For non-even p, every minimizer of the p-frame energy on the sphere has support with empty interior, and for potentials with finitely many positive Gegenbauer coefficients a discrete minimizer always exists.
Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials
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abstract
In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive-attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the $\lambda-$Wasserstein metric topology with $1\le \lambda<\infty$, provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of $\infty$-Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.
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math.CA 1years
2019 1verdicts
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Energy on spheres and discreteness of minimizing measures
For non-even p, every minimizer of the p-frame energy on the sphere has support with empty interior, and for potentials with finitely many positive Gegenbauer coefficients a discrete minimizer always exists.