论文标题
在3D中ohta-kawasaki Energy的最佳配置的分区数量的统一数量
Uniform bound on the number of partitions for optimal configurations of the Ohta-Kawasaki energy in 3D
论文作者
论文摘要
我们研究了一个3D三元系统，该系统作为三嵌段共聚物中nakazawa-Ohta密度功能理论的尖锐地面极限，该系统将界面能量与远距离相互作用项相结合。尽管对2D和3D中的二进制案例以及2D中的三元案例都进行了很好的研究，但对于3D中的三元案例知之甚少。特别是，甚至还不清楚最小化器是否由有限的许多组件制成。在本文中，我们通过证明最小化器中的组件数量仅取决于总质量和相互作用系数，从而从上面的数量界定了一个积极的答案。一个关键的困难是，3D结构使我们无法从外围术语中解脱出类似库仑的远距离相互作用，因此最小化的实际形状是未知的，甚至不是小群体。这是由于缺乏定量的等值不平等，在3D中有两个质量限制，这使得竞争对手的构建明显更加精致。
We study a 3D ternary system derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers, which combines an interface energy with a long range interaction term. Although both the binary case in 2D and 3D, and the ternary case in 2D, are quite well studied, very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by some quantity depending only on the total masses and the interaction coefficients. One key difficulty is that the 3D structure prevents us from uncoupling the Coulomb-like long range interaction from the perimeter term, hence the actual shape of minimizers is unknown, not even for small masses. This is due to the lack of a quantitative isoperimetric inequality with two mass constraints in 3D, and it makes the construction of competitors significantly more delicate.
