Skip to main content

To a nonlocal generalization of the Dirichlet problem


A mixed problem with a boundary Dirichlet condition and nonlocal integral condition is considered for a two-dimensional elliptic equation.The existence and uniqueness of a weak solution of this problem are proved in a weighted Sobolev space.



  1. 1.

    Cannon JR: The solution of the heat equation subject to the specification of energy. Quarterly of Applied Mathematics 1963,21(2):155–160.

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Chipot M, Lovat B: Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Analysis 1997,30(7):4619–4627. 10.1016/S0362-546X(97)00169-7

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ciarlet PG: The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications. Volume 4. North-Holland, Amsterdam; 1978:xix+530.

    Google Scholar 

  4. 4.

    De Schepper H, Slodička M: Recovery of the boundary data for a linear second order elliptic problem with a nonlocal boundary condition. ANZIAM Journal 2000, 42: Part C, C518-C535.

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Gordeziani D, Avalishvili G: Investigation of the nonlocal initial boundary value problems for some hyperbolic equations. Hiroshima Mathematical Journal 2001,31(3):345–366.

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Gushchin AK, Mikhaĭlov VP: On the solvability of nonlocal problems for a second-order elliptic equation. Matematicheskiĭ Sbornik 1994,185(1):121–160. translated in Russian Acad. Sci. Sb. Math. 81 (1995), no. 1, 101–136 translated in Russian Acad. Sci. Sb. Math. 81 (1995), no. 1, 101–136

    MathSciNet  Google Scholar 

  7. 7.

    Kufner A, Sändig A-M: Some Applications of Weighted Sobolev Spaces, Teubner-Texte zur Mathematik. Volume 100. BSB B. G. Teubner, Leipzig; 1987:268.

    Google Scholar 

  8. 8.

    Martynyuk AE: Some new applications of methods of Galerkin type. Matematicheskiĭ Sbornik 1959, 49 (91): 85–108.

    MathSciNet  Google Scholar 

  9. 9.

    Mesloub S, Bouziani A, Kechkar N: A strong solution of an evolution problem with integral conditions. Georgian Mathematical Journal 2002,9(1):149–159.

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Nekvinda A, Pick L: A note on the Dirichlet problem for the elliptic linear operator in Sobolev spaces with weight. Commentationes Mathematicae Universitatis Carolinae 1988,29(1):63–71.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Paneyakh BP: Some nonlocal boundary value problems for linear differential operators. Matematicheskie Zametki 1984,35(3):425–434.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Petryshyn WV: On a class of and nonoperators and operator equations. Journal of Mathematical Analysis and Applications 1965,10(1):1–24. 10.1016/0022-247X(65)90142-3

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Sapagovas MP: A difference scheme for two-dimensional elliptic problems with an integral condition. Litovsk. Mat. Sb. 1983,23(3):155–159.

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Skubachevskiĭ AL, Steblov GM: On the spectrum of differential operators with a domain that is not dense in. Doklady Akademii Nauk SSSR 1991,321(6):1158–1163. translated in Soviet Mathematics Doklady 44 (1992), no. 3, 870–875 translated in Soviet Mathematics Doklady 44 (1992), no. 3, 870–875

    MATH  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Givi Berikelashvili.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Berikelashvili, G. To a nonlocal generalization of the Dirichlet problem. J Inequal Appl 2006, 93858 (2006).

Download citation


  • Weak Solution
  • Sobolev Space
  • Dirichlet Problem
  • Boundary Dirichlet Condition
  • Integral Condition