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

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 $s → (d − 2)^+$.