Constructive solution of a certain class of Riemann–Hilbert problems on multiply connected circular regions

A class of Riemann–Hilbert (RH) problems on multiply connected circular regions is treated iteratively by a method of successive conjugation (SC), which solves the problem on a single boundary circle each time, running through all circles in each iterative step. It is shown that the method converges. Convergence is linear with a rate which can be calculated with an eigenvalue problem. The RH problems considered have negative index. In order to fulfill the solvability conditions, free parameters are introduced in form of polynomials. The proof of convergence of the SC method shows at the same time that the RH problems with this specification of the free parameters have unique solutions. RH problems of this kind are of practical use in conformal mapping problems of multiply connected regions to circular regions. They play a role in the mapping of nearly circular regions (Lavrentevʹs principle). There is also a close relationship to Koebeʹs iterative method for the calculation of conformal mappings. The properties of RH problems are different for unbounded and bounded regions. The SC iteration must be adapted accordingly. The performance of the SC iteration is illustrated with several examples.