Green energy and extremal decompositions
Vladimir Nikolaevich Dubinin · Problemy analiza · 2019
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.