GREEN ENERGY AND EXTREMAL DECOMPOSITIONS
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 P´olya-Schur inequa...
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| Format: | Article |
| Language: | English |
| Published: |
Petrozavodsk State University
2019-10-01
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| Series: | Проблемы анализа |
| Subjects: | |
| Online Access: | http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=6730&lang=en |
| Summary: | 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 P´olya-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.
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| ISSN: | 2306-3424 2306-3432 |