arXiv 2012.125592020-12-23math.AP

周期性介质中的拓扑奇异性：Ginzburg-Landau和核半径方法

Topological singularities in periodic media: Ginzburg-Landau and core-radius approaches

Roberto Alicandro, Andrea Braides, Marco Cicalese, Lucia De Luca, Andrey Piatnitski

DOI：: 10.1007/s00205-021-01731-7

摘要：

我们描述了出现在周期性介质内的Ginzburg-Landau模型和核心半径的方法拓扑奇点。两个模型的能量泛函都用$E_{\varepsilon,δ}$表示，其中$\varepsilon$表示相干长度（在Ginzburg-Landau模型中）或核半径大小（在核半径方法中），$δ$表示周期尺度。在$|\log\varepsilon|$标度区域中，我们对$E_{\varepsilon,δ}$作为$\varepsilon\to 0$和$δ=δ_{\varepsilon}\to 0$进行了$Γ$ -收敛分析，结果表明，$Γ$ -极限包含在整数阶的多个涡状点奇点的能量代价中.在引入尺度参数（在提取的双折射）$$ λ=\min\Bigl\{1,\lim_{\varepsilon\to0} {|\log δ_{\varepsilon}|\over|\log{\varepsilon}|}\Bigr\}, $$，我们表明，在某种意义上，我们总是有一个分离的规模效应：在尺度小于$\varepsilon^λ$，我们首先有一个浓度过程周围的一些涡的位置随后优化，而尺度大于$\varepsilon^λ$的浓度过程发生“后”均匀化。

英文摘要：

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.

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