Journal of Mathematical Sciences

, Volume 237, Issue 1, pp 1–29 | Cite as

Boundary-Value Problems with Shift and Beltrami Systems

  • G. Akhalaia
  • G. Giorgadze
  • N. Manjavidze


In this paper, we consider boundary-value problems with shift and show that such problems are equivalent to boundary-value problems for generalized analytic functions. We interpret the shift as a change of the complex structure on the complex plane with a given closed curve.


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  1. 1.
    L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Math. Stud. (1966)Google Scholar
  2. 2.
    G. Akhalaia, G. Giorgadze, V. Jikia, N. Kaldani, G. Makatsaria, abd N. Manjavidze, “Elliptic systems on Riemann surfaces,” Lect. Notes TICMI, 13, 3–167 (2012).Google Scholar
  3. 3.
    G. Akhalaia and G. Manjavidze, “Generalized analytic vectors,” in: Complex Methods for Partial Differential Equations, Kluwer Academic (1999), pp. 57–97.Google Scholar
  4. 4.
    L. Bers, Theory of Pseudo-Analytic Functions, Courant Inst., New York (1953).zbMATHGoogle Scholar
  5. 5.
    B. Bojarski, “Generalized solutions of a system of differential equations of first order and elliptic type with discontinuous coefficients,” Mat. Sb., 43 (85), No. 4, 451–563 (1957).MathSciNetGoogle Scholar
  6. 6.
    B. Bojarski, “On Hilbert generalized boundary-value problem,” Bull. Akad. Nauk Gruz. SSR, 25, No, 4, 385–390 (1960).Google Scholar
  7. 7.
    B. Bojarski, “Analysis of solvability of the boundary-value problems of the theory of functions,” in: Modern Problems of the Theory of Functions of Complex Variables [in Russian], Fizmatlit, Moscow (1961), pp. 57–79.Google Scholar
  8. 8.
    B. Bojarski, “Theory of generalized analytic vector,” Acad. Polon. Math., 17, No. 3, 281–320 (1966).MathSciNetCrossRefGoogle Scholar
  9. 9.
    B. Bojarski and G. Giorgadze, “Some analytical and geometric aspects of stable partial indices,” Proc. Vekua Inst. Appl. Math., 61-62, 14–32 (2011/12).Google Scholar
  10. 10.
    A. P. Calderon and A. Zygmund, “On singular integrals,” Am. J. Math., 79, No. 2, 289–307 (1956).CrossRefGoogle Scholar
  11. 11.
    G. Giorgadze, “Moduli space of complex structures,” J. Math. Sci. (N.Y), 160, No. 6, 697–716 (2009).MathSciNetCrossRefGoogle Scholar
  12. 12.
    G. Giorgadze, “On monodromy of generalized analytic functions,” J. Math. Sci. (N.Y), 132, No. 6, 716–738 (2006).MathSciNetCrossRefGoogle Scholar
  13. 13.
    G. Giorgadze, “Geometric aspects of generalized analytic functions,” in: Topics in Analysis and Its Application, Kluwer Academic (2004), pp. 69–81.Google Scholar
  14. 14.
    G. Giorgadze and V. Jikia, “On some properties of generalized analytic functions induced from irregular Carleman–Bers–Vekua equations,” Complex Var. Elliptic Equ., 58, No. 9, 1183–1194 (2013).MathSciNetCrossRefGoogle Scholar
  15. 15.
    G. Giorgadze and G. Khimshiashvili, “The Riemann–Hilbert problem in loop spaces,” Dokl. Math., 73, No. 2, 258–260 (2006).CrossRefGoogle Scholar
  16. 16.
    G. Giorgadze and N. Manjavidze, “On some constructive methods for the matrix Riemann–Hilbert boundary-value problem,” J. Math. Sci. (N.Y.), 195, No. 2, 146–174 (2013).MathSciNetCrossRefGoogle Scholar
  17. 17.
    G. Litvinchuk, Boundary-Value Problems and Singular Integral Equations with Shift, [in Russian], Nauka, Moscow (1977).zbMATHGoogle Scholar
  18. 18.
    G. Manjavidze, Boundary-Value Problems with Shift for Analytic and Generalized Analytic Functions, Tbilisi Univ. Press (1990).Google Scholar
  19. 19.
    G. Manjavidze, “Boundary-value problem of linear conjugation with piecewise-continuous matrix coefficients,” in: Mechanics of Solid Media and Related Problems of Analysis [in Russian], Nauka, Moscow (1972), pp. 297–304.Google Scholar
  20. 20.
    G. Manjavidze, “Application of the theory of generalized analytic functions in the study of boundary value problem of linear conjugation with displacement,” in: Differential and Integral Equations and Boundary-Value Problems, Tbilisi Univ. Press (1979).Google Scholar
  21. 21.
    G. Manjavidze, “Some properties of boundary-value problems of linear conjugation,” Proc. Vekua Inst. Appl. Math., 45–52 (1980).Google Scholar
  22. 22.
    G. Manjavidze, “Boundary-value problems of linear cojugation with displacement,” in: Complex Analysis and Application, Sofia (1984), pp. 375–382.Google Scholar
  23. 23.
    G. Manjavidze and B. Khvedelidze, “On Riemann–Privalov problem with continuous coefficients,” Dokl. Akad. Nauk SSSR, 123, No. 5, 791–194 (1958).MathSciNetGoogle Scholar
  24. 24.
    G. Manjavidze and B. Khvedelidze, “Boundary-value problem of linear conjugation and singular integral equations with Cauchy kernel with continuous coefficients,” Proc. Tbilisi Math. Inst., 28, 85–105 (1962).Google Scholar
  25. 25.
    G. Manjavidze and N. Manjavidze, “Boundary-value problems for analytic and generalized analytic functions,” J. Math. Sci. (N.Y.), 160, No. 6, 745–821 (2009).MathSciNetCrossRefGoogle Scholar
  26. 26.
    N. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, Leyden (1953).Google Scholar
  27. 27.
    I. Vekua, Generalized Analytic Functions, Pergamon, Oxford (1962).zbMATHGoogle Scholar
  28. 28.
    E. Zverovich and G. Litvinchuk, “Boundary-value problems with shift for analytic functions and singular functional equations,” Usp. Mat. Nauk, 23, No. 3, 67–121 (1968).MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.I. Vekua Institute of Applied MathematicsTbilisi State UniversityTbilisiGeorgia
  2. 2.Tbilisi State UniversityTbilisiGeorgia
  3. 3.Ilia State UniversityTbilisiGeorgia

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