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This article is cited in 35 scientific papers (total in 35 papers)
Elliptic boundary value problems with periodic coefficients in a cylinder
S. A. Nazarov
Abstract:
In a domain with periodically varying cross-section, this paper studies boundary value problems, elliptic in the Douglis–Nirenberg sense, in which the coefficients are periodic functions with the same period. Necessary and sufficient conditions for the unique solvability of these problems in function spaces with weighted norms are proved, and theorems on the Noether property and on the asymptotics of the solutions of boundary value problems with exponentially small perturbations of the coefficients are adduced.
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Mathematics of the USSR-Izvestiya, 1982, 18:1, 89–98
Bibliographic databases:
   
UDC:
517.948
MSC: Primary 35J55, 35A05; Secondary 35B20, 35B40
Received: 07.02.1980
Citation:
S. A. Nazarov, “Elliptic boundary value problems with periodic coefficients in a cylinder”, Izv. Akad. Nauk SSSR Ser. Mat., 45:1 (1981), 101–112; Math. USSR-Izv., 18:1 (1982), 89–98
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/izv1549 http://mi.mathnet.ru/eng/izv/v45/i1/p101
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P. A. Kuchment, “Floquet theory for partial differential equations”, Russian Math. Surveys, 37:4 (1982), 1–60
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M.C Shen, D Ebel, “Asymptotic methods for peristaltic transport of a heat-conducting fluid”, Journal of Mathematical Analysis and Applications, 127:1 (1987), 49
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S. A. Nazarov, “Asymptotics of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle”, Math. USSR-Sb., 73:1 (1992), 79–110
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F Blanc, S.A Nazarov, “Asymptotics of solutions to the Poisson problem in a perforated domain with corners”, Journal de Mathématiques Pures et Appliquées, 76:10 (1997), 893
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S. A. Nazarov, A. S. Slutskij, “Asymptotic behaviour of solutions of boundary-value problems for equations with rapidly oscillating coefficients in a domain with a small cavity”, Sb. Math., 189:9 (1998), 1385–1422
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S. A. Nazarov, “The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes”, Russian Math. Surveys, 54:5 (1999), 947–1014
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S. A. Nazarov, A. S. Slutskii, “Saint-venant principle for paraboloidal elastic bodies”, Journal of Mathematical Sciences (New York), 98:6 (2000), 717
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S. A. Nazarov, A. S. Slutskij, “Averaging of an elliptic system under condensing perforation of a domain”, St. Petersburg Math. J., 17:6 (2006), 989–1014
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S. A. Nazarov, “Concentration of trapped modes in problems of the linearized theory of water waves”, Sb. Math., 199:12 (2008), 1783–1807
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Sergey A. Nazarov, “A gap in the continuous spectrum of an elastic waveguide”, Comptes Rendus Mécanique, 336:10 (2008), 751
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S. A. Nazarov, “Opening a gap in the essential spectrum of the elasticity problem in a periodic semi-layer”, St. Petersburg Math. J., 21:2 (2010), 281–307
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S. A. Nazarov, “Gap detection in the spectrum of an elastic periodic waveguide with a free surface”, Comput. Math. Math. Phys., 49:2 (2009), 323–333
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S. A. Nazarov, “An example of multiple gaps in the spectrum of a periodic waveguide”, Sb. Math., 201:4 (2010), 569–594
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S. A. Nazarov, “Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide”, Math. Notes, 87:5 (2010), 738–756
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S. A. Nazarov, “Formation of gaps in the spectrum of the problem of waves on the surface of a periodic channel”, Comput. Math. Math. Phys., 50:6 (2010), 1038–1054
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S. A. Nazarov, “Discrete spectrum of cranked, branchy, and periodic waveguides”, St. Petersburg Math. J., 23:2 (2012), 351–379
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S. A. Nazarov, “On the spectrum of the Laplace operator on the infinite Dirichlet ladder”, St. Petersburg Math. J., 23:6 (2012), 1023–1045
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Nazarov S.A., “Lokalizovannye uprugie polya v periodicheskikh volnovodakh s defektami”, Prikladnaya mekhanika i tekhnicheskaya fizika, 52:2 (2011), 183–194
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Nazarov S.A., “Asimptotika sobstvennykh chastot, poyavlyayuschikhsya vnutri lakun pri vozmuschenii periodicheskogo volnovoda”, Doklady akademii nauk, 447:4 (2012), 382–382
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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
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S. A. Nazarov, “Umov-Mandelshtam radiation conditions in elastic periodic waveguides”, Sb. Math., 205:7 (2014), 953–982
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S. A. Nazarov, “Almost standing waves in a periodic waveguide with a resonator and near-threshold eigenvalues”, St. Petersburg Math. J., 28:3 (2017), 377–410
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F. L. Bakharev, S. A. Nazarov, “Open waveguides in doubly periodic junctions of domains with different limit dimensions”, Siberian Math. J., 57:6 (2016), 943–956
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S. A. Nazarov, “The asymptotic behaviour of the scattering matrix in a neighbourhood of the endpoints of a spectral gap”, Sb. Math., 208:1 (2017), 103–156
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S. A. Nazarov, “Open waveguides in a thin Dirichlet ladder: I. Asymptotic structure of the spectrum”, Comput. Math. Math. Phys., 57:1 (2017), 156–174
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S. A. Nazarov, “Open waveguides in a thin Dirichlet lattice: II. Localized waves and radiation conditions”, Comput. Math. Math. Phys., 57:2 (2017), 236–252
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Gomez D. Nazarov S.A. Perez M.E., “Homogenization of Winkler-Steklov Spectral Conditions in Three-Dimensional Linear Elasticity”, Z. Angew. Math. Phys., 69:2 (2018), 35
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S. A. Nazarov, “The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section”, Sb. Math., 209:9 (2018), 1287–1336
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S. A. Nazarov, “Asymptotics of eigenvalues in spectral gaps of periodic waveguides with small singular perturbations”, J. Math. Sci. (N. Y.), 243:5 (2019), 746–773
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S. A. Nazarov, “Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity”, Trans. Moscow Math. Soc., 80 (2019), 1–51
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V. Kozlov, J. Taskinen, “Floquet problem and center manifold reduction for ordinary differential operators with periodic coefficients in Hilbert spaces”, Algebra i analiz, 32:3 (2020), 191–218
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S. A. Nazarov, “Homogenization of Kirchhoff plates with oscillating edges and point supports”, Izv. Math., 84:4 (2020), 722–779
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S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. Math., 84:6 (2020), 1105–1160
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