Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets

الأصل · EN

Given a compact d-rectifiable set A embedded in Euclidean space and a distribution ρ(x) with respect to d-dimensional Hausdorff measure on A, we address the following question: how can one generate optimal configurations of N points on A that are "well-separated" and have asymptotic distribution ρ(x) as N→ ∞? For this purpose we investigate minimal weighted Riesz energy points, that is, points interacting via the weighted power law potential V=w(x,y)|x-y|⁻ˢ, where s>0 is a fixed parameter and w is suitably chosen. In the unweighted case (w≡ 1) such points for N fixed tend to the solution of the best-packing problem on A as the parameter s→ ∞.