Geodesic
Spectral properties of a thin layer with a doubly periodic family of thinning regions
Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 3, pp. 398-415 Cet article a éte moissonné depuis la source Math-Net.Ru

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We show that the spectrum of the Dirichlet problem for the Laplace operator in a layer with a doubly periodic structure has gaps and determine several characteristics of their location. The result is obtained by asymptotic analysis of a model spectral problem on the periodicity cell.
Keywords: Dirichlet problem in a doubly periodic layer, asymptotic behavior, eigenvalue localization, spectral gap. 
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S. A. Nazarov. Spectral properties of a thin layer with a doubly periodic family of thinning regions. Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 3, pp. 398-415. http://geodesic.mathdoc.fr/item/TMF_2013_174_3_a3/

[1] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Ucheb. posobie, Izd-vo LGU, L., 1980 | MR | MR | Zbl

[2] P. A. Kuchment, UMN, 37:4(226) (1982), 3–52 | DOI | MR | Zbl

[3] M. M. Skriganov, Geometricheskie i arifmeticheskie metody v spektralnoi teorii mnogomernykh periodicheskikh operatorov, v. 171, Trudy MIAN, Nauka, L., 1985 | Zbl

[4] P. Kuchment, Floquet Theory for Partial Differential Equations, Operator Theory: Advances and Applications, 60, Birchäuser, Basel, 1993 | MR | Zbl

[5] I. M. Gelfand, Dokl. AN SSSR, 73 (1950), 1117–1120 | MR | Zbl

[6] O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR | MR | Zbl

[7] I. V. Kamotskii, S. A. Nazarov, “O sobstvennykh funktsiyakh, lokalizovannykh okolo kromki tonkoi oblasti”, Problemy matem. analiza, vyp. 19, Nauchn. kniga, Novosibirsk, 1999, 105–148

[8] S. A. Nazarov, Math. Bohem., 127:2 (2002), 283–292 | MR | Zbl

[9] G. Cardone, T. Durante, S. A. Nazarov, SIAM J. Math. Anal., 42:6 (2010), 2581–2609 | DOI | MR | Zbl

[10] L. Friedlander, M. Solomyak, Russ. J. Math. Phys., 15:2 (2008), 238–242 | DOI | MR | Zbl

[11] L. Friedlander, M. Solomyak, Israel J. Math., 170:1 (2009), 337–354 | DOI | MR | Zbl

[12] K. Yoshitomi, J. Differ. Equations, 142:1 (1998), 123–166 | DOI | MR | Zbl

[13] V. Kozlov, J. Differ. Equations, 230:2 (2006), 532–555 | DOI | MR | Zbl

[14] J. Hadamard, “Mémoire sur le problème d'analyse relatifà l'équilibre des plaques èlastiques encastrés”, {ØE}uvres, v. 2, Editions du Centre National de la Recherche Scientifique, Paris, 1968, 515–631 | MR | Zbl

[15] L. A. Ivanov, L. A. Kotko, S. G. Krein, Kraevye zadachi v peremennykh oblastyakh, Differentsialnye uravneniya i ikh primenenie, 19, In-t matematiki i kibernetiki AN LitSSR, Vilnyus, 1977

[16] M. I. Vishik, L. A. Lyusternik, UMN, 12:5(77) (1957), 3–122 | MR | Zbl

[17] S. A. Nazarov, Asimptoticheskaya teoriya tonkikh plastin i sterzhnei. Ponizhenie razmernosti i integralnye otsenki, Nauchn. kniga, Novosibirsk, 2002 | Zbl

[18] W. G. Mazja, S. A. Nazarov, B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten, v. 1, Mathematische Lehrbücher und Monographien, 82, Akademie, Berlin, 1991 | MR | Zbl

[19] S. A. Nazarov, M. V. Olyushin, Algebra i analiz, 5:2 (1993), 169–188 | MR | Zbl

[20] S. A. Nazarov, “Ob optimizatsii zaplaty”, Problemy matem. analiza, vyp. 42, Nauchn. kniga, Novosibirsk, 2009, 65–82

[21] S. A. Nazarov, K. Ruotsalainen, Ya. Taskinen, “Spektralnye lakuny v zadachakh Dirikhle i Neimana na ploskosti, perforirovannoi dvoyakoperiodicheskim semeistvom krugovykh otverstii”, Problemy matem. analiza, vyp. 62, Nauchn. kniga, Novosibirsk, 2011, 51–100