Russian Journal of Mathematical Physics

, Volume 25, Issue 1, pp 73–87 | Cite as

Spectrum of a Problem of Elasticity Theory in the Union of Several Infinite Layers

  • S. A. NazarovEmail author


The essential spectrum of the Dirichlet problem for the system of Lamé equations in a three-dimensional domain formed by three mutually perpendicular elastic layers occupies the ray [Λ,+∞). The lower bound Λ > 0 is the least eigenvalue (its existence is established) of the problem of elasticity theory in an infinite two-dimensional cross-shaped waveguide. It is proved that the discrete spectrum of the spatial problem is nonempty. Other configurations of layers and the scalar problem of the junction of quantum waveguides are also considered.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Problems (Izdatatel’stvo Nauka, Moscow, 1973; Springer-Verlag, New York, 1985).Google Scholar
  2. 2.
    G. Fichera, “Existence Theorems in Elasticity Theory,” in: Linear Theories of Elasticity and Thermoelasticity (Springer-Verlag, Berlin–Heidelberg, 1973; Existence Theorems in Elasticity Theory, Mir, Moscow, 1974).Google Scholar
  3. 3.
    M. Sh. Birman and M. Z. Solomyak [Solomjak], Spectral Theory of Self-adjoint Operators in Hilbert Spaces (Leningrad. Univ., Leningrad, 1980; D. Reidel Publishing Co., Dordrecht, 1987).zbMATHGoogle Scholar
  4. 4.
    G. P. Cherepanov, Mechanics of Brittle Fracture (Nauka, Moscow, 1974; McGraw-Hill, New York, 1979).Google Scholar
  5. 5.
    Y. Avishai, D. Bessis, B.G. Giraud, and G. Mantica, “Quamtum Bound States in Open Geometries,” Phys. Rev. B 44 (15), 8028–8034 (1991).ADSCrossRefGoogle Scholar
  6. 6.
    S. A. Nazarov, “The Eigenfrequencies of a Slightly Curved Isotropic Strip Clamped between Absolutely Rigid Profiles,” Prikl. Mat. Mekh. 78 (4), 527–541 (2014) [J. Appl. Math. Mech. 78, (4) 374–383 (2014)].MathSciNetGoogle Scholar
  7. 7.
    S. A. Nazarov, “Discrete Spectrum of Cranked Quantum and Elastic Wavegiodes,” Zh. Vychisl. Mat. Mat. Fiz. 56 (5), 879–895 (2016) [Comput. Math. Math. Phys. 56, (5) 864–880 (2016)].Google Scholar
  8. 8.
    R.L. Shult, D.G. Ravenhall, and H.D. Wyld, “Quamtum Bound States in a Classically Unbounded System of Crossed Wires,” Phys. Rev. B 39 (8), 5476–5479 (1989).ADSCrossRefGoogle Scholar
  9. 9.
    S. A. Nazarov, “The Spectrum of Rectangular Lattices of Quantum Waveguides,” Izv. Ross. Akad. Nauk Ser. Mat. 81 (1), 3–64 (2017) [Izv. Math. 81, (1) 29–90 (2017)].MathSciNetGoogle Scholar
  10. 10.
    I. V. Kamotskii and S. A. Nazarov, “Elastic Waves Localized near Periodic Families of Defects,” Dokl. Ross. Akad. Nauk 368 (6), 771–773 (1999) [Dokl. Math. 44, (10) 715–717 (1999)].MathSciNetzbMATHGoogle Scholar
  11. 11.
    I. V. Kamotskii and S. A. Nazarov, “Exponentially Decreasing Solutions of Diffraction Problems on a Rigid Periodic Boundary,” Mat. Zametki 73 (1), 138–140 (2003) [Math. Notes 73, (1) 129–131 (2003)].MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    R. D. Mindlin, Waves and Vibration in Isotropic Elastic Plate (Structural Mechanics, Proceedings of the 1St Symposium of Naval Structural Mechanics. Pergamon Press, 1960).Google Scholar
  13. 13.
    I. I. Vorovich and V. A. Babeshko, Dynamic Mixed Problems of Elasticity for Nonclassical Domains (Nauka, Moscow, 1979).zbMATHGoogle Scholar
  14. 14.
    V. A. Babeshko, E. V. Glushakov, and Zh. F. Zinchenko, Dynamics of Inhomogeneous Elastic Media (Nauka, Moscow, 1989).Google Scholar
  15. 15.
    M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations (Nauka, Moscow, 1969).zbMATHGoogle Scholar
  16. 16.
    M. S. Agranivich and M. I. Vishik, “Elliptic Problems with a Parameter and Parabolic Problems of General Type,” Uspekhi Mat. Nauk 19 (3), 53–161 (1964) [Russian Math. Surveys 19, (3) 53–157 (1964)].MathSciNetGoogle Scholar
  17. 17.
    S. A. Nazarov and A. V. Shanin, “Trapped Modes in Angular Joints of 2D Waveguides,” Appl. Anal. 93 (3), 572–582 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    D. S. Jones, “The Eigenvalues of ∇2 u + λu = 0 When the Boundary Conditions Are Given on Semi-Infinite Domains,” Proc. Camb. Phil. Soc. 49, 668–684 (1953).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    S. A. Nazarov, “Discrete Spectrum of Cranked, Branching, and PeriodicWaveguides,” Algebra i Analiz 23 (2), 206–247 (2011) [St. Petersburg Math. J. 23, (2) 351–379 (2012)].Google Scholar
  20. 20.
    M. Dauge and N. Raymond, “Plane Waveguides with Corners in the Small Angle Limit,” J. Math. Phys. 53 (2012) DOI: 10.1063/1.4769993.Google Scholar
  21. 21.
    S. A. Nazarov, Asymptotic Theory of Thin Plates and Rodes. Dimension Reduction and Integral Estimates (Nauchnaya Kniga, Novosibirsk, 2002).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Mathematics and Mechanics FacultySt. Petersurg State UniversitySt. PetersburgRussia
  2. 2.Peter the Great St. Petersburg State Polytechnic UniversitySt. PetersburgRussia
  3. 3.Institute of Problems of Mechanical Engineering of the Russian Academy of SciencesSt. PetersburgRussia

Personalised recommendations