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.

中文翻译：

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具有轻度排斥势的相互作用能的局部极小值的唯一性和特征

在本文中，我们关注一维受幂律类型的排斥力-吸引力的相互作用能的局部最小化器。我们证明两个Dirac质量之和是\（\ lambda \）- Wasserstein度量拓扑与\（1 \ le \ lambda <\ infty \）下唯一的局部极小值，前提是Dirac delta的质量和距离相等且为一个，分别。此外，在\（\ infty \）- Wasserstein度量的情况下，我们根据相互作用势的幂来表征稳态解的稳定性。