Plateau's problem

Joseph Plateau spent years dipping wire frames into soapy water and recording the films that formed. The mathematical question his experiments pose is disarmingly simple to state: given a closed curve in space, is there a surface of least area that spans it, and what does it look like? That is Plateau's problem. The soap film answers it physically every time; proving that a mathematical minimizer always exists, and is smooth, took more than a century.

Formally: given a Jordan curve Gamma in R^3 (or a more general boundary), find a surface Sigma with boundary Gamma that minimizes area among all such spanning surfaces. The minimizer, when smooth, is a minimal surface (H = 0). The first rigorous solutions, by Jesse Douglas (who won one of the first Fields Medals for it) and independently Tibor Rado around 1930, took the disk-type approach: parametrize candidate surfaces by maps from a disk, and minimize the Dirichlet energy instead of area directly — by conformal invariance a Dirichlet-energy minimizer is automatically area-minimizing and conformally parametrized, sidestepping the parametrization ambiguity. The method walks through three steps: minimize energy in a function space, extract a limit by compactness, and prove the limit is regular. Later, geometric measure theory (Federer, Fleming, De Giorgi, Almgren) gave a far more general solution using currents and varifolds — generalized surfaces with no fixed topology — that allows the branching, films-meeting-along-curves, and singularities real soap films exhibit.

Plateau's problem is a founding problem of geometric analysis precisely because its 'obvious' physical answer is mathematically deep at every step: existence needs compactness in the right function space, regularity needs elliptic PDE estimates, and the most physical version (with triple junctions where films meet at 120-degree angles) needs the singular structure theory of geometric measure theory. The honest caveats. The Douglas-Rado solution produces a disk-type surface, which need not be embedded and can have branch points; it does not by itself capture films with the topology of a Mobius band or with interior singularities. And uniqueness fails in general — a single wire can bound several distinct minimal films, and which one a real soap film picks depends on how you dip it. 'Solved' means existence and regularity are understood, not that the answer is unique or always a smooth embedded surface.