On compact classes of solutions of the Dirichlet problem with integral restrictions

TL;DR

This paper establishes the compactness of classes of solutions to the Dirichlet problem with integral restrictions, focusing on Beltrami equation solutions with specific boundary and support conditions.

Contribution

It introduces new theorems on the compactness of solution classes for the Beltrami equation with integral constraints, impacting Dirichlet problem solutions.

Findings

Proved compactness theorems for Beltrami equation solutions

Established results for Dirichlet problems in Jordan domains

Analyzed solutions with integral-type characteristic constraints

Abstract

We have proved theorems on compact classes of homeomorphisms with hydrodynamic normalization that are solutions of the Beltrami equation, whose characteristics are compactly supported and satisfy certain constraints of an integral type. As a consequence, we obtained results on compact classes of solutions of corresponding Dirichlet problems considered in some Jordan domain.