Harmonic measure of radial line segments and symmetrization

A. Yu. Solynin

Sbornik. Mathematics, Tome 189 (1998) no. 11, pp. 1701-1718

Résumé

Let $l_k=\{z:\operatorname {arg}z=\alpha _k,\ r_1\leqslant |z|\leqslant r_2\}$ for $k=1,\dots,n$, $0$, and $\alpha _k\in \mathbb R$; let $E=\bigcup _{k=1}^nl_k$, let $E^*=\{z:\operatorname {arg}z^n=0,\ r_1\leqslant |z|\leqslant r_2\}$; and let $\omega _E(z)$ be the harmonic measure of $E$ with respect to the domain $\{z:|z|1\}\setminus E$. The inequality $\omega _E(0)\leqslant \omega _{E^*}(0)$ is established, which solves the problem of Gonchar on the harmonic measure of radial slits. The proof uses the dissymmetrization method of Dubinin and the method of the extremal metric in the form of the problem of extremal partitioning into non-overlapping domains.