Seminar in Numerical Analysis: Wolfgang Wendland (Universität Stuttgart)

Collocation methods for nonlinear Riemann-Hilbert problems on doubly connected domains

As a special case of nonlinear Rieman--Hilbert problems with closed boundary data in multiply connected domains, here a doubly connected domain like an annulus is considered.

The nonlinear boundary conditions for the desired holomorphic solutions lead to nonlinear singular integral equations on the boundary which belong to the class of quasiruled Fredholm maps defined on quasicylindrical domains in appropriate separable Banach spaces.

The closed boundary data give a priori estimates for the modulus of solutions which in turn implies a priori estimates in the Sobolev spaces considered here. For this class of problems, the Shnirelman--Efendiev degree of mappings can be defined which allows to investigate the existence of solutions if the boundary conditions satisfy some topological assumptions.

The lifting of the boundary  value problem via holomorphic transformation onto the universal covering of the unit disc allows to construct a homotopic deformation of the lifted nonlinear singular integral equations to a uniquely solvable case  which implies that the degree of mapping is 1 and existence of (in fact at least two) solutions follows.

If the nonlinear integral equations on the boundary are appoximated by trigonometric point collocation then the theory also implies that approximate solutions exist and converge asymptotically.

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