On the Riemann-Hilbert problem IV

TL;DR

This paper proves the existence of solutions for the Riemann-Hilbert problem in general Jordan domains with measurable data, using harmonic measure and asymptotic values, and shows the solution space is infinite-dimensional.

Contribution

It establishes existence criteria for the Riemann-Hilbert problem without index restrictions in broad settings, including arbitrary Jordan domains and measurable boundary data.

Findings

Existence of solutions in general Jordan domains.

Reinforced criteria for domains with rectifiable boundaries.

Infinite-dimensionality of solution spaces.

Abstract

With no criteria of the index type, it is proved the existence of a solution for the Riemann-Hilbert problem in the fairly general setting of arbitrary Jordan domains, 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 arbitrary rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Moreover, it is shown that the dimension of the spaces of solutions is infinite.