On certain quasiconformal and elliptic mappings

TL;DR

This paper characterizes elliptic mappings satisfying Poisson's equation in the unit disk and establishes sharp distortion theorems for such mappings with finite perimeter and radial length, extending classical results.

Contribution

It provides new characterizations and sharp distortion theorems for elliptic mappings solving Poisson's equation, extending classical results in the field.

Findings

Characterizations of elliptic mappings satisfying Poisson's equation

Sharp distortion theorems for elliptic mappings with finite perimeter

Extension of classical results to broader classes of mappings

Abstract

Let $\overline{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\overline{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings $f$ satisfying the Poisson's equation $\Delta f=g$ in $\mathbb{D}$, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results.