Minimal Riesz energy on the sphere for axis-supported external fields

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Date
2009MFO Scientific Program
OWLF 2008Series
Oberwolfach Preprints;2009;04Author
Brauchart, Johann S.
Dragnev, Peter D.
Saff, Edward B.
Metadata
Show full item recordOWP-2009-04
Abstract
Abstract. We investigate the minimal Riesz $s$-energy problem for positive
measures on the d-dimensional unit sphere $\mathbb{S}^d$
in the presence of an external
field induced by a point charge, and more generally by a line charge. The model
interaction is that of Riesz potentials $|x−y|^{−s}$ with $d−2 ≤ s < d$. For a given
axis-supported external field, the support and the density of the corresponding
extremal measure on $\mathbb{S}^d$
is determined. The special case $s = d − 2$ yields
interesting phenomena, which we investigate in detail. A weak$^∗$ asymptotic
analysis is provided as $s → (d − 2)^+$.