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

  • Vladimir MityushevEmail author
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)



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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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