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 Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 3, Pages 103–158 (Mi izv2600)

Asymptotics of solutions and modelling the problems of elasticity theory in domains with rapidly oscillating boundaries

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: We obtain explicit formulae for two terms of asymptotics of solutions of the Neumann and Dirichlet problems for the system of two-dimensional equations of elasticity theory in a domain with rapidly oscillating boundary and suggest an algorithm for constructing complete asymptotic expansions. We justify the asymptotic representations of solutions using Korn's inequality in singularly perturbed domains. We discuss two methods of modelling these problems of elasticity theory by constructing new, simpler, boundary-value problems whose solutions provide two-term asymptotics of solutions of the original problems. The first method is based on the introduction of the so-called wall laws containing a small parameter in the higher derivatives. The second method is based on the use of the concept of a smooth image of the singularly perturbed boundary.

DOI: https://doi.org/10.4213/im2600

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English version:
Izvestiya: Mathematics, 2008, 72:3, 509–564

Bibliographic databases:

UDC: 517.946+539.3
MSC: 35J45, 35R30, 35J60, 35B40

Citation: S. A. Nazarov, “Asymptotics of solutions and modelling the problems of elasticity theory in domains with rapidly oscillating boundaries”, Izv. RAN. Ser. Mat., 72:3 (2008), 103–158; Izv. Math., 72:3 (2008), 509–564

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv2600
• https://doi.org/10.4213/im2600
• http://mi.mathnet.ru/eng/izv/v72/i3/p103

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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. S. A. Nazarov, “Asymptotic modeling of a problem with contrasting stiffness”, J. Math. Sci. (N. Y.), 167:5 (2010), 692–712
2. V. A. Kozlov, S. A. Nazarov, “The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary”, St. Petersburg Math. J., 22:6 (2011), 941–983
3. Nazarov S.A., Sokolowski J., Specovius-Neugebauer M., “Polarization matrices in anisotropic heterogeneous elasticity”, Asymptot. Anal., 68:4 (2010), 189–221
4. Denis Borisov, Giuseppe Cardone, Luisa Faella, Carmen Perugia, “Uniform resolvent convergence for strip with fast oscillating boundary”, Journal of Differential Equations, 255:12 (2013), 4378–4402
5. S. A. Nazarov, “Nonreflecting distortions of an isotropic strip clamped between rigid punches”, Comput. Math. Math. Phys., 53:10 (2013), 1512–1522
6. S. A. Nazarov, “Bounded solutions in a $\mathrm{T}$-shaped waveguide and the spectral properties of the Dirichlet ladder”, Comput. Math. Math. Phys., 54:8 (2014), 1261–1279
7. Cardone G., “Waveguides With Fast Oscillating Boundary”, Nanosyst.-Phys. Chem. Math., 8:2 (2017), 160–165
8. Gomez D. Nazarov S.A. Perez M.E., “Homogenization of Winkler-Steklov Spectral Conditions in Three-Dimensional Linear Elasticity”, Z. Angew. Math. Phys., 69:2 (2018), 35
9. S. A. Nazarov, “Osrednenie plastin Kirkhgofa, soedinennykh zaklepkami, kotorye modeliruyutsya tochechnymi usloviyami Soboleva”, Algebra i analiz, 32:2 (2020), 143–200
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