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

This article is cited in 9 scientific papers (total in 9 papers)

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.


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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
Received: 19.12.2006

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
\by S.~A.~Nazarov
\paper Asymptotics of solutions and modelling the problems of elasticity theory in domains with rapidly oscillating boundaries
\jour Izv. RAN. Ser. Mat.
\yr 2008
\vol 72
\issue 3
\pages 103--158
\jour Izv. Math.
\yr 2008
\vol 72
\issue 3
\pages 509--564

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    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  mathnet  crossref  elib
    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  mathnet  crossref  mathscinet  zmath  isi
    3. Nazarov S.A., Sokolowski J., Specovius-Neugebauer M., “Polarization matrices in anisotropic heterogeneous elasticity”, Asymptot. Anal., 68:4 (2010), 189–221  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    5. S. A. Nazarov, “Nonreflecting distortions of an isotropic strip clamped between rigid punches”, Comput. Math. Math. Phys., 53:10 (2013), 1512–1522  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. Cardone G., “Waveguides With Fast Oscillating Boundary”, Nanosyst.-Phys. Chem. Math., 8:2 (2017), 160–165  crossref  mathscinet  isi
    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  crossref  mathscinet  zmath  isi  scopus
    9. S. A. Nazarov, “Osrednenie plastin Kirkhgofa, soedinennykh zaklepkami, kotorye modeliruyutsya tochechnymi usloviyami Soboleva”, Algebra i analiz, 32:2 (2020), 143–200  mathnet
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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