带周长惩罚的吸引-排斥能量的球的稳定性与极小性
Stability and minimality of the ball for attractive-repulsive energies with perimeter penalization
AI总结：
本文研究带周长惩罚的吸引-排斥能量中球的稳定性与极小性，证明参数空间中存在分离稳定与极小区域的曲线，并发现小扰动下球稳定区域至少有两个不连通部分。
摘要：
我们考虑一类吸引-排斥能量的周长扰动，该能量由两个具有幂律核的非局部相互作用之和给出，定义在具有固定测度的集合上。我们证明在扰动-体积参数空间中存在曲线，这些曲线将球的稳定性/不稳定性和全局极小性/非极小性区域分离开来，并对某些相互作用核给出了这些曲线的精确描述。特别地，我们证明在小扰动机制下，质量参数至少存在两个不连通区域，使得球在其中是稳定的，且这两个区域被一个不稳定区域所分隔。
英文摘要：
We consider perimeter perturbations of a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. We prove that there exists curves in the perturbation-volume parameters space that separate stability/instability and global minimality/non-minimality regions of the ball, and provide a precise description of these curves for certain interaction kernels. In particular, we show that in small perturbation regimes there are (at least) two disconnected regions for the mass parameter in which the ball is stable, separated by an instability region.
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