We prove existence of partitions of an open set $\Omega$ 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：

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平面中的非局部最小聚类

我们证明开放集的分区的存在 $\Omega$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。