The Riemann-Hilbert problem for degenerate elliptic complex equations and its application

Abstract

This article deals with some elliptic complex equations of first order, i.e. the generalized Beltrami equation with two degenerate lines in multiply connected domains. We first propose the Riemann–Hilbert problem and give estimates of solutions for the boundary value problem. Afterwards by using the method of parameter extension, the existence of continuous solutions for the generalized Beltrami equation is verified. Finally, we shall prove the existence of a homeomorphic solution of the Beltrami equation with two degenerate lines from the multiply connected domain D onto the rectilinear slit domain. In the article, the Hlder continuity of a singular double integral is applied. The above problem possesses important applications to boundary value problem of mixed type equations.

Keywords:

• elliptic complex equations
• two degenerate lines
• Riemann–Hilbert problem
• existence of solutions
• Hölder continuity of singular integral
• quasi-conformal mapping

AMS Subject Classifications:

• 35J70
• 35J55
• 30C60

Acknowledgments

The author would like to thank the referees for their valuable opinions and improve the quality of this article.