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This article is cited in 13 scientific papers (total in 13 papers)
Harmonic measure of radial line segments and symmetrization
A. Yu. Solynin St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract:
Let $l_k=\{z:\operatorname {arg}z=\alpha _k,\ r_1\leqslant |z|\leqslant r_2\}$ for
$k=1,\dots,n$, $0<r_1<r_2\leqslant 1$, 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.
Received: 18.11.1997
Citation:
A. Yu. Solynin, “Harmonic measure of radial line segments and symmetrization”, Mat. Sb., 189:11 (1998), 121–138; Sb. Math., 189:11 (1998), 1701–1718
Linking options:
https://www.mathnet.ru/eng/sm361https://doi.org/10.1070/sm1998v189n11ABEH000361 https://www.mathnet.ru/eng/sm/v189/i11/p121
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Abstract page: | 534 | Russian version PDF: | 230 | English version PDF: | 15 | References: | 86 | First page: | 1 |
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