Abstract
We establish that every K-quasiconformal mapping w of the unit disk \(\mathbb{D}\) onto a C2-Jordan domain Ω is Lipschitz provided that Δw ∈ Lp(\(\mathbb{D}\)) for some p > 2. We also prove that if in this situation K → 1 with ||Δw||Lp(\(\mathbb{D}\)) → 0, and Ω→\(\mathbb{D}\) in C1,α-sense with α > 1/2, then the bound for the Lipschitz constant tends to 1. In addition, we provide a quasiconformal analogue of the Smirnov theorem on absolute continuity over the boundary.
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A part of this project was finished during the visit of the first author to Helsinki University in April 2014.
Supported by the Finnish CoE in Analysis and Dynamics Research, and by the Academy of Finland, projects 113826 and 118765.
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Kalaj, D., Saksman, E. Quasiconformal maps with controlled Laplacian. JAMA 137, 251–268 (2019). https://doi.org/10.1007/s11854-018-0072-5
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DOI: https://doi.org/10.1007/s11854-018-0072-5