TY - JOUR

T1 - Extremal configurations of certain problems on the capacity and harmonic measure

AU - Solynin, A. Yu

N1 - Funding Information:
The lemma is proved. \[\] This work was supported by the Russian Foundation for Basic Research, Grant No. 94-01-00911.

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=54749143300&partnerID=8YFLogxK

U2 - 10.1007/BF02358540

DO - 10.1007/BF02358540

M3 - Article

AN - SCOPUS:54749143300

VL - 89

SP - 1031

EP - 1049

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 1

ER -