通过自由不连续泛函逼近拓扑奇点：临界与超临界情形
Approximation of topological singularities through free discontinuity functionals: the critical and super-critical regimes
AI总结：
该研究拓展了Mumford-Shah型自由不连续泛函逼近拓扑奇点的方法，证明二维反平面螺位错能量在临界与超临界区间下三类泛函的变分等价性，并通过新的稠密性结果去除了序参数跳跃集为紧集的先验假设。
摘要：
我们进一步研究了文献[DLSVG]中提出的一种通过Mumford-Shah型自由不连续泛函处理拓扑奇点的方法的性质。我们证明了在相关的能量区间$|\log \varepsilon|^a$（$a\geq 1$，其中$\varepsilon$表示缺陷附近过程区的线性尺寸）下，二维反平面螺位错能量的这类泛函、Ginzburg-Landau与Core-Radius之间的变分等价性。此外，我们去除了逼近序参数具有紧跳跃集的先验限制性假设。这一结果是通过证明$\u212a^1$值$SBV^p$函数的一个新的稠密性定理得到的，这类函数可由具有本质闭跳跃集的函数在强$BV$范数下逼近。
英文摘要：
We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in \cite{DLSVG}. We prove the variational equivalence between such energies, Ginzburg-Landau, and Core-Radius for anti-plane screw dislocations energies in dimension two, in the relevant energetic regimes $|\log \varepsilon|^a$, $a\geq 1$, where $\varepsilon$ denotes the linear size of the process zone near the defects. Further, we remove the \emph{a priori} restrictive assumptions that the approximating order parameters have compact jump set. This is obtained by proving a new density result for $\mathbb S^1$-valued $SBV^p$ functions, approximated through functions with essentially closed jump set, in the strong $BV$ norm.
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