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Russia Ufa; Yekaterinburg
Year
2023
Volume
33
Issue
1
Pages
54-65
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Section Mathematics
Title On eigenelements of a two-dimensional Steklov-type boundary value problem for the Lamé operator
Author(-s) Davletov D.B.ab, Davletov O.B.c, Davletova R.R.d, Ershov A.A.ef
Affiliations Bashkir State Pedagogical Universitya, Ufa University of Science and Technologyb, Ufa State Petroleum Technological Universityc, Financial University under the Government of the Russian Federation, Ufa Branchd, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencese, Ural Federal Universityf
Abstract In this paper, we study a two-dimensional Steklov-type boundary value problem for the Lamé operator in a half-strip, which is the limiting problem for a singularly perturbed boundary-value problem in a half-strip with a small hole. A theorem on the existence of eigenelements of the boundary value problem under study is proved. In particular, we obtain estimates for the eigenvalues expressed in terms of the Lamé constants and a parameter that determines the width of the half-strip, and refine the structure of the corresponding eigenfunctions, which determines their behavior as their argument move away from the base of the half-strip. Moreover, explicit expressions for the eigenvalues of the limiting boundary value problem are found up to the solution of a system of algebraic equations. The results obtained in this paper will make it possible to construct and rigorously justify an asymptotic expansion of the eigenvalue of a singularly perturbed boundary value problem in a half-strip with a small round hole in powers of a small parameter that determines the diameter of the hole.
Keywords boundary value problem, Steklov spectral condition, Lamé operator, eigenelements
UDC 517.929.7, 517.929.8, 517.984
MSC 35J25, 35P20
DOI 10.35634/vm230104
Received 13 September 2022
Language Russian
Citation Davletov D.B., Davletov O.B., Davletova R.R., Ershov A.A. On eigenelements of a two-dimensional Steklov-type boundary value problem for the Lamé operator, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 1, pp. 54-65.
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