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

TL;DR

This paper compares various calibration concepts used to verify minimality in planar minimal partitions and Steiner problems, discussing their differences, limitations, and extensions.

Contribution

It provides a systematic comparison of calibration notions for minimal partitions and Steiner problems in the plane, including new insights on convexification and calibration existence.

Findings

Different calibration notions have distinct applicability and limitations.

Calibrations can be non-existent in certain configurations.

Extensions to calibration families are discussed.

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.