Plateau's problem
Encyclopedia
In mathematics
, Plateau's problem is to show the existence of a minimal surface
with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who was interested in soap film
s. The problem is considered part of the calculus of variations
. The existence and regularity problems are part of geometric measure theory
.
Various specialized forms of the problem were solved, but it was only in 1930 that general solutions were found independently by Jesse Douglas
and Tibor Rado
. Their methods were quite different; Rado's work built on the previous work of 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 Rado minimized the "energy". Douglas went on to be awarded the Fields medal
in 1936 for his efforts.
The extension of the problem to higher dimension
s (that is, for k-dimensional surfaces in n-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 k ≤ n − 2. In the hypersurface
case where k = n − 1, singularities occur only for n ≥ 8.
To solve the extended problem, the theory of perimeters (De Giorgi
) for boundaries and the theory of rectifiable currents (Federer
and Fleming) have been developed.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Plateau's problem is to show the existence of a minimal surface
Minimal surface
In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who was interested in soap film
Soap film
Soap films are thin layers of liquid surrounded by air. For example, if two soap bubbles enters in contact, they merged and a thin film is created in between. Thus, foams are composed of a network of films connected by Plateau borders...
s. The problem is considered part of the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. The existence and regularity problems are part of geometric measure theory
Geometric measure theory
In mathematics, geometric measure theory is the study of the geometric properties of the measures of sets , including such things as arc lengths and areas. It uses measure theory to generalize differential geometry to surfaces with mild singularities called rectifiable sets...
.
Various specialized forms of the problem were solved, but it was only in 1930 that general solutions were found independently by Jesse Douglas
Jesse Douglas
Jesse Douglas was an American mathematician. He was born in New York and attended Columbia College of Columbia University from 1920–1924. Douglas was one of two winners of the first Fields Medals, awarded in 1936. He was honored for solving, in 1930, the problem of Plateau, which asks whether a...
and Tibor Rado
Tibor Radó
Tibor Radó was a Hungarian mathematician who moved to the USA after World War I. He was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute. In World War I, he became a First Lieutenant in the Hungarian Army and was captured on the Russian Front...
. Their methods were quite different; Rado's work built on the previous work of 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 Rado minimized the "energy". Douglas went on to be awarded the Fields medal
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
in 1936 for his efforts.
The extension of the problem to higher dimension
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
s (that is, for k-dimensional surfaces in n-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
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
if k ≤ n − 2. In the hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
case where k = n − 1, singularities occur only for n ≥ 8.
To solve the extended problem, the theory of perimeters (De Giorgi
Ennio de Giorgi
-References:. The first paper about SBV functions and related variational problems.. The first note published by De Giorgi describing his approach to Caccioppoli sets.. The first complete exposition by De Giorgi of the theory of Caccioppoli sets.. An advanced text, oriented to the theory of minimal...
) for boundaries and the theory of rectifiable currents (Federer
Herbert Federer
Herbert Federer was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.-Career:...
and Fleming) have been developed.