Plateau's problem

Various specialized forms of the problem were solved, but it was only in 1930 that general solutions were found in the context of mappings (immersions) independently by Jesse Douglas and Tibor Radó.

Their methods were quite different; Radó's work built on the previous work of René Garnier and held only for rectifiable simple closed curves, whereas Douglas used completely new ideas with his result holding for an arbitrary simple closed curve.

that was first described by Jim Simons and was shown to be an area minimizer by Bombieri, De Giorgi and Giusti.

[1] To solve the extended problem in certain special cases, the theory of perimeters (De Giorgi) for codimension 1 and the theory of rectifiable currents (Federer and Fleming) for higher codimension have been developed.

The theory guarantees existence of codimension 1 solutions that are smooth away from a closed set of Hausdorff dimension

In the case of higher codimension Almgren proved existence of solutions with singular set of dimension at most

S. X. Chang, a student of Almgren, built upon Almgren’s work to show that the singularities of 2-dimensional area minimizing integral currents (in arbitrary codimension) form a finite discrete set.

In particular, they solve the anisotropic Plateau problem in arbitrary dimension and codimension for any collection of rectifiable sets satisfying a combination of general homological, cohomological or homotopical spanning conditions.

-minimal sets of Frederick Almgren, but the lack of a compactness theorem makes it difficult to prove the existence of an area minimizer.

In this context, a persistent open question has been the existence of a least-area soap film.

Ernst Robert Reifenberg solved such a "universal Plateau's problem" for boundaries which are homeomorphic to single embedded spheres.