Problem of the extremal decomposition of a closed plane

Abstract

One considers the problem of the maximum of the product of powers of conformal radii of nonoverlapping domains in the following formulation. Let A=a₁, ..., a_n and B=(b₁, ..., b_m be systems of distinguished points in ¯C and let α={α₁,..., α_m} be a system of positive numbers. ByU(D_ℓ,b _ℓ) we denote the reduced modulus of the simply connected domain D_ℓ relative to the pointb _ℓ∈D_ℓ. Find the maximum of the sum$\sum\limits_{\ell = 1}^m {\alpha _\ell ^2 \mathcal{U}(\mathcal{D}_\ell ,b_\ell )}$ in the familyD of all systems of nonoverlapping simply connected domains D_j, j=1, ..., m, satisfying the following condition: the domain D_j does not contain points b_i ∃ B, different from b_j, and some collection A_j, for each domain, of points from A, ⋃ ^m _j=1 A _j =A. The solution of this problem is obtained by the simultaneous use of the method of variation and of the method of the moduli of families of curves and is given by Theorem 1 of the present paper. As consequences of Theorem 1 one obtains Theorems 2 and 3, strengthening the corresponding results of a previous paper of the author.