Journal of Mathematical Sciences

, Volume 237, Issue 1, pp 30–109 | Cite as

Singular Generalized Analytic Functions

  • G. GiorgadzeEmail author
  • V. Jikia
  • G. Makatsaria


In this paper, we consider solution spaces for some class of singular elliptic systems on Riemann surfaces and boundary-value problems for solution spaces of such systems. We also discuss some relations for the kernels of the Carleman–Vekua equation. In particular, representations of these kernels in the form of generalized power functions are completely analogous to the classical Cauchy kernel expansion. The obtained results are applied to some problems of the theory of generalized analytic functions.


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  1. 1.
    L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, New York (1966).zbMATHGoogle 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, G. Makatsaria, and N. Manjavidze, “Some problems for elliptic systems on the plane,” in: Further Progress in Analysis, World Scientific, Hackensack, New Jersey (2009), pp. 303–310.Google Scholar
  4. 4.
    G. Akhalaia, G. Makatsaria, and N. Manjavidze, “On some qualitative issues for the first order elliptic systems in the plane,” in: Progress in Analysis and Its Applications, World Scientific, Hackensack, New Jersey (2010), pp. 67–73.Google Scholar
  5. 5.
    K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Univ. Press, Princeton, New Jersey (2009).zbMATHGoogle Scholar
  6. 6.
    L. Bers, Theory of Pseudo-Analytic Functions, New York Univ. (1950).Google Scholar
  7. 7.
    H. Begehr, Complex Analytic Methods for Partial Differential Equation. An Introductory Text, World Scientific, Singapore (1994).CrossRefGoogle Scholar
  8. 8.
    H. Begehr and D.-Q. Dai, “On continuous solutions of a generalized Cauchy–Riemann system with more than one singularity,” J. Differ. Equ., 196, 1, 67–90 (2004).MathSciNetCrossRefGoogle Scholar
  9. 9.
    N. K. Bliev, Generalized Analytic Functions in Fractional Spaces, Longman, Harlow (1997).zbMATHGoogle Scholar
  10. 10.
    B. Bojarski, “The geometry of the Riemann–Hilbert problem,” Contemp. Math., 242, 25–33 (1999).MathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Bojarski, “Homeomorphic solutions of the Beltrami systems,” Dokl. Akad. Nauk SSSR, 102, 661–664 (1955).MathSciNetGoogle Scholar
  12. 12.
    B. Bojarski, “On solutions of elliptic systems in the plane,” Dokl. Akad. Nauk SSSR, 102 871–874 (1955).MathSciNetGoogle Scholar
  13. 13.
    B. Bojarski, “Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients,” Mat. Sb., 43, 451–503 (1957).MathSciNetGoogle Scholar
  14. 14.
    B. Bojarski, “Quasiconformal mappings and general structural properties of systems of nonlinear equations elliptic in the sense of Lavrentiev,” in: Symp. Math., 18, Academic Press, London, (1976), pp. 485–499.Google Scholar
  15. 15.
    B. Bojarski, “Old and new on Beltrami equation,” in: Functional Analytic Methods in Complex Analysis and Applications to Partial Differential Equations, World Scientific, River Edge (1990).Google Scholar
  16. 16.
    B. Bojarski, “Geometry of the general Beltrami equations,” in: Complex Analysis and Potential Theory (T. Aliyev Azeroglu and P. M. Tamrazov, eds.), World Scientific, Singapore (2007), pp. 66–83.Google Scholar
  17. 17.
    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
  18. 18.
    A. P. Calderón and A. Zygmund, “On the existence of certain singular integrals,” Acta Math., 88, 85–139 (1952).MathSciNetCrossRefGoogle Scholar
  19. 19.
    E M. Chirka, Riemann Surfaces [in Russian], Moscow (2006).Google Scholar
  20. 20.
    R. Gilbert and J. Buchanan, First-Order Elliptic Systems: A Function Theoretic Approach, Academic Press (1983).Google Scholar
  21. 21.
    G. Giorgadze, “Geometric aspects of generalized analytic functions,” in: Topics in Analysis and Its Applications, NATO Sci. Ser., 147 (2004), pp. 69–81.Google Scholar
  22. 22.
    G. Giorgadze, “On monodromy of generalized analytic functions,” J. Math. Sci. (N.Y.), 132, No. 6, 716–738 (2006).MathSciNetCrossRefGoogle Scholar
  23. 23.
    G. Giorgadze, “Moduli space of complex structures,” J. Math. Sci. (N.Y.), 160, No. 6, 697–716 (2009).MathSciNetCrossRefGoogle Scholar
  24. 24.
    G. Giorgadze, “G-systems and holomorphic principal bundles on Riemann surfaces,” J. Dynam. Control Syst., 8, No. 2, 245–291 (2002).MathSciNetCrossRefGoogle Scholar
  25. 25.
    G. Giorgadze, “Regular systems on Riemann surfaces,” J. Math. Sci. (N.Y.), 132, No. 5, 5347–5399 (2003).MathSciNetCrossRefGoogle Scholar
  26. 26.
    G. Giorgadze and V. Jikia, “On some properties of generalized analytic functions induced from irregular Carleman–Bers–Vekua equations,” Compl. Var. Elliptic Equ., 58, No. 9, 1183–1194 (2013).MathSciNetCrossRefGoogle Scholar
  27. 27.
    G. Giorgadze and G. Khimshiashvili, “The Riemann–Hilbert problem in loop spaces,” Dokl. Math., 73, No. 2, 258–260 (2006).CrossRefGoogle Scholar
  28. 28.
    P. G. Grinevich and R. G. Novikov, “Generalized analytic functions, Moutard-type transforms and holomorphic maps,” ArXiv: Math.CV 1520/00343 (2015).Google Scholar
  29. 29.
    N. Kaldani, “Generalized power functions,” J. Math. Sci. (N.Y.), 15, No. 2, 181–197 (2013).MathSciNetCrossRefGoogle Scholar
  30. 30.
    V. Kravchenko, Applied Pseudoanalytic Function Theory, Birkhäuser Verlag, Basel (2009).CrossRefGoogle Scholar
  31. 31.
    V. Kravchenko and A. Meziani, “On the two-dimensional stationary Schrödinger equation with a singular potential,” J. Math. Anal. Appl., 377, 420–427 (2011).MathSciNetCrossRefGoogle Scholar
  32. 32.
    R. Kühnau, Handbook of Complex Analysis: Geometric Function Theory, Vols. 1-2, Elsevier, Amsterdam (2002, 2005).Google Scholar
  33. 33.
    O. Lehto, Univalent Functions and Teichmüller Spaces, Springer-Verlag, Berlin (1987).CrossRefGoogle Scholar
  34. 34.
    M. Lukomskaya, “Solution of some systems of partial differential equations by means of inclusion in a cycle,” Prikl. Mat. Mekh., 17, 745–747 (1953).MathSciNetGoogle Scholar
  35. 35.
    G. Makatsaria, “Singular points of solutions of some elliptic systems on the plane,” J. Math. Sci. (N.Y.), 160, No. 6, 737–744 (2009).MathSciNetCrossRefGoogle Scholar
  36. 36.
    G. Makatsaria, “The behaviour of solutions of the Carleman–Vekua equation with polar singularities in the neighbourhood of fixed singular point,” Bull. Akad. Nauk. Gruz. SSR, 107, No. 3, 473–476 (1982).Google Scholar
  37. 37.
    G. Makatsaria, “Correct boundary value problems for some classes of singular elliptic differential equations on a plane,” Mem. Differ. Eq. Math. Phys., 34, 115–134 (2005).MathSciNetzbMATHGoogle Scholar
  38. 38.
    G. Makatsaria and N. Manjavidze, “Riemann–Hilbert boundary value problem for Carleman–Vekua equation with polar singularities,” Bull. Georgian Nat. Acad. Sci., 9, No. 3, 12–19 (2015).MathSciNetzbMATHGoogle Scholar
  39. 39.
    V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer-Verlag (1991).Google Scholar
  40. 40.
    L. G. Mikhailov, A New Class of Singular Equations and Its Application to Differential Equations with Singular Coefficients, Akademik-Verlag, Berlin (1970).zbMATHGoogle Scholar
  41. 41.
    N. I. Muskhelishvili, Singular Integral Equations, P. Noordhoff, Groningen (1953).zbMATHGoogle Scholar
  42. 42.
    M. H. Protter, “The periodicity problem for pseudoanalytic functions,” Ann. Math., 64, No. 2, 154–174 (1956).MathSciNetCrossRefGoogle Scholar
  43. 43.
    G. N. Polozhy, Generalization of the Theory of Analytic Functions of Complex Variables: p-Analytic and (p, q)-Analytic Functions and Some Aplications [in Russian], Kiev Univ. (1965).Google Scholar
  44. 44.
    M. Reissig and A. Timofeev, “Special Vekua equations with singular coefficients,” Appl. Anal., 73, 187–199 (1999).MathSciNetCrossRefGoogle Scholar
  45. 45.
    H. Renelt, Elliptic Systems and Quasiconformal Mappings, Wiley (1988).Google Scholar
  46. 46.
    Yu. Rodin and A. Turakulov, “Boundary-value problem for generalized analytic functions with singular coefficients on the compact Riemann surfaces,” Bull. Acad. Sci. Georgian SSR, 96, No. 1, 21–24 (1979).MathSciNetzbMATHGoogle Scholar
  47. 47.
    Yu. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Reidel, Dordrecht (1988).CrossRefGoogle Scholar
  48. 48.
    R. Saks, “Riemann–Hilbert problem for new class of model Vekua equations with singular degeneration,” Appl. Anal., 73, Nos. 1-2, 201–211 (1999).MathSciNetCrossRefGoogle Scholar
  49. 49.
    A. Suzko and G. Giorgadze, “Darboux transformations for the generalized Schr¨odinger equation,” Phys. Atom. Nucl., 70, No. 3, 607–610 (2007).CrossRefGoogle Scholar
  50. 50.
    A. Yu. Timofeev, “Dirichlet problem for generalized Cauchy–Riemann system,” in: Recent Developments in Generalized Analytic Functions [in Russian], Tbilisi State University (2011), pp. 143–147.Google Scholar
  51. 51.
    E. Titchmarsch, The Theory of Functions, Oxford Univ. Press (1939).Google Scholar
  52. 52.
    A. Tungatarov, “A class of Carleman–Vekua equations with a singular point,” Izv. Akad. Nauk Kazakh. SSR, Ser. Fiz.-Mat., 5, 58–60 (1988).MathSciNetzbMATHGoogle Scholar
  53. 53.
    Z. D. Usmanov, Generalized Cauchy–Riemann Systems with a Singular Point, Longman, Harlow (1997).zbMATHGoogle Scholar
  54. 54.
    I. N. Vekua, Generalized Analytic Functions [in Russian], Nauka (1988).Google Scholar
  55. 55.
    I. N. Vekua, “On one class of the elliptic systems with singularities,” in: Proc. Int. Conf. on Functional Analysis and Related Topics, Tokyo (1969), pp. 142–147.Google Scholar
  56. 56.
    V. S. Vinogradov, “Liouville’s theorem for generalized analytic functions,” Dokl. Akad. Nauk SSSR, 183, No. 3, 503–508 (1968).MathSciNetzbMATHGoogle Scholar
  57. 57.
    W. L. Wendland, Elliptic Systems in the Plane, Pitman (1979).Google Scholar

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Authors and Affiliations

  1. 1.Iv. Javakhishvili Tbilisi State UniversityTbilisiGeorgia
  2. 2.I. Vekua Institute of Applied MathematicsTbilisi State UniversityTbilisiGeorgia
  3. 3.Saint Andrew The First-Called Georgian University of Patriarchate of GeorgiaTbilisiGeorgia

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