The method of extremal metric in extremal decomposition problems with free parameters
Let \(R(D_k,a_k)\) be the conformal radius of a domain \(D_k\) with respect to the point \( a_k \in D_k, \) and let \(\mathcal D \) be a family \( \{ D_1,D_2,\dots,D_n \} \) of non-overlapping simply connected domains. The topic of the paper under review is the problem to maximize \[ \sum_{k=1}^n \alpha_kM(D_k,a_k) \] in \(\mathcal D \) with \( M(D_k,a_k)= \frac{1}{2\pi}\log R(D_k,a_k) \) if \(a_k \not = \infty \) and \ \( M(D_k,a_k)= - \frac{1}{2\pi}\log R(D_k,a_k) \) if \(a_k = \infty. \) The method of extremal metric is an essential tool for solving such problems. A comprehensive treatment of this method is given by [\textit{J. A. Jenkins}, The method of extremal metric, in: Kühnau, R. (ed.), Handbook of complex analysis: geometric function theory. Volume 1. Amsterdam: North Holland. 393--456 (2002; Zbl 1064.30014)]. Based on an approach given by \textit{V. N. Dubinin} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, Vol. 168, 48--66 (1988; Zbl 0684.30017)] and used by \textit{E. G. Emel'yanov} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 160, 91--98 (1987; Zbl 0661.05009)], the above mentioned problem is solved for a system of alternating points of the circle \(|z|=1\). For the extremal system of domains a quadratic differential is given describing these domains and also the maximum value is given.
scientific article; zbMATH DE number 5312792