Differential Equations

, Volume 45, Issue 1, pp 86–101 | Cite as

Boundary value problem for the Laplace equation outside cuts on the plane with different conditions of the third kind on opposite sides of the cuts

  • P. A. Krutitskii
Partial Differential Equations


We consider a boundary value problem for the Laplace equation outside cuts on a plane. Boundary conditions of the third kind, which are in general different on different sides of each cut, are posed on the cuts. We show that the classical solution of the problem exists and is unique. We obtain an integral representation for the solution of the problem in the form of potentials whose densities are found from a uniquely solvable system of Fredholm integral equations of the second kind.


Operator Mapping Laplace Equation Compact Operator Singular Integral Equation Neumann Problem 
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  1. 1.
    Muskhelishvili, N.I., Singulyarnye integral’nye uravneniya (Singular Integral Equations), Moscow: Nauka, 1968.Google Scholar
  2. 2.
    Vainikko, G.M., Lifanov, I.K., and Poltavskii, L.N., Chislennye metody v gipersingulyarnykh integral’nykh uravneniyakh i ikh prilozheniya (Numerical Methods for Hypersingular Integral Equations and Their Applications), Moscow, 2001.Google Scholar
  3. 3.
    Krutitskii, P.A., Z. Anal. Anwendungen, 1997, vol. 16, no. 3, pp. 739–747.MATHMathSciNetGoogle Scholar
  4. 4.
    Krutitskii, P.A., IMA J. Appl. Math., 1998, vol. 60, pp. 285–297.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Krutitskii, P.A., Chikilev, A.O., Krutitskaya, N.Ch., and Kolybasova, V.V., Math. Methods Appl. Sci., 2005, vol. 28, pp. 593–606.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Krutitskii, P.A., Differ. Uravn., 1997, vol. 33, no. 9, pp. 1181–1190.MathSciNetGoogle Scholar
  7. 7.
    Krutitskii, P.A., Zh. Vychisl. Mat. Mat. Fiz., 1994, vol. 34, No. 8–9, pp. 1237–1257.MathSciNetGoogle Scholar
  8. 8.
    Krutitskii, P.A., Zh. Vychisl. Mat. Mat. Fiz., 1994, vol. 34, No. 11, pp. 1652–1665.MathSciNetGoogle Scholar
  9. 9.
    Lifanov, I.K., Metod singulyarnykh uravnenii i chislennyi eksperiment (The Method of Singular Integral Equations and Numerical Experiment), Moscow: Yanus, 1995.MATHGoogle Scholar
  10. 10.
    Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1981.Google Scholar
  11. 11.
    Gabov, S.A., Mat. Sb., 1977, vol. 103(145), no. 4, pp. 490–504.MathSciNetGoogle Scholar
  12. 12.
    Krutitskii, P.A. and Kolybasova, V.V., Differ. Uravn., 2005, vol. 41, no. 9, pp. 1155–1165.MathSciNetGoogle Scholar
  13. 13.
    Mikhlin, S.G., Kurs matematicheskoi fiziki (A Course in Mathematical Physics), Moscow: Nauka, 1968.Google Scholar
  14. 14.
    Krutitskii, P.A., Math. Methods Appl. Sci., 1995, vol. 18, pp. 897–925.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Nauka, 1981.Google Scholar
  16. 16.
    Mat. entsikl. (Mathematical Encyclopedia), Moscow, 1977, vol. 1.Google Scholar
  17. 17.
    Funktsional’nyi analiz (Functional Analysis), Krein, S.G., Ed., Moscow: Nauka, 1964.Google Scholar
  18. 18.
    Kantorovich, A.V. and Akilov, G.P., Funktsional’nyi analiz (Functional Analysis), Moscow: Nauka, 1984.Google Scholar
  19. 19.
    Trenogin, V.A., Funktsional’nyi analiz (Functional Analysis), Moscow: Nauka, 1980.Google Scholar
  20. 20.
    Smirnov, V.I., Kurs vysshei matematiki (A Course of Higher Mathematics), Moscow: Gosudarstv. Izdat. Tehn.-Teor. Lit., 1951, vol. 4.Google Scholar

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© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  • P. A. Krutitskii
    • 1
  1. 1.Institute for Applied MathematicsRussian Academy of SciencesMoscowRussia

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