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

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in Buch/BerichtForschungBegutachtung

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.

Originalspracheenglisch
TitelContemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan
Herausgeber (Verlag)Springer International Publishing AG
Seiten179-203
Seitenumfang25
ISBN (elektronisch)9783319724560
ISBN (Print)9783319724553
DOIs
PublikationsstatusVeröffentlicht - 23 Mai 2018

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Logarithmic
Charge
External Field
Logarithmic Potential
Riesz Potential
Equilibrium Measure
Minimal Energy
Point Source
Unit Sphere
Euclidean Distance
Energy
Euclidean space
Union
Complement
Radius
Numerical Examples
Cap

ASJC Scopus subject areas

  • !!Mathematics(all)

Fields of Expertise

  • Information, Communication & Computing

Treatment code (Nähere Zuordnung)

  • Basic - Fundamental (Grundlagenforschung)

Dies zitieren

Brauchart, J. S., Dragnev, P. D., Saff, E. B., & Womersley, R. S. (2018). Logarithmic and riesz equilibrium for multiple sources on the sphere: The exceptional case. in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan (S. 179-203). Springer International Publishing AG . https://doi.org/10.1007/978-3-319-72456-0_10

Logarithmic and riesz equilibrium for multiple sources on the sphere : The exceptional case. / Brauchart, Johann S.; Dragnev, Peter D.; Saff, Edward B.; Womersley, Robert S.

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer International Publishing AG , 2018. S. 179-203.

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in Buch/BerichtForschungBegutachtung

Brauchart, JS, Dragnev, PD, Saff, EB & Womersley, RS 2018, Logarithmic and riesz equilibrium for multiple sources on the sphere: The exceptional case. in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer International Publishing AG , S. 179-203. https://doi.org/10.1007/978-3-319-72456-0_10
Brauchart JS, Dragnev PD, Saff EB, Womersley RS. Logarithmic and riesz equilibrium for multiple sources on the sphere: The exceptional case. in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer International Publishing AG . 2018. S. 179-203 https://doi.org/10.1007/978-3-319-72456-0_10
Brauchart, Johann S. ; Dragnev, Peter D. ; Saff, Edward B. ; Womersley, Robert S. / Logarithmic and riesz equilibrium for multiple sources on the sphere : The exceptional case. Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer International Publishing AG , 2018. S. 179-203
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