Abstract
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.
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Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 170–195.
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Solynin, A.Y. Extremal configurations of certain problems on the capacity and harmonic measure. J Math Sci 89, 1031–1049 (1998). https://doi.org/10.1007/BF02358540
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DOI: https://doi.org/10.1007/BF02358540