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.
Citation:
S. A. Nazarov, “Asymptotics of solutions and modelling the problems of elasticity theory in domains with rapidly oscillating boundaries”, Izv. Math., 72:3 (2008), 509–564
\Bibitem{Naz08}
\by S.~A.~Nazarov
\paper Asymptotics of solutions and modelling the problems of elasticity theory in domains with rapidly oscillating boundaries
\jour Izv. Math.
\yr 2008
\vol 72
\issue 3
\pages 509--564
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Linking options:
https://www.mathnet.ru/eng/im2600
https://doi.org/10.1070/IM2008v072n03ABEH002410
https://www.mathnet.ru/eng/im/v72/i3/p103
This publication is cited in the following 19 articles:
D. I. Borisov, R. R. Suleimanov, “Operator estimates for elliptic equations in multidimensional domains with strongly curved boundaries”, Sb. Math., 216:1 (2025), 25–53
S. A. Nazarov, J. Taskinen, “Model of a Plane Strain-State of a Two-Dimensional Plate with Small Periodic Areas of Fixed Edge”, J Math Sci, 283:4 (2024), 586
D. Gómez, S. A. Nazarov, M.-E. Pérez-Martínez, “Pointwise Fixation along the Edge of a Kirchhoff Plate”, J Math Sci, 277:4 (2023), 545
Giuseppe Cardone, Sergey A. Nazarov, Jari Taskinen, “Asymptotic Expansions of Solutions to the Poisson Equation with Alternating Boundary Conditions on an Open Arc”, SIAM J. Math. Anal., 55:6 (2023), 6940
S. A. Nazarov, “Parasitic eigenvalues of spectral problems for the Laplacian with third-type boundary conditions”, Comput. Math. Math. Phys., 63:7 (2023), 1237–1253
S. A. Nazarov, Ya. Taskinen, “Model ploskogo deformirovannogo sostoyaniya dvumernoi plastiny s melkimi pochti periodicheskimi uchastkami zaschemleniya kraya”, Matematicheskie voprosy teorii rasprostraneniya voln. 51, Zap. nauchn. sem. POMI, 506, POMI, SPb., 2021, 130–174
S. A. Nazarov, “Homogenization of Kirchhoff plates joined by rivets which are modeled by the Sobolev point conditions”, St. Petersburg Math. J., 32:2 (2021), 307–348
D. Gomes, S. A. Nazarov, M.-E. Peres, “Tochechnoe kreplenie plastiny Kirkhgofa vdol ee kromki”, Matematicheskie voprosy teorii rasprostraneniya voln. 50, Posvyaschaetsya devyanostoletiyu Vasiliya Mikhailovicha BABIChA, Zap. nauchn. sem. POMI, 493, POMI, SPb., 2020, 107–137
Gomez D., Nazarov S.A., Perez-Martinez M.-E., “Asymptotics For Spectral Problems With Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation”, J. Elast., 142:1 (2020), 89–120
Delfina Gómez, Sergey A. Nazarov, Maria-Eugenia Pérez-Martínez, Computational and Analytic Methods in Science and Engineering, 2020, 127
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
Cardone G., “Waveguides With Fast Oscillating Boundary”, Nanosyst.-Phys. Chem. Math., 8:2 (2017), 160–165
S. A. Nazarov, “Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder”, Comput. Math. Math. Phys., 54:8 (2014), 1261–1279
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
S. A. Nazarov, “Nonreflecting distortions of an isotropic strip clamped between rigid punches”, Comput. Math. Math. Phys., 53:10 (2013), 1512–1522
D. Gómez, S. A. Nazarov, E. Pérez, Integral Methods in Science and Engineering, 2011, 159
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
Nazarov S.A., Sokolowski J., Specovius-Neugebauer M., “Polarization matrices in anisotropic heterogeneous elasticity”, Asymptot. Anal., 68:4 (2010), 189–221
S. A. Nazarov, “Asymptotic modeling of a problem with contrasting stiffness”, J. Math. Sci. (N. Y.), 167:5 (2010), 692–712
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