On compact classes of solutions of Dirichlet problem in simply connected domains

TL;DR

This paper investigates the compactness of solutions to the Dirichlet problem for the Beltrami equation in simply connected domains, providing detailed results under integral constraints on maximal dilations and analyzing the behavior of related mappings.

Contribution

It offers new theorems on the compactness and behavior of solutions to the Dirichlet problem for the Beltrami equation, with detailed prime end analysis and modulus conditions.

Findings

Proved compactness results for solutions under integral constraints.

Established local and global behavior theorems for plane and spatial mappings.

Analyzed the influence of maximal dilations on solution properties.

Abstract

The article is devoted to questions concerning the problems of compactness of solutions of the Dirichlet problem for the Beltrami equation in some simply connected domain. In terms of prime ends, we have proved results of a detailed form for the case when the maximal dilations of these solutions satisfy certain integral constraints. In addition, in this article we have proved theorems on the local and global behavior of plane and spatial mappings with direct and inverse modulus conditions.