Existence and regularity results for minimal sets; Plateau problem

TL;DR

This paper advances the understanding of minimal sets by classifying boundary cones in half-spaces, introducing new minimal cones, and employing calibration techniques to establish their minimality within the context of the Plateau problem.

Contribution

It provides a classification of boundary minimal cones in half-spaces, introduces four new two-dimensional minimal cones, and extends calibration methods to higher dimensions.

Findings

Classified one-dimensional minimal cones in the half-plane.

Discovered four new two-dimensional minimal cones in three-dimensional half-space.

Used paired calibrations and numerical simulations to verify minimality.

Abstract

Solving the Plateau problem means to find the surface with minimal area among all surfaces with a given boundary. Part of the problem actually consists of giving a suitable definition to the notions of 'surface', 'area' and 'boundary'. In our setting the considered objects are sets whose Hausdorff area is locally finite. The sliding boundary condition is given in term of a one parameter family of compact deformations which allows the boundary of the surface to moove along a closed set. The area functional is related to capillarity and free-boundary problems, and is a slight modification of the Hausdorff area. We focused on minimal boundary cones; that is to say tangent cones on boundary points of sliding minimal surfaces. In particular we studied cones contained in an half-space and whose boundary can slide along the bounding hyperplane. After giving a classification of one-dimensional minimal cones in the half-plane we provided four new two-dimensional minimal cones in the three-dimensional half space (which cannot be obtained as the Cartesian product of the real line with one of the previous cones). We employed the technique of paired calibrations and in one case could also generalise it to higher dimension. In order to prove that the provided list of minimal cones is complete, we started the classification of cones satisfying the necessary conditions for the minimality, and with numeric simulations we obtained better competitors for these new candidates