Mixed Riemann-Hilbert boundary value problem 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 Hölder continuous function $f$ on $\overline{\Delta }$, holomorphic on $\Delta$, such that $\mathrm{Re}(\overline{a(\xi )}f(\xi ))=0$ on $L$ and $f(\xi )\in {\gamma }_{\xi }$ on $\partial \Delta \setminus \mathring{L}$.

27 pages