On the theory of convergence and compactness for Beltrami equations with constraints of set-theoretic type
Let \(D\) be a domain in the complex plane \(\mathbb C\), let \(\mu (z) : D \rightarrow \mathbb C\) be a measurable function such that \(|\mu (z)| < 1\) almost everywhere, \(z = x+iy\), \(f\overline{z} =\frac{ f_x +if_y}{2}\), \(f_z =\frac{f_x - if_y}{2}\) and \(K_{\mu} (z) = (1 + |\mu (z)|)/(1 - |\mu (z)|)\). Then the equation \(f\overline{z} = \mu (z)f_z\) is called a Beltrami equation. In particular, if \(\text{esssup}K_{\mu} (z) = \infty\), then it is called degenerate. In the paper under review, the author establishes several theorems on the convergence and compactness for the regular solutions of the degenerate Beltrami equations with constraints of the set-theoretic type imposed on the complex coefficient \(\mu (z)\). As applications, variations for the classes of regular solutions of degenerate Beltrami equations are constructed.
scientific article; zbMATH DE number 6110037
scientific article