Nonlocal minimal clusters in the plane

Abstract

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.