Extremal configurations in some problems on the capacity and harmonic measure

A. Yu. Solynin

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 13, Tome 226 (1996), pp. 170-195

Résumé

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.