论文标题
谐波图的远场扩展和列悬浮液中的静电类比
Far-Field Expansions for Harmonic Maps and the Electrostatics Analogy in Nematic Suspensions
论文作者
论文摘要
对于平滑界的域$ g \ subset \ mathbb {r}^3 $，我们考虑映射$ n \ colon \ colon \ mathbb r^3 \ setMinus g \ to \ setbb s^2 $最小化能量$ e（n）= \ int = \ int _ { +f_s（n _ {\ lfloor \ partial g}）$中的$ \ mathbb s^2 $ - 值地图，以至于$ n（x）\ lot n_0 $ as $ | x | x | \ to \ infty $。这是浸入列液晶的粒子$ g $的模型。表面能$ f_s $描述了粒子的锚定特性，并且可以非常笼统。我们证明，这种最小化地图$ n $的渐近扩展为$ 1/r $。此外，我们表明，通过将其与$ n_0 $相对于最小能量的梯度，几乎所有$ n_0 \ in \ mathbb s^2 $中的远场条件$ n_0 $ n_0 $唯一确定了领先的订单$ 1/r $项。我们在出于身体动机的情况下得出了这种关系的各种后果：当粒子$ g $的方向相对于规定的远场对齐$ n_0 $稳定；当粒子$ g $具有一些旋转对称时。特别是，这些推论证明了在物理文献中可以找到的一些近似值，可以通过所谓的静电类比来描述列的悬浮液。
For a smooth bounded domain $G\subset\mathbb{R}^3$ we consider maps $n\colon\mathbb R^3\setminus G\to\mathbb S^2$ minimizing the energy $E(n)=\int_{\mathbb R^3\setminus G}|\nabla n|^2 +F_s(n_{\lfloor\partial G})$ among $\mathbb S^2$-valued map such that $n(x)\approx n_0$ as $|x|\to\infty$. This is a model for a particle $G$ immersed in nematic liquid crystal. The surface energy $F_s$ describes the anchoring properties of the particle, and can be quite general. We prove that such minimizing map $n$ has an asymptotic expansion in powers of $1/r$. Further, we show that the leading order $1/r$ term is uniquely determined by the far-field condition $n_0$ for almost all $n_0\in\mathbb S^2$, by relating it to the gradient of the minimal energy with respect to $n_0$. We derive various consequences of this relation in physically motivated situations: when the orientation of the particle $G$ is stable relative to a prescribed far-field alignment $n_0$; and when the particle $G$ has some rotational symmetries. In particular, these corollaries justify some approximations that can be found in the physics literature to describe nematic suspensions via a so-called electrostatics analogy.
