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 Izv. RAN. Ser. Mat., 1996, Volume 60, Issue 1, Pages 3–36 (Mi izv60)

Asymptotic analysis of problems on junctions of domains of different limit dimensions. A body pierced by a thin rod

I. I. Argatov, S. A. Nazarov

Abstract: We consider the junction problem on the union of two bodies: a thin cylinder $Q_\varepsilon$ and a massive body $\Omega(\varepsilon)$ with an opening into which this cylinder has been inserted. The equations on $Q_\varepsilon$ and $\Omega(\varepsilon)$ contain the operators $\mu\Delta$ and $\Delta$ (where $\mu =\mu (\varepsilon)$ is a large parameter and $\Delta$ is the Laplacian): Dirichlet conditions are imposed on the ends of $Q_\varepsilon$ and Neumann conditions on the remainder of the exterior boundary. We study the asymptotic behaviour of a solution $\{u_Q,u_\Omega\}$ as $\varepsilon\to+0$. The principal asymptotic formulae are as follows: $u_Q\sim w$ on $Q_\varepsilon$ and $u_\Omega\sim v$ on $\Omega(\varepsilon)$, where $v$ is a solution of the Neumann problem in $\Omega$ and the Dirac function is distributed along the interval $\Omega\setminus\Omega(0)$ with density $\gamma$. The functions $w$ and $\gamma$, depending on the axis variable of the cylinder, are found as solutions of a so-called resulting problem, in which a second-order differential equation and an integral equation (principal symbol of the operator $(2\pi)^{-1}\ln|\xi|$) are included. In the resulting problem the large parameter $\lvert\ln\varepsilon\rvert$ remains. Various methods of constructing its asymptotic solutions are discussed. The most interesting turns out to be the case $\mu(\varepsilon)=O(\varepsilon^{-2}\lvert\ln\varepsilon\rvert^{-1})$) (even the principal terms of the functions $w$ and $\gamma$ are not found separately). All the asymptotic formulae are justified; the remainders are estimated in the energy norm.

DOI: https://doi.org/10.4213/im60

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English version:
Izvestiya: Mathematics, 1996, 60:1, 1–37

Bibliographic databases:

MSC: Primary 35J25, 35B40, 73C35; Secondary 35A35, 35C10

Citation: I. I. Argatov, S. A. Nazarov, “Asymptotic analysis of problems on junctions of domains of different limit dimensions. A body pierced by a thin rod”, Izv. RAN. Ser. Mat., 60:1 (1996), 3–36; Izv. Math., 60:1 (1996), 1–37

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Aldoshina I.A., Nazarov S.A., “Asymptotically exact joining conditions at the junction of plates with very different characteristic”, Pmm Journal of Applied Mathematics and Mechanics, 62:2 (1998), 253–261
2. S. A. Nazarov, “The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes”, Russian Math. Surveys, 54:5 (1999), 947–1014
3. V. G. Maz'ya, A. B. Movchan, “Dynamic singular perturbation problems for multi-structures”, Appl Stochastic Models Bus Ind, 16:4 (2000), 249
4. S. A. Nazarov, “Weighted anisotropic Korn's inequality for a junction of a plate and a rod”, Sb. Math., 195:4 (2004), 553–583
5. S. A. Nazarov, J. Sokolowski, “The topological derivative of the Dirichlet integral under formation of a thin ligament”, Siberian Math. J., 45:2 (2004), 341–355
6. I. I. Argatov, “Asymptotics of the reduced logarithmic capacity of a narrow cylinder”, Math. Notes, 77:1 (2005), 15–25
7. Movchan A.B., “Multi-structures: asymptotic analysis and singular perturbation problems”, European Journal of Mechanics A-Solids, 25:4 (2006), 677–694
8. Andrianov I.V., Awrejcewicz J., Weichert D., “Load-Transfer from an Elastic Fibre to Isotropic Half-Space with Coating”, Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems - State-of-the-Art, Perspectives and Applications, 2009, 1–11
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