On the Dirichlet problem for degenerate Beltrami equations

TL;DR

This paper studies solutions to degenerate Beltrami equations, showing they are lower Q-homeomorphisms and establishing the existence of boundary and Dirichlet problem solutions in various complex domains.

Contribution

It introduces the concept of lower Q-homeomorphisms for solutions of degenerate Beltrami equations and develops their boundary behavior theory, proving existence of solutions in complex domains.

Findings

Solutions are lower Q-homeomorphisms with tangent dilatation

Existence of regular solutions in Jordan domains

Existence of pseudoregular and multi-valued solutions in finitely connected domains

Abstract

We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ to a Beltrami equation $\bar{\partial}f=\mu \partial f$ in a domain $D\subset\Bbb C$ is the so--called lower $Q-$homeomorphism with $Q(z)=K^T_{\mu}(z, z_0)$ where $K^T_{\mu}(z, z_0)$ is the tangent dilatation of $f$ with respect to an arbitrary point $z_0\in {\bar{D}}$ and develop the theory of the boundary behavior of such solutions. Then, on this basis, we show that, for wide classes of degenerate Beltrami equations $\bar{\partial}f=\mu \partial f$, there exist regular solutions of the Dirichlet problem in arbitrary Jordan domains in $\Bbb C$ and pseudoregular and multi-valued solutions in arbitrary finitely connected domains in $\Bbb C$ bounded by mutually disjoint Jordan curves.