Plateau's problem is to show the existence of a minimal surface with a given boundary. It is named after Joseph Plateau, who was interested in soap films, but was raised by Joseph Louis Lagrange in 1760. The problem is considered part of the calculus of variations.
Various specialized forms of the problem were solved, but it was only in 1930 that general solutions was found independently by Jesse Douglas and Tibor Rado. Their methods were quite different; Rado's work build on the previous work of Garnier and held only for piecewise-differentiable simple closed curves, whereas Douglas used completely new ideas with his result holding for an arbitrary simple closed curve. Douglas went on to be awarded the Fields medal in 1936 for his efforts.
References
- Douglas, Jesse, Solution of the problem of Plateau.Trans. Amer. Math. Soc. 33 (1931), no. 1, 263--321.
- Tibor Rado, On Plateau's problem. Ann. of Math. (2) 31 (1930), no. 3, 457--469.