Riemann–Hilbert problem for variable exponent poly-Smirnov space on a piecewise Lyapunov boundary

Abstract

In this paper we study the Riemann–Hilbert problem for variable exponent poly-Smirnov space on a piecewise Lyapunov boundary. At first we introduce the poly-Smirnov function space with variable exponent, we obtain a decomposition of this space and verify the existence of the angular boundary value of function in this space. Then we investigate Dirichlet problem for poly-Smirnov function space. At last we consider Riemann–Hilbert problem for variable exponent poly-Smirnov space on a piecewise Lyapunov boundary. The method is transforming this problem into independent homogeneous Dirichlet problems, and the conditions of solvability as well as the explicit solutions are obtained.

Keywords:

• Riemann–Hilbert problem
• variable exponent
• poly-Smirnov space
• Dirichlet problem
• piecewise Lyapunov curve

AMS Subject Classifications:

• 30E20
• 30E25
• 35Q15
• 30H15
• 42B30
• 46E30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Author contributions

All authors contributed to the study conception and design, all authors read and approved the final manuscript.

Data availability statements

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Funding

This work was supported by National Natural Science Foundation of China [11601525].