For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15–19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. B. Vasil’eva, and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations [in Russian], Vysshaya Shkola, Moscow (1990).
A. L. Gol’denveizer, Theory of Elastic Thin Shells [in Russian], Nauka, Moscow (1976).
S. A. Nazarov, Asymptotic Analysis of Thin Plates and Bars [in Russian], Vol. 1, Nauchnaya Kniga, Novosibirsk (2002).
D. Caillerie, “Thin elastic and periodic plates,” Math. Meth. Appl. Sci., 6, 159–191 (1984).
T. Lewinsky and J. Telega, “Plates, laminates and shells,” in: Asymptotic Analysis and Homogenization, World Scientific, Singapore (2000).
A. L. Gol’denveizer, “Construction of an approximate theory of bending of a plate by the method of asymptotic integration of equations of the theory of elasticity,” Prikl. Mat. Mekh., 26, No. 4, 668–686 (1962).
M. G. Dzhavadov, “Asymptotics of a solution of the boundary-value problem for elliptic equations of the second order in thin domains,” Differents. Uravn., 4, No. 10, 1901–1909 (1968).
S. A. Nazarov, “Structure of solutions of the boundary-value problem for elliptic equations in thin domains,” Vestn. Leningrad. Univ., Ser. Mat. Mekh., Astronom., Issue 2, 65–68 (1982).
S. N. Leora, S. A. Nazarov, and A. V. Proskura, “Computer-aided derivation of limit equations for elliptic problems in thin domains,” Zh. Vychisl. Mat. Mat. Fiz., 26, No. 7, 1032–1048 (1986).
S. A. Nazarov, “General scheme of self-adjoint elliptic systems in multidimensional domains,” Alg. Analiz, 7, No. 5, 1–92 (1995).
P. Ciarlet and S. Kesavan, “Two-dimensional approximations of three-dimensional eigenvalue problem in plate theory,” Comput. Meth. Appl. Mech. Eng., 26, 145–172 (1981).
R.V. Korn and M. Vogelius, “A new model for thin plates with rapidly varying thickness. II: A convergence proof,” Quart. Appl. Math., 18, No. 1, 1–22 (1985).
M. I. Vishik and L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with parameter,” Usp. Mat. Nauk, 12, No. 5, 3–192 (1957).
G. P. Panasenko and M. V. Reztsov, “Homogenization of a three-dimensional problem of the theory of elasticity in an inhomogeneous plate,” Dokl. Akad. Nauk SSSR, 294, No. 5, 1061–1065 (1987).
T. A. Mel’nik, “Homogenization of elliptic equations that describe processes in strongly inhomogeneous thin perforated domains with rapidly varying thickness,” Dopov. Akad. Nauk Ukr., No. 10, 15–19 (1991).
A. G. Kolpakov, “Defining equations of a thin elastic stressed bar of periodic structure,” Prikl. Mat. Mekh., 63, Issue 3, 513–523 (1999).
G. A. Chechkin and E. A. Pichugina, “Weighted Korn’s inequality for a thin plate with a rough surface,” Russian J. Math. Phys., 7, No. 3, 375–383 (2000).
O. A. Oleinik, G. A. Iosif’yan, and A. S. Shamaev, Mathematical Problems of the Theory of Strongly Inhomogeneous Elastic Media [in Russian], Moscow University (1990).
N. S. Bakhvalov and G. P. Panasenko, Homogenization of Processes in Periodic Media [in Russian], Moscow, Nauka (1984).
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1983).
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the Boundary for Solutions of Elliptic Partial Difference Equations Satisfying General Boundary Conditions. I, Interscience, New York (1959).
T. A. Mel’nik, “Asymptotic expansions of eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube,” Tr. Sem. Im. I. G. Petrovskogo, Issue 17, 51–88 (1994).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 13, No. 1, pp. 50–74, January–March, 2010.
Rights and permissions
About this article
Cite this article
Mel’nyk, T.A., Popov, A.V. Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness. Nonlinear Oscill 13, 57–84 (2010). https://doi.org/10.1007/s11072-010-0101-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11072-010-0101-5