Ring homeomorphisms and prime ends

TL;DR

This paper demonstrates that solutions to certain Beltrami equations are ring homeomorphisms with specific dilatation properties and develops boundary behavior theory to establish existence of various solutions to the Dirichlet problem in complex domains.

Contribution

It introduces the concept of ring Q-homeomorphisms related to Beltrami equations and analyzes their boundary behavior using prime ends, ensuring solution existence in complex domains.

Findings

Solutions are ring Q-homeomorphisms with tangent dilatation quotient.

Boundary behavior theory for ring Q-homeomorphisms with prime ends.

Existence of regular, pseudoregular, and multivalent solutions to the Dirichlet problem.

Abstract

We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ of a Beltrami equation $\overline{\partial}f=\mu\,\partial f$ in a domain $D\subseteq\Bbb C$ is the so--called ring $Q-$homeomorphism with $Q(z)=K^T_{\mu}(z, z_0)$ where $K^T_{\mu}(z, z_0)$ is the tangent (angular) dilatation quotient of the equation with respect to an arbitrary point $z_0\in {\overline{D}}$. In this connection, we develop the theory of the boundary behavior of the ring $Q-$homeomorphisms with respect to prime ends. On this basis, we show that, for wide classes of degenerate Beltrami equations $\overline{\partial}f=\mu\,\partial f$, there exist regular solutions of the Dirichlet problem in arbitrary simply connected domains in $\Bbb C$ and pseudoregular and multivalent solutions in arbitrary finitely connected domains in $\Bbb C$ with boundary datum $\varphi$ that are continuous with respect to the topology of prime ends.