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ЖУРНАЛЫ // Проблемы анализа — Issues of Analysis // Архив

Пробл. анал. Issues Anal., 2019, том 8(26), выпуск 3, страницы 38–44 (Mi pa270)

Эта публикация цитируется в 5 статьях

Green energy and extremal decompositions

V. N. Dubininab

a Far Eastern Federal University, 8 Sukhanov str., Vladivostok, 690090, Russia
b Institute of Applied Mathematics, Far Eastern Branch of RAS, 7 Radio str., Vladivostok, 690041, Russia

Аннотация: We give two precise estimates for the Green energy of a discrete charge, concentrated in an even number of points on the circle, with respect to the concentric ring. The lower estimate for the Green energy is attained for the points with a nonstandard symmetry. The well-known Pólya-Schur inequality for the logarithmic energy is a special case of this estimate. The proof is based on the application of dissymmetrization and an asymptotic formula for the conformal capacity of a generalized condenser in the case when some of its plates contract to given points. The upper bound is established for a charge that takes values of opposite signs. Its proof reduces to solving a problem on the so-called extremal decomposition of a circular ring with free poles on a circle.

Ключевые слова: Green energy, discrete charge, dissymmetrization, extremal decompositions.

УДК: 517.956.224

MSC: 31A15

Поступила в редакцию: 18.07.2019
Исправленный вариант: 30.10.2019
Принята в печать: 14.10.2019

Язык публикации: английский

DOI: 10.15393/j3.art.2019.6730



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