On spectrum gaps of some divergent elliptic operators with periodic coefficients

Zhikov, V.

doi:10.1090/S1061-0022-05-00878-2

On spectrum gaps of some divergent elliptic operators with periodic coefficients
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by V. V. Zhikov
Translated by: B. M. Bekker

St. Petersburg Math. J. 16 (2005), 773-790

DOI: https://doi.org/10.1090/S1061-0022-05-00878-2

Published electronically: September 21, 2005

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References

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystals. Molding the flow of light, Princeton Univ. Press, Princeton, NJ, 1995.
A. Figotin and P. Kuchment, Spectral properties of classical waves in high-contrast periodic media, SIAM J. Appl. Math. 58 (1998), no. 2, 683–702. MR 1617610, DOI 10.1137/S0036139996297249
Peter Kuchment and Leonid A. Kunyansky, Spectral properties of high contrast band-gap materials and operators on graphs, Experiment. Math. 8 (1999), no. 1, 1–28. MR 1685034
P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals, Adv. Comput. Math. 16 (2002), no. 2-3, 263–290. Modeling and computation in optics and electromagnetics. MR 1892252, DOI 10.1023/A:1014481629504
V. V. Zhikov, On an extension and an application of the two-scale convergence method, Mat. Sb. 191 (2000), no. 7, 31–72 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 7-8, 973–1014. MR 1809928, DOI 10.1070/SM2000v191n07ABEH000491
Rainer Hempel and Karsten Lienau, Spectral properties of periodic media in the large coupling limit, Comm. Partial Differential Equations 25 (2000), no. 7-8, 1445–1470. MR 1765136, DOI 10.1080/03605300008821555
L. Friedlander, On the density of states of periodic media in the large coupling limit, Preprint ESI no. 1038, Vienna, 2001.
Michael Birman and Tatyana Suslina, Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 71–107. MR 1882692
Grégoire Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), no. 6, 1482–1518. MR 1185639, DOI 10.1137/0523084
J. H. Albert, Genericity of simple eigenvalues for elliptic PDE’s, Proc. Amer. Math. Soc. 48 (1975), 413–418. MR 385934, DOI 10.1090/S0002-9939-1975-0385934-4
Todd Arbogast, Jim Douglas Jr., and Ulrich Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal. 21 (1990), no. 4, 823–836. MR 1052874, DOI 10.1137/0521046
Ulrich Hornung (ed.), Homogenization and porous media, Interdisciplinary Applied Mathematics, vol. 6, Springer-Verlag, New York, 1997. MR 1434315, DOI 10.1007/978-1-4612-1920-0
G. V. Sandrakov, Averaging of nonstationary equations with contrast coefficients, Dokl. Akad. Nauk 355 (1997), no. 5, 605–608 (Russian). MR 1600386
V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, Usrednenie differentsial′nykh operatorov, “Nauka”, Moscow, 1993 (Russian, with English and Russian summaries). MR 1318242
V. V. Zhikov, Averaging of problems in the theory of elasticity on singular structures, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 2, 81–148 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 2, 299–365. MR 1918845, DOI 10.1070/IM2002v066n02ABEH000380
A. A. Kotel′nikova and V. V. Zhikov, Spectral aspects in double-porosity models, Abstracts of the International Workshop ANASTHEM, St. Petersburg, Russia, 2000, pp. 38-41.
A. A. Kotel′nikova, On gaps in the spectrum of an elliptic operator on a periodic grid, Uspekhi Mat. Nauk 59 (2004), no. 5(359), 153–154 (Russian); English transl., Russian Math. Surveys 59 (2004), no. 5, 963–965. MR 2125936, DOI 10.1070/RM2004v059n05ABEH000780
M. Sh. Birman and T. A. Suslina, Periodic second-order differential operators. Threshold properties and averaging, Algebra i Analiz 15 (2003), no. 5, 1–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639–714. MR 2068790, DOI 10.1090/S1061-0022-04-00827-1