Compactness of classes of solutions of the Dirichlet problem for Beltrami equations (Q2901729)

The present paper is devoted to the study of the Beltrami equation with degeneration. The technique of the research here is closely related to the so-called ring \(Q\)-homeomorphisms introduced recently by V. Ryazanov, U. Srebro and E. Yakubov. It is proved that a limit of a sequence of regular homeomorphisms in the class ACL that map the unit disk \({\mathbb D}\) onto itself fixing 0 and such that the maximal dilatations \(K_{\mu}(z)\) have a general majorant \(Q(z)\in L^1({\mathbb D})\) is a regular homeomorphism of the same class. Moreover, the above class of homeomorphisms is shown to be a compact family provided that \(Q(z)\) satisfies some additional conditions. As an application, compactness results for classes of regular solutions of the Dirichlet problem are obtained. More precisely, all investigations of the author hold in arbitrary domains in \({\mathbb C}\). The interesting and very important statement of the paper is Lemma 1 presenting a lower estimate of the length of some curve trough the diameter of a given domain \(D\). Lemma 2 states a lower estimate of the Euclidean distance between two points \(f(z_1), f(z_2)\), where \(z_1\) and \(z_2\) are in the unit disk \({\mathbb D}\) and \(f\) is a regular homeomorphism \(f:{\mathbb D}\rightarrow {\mathbb D}\) satisfying the conditions \(f({\mathbb D})={\mathbb D}\), \(f(0)=0\) and \(K_{\mu}\in L^{1}({\mathbb D})\). The above estimate is uniform in \(f\); here, we use the notion of \(K_{\mu} (z)\) for the maximal dilatation of the mapping \(f\).NEWLINENEWLINEA homeomorphism \(f\) is said to be regular if \(f\in W_{loc}^{1,1}\) and its Jacobian is non-degenerate almost everywhere. Theorem 1 states the closeness of the class of all regular homeomorphisms satisfying all the conditions of Lemma 2 and having a general majorant that is integrable in \({\mathbb D}\). The most important statement is Theorem 2 stating the following. For every continuous function \(\varphi: \partial D \rightarrow {\mathbb R}\), the class of all regular solutions of the Dirichlet problem NEWLINE\[NEWLINEf_{\overline z}=\mu(z)\cdot f_z, \quad f| {_{\partial {\mathbb D} }}=\varphiNEWLINE\]NEWLINE for the Beltrami equation \(f_{\overline z}=\mu(z)\cdot f_z\) in \({\mathbb D}\) is a compact family of mappings provided that the above class has a general majorant satisfying finite mean oscillation at every point \(z_0\in \overline{{\mathbb D}}\).