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arXiv · 1310.2765

Riesz external field problems on the hypersphere and optimal point separation

Johann S. Brauchart·Peter D. Dragnev·Edward B. Saff
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Abstract

We consider the minimal energy problem on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field $Q$, where the energy arises from the Riesz potential $1/r^s$ (where $r$ is the Euclidean distance and $s$ is the Riesz parameter) or the logarithmic potential $\log(1/r)$. Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range $d-2 \leq s < d - 1.$ The proof uses a maximum principle for measures supported on $\mathbb{S}^d$. When $Q$ is the Riesz $s$-potential of a signed measure and $d-2 \leq s d$, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.

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Johann S. Brauchart, Peter D. Dragnev, Edward B. Saff. 2013-10-10. Riesz external field problems on the hypersphere and optimal point separation. https://doi.org/10.1007/s11118-014-9387-8

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