The support of the limit distribution of optimal Riesz energy points on sets of revolution in $\mathbb{R}^{3}$

TL;DR

This paper studies the support of equilibrium measures for minimal Riesz energy points on sets of revolution in three-dimensional space, revealing differences from classical Coulomb and logarithmic cases.

Contribution

It demonstrates that the support of the equilibrium measure can be incomplete for certain sets, and that it becomes full as the set expands, contrasting with known cases.

Findings

Support may not be the full outer boundary for certain sets.

Support becomes full as the set expands to infinity.

Differences from Coulomb and logarithmic potential cases.

Abstract

Let A be a compact set in the right-half plane and $\Gamma(A)$ the set in $\mathbb{R}^{3}$ obtained by rotating A about the vertical axis. We investigate the support of the limit distribution of minimal energy point charges on $\Gamma(A)$ that interact according to the Riesz potential 1/r^{s}, 0<s<1, where r is the Euclidean distance between points. Potential theory yields that this limit distribution coincides with the equilibrium measure on $\Gamma(A)$ which is supported on the outer boundary of $\Gamma(A)$. We show that there are sets of revolution $\Gamma(A)$ such that the support of the equilibrium measure on $\Gamma(A)$ is {\bf not} the complete outer boundary, in contrast to the Coulomb case s=1. However, the support of the limit distribution on the set of revolution $\Gamma(R+A)$ as R goes to infinity, is the full outer boundary for certain sets A, in contrast to the logarithmic case (s=0).