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

TL;DR

This paper studies the minimal Riesz energy configurations on the sphere under external fields from point and line charges, analyzing the support, density, and asymptotic behavior of extremal measures for various s.

Contribution

It characterizes the extremal measures' support and density on the sphere under axis-supported external fields, especially for the critical case s=d-2.

Findings

Support and density of extremal measures are determined for various external fields.

Special phenomena occur at the critical case s=d-2, analyzed in detail.

Asymptotic behavior of measures as s approaches (d-2)^+ is described.

Abstract

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere 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 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 goes to (d-2)^+.