论文标题
球的刚性对于等等的问题，具有强稳定性排斥力
Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion
论文作者
论文摘要
我们考虑一个涉及表面张力和电荷排斥之间竞争的变分问题。我们表明，与弱（短距离）相互作用的情况相反，我们在上一篇论文中证明了问题的不良性，而当排斥力更强时，周长在小尺度上占主导地位。特别是，我们证明存在最小化的小额费用及其规律性。将其与小$ c^{1，γ} $扰动下的球的稳定性相结合，最终导致球的最低限度的小电荷。我们尤其介绍了$ 1- $容量的边界案例，其中能量中的两个条款均为相同的顺序。
We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small $C^{1,γ}$ perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the $1-$capacity where both terms in the energy are of the same order.
