Quasiconformal maps with controlled Laplacian

Abstract

We establish that every K-quasiconformal mapping w of the unit disk D onto a C 2 -Jordan domain Ω is Lipschitz provided that ∆w ∈ L p (D) for some p > 2. We also prove that if in this situation K → 1 with ∆w L p (D) → 0, and Ω → D in C 1,α -sense with α > 1/2, then the bound for the Lipschitz constant tends to 1. In addition, we provide a quasiconformal analogue of the Smirnov theorem on absolute continuity over the boundary.

On certain quasiconformal and elliptic mappings

“…Then f is a quasiconformal self-homeomorphism of D. However, f is not Lipschitz at the origin (cf. [23]).…”

On Heinz type inequality and Lipschitz characteristic for mappings satisfying polyharmonic equations

“…[4]). But quasiconformal mappings are not necessarily bi-Lipschitz, not even Lipschitz (see [14,22,27]). Hence it is nature to consider the following question.…”

“…Astala and Manojlović [5], and Kalaj [26] developed different approaches to investigate the Lipschitz-property of harmonic quasiconformal mappings in R n . On the discussion of the related topic, we refer to [1,2,3,10,22,27,28,29,30,31] and the references therein. By using Theorem 1.1, Green's potential theory and partial proof methods in [26], we will discuss the Lipschitz characteristic of quasiconformal mappings satisfying the polyharmonic equations.…”

Quasiconformal Harmonic Mappings Between Domains Containing Infinity

“…The same conclusion was obtained in [2] by Božin and Mateljević for merely C 1,α domains. Further results in the two-dimensional case can be found in [13] and for several dimensions in [1] and [14]. For a different setting for the class of quasiconformal harmonic mappings, we refer to the papers [16,18].…”