论文标题
球的稳定性用于有吸引力的抑制能量
Stability of the ball for attractive-repulsive energies
论文作者
论文摘要
我们考虑了一类吸引人的抑制能量，由两个非局部相互作用与幂律内核的总和给出，并定义了以固定度量的设置。 R. Frank和E. Lieb最近证明了球是足够大的质量的独特（翻译）全球最小化器。我们关注球的稳定性问题，这是在能量在球边界的平滑扰动方面的第二个变化的积极性的意义上。我们表征了第二个变化为正定义（大质量）或负定质量（小质量）的质量范围。此外，我们证明了球的稳定性意味着在Hausdorff距离足够接近的集合之间的局部最小值，但不在$ l^1 $ - sense中。
We consider a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. It has recently been proved by R. Frank and E. Lieb that the ball is the unique (up to translation) global minimizer for sufficiently large mass. We focus on the issue of the stability of the ball, in the sense of the positivity of the second variation of the energy with respect to smooth perturbations of the boundary of the ball. We characterize the range of masses for which the second variation is positive definite (large masses) or negative definite (small masses). Moreover, we prove that the stability of the ball implies its local minimality among sets sufficiently close in the Hausdorff distance, but not in $L^1$-sense.
