Calibration for minimal surfaces with free boundary and Cheeger-type problems

TL;DR

This paper investigates minimal surfaces with free boundaries through a non-convex minimization approach, introducing a calibration method and relating it to a Cheeger-type problem to characterize optimal solutions.

Contribution

It develops a calibration framework for free boundary minimal surfaces and explicitly solves a Cheeger-type problem to establish optimality.

Findings

Explicit solution to a Cheeger-type problem

Construction of a cut-locus potential for optimality proof

Comparison between the Cheeger variant and original problem

Abstract

We study a problem of minimal surfaces with free boundary written in the form of a non convex minimization problem. Our aim is to characterize optimal solutions by finding a suitable calibration field. A natural upper bound of the infimum is given by a variant of the Cheeger problem that we solve explicitly proving the optimality thanks to the construction of a cut-locus potential. The comparison with the original problem is then discussed in detail.