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Univalent harmonic exterior mappings as solutions of an optimization problem. (English) Zbl 0708.30013

In the paper under review a region \(\Omega \subset {\mathbb{R}}^*\) is called costarlike if the complement of \(\Omega\) is bounded and starlike with respect to the origin. Let \(\Omega\) be a costarlike region and \(\Delta =\{z:| z| >1\}\) the exterior of the unit circle. The authors study mappings f: \(\Delta\to \Omega\), univalent, harmonic, orientation preserving and keeping \(\infty\) fixed. According to a paper by W. Hengartner and G. Schober [Trans. Am. Math. Soc. 299, 1-31 (1987; Zbl 0613.30020)] these mappings must have the form \[ f(z,\bar z)=Az+\overline{Bz}+C \log | z|^ 2+h(z)+\overline{g(z)}, \] where h, g are holomorphic on \(\Delta\), and f satisfies a certain first order partial differential equation of Beltrami type. In a first step the authors show that under certain conditions the above mapping problem has a unique solution F and in a second step they show, that F is a solution of an extremal problem which gives rise to a numerical procedure for approximating F involving nonlinear optimization techniques. There is a short numerical example where \(\Omega\) is bounded by an ellipse.
Reviewer: G.Opfer

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions

Citations:

Zbl 0613.30020
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