AI 中文总结
该研究在Landau–de Gennes Q张量框架下，首次严格处理向列相薄膜的同步均匀化与降维极限，锚定强度作为标度参数产生不同有效模型，有效弹性响应依赖于标度比ρ。
AI 中文摘要
我们在Landau–de Gennes Q张量框架下，研究了非均匀向列相液晶薄膜的均匀化（ε→0）与降维（h→0）的同步极限问题。弹性能量密度由张量A(x'/ε,x₃/h)控制，该张量在面内快变量上是周期的，在归一化厚度变量上是可测的。上下表面的表面锚定由强度为h^γ（γ≥0）的弱锚定能建模，该框架涵盖了主要的经典锚定几何结构，属于一般集值框架。同步极限揭示了两个主要特征：第一，锚定标度根据γ产生了有效行为的层级：对于0≤γ<1，为硬约束Q∈H¹(ω;𝒜)；对于γ=1，为有限表面密度贡献；对于γ>1，为消失的锚定。当0≤γ<1且为单轴锚定时，均质化能量在约束类上简化为各向异性的Oseen–Frank和Ericksen能量。第二，有效弹性响应取决于标度比ρ=lim h/ε∈[0,∞]：每种情况都有不同的校正子结构，并产生具有均质化张量A^hom_ρ的二维LdG能量，其特征是依赖于情况的胞元问题。据我们所知，这是首次对Landau–de Gennes能量的同步均匀化和降维极限进行严格处理，也是首次将锚定强度作为标度参数引入，产生了一系列性质截然不同的有效模型。
英文摘要
We study the simultaneous limits of homogenisation $(\varepsilon\to0)$ and dimension reduction $(h\to0)$ for thin heterogeneous nematic liquid crystal films in the Landau--de Gennes $Q$-tensor framework. The elastic energy density is governed by a tensor $\mathbf{A}(x'/\varepsilon,x_3/h)$ that is periodic in the in-plane fast variable and measurable in the normalised thickness variable. Surface anchoring on the top and bottom faces is modelled by a weak anchoring energy of strength $h^γ$, $γ\geq0$, in a general set-valued framework covering the principal classical anchoring geometries. The simultaneous limit reveals two principal features. First, the anchoring scaling yields a hierarchy of effective behaviours depending on $γ$: a hard constraint $\mathcal{Q}\in H^1(ω;\mathscr{A})$ for $0\leqγ<1$, a finite surface density contribution for $γ=1$, and vanishing anchoring for $γ>1$. For $0\leq γ<1$ with uniaxial anchoring, the homogenised energy reduces on the constrained class to anisotropic Oseen--Frank and Ericksen energies. Second, the effective elastic response depends on the scale ratio $ρ=\lim h/\varepsilon\in[0,\infty]$: each regime has a distinct corrector structure and yields a two-dimensional LdG energy with homogenised tensor $\mathbf{A}^{\mathrm{hom}}_ρ$, characterised by a regime-dependent cell problem. To the best of our knowledge, this is the first rigorous treatment of the simultaneous homogenisation and dimension reduction limit for the Landau--de Gennes energy, and the first in which the anchoring strength enters as a scaling parameter, producing a hierarchy of qualitatively distinct effective models.
Comments45 pages, 1 figure
