentrated masses on the boundary. We study the asymptotic behaviour of the frequencies of eigenvibrations of the membrane as the small parameter (which characterizes the diameter and density of the concentrated masses) tends to zero. We construct asymptotic expansions of eigenelements of the corresponding problems and carefully justify these expansions.

DOI: https://doi.org/10.4213/im652

Full text: PDF file (3090 kB)
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English version:
Izvestiya: Mathematics, 2005, 69:4, 805–846

Bibliographic databases:

UDC: 517.956.226
MSC: 35J25, 35B25, 35B27, 35B40
Received: 02.06.2004

Citation: G. A. Chechkin, “Asymptotic expansions of eigenvalues and eigenfunctions of an elliptic operator in a domain with many “light” concentrated masses situated on the boundary. Two-dimensional case”, Izv. RAN. Ser. Mat., 69:4 (2005), 161–204; Izv. Math., 69:4 (2005), 805–846

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Amirat Y., Chechkin G.A., Gadyl'shin R.R., “Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues”, Appl. Anal., 86:7 (2007), 873–897  crossref  mathscinet  zmath  isi
    2. Chechkin G.A., Koroleva Yu.O., Meidell A., Persson L.-E., “On the Friedrichs inequality in a domain perforated aperiodically along the boundary. Homogenization procedure. Asymptotics for parabolic problems”, Russ. J. Math. Phys., 16:1 (2009), 1–16  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Gadyl'shin R.R., Koroleva Yu.O., Chechkin G.A., “On the convergence of solutions and eigenelements of a boundary value problem in a domain perforated along the boundary”, Diff. Eq., 46:5 (2010), 667–680  crossref  mathscinet  zmath  isi  elib  elib  scopus
    4. G. A. Chechkin, Yu. O. Koroleva, L.-E. Persson, P. Wall, “A new weighted Friedrichs-type inequality for a perforated domain with a sharp constant”, Eurasian Math. J., 2:1 (2011), 81–103  mathnet  mathscinet  zmath
    5. Gregory A. Chechkin, Taras A. Mel'nyk, “Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses”, Applicable Analysis, 2011, 1  crossref  mathscinet  isi  scopus
    6. G.A. Chechkin, D. Cioranescu, A. Damlamian, A.L. Piatnitski, “On boundary value problem with singular inhomogeneity concentrated on the boundary”, Journal de Mathématiques Pures et Appliquées, 2011  crossref  mathscinet  isi  scopus
    7. Gadylshin R.R., Korolëva Yu.O., Chechkin G.A., “Ob asimptotike resheniya kraevoi zadachi v oblasti, perforirovannoi vdol granitsy”, Vestnik Chelyabinskogo gosudarstvennogo universiteta, 2011, no. 27, 27–36  mathscinet  elib
    8. R. R. Gadyl'shin, Yu. O. Koroleva, G. A. Chechkin, “On the asymptotic behavior of a simple eigenvalue of a boundary value problem in a domain perforated along the boundary”, Differ. Equ., 47:6 (2011), 822–831  crossref  mathscinet  zmath  isi  elib  scopus
    9. R. R. Gadylshin, Yu. O. Koroleva, G. A. Chechkin, “Ob asimptotike resheniya kraevoi zadachi v oblasti, perforirovannoi vdol granitsy”, Vestnik ChelGU, 2011, no. 14, 27–36  mathnet
    10. T. A. Mel’nik, G. A. Chechkin, “On new types of vibrations of thick cascade junctions with concentrated masses”, Dokl. Math, 87:1 (2013), 102  crossref  mathscinet  zmath  isi  scopus
    11. G. A. Chechkin, T. A. Mel'nyk, “Spatial-skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses”, Math. Meth. Appl. Sci, 2013, n/a  crossref  mathscinet  isi  scopus
    12. Holovatyi Yu.D., Hut V.M., “Vibrating Systems with Rigid Light-Weight Inclusions: Asymptotics of the Spectrum and Eigenspaces”, Ukr. Math. J., 64:10 (2013), 1495–1513  crossref  mathscinet  zmath  isi  elib  scopus
    13. N. N. Abdullazade, G. A. Chechkin, “Perturbation of the Steklov Problem on a Small Part of the Boundary”, J Math Sci, 2014  crossref  mathscinet  scopus
    14. G.A.. Chechkin, T.A.. Mel'nyk, “High-frequency cell vibrations and spatial skin effect in thick cascade junction with heavy concentrated masses”, Comptes Rendus Mécanique, 2014  crossref  scopus
    15. A. R. Bikmetov, T. R. Gadyl’shin, I. Kh. Khusnullin, “Perturbation by Slender Potential of Operators Associated with Sectorial Forms”, J Math Sci, 2014  crossref  mathscinet  scopus
    16. T. F. Sharapov, “On the resolvent of multidimensional operators with frequently changing boundary conditions in the case of the homogenized Dirichlet condition”, Sb. Math., 205:10 (2014), 1492–1527  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    17. Giuseppe Cardone, Andrii Khrabustovskyi, “Neumann spectral problem in a domain with very corrugated boundary”, Journal of Differential Equations, 2015  crossref  mathscinet  scopus
    18. T. A. Mel'nik, G. A. Chechkin, “Eigenvibrations of thick cascade junctions with ‘very heavy’ concentrated masses”, Izv. Math., 79:3 (2015), 467–511  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. A. G. Chechkina, V. A. Sadovnichy, “Degeneration of Steklov–type boundary conditions in one spectral homogenization problem”, Eurasian Math. J., 6:3 (2015), 13–29  mathnet
    20. Koroleva Yu., “Spectral Analysis of a Nonlinear Boundary-Value Problem in a Perforated Domain. Applications to the Friedrichs Inequality in Lp”, Diff. Equat. Appl., 8:4 (2016), 437–458  crossref  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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