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Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2019, Volume 162, Pages 3–14 (Mi into436)  

On lacunas in the spectrum of the Laplacian with the Dirichlet boundary condition in a strip with oscillating boundary

D. I. Borisovabc

a Institute of Mathematics with Computing Centre — Subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa
b Bashkir State Pedagogical University, Ufa
c University of Hradec Králové

Abstract: In this paper, we consider the Laplace operator in a flat strip whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.

Keywords: Bethe–Sommerfeld hypothesis, Laplacian, strip, oscillating boundary

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00046_а
This work was supported by the Russian Foundation for Basic Research (project No. 18-01-00046).


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Bibliographic databases:
UDC: 517.958, 517.984, 519.21
MSC: 35P05, 47A10, 35B27

Citation: D. I. Borisov, “On lacunas in the spectrum of the Laplacian with the Dirichlet boundary condition in a strip with oscillating boundary”, Complex Analysis. Mathematical Physics, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 162, VINITI, Moscow, 2019, 3–14

Citation in format AMSBIB
\Bibitem{Bor19}
\by D.~I.~Borisov
\paper On lacunas in the spectrum of the Laplacian with the Dirichlet boundary condition in a strip with oscillating boundary
\inbook Complex Analysis. Mathematical Physics
\serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz.
\yr 2019
\vol 162
\pages 3--14
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/into436}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3981812}


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