A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary

  • Tynysbek Sh. Kal’menov
  • Berikbol T. TorebekEmail author


In this paper a nonlocal problem for the elliptic equation in a cylindrical domain is considered. It is shown that this problem is ill-posed as well as the Cauchy problem for the Laplace equation. The method of spectral expansion in eigenfunctions of the nonlocal problem for equations with involution establishes a criterion of the strong solvability of the considered nonlocal problem. It is shown that the ill-posedness of the nonlocal problem is equivalent to the existence of an isolated point of the continuous spectrum for a nonself-adjoint operator with involution.


Elliptic operator Nonlocal boundary conditions Operator with involution Criterion of well-posedness Riesz basis 

Mathematics Subject Classification

Primary 35J25 35C10 Secondary 35P10 


  1. 1.
    Hadamard, J.: Lectures on the Cauchy Problem in Linear Differential Equations. Yale University Press, New Haven (1923)zbMATHGoogle Scholar
  2. 2.
    Lavrentiev, M.M.: On a Cauchy problem for the Poisson equation. Izv. Akad. Nauk SSSR. Ser. Mat. 20(6), 819–842 (1955)Google Scholar
  3. 3.
    Tikhonov, A.N., Arsenin, V.Ya.: Methods for sSolving Ill-Posed Problems. Nauka, Moskow (1979)Google Scholar
  4. 4.
    Qian, Z., Fu, C.L., Li, Z.P.: Two regularization methods for a Cauchy problem for the Laplace equation. J. Math. Anal. Appl. 338(1), 479–489 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Klibanov, M.V.: Carleman estimates for the regularization of ill-posed Cauchy problems. Appl. Numer. Math. 94, 46–74 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kal’menov, T.S., Iskakova, U.A.: A criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation. Dokl. Math. 75(3), 370–373 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kal’menov, T.S., Iskakova, U.A.: Criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation. Differ. Equ. 45(10), 1460–1466 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Torebek, B.T.: A method for solving ill-posed Robin–Cauchy problems for secondorder elliptic equations in multi-dimensional cylindrical domains. Electron. J. Differ. Equ. 2016(252), 1–9 (2016)MathSciNetGoogle Scholar
  9. 9.
    Kal’menov, T.S., Sadybekov, M.A., Iskakova, U.A.: On a criterion for the solvability of one ill-posed problem for the biharmonic equation. J. Inverse Ill-Posed Probl. 24(6), 777–783 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kal’menov, T.S., Torebek, B.T.: On an ill-posed problem for the Laplace operator with nonlocal boundary condition. Eurasian Math. J. 8(1), 50–57 (2017)MathSciNetGoogle Scholar
  11. 11.
    Wiener, J.: Generalized Solutions of Functional–Differential Equations. World Scientific Publishing, New Jersey (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    Naimark, M.A.: Linear Differential Operators. Part II. Ungar, New York (1968)zbMATHGoogle Scholar
  13. 13.
    Shkalikov, A.A.: On the basis problem of the eigenfunctions of an ordinary differential operator. Russ. Math. Surv. 34(34), 249–250 (1979)CrossRefzbMATHGoogle Scholar
  14. 14.
    Titchmarsh, E.: The Theory of Functions. Nauka, Moskow (1980)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Differential EquationsInstitute of Mathematics and Mathematical ModelingAlmatyKazakhstan
  2. 2.Al-Farabi Kazakh National UniversityAlmatyKazakhstan

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