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 Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 6, Pages 37–64 (Mi izv8674)

On the asymptotic behaviour of eigenvalues of a boundary-value problem in a planar domain of Steklov sieve type

R. R. Gadyl'shinab, A. L. Piatnitskicd, G. A. Chechkine

a Bashkir State Pedagogical University, Ufa
b Bashkir State University, Ufa
c Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
d The Arctic University of Norway, Narvik, Norway
e Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider a two-dimensional spectral problem of Steklov type for the Laplace operator in a domain divided into two parts by a perforated partition with a periodic microstructure. The Steklov boundary condition is imposed on the lateral sides of the perforation, the Neumann condition on the remaining part of the boundary, and the Dirichlet and Neumann conditions on the outer boundary of the domain. We construct and justify two-term asymptotic expressions for the eigenvalues of this problem. We also construct a two-term asymptotic formula for the corresponding eigenfunctions.

Keywords: asymptotic behaviour of eigenvalues, spectral problem, Steklov problem, homogenization of spectral problems.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00046 The investigation of the third author was carried out with the support of the Russian Foundation for Basic Research (grant no. 18-01-00046).

DOI: https://doi.org/10.4213/im8674

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English version:
Izvestiya: Mathematics, 2018, 82:6, 1108–1135

Bibliographic databases:

UDC: 517.956.226
MSC: Primary 35B27; Secondary 35C20, 35J05, 35J25
Revised: 23.02.2018

Citation: R. R. Gadyl'shin, A. L. Piatnitski, G. A. Chechkin, “On the asymptotic behaviour of eigenvalues of a boundary-value problem in a planar domain of Steklov sieve type”, Izv. RAN. Ser. Mat., 82:6 (2018), 37–64; Izv. Math., 82:6 (2018), 1108–1135

Citation in format AMSBIB
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