Riesz External Field Problems on the Hypersphere and Optimal Point Separation |
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Authors: | Johann S Brauchart Peter D Dragnev Edward B Saff |
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Institution: | 1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia 2. Department of Mathematical Sciences, Indiana University-Purdue University Fort Wayne, Fort Wayne, IN, 46805, USA 3. Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA
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Abstract: | We consider the minimal energy problem on the unit sphere ?? d in the Euclidean space ? d+1 in the presence of an external field Q, where the energy arises from the Riesz potential 1/r s (where r is the Euclidean distance and s is the Riesz parameter) or the logarithmic potential log(1/r). Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range d ? 2 ≤ s < d ? 1. The proof uses a maximum principle for measures supported on ?? d . When Q is the Riesz s-potential of a signed measure and d ? 2 ≤ s < d, our results lead to explicit point-separation estimates for (Q,s)-Fekete points, which are n-point configurations minimizing the Riesz s-energy on ?? d with external field Q. In the hyper-singular case s > d, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem. |
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