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

Abstract

For $K\geq1$, suppose that $f$ is a $K$-quasiconformal self-mapping of the unit ball $\mathbb{B}^{n}$, which satisfies the following: $(1)$ the polyharmonic equation $\Delta^{m}f=\Delta(\Delta^{m-1} f)$$=\varphi_{m}$ $(\varphi_{m}\in\mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n}))$, (2) the boundary conditions $\Delta^{m-1}f|_{\mathbb{S}^{n-1}}=\varphi_{m-1},~\ldots,~\Delta^{1}f|_{\mathbb{S}^{n-1}}=\varphi_{1}$ ($\varphi_{k}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$ for $j\in\{1,\ldots,m-1\}$ and $\mathbb{S}^{n-1}$ denotes the unit sphere in $\mathbb{R}^{n}$), and $(3)$ $f(0)=0$, where $n\geq3$ and $m\geq2$ are integers. We first establish a Heinz type inequality on mappings satisfying the polyharmonic equation. Then we use the obtained results to show that $f$ is Lipschitz continuous, and the estimate is asymptotically sharp as $K\to 1$ and $\|\varphi_{j}\|_{\infty}\to 0$ for $j\in\{1,\ldots,m\}$.Comment: 28 page