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

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

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English version:
Izvestiya: Mathematics, 2005, 69:4, 805–846

Bibliographic databases:

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

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
Related articles on Google Scholar: Russian articles, English articles

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
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
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
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
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
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
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
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
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
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
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
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
13. N. N. Abdullazade, G. A. Chechkin, “Perturbation of the Steklov Problem on a Small Part of the Boundary”, J Math Sci, 2014
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
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
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
17. Giuseppe Cardone, Andrii Khrabustovskyi, “Neumann spectral problem in a domain with very corrugated boundary”, Journal of Differential Equations, 2015
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
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
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
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