Logarithmic Equilibrium on the Sphere in the Presence of Multiple Point Charges
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Logarithmic Equilibrium on the Sphere in the Presence of Multiple Point Charges A. R. Legg1 · P. D. Dragnev1 Received: 23 December 2019 / Revised: 2 June 2020 / Accepted: 25 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract With the sphere S2 ⊂ R3 as a conductor holding a unit charge with logarithmic interactions, we consider the problem of determining the support of the equilibrium measure in the presence of an external field consisting of finitely many point charges on the surface of the sphere. We determine that for any such configuration, the complement of the equilibrium support is the stereographic preimage from the plane of a union of classical quadrature domains, whose orders sum to the number of point charges. Keywords Quadrature domain · Equilibrium measure · Schwarz function · Balayage Mathematics Subject Classification 30C40 · 30E20 · 31A05 · 74G05 · 74G65
1 Introduction to the Problem Consider the unit sphere S2 ⊂ R3 as a conductor, carrying a unit positive electric charge which is free to distribute into the Borel measure which will uniquely minimize logarithmic energy. With no other external field present, we of course intuit that the equilibrium state is uniform over the whole sphere. But what happens in the presence of an added field? The case of an external field consisting of a single point charge has been considered in [12], with the conclusion that the equilibrium support is the complement of a perfect
Communicated by Arno Kuijlaars. P. D. Dragnev: The research of this author was supported in part by a Simons Foundation CGM No. 282207.
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A. R. Legg [email protected] P. D. Dragnev [email protected]
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Department of Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne, IN 46805, USA
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Constructive Approximation
spherical cap centered at the point charge. That is to say, a single point charge tends to repel the charge on the sphere, so that a perfect cap is swept clean of charge. The radius of the cap can be explicitly calculated based on the intensity of the point charge, and the result can be extended to Riesz energies of various exponent, and even to higher dimensions (see [13] and [5]). In [6], the case of multiple point charges is undertaken and the authors demonstrate that, similar to the single-point-charge case, the equilibrium support is the complement of the union of spherical caps centered at the various point charges, with the caveat that this holds only in case the interiors of these “caps of influence” do not overlap. Numerically generated graphics are shown there that illustrate the case when two point charges’ caps of influence do overlap, and what arises is an apparently smooth lobe-shaped equilibrium support excluding both of the individual caps of influence. The open question raised there, then, is how exactly to characterize the equilibrium support when multiple charges are present, and the charges are close enough or strong enough that their individual caps of influence overlap. We do so by means of classical planar qu
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