Abstract
For two constants \(K\ge 1\) and \(K'\ge 0\), suppose that f is a \((K,K')\)-quasiconformal self-mapping of the unit disk \({\mathbb {D}}\), which satisfies the following: (1) the inhomogeneous polyharmonic equation \(\Delta ^{n}f=\Delta (\Delta ^{n-1} f)=\varphi _{n}\) in \({\mathbb {D}}\) \((\varphi _{n}\in {\mathcal {C}}(\overline{{\mathbb {D}}}))\), (2) the boundary conditions \(\Delta ^{n-1}f=\varphi _{n-1},~\ldots ,~\Delta ^{1}f=\varphi _{1}\) on \({\mathbb {T}}\) (\(\varphi _{j}\in {\mathcal {C}}({\mathbb {T}})\) for \(j\in \{1,\ldots ,n-1\}\) and \({\mathbb {T}}\) denotes the unit circle), and (3) \(f(0)=0\), where \(n\ge 2\) is an integer. The main aim of this paper is to prove that f is Lipschitz continuous, and, further, it is bi-Lipschitz continuous when \(\Vert \varphi _{j}\Vert _{\infty }\) are small enough for \(j\in \{1,\ldots ,n\}\). Moreover, the estimates are asymptotically sharp as \(K\rightarrow 1^{+}\), \(K'\rightarrow 0^{+}\) and \(\Vert \varphi _{j}\Vert _{\infty }\rightarrow 0^{+}\) for \(j\in \{1,\ldots ,n\}\).
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Acknowledgements
We are grateful to the referee for her/his comments and suggestions. This research was partly supported by the exchange project for the third regular session of the China-Montenegro Committee for Cooperation in Science and Technology (No. 3-13), the Hunan Provincial Education Department Outstanding Youth Project (No. 18B365), the Science and Technology Plan Project of Hengyang City (No. 2018KJ125), the National Natural Science Foundation of China (No. 11571216), the Science and Technology Plan Project of Hunan Province (No. 2016TP1020), the Science and Technology Plan Project of Hengyang City (No. 2017KJ183), and the Application-Oriented Characterized Disciplines, Double First-Class University Project of Hunan Province (Xiangjiaotong [2018]469).
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Chen, S., Kalaj, D. On Asymptotically Sharp Bi-Lipschitz Inequalities of Quasiconformal Mappings Satisfying Inhomogeneous Polyharmonic Equations. J Geom Anal 31, 4865–4905 (2021). https://doi.org/10.1007/s12220-020-00460-9
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DOI: https://doi.org/10.1007/s12220-020-00460-9