Riesz external field problems on the hypersphere and optimal point separation

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 $d-2\le s<d-1.$ 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 $d-2\le s<d$, our results lead to explicit point-separation estimates for $(Q,s)$-Fekete points, which are $n$-point configurations minimizing the Riesz $s$-energy on ${\mathbb{S}}^d$ with external field $Q$. In the hyper-singular case $s>d$, 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.

35 pages, 4 figures

References in corpus (2)

• Minimal Riesz energy on the sphere for axis-supported external fields
• A Fascinating Polynomial Sequence arising from an Electrostatics Problem on the Sphere

Cited by in corpus (1)

• Coulomb and Riesz gases: The known and the unknown