Gap in the essential spectrum of the Neumann problem for an elliptic system in a periodic domain. (English. Russian original) Zbl 1271.35025

Nazarov, S. A.

Funct. Anal. Appl. 43, No. 3, 239-241 (2009); translation from Funkts. Anal. Prilozh. 43, No. 3, 92-95 (2009).

Summary: We establish the existence of a gap in the essential spectrum of the Neumann problem for an elliptic formally self-adjoint system of second-order differential equations on a quasi-cylinder (a domain with periodically varying cross-section).

MSC:

35J57	Boundary value problems for second-order elliptic systems
35P05	General topics in linear spectral theory for PDEs

Keywords:

gap in essential spectrum; Neumann problem for elliptic system; periodic domain; quasi-cylinder

References:

[1]	J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson-Academia, Paris-Prague, 1967.
[2]	S. A. Nazarov, Uspekhi Mat. Nauk, 54:5 (1999), 77–142. · doi:10.4213/rm204
[3]	M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Selfadjoint Operators in Hilbert Space [in Russian], Leningrad State University, Leningrad, 1980.
[4]	P. A. Kuchment, Uspekhi Mat. Nauk, 37:4 (1982), 3–52.
[5]	P. Kuchment, Floquet theory for partial differential equations, Birchäuser, Basel, 1993. · Zbl 0789.35002
[6]	S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains with Piecewise Smooth Boundary [in Russian], Nauka, Moscow, 1991.
[7]	I. M. Gelfand, Dokl. Akad. Nauk SSSR, 73 (1950), 1117–1120.
[8]	P. Kuchment, in: Mathematical Modelling in Optical Science, Frontiers in Applied Mathematics, vol. 22, SIAM, Philadelphia, PA, 2001, 207–272.
[9]	V. V. Zhikov, Algebra and Analiz, 6:5 (2004), 34–58.
[10]	N. Filonov, Comm. Math. Physics, 6:1–2 (2003), 161–170. · Zbl 1037.35051 · doi:10.1007/s00220-003-0904-7
[11]	S. A. Nazarov, Uspekhi Mat. Nauk, 63:5 (2008), 37–110. · doi:10.4213/rm8545