屏蔽型二嵌段共聚物熔体模型中的德洛内型界面
Delaunay-type interface in a screened model of diblock copolymer melts
- Brandenburgische Technische Universität Cottbus–Senftenberg（勃兰登堡工业大学科特布斯-森夫滕贝格分校）
- African Institute for Mathematical Sciences in Senegal（塞内加尔非洲数学科学研究院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究针对屏蔽型二嵌段共聚物熔体的界面平衡问题，通过分析平圆柱附近的线性化并运用Crandall-Rabinowitz分歧定理，证明了无穷多德洛内型光滑周期无界域平衡界面的存在性。
AI中文摘要：
二嵌段共聚物是一种软物质，由两段化学性质不同的重复单体嵌段在端-端连接处共价键合形成单条聚合物链。本文在$\mathbb{R}^3$中证明了无穷多德洛内型光滑周期无界域图案的存在性，这类图案可优化二嵌段共聚物熔体中的能量分布。我们强调，平衡态下的图案域对应屏蔽型太田-川崎自由能泛函的平稳集，该泛函为\begin{align*} \mathcal{P}_γ(Ω) := |\partialΩ| + γ\int_Ω\int_Ω G_κ(|x-y|) \\,\mathrm{d}x\mathrm{d}y, \end{align*} 其中$γ>0$、$κ>0$与$G_κ(r)=\frac{1}{r} e^{-κr}$为排斥性汤川势。等价地，这些平衡态满足对应的欧拉-拉格朗日方程\begin{align*} \mathcal{H}_Ω(x):= H_{\partialΩ}(x) + γ\int_Ω G_κ(|x-y|) \mathrm{d}y = \textrm{Const} \quad \text{在 } \partialΩ \text{ 上}, \end{align*} 式中$H_{\partialΩ}$表示曲面$\partialΩ$的平均曲率。通过分析$Ω\mapsto \mathcal{H}_Ω$在平圆柱附近的线性化，并运用克兰德尔-拉比诺维茨分歧定理，对任意$κ> 0$及充分小的$γ> 0$，我们证明了非平凡的、$2π$周期的德洛内型平衡圆柱界面的存在性，其形状接近常平均曲率的德洛内波状体曲面。
英文摘要：
A diblock copolymer is a soft-matter composed of two chemically distinct block of repeating monomers covalently bonded together at an end-to-end junction to form a single polymer chain. In this paper, we establish the existence of infinitely many smooth periodic unbounded domain patterns of Delaunay-type in $\mathbb{R}^3$ that optimize the energy distribution in diblock copolymer melts. We emphasize that pattern domains at the equilibrium correspond to stationary sets of the screened Ohta--Kawasaki free energy functional \begin{align*} \mathcal{P}_γ(Ω) := |\partialΩ| + γ\int_Ω\int_Ω G_κ(|x-y|) \,\mathrm{d}x\mathrm{d}y, \end{align*} where $γ>0$, $κ>0$ and $G_κ(r)=\frac{1}{r} e^{-κr}$ is the repulisive Yukawa potential. Equivalently, these equilibria satisfy the corresponding Euler--Lagrange equation \begin{align*} \mathcal{H}_Ω(x):= H_{\partialΩ}(x) + γ\int_Ω G_κ(|x-y|) \mathrm{d}y = \textrm{Const} \quad \text{on } \partialΩ, \end{align*} where $H_{\partialΩ}$ denotes the mean curvature of the surface $\partialΩ$. By analyzing the linearization of $Ω\mapsto \mathcal{H}_Ω$ around flat cylinders and applying the Crandall--Rabinowitz bifurcation theorem, for any $κ> 0$ and sufficiently small $γ> 0$, we prove the existence of non-trivial, $2π$-periodic Delaunay-type equilibrium cylinder interfaces with shapes close to a Delaunay unduloid surface of constant mean curvature.
