Modern Problems in Applied Analysis pp 135-152 | Cite as

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

## Abstract

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.

## Keywords

Mixed boundary value problem Keldysh-Sedov formulae Riemann-Hilbert problem Multiply connected domain Iterative functional equation## Mathematics Subject Classification (2010)

30E25## References

- 1.I.A. Aleksandrov, A.S. Sorokin, The problem of Schwarz for multiply connected domains. Sib. Math. Zh.
**13**, 971–1001 (1972)zbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 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.M.V. Keldysh, L.I. Sedov, Effective solution to some boundary value problems for harmonic functions. Dokl. Akad. Nauk SSSR
**16**, 7–10 (1937)zbMATHGoogle Scholar - 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.V. Mityushev, Hilbert boundary value problem for multiply connected domains. Complex Var.
**35**, 283–295 (1998)MathSciNetzbMATHGoogle Scholar - 8.V. Mityushev, Riemann-Hilbert problems for multiply connected domains and circular slit maps. Comput. Methods Funct. Theory
**11**, 575–590 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.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.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.N. Rylko, Fractal local fields in random composites. Appl. Math. Comput.
**69**, 247–254 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 14.N. Rylko, Edge effects for heat flux in fibrous composites. Appl. Math. Comput.
**70**, 2283–2291 (2015)MathSciNetCrossRefGoogle Scholar - 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)zbMATHGoogle Scholar - 16.A.S. Sorokin, The Keldysh-Sedov problem for multiply connected circular domains. Sibirsk. Mat. Zh.
**36**, 186–202 (1995)MathSciNetzbMATHGoogle Scholar