Extremal configurations in some problems on the capacity and harmonic measure

We study certain extremal problems concerning the capacity of a condenser and the harmonic measure of a compact set. In particular, we answer in the negative Tamrazov's question on the minimum of the capacity of a condenser. We find the solution to Dubinin's problem on the maximum of the harmonic measure of a boundary set in the family of domains containing no “long” segments of given inclination. It is also shown that the segment $[1-L,1]$ has the maximal harmonic measure at the point $z=0$ among all curves $\gamma=\{z=z(t),\ 0\le t\le1\}$, $z(0)=1$, that lie in the unit disk and have given length $L$, $0<L<1$. The proofs are based on Baernstein's method of $*$-functions, Dubinin's dissymmetrization method, and the method of extremal metrics. Bibl. 21 titles.