Energy on spheres and discreteness of minimizing measures

In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the p-fra...

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Bibliographic Details
Published in:Journal of functional analysis 2021-06, Vol.280 (11), p.108995, Article 108995
Main Authors: Bilyk, Dmitriy, Glazyrin, Alexey, Matzke, Ryan, Park, Josiah, Vlasiuk, Oleksandr
Format: Article
Language:English
Subjects:
Potential energy minimization
Spherical codes
Spherical designs
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Summary:In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the p-frame energy has empty interior whenever p is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2021.108995