Riemann boundary value problem on nonrectifiable curves for certain Beltrami equations

Abstract

In this paper, we obtain a Cauchy‐type integral representation to solve certain Beltrami equations and apply it for research of the Riemann boundary value problem for that equations on nonrectifiable contours. While more or less studied for piecewise smooth contours, for the case of nonrectifiable curve, this is a pioneer result. There is only one relatively simple type of Beltrami equation that is solved under such condition (see below). Our next goal is to achieve solutions for more types of the Beltrami equations on nonrectifiable curves.

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

“…Finally, by decomposition theorem for solutions of higher order iterated Beltrami equations, we transform the original problems to the vector-valued Riemann-Hilbert problems with shift for β-analytic functions, and further construct canonical matrix of triangular matrix functions explicitly, and thus obtain the explicit solutions and conditions of solvability. The main results extend those of [18,19,22,27,29,30,32,[34][35][36][37][38].…”

“…In this section, we recall some basic notions, notations, and facts in [30,32] which will be used in the sequel, and establish a decomposition theorem for a class of higher order iterated Beltrami equations.…”

“…In recent years, Blaya and his collaborators studied some jump Riemann BVPs for solutions of a special Beltrami equation with good properties, which are called β-analytic functions (see [26,27]) and were firstly proposed by Tungatarov [26]. Afterwards, some variations of these BVPs and related problems aroused a great deal of interest in some fields such as harmonic analysis, infinitesimal geometry, and fractal theory [27][28][29][30][31][32][33]. More recently, Kats made a deep study on jump Riemann BVPs for a wider class on non-rectifiable curves in [30,32].…”

“…Afterwards, some variations of these BVPs and related problems aroused a great deal of interest in some fields such as harmonic analysis, infinitesimal geometry, and fractal theory [27][28][29][30][31][32][33]. More recently, Kats made a deep study on jump Riemann BVPs for a wider class on non-rectifiable curves in [30,32]. They gave explicit solutions and conditions of solvability by introducing integration over non-rectifiable paths and establishing the corresponding Cauchy-Pompeiu formula.…”