Inserted: 8 oct 2019
Last Updated: 9 oct 2019
We show existence of nonlocal minimal cluster with Dirichlet boundary data. In two dimensions we show that, if the fractional parameter $s$ is sufficiently close to $1$, the only singular minimal cone consists of three half-lines meeting at $120$ degrees at a common end-point. In the case of fractional perimeter with weights, we show that there exists a unique minimal cone with three phases.