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Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

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arxiv 1907.07004 v3 pith:UUGWPMQI submitted 2019-07-16 math.PR math.APmath.OC

classification math.PRmath.APmath.OC
keywords interactionlocalpotentialsdiracenergyinftymassesmetric
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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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  1. Energy on spheres and discreteness of minimizing measures

    math.CA 2019-08 conditional novelty 7.0 of 10

    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.

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