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Equilibria of Riesz Potentials Generated by Point Charges at the Roots of Unity

Equilibria of Riesz Potentials Generated by Point Charges at the Roots of Unity We consider a special case of Maxwell’s problem on the number of equilibrium points of the Riesz potential $$1/r^{2\beta }$$ 1 / r 2 β (where $$r$$ r is the Euclidean distance and $$\beta $$ β is the Riesz parameter) for positive unit point charges placed at the vertices of a regular polygon. We show that the equilibrium points are located on the perpendicular bisectors to the sides of the regular polygon, and study the asymptotic behavior of the equilibrium points with regard to the number of charges $$n$$ n and the Riesz parameter $$\beta $$ β . Finally, we prove that for values of $$\beta $$ β in a small left-hand side neighborhood of $$\beta =1$$ β = 1 , the Riesz potential has only one equilibrium point different from the origin on each perpendicular bisector, and one equilibrium point at the origin. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Equilibria of Riesz Potentials Generated by Point Charges at the Roots of Unity

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References (21)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-015-0115-1
Publisher site
See Article on Publisher Site

Abstract

We consider a special case of Maxwell’s problem on the number of equilibrium points of the Riesz potential $$1/r^{2\beta }$$ 1 / r 2 β (where $$r$$ r is the Euclidean distance and $$\beta $$ β is the Riesz parameter) for positive unit point charges placed at the vertices of a regular polygon. We show that the equilibrium points are located on the perpendicular bisectors to the sides of the regular polygon, and study the asymptotic behavior of the equilibrium points with regard to the number of charges $$n$$ n and the Riesz parameter $$\beta $$ β . Finally, we prove that for values of $$\beta $$ β in a small left-hand side neighborhood of $$\beta =1$$ β = 1 , the Riesz potential has only one equilibrium point different from the origin on each perpendicular bisector, and one equilibrium point at the origin.

Journal

Computational Methods and Function Theory – Springer Journals

Published: May 5, 2015

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