论文标题
基于本地配对校准的概念，界面长度的局部最小化器功能
Local minimizers of the interface length functional based on a concept of local paired calibrations
论文作者
论文摘要
我们确定，相对于$ l^1 $扰动的界面能量，常规平面分区正在局部最小化。常规的平面分区是$ \ mathbb {r}^2 $中的开放式套件的分区，其接口网络由有限的许多直线段组成，其中有一个有限的单数组成，由有限的许多三重连接组成，其中满足了鲱鱼角条件。该结果不仅适用于周长功能的情况，而且适用于一般的表面张力矩阵。我们的证明依赖于配对校准方法的局部版本，该方法由Lawlor和Morgan（Pac。J.Appl。Math。，166（1），1994年）与相对能量功能相结合，可精确地捕获经典校准估算的次级临时性。反之亦然，我们表明，长度功能的任何固定点（从指标空间的意义上）都必须是常规平坦分区。
We establish that regular flat partitions are locally minimizing for the interface energy with respect to $L^1$ perturbations of the phases. Regular flat partitions are partitions of open sets in $\mathbb{R}^2$ whose network of interfaces consists of finitely many straight segments with a singular set made up of finitely many triple junctions at which the Herring angle condition is satisfied. This result not only holds for the case of the perimeter functional but for a general class of surface tension matrices. Our proof relies on a localized version of the paired calibration method which was introduced by Lawlor and Morgan (Pac. J. Appl. Math., 166(1), 1994) in conjunction with a relative energy functional that precisely captures the suboptimality of classical calibration estimates. Vice versa, we show that any stationary point of the length functional (in a sense of metric spaces) has to be a regular flat partition.
