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Solving the Plateau problem means to find the surface with minimal area amongall surfaces with a given boundary. Part of the problem actually consists ofgiving a suitable definition to the notions of 'surface', 'area' and'boundary'. In our setting the considered objects are sets whose Hausdorff areais locally finite. The sliding boundary condition is given in term of a oneparameter family of compact deformations which allows the boundary of thesurface to moove along a closed set. The area functional is related tocapillarity and free-boundary problems, and is a slight modification of theHausdorff area. We focused on minimal boundary cones; that is to say tangentcones on boundary points of sliding minimal surfaces. In particular we studiedcones contained in an half-space and whose boundary can slide along thebounding hyperplane. After giving a classification of one-dimensional minimalcones in the half-plane we provided four new two-dimensional minimal cones inthe three-dimensional half space (which cannot be obtained as the Cartesianproduct of the real line with one of the previous cones). We employed thetechnique of paired calibrations and in one case could also generalise it tohigher dimension. In order to prove that the provided list of minimal cones iscomplete, we started the classification of cones satisfying the necessaryconditions for the minimality, and with numeric simulations we obtained bettercompetitors for these new candidates

Existence and regularity results for minimal sets; Plateau problem