On the Riemann-Hilbert problem for the Beltrami equations in quasidisks
Let \(D\) be a domain in the complex plane \(\mathbb C\) and let \(\mu:D\to\mathbb C\) be a measurable function with \(|\mu(z)|<1\) a.e. The Beltrami equation in \(D\) with coefficient \(\mu\) is the equation \[ f_{\bar{z}} =\mu(z) f_z, \tag{1} \] where \(f_{\bar{z}} =\bar{d}f=(f_x+if_y )/2\), \( f_z=df=(f_x-if_y )/2\), \(z=x+iy\), \(f_x\) and \(f_y\) are the partial derivatives of the function \(f:D\to\mathbb C \) with respect to \(x\) and \(y\), respectively. Equation (1) is called non-degenerate if \(\|\mu\|_\infty<1\). The homeomorphic solutions of the non-degenerate Beltrami equations (1) with generalized derivatives are called quasi-conformal mappings. The images of the unit circle \(D=\{z\in C; |z|<1\}\) under quasi-conformal mappings of \(\mathbb C\) onto itself are called quasidisks, and their boundaries are called quasicircles or quasi-conformal curves. In the paper under review, for the nondegenerate Beltrami equation in quasidisks, the existence of regular solutions of the Riemann-Hilbert problem with coefficients of bounded variation and boundary data that are measurable with respect to the absolute harmonic measure (logarithmic capacity) is proved.
scientific article; zbMATH DE number 6532884
scientific article