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 $N\to\infty$ of $N$-point configurations minimizing the energy arising from a Riesz $s$-potential $1/r^s$ and an external field in the hypersingular case $s\geq d$. Formulas for the weak$^*$ 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.