Quasiconformal and harmonic mappings between Jordan domains

Abstract

Let and 1 be Jordan domains, let µ ∈ (0, 1], and let f : → 1 be a harmonic homeomorphism. The object of the paper is to prove the following results: (a) If f is q.c. and ∂ , ∂ 1 ∈ C 1,µ , then f is Lipschitz; (b) if f is q.c., ∂ , ∂ 1 ∈ C 1,µ and 1 is convex, then f is bi-Lipschitz; and (c) if is the unit disk, 1 is convex, and ∂ 1 ∈ C 1,µ , then f is quasiconformal if and only if its boundary function is bi-Lipschitz and the Hilbert transform of its derivative is in L ∞ . These extend the results of Pavlović (Ann. Acad. Sci. Fenn. 27:365-372, 2002).

On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation

“…O. Martio [17] was the first who considered harmonic quasiconformal mappings on the complex plane. Recent papers [10], [12], [14], [21] and [13] bring much light on the topic of quasiconformal harmonic mappings on the plane. See also [11] for the extension of the problem on the space.…”

On Certain Nonlinear Elliptic PDE and Quasiconformal Maps Between Euclidean Surfaces

“…harmonic mappings in the plane we refer to the papers [3][4][5][6][7][8][9][11][12][13][14][15][16].…”

Harmonic quasiconformal self-mappings and Möbius transformations of the unit ball

“…Martio [1968] was the first to consider harmonic quasiconformal mappings on the complex plane. Recent papers [Kalaj 2004[Kalaj , 2008Kalaj and Mateljević 2006;Pavlović 2005, 2009;Manojlović 2009;Pavlović 2002] shed much light on this topic. Proposition 1.1 [Kalaj 2009].…”