Riemann–Hilbert problems with shift on the Lyapunov curve for null-solutions of iterated Beltrami equations

Abstract

Abstract In this article, we study some Riemann–Hilbert problems with shift on the Lyapunov curve for generalized polyanalytic functions, which are null-solutions of a class of iterated Beltrami equations. Firstly, we obtain two integral representations of these functions by using Cauchy formula associated with the Beltrami equations and explicitly constructing two weakly singular kernels. Next, we develop a theory of matrix factorization for triangular matrix functions with shift in the frame of Beltrami equations and solve the Riemann–Hilbert problems by using them and the decomposition theorem for the iterated Beltrami equations. Finally, we get explicit formulae of solutions and conditions of solvability for the Riemann–Hilbert problems.

Keywords

• Riemann–Hilbert problem
• Polynomial Beltrami equation
• Integral representation
• Canonical matrix
• Triangular matrix