mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by
Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaJoseph Plateau who experimented with soap films. The problem is considered part of the
calculus of variations
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of f ...
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
Jesse Douglas (July 3, 1897 – September 7, 1965) was an American mathematician and Fields Medalist known for his general solution to Plateau's problem.
Life and career
He was born to a Jewish family in New York City, the son of Sarah ...
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. Both relied on setting up minimization problems; Douglas minimized the now-named Douglas integral while Radó minimized the "energy". Douglas went on to be awarded the
Fields Medal
The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...
in 1936 for his efforts.
In higher dimensions
The extension of the problem to higher
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
s (that is, for -dimensional surfaces in -dimensional space) turns out to be much more difficult to study. Moreover, while the solutions to the original problem are always regular, it turns out that the solutions to the extended problem may have singularities if . In the
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
case where , singularities occur only for . An example of such singular solution of the Plateau problem is the Simons cone, a cone over in that was first described by Jim Simons and was shown to be an area minimizer by Bombieri, De Giorgi and Giusti. 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 mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line ...
. In the case of higher codimension Almgren proved existence of solutions with singular set of dimension at most in his regularity theorem. 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.
The axiomatic approach of
Jenny Harrison
Jenny Harrison is a professor of mathematics at the University of California, Berkeley.
Education and early career
Harrison grew up in Tuscaloosa, Alabama, and earned her undergraduate degree from the University of Alabama. Awarded a Marshall S ...
and Harrison Pugh treats a wide variety of special cases. 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. A different proof of Harrison-Pugh's results were obtained by Camillo De Lellis, Francesco Ghiraldin and Francesco Maggi.
Physical applications
Physical soap films are more accurately modeled by the -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.