Asymptotic Expansions of Eigenfunctions and Eigenvalues of the Steklov Spectral Problem in Thin Perforated Domains with Rapidly Varying Thickness and Different Limit Dimensions
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We consider a Steklov spectral problem for an elliptic differential equation with rapidly oscillating coefficients for thin perforated domains with rapidly varying thickness. We describe asymptotic algorithms for the solution of problems of this kind for thin perforated domains with different limit dimensions. We also establish asymptotic estimates for eigenvalues of the Steklov spectral problem for thin perforated domains with different limit dimensions. For certain symmetry conditions imposed on the structure of thin perforated domain and on the coefficients of differential operators, we construct and substantiate asymptotic expansions for eigenfunctions and eigenvalues.
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- 5.A. L. Gol’denveizer, Theory of Elastic Thin Shells [in Russian], Nauka, Moscow (1976).Google Scholar
- 9.T. A. Mel’nyk, “Averaging of elliptic equations that describe processes in strongly inhomogeneous thin perforated domains with rapidly varying thickness,” Dop. Akad. Nauk Ukr., No. 10, 15–19 (1991).Google Scholar
- 10.T. A. Mel’nik, “Asymptotic expansions of the eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube,” in: Proc. of the Pertovskii Seminar [in Russian], Issue 17 (1994), pp. 51–88.Google Scholar
- 12.S. A. Nazarov, Asymptotic Analysis of Thin Plates and Rods [in Russian], Vol. 1, Nauchnaya Kniga, Novosibirsk (2002).Google Scholar
- 13.S. A. Nazarov, “General scheme of averaging of self-adjoint elliptic systems in multidimensional domains including thin domains,” Algebra Analiz, 7, No. 5, 1–92 (1995).Google Scholar
- 14.S. A. Nazarov, “Structure of solutions of the boundary-value problems for elliptic equations in thin domains,” Vestn. Leningrad. Univ., Ser. Mat., Mekh. Astronom., Issue 2, 65–68 (1982).Google Scholar
- 15.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, Moscow (1990).Google Scholar
- 22.T. A. Mel’nyk and A. V. Popov, “Asymptotic analysis of the Dirichlet spectral problems in thin perforated domains with rapidly varying thickness and different limit dimensions,” in: Mathematics and Life Sciences, De Gruyter, Berlin (2012), pp. 89–109.Google Scholar