On different notions of calibrations for minimal partitions and minimal networks in $\mathbb{R}^2$

Abstract

Calibrations are a possible tool to validate the minimality of a certain candidate. They have been introduced in the context of minimal surfaces and adapted to the case of Steiner problem in several variants. Our goal is to compare the different notions of calibrations for the Steiner Problem and for planar minimal partitions. The paper is then complemented with remarks on the convexification of the problem, on non-existence of calibrations and on calibrations in families.