Differential Equations

, Volume 55, Issue 10, pp 1349–1361 | Cite as

Basis Property of the System of Root Functions of the Oblique Derivative Problem for the Laplace Operator in a Disk

  • A. B. KostinEmail author
  • V. B. SherstyukovEmail author
Partial Differential Equations


We study the spectral oblique derivative problem for the Laplace operator in a disk D. The asymptotic properties of the eigenvalues are established, and the basis property with parentheses in the space L2(D) is proved for the system of root functions of the above problem.


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This work was supported in part by the Competitiveness Enhancement Program of National Nuclear Research University MEPhI, project no. 02.a03.21.0005. The research by V.B. Sherstyukov was supported by the Russian Foundation for Basic Research, project no. 18-01-00236.


  1. 1.
    Agmon, S., On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 1962, vol. 15, pp. 119–147.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Moiseev, E.I., On the completeness of the eigenfunctions of some boundary value problems, Dokl. Akad. Nauk SSSR, 1976, vol. 226, no. 5, pp. 1012–1014.MathSciNetGoogle Scholar
  3. 3.
    Il’in, V.A. and Moiseev, E.I., On the absence of the basis property of a system of root functions of a problem with an oblique derivative, Differ. Equations, 1994, vol. 30, no. 1, pp. 119–132.MathSciNetGoogle Scholar
  4. 4.
    Kostin, A.B. and Sherstyukov, V.B., On complex roots of an equation arising in the oblique derivative problem, J. Phys. Conf. Ser., 2017, vol. 788, article ID 012052, pp. 1–7, doi: Scholar
  5. 5.
    Kostin, A.B. and Sherstyukov, V.B., The basis property of the system of root functions of the oblique derivative problem, Dokl. Math., 2018, vol. 98, no. 2, pp. 409–412.CrossRefGoogle Scholar
  6. 6.
    Gokhberg, I.C. and Krein, M.G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gil’bertovom prostranstve (Introduction to the Theory of Linear Nonself-Adjoint Operators in Hilbert Space), Moscow: Nauka, Fizmatlit, 1965.Google Scholar
  7. 7.
    Agranovich, M.S., On series in terms of the root vectors of operators defined by forms with a self-adjoint leading part, Funkts. Anal. Ego Pril., 1994, vol. 28, no. 3, pp. 1–21.Google Scholar
  8. 8.
    Shkalikov, A.A., On the basis property of root vectors of a perturbed self-adjoint operator, Proc. Steklov Inst. Math., 2010, vol. 269, no. 1, pp. 284–298.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1988.Google Scholar
  10. 10.
    Dalzell, D.P., On the completeness of Dini’s series, J. London Math. Soc., 1945, vol. 20, pp. 213–218.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Watson, G.N., A Treatise on the Theory of Bessel Functions, Cambridge: Cambridge Univ., 1922, 2nd ed. Translated under the title Teoriya besselevykh funktsii, Vol. 1. Moscow: Izd. Inostr. Lit., 1949.zbMATHGoogle Scholar
  12. 12.
    Kostin, A.B. and Sherstyukov, V.B., Calculation of Rayleigh type sums for zeros of the equation arising in spectral problem, J. Phys. Conf. Ser., 2017, vol. 937, article ID 012022, pp. 1–9, doi: Scholar
  13. 13.
    Bari, N.K., Biorthogonal systems and bases in a Hilbert space, Uch. Zap. Mosk. Gos. Univ. Mat., 1951, vol. 4(148), pp. 69–107.MathSciNetGoogle Scholar
  14. 14.
    Kapitsa, P.L., Calculation of sums of negative even powers of the roots of Bessel functions, Dokl. Akad. Nauk SSSR, 1951, vol. 77, no. 4, pp. 561–564.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Meiman, N.N., On recurrent formulas for the power sums of zeros of Bessel functions, Dokl. Akad. Nauk SSSR, 1956, vol. 108, no. 2, pp. 190–193.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kerimov, M.K., The Rayleigh function: Theory and calculation methods, Comput. Math. Math. Phys., 1999, vol. 39, no. 12, pp. 1883–1925.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kerimov, M.K., Overview of some new results concerning the theory and applications of the Rayleigh special function, Comput. Math. Math. Phys., 2008, vol. 48, no. 9, pp. 1454–1507.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Shkalikov, A.A., Basis property of eigenfunctions of a problem on antiplanar vibrations of a cylinder with external friction, Math. Notes, 1993, vol. 53, no. 2, pp. 223–234.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Shkalikov, A.A., Perturbations of self-adjoint and normal operators with discrete spectrum, Russ. Math. Surv., 2016, vol. 71, no. 5, pp. 907–964.MathSciNetCrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.National Nuclear Research University MEPhIMoscowRussia

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