Mixed Riemann-Hilbert boundary value problem with simply connected fibers

TL;DR

This paper proves the existence of solutions to a mixed Riemann-Hilbert boundary value problem with simply connected fibers on the unit disk, under specific geometric and boundary conditions.

Contribution

It establishes new existence results for mixed Riemann-Hilbert problems with simply connected fibers, extending previous work to more general boundary configurations.

Findings

Existence of Hölder continuous solutions under geometric conditions.

Solutions satisfy boundary conditions involving complex curves.

Results applicable to boundary value problems with simply connected fibers.

Abstract

We study the existence of solutions of mixed Riemann-Hilbert or Cherepanov boundary value problem with simply connected fibers on the unit disk ${\Delta}$. Let $L$ be a closed arc on $\partial{\Delta}$ with the end points $\omega_{-1}, \omega_1$ and let $a$ be a smooth function on $L$ with no zeros. Let $\gamma_{\xi}$, $\xi\in\partial{\Delta}\setminus\mathring{L}$, be a smooth family of smooth Jordan curves in the complex plane which all contain point $0$ in their interiors and such that $\gamma_{\omega_{-1}}$, $\gamma_{\omega_{1}}$ are strongly starshaped with respect to $0$. Then under condition that for each $w\in\gamma_{\omega_{\pm 1}}$ the angle between $w$ and the normal to $\gamma_{\omega_{\pm 1}}$ at $w$ is less than $\frac{\pi}{10}$, there exists a H\"{o}lder continuous function $f$ on $\overline{\Delta}$, holomorphic on ${\Delta}$, such that ${\rm Re}(\overline{a(\xi)} f(\xi)) = 0$ on $L$ and $f(\xi)\in\gamma_{\xi}$ on $\partial{\Delta}\setminus\mathring{L}$.