On compactness of one class of solutions of the Dirichlet problem

TL;DR

This paper proves the compactness of a class of homeomorphic solutions to the Dirichlet problem for the Beltrami equation in simply connected domains, using prime ends and set-theoretic constraints.

Contribution

It establishes the compactness of solutions class under broad conditions, extending understanding of solution behavior for the Beltrami equation.

Findings

Class of solutions is compact in prime ends topology.

Compactness holds for arbitrary continuous Dirichlet data.

Results apply to solutions with set-theoretic constraints on complex characteristics.

Abstract

We consider the Dirichlet problem for the Beltrami equation in some simply connected domain. We consider the class of all homeomorphic 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.