Logarithmic and riesz equilibrium for multiple sources on the sphere: The exceptional case

Johann S. Brauchart, Peter D. Dragnev, Edward B. Saff*, Robert S. Womersley

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

We consider the minimal discrete and continuous energy problems on the unit sphere Sd in the Euclidean space ℝd+1 in the presence of an external field due to finitely many localized charge distributions on Sd, 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.

Original languageEnglish
Title of host publicationContemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan
PublisherSpringer International Publishing AG
Pages179-203
Number of pages25
ISBN (Electronic)9783319724560
ISBN (Print)9783319724553
DOIs
Publication statusPublished - 23 May 2018

ASJC Scopus subject areas

  • Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing

Treatment code (Nähere Zuordnung)

  • Basic - Fundamental (Grundlagenforschung)

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