Calculus of Variations and Geometric Measure Theory

A. Cesaroni - M. Novaga

Nonlocal minimal clusters in the plane

created by novaga on 08 Oct 2019
modified on 06 May 2020


Published Paper

Inserted: 8 oct 2019
Last Updated: 6 may 2020

Journal: Nonlinear Analysis
Volume: 199
Pages: 111945
Year: 2020

ArXiv: 1910.03429 PDF


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.