论文标题
定期媒体中的拓扑奇异性：金茨堡 - 兰道和核心 - 拉迪乌斯的方法
Topological singularities in periodic media: Ginzburg-Landau and core-radius approaches
论文作者
论文摘要
我们描述了金茨堡 - 兰多模型和核心 - 拉迪乌斯方法中周期性媒体中拓扑奇异性的出现。两种模型的能量功能用$ e _ {\ varepsilon，δ} $表示，其中$ \ varepsilon $代表连贯的长度（在金兹堡 - 兰道模型中）或核心 - 拉迪乌斯大小（在核心 - 拉迪乌斯方法中）和$δ$表示周期性尺度。我们对$ e _ {\ varepsilon，δ} $进行$γ$ -Convergence分析，为$ \ varepsilon \ to 0 $和$δ=δ=δ_ {\ varepsilon} \ to $ | | | | | \ varepsilon consistion $ scalist $ contistion $ consistion $ - log \ varepsilon | lim-lim-lim---- $ - 整数学位的涡流状点奇点。引入了比例参数（在提取子序列时）$λ= \ min \ min \ bigl \ {1，\ lim _ {\ varepsilon \ to0} {| \logΔ__{\logΔ_{\ varepsilon}尺度分离效应：在小于$ \ varepsilon^λ$的尺度上，我们首先在某些涡流周围具有浓度过程，其位置随后被优化，而对于大于$ \ varepsilon^λ$的量表，浓度浓度是在“均质化之后”之后的。
We describe the emergence of topological singularities in periodic media within the Ginzburg-Landau model and the core-radius approach. The energy functionals of both models are denoted by $E_{\varepsilon,δ}$, where $\varepsilon$ represent the coherence length (in the Ginzburg-Landau model) or the core-radius size (in the core-radius approach) and $δ$ denotes the periodicity scale. We carry out the $Γ$-convergence analysis of $E_{\varepsilon,δ}$ as $\varepsilon\to 0$ and $δ=δ_{\varepsilon}\to 0$ in the $|\log\varepsilon|$ scaling regime, showing that the $Γ$-limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter (upon extraction of subsequences) $$ λ=\min\Bigl\{1,\lim_{\varepsilon\to0} {|\log δ_{\varepsilon}|\over|\log{\varepsilon}|}\Bigr\}, $$ we show that in a sense we always have a separation-of-scale effect: at scales less than $\varepsilon^λ$ we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than $\varepsilon^λ$ the concentration process takes place "after" homogenization.
