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

Let w be a K -quasiconformal self-mapping of the unit disk D satisfying the Dirichlet–Neumann problem: ( ∂ z ∂ z ¯ ) 2 w = g in D , w = γ 0 and ∂ ν ∂ z ∂ z ¯ w = γ on T (the unit circle), 1 2 π i ∫ T w ζ ζ ¯ ( ζ ) d ζ ζ = c , where ∂ ν denotes the differentiation in the outward normal direction. In addition, suppose that the data for w satisfy the following two conditions: 1 2 π ∫ 0 2 π γ ( e it ) d t = 2 π ∫ D g ( ζ ) d A ( ζ ) 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 |, ‖ γ ‖ ∞ and ‖ g ‖ ∞ are small enough. Moreover, the estimates are asymptotically sharp as K → 1 , | c | → 0 , ‖ γ ‖ ∞ → 0 and ‖ g ‖ ∞ → 0 , and so, such a mapping w behaves almost like a rotation for sufficiently small K , | c |, ‖ γ ‖ ∞ and ‖ g ‖ ∞ .