Bi-Lipschitz characteristic of quasiconformal self-mappings of the unit disk satisfying bi-harmonic equation

TL;DR

This paper proves that certain quasiconformal self-mappings of the unit disk satisfying a biharmonic equation are Lipschitz continuous, and bi-Lipschitz when parameters are small, with asymptotically sharp estimates as parameters approach zero.

Contribution

It establishes bi-Lipschitz continuity of quasiconformal solutions to a biharmonic equation with boundary conditions, extending understanding of their geometric behavior.

Findings

Mappings are Lipschitz continuous.

Mappings are bi-Lipschitz when parameters are small.

Estimates are asymptotically sharp as parameters approach zero.

Abstract

Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk $\mathbb{D}$, which satisfies the following: $(1)$ the biharmonic equation $\Delta(\Delta f)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary condition $\Delta f=\varphi$ ($\varphi\in\mathcal{C}(\mathbb{T})$ and $\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$. The purpose of this paper is to prove that $f$ is Lipschitz continuos, and, further, it is bi-Lipschitz continuous when $\|g\|_{\infty}$ and $\|\varphi\|_{\infty}$ are small enough. Moreover, the estimates are asymptotically sharp as $K\to 1$, $\|g\|_{\infty}\to 0$ and $\|\varphi\|_{\infty}\to 0$, and thus, such a mapping $f$ behaves almost like a rotation for sufficiently small $K$, $\|g\|_{\infty}$ and $\|\varphi\|_{\infty}$.