Mixed Riemann-Hilbert boundary value problem with simply connected fibers
We study the existence of solutions of mixed Riemann-Hilbert or Cherepanov boundary value problem with simply connected fibers on the unit disk $Δ$. Let $L$ be a closed arc on $\partialΔ$ with the end points $ω_{-1}, ω_1$ and let $a$ be a smooth function on $L$ with no zeros. Let $γ_ξ$, $ξ\in\partialΔ\setminus\mathring{L}$, be a smooth family of smooth Jordan curves in the complex plane which all contain point $0$ in their interiors and such that $γ_{ω_{-1}}$, $γ_{ω_{1}}$ are strongly starshaped with respect to $0$. Then under condition that for each $w\inγ_{ω_{\pm 1}}$ the angle between $w$ and the normal to $γ_{ω_{\pm 1}}$ at $w$ is less than $\fracπ{10}$, there exists a Hölder continuous function $f$ on $\overlineΔ$, holomorphic on $Δ$, such that ${\rm Re}(\overline{a(ξ)} f(ξ)) = 0$ on $L$ and $f(ξ)\inγ_ξ$ on $\partialΔ\setminus\mathring{L}$.