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 Mat. Sb., 2009, Volume 200, Number 3, Pages 49–74 (Mi msb4500)

Asymptotic analysis of boundary-value problems in thick three-dimensional multi-level junctions

T. A. Mel'nika, G. A. Chechkinbc

a National Taras Shevchenko University of Kyiv, The Faculty of Mechanics and Mathematics
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
c Narvik University College

Abstract: We consider homogenization problems in a singularly perturbed three-dimensional domain of multi-level-junction type which consists of the junction body and a large number of alternating thin curvilinear cylinders that belong to two classes. Under the assumption that one class consists of cylinders of finite height, and the second class of cylinders of infinitesimal height, and that different inhomogeneous boundary conditions of the third kind with perturbed coefficients are given on the boundaries of the thin curvilinear cylinders, we prove the homogenization theorems and the convergence of the energy integrals.
Bibliography: 42 titles.

Keywords: homogenization, thick junctions, asymptotics.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm4500

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English version:
Sbornik: Mathematics, 2009, 200:3, 357–383

Bibliographic databases:

UDC: 517.956.225+517.956.8
MSC: Primary 35B40; Secondary 35B25, 35B27, 35J25

Citation: T. A. Mel'nik, G. A. Chechkin, “Asymptotic analysis of boundary-value problems in thick three-dimensional multi-level junctions”, Mat. Sb., 200:3 (2009), 49–74; Sb. Math., 200:3 (2009), 357–383

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb4500
• https://doi.org/10.4213/sm4500
• http://mi.mathnet.ru/eng/msb/v200/i3/p49

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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. Chechkin G.A., Chechkina T.P., D'Apice C., De Maio U., Mel'nyk T.A., “Homogenization of 3D thick cascade junction with a random transmission zone periodic in one direction”, Russ. J. Math. Phys., 17:1 (2010), 35–55
2. T. P. Chechkina, “Convergence of solutions of a boundary-value problem in a thick cascade junction with oscillating boundary of the transmission zone in the case of Neumann conditions at the boundary”, Russian Math. Surveys, 65:5 (2010), 982–983
3. G. A. Chechkin, C. D'Apice, U. De Maio, “On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary”, Math. Meth. Appl. Sci., 33:17 (2010), 2019–2036
4. T. P. Chechkina, “Averaging in cascade junctions with a “wide” transmission domain”, Journal of Mathematical Sciences, 190:1 (2013), 157–169
5. Chechkin G.A., Mel'nyk T.A., “Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses”, Appl. Anal., 91:6 (2012), 1055–1095
6. Mel'nyk T.A., Sadovyi D.Yu., “Homogenization of a quasilinear parabolic problem with different alternating nonlinear Fourier boundary conditions in a two-level thick junction of the type 3:2:2”, Ukr. Math. J., 63:12 (2012), 1855–1882
7. Durante T., Mel'nyk T.A., “Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3:2:1”, ESAIM Control Optim. Calc. Var., 18:2 (2012), 583–610
8. Mel'nik T.A., Chechkin G.A., “On new types of vibrations of thick cascade junctions with concentrated masses”, Dokl. Math., 87:1 (2013), 102
9. G. A. Chechkin, T. A. Mel'nyk, “Spatial-skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses”, Math. Methods Appl. Sci., 37:1 (2014), 56–74
10. G. A. Chechkin, C. D'Apice, U. De Maio, A. L. Piatnitski, “On the rate of convergence of solutions in domain with random multilevel oscillating boundary”, Asymptot. Anal., 87:1-2 (2014), 1–28
11. Zhao J., Wang J., “Convergence Rates in Homogenization of the Mixed Boundary Value Problems”, Math. Probl. Eng., 2019, 2680657
12. Zhao J., Wang J., “Homogenization of Nonlinear Equations With Mixed Boundary Conditions”, J. Math. Phys., 60:8 (2019), 081512
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