The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. Droplet Arrangement via the Renormalized Energy
The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. Droplet Arrangement via the Renormalized Energy
复制标题
通过重正化能量的二维 Ohta-Kawasaki 能量的 Γ 极限。
DOI：
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
S. Serfaty
中科院分区：
文献类型：
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作者：
D. Goldman;C. Muratov;S. Serfaty
这是一系列论文中的第二篇，在这些论文中，我们推导了与二嵌段共聚物系统有关的二维非局部金兹堡-朗道能量与库仑排斥的Γ-expansion，即大田-川崎模型。在这个模型中，出现了两个相，它们通过非局部库仑型能量相互作用。在这里，我们关注的是这种能量的尖锐界面版本，其中一个相具有非常小的体积分数，从而在多数相的“海洋”中产生少数相的小“水滴”。在我们之前的论文中，我们计算了阶能量的Γ-limit，它产生了几乎最小的平均行为，即液滴的密度应该是均匀的。这里我们进入下一阶并推导出下一阶Γ-limit能量，这正是Sandier和Serfaty在磁金兹堡-朗道模型中作为涡的极限相互作用能而得到的库仑重归一化能量。该推导基于Sandier-Serfaty的抽象格式，该格式用于获得二尺度能量的下界，并通过多参数遍历定理通过模式上的一些概率来表示。因此，在不使用欧拉-拉格朗日方程的情况下，我们建立了所有具有“几乎最小能量”的构型的液滴的渐近圆度和半径，以及它们渐近收缩到其排列在某种平均意义上最小化重整化能量的点的事实。通过Γ-equivalence的推导，得到了最小能量的展开式和原始大田-川崎能量的极小值的零超水平集的表征。这导致人们期望看到三角形的液滴晶格作为能量最小化器。
This is the second in a series of papers in which we derive a Γ-expansion for the two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion known as the Ohta–Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small “droplets” of the minority phase in a “sea” of the majority phase. In our previous paper, we computed the Γ-limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order Γ-limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg–Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Thus, without appealing to the Euler–Lagrange equation, we establish for all configurations which have “almost minimal energy” the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of Γ-equivalence, the obtained results also yield an expansion of the minimal energy and a characterization of the zero super-level sets of the minimizers for the original Ohta–Kawasaki energy. This leads to the expectation of seeing triangular lattices of droplets as energy minimizers.
具有长程相互作用的等周问题的均匀能量分布
影响因子：
3.9
作者：
G. Alberti;R. Choksi;F. Otto
通讯作者：
F. Otto