Generating Point Configurations via Hypersingular Riesz Energy With an External Field

For a compact $d$-dimensional rectifiable subset of ${\mathbb{R}}^p$ we study asymptotic properties as $N\to \mathrm{\infty }$ of $N$-point configurations minimizing the energy arising from a Riesz $s$-potential $1\mathrm{/}{r}^s$ and an external field in the hypersingular case $s\ge d$. Formulas for the weak${}^{\ast }$ limit of normalized counting measures of such optimal point sets and the first-order asymptotic values of minimal energy are obtained. As an application, we derive a method for generating configurations whose normalized counting measures converge to a given absolutely continuous measure supported on a rectifiable subset of ${\mathbb{R}}^p$. Results on separation and covering properties of discrete minimizers are given. Our theorems are illustrated with several numerical examples.