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The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions. (English) Zbl 1157.45303
Summary: This paper presents and studies Fredholm integral equations associated with the linear Riemann-Hilbert problems on multiply connected regions with smooth boundary curves. The kernel of these integral equations is the generalized Neumann kernel. The approach is similar to that for simply connected regions [see R. Wegmann, A. H. M. Murid and M. M. S. Nasser, J. Comput. Appl. Math. 182, No. 2, 388–415 (2005; Zbl 1070.30017)]. There are, however, several characteristic differences, which are mainly due to the fact that the complement of a multiply connected region has a quite different topological structure. This implies that there is no longer perfect duality between the interior and exterior problems. We investigate the existence and uniqueness of solutions of the integral equations. In particular, we determine the exact number of linearly independent solutions of the integral equations and their adjoints. The latter determine the conditions for solvability. An analytic example on a circular annulus and several numerically calculated examples illustrate the results.
##### MSC:
 4.5e+11 Integral equations of the convolution type 3e+26 Boundary value problems, complex analysis