论文标题
通过雅各比弱概念来实现有限变化的功能，一种拓扑奇异的新方法
A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation
论文作者
论文摘要
我们引入了$ 2 \ times 2 $ - $ bv $函数子类的梯度的薄弱概念。对于$ bv（\ mathbb {r}^2; \ mathbb {r}^2）的地图，这样的概念将jacobian决定因素的标准定义扩展到了非Sobolev地图。 我们使用这种分布的雅各布来证明紧凑性和$γ$ - 融合的结果，这是一个新模型，以吉兹堡 - 兰道和核心 - 拉迪乌斯方法的精神来描述拓扑奇异性的出现。在我们的框架内，订单参数是$ sbv $ map $ u $，以$ \ mathbb {s}^1 $获取值，并且能量是由$ \ nabla U $的平方$ l^2 $ norm of $ \ nabla u $的总和（$ $ u $ u $ u $ $ $ $ $ $ $ us $ \ frac frac 1 frac 1 \ varepsilon $ \ varepsilon $ \ varepsilon $ \ v v v able）的总和。在这里，$ \ varepsilon $是一个长度尺度参数。我们表明，在$ | \ log \ varepsilon | $制度中，雅各布分布收敛为$ \ varepsilon \ to 0^+$，以有限的$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $。
We introduce a weak notion of $2\times 2$-minors of gradients of a suitable subclass of $BV$ functions. In the case of maps in $BV(\mathbb{R}^2;\mathbb{R}^2)$ such a notion extends the standard definition of Jacobian determinant to non-Sobolev maps. We use this distributional Jacobian to prove a compactness and $Γ$-convergence result for a new model describing the emergence of topological singularities in two dimensions, in the spirit of Ginzburg-Landau and core-radius approaches. Within our framework, the order parameter is an $SBV$ map $u$ taking values in $\mathbb{S}^1$ and the energy is made by the sum of the squared $L^2$ norm of $\nabla u$ and of the length of (the closure of) the jump set of $u$ multiplied by $\frac 1 \varepsilon$. Here, $\varepsilon$ is a length-scale parameter. We show that, in the $|\log\varepsilon|$ regime, the Jacobian distributions converge, as $\varepsilon\to 0^+$, to a finite sum $μ$ of Dirac deltas with weights multiple of $π$, and that the corresponding effective energy is given by the total variation of $μ$.
