Harmonic measure of radial line segments and symmetrization

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Abstract

Let lk = {z : arg z = αk, r1 ≤ |z\ ≤ r2} for k = 1, . . ., n, 0 < r1 < r2 ≤ 1, and αk ∈ ℝ; let E = Unk=1 lk; let E* = {z : arg zn = 0, r1 ≤ |z| ≤ r2}; and let ωE(z) be the harmonic measure of E with respect to the domain {z : \z\ <1} \ E. The inequality ωE(0) ≤ ωE* (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.

Original languageEnglish
Pages (from-to)1701-1718
Number of pages18
JournalSbornik Mathematics
Volume189
Issue number11-12
DOIs
StatePublished - 1998

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