On compactness of classes of solutions of Beltrami equations with theoretic-set type constraints
The authors study the Beltrami equation \(f_{\overline z}=\mu(z)\cdot f_z\) and investigate the problems of compactness of the classes of its solutions. The Beltrami equation is a classical and well studied topic in geometrical function theory. The research methods are closely connected with those for different problems of space and plane mappings.NEWLINENEWLINE Denote by \(\Delta_z\) the disk centered at the origin with radius \(q(z)\). For a given mapping \(f\) we define NEWLINE\[NEWLINE\begin{cases} \mu(z)=\mu_f(z)=f_{\overline{z}}/f_z &\text{ if } f_z\neq 0\\ \mu(z)=\mu_f(z)=0 &\text{ if } f_z=0.\end{cases}NEWLINE\]NEWLINE The main results of the work under review are the following. The family of all plane regular homeomorphisms normalized by \(f(0)=0\), \(f(1)=1\) and \(f(\infty)=\infty\) and having a majorant which is integrable in \({\mathbb C}\) is closed. Furthermore, the set of all homeomorphisms \(f:\overline{{\mathbb C}}\rightarrow \overline{{\mathbb C}}\) in the class \(W_{loc}^{1,1}\) normalized by \(f(0)=0\), \(f(1)=1\) and \(f(\infty)=\infty\), whose complex dilatations \(\mu(z)\) lie in the set \(M(z)\subset \Delta_{q(z)}\) for a.e. \(z\), provided that a maximal dilatation \(K_{\mu}(z):=\frac{1+| \mu(z)| }{1-| \mu(z)| }\) has a majorant \(Q(z)\) which satisfies the FMO condition, is compact.
scientific article; zbMATH DE number 6022827