×

Connections between different problems on extremal decomposition. (English. Russian original) Zbl 0995.30018

J. Math. Sci., New York 105, No. 4, 2197-2209 (2001); translation from Zap. Nauchn. Semin. POMI 254, 116-131 (1998).
Let \(R(D,c)\) denote the inner conformal radius of a simply-connected domain \(D\) in the plane with respect to a point \(c\in D\). The author treats several problems of mazimizing a product \(\prod^n_{k=1} R(D_k,c_k)\) for non-overlapping domains \(D_k\). From the reviewer’s fundamental theorem for modules of multiple curve families it follows that the solution of such a problem is associated with a quadratic differential [Ann. Math., II. Ser. 66, 440-453 (1957; Zbl 0082.06301)].
The first problem is, given \(r>0\) for points \(a_k,b_k,k=1, \dots, n\), with \(a_k=r^{-1} e^{i\pi\varphi_k}\), \(b_k=re^{i\pi \psi_k}\) and \(\varphi_1 <\psi_1< \varphi_2<\psi_2 <\cdots< \varphi_n< \psi_n< \varphi_{n+1} \) to maximize \(\prod^n_{k=1} R(D_k^{(1)},a_k) R(D_k^{(2)},b_k)\) with all \(D_k^{(1)}, D_k^{(2)}\) non-overlapping.
The second problem with \(r>0\), \(\alpha \in (0,\pi/2]\), \(c_1=-1/r\), \(c_2=1/r\), \(c_3=re^{i\alpha}\), \(c_4=re^{-i \alpha}\), \(D_k\), \(k=1,\dots,4\), non-overlapping is to maximize \(\prod^4_{k=1} R(D_k,c_k)\). The third problem is to find the maximum of \(\prod^3_{k=1} R(D_k, c_k)\) where \(D_k\), \(k=1,2,3\), are non-overlapping simply-connected domains in the unit disc, \(c_k\in D_k\), \(k=1,2,3\).

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

Citations:

Zbl 0082.06301
PDFBibTeX XMLCite
Full Text: DOI