On boundary behavior of mappings with finite distortion in the plane

TL;DR

This paper investigates the boundary behavior of lower Q-homeomorphisms, a generalization of quasiconformal mappings, providing conditions for their extension to the boundary and applying the results to various classes of mappings.

Contribution

It establishes effective boundary extension conditions for lower Q-homeomorphisms and applies these to finite distortion mappings, Beltrami solutions, and bi-Lipschitz mappings.

Findings

Derived conditions for boundary extension via prime ends.

Extended the class of mappings with known boundary behavior.

Applied theory to finite distortion and Beltrami equation solutions.

Abstract

In the present paper, it was studied the boundary behavior of the so-called lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec, also to solutions of the Beltrami equations, as well as to finitely bi--Lipschitz mappings that a far--reaching extension of the known classes of isometric and quasiisometric mappings.