The method of extremal metric in extremal decomposition problems with free parameters

Let $a_1,\dots,a_n$ be a system of distinct points on the $z$-sphere $\overline{\mathbb C}$, and let $\mathcal D$ be a system of all non-overlapping simply-connected domains $D_1,\dots,D_n$ on $\overline{\mathbb C}$ such that $a_k\in D_k$, $k=1,\dots, n$. Let $M(D_k, a_k)$ be the reduced module of the domain Dk with respect to the point $a_k\in D_k$. In the present paper, we solve some problems concerning the maximum of weighted sums of the reduced modules $M(D_k, a_k)$ in certain families of systems of domains $\{D_k\}$ described above, where the systems of points $\{a_k\}$ satisfy prescribed symmetry conditions. In each case, the proof is based on an explicit construction of an admissible metric of the module problem, which is equivalent to the extremal problem under consideration, from known extremal metrics of simpler module problems.