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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets


Authors: S. V. Borodachov, D. P. Hardin and E. B. Saff
Journal: Trans. Amer. Math. Soc. 360 (2008), 1559-1580
MSC (2000): Primary 11K41, 70F10, 28A78; Secondary 78A30, 52A40
Published electronically: October 17, 2007
MathSciNet review: 2357705
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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\vert x-y\right\vert^{-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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Additional Information

S. V. Borodachov
Affiliation: Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240
Address at time of publication: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: sergiy.v.borodachov@vanderbilt.edu, borodasv@math.gatech.edu

D. P. Hardin
Affiliation: Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240
Email: doug.hardin@vanderbilt.edu

E. B. Saff
Affiliation: Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240
Email: edward.b.saff@vanderbilt.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04416-9
PII: S 0002-9947(07)04416-9
Keywords: Minimal discrete Riesz energy, best-packing, Hausdorff measure, rectifiable sets, non-uniform distribution of points, power law potential, separation radius
Received by editor(s): February 10, 2006
Published electronically: October 17, 2007
Additional Notes: The research of the first author was conducted as a graduate student under the supervision of E.B. Saff and D. P. Hardin at Vanderbilt University.
The research of the second author was supported, in part, by the U. S. National Science Foundation under grants DMS-0505756 and DMS-0532154.
The research of the third author was supported, in part, by the U. S. National Science Foundation under grant DMS-0532154.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.