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Izv. Akad. Nauk SSSR Ser. Mat., 1981, Volume 45, Issue 1, Pages 101–112 (Mi izv1549)  

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.
Bibliography: 15 titles.

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English version:
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
\by S.~A.~Nazarov
\paper Elliptic boundary value problems with periodic coefficients in a~cylinder
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1981
\vol 45
\issue 1
\pages 101--112
\jour Math. USSR-Izv.
\yr 1982
\vol 18
\issue 1
\pages 89--98

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    This publication is cited in the following articles:
    1. P. A. Kuchment, “Floquet theory for partial differential equations”, Russian Math. Surveys, 37:4 (1982), 1–60  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. 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  crossref
    3. 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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. 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  crossref  elib
    5. 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  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    7. S. A. Nazarov, A. S. Slutskii, “Saint-venant principle for paraboloidal elastic bodies”, Journal of Mathematical Sciences (New York), 98:6 (2000), 717  crossref  mathscinet  elib
    8. 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  mathnet  crossref  mathscinet  zmath  elib
    9. S. A. Nazarov, “Concentration of trapped modes in problems of the linearized theory of water waves”, Sb. Math., 199:12 (2008), 1783–1807  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. Sergey A. Nazarov, “A gap in the continuous spectrum of an elastic waveguide”, Comptes Rendus Mécanique, 336:10 (2008), 751  crossref
    11. 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  mathnet  crossref  mathscinet  zmath  isi
    12. 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  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    13. S. A. Nazarov, “An example of multiple gaps in the spectrum of a periodic waveguide”, Sb. Math., 201:4 (2010), 569–594  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. S. A. Nazarov, “Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide”, Math. Notes, 87:5 (2010), 738–756  mathnet  crossref  crossref  mathscinet  isi  elib
    15. 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  mathnet  crossref  mathscinet  adsnasa  isi  elib
    16. S. A. Nazarov, “Discrete spectrum of cranked, branchy, and periodic waveguides”, St. Petersburg Math. J., 23:2 (2012), 351–379  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    17. S. A. Nazarov, “On the spectrum of the Laplace operator on the infinite Dirichlet ladder”, St. Petersburg Math. J., 23:6 (2012), 1023–1045  mathnet  crossref  mathscinet  isi  elib  elib
    18. Nazarov S.A., “Lokalizovannye uprugie polya v periodicheskikh volnovodakh s defektami”, Prikladnaya mekhanika i tekhnicheskaya fizika, 52:2 (2011), 183–194  elib
    19. Nazarov S.A., “Asimptotika sobstvennykh chastot, poyavlyayuschikhsya vnutri lakun pri vozmuschenii periodicheskogo volnovoda”, Doklady akademii nauk, 447:4 (2012), 382–382  elib
    20. 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  mathnet  crossref
    21. S. A. Nazarov, “Umov-Mandelshtam radiation conditions in elastic periodic waveguides”, Sb. Math., 205:7 (2014), 953–982  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    22. Antonio Gaudiello, Grigory Panasenko, Andrey Piatnitski, “Asymptotic analysis and domain decomposition for a biharmonic problem in a thin multi-structure”, Commun. Contemp. Math, 2015, 1550057  crossref
    23. S.A.. Nazarov, Jari Taskinen, “Spectral gaps for periodic piezoelectric waveguides”, Z. Angew. Math. Phys, 2015  crossref
    24. 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  mathnet  crossref  mathscinet  isi  elib
    25. 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  mathnet  crossref  crossref  isi  elib
    26. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    27. 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  mathnet  crossref  crossref  isi  elib
    28. 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  mathnet  crossref  crossref  isi  elib
    29. 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  crossref  isi
    30. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    31. 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  mathnet  crossref
    32. 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  mathnet  crossref  elib
    33. 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  mathnet
    34. S. A. Nazarov, “Homogenization of Kirchhoff plates with oscillating edges and point supports”, Izv. Math., 84:4 (2020), 722–779  mathnet  crossref  crossref  mathscinet  isi
    35. S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. Math., 84:6 (2020), 1105–1160  mathnet  crossref  crossref  mathscinet  isi
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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