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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)

Abstract

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.

Keywords

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

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Copyright information

© Springer Basel 2012

Authors and Affiliations

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

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