Asymptotic Results for the minimum energy and Best-Packing Problems on Rectifiable Sets

Abstract

This work studies the behavior of the minimal discrete Riesz s-energy and best-packing distance on rectifiable sets as the cardinality N of point configurations gets large.

We extend known asymptotic results for the minimal s-energy on d-dimensional rectifiable manifolds (s>d, d is an integer) to the case of d-dimensional countably rectifiable sets whose d-dimensional Minkowski content equals the d-dimensional Hausdorff measure, and show that these results fail for countably rectifiable sets not satisfying this condition and s sufficiently large.

We also show that the asymptotic behavior of the best-packing distance on a d-dimensional countably rectifiable set A is the same as on a d-dimensional cube of the same d-dimensional Hausdorff measure if and only if this measure and d-dimensional Minkowski content of A are equal. Uniformity of the asymptotic distribution of optimal points is proved on d-rectifiable sets both for the minimum s-energy problem (s>d) and best-packing.

For smooth Jordan curves we get the next order term of the minimal Riesz s-energy for s>1. Known lower estimate of the minimal pairwise distance between minimum s-energy points on d-dimensional rectifiable manifolds (s>d) is extended to arbitrary compact sets of positive d-dimensional Hausdorff measure.

We also consider the problem of minimizing energy of particles interacting via the Riesz potential multiplied by a weight depending on positions of both points. For s>d, closed d-rectifiable sets and a bounded weight which is continuous and positive on the diagonal, we obtain asymptotic behavior of the minimal energy and limit distribution of optimal configurations. We also prove separation estimates for the minimal weighted energy problem.