×

Problems of the extremal decomposition of the Riemannian sphere. III. (Russian, English) Zbl 1081.30028

Zap. Nauchn. Semin. POMI 314, 124-141, 287 (2004); translation in J. Math. Sci., New York 133, No. 6, 1676-1685 (2006).
Let \(\mathbb{D}(a)\) denote a system of all disjoint simply connected domains \(\{D_1,\dots, D_n\}\), \(a_k\in D_k\) for \(k= 1,2,\dots, n\), where \({\mathbf a}= \{a_1,\dots, a_n\}\) is a system of different points on the plane \(\overline{\mathbb{C}}\), \(\alpha= \{\alpha_1,\dots, \alpha_n\}\) – a system of real positive numbers. Let \(R(D_k, a_k)\) denote the radius of conformality of the domain \(D_k\) at the point \(a_k\). In many papers one can find the problems: to determine the maximum of the quantities: \[ \mathbb{R}= \sideset\and{^\prime}\to\prod^n_{k=1} R^{\alpha^2_k}(D_k, \alpha_k)\tag{1} \] (where \(R(D_k,\infty)\) in (1) we replace with \(R^{-1}(D_k,\infty)\)); \[ \mathbb{M}= \sum^n_{k=1} \alpha^2_k M(D_k, \alpha_k)\tag{2} \] (where \(M(D,a)= {1\over 2\pi}\log R(D,a)\)); \[ \log\mathbb{J}(a_1,\dots, a_n)= 2\pi\mathbb{M}(a_1,\dots, a_n)- {2\over n- 1}\log\prod_{1\leq k\leq l\leq n}|a_k- a_l| \] (where \(\mathbb{M}(a_1,\dots, a_n)= \mathbb{M}(a_1,\dots, a_n; 1,\dots, 1)\); the maximum of the quantity (2) does not depend on the domains).
The paper belongs to the mentioned publication cycle and it is a continuation of the author’s research from 1990, 2001–2003. There is proved the theorem on the maximum of \(\mathbb{J}(a_1,\dots,a_5)\) for an arbitrary system of points.

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
PDFBibTeX XMLCite
Full Text: EuDML