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Abstract
We consider the minimal discrete and continuous energy problems on the unit sphere Sd in the Euclidean space ℝ^{d+1} in the presence of an external field due to finitely many localized charge distributions on S^{d}, where the energy arises from the Riesz potential 1/r^{s} (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 language  English 

Title of host publication  Contemporary Computational Mathematics  A Celebration of the 80th Birthday of Ian Sloan 
Publisher  Springer International Publishing AG 
Pages  179203 
Number of pages  25 
ISBN (Electronic)  9783319724560 
ISBN (Print)  9783319724553 
DOIs  
Publication status  Published  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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Dive into the research topics of 'Logarithmic and riesz equilibrium for multiple sources on the sphere: The exceptional case'. Together they form a unique fingerprint.Projects
 1 Finished

FWF  Self WW  Self organization by local interaction: minimal energy, external fields, and numerical integration
1/10/16 → 28/02/19
Project: Research project