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 Mat. Sb., 2005, Volume 196, Number 5, Pages 83–120 (Mi msb1359)

Asymptotics of the eigenelements of the Laplacian with singular perturbations of boundary conditions on narrow and thin sets

M. Yu. Planida

Bashkir State Pedagogical University

Abstract: Perturbations of the three-dimensional Dirichlet problem in a bounded domain are studied. One type of perturbation is change of type of the boundary condition on a narrow strip contracting to a closed curve on the boundary. The second type of perturbation is effected by cutting out in the domain a thin ‘toroidal’ body, also contracting to a closed curve (but now contained inside the domain) and imposing a Neumann boundary condition at the boundary of this thin body. For these problems the method of matched asymptotic expansions is used to construct complete asymptotics (in a small parameter) of the eigenvalues, converging to the simple eigenvalues of the unperturbed problem, and of the corresponding eigenfunctions. The small parameter is the width of the strip and the diameter of a section of the torus, respectively.

DOI: https://doi.org/10.4213/sm1359

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English version:
Sbornik: Mathematics, 2005, 196:5, 703–741

Bibliographic databases:

UDC: 517.956
MSC: Primary 35P05; Secondary 35B20, 35B40, 35J25

Citation: M. Yu. Planida, “Asymptotics of the eigenelements of the Laplacian with singular perturbations of boundary conditions on narrow and thin sets”, Mat. Sb., 196:5 (2005), 83–120; Sb. Math., 196:5 (2005), 703–741

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb1359
• https://doi.org/10.4213/sm1359
• http://mi.mathnet.ru/eng/msb/v196/i5/p83

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This publication is cited in the following articles:
1. Gadyl'shin R.R., Il'in A.M., “On the spectrum of the Neumann problem for Laplace equation in a domain with a narrow slit”, Asymptotic Anal., 67:3-4 (2010), 167–189
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