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

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 $\Delta f=\phi$ ($\phi \in \mathcal{C}(\mathbb{T})$ 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 $\Vert g{\Vert }_{\infty }$ and $\Vert \phi {\Vert }_{\infty }$ are small enough. Moreover, the estimates are asymptotically sharp as $K\to 1$, $\Vert g{\Vert }_{\infty }\to 0$ and $\Vert \phi {\Vert }_{\infty }\to 0$, and thus, such a mapping $f$ behaves almost like a rotation for sufficiently small $K$, $\Vert g{\Vert }_{\infty }$ and $\Vert \phi {\Vert }_{\infty }$.

arXiv comment26 pages. Indiana University Mathematics Journal, 2020

Cited by in corpus (1)

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