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Smoothness to the boundary of conformal maps. (English) Zbl 0626.30005

A self contained and elementary proof of the following classical result is given. Theorem. Let \(\Omega\subseteq {\mathbb{C}}\) be a bounded simply connected domain with \(C^{\infty}\) smooth boundary. Let \(f: \Omega\to \Delta\) be a one-to-one conformal map of \(\Omega\) onto the unit disc, as provided by the Riemann mapping theorem. Then f continues up to the boundary to be a diffeomorphism between the closure of \(\Omega\) and the closure of \(\Delta\).
A brief history of the theorem is given, together with applications to the Dirichlet problem.

MSC:

30C35 General theory of conformal mappings
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
30C40 Kernel functions in one complex variable and applications
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