This note concerns the problem of minimizing a certain family of non-local energy functionals over measures on ${\mathbb{R}}^n$, subject to a mass constraint, in a strong attraction limit. In these problems, the total energy is an integral over pair interactions of attractive-repulsive type. The interaction kernel is a sum of competing power law potentials with attractive powers $\alpha \in (0,\infty )$ and repulsive powers associated with Riesz potentials. The strong attraction limit $\alpha \to \infty$ is addressed via Gamma-convergence, and minimizers of the limit are characterized in terms of an isodiametric capacity problem. We also provide evidence for symmetry-breaking of minimizers in high dimensions.

本说明涉及的问题是，通过以下措施将某些非本地能源功能族最小化： ${\mathbb{[R}}^{ñ}$受到质量约束，并且具有很强的吸引力限制。在这些问题中，总能量是吸引-排斥类型的成对相互作用的积分。交互核是具有吸引力的竞争幂律势的总和$\alpha \in （0，\infty ）$和与里兹电势相关的排斥力。强大的吸引力极限$\alpha \to \infty$通过伽马收敛解决问题，并且通过等径容量问题来表征极限的最小值。我们还为高尺寸的最小化器的对称破坏提供了证据。