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Mat. Sb., 2010, Volume 201, Number 4, Pages 99–124 (Mi msb7547)  

This article is cited in 14 scientific papers (total in 14 papers)

An example of multiple gaps in the spectrum of a periodic waveguide

S. A. Nazarov

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: A periodic waveguide is constructed, whose shape depends on a small parameter $h>0$, in which the essential spectrum of the operators for some boundary-value problems (Dirichlet, Neumann, and mixed under certain restrictions) for a formally self-adjoint elliptic system of second-order differential equations acquires any pre-assigned number of gaps. The geometric shape of the waveguide can be interpreted as an infinite periodic set of beads connected by thin, short ligaments. The proof of that gaps appear is based on an application of the max-min principle and the weighted Korn inequality.
Bibliography: 43 titles.

Keywords: gaps in the essential spectrum, formally self-adjoint elliptic system of differential equations with polynomial property.


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English version:
Sbornik: Mathematics, 2010, 201:4, 569–594

Bibliographic databases:

UDC: 517.956.227+517.958+631.372.8
MSC: Primary 35J57; Secondary 35J20, 35P15
Received: 04.03.2009

Citation: S. A. Nazarov, “An example of multiple gaps in the spectrum of a periodic waveguide”, Mat. Sb., 201:4 (2010), 99–124; Sb. Math., 201:4 (2010), 569–594

Citation in format AMSBIB
\by S.~A.~Nazarov
\paper An example of multiple gaps in the spectrum of a~periodic waveguide
\jour Mat. Sb.
\yr 2010
\vol 201
\issue 4
\pages 99--124
\jour Sb. Math.
\yr 2010
\vol 201
\issue 4
\pages 569--594

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    This publication is cited in the following articles:
    1. S. A. Nazarov, “On the spectrum of the Laplace operator on the infinite Dirichlet ladder”, St. Petersburg Math. J., 23:6 (2012), 1023–1045  mathnet  crossref  mathscinet  isi  elib  elib
    2. S. A. Nazarov, J. Taskinen, “Structure of the spectrum of the periodic family of identical cells connected through apertures of reducing sizes”, J. Math. Sci. (N. Y.), 194:1 (2013), 72–82  mathnet  crossref  mathscinet
    3. Nazarov S.A., Slutskij A.S., Sweers G.H., “Korn inequalities for a reinforced plate”, J. Elasticity, 106:1 (2012), 43–69  crossref  mathscinet  zmath  isi  elib  scopus
    4. D. I. Borisov, K. V. Pankrashin, “Gap Opening and Split Band Edges in Waveguides Coupled by a Periodic System of Small Windows”, Math. Notes, 93:5 (2013), 660–675  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. Bakharev F.L., Nazarov S.A., Ruotsalainen K.M., “A Gap in the Spectrum of the Neumann-Laplacian on a Periodic Waveguide”, Appl. Anal., 92:9 (2013), 1889–1915  crossref  mathscinet  zmath  isi  elib  scopus
    6. S. A. Nazarov, “Gap opening around a given point of the spectrum of a cylindrical waveguide by means of gentle periodic perturbation of walls”, J. Math. Sci. (N. Y.), 206:3 (2015), 288–314  mathnet  crossref
    7. A. Khrabustovskyi, “Opening up and control of spectral gaps of the Laplacian in periodic domains”, J. Math. Phys., 55:12 (2014), 121502  crossref  mathscinet  zmath  isi  elib  scopus
    8. S. A. Nazarov, J. Taskinen, “Spectral gaps for periodic piezoelectric waveguides”, Z. Angew. Math. Phys., 66:6 (2015), 3017–3047  crossref  mathscinet  zmath  isi  elib  scopus
    9. S. A. Nazarov, “Eigenmodes of a thin elastic layer between periodic rigid profiles”, Comput. Math. Math. Phys., 55:10 (2015), 1684–1697  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    10. Nazarov S.A., Taskinen J., “Elastic and piezoelectric waveguides may have infinite number of gaps in their spectra”, C. R. Mec., 344:3 (2016), 190–194  crossref  isi  scopus
    11. Borisov D.I., “Creation of spectral bands for a periodic domain with small windows”, Russ. J. Math. Phys., 23:1 (2016), 19–34  crossref  mathscinet  zmath  isi  scopus
    12. Cardone G. Khrabustovskyi A., “Spectrum of a Singularly Perturbed Periodic Thin Waveguide”, J. Math. Anal. Appl., 454:2 (2017), 673–694  crossref  mathscinet  zmath  isi  scopus
    13. Delourme B., Fliss S., Joly P., Vasilevskaya E., “Trapped Modes in Thin and Infinite Ladder Like Domains. Part 1: Existence Results”, Asymptotic Anal., 103:3 (2017), 103–134  crossref  mathscinet  zmath  isi  scopus
    14. Cardone G., Khrabustovskyi A., “Delta `-Interaction as a Limit of a Thin Neumann Waveguide With Transversal Window”, J. Math. Anal. Appl., 473:2 (2019), 1320–1342  crossref  mathscinet  zmath  isi  scopus
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