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Nonlocal minimal clusters in the plane
Nonlinear Analysis ( IF 1.3 ) Pub Date : 2020-05-04 , DOI: 10.1016/j.na.2020.111945
Annalisa Cesaroni , Matteo Novaga

We prove existence of partitions of an open set Ω with a given number of phases, which minimize the sum of the fractional perimeters of all the phases, with Dirichlet boundary conditions. In two dimensions we show that, if the fractional parameter s is sufficiently close to 1, the only singular minimal cone, that is, the only minimal partition invariant by dilations and with a singular point, is given by three half-lines meeting at 120 degrees. In the case of a weighted sum of fractional perimeters, we show that there exists a unique minimal cone with three phases.



中文翻译:

平面中的非局部最小聚类

我们证明开放集的分区的存在 Ω具有给定数量的相,这在Dirichlet边界条件下使所有相的分数周长之和最小。我们在二维中表明,如果分数参数s当s足够接近1时,唯一的奇异最小锥(即,唯一由扩张和奇异点不变的最小分区)由三个相交于120度的半线给出。在分数周长的加权和的情况下,我们表明存在一个唯一的具有三个相位的最小圆锥。

更新日期:2020-05-04
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