Nonlinear Riemann–Hilbert problems with circular target curves

Abstract

The paper gives a systematic and self-contained treatment of the nonlinear Riemann-Hilbert problem with circular target curves |w − c| = r, sometimes also called the generalized modulus problem. We assume that c and r are Hölder continuous functions on the unit circle and describe the complete set of solutions w in the disk algebra H ∞ ∩ C and in the Hardy space H ∞ of bounded holomorphic functions. The approach is based on the interplay with the Nehari problem of best approximation by bounded holomorphic functions. It is shown that the considered problems fall into three classes (regular, singular, and void) and we give criteria which allow to classify a given problem.For regular problems the target manifold is covered by the traces of solutions with winding number zero in a schlicht manner. Counterexamples demonstrate that this need not be so if the boundary condition is merely continuous.Paying special attention to constructive aspects of the matter we show how the Nevanlinna parametrization of the full solution set can be obtained from one particular solution of arbitrary winding number.