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.
Bilyk, Dmitriy, Alexey Glazyrin, Ryan Matzke, Josiah Park, and Oleksandr Vlasiuk. 2021. “Energy on Spheres and Discreteness of Minimizing Measures.” Journal of Functional Analysis 280 (11): 108995. https://doi.org/10.1016/j.jfa.2021.108995.
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Journal of Functional Analysis