Extremal configurations of certain problems on the capacity and harmonic measure

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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 γ = {z = z(t) 0 ≤ t ≤ 1}, 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. Bibliography: 21 titles.

Original languageEnglish
Pages (from-to)1031-1049
Number of pages19
JournalJournal of Mathematical Sciences
Issue number1
StatePublished - Jan 1 1998


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