Properties of Spectra of Boundary Value Problems in Cylindrical and Quasicylindrical Domains

  • Sergey Nazarov
Part of the International Mathematical Series book series (IMAT, volume 9)


General formally self-adjoint boundary value problems with spec tral parameter are investigated in domains with cylindrical and quasicylin drical (periodic) outlets to infinity. The structure of the spectra is studied for operators generated by the corresponding sesquilinear forms. In addition to general results, approaches and methods are discussed to get a piece of infor mation on continuous, point, and discrete spectra, in particular, for specific problems in the mathematical physics.


Continuous Spectrum Discrete Spectrum Essential Spectrum Periodicity Cell Integral Identity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Birman, M.S., Solomyak, M.Z.: Spectral Theory of Self-Adjoint operators in Hilbert Space. Reidel Publ. Company, Dordrecht (1986)Google Scholar
  2. 2.
    Bonnet-Bendhia, A.S., Duterte, J., Joly, P.: Mathematical analysis of elastic surface waves in topographic waveguides. Math. Meth. Appl. Sci.9, no. 5, 755–798 (1999)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bonnet-Bendhia, A.S., Starling, F.: Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Meth. Appl. Sci.77, 305–338 (1994)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Cherepanov, G.P.: Crack propagation in continuous media (Russian). Prikl. Mat. Mekh.31, 476–488 (1967); English transl.: J. Appl. Math. Mech.31, 503–512 (1967)Google Scholar
  5. 5.
    Costabel, M., Dauge, M.: Crack singularities for general elliptic systems. Math. Nachr.235, 29–49 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Duduchava, R., Wendland, W.L.: The Wiener-Hopf method for systems of pseudod ifferential equations with an application to crack problems. Integral Eq. Operator Theory23, no. 3, 294–335 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Duvaut, G., Lions. J.L.: Les inéquations en mćanique et en physique. Dunod, Paris (1972)Google Scholar
  8. 8.
    Evans, D.V., Levitin, M., Vasil'ev, D.: Existence theorems for trapped modes. J. Fluid Mech.261, 21–31 (1994)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Figotin, A., Kuchment, P.: Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model. SIAM J. Appl. Math.56, 68–88 (1996); II. Two-dimensional photonic crystals. ibid.56, 1561–1620 (1996)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Filonov, N.: Gaps in the spectrum of the Maxwell operator with periodic coefficients. Commun. Math. Phys.240, no. 1–2, 161–170 (2003)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Friedlander, L.: On the density of states of periodic media in the large coupling limit. Commun. Partial Differ. Equ.27, 355–380 (2002)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gel'fand, I.M.: Expansion in characteristic functions of an equation with periodic coefficients (Russian). Dokl. Akad. Nauk SSSR73, 1117–1120 (1950)MATHGoogle Scholar
  13. 13.
    Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Am. Math. Soc., Providence, RI (1969)Google Scholar
  14. 14.
    Gohberg, I.C., Sigal, E.I.: An operator generalization of the logarithmic residue the orem and the theorem of Rouché (Russian). Mat. Sb.84, 607–629. (1971); English transl.: Math. USSR-Sb.13, 603–625 (1971)MathSciNetGoogle Scholar
  15. 15.
    Green, E.L.: Spectral theory of Laplace-Beltrami operators with periodic metrics. J. Differ. Equations133, 15–29 (1997)MATHCrossRefGoogle Scholar
  16. 16.
    Grinchenko, V.T., Ulitko, A.F., Shul'ga, N.A.: Mechanics of Coupled Fields in Struc tural Elements (Russian). Naukova Dumka, Kiev (1989)Google Scholar
  17. 17.
    Hempel, R., Lineau, K.: Spectral properties of the periodic media in large coupling limit. Commun. Partial Differ. Equ.25, 1445–1470 (2000)MATHCrossRefGoogle Scholar
  18. 18.
    Jones, D.S.: 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)MATHCrossRefGoogle Scholar
  19. 19.
    Kamotskii, I.V., Nazarov, S.A.: Elastic waves localized near periodic sets of flaws (Russian). Dokl. Ross. Akad. Nauk.368, no. 6, 771–773 (1999); English transl.: Dokl. Physics44, no. 10, 715–717 (1999)MathSciNetGoogle Scholar
  20. 20.
    Kamotskii, I.V., Nazarov, S.A.: On eigenfunctions localized in a neighborhood of the lateral surface of a thin domain (Russian). Probl. Mat. Anal.19, 105–148 (1999); English transl.: J. Math. Sci.101, no. 2, 2941–2974 (2000)Google Scholar
  21. 21.
    Kamotskii, I.V., Nazarov, S.A.: Exponentially decreasing solutions of diffraction prob lems on a rigid periodic boundary (Russian). Mat. Zametki73, no. 1, 138–140 (2003); English transl.: Math. Notes.73, no. 1, 129–131 (2003)MathSciNetGoogle Scholar
  22. 22.
    Kato, T.: Perturbation Theory for Linear Operators. Springer (1966)Google Scholar
  23. 23.
    Khallaf, N.S.A., Parnovski, L., Vassiliev, D.: Trapped modes in a waveguide with a long obstacle. J. Fluid Mech.403, 251–261 (2000)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Knowles, J.K., Sternberg, E.: On a class of conservation laws in linearizad and finite elastostatics. Arch. Ration. Mech. Anal.44, no. 3, 187–211 (1972)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kondratiev, V.A.: Boundary problems for elliptic equations in domains with conical or angular points (Russian). Tr. Mosk. Mat. Obshch.16, 209–292 (1967); English transl.: Trans. Mosc. Math. Soc.16, 227–313 (1967)Google Scholar
  26. 26.
    Kondratiev, V.A., Oleinik, O.A.: Boundary value problems for the system of elasticity theory in unbounded domains. Korn's inequalities (Russian). Uspekhi Mat. Nauk43, no. 5, 55–98 (1988); English transl.: Russian Math. Surveys43, no. 5, 65–119 (1988)MATHMathSciNetGoogle Scholar
  27. 27.
    Korn, A.: Solution générale du probléme déquilibre dans la théorie lélasticité dans le cas ou` les efforts sont donnés à la surface. Ann. Univ. Toulouse, 165–269 (1908)Google Scholar
  28. 28.
    Kozlov, V.A., Maz'ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities. Am. Math. Soc., Providence, RI (1997)Google Scholar
  29. 29.
    Kuchment, P.A.: Floquet theory for partial differential equations (Russian). Uspekhi Mat. Nauk37, no. 4, 3–52 (1982); English transl.: Russian Math. Surveys37, no. 4, 1–60 (1982)MathSciNetGoogle Scholar
  30. 30.
    Kuchment, P.: Floquet Theory for Partial Differential Equations. Birch¨auser, Basel (1993)MATHGoogle Scholar
  31. 31.
    Kuchment, P.: The mathematics of photonic crystals. In: Mathematical Modeling in Optical Science, pp. 207–272. SIAM (2001)Google Scholar
  32. 32.
    Kulikov, A.A., Nazarov, S.A.: Cracks in piezoelectric and electro-conductive bodies (Russian). Sib. Zh. Ind. Mat.8, no. 1, 70–87 (2005)MathSciNetGoogle Scholar
  33. 33.
    Ladyzhenskaya, O.A.: Boundary Value Problems of Mathematical Physics. Springer-Verlag, New-York (1985)Google Scholar
  34. 34.
    Lekhnitskij, S.G.: Theory of Elasticity of an Anisotropic Body (Russian). Nauka, Moscow (1977); English transl.: Mir Publishers, Moscow, 1981.Google Scholar
  35. 35.
    Lions. J.L., Magenes. E.:Non-Homogeneous Boundary Value Problems and Ap plications(French). Dunod, Paris (1968); English transl.: Springer-Verlag, Berlin-Heidelberg-New York (1972)Google Scholar
  36. 36.
    Maz'ja, V.G., Plamenevskii, B.A.: On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Math. Nachr.76, 29–60 (1977); English transl.: Am. Math. Soc. Transl. (Ser. 2)123, 57–89 (1984)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Maz'ja, V.G., Plamenevskii, B.A.: Estimates inLpand Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value prob lems in domains with singular points on the boundary. Math. Nachr.81, 25–82 (1978); English transl.: Am. Math. Soc. Transl. (Ser. 2)123, 1–56 (1984)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Nazarov, S.A.: Elliptic boundary value problems with periodic coefficients in a cylin der (Russian). Izv. Akad. Nauk SSSR. Ser. Mat.45, no. 1, 101–112 (1981); English transl.: Math. USSR.-Izv.18, no. 1, 89–98 (1982)MATHMathSciNetGoogle Scholar
  39. 39.
    Nazarov, S.A.: A general scheme for averaging self-adjoint elliptic systems in multi dimensional domains, including thin domains (Russian). Algebra Anal.7, no. 5, 1–92 (1995); English transl.: St. Petersbg. Math. J.7, no. 5, 681–748 (1996)MATHGoogle Scholar
  40. 40.
    Nazarov, S.A.: Korn's inequalities for junctions of spatial bodies and thin rods. Math. Methods Appl. Sci.20, no. 3, 219–243 (1997)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Nazarov, S.A.: Self-adjoint elliptic boundary-value problems. The polynomial prop erty and formally positive operators (Russian). Probl. Mat. Anal.16, 167–192 (1997); English transl.: J. Math. Sci.92, no. 6, 4338–4353 (1998)Google Scholar
  42. 42.
    Nazarov, S.A.: Weight functions and invariant integrals (Russian). Vychisl. Mekh. Deform. Tverd. Tela.1, 17–31 (1990)Google Scholar
  43. 43.
    Nazarov, S.A.: The interface crack in anisotropic bodies. Stress singularities and invariant integrals (Russian). Prikl. Mat. Mekh.62, no. 3, 489–502 (1998); English transl.: J. Appl. Math. Mech.62, no. 3, 453–464 (1998)Google Scholar
  44. 44.
    Nazarov, S.A.: The polynomial property of self-adjoint elliptic boundary-value prob lems and the algebraic description of their attributes (Russian). Uspekhi Mat. Nauk54, no. 5, 77–142 (1999); English transl.: Russian Math. Surveys54, no. 5, 947–1014 (1999)MathSciNetGoogle Scholar
  45. 45.
    Nazarov, S.A.: Asymptotic Theory of Thin Plates and Rods. Vol.1. Dimension Reduc tion and Integral Estimates (Russian). Nauchnaya Kniga IDMI, Novosibirsk (2002)Google Scholar
  46. 46.
    Nazarov, S.A.: Korn's inequality for an elastic junction of a body with a rod (Russian). In: Problems of Mechanics of Solids, pp. 234–240. St.-Petersburg State Univ. Press, St.-Petersburg (2002)Google Scholar
  47. 47.
    Nazarov, S.A.: Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate (Russian). Probl. Mat. Anal.25, 99–188 (2003); English transl.: J. Math. Sci.114, no. 5, 1657–1725 (2003)MATHGoogle Scholar
  48. 48.
    Nazarov, S.A.: Trapped modes in a cylindrical elastic waveguide with a damping gasket (Russian). Zh. Vychisl. Mat. Mat. Fiz.48, no. 5, 863–881 (2008); English transl.: Comput. Math. Math. Phys.48, no. 5, 816–833 (2008)MathSciNetGoogle Scholar
  49. 49.
    Nazarov, S.A.: Korn inequalities for elastic junctions of massive bodies, thin plates and rods. inequalities (Russian). Uspekhi Mat. Nauk63, no. 5, 37–110 (2008); English transl.: Russian Math. Surveys63, no. 5 (2008)Google Scholar
  50. 50.
    Nazarov, S.A.: The Rayleigh waves in an elastic semi-layer with periodic boundary. Dokl. Ross. Akad. Nauk [Submitted]Google Scholar
  51. 51.
    Nazarov, S.A.: On the essential spectrum of boundary value problems for systems of differential equations in a bounded domain with a peak. Funkt. Anal. Pril. [To appear]Google Scholar
  52. 52.
    Nazarov, S.A.: Opening gaps in the continuous spectrum of periodic elastic waveguide with traction-free boundary. Zh. Vychisl. Mat. Mat. Fiz. [Submitted]Google Scholar
  53. 53.
    Nazarov, S.A., Plamenevskii, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin, New York (1994)MATHGoogle Scholar
  54. 54.
    Nečas. J.: Les méthodes in théorie des equations elliptiques. Masson-Academia, Paris-Prague (1967)Google Scholar
  55. 55.
    Parton, V.Z., Kudryavtsev, B.A.: Electromagnetoelasticity of Piezoelectronics and Conductive Solids (Russian). Nauka, Moscow (1988)Google Scholar
  56. 56.
    Rellich, F.: Über das asymptotische Verhalten der Lösungen von ▿2 u+ λu= 0 in unendlichen Gebieten. J. Dtsch. Math.-Ver.53, no. 1, 57–65 (1943)MathSciNetGoogle Scholar
  57. 57.
    Rice, J.R.: A path independent integral and the approximate analysis of strain con centration by notches and cracks. J. Appl. Mech.35, no. 2, 379–386 (1968)MathSciNetGoogle Scholar
  58. 58.
    Roitberg, I., Vassiliev, D., Weidl, T.: Edge resonance in an elastic semi-strip. Quart. J. Appl. Math.51, no. 1, 1–13 (1998)MATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Skriganov, M.M.: Geometric and arithmetical methods in the spectral theory of mul tidimensional periodic operators (Russian). Tr. Mat. Inst. Steklova171(1985)Google Scholar
  60. 60.
    Sobolev, S.L.: Some Applications of Functional Analysis in Mathematical Physics (Russian). 3rd Ed. Nauka, Moscow (1988); English transl.: Am. Math. Soc, Provi dence, RI (1991)Google Scholar
  61. 61.
    Sukhinin, S.V.: Waveguide, anomalous, and whispering properties of a periodic chain of obstacles (Russian). Sib. Zh. Ind. Mat.1, no. 2, 175–198 (1998)MATHMathSciNetGoogle Scholar
  62. 62.
    Zhikov, V.V.: Gaps in the spectrum of some elliptic operators in divergent form with periodic coefficients (Russian). Algebra Anal.16, no. 5, 34–58 (2004); English transl.: St. Petersbg. Math. J.16, no. 5, 773–790 (2005)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Institute of Problems in Mechanical EngineeringRussian Acad. Sci.Russia

Personalised recommendations