Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere: The Exceptional Case

  • Johann S. Brauchart
  • Peter D. Dragnev
  • Edward B. SaffEmail author
  • Robert S. Womersley


We consider the minimal discrete and continuous energy problems on the unit sphere \(\mathbb {S}^d\) in the Euclidean space \(\mathbb {R}^{d+1}\) in the presence of an external field due to finitely many localized charge distributions on \(\mathbb {S}^d\), where the energy arises from the Riesz potential 1∕rs (r is the Euclidean distance) for the critical Riesz parameter s = d − 2 if d ≥ 3 and the logarithmic potential \(\log (1/r)\) if d = 2. Individually, a localized charge distribution is either a point charge or assumed to be rotationally symmetric. The extremal measure solving the continuous external field problem for weak fields is shown to be the uniform measure on the sphere but restricted to the exterior of spherical caps surrounding the localized charge distributions. The radii are determined by the relative strengths of the generating charges. Furthermore, we show that the minimal energy points solving the related discrete external field problem are confined to this support. For d − 2 ≤ s < d, we show that for point sources on the sphere, the equilibrium measure has support in the complement of the union of specified spherical caps about the sources. Numerical examples are provided to illustrate our results.



The research of Johann S. Brauchart was supported, in part, by the Austrian Science Fund FWF project F5510 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”) and was also supported by the Meitner-Programme M2030 “Self organization by local interaction” funded by the Austrian Science Fund FWF. The research of Peter D. Dragnev was supported by the Simon’s Foundation grant no. 282207. The research of Edward B. Saff was supported by U.S. National Science Foundation grant DMS-1516400. The research of Robert S. Womersley was supported by IPFW Scholar-in-Residence program. All the authors acknowledge the support of the Erwin Schrödinger Institute in Vienna, where part of the work was carried out. This research includes computations using the Linux computational cluster Katana supported by the Faculty of Science, UNSW Sydney.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Johann S. Brauchart
    • 1
  • Peter D. Dragnev
    • 2
  • Edward B. Saff
    • 3
    Email author
  • Robert S. Womersley
    • 4
  1. 1.Institute of Analysis and Number TheoryGraz University of TechnologyGrazAustria
  2. 2.Department of Mathematical SciencesIndiana University - Purdue UniversityFort WayneUSA
  3. 3.Center for Constructive Approximation, Department of MathematicsVanderbilt UniversityNashvilleUSA
  4. 4.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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