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A gap in the spectrum of the Neumann-Laplacian on a periodic waveguide. (English) Zbl 1302.35266
Summary: We study a spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder which contains a periodic arrangement of inclusions. On the boundary of the waveguide, we consider both Neumann and Dirichlet conditions. We prove that provided the diameter of the inclusion is small enough the spectrum of Laplace operator contains band gaps, i.e. there are frequencies that do not propagate through the waveguide. The existence of the band gaps is verified using the asymptotic analysis of elliptic operators.

MSC:
35P05 General topics in linear spectral theory for PDEs
34E05 Asymptotic expansions of solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
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References:
[1] Wilcox CH, Scattering Theory for Diffraction Gratings (1979)
[2] DOI: 10.1007/978-94-009-4586-9 · doi:10.1007/978-94-009-4586-9
[3] Gelfand IM, Dokl. Acad. Nauk SSSR 73 pp 1117– (1950)
[4] DOI: 10.1007/978-3-0348-8573-7 · doi:10.1007/978-3-0348-8573-7
[5] Nazarov SA, Elliptic Problems in Domains With Piecewise Smooth Boundaries (1991)
[6] DOI: 10.1137/S0036139994263859 · Zbl 0852.35014 · doi:10.1137/S0036139994263859
[7] DOI: 10.1081/PDE-120002790 · Zbl 1055.35083 · doi:10.1081/PDE-120002790
[8] DOI: 10.1006/jdeq.1996.3204 · Zbl 0924.35087 · doi:10.1006/jdeq.1996.3204
[9] DOI: 10.1080/03605300008821555 · Zbl 0958.35089 · doi:10.1080/03605300008821555
[10] Zhikov VV, Algebra i Analiz 16 (5) pp 34– (2004)
[11] DOI: 10.1007/s00220-003-0904-7 · Zbl 1037.35051 · doi:10.1007/s00220-003-0904-7
[12] DOI: 10.1134/S0012266110050125 · Zbl 1204.35083 · doi:10.1134/S0012266110050125
[13] Cardone G, Math. Nachr. 24 pp 1222– (2009)
[14] DOI: 10.1134/S1061920808020076 · Zbl 1180.35392 · doi:10.1134/S1061920808020076
[15] DOI: 10.1134/S0001434610050123 · Zbl 1291.35143 · doi:10.1134/S0001434610050123
[16] DOI: 10.1006/jdeq.1997.3337 · Zbl 0901.35066 · doi:10.1006/jdeq.1997.3337
[17] DOI: 10.4213/faa2966 · doi:10.4213/faa2966
[18] DOI: 10.4213/sm7547 · doi:10.4213/sm7547
[19] DOI: 10.1080/00036810903479715 · Zbl 1186.35123 · doi:10.1080/00036810903479715
[20] Mazya VG, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains (2000)
[21] Gadyl’shin RR, Differents. Uravneniya 22 pp 640– (1986)
[22] Kamotskii IV, Trans. Am. Math. Soc. Ser. 2. 199 pp 127– (2000) · doi:10.1090/trans2/199/04
[23] Mazya VG, Izv. Akad. Nauk SSSR. Ser. Mat. 48 (2) pp 347– (1984)
[24] Mazya VG, Sibirsk. Mat. Zh. 30 (3) pp 52– (1989)
[25] Nazarov SA, Asymptotic Anal. 56 (3) pp 159– (2008)
[26] Nazarov SA, Control Cybern. 37 (4) pp 999– (2008)
[27] Ozawa S, Osaka J. Math. 22 (4) pp 39– (1985)
[28] Pólya G, Isoperimetric Inequalities in Mathematical Physics (1951) · Zbl 0044.38301 · doi:10.1515/9781400882663
[29] Visik MI, Am. Math. Soc. Transl. 20 pp 239– (1962) · Zbl 0122.32402 · doi:10.1090/trans2/020/06
[30] Campbell A, RAIRO Model. Math. Anal. Num. 32 (5) pp 579– (1998) · Zbl 0905.73029 · doi:10.1051/m2an/1998320505791
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