Mixed Riemann-Hilbert boundary value problem with simply connected fibers

Černe, Miran (author)

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 $\mathbb C$ 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 ${\rm Re}(\overline{a(\xi)} f(\xi)) = 0$ on $L$ and $f(\xi)\in\gamma_{\xi}$ on $\partial\Delta\setminus\mathring{L}$.

• Language: English
• Keywords: boundary value problems, mixed Riemann-Hilbert problem, Cherepanov problem
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 20 str.
• Numbering: Vol. 529, iss. 1, art. 127557
• PID: 20.500.12556/RUL-148423
• UDC: 517.9
• ISSN on article: 0022-247X
• DOI: 10.1016/j.jmaa.2023.127557
• COBISS.SI-ID: 158869763
• Publication date in RUL: 22.08.2023

Record is a part of a journal

• Title: Journal of mathematical analysis and applications
• Shortened title: J. math. anal. appl.
• Publisher: Elsevier
• ISSN: 0022-247X
• COBISS.SI-ID: 3081231

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: robni problem, mešani Riemann-Hilbertov problem, problem Čerepanova