On extremal decomposition problems

The capacity approach and symmetrization method arc adapted to some extremal decomposition problems on the unit disk or an annulus. The problems on the maximum product of the interior radii of pairwise nonoverlapping domains and the maximum product of the Robin radii of such domains are considered. New invariants with respect to the M?bius transformations of the Riemann sphere are introduced. In particular, for these invariants problems on extremal decomposition with free poles on the unit circle are investigated. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 54–74.