二嵌段共聚物问题的小体积分数极限：I。Sharp接口功能
Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp Interface Functional
摘要：
我们提出的第一个两篇文章的小体积分数限制的非本地Cahn-Hilliard功能引入模型的微相分离的二嵌段共聚物。在这里，我们把注意力集中在尖锐的界面版本的功能，并考虑一个限制，其中体积分数趋于零，但少数相（称为粒子）的数量仍然是O（1）。使用伽玛收敛的语言，我们专注于这种收敛的两个层次，并推导出一阶和二阶有效能量，其能量景观更简单，更透明。这些极限能量仅在δ函数的加权和上是有限的，对应于质量集中到“点粒子”中。在最高水平上，有效能量完全是局部的，包含关于每个粒子结构的信息，但没有关于它们空间分布的信息。在下一个层次上，我们遇到了粒子之间的库仑式相互作用，这是图案形成的原因。我们在这里提出的结果在三维和二维。
英文摘要：
We present the first of two articles on the small volume fraction limit of a nonlocal Cahn-Hilliard functional introduced to model microphase separation of diblock copolymers. Here we focus attention on the sharp-interface version of the functional and consider a limit in which the volume fraction tends to zero but the number of minority phases (called particles) remains O(1). Using the language of Gamma-convergence, we focus on two levels of this convergence, and derive first and second order effective energies, whose energy landscapes are simpler and more transparent. These limiting energies are only finite on weighted sums of delta functions, corresponding to the concentration of mass into `point particles'. At the highest level, the effective energy is entirely local and contains information about the structure of each particle but no information about their spatial distribution. At the next level we encounter a Coulomb-like interaction between the particles, which is responsible for the pattern formation. We present the results here in both three and two dimensions.
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