Mixed Problem for Laplace’s Equation in an Arbitrary Circular Multiply Connected Domain

Conference paper
Part of the Trends in Mathematics book series (TM)


Mixed boundary value problems for the two-dimensional Laplace’s equation in a domain D are reduced to the Riemann-Hilbert problem Re \(\overline {\lambda (t)}\psi (t) = 0\), t ∈ ∂D, with a given Hölder continuous function λ(t) on ∂D except at a finite number of points where a one-sided discontinuity is admitted. The celebrated Keldysh-Sedov formulae were used to solve such a problem for a simply connected domain. In this paper, a method of functional equations is developed to mixed problems for multiply connected domains. For definiteness, we discuss a problem having applications in composites with a discontinuous coefficient λ(t) on one of the boundary components. It is assumed that the domain D is a canonical domain, the lower half-plane with circular holes. A constructive iterative algorithm to obtain an approximate solution in analytical form is developed in the form of an expansion in the radius of the holes.


Mixed boundary value problem Keldysh-Sedov formulae Riemann-Hilbert problem Multiply connected domain Iterative functional equation 

Mathematics Subject Classification (2010)



  1. 1.
    I.A. Aleksandrov, A.S. Sorokin, The problem of Schwarz for multiply connected domains. Sib. Math. Zh. 13, 971–1001 (1972)MATHGoogle Scholar
  2. 2.
    R. Balu, T.K. DeLillo, Numerical methods for Riemann-Hilbert problems in multiply connected circle domains. J. Comput. Appl. Math. 307, 248–261 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    T.K. DeLillo, A.R. Elcrat, E.H. Kropf, J.A. Pfaltzgraff, Efficient calculation of Schwarz-Christoffel transformations for multiply connected domains using Laurent series. Comput. Methods Funct. Theory 13, 307–336 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    F.D. Gakhov, Boundary Value Problems, 3rd edn. (Nauka, Moscow, 1977, in Russian); Engl. transl. of 1st edn. (Pergamon Press, Oxford, 1966)Google Scholar
  5. 5.
    M.V. Keldysh, L.I. Sedov, Effective solution to some boundary value problems for harmonic functions. Dokl. Akad. Nauk SSSR 16, 7–10 (1937)MATHGoogle Scholar
  6. 6.
    V. Mityushev, Solution of the Hilbert boundary value problem for a multiply connected domain. Slupskie Prace Mat-Przyr. 9a, 33–67 (1994)Google Scholar
  7. 7.
    V. Mityushev, Hilbert boundary value problem for multiply connected domains. Complex Var. 35, 283–295 (1998)MathSciNetMATHGoogle Scholar
  8. 8.
    V. Mityushev, Riemann-Hilbert problems for multiply connected domains and circular slit maps. Comput. Methods Funct. Theory 11, 575–590 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    V. Mityushev, Scalar Riemann-Hilbert problem for multiply connected domains, in Functional Equations in Mathematical Analysis, ed. by Th.M. Rassias, J. Brzdȩk. Springer Optimization and its Applications, vol. 52. (Springer Science+Business Media, LLC, New York, 2012), pp. 599–632. https://doi.org/10.1007/978-1-4614-0055-438
  10. 10.
    V. Mityushev, \(\mathbb R\) -Linear and Riemann-Hilbert Problems for Multiply Connected Domains, ed. by S.V. Rogosin, A.A. Koroleva. Advances in Applied Analysis (Birkhäuser, Basel, 2012), pp. 147–176Google Scholar
  11. 11.
    V. Mityushev, Poincare α-Series for Classical Schottky Groups and its Applications, ed. by G.V. Milovanović, M.Th. Rassias. Analytic Number Theory, Approximation Theory, and Special Functions (Springer, Berlin, 2014), pp. 827–852Google Scholar
  12. 12.
    V.V. Mityushev, S.V. Rogosin, Constructive Methods to Linear and Non-linear Boundary Value Problems of the Analytic Function. Theory and Applications. Monographs and Surveys in Pure and Applied Mathematics (Chapman & Hall/CRC, Boca Raton, 2000)Google Scholar
  13. 13.
    N. Rylko, Fractal local fields in random composites. Appl. Math. Comput. 69, 247–254 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    N. Rylko, Edge effects for heat flux in fibrous composites. Appl. Math. Comput. 70, 2283–2291 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    A.S. Sorokin, The homogeneous Keldysh-Sedov problem for circular multiply connected circular domains in Muskhelishvili’s class h 0. Differ. Uravn. 25, 283–293 (1989)MATHGoogle Scholar
  16. 16.
    A.S. Sorokin, The Keldysh-Sedov problem for multiply connected circular domains. Sibirsk. Mat. Zh. 36, 186–202 (1995)MathSciNetMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Computer SciencesPedagogical UniversityKrakowPoland

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