Bi-Lipschitz Continuity of Quasiconformal Solutions to a Biharmonic Dirichlet–Neumann Problem in the Unit Disk

Abstract

Let w be a K-quasiconformal self-mapping of the unit disk ${{\mathbb {D}}}$ satisfying the Dirichlet–Neumann problem: $(\partial _z\partial _{\overline{z}})^2 w=g$ in ${{\mathbb {D}}}$, $w=\gamma _0$ and $\partial _{\nu }\partial _z\partial _{\overline{z}}w=\gamma$ on $\mathbb {T}$ (the unit circle), $\frac{1}{2\pi i}\int _{\mathbb {T}}w_{\zeta \overline{\zeta }}(\zeta )\frac{d\zeta }{\zeta }=c$, where $\partial _\nu$ denotes the differentiation in the outward normal direction. In addition, suppose that the data for w satisfy the following two conditions: $\frac{1}{2\pi }\int ^{2\pi }_{0}\gamma (e^{it})\,dt=\frac{2}{\pi }\int _{{{\mathbb {D}}}}g(\zeta )\,d A(\zeta )$ and $w(0)=0$. The aim of this paper is to prove that w is Lipschitz continuous and, furthermore, it is bi-Lipschitz continuous when |c|, $\Vert \gamma \Vert _{\infty }$ and $\Vert g\Vert _{\infty }$ are small enough. Moreover, the estimates are asymptotically sharp as $K\rightarrow 1$, $|c|\rightarrow 0$, $\Vert \gamma \Vert _{\infty }\rightarrow 0$ and $\Vert g\Vert _{\infty }\rightarrow 0$, and so, such a mapping w behaves almost like a rotation for sufficiently small K, |c|, $\Vert \gamma \Vert _{\infty }$ and $\Vert g\Vert _{\infty }$.