### 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.

Originalsprache | englisch |
---|---|

Titel | Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan |

Herausgeber (Verlag) | Springer International Publishing AG |

Seiten | 179-203 |

Seitenumfang | 25 |

ISBN (elektronisch) | 9783319724560 |

ISBN (Print) | 9783319724553 |

DOIs | |

Publikationsstatus | Veröffentlicht - 23 Mai 2018 |

### Fingerprint

### ASJC Scopus subject areas

- !!Mathematics(all)

### Fields of Expertise

- Information, Communication & Computing

### Treatment code (Nähere Zuordnung)

- Basic - Fundamental (Grundlagenforschung)

### Dies zitieren

*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.

Publikation: Beitrag in Buch/Bericht/Konferenzband › Beitrag in Buch/Bericht › Forschung › Begutachtung

*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

}

TY - CHAP

T1 - Logarithmic and riesz equilibrium for multiple sources on the sphere

T2 - The exceptional case

AU - Brauchart, Johann S.

AU - Dragnev, Peter D.

AU - Saff, Edward B.

AU - Womersley, Robert S.

PY - 2018/5/23

Y1 - 2018/5/23

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85053954928&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-72456-0_10

DO - 10.1007/978-3-319-72456-0_10

M3 - Chapter

SN - 9783319724553

SP - 179

EP - 203

BT - Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

PB - Springer International Publishing AG

ER -