On boundary behavior and Dirichlet problem for Beltrami equations

TL;DR

This paper investigates the boundary behavior of homeomorphic solutions to Beltrami equations, establishing criteria for the existence of various types of solutions to the Dirichlet problem in complex domains.

Contribution

It develops a comprehensive theory linking moduli inequalities to boundary behavior and solves the Dirichlet problem for Beltrami equations in complex domains.

Findings

Established moduli inequalities for solutions

Derived criteria for existence of regular and multi-valued solutions

Extended results to arbitrary Jordan and finitely connected domains

Abstract

We show that arbitrary homeomorphic solutions to the Beltrami equations with generalized derivatives satisfy certain moduli inequalities. On this basis, we develope the theory of the boundary behavior of such solutions and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for the Dirichlet problem to the Beltrami equations in arbitrary Jordan domains and in arbitrary finitely connected domains bounded by mutually disjoint Jordan curves, correspondingly.