Journal of Convex Analysis 33 (2026), No. 3&4, 691--736
Copyright Heldermann Verlag 2026
Calibrations for Minimal Surfaces with Free Boundary and Cheeger-Type Problems
Guy Bouchitté
Lab IMATH, Université de Toulon, La Garde, France
bouchitte@univ-tln.fr
Minh Phan
The University of Danang, University of Science and Education, Danang, Vietnam
ptdminh@ued.udn.vn
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.
Keywords: Free boundary problems, calibrations, minimal surfaces, shape derivative, Cheeger sets, cut-locus potential.
MSC: 49J45, 49N15, 49Q10, 65K10, 90C26.
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