On the Compactness of One Class of Solutions for the Dirichlet Problem

We consider the Dirichlet problem for the Beltrami equation in an arbitrary bounded simply connected domain in the complex plane C. Namely, we study the class of all regular solutions of such a problem with a normalization condition and set-theoretic constraints on their complex characteristics. We have proved the compactness of this class in terms of prime ends for an arbitrary continuous function in the Dirichlet condition.