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Inner estimate and quasiconformal harmonic maps between smooth domains. (English) Zbl 1173.30311
Summary: We prove a type of “inner estimate” for quasi-conformal diffeomorphisms, which satisfies a certain estimate concerning their Laplacian. This, in turn, implies that quasiconformal harmonic mappings between smooth domains (with respect to an approximately analytic metric), have bounded partial derivatives; in particular, these mappings are Lipschitz. We discuss harmonic mappings with respect to (a) spherical and Euclidean metrics (which are approximately analytic) (b) the metric induced by a holomorphic quadratic differential.

MSC:
30C62 Quasiconformal mappings in the complex plane
30C25 Covering theorems in conformal mapping theory
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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