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 $ Ntoinfty $ of $N$-point configurations minimizing the energy arising from a Riesz $ s $-potential $ 1/r^s $ and an external field in the hypersingular case $ sgeq 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.