ℝ-linear and Riemann–Hilbert Problems for Multiply Connected Domains

  • Vladimir V. MityushevEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


The ℝ-linear problem with constant coefficients for arbitrary multiply connected domains has been solved. The method is based on reduction of the problem to a system of functional equations for a circular domain and to integral equations for a general domain. In previous works, theℝ-linear problem and its partial cases such as the Riemann –Hilbert problem and the Dirichlet problem were solved under geometrical restrictions to the domains. In the present work, the solution is constructed for any circular multiply connected domain in the form of modified Poincar ´e series. Moreover, the modified alternating Schwarz method has been justified for an arbitrary multiply connected domain. This extends application of the alternating Schwarz method, since in the previous works geometrical restrictions were imposed on locations of the inclusions. The same concerns Grave’s method which was worked out before only for simple closed algebraic boundaries or for a collection of confocal boundaries.


ℝ-linear problem Riemann-Hilbert problem multiply connected domain Poincaré series Schwarz alternating method Grave method. 


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  1. [Akaza (1966)] Akaza, T. (1966). Singular sets of some Kleinian groups, Nagoya Math. J., 26, pp. 127-143. 155Google Scholar
  2. [Akaza and Inoue (1984)] Akaza, T. and Inoue, K. (1984). Limit sets of geometrically finite free Kleinian groups, Tohoku Math. J., 36, pp. 1-16. 155Google Scholar
  3. [Akhiezer (1990)] Akhiezer, N.I. (1990). Elements of the Theory of Elliptic Functions.(AMS, Providence). 148Google Scholar
  4. [Aleksandrov and Sorokin (1972)] Aleksandrov, I.A. and Sorokin, A.S. (1972). The problem of Schwarz for multiply connected domains, Sib. Math. Zh., 13, pp. 971-1001 (inRussian). 153Google Scholar
  5. [Apel’tsin (2000)] Apel’tsin, F.A. (2000). A generalization of D.A. Grave’s method for plane boundary-value problems in harmonic potentials theory, Comput. Math. Modeling, 11, pp. 1-14. 173Google Scholar
  6. [Baker (1996)] Baker, H.F. (1996). Abelian Functions, (Cambridge University Press). 148Google Scholar
  7. [Bancuri (1975)] Bancuri, R.D. (1975). On the Riemann-Hilbert problem for doubly connected domains, Soobsch. AN GruzSSR, 80, pp. 549-552 (in Russian). 149Google Scholar
  8. [Bojarski (1958)] Bojarski, B. (1958). On a boundary value problem of the theory ofanalytic functions, Dokl. AN SSSR, 119, pp. 199-202 (in Russian).Google Scholar
  9. [Bojarski (1960)] Bojarski, B. (1960). On generalized Hilbert boundary value problem,Soobsch. AN GruzSSR, 25, pp. 385-390 (in Russian). 151Google Scholar
  10. [Burnside (1891)] Burnside, W. (1891). On a Class of Automorphic Functions, Proc.London Math. Soc. 23, pp. 49-88. 155Google Scholar
  11. [Koppenfels and Stallman (1959)] Koppenfels, W. and Stallman, F. (1959). Practice ofconformal mapping, (Springer Verlag). 148Google Scholar
  12. [Chibrikova (1977)] Chibrikova, L.I. (1977). Basic Boundary-Value Problems for AnalyticFunctions, (Kazan Univ. Publ.)(in Russian).Google Scholar
  13. [Crowdy (2009)] Crowdy, D. (2009). Explicit solution of a class of Riemann-Hilbert prob-lems, Annales Universitatis Paedagogicae Cracoviensis, 8, pp. 5-18. 148Google Scholar
  14. [Crowdy (2008a)] Crowdy, D. (2008). The Schwarz problem in multiply connected do-mains and the Schottky-Klein prime function, Complex Variables and Elliptic Equa-tions, 53, pp. 221-236. 148Google Scholar
  15. [Crowdy (2008b)] Crowdy, D. (2008) Geometric function theory: a modern view of aclassical subject, Nonlinearity, 21, pp. T205-T219. 148, 155Google Scholar
  16. [Dunduchenko (1966)] Dunduchenko, L.E. (1966). On the Schwarz formula for an n-connected domain, Dopovedi AN URSR, n. 5, pp. 1386-1389 (in Ukranian). 153Google Scholar
  17. [Gakhov (1977)] Gakhov, F.D. (1977). Boundary Value Problems, 3rd edn. (Nauka,Moscow) (in Russian); Engl. transl. of 1st edn. (Pergamon Press). 147, 148, 149,157, 169, 170Google Scholar
  18. [Golusin (1934)] Golusin, G.M. (1934). Solution of basic plane problems of mathematicalphysics for the case of Laplace equation and multiply connected domains boundedby circles (method of functional equations), Math. zb. 41, pp. 246-276. 153Google Scholar
  19. [Golusin (1935)] Golusin, G.M. (1935). Solution of plane heat conduction problem formultiply connected domains enclosed by circles in the case of isolated layer, Math.zb. 42, pp. 191-198. 151, 153Google Scholar
  20. [Golusin (1969)] Golusin, G.M. (1969). Geometric Theory of Functions of Complex Variable. (AMS, Providence). 150Google Scholar
  21. [Kantorovich and Krylov (1958)] Kantorovich, L.V. and Krylov, V.I. (1958). Approxi-mate methods of higher analysis, (Groningen, Noordhoff ). 153Google Scholar
  22. [Krasnosel’skii et al. (1969)] Krasnosel’skii, M.A., Vainikko, G.M., Zabreiko, P.P., Rutic-kii, Ja.B. and Stecenko, V.Ja. (1969). Approximate Methods for Solution of OperatorEquations, (Nauka, Moscow) (in Russian); Engl. transl. (Wolters-Noordhoff Publ).157, 171Google Scholar
  23. [Kveselava (1945)] Kveselava, D.A. (1945). Riemann-Hilbert problem for multiply con-nected domain, Soobsch. AN GruzSSR 6, pp. 581-590 (in Russian). 149Google Scholar
  24. [Kuczma et al. (1990)] Kuczma, M., Chosewski, B. and Ger, R. (1990). Iterative func-tional equations, (Cambridge University Press). 148Google Scholar
  25. [Markushevich (1946)] Markushevich, A.I. (1946) On a boundary value problem of ana-lytic function theory, Uch. zapiski MGU, 1, n. 100, pp. 20-30 (in Russian). 151Google Scholar
  26. [Mikhailov (1961)] Mikhailov, L.G. (1961). On a boundary value problem, DAN SSSR,139, pp. 294-297 (in Russian). 152Google Scholar
  27. [Mikhailov (1963)] Mikhailov, L.G. (1963). New Class of Singular Integral Equations andits Applications to Differential Equations with Singular Coefficients. (AN TadzhSSR,Dushanbe) (in Russian); English transl.: (Akademie Verlag). 150, 151, 152, 169Google Scholar
  28. [Mikhlin (1964)] Mikhlin, S.G. (1964). Integral Equations, (Pergamon Press). 153, 164,169, 173Google Scholar
  29. [Mityushev (1994)] Mityushev, V.V. (1994). Solution of the Hilbert boundary value prob-lem for a multiply connected domain, Slupskie Prace Mat.-Przyr. 9a pp. 37-69. 150Google Scholar
  30. [Mityushev (1994)] Mityushev, V.V. (1995). Generalized method of Schwarz and additiontheorems in mechanics of materials containing cavities, Arch. Mech. 47, pp. 1169-1181. 153Google Scholar
  31. [Mityushev (1998)] Mityushev, V.V. (1998). Convergence of the Poincaré series for clas-sical Schottky groups, Proceedings Amer. Math. Soc. 126 pp. 2399-2406. 148, 155,156Google Scholar
  32. [Mityushev (1998)] Mityushev, V.V. (1998). Hilbert boundary value problem for multiplyconnected domains, Complex Variables 35 pp. 283-295. 150Google Scholar
  33. [Mityushev and Rogosin (2000)] Mityushev, V.V. and Rogosin, S.V. (2000). Constructivemethods to linear and non-linear boundary value problems of the analytic function.Theory and applications, (Chapman & Hall / CRC, Monographs and Surveys in Pureand Applied Mathematics). 147, 148, 150, 151, 153, 155, 157, 158, 164, 165, 169, 171Google Scholar
  34. [Mityushev et al. (2008)] Mityushev, V.V., Pesetskaya, E. and Rogosin S.V. (2008). An-alytical Methods for Heat Conduction in Composites and Porous Media in Cellularand Porous Materials: Thermal Properties Simulation and Prediction, (Wiley, An-dreas) Ö chsner, Graeme E. Murch, Marcelo J.S. de Lemos (eds.). 173[Mityushev (2009)] Mityushev, V. (2009). Conductivity of a two-dimensional compositecontaining elliptical inclusions, Proc. R. Soc. A465, pp. 2991-3010. 172Google Scholar
  35. [Myrberg (1916)] Myrberg, P.J. (1916) Zur Theorie der Konvergenz der PoincaréschenReihen, Ann. Acad. Sci. Fennicae, A9, 4, pp. 1-75. 155Google Scholar
  36. [Muskhelishvili (1932)] Muskhelishvili, N.I. (1932). To the problem of torsion and bending of beams constituted from different materials, Izv. AN SSSR, n. 7, pp. 907-945Google Scholar
  37. [Muskhelishvili (1968)] Muskhelishvili, N.I. (1968). Singular Integral Equations, 3th edn.; English translation of the 1st edn. (P. Noordhoff N.V., Groningen). 147, 148, 149, 151, 169Google Scholar
  38. [Muskhelishvili (1966)] Muskhelishvili, N.I. (1966) Some Basic Problems of Mathematical; English translation of the 1st edn. (Noordhoff, Groningen). 151, 169Google Scholar
  39. [Poincaré (1916)] Poincaré, H. (1916). Oeuvres, (Gauthier-Villars, Paris). 155Google Scholar
  40. [Smith et al. (1996)] Smith, B., Björstad, P. and Gropp, W. (1996) Domain decomposi-tion. Parallel multilevel methods for elliptic partial differential equations. (Cambridge University Press). 153Google Scholar
  41. [Vekua and Rukhadze (1933)] Vekua, I.N. and Rukhadze, A.K. (1933). The problem ofthe torsion of circular cylinder reinforced by transversal circular beam. Izv. AN SSSR, n. 3, pp. 373-386. 151Google Scholar
  42. [Vekua and Rukhadze (1933)] Vekua, I.N. and Rukhadze, A.K. (1933). Torsion andtransversal bending of the beam compounded by two materials restricted by confocalellipses, Prikladnaya Matematika i Mechanika (Leningrad) 1, 2, pp. 167-178. 151Google Scholar
  43. [Vekua (1988)] Vekua, I.N. (1988). Generalized Analytic Functions, (Nauka, Moscow)(inRussian). 147, 149, 169Google Scholar
  44. [Vekua (1967)] Vekua, N.P. (1967). Systems of Singular Integral Equations, 2nd edn.(Noordhoff, Groningen). 151Google Scholar
  45. [Zmorovich (1958)] Zmorovich, V.A. (1958). On a generalization of the Schwarz integralformula on -connected domains, Dopovedi URSR, n. 5, pp. 489-492 (in Ukranian).Google Scholar
  46. [Zverovich (1971)] Zverovich, E.I. (1971). Boundary value problems of analytic functions in Hölder classes on Riemann surfaces, Uspekhi Mat. nauk, 26 (1), pp. 113-179 (in Russian). 148, 150Google Scholar

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© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Computer Sciences and Computer MethodsPedagogical UniversityKrakowPoland

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