The Riemann and Dirichlet Problems with Data from the Grand Lebesgue Spaces

Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 229)


In Section 1, we present a solution of the following boundary value problem: find an analytic function Φon the plane cut along a closed piecewisesmooth curve Γ which is represented by a Cauchy type integral with a density from the Grand Lebesgue Space \( L^{p)},\theta(\Gamma)(1 < p < \infty, 0 < \theta < \infty) \) and whose boundary values satisfy the conjugacy condition
$$ \Phi^{+}(t)=G(t)\Phi^{-}(t)+g(t),\quad t \in \Gamma $$
Here G and g are functions defined on Γ such that G is a piecewise continuous function, \( G(t)\neq 0 \) and \( g \in L^{{p}),\theta}(\Gamma) \) The conditions for the problem to be solvable are established and the solutions are constructed in explicit form.

In Section 2, the Dirichlet problem for harmonic functions, real parts of Cauchy type integrals with densities from weighted generalized Grand Lebesgue Spaces is studied when boundary data belong to the same space.


Boundary value problem analytic function Cauchy type integral the Grand Lebesgue Space piecewise smooth boundary piecewise continuous coefficient Dirichlet problem Lyapunov contour. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H.G.W. Begehr, Complex analytic methods for partial differential equations. An introductory text. World Scientific Publishing Co., Inc. River Edge, NJ, 1994.Google Scholar
  2. [2]
    A. Böttcher and Y.I. Karlovich, Carleson curves, Muckenhoupt weights, and Toeplitz operators. Progress in Mathematics, 154. Birkhäuser Verlag, Basel, 1997.Google Scholar
  3. [3]
    L. Diening, P. Harjulehto, P. Hästo, and M. Růžička, Lebesgue and Sobolev spaces with variable exponent. Lecture Notes in Math., vol. 2017, Springer-Verlag, 2011.Google Scholar
  4. [4]
    P.L. Duren, Theory of H p spaces. Pure and Applied Mathematics, Vol. 38. Academic Press, New York–London, 1970.Google Scholar
  5. [5]
    E.M. Dynkin, Methods of the theory of singular integrals. (Hilbert transforms and Calderón–Zygmund theory.) (Russian) Itogi Nauki i Tekhniki, Sovrem. Probl. Mat. Fund. Naprav. 15, Moscow, 1987, 197–292.Google Scholar
  6. [6]
    A. Fiorenza, B. Gupta and P. Jain, The maximal theorem in weighted grand Lebesgue spaces, Studia Math. 188(2008), No. 2, 123–133.Google Scholar
  7. [7]
    A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right-hand side in L 1.Studia Math. 127 (1998), no. 3, 223–231.MathSciNetMATHGoogle Scholar
  8. [8]
    F.D. Gakhov, Boundary value problems. (Russian) Third edition, revised and augmented Izdat. “Nauka”, Moscow, 1977. English Translation of the second edition edited by I.N. Sneddon, Pergamon Press International Series of Monographs in Pure and Applied Mathematics, Volume 85 Addison Wesley 1965.Google Scholar
  9. [9]
    G.M. Goluzin, Geometrical theory of functions of a complex variable. (Russian) Second edition. “Nauka”, Moscow, 1966.Google Scholar
  10. [10]
    L. Greco, T. Iwaniec, and C. Sbordone, Inverting the p-harmonic operator. Manuscripta Math. 92 (1997), no. 2, 249–258.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses. Arch. Rational Mech. Anal. 119 (1992), no. 2, 129–143.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    G. Khuskivadze, V. Kokilashvili, and V. Paatashvili, Boundary value problems for analytic and harmonic functions in domains with nonsmooth boundaries. Applications to conformal mappings. Mem. Differential Equations Math. Phys. 14 (1998), 195 pp.Google Scholar
  13. [13]
    G. Khuskivadze, V. Kokilashvili, and V. Paatashvili, The Dirichlet problem for variable exponent Smirnov class harmonic functions in doubly-connected domains. Mem. Differential Equations Math. Phys. 52 (2011), 131–156.MathSciNetMATHGoogle Scholar
  14. [14]
    B.V. Khvedelidze, Linear discontinuous boundary problems in the theory of functions, singular integral equations and some of their applications. (Russian) Trudy Tbiliss. Mat. Inst. Razmadze 23 (1956), 3–158.Google Scholar
  15. [15]
    B.V. Khvedelidze, The method of Cauchy type integrals in discontinuous boundary value problems of the theory of holomorphic functions of a complex variable. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat., 7, VINITI, Moscow, 1975, 5–162; translation in J. Soviet Math. 7 (1977), no. 3, 309–414.Google Scholar
  16. [16]
    V. Kokilashvili, V. Paatashvili, and S. Samko, Boundary value problems for analytic functions in the class of Cauchy-type integrals with density in L p(·)(Γ).Bound. Value Probl. 2005, no. 1, 43–71.Google Scholar
  17. [17]
    V. Kokilashvili and V. Paatashvili, The Riemann–Hilbert problem in weighted classes of Cauchy type integrals with density from L p(·)(Γ).Complex Anal. Oper. Theory 2 (2008), no. 4, 569–591.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    V. Kokilashvili, The Riemann boundary value problem for analytic functions in the frame of grand L p spaces. Bull. Georgian Natl. Acad. Sci. (N.S.) 4 (2010), no. 1, 5–7.Google Scholar
  19. [19]
    V. Kokilashvili, Boundedness criteria for singular integrals in weighted grand Lebesgue spaces. J. Math. Sci. 170 (2010), no. 1, 20–23.MathSciNetCrossRefGoogle Scholar
  20. [20]
    V. Kokilashvili and S. Samko, Boundedness of weighted singular integral operators on Carleson curves in Grand Lebesgue spaces. In: ICNAAM 2010: Intern. Conf. Nam. Anal. and Appl. Math. 1281, pp. 490–493. ATP Conf. Proc., 2010.Google Scholar
  21. [21]
    V. Kokilashvili and S. Samko, Boundedness of weighted singular integral operators in Grand Lebesgue spaces. Georgian Math. J. 18 (2011), 259–269.MathSciNetMATHGoogle Scholar
  22. [22]
    V. Kokilashvili and V. Paatashvili, The Dirichlet problem for harmonic functions from variable exponent Smirnov classes in domains with piecewise smooth boundary. J. Math. Sci 172 (2011), no. 3, 1–21.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Z. Meshveliani, V. Paatashvili, On Smirnov classes of harmonic functions, and the Dirichlet problem. Proc. A. Razmadze Math. Inst. 123 (2000), 61–91.MathSciNetMATHGoogle Scholar
  24. [24]
    N.I. Muskhelishvili, Singular integral equations. (Russian) Third, corrected and augmented edition. Izdat. “Nauka”, Moscow, 1968. English Translation edited by J.R. Radok, Noordhoff International Publishing Leyden, 1977.Google Scholar
  25. [25]
    B. Riemann, Grundlagen Für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse. Worke, Leipzig, 1867.Google Scholar
  26. [26]
    J.L. Rubio de Francia, Factorization and extrapolation of weights. Bull. Amer. Math. Soc. (N.S.), 7(1982), 393–395.Google Scholar
  27. [27]
    R. Wegmann, A.H.M. Murid, M.M.S. Nasser, The Riemann-Hilbert problem and the generalized Neumann kernel. Journal of Computational and Applied Mathematics. 182 (2005), 388–415.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710.Google Scholar
  29. [29]
    V.V. Zhikov, S.E. Pastukhova, On the improved integrability of the gradient of solutions of elliptic equations with a variable nonlinearity exponent. (Russian) Mat. Sb. 199 (2008), no. 12, 19–52; translation in Sb. Math. 199 (2008), no. 11-12, 1751–1782.Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.A. Razmadze Mathematical Institute I. Javakhishvili Tbilisi State UniversityTbilisiGeorgia

Personalised recommendations