Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets

Borodachov, S.; Hardin, D.; Saff, E.

doi:10.1090/S0002-9947-07-04416-9

Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets
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by S. V. Borodachov, D. P. Hardin and E. B. Saff PDF

Trans. Amer. Math. Soc. 360 (2008), 1559-1580 Request permission

Abstract:

Given a closed $d$-rectifiable set $A$ embedded in Euclidean space, we investigate minimal weighted Riesz energy points on $A$; that is, $N$ points constrained to $A$ and interacting via the weighted power law potential $V=w(x,y)\left |x-y\right |^{-s}$, where $s>0$ is a fixed parameter and $w$ is an admissible weight. (In the unweighted case ($w\equiv 1$) such points for $N$ fixed tend to the solution of the best-packing problem on $A$ as the parameter $s\to \infty$.) Our main results concern the asymptotic behavior as $N\to \infty$ of the minimal energies as well as the corresponding equilibrium configurations. Given a distribution $\rho (x)$ with respect to $d$-dimensional Hausdorff measure on $A$, our results provide a method for generating $N$-point configurations on $A$ that are “well-separated” and have asymptotic distribution $\rho (x)$ as $N\to \infty$.

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