Abstract
We consider the minimal energy problem on the unit sphere \({\mathbb {S}}^2\) in the Euclidean space \({\mathbb {R}}^3\) immersed in an external field Q, where the charges are assumed to interact via Newtonian potential 1/r, r being the Euclidean distance. The problem is solved by finding the support of the extremal measure, and obtaining an explicit expression for the equilibrium density. We then apply our results to an external field generated by a point charge, and to a quadratic external field.
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Acknowledgments
The author would like to thank his doctoral advisor Prof. Igor E. Pritsker for suggesting the problem and stimulating discussions. The author expresses gratitude to Prof. Edward B. Saff for his kind permission to use a reference from the forthcoming book [1], and the referee for helpful comments and suggestions.
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Bilogliadov, M. Weighted energy problem on the unit sphere. Anal.Math.Phys. 6, 403–424 (2016). https://doi.org/10.1007/s13324-016-0125-9
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DOI: https://doi.org/10.1007/s13324-016-0125-9