On asymptotically sharp bi-Lipschitz inequalities of quasiconformal mappings satisfying inhomogeneous polyharmonic equations

Abstract

Suppose that $f$ is a $K$-quasiconformal ($(K,K^{\prime })$-quasiconformal resp.) self-mapping of the unit disk $\mathbb{D}$, which satisfies the following: $(1)$ the inhomogeneous polyharmonic equation ${\Delta }^{n}f=\Delta ({\Delta }^{n-1}f)={\phi }_n$ $({\phi }_n\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary conditions ${\Delta }^{n-1}f{|}_{\mathbb{T}}={\phi }_{n-1},\dots ,{\Delta }^{1}f{|}_{\mathbb{T}}={\phi }_1$ (${\phi }_j\in \mathcal{C}(\mathbb{T})$ for $j\in \{1,\dots ,n-1\}$ and $\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$, where $n\ge 2$ is an integer and $K\ge 1$ ($K^{\prime }\ge 0$ resp.). The main aim of this paper is to prove that $f$ is Lipschitz continuous, and,further, it is bi-Lipschitz continuous when $\Vert {\phi }_j{\Vert }_{\infty }$ are small enough for $j\in \{1,\dots ,n\}$. Moreover, the estimates are asymptotically sharp as $K\to 1$ ($K^{\prime }\to 0$ resp.) and $\Vert {\phi }_j{\Vert }_{\infty }\to 0$ for $j\in \{1,\dots ,n\}$, and thus, such a mapping $f$ behaves almost like a rotation for sufficiently small $K$ ($K^{\prime }$ resp.) and $\Vert {\phi }_j{\Vert }_{\infty }$ for $j\in \{1,\dots ,n\}$.

37 pages. To appear in Journal of Geometric Analysis. arXiv admin note: substantial text overlap with arXiv:1801.04428