Plateau's problem as a singular limit of capillarity problems

Abstract

Soap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a homotopic spanning condition. This point of view introduces a length scale in the classical Plateau's problem, which is in turn recovered in the vanishing volume limit. This approximation of area minimizing hypersurfaces leads to an energy based selection principle for Plateau's problem, points at physical features of soap films that are unaccessible by simply looking at minimal surfaces, and opens several challenging questions.

arXiv comment53 pages, 13 figures. In v3, Lemma 2.5 has been modified to address a possibly occurring situation (see Figure 2.1(b)) which was not discussed in previous versions

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• Regularity for Minimizers of a Planar Partitioning Problem with Cusps