Open Access

To a nonlocal generalization of the Dirichlet problem

Journal of Inequalities and Applications20062006:93858

DOI: 10.1155/JIA/2006/93858

Received: 20 August 2004

Accepted: 22 September 2004

Published: 23 January 2006


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.


Authors’ Affiliations

A. Razmadze Mathematical Institute, Georgian Academy of Sciences


  1. Cannon JR: The solution of the heat equation subject to the specification of energy. Quarterly of Applied Mathematics 1963,21(2):155–160.MathSciNetMATHGoogle Scholar
  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)00169-7MathSciNetView ArticleMATHGoogle Scholar
  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. 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.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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, 101–136MathSciNetGoogle Scholar
  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. Martynyuk AE: Some new applications of methods of Galerkin type. Matematicheskiĭ Sbornik 1959, 49 (91): 85–108.MathSciNetGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  11. Paneyakh BP: Some nonlocal boundary value problems for linear differential operators. Matematicheskie Zametki 1984,35(3):425–434.MathSciNetMATHGoogle Scholar
  12. Petryshyn WV: On a class of and non operators and operator equations. Journal of Mathematical Analysis and Applications 1965,10(1):1–24. 10.1016/0022-247X(65)90142-3MathSciNetView ArticleMATHGoogle Scholar
  13. Sapagovas MP: A difference scheme for two-dimensional elliptic problems with an integral condition. Litovsk. Mat. Sb. 1983,23(3):155–159.MathSciNetMATHGoogle Scholar
  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, 870–875MATHGoogle Scholar


© Berikelashvili 2006

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