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Boundary-value problems for analytic and generalized analytic functions

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Abstract

This paper deals with boundary value problems of linear conjugation with shift for analytic functions in the case of piecewise continuous coefficients. Int main goal is the construction of a canonical matrix for these problems. Boundary value problems with shift for generalized analytic functions and vectors as well as differential boundary value problems are studied.

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References

  1. L. V. Ahlfors, Lectures on Quasi-Conformal Mappings Van Nostrand Math. Stud. (1966).

  2. G. Airapetian, “Boundary-value problem of linear conjugation with displacement in the class L 1,” Izv. Akad. Nauk Arm. SSR, Ser. Mat., 22, No. 3, 238–252 (1987).

    Google Scholar 

  3. G. Alexandria, “On one general boundary-value problem of linear conjugation with given displacements in the class of generalized analytic functions,” Bull. Acad. Sci. Georgia, 21, No. 3, 257–262 (1958).

    Google Scholar 

  4. G. Alexandria, “Some boundary-value problems with different displacement in the class of generalized analytic function,” Proc. Tbilisi Univ., 86, 321–331 (1960).

    Google Scholar 

  5. G. Alexandria, “On one boundary-value problem for generalized analytic functions,” Proc. Tbilisi Univ., 56, 135–139 (1995).

    Google Scholar 

  6. G. Akhalaia and G. Manjavidze, “Differential boundary-value problems for generalized analytic functions,” Proc. Vekua Inst. Appl. Math., 21, 5–17 (1987).

    Google Scholar 

  7. G. Akhalaia and G. Manjavidze, “Generalized analytic vectors,” in: Complex Methods for Partial Differential Equations, Kluwer Academic Publ. (1999), pp. 57–97.

  8. F. Bagemill and G. Piranian, “Absolutely convergent power series,” Ann. Univ. Sci. Budapest Eotvos Sec. Math., 51, Nos. 3–4, 27–34 (1960).

    Google Scholar 

  9. R. Bantsuri, “Contact problem for wedge with elastic strengthening,” Dokl. Akad. Nauk SSSR, 211, No. 4, 797–800 (1977).

    MathSciNet  Google Scholar 

  10. R. Bantsuri and R. Isakhanov, “On one mixed problem of the theory of analytic functions,” Proc. Vekua Inst. Appl. Math., 2, No. 1, 19–22 (1986).

    Google Scholar 

  11. H. Begehr, Complex Analytic Methods for Partial Differential Equations, World Scientific (1994).

  12. L. Bers, Theory of Pseudoanalytic Functions, Courant Inst., New York (1953).

    Google Scholar 

  13. L. Bers and L. Nirenberg, “On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications,” in: Convegno Int. sulle Equaz. Lin. alle Deriv. Parz., Triest (1954), pp. 111–140.

  14. I. Bikchantaev, “Linear conjugation problems for systems of differential equations of first order of elliptic type with derivatives in the boundary conditions,” Differ. Equ., 21, No. 6, 1007–1013 (1985).

    MathSciNet  Google Scholar 

  15. I. Bikchantaev, “Hilbert problem with derivative for generalized analytic functions,” Proc. Semin. Boundary-Value Probl. Kazan. Univ., 22, 54–59 (1985).

    MATH  MathSciNet  Google Scholar 

  16. G. D. Birkhoff, “A theorem on matrices of analytic functions,” Math. Ann., 74, No. 1, 122–133 (1913); Coll. Math. Papers, 1, 240–251 (1950).

    Article  MathSciNet  Google Scholar 

  17. G. D. Birkhoff, “The generalized Riemann problem for linear differential equations and the apllied problems for linear difference and q-difference equations,” Proc. Amer. Acad. Arts Sci., 49, 521–568 (1913); Coll. Math. Papers, 1, 259–306 (1950).

    Google Scholar 

  18. A. Bitsadze, Boundary-Value Problems for Elliptic Equations of Second Order [Russian], Nauka, Moscow (1966).

    Google Scholar 

  19. A. Bitsadze, Some Classes of Partial Differential Equations [Russian], Nauka, Moscow (1981).

    MATH  Google Scholar 

  20. N. Bliev, Generalized Analytic Functions in Fractional Spaces, Addison-Wesley, Longman, Harlow (1998).

    Google Scholar 

  21. B. Bojarski, “Generalized solutions of a system of elliptic differential equations of first order with discontinuous coefficients,” Mat. Sb., 43, No. 4, 451–563 (1957).

    MathSciNet  Google Scholar 

  22. B. Bojarski, “On Hilbert generalized boundary-value problem,” Bull Akad. Nauk Georgia, 25, No. 4, 385–390 (1960).

    MathSciNet  Google Scholar 

  23. B. Bojarski, “Analysis of solvability of boundary-value problems of the theory of functions,” in: Modern Problems of the Theory of Functions of Complex Variable [in Russian] (A. Markushevich, ed.), Fizmatlit, Moscow (1961), pp. 57–79.

    Google Scholar 

  24. B. Bojarski, “Theory of generalized analytic vectors,” Acad. Pol. Math., 17, No. 3, 281–320 (1966).

    MathSciNet  Google Scholar 

  25. V. Cabaila, “Certain properties of the functions of class H’ p and interpolation problems,” Lit. Mat. Sb., 10, No. 4, 471–490 (1970).

    Google Scholar 

  26. A. P. Calderon, “Cauchy integrals on Lipschits curves and related operator,” Proc. Natl. Acad. Sci. U.S.A., 74, No. 4, 1324–1327 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  27. A. P. Calderon, C. P. Calderon, E. B. Fabes, M. Jodient, and I. V. Riviere, “Applications of the Cauchy integral along Lipschitz curves,” Bull. Am. Math. Soc., 84, No. 2, 287–290 (1978).

    Article  MATH  Google Scholar 

  28. A. P. Calderon and A. Zygmund, “On singular integrals,” Am. J. Math., 79, No. 2, 289–307 (1956).

    Article  MathSciNet  Google Scholar 

  29. T. Carleman, “Sur la theorie des equations integrals et ses aplications,” Verband des Int. Math. Kongr., 1, Zurich (1932), pp. 138–151.

    Google Scholar 

  30. Yu. Chersi, “Singular integral equations with shift,” Dokl. Akad Nauk Ukr. SSR, Ser. A, 12, 15–18 (1980).

    Google Scholar 

  31. L. Chibrikova, “On boundary-value problems for rectangle,” Uch. Zap. Kazan Univ., 123, No. 9, 3–14 (1964).

    Google Scholar 

  32. T. Chochiev, “On one boundary-value problem of the theory of functions,” Bull. Acad. Sci. Georgia, 27, No. 3, 263–269 (1961).

    Google Scholar 

  33. K Clancy and I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Birkhäuser, Basel–Boston–Stuttgart (1981).

    Google Scholar 

  34. R. Coifman, A. Macintosh, and Y. Meyer, “L’-Integrals de Cauchy definit un operator bourne sur L 2 peur les courves Lipschitzeunes,” Ann. Math., 116, No. 2, 361–389 (1986).

    Article  Google Scholar 

  35. I. Danilyuk, “On the problem of oblique derivative,” Sib. Mat. Zh., 3, No. 1, 17–55 (1962).

    MATH  Google Scholar 

  36. I. Danilyuk, Nonregular Boundary-Value Problems on a Plane [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  37. G. David, “Operateurs integraux singuliereur certaines courbec du plou complex,” Ann. Sci. Ec. Norm. Super., Ser. 4, 157–189 (1984).

    Google Scholar 

  38. A. Douglis, “A function-theoretic approach to elliptic systems of equations in two variables,” Commun. Pure. Appl. Math., 6, 259–289 (1953).

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Drachinski, “On Hilbert–Privalov problem with given displacement in the space of summarized functions,” Proc. Georgian Polytech. Inst., 8, 53–56 (1963).

    Google Scholar 

  40. P. L. Duren, Theory of H p -Spaces, Academic Press, New York–London (1970).

    MATH  Google Scholar 

  41. E. Dynkin, “Methods of the theory of singular integrals. Hilbert transformation and the theory of Calderon–Zygmund,” in: Sovr. Probl. Mat., VINITI, Moscow (1987), pp. 197–292.

    Google Scholar 

  42. E. Dynkin and B. Osilenker, “Weight estimate of singular integrals and its applications,” in: Sovr. Probl. Mat., 21, VINITI, Moscow (1983), pp. 42–129.

    Google Scholar 

  43. A. Dzhuraev, Systems of Equations of Compound Type [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  44. A. Dzhuraev, Method of Singular Integral Equations, Longman, Harlow (1992).

    Google Scholar 

  45. F. D. Gakhov, “Riemann problem for the system and for the pair of functions,” Usp. Mat. Nauk, 7, No. 4, 3–54 (1952).

    MATH  Google Scholar 

  46. F. D. Gakhov, Boundary-Value Problems, Pergamon, Oxford (1962).

    Google Scholar 

  47. F. Gantmacher, Theory of Matrices, Berlin (1958).

  48. I. Garcia-Cuevra and I. L. Rubio de Francia, “Weighted norm inequalities and related topics,” Math. Stud., 116, 604 (1986).

    Google Scholar 

  49. C. Gattegno and R. Ostrowski, “Representation conforme à la frontiére; domains particuliers,” in: Mem. Sci. Math., 110, Paris (1949).

  50. R. P. Gilbert and I. L. Buchanan, First-Order Elliptic Systems. A Function-Theoretic Approach, Academic Press, New York–London (1983).

    MATH  Google Scholar 

  51. G. Giorgadze, “On monodromy of generalized analytic functions,” J. Math. Sci., 132, No. 6, 716–738 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  52. G. Giorgadze, “Geometric aspects of generalized analytic functions,” in: Topics in Analysis and Its Application, Kluwer Acad. Publ. (2004), pp. 69–81.

  53. I. Gohberg, “On linear operators analytically depending on parameters,” Dokl. Akad. Nauk SSSR, 78, No. 4, 629–832 (1951).

    Google Scholar 

  54. I. Gohberg and M. Krein, “Systems of integral equations on the half-line with kernels depending on the difference of arguments,” Usp. Mat. Nauk, 13, No. 2, 3–72 (1958).

    MathSciNet  Google Scholar 

  55. I. Gohberg and N. Krupnik, “Systems of singular integral equations in weighted L p spaces,” Dokl. Akad. Nauk SSSR, 186, No. 5, 998–1001 (1969).

    MathSciNet  Google Scholar 

  56. I. Gohberg and N. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Equations [in Russian], Shtiintsa, Kishinev (1973).

    Google Scholar 

  57. B. Goldschmith, “Funktionen theoretische Eigenschaften verallgemeinerter analytischer vektors,” Math. Nachrichten, 90, 57–90 (1979).

    Article  Google Scholar 

  58. G. Golusin, Geometric Theory of Functions of Complex Variables [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  59. E. Gordadze, “On singular integrals with Cauchy kernels,” Bull. Acad. Sci. Georgia, 27, No. 3, 521–526 (1965).

    MathSciNet  Google Scholar 

  60. S. Grudski, Compactness of one singular operator [in Russian], Rostov Univ., Deposited at VINITI, No. 4856-80 (1982)

  61. S. Grudski and V. Dibin, “Riemann boundary problem with discontinuous almost periodic coefficients,” Dokl. Akad. Nauk SSSR, 237, No. 1, 21–24 (1977).

    MathSciNet  Google Scholar 

  62. C. Hasemann, Anwendung der Theorie der Integralgleichungen auf lineare Randwertaufgaben, Gottingen (1907).

  63. R. A. Hunt, B. Muchenhoupt, and R. L.Wheeden, “Weighted norm inequalities for the conjugate function and Hilbert transformation,” Trans. Am. Math. Soc., 176, 227–251 (1973).

    Article  MATH  Google Scholar 

  64. R. Isakhanov, “On one class of singular integral equations,” Bull. Acad. Sci. Georgia, 20, No. 1, 9–21 (1958).

    Google Scholar 

  65. R. Isakhanov, “On some boundary-value problems of linear conjugation,” Proc. Tbilisi Math. Inst., 28, 73–84 (1962).

    MATH  Google Scholar 

  66. R. Isakhanov, “Boundary-value problem of linear conjugation for finite number of domains,” Proc. Tbilisi Math. Inst., 44, 79–99 (1973).

    Google Scholar 

  67. R. Isakhanov, “General boundary-value problem for holomorphic functions,” Proc. Tbilisi Math. Inst., 65, 99–109 (1980).

    MATH  MathSciNet  Google Scholar 

  68. I. L. Journe, Calderon–Zigmund Operators, Pseudo-Differential Operators, and the Cauchy Integral of Calderon, Lect. Notes Math., 994, Springer-Verlag (1983).

  69. G. Kapanadze, “General Hilbert problem with displacements for several unknown functions,” Proc. Tbilisi Univ., 239, No. 5, 39–156 (1985).

    Google Scholar 

  70. G. Kapanadze, “General Hilbert problem with displacment for piecewise-holomorphic vectors,” Proc. Vekua Inst. Appl. Math., 1, No. 1, 94–97 (1985).

    Google Scholar 

  71. N. Karapetiants and S. Samko, Equations with Involutive Operators and Their Applications [in Russian], Rostov State Univ. (1988).

  72. Yu. Karlowich and V. Kravchenko, “Systems of singular integral equations with shift,” Mat. Sb., 116, No. 1, 87–110 (1981).

    MathSciNet  Google Scholar 

  73. Yu. Karlowich, V. Kravchenko, and G. Litvinchuk, “Noether theory of singular integral opertors with shift,” Izv. Vyssh. Ucheb. Zaved. Ser. Mat., 4, 3–27 (1983).

    Google Scholar 

  74. V. Khavin, “Boundary-value properties of Cauchy-type integrals and harmonically conjugate functions in domains with rectifiable boundaries,” Mat. Sb., 68, No. 4, 499–514 (1965). (Russian)

    MathSciNet  Google Scholar 

  75. G. Khimshiashvili, “Geometric aspects of Riemann–Hilbert problems,” Mem. Differ. Equations Math. Phys., 27, 1–114 (2002).

    MATH  MathSciNet  Google Scholar 

  76. B. Khvedelidze, “Linear discontinuous boundary-value problems of the theory of functions, singular integral equations and some applications,” Proc. Tbilisi Math. Inst., 23, 3–158 (1957).

    Google Scholar 

  77. B. Khvedelidze, “On Reimann–Hilbert discontinous problem with given displacement,” Bull. Acad. Sci. Georgia, 21, No. 4, 385–389 (1957).

    Google Scholar 

  78. B. Khvedelidze, “The method of Cauchy type integrals in discontinuous boundary value problems of the theory of holomorphic functions of one complex variable,” in: Sovr. Probl. Mat., VINITI, Moscow (1975), pp. 5–162.

    Google Scholar 

  79. V. Kokilashvili, “On maximal singular integral operators with Cauchy kernels,” Proc. Tbilisi Math. Inst., 55, 38–58 (1977).

    Google Scholar 

  80. V. Kokilashvili, “Singular and bisingular operators in weighted spaces,” Proc. Tbilisi Math. Inst., 69, 51–79 (1982).

    MATH  MathSciNet  Google Scholar 

  81. R. Kordadze, “On one class of singular integral equations with shift,” Dokl. Akad. Nauk SSSR, 168, No. 6, 1245–1248 (1966).

    MathSciNet  Google Scholar 

  82. R. Kordadze, “Index of singular integro-differencial operators,” Dokl. Akad. Nauk SSSR, 185, No. 4, 753–756 (1969).

    MathSciNet  Google Scholar 

  83. M. Krasnoselski, P. Zabreiko, E. Pustilnik, and P. Sobolevski, Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  84. V. Kravchenko and G. Litvinchuk, Singular Integral Equations with Non-Carleman Shift and Some Applications in the Theory of Boundary-Value Problems for Differential Equations of Mixed Type [in Russian], Metsniereba, Tbilisi (1973).

    Google Scholar 

  85. D. Kveselava, “Solution of one boundary-value problem of the theory of functions,” Dokl. Akad. Nauk SSSR, 53, No. 8, 683–686 (1946).

    MathSciNet  Google Scholar 

  86. D. Kveselava, “On one boundary-value problem of the theory of functions,” Bull. Acad. Sci. Georgia, 7, Nos. 9–10, 609–614 (1946).

    Google Scholar 

  87. D. Kveselava, “Solution of one boundary-value problem of Carleman,” Dokl. Akad. Nauk SSSR, 55, No. 8, 683–686 (1947).

    Google Scholar 

  88. D. Kveselava, “Some boundary-value problems of the theory of functions,” Proc. Tbilisi Math. Inst., 16, 39–80 (1948).

    MathSciNet  Google Scholar 

  89. Yu. Latushkin, G. Litvichuk, and I. Spitkovski, “On the theory of one boundary-value problem of I. N. Vekua,” Proc. Tbilisi Math. Inst., 259, Nos. 19–20, 161–188 (1986).

    Google Scholar 

  90. N. Lisovets, “Four-element boundary-value problem with discontinuous coefficients with shift on compound contour,” Ukr. Mat. Zh., 35, No. 6, 764–767 (1983).

    MathSciNet  Google Scholar 

  91. G. Litvinchuk, Boundary-Value Problems and Singular Integral Equations with Shift [in Russian], Nauka, Moscow (1977).

    MATH  Google Scholar 

  92. G. Litvinchuk, “On the operator approach to the theory of boundary-value problems with shift for functions analytic in domains,” in: Proc. Sci. Semin. BVP, Minsk (1985).

  93. G. Litvinchuk and I. Spitkovski, Factorization of Measurable Matrix Functions, Akademie Verlag, Berlin (1987).

    Google Scholar 

  94. L. Magnaradze, “On one system of linear singular integral equations and on a linear boundaryvalue problem,” Bull. Acad. Sci. Georgia, 4, No. 1, 3–9 (1943).

    MATH  MathSciNet  Google Scholar 

  95. L. Magnaradze, “Theory of one class of linear singular integro-differential equations and its applications in problems of wing oscillation of aeroplane with finite amplitude, shock of water surface, and in analoguous problems,” Bull. Acad. Sci. Georgia, 4, No. 2, 108–110 (1943).

    Google Scholar 

  96. G. Manjavidze, Boundary-Value Problems with Shift for Analytic and Generalized Analytic Functions, Tbilisi Univ. Press (1990).

  97. G. Manjavidze, “On one system of singular integral equations with discontinous coefficients,” Bull. Acad. Sci. Georgia, 11, No. 6, 351–356 (1950).

    Google Scholar 

  98. G. Manjavidze, “Singular integral equations as a tool for the solution of mixed problems of planar theory of elasticity,” in: Proc. Tbilisi Int. Symp., Sept. 17–23, 1963, Tbilisi [in Russian], Vol. 1, Nauka, Moscow (1965), pp. 237–247.

    Google Scholar 

  99. G. Manjavidze, “On a boundary-value problem of linear conjugation and its relation to the theory of generalized analytic functions,” Proc. Tbilisi Math. Inst., 33, 82–87 (1967).

    Google Scholar 

  100. G. Manjavidze, “On the behavior of solutions of the boundary-value problem of linear conjugation,” Proc. Tbilisi Math. Inst., 35, 173–182 (1969).

    Google Scholar 

  101. G. Manjavidze, “On a boundary-value problem of linear conjugation with piecewise-continuous matrix coefficients,” in: Mechanics of Solid Medium and Related Problems of Analysis [in Russian], Nauka, Moscow (1972), pp. 297–304.

    Google Scholar 

  102. G. Manjavidze, “Application of the theory of generalized analytic functions in the study of boundary-value problems of linear conjugation with displacement,” in: Differential and Integral Equations and Boundary-Value Problems [in Russian], Tbilisi Univ. Press (1979).

  103. G. Manjavidze, “Some properties of the boundary-value problems of linear conjugation,” Proc. Vekua Inst. Appl. Math., 45–52 (1980).

  104. G. Manjavidze, “The boundary-value problems of linear cojugation with displacement,” in: Complex Analysis and Applications, Sofia (1984), pp. 375–382.

  105. G. Manjavidze and B. Khvedelidze, “On the Riemann–Privalov problem with continuous coefficients,” Dokl. Akad. Nauk SSSR, 123, No. 5, 791–194 (1958).

    MathSciNet  Google Scholar 

  106. G. Manjavidze and B. Khvedelidze, “Boundary-value problems of linear conjugation and singular integral equations with Cauchy kernels and continuous coefficients,” Proc. Tbilisi Math. Inst., 28, 85–105 (1962).

    Google Scholar 

  107. G. Manjavidze and V. L. Ngo, “Differential boundary-value problems for generalized analytic vectors,” in: Modern Problems of Mathematical Physics [in Russian], Tbilisi Univ. Press, Tbilisi (1987), pp. 74–84.

    Google Scholar 

  108. N. Manjavidze, “The Riemann–Hilbert boundary-value problem on a cut plane,” in: Topics in Analysis and Its Application, Kluwer Acad. Publ. (2004), pp. 83–90.

  109. L. Mikhailov, A New Class of Singular Integral Equations and Its Applications to Differential Equations with Singular Coefficients, Wolters-Noordhoff, Groningen (1970).

    Google Scholar 

  110. S. Mikhlin, Multidimensional Singular Integrals and Integral Equations [in Russian], Fizmatizdat, Moscow (1962).

    Google Scholar 

  111. S. Mikhlin and S. Prossdorf, Singular Integral Operators, Akademie-Verlag, Berlin (1986).

    Google Scholar 

  112. V. Monakhov, Boundary-Value Problems with Free Boundaries for Elliptic Systems, Am. Math. Soc., Providence, Rhode Island (1983).

    MATH  Google Scholar 

  113. N. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen–Leyden (1953).

    MATH  Google Scholar 

  114. M. H. A. Neumann, Elements of the Topology of Plane Sets of Points, Cambridge Univ. Press (1938).

  115. V. L. Ngo, “General boundary-value problem of linear conjugation with displacements for Qholomorphie vectors,” Bull. Acad. Sci. Georgia, 57, No. 3, 549–552 (1970).

    MATH  Google Scholar 

  116. V. L. Ngo, “On one differential boundary-value problem of linear conjugation for analytic vectors,” Proc. Vekua Inst. Appl. Math., 2, No. 1, 50–53 (1982).

    MathSciNet  Google Scholar 

  117. V. L. Ngo, “General differential boundary-value problems for generalized analytic vectors,” Proc. Vekua Inst. Appl. Math., 28, 20–131 (1988).

    MATH  Google Scholar 

  118. S. Nikolski, “Linear equations in linear normed spaces,” Izv. Akad. Nauk. SSSR, Ser. Mat., 7, No. 3, 147–166 (1943).

    Google Scholar 

  119. V. Paatashvili, “On Cauchy singular integrals,” Bull. Acad. Sci. Georgia, 53, No. 3, 529–532 (1969).

    MATH  Google Scholar 

  120. V. Paatashvili, “On affiliation to the analytic function classes E p representable by Cauchy type integral,” Proc. Tbilisi Math. Inst., 42, 87–94 (1972).

    MATH  Google Scholar 

  121. J. Plemely, “Riemannsche Functionenscharen mit gegebener Monodromiargruppe,” Monatsh. Math. Phys., 19, 211–245 (1908).

    Article  MathSciNet  Google Scholar 

  122. G. Poloji, Theory and Application of p-Analytic and p, q-Analytic Functions [in Russian], Naukova Dumka, Kiev (1973).

    Google Scholar 

  123. I. Privalov, Boundary Properties of Analytic Functions [in Russian], Gostekhizdat, Moscow (1950).

    Google Scholar 

  124. S. Prossdorf, Some Classes of Singular Equations, North-Holland, Amsterdam–New York–Oxford (1978).

  125. S. Prossdorf and L. von Wolerstorf, “Zur Theorie der Noetherchen Operatoren und einiger singularer Integral und Integrodifferentialgleichungen,” Wiss. Z. Tech. Nochschule Karl-Marx-Stadt, 13, 103–116 (1971).

    Google Scholar 

  126. D. Sherman, “On a general problem of potential theory,” Izv. Akad. Nauk SSSR, Ser. Mat., 10, No. 2, 121–134 (1946).

    MATH  MathSciNet  Google Scholar 

  127. I. Simonenko, Riemann Boundary-Value Problem and the Riemann–Hasemann Problem with Continuous Coefficients [in Russian], Fizmatizdat, Moscow (1961).

    Google Scholar 

  128. I. Simonenko, “Riemann boundary-value problem for n-tuple functions with measurable coefficients and its application in investigation of singular integrals in weighted spaces,” Izv. Akad. Nauk SSSR, Ser. Mat., 28, No. 2, 277–306 (1964).

    MATH  MathSciNet  Google Scholar 

  129. I. Simonenko, “Stability properties os weight of Cauchy singular integrals,” Mat. Zametki, 133, No. 3, 409–416 (1983).

    MathSciNet  Google Scholar 

  130. N. Skhirtladze, “On a boundary-value problem of the theory of analytic functions with displacements and its application in the plane elasticity theory,” Bull. Acad. Sci. Georgia, 105, No. 2, 249–252 (1982).

    MATH  MathSciNet  Google Scholar 

  131. N. Skhirtladze, “Boundary-value problems of plane elasticity theory for bodies with curvelinear cuts,” Proc. Vekua Inst. Appl. Math., 28, 132–162 (1988).

    MATH  MathSciNet  Google Scholar 

  132. S. Skorokhod, “On some boundary-value problems with multi-element boundary with non-Carleman shift for analytic functions in domains,” Bull. Acad. Sci. Georgia, 100, No. 1, 21–24 (1989).

    MathSciNet  Google Scholar 

  133. A. Soldatov, “Hyperanalytic functions and their application,” J. Math. Sci., 132, No. 6, 827–882 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  134. A. Soldatov, “Boundary-value problem of conjugation with shift on piecewise-smooth curves in weighted spaces,” Dokl. Akad. Nauk SSSR, 253, No. 5, 1050–1055 (1980).

    MathSciNet  Google Scholar 

  135. N. Tovmasyan, “Integral representations of holomorphic functions by holomorphic densities and their applications,” Lect. Notes Math., 798, 439–445 (1980).

    Article  MathSciNet  Google Scholar 

  136. N. Vasilevski and M. Shapiro, “On an algebra generated by singular integral operators with Carlemann shift and piecewise-continuous coefficients,” Ukr. Mat. Zh., 27, No. 2, 216–223 (1975).

    Article  Google Scholar 

  137. I. N. Vekua, New Methods of the Solutions of Elliptic Equations [in Russian], Gostekhizdat, Moscow (1948).

    Google Scholar 

  138. I. N. Vekua, Generalized Analytic Functions, Pergamon Press, Oxford (1962).

    MATH  Google Scholar 

  139. N. Vekua, Systems of Singular Integral Equations [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  140. N. Vekua, “Hilbert boundary-value problem for several unknown functions with rational coefficients,” Proc. Tbilisi Math. Inst., 89, 64–68 (1989).

    MathSciNet  Google Scholar 

  141. V. Vinogragov, “Boundary-value problem for elliptic systems of first order on the plane,” Differ. Uravn., 7, No. 8, 1440–1448 (1971).

    Google Scholar 

  142. W. Wendland, Elliptic Systems in the Plane, Pitman, London (1979).

    MATH  Google Scholar 

  143. A. Zigmund, Trigonometric Series, Vol. 1, Cambridge (1959).

  144. E. Zverovich, “Boundary-value problems with shift on abstract Riemann surfaces,” Sib. Mat. Zh., 7, No. 4, 804–819 (1966).

    MATH  Google Scholar 

  145. 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).

    Google Scholar 

  146. L. Zverovich, “Boundary-value problems with shift for the vector on Riemann surface with edges,” Izv. Vyssh. Uchebn. Zaved. Ser. Mat., 5, 31–39 (1975).

    Google Scholar 

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.

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Manjavidze, G., Manjavidze, N. Boundary-value problems for analytic and generalized analytic functions. J Math Sci 160, 745–821 (2009). https://doi.org/10.1007/s10958-009-9526-8

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