Green energy and extremal decompositions

Abstract

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.

Optimal Green energy points on the circles in d-space

“…The proofs of Theorems 1,2 hinge on the theory of condenser capacity and dissymmetrization [7], [8]. These proofs are related conceptually with the solutions of the so-called extremal decomposition problems [9], [10], [14], [15]. In the recent paper [9], analogues of Theorems 1,2 for the case of the plane and one circle and a concentric ring have been presented.…”

“…These proofs are related conceptually with the solutions of the so-called extremal decomposition problems [9], [10], [14], [15]. In the recent paper [9], analogues of Theorems 1,2 for the case of the plane and one circle and a concentric ring have been presented. The proof of Theorem 1 of this paper follows the same line of argument as the one presented in [9] with modifications related to the use of dissymmetrization [7] and the asymptotic formula for the capacity of the spacial rather than plane condenser [10].…”

“…In the recent paper [9], analogues of Theorems 1,2 for the case of the plane and one circle and a concentric ring have been presented. The proof of Theorem 1 of this paper follows the same line of argument as the one presented in [9] with modifications related to the use of dissymmetrization [7] and the asymptotic formula for the capacity of the spacial rather than plane condenser [10]. The proof of an analogue of Theorem 2 for the plane case [9] is based on the radial averaging transformation and conformal mapping.…”

“…The proof of Theorem 1 of this paper follows the same line of argument as the one presented in [9] with modifications related to the use of dissymmetrization [7] and the asymptotic formula for the capacity of the spacial rather than plane condenser [10]. The proof of an analogue of Theorem 2 for the plane case [9] is based on the radial averaging transformation and conformal mapping. This method is not applicable in the Euclidean space due to absence of the suitable conformal mappings.…”

Об Оптимальной Дискретной Энергии Неймана В Шаре И Круговом Кольце

“…Существует много исследований, связанных с экстремальными задачами для различных видов энергий дискретного заряд (см., например, работы [3], [4], [5] и ссылки в них). В [6] получены две оценки дискретной энергии функции Грина кругового кольца на плоскости в случае точек, расположенных на некоторой окружности. Эти результаты были распространены в евклидово пространство в [7].…”