The Riemann-Hilbert problem for elliptic vector fields with degeneracies

TL;DR

This paper investigates the Riemann-Hilbert boundary value problem for specific planar complex vector fields with degeneracies, utilizing first integrals to transform it into classical problems with discontinuous data.

Contribution

It introduces a method to reduce complex vector field problems with degeneracies to classical Riemann-Hilbert problems using first integrals.

Findings

Reduction of complex vector field problems to classical Riemann-Hilbert problems.

Handling of discontinuous data in boundary value problems.

Framework applicable to vector fields with degeneracies.

Abstract

The Riemann-Hilbert boundary value problem is studied for a class of planar complex vector fields $L$ in a simply connected open set $\Om\subset\R^2$. The first integrals of $L$ are used to reduce the problem into a collection of classical Riemann-Hilbert problems with discontinuous data.