Abstract
A rigorous method for constructing asymptotic approximations in thin domains was first proposed by Gol’denveizer (The Theory of Elastic Thin Shells, Nauka, Moscow, 1976; Prikl. Mat. Meh. 26(4):668–686, 1962); it was further developed for thin domains of cylindrical type in (Dzhavadov, Differ. Urav. 4(10):1901–1909, 1968; Caillerie, Math. Math. Appl. Sci. 6:159–191, 1984; Vasil’eva and Butuzov, Asymptotic Methods in the Theory of Singular Perturbations, Vyssh. Shkola, Moscow, 1990; Nazarov, Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron. 2:65–68, 1982). These authors considered thin domains of cylindrical type and the main approach to asymptotic analysis was to make a special change of coordinates after which the scaled domain was independent of the small parameter. Then a small parameter appeared in the higher derivatives of the differential equations and the Lyusternik–Vishik method (Vishik and Lyusternik, Usp. Mat. Nauk 12(5):3–192, 1957) was used.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Gol’denveizer, A. L.: The Theory of Elastic Thin Shells, Nauka, Moscow (1976)
Gol’denveizer, A.L.: Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity, Prikl. Mat. Meh. 26, No. 4, pp. 668–686 (1962);
Dzhavadov, M.G.: Asymptotics of solutions of a boundary-value problem for second-order elliptic equations in thin domains, Differ. Urav. 4, No. 10, pp. 1901–1909 (1968);
Caillerie, D.: Thin elastic and periodic plates, Math. Math. Appl. Sci. 6, pp. 159–191 (1984)
Vasil’eva, A.B., Butuzov, V.F., Asymptotic Methods in the Theory of Singular Perturbations, Vyssh. Shkola, Moscow (1990)
Nazarov, S.A., The structure of the solutions of elliptic boundary value problems in thin domains, Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron., No. 2, pp. 65–68 (1982)
Vishik, M.I., Lyusternik, L.A.: Regular degeneralization and boundary layer for linear differential equations with parameter, Usp. Mat. Nauk 12, No. 5, 3–192 (1957)
Panasenko, G.P., Reztsov, M.V.: Averaging a three-dimensional problem of elasticity theory in an inhomogeneous plate, Dokl. Akad. SSSR 294, No. 5, 1061–1065 (1987);
Mel’nik, T.A.: Averaging of elliptic equations describing processes in strongly inhomogeneous thin perforated domains with rapidly changing thickness, Akad. Nauk Ukr. SSR 10, 15–18 (1991)
Mel’nyk, T.A.: Asymptotic expansions of eigenvalues and eigenfunctions for elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube, Tr. Semin. Im. Petrovskogo 17, 51–88 (1994);
Akimova, E.A., Nazarov, S.A., Chechkin, G.A.: Asymptotics of the solution of the problem of deformation of an arbitrary locally periodic thin plate, Tr. Mosk. Mat. O.-va 65, 3–34 (2004);
Cioranescu, D., Chechkin, G.A.: Vibration of a thin plate with a ’rough’ surface In: Nonlinear Partial Differential Equations and their Applications. Coll‘ege de France Seminar. Volume XIV. Studies in Mathematics and its Applications, Elsevier, Amsterdam etc. pp. 147–169 (2002)
Nazarov, S.A.: Asymptotic Analysis of Thin Plates and Bars, Vol. 1, Nauchnaya Kniga, Novosibirsk (2002)
Isakov, R.V: Asymptotics of the spectral series of the Steklov problem for the Laplace equation in a ’thin’ domain with nonsmooth boundary, Mat. Zametki, 44:5, pp. 694–696 (1988)
Kolpakov, A.G.: The governing equations of a thin elastic stressed beam with a periodic structure, Prikl. Mat. Mekh. 63, No. 3, 513–523 (1999);
Korn, R.V., Vogelius, V.: A new model for thin plates with rapidly varying thickness.II: A convergence proof, Quart. Appl. Math. 18, No. 1, 1–22, (1985)
Oleinik, O.A., Shamaev, A.S., Yosifyan, G.A.: Mathematical Problems in the Theory of Strongly Inhomogeneous Elastic Media, Moscow Univ. Press, Moscow (1990)
Bakhvalov, N.S., Panasenko, G.P.: Homogenization of Processes in Periodic Media, Nauka, Moscow (1984)
Mel’nyk, T.A., Popov, A.V.: Asymptotic analysis of the Dirichlet spectral problems in thin perforated domains with rapidly varying thickness and different limit dimensions. In: Roderick V. N. Melnik, Alexandra V. Antoniouk (eds.) Mathematics and Life Sciences, pp. 90–109. De Gruyter, Berlin. 89–107 (2012)
Mel’nyk, T.A., Popov, A.V.: Asymptotic analysis of boundary-value and spectral problems in thin perforated regions with rapidly changing thickness and different limiting dimensions, Matem. Sbornik, 203:8, 97–124 (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Popov, A. (2015). Asymptotic Analysis of the Steklov Spectral Problem in Thin Perforated Domains with Rapidly Varying Thickness and Different Limit Dimensions. In: Constanda, C., Kirsch, A. (eds) Integral Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-16727-5_41
Download citation
DOI: https://doi.org/10.1007/978-3-319-16727-5_41
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-16726-8
Online ISBN: 978-3-319-16727-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)