On multivalent solutions of Riemann-Hilbert problem

TL;DR

This paper proves the existence of multivalent solutions to the Riemann-Hilbert problem in complex domains with measurable data, establishing conditions for solutions and analyzing their dimensionality.

Contribution

It introduces new existence theorems for multivalent solutions in general finitely connected domains with measurable coefficients and boundary data.

Findings

Solutions exist under broad measurable conditions.

The dimension of solution spaces is infinite.

Criteria are provided for domains with rectifiable boundaries.

Abstract

It is proved the existence of multivalent solutions for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthemore, it is shown that the dimension of the spaces of these solutions is infinite.