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arXiv · 0708.0568

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

J. S. Brauchart·D. P. Hardin·E. B. Saff
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Abstract

Let A be a compact set in the right-half plane and $Γ(A)$ 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 $Γ(A)$ 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 $Γ(A)$ which is supported on the outer boundary of $Γ(A)$. We show that there are sets of revolution $Γ(A)$ such that the support of the equilibrium measure on $Γ(A)$ 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 $Γ(R+A)$ as R goes to infinity, is the full outer boundary for certain sets A, in contrast to the logarithmic case (s=0).

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J. S. Brauchart, D. P. Hardin, E. B. Saff. 2007-08-03. The support of the limit distribution of optimal Riesz energy points on sets of revolution in $\mathbb{R}^{3}$. https://doi.org/10.1063/1.2817823

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