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arXiv · 1908.10354

Energy on spheres and discreteness of minimizing measures

Dmitriy Bilyk·Alexey Glazyrin·Ryan Matzke·Josiah Park·Oleksandr Vlasiuk
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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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Dmitriy Bilyk, Alexey Glazyrin, Ryan Matzke, Josiah Park, Oleksandr Vlasiuk. 2019-08-27. Energy on spheres and discreteness of minimizing measures. https://arxiv.org/abs/1908.10354

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