Energy on spheres and discreteness of minimizing measures

2019/08/27 by Bilyk, Dmitriy, Glazyrin, Alexey, Matzke, Ryan +2 · 1 citation
#31E05 #52A40 #58C35 #90C26 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1908.10354

Abstract

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.

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• Real and complex spherical designs and their Gramian

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