论文标题
拓扑奇异的矢量值图，ii：$γ$ - 金茨堡 - 兰道型功能
Topological singular set of vector-valued maps, II: $Γ$-convergence for Ginzburg-Landau type functionals
论文作者
论文摘要
我们证明了一类带有$ \ MATHCAL {N} $的Ginzburg-Landau类型功能的$γ$ -Convergence-well潜力，其中$ \ Mathcal {n} $是封闭的，$（k-2）$（k-2）$ - 连接的submanifold of $ \ submanifold of $ \ mathbb {r} r} r}^m $ $ $ $ $，IN仲裁dimension in mutiverary dimemens。例如，该类包括用于列液晶的Landau-de Gennes自由能。在差异的边界条件下，最小化器的能量密度会收敛到广义的表面（更确切地说，是$π_{k-1}（\ Mathcal {n}）$中的系数的扁平链，该链条解决了Codimension $ k $中的平台问题。该分析至关重要地依赖于拓扑奇异性的集合，即我们在同伴论文Arxiv：1712.10203中引入的操作员$ \ Mathbf {S} $。
We prove a $Γ$-convergence result for a class of Ginzburg-Landau type functionals with $\mathcal{N}$-well potentials, where $\mathcal{N}$ is a closed and $(k-2)$-connected submanifold of $\mathbb{R}^m$, in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of minimisers, subject to Dirichlet boundary conditions, converges to a generalised surface (more precisely, a flat chain with coefficients in $π_{k-1}(\mathcal{N})$) which solves the Plateau problem in codimension $k$. The analysis relies crucially on the set of topological singularities, that is, the operator $\mathbf{S}$ we introduced in the companion paper arXiv:1712.10203.
