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In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Plateau's problem is to show the existence of a
minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
with a given boundary, a problem raised by
Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[Joseph Plateau
Joseph Antoine Ferdinand Plateau (14 October 1801 – 15 September 1883) was a Belgian physicist and mathematician. He was one of the first people to demonstrate the illusion of a moving image. To do this, he used counterrotating disks with repea ...](_blank)
who experimented with
soap film
Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Platea ...
s. 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 functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. The existence and regularity problems are part of
geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
.
History
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 (3 July 1897 – 7 September 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 (née ...
and
Tibor Radó
Tibor Radó (June 2, 1895 – December 29, 1965) was a Hungarian mathematician who moved to the United States after World War I.
Biography
Radó was born in Budapest and between 1913 and 1915 attended the Polytechnic Institute, studying civ ...
. 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 the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...
in 1936 for his efforts.
In higher dimensions
The extension of the problem to higher
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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 Euclidean ...
case where
, singularities occur only for
. An example of such singular solution of the Plateau problem is the
Simons cone Simons is a surname of Scandinavia, Scandinavian origins and a variant of Sigmund, Sigmundsson, a patronymic surname with roots in proto-Germanic wikt:Reconstruction:Proto-Germanic/segaz, ''*segaz'' and wikt:Reconstruction:Proto-Germanic/mundō, ''* ...
, 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 current
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Re ...
s (
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 first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a ...
. In the case of higher codimension
Almgren proved existence of solutions with
singular set
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...
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 career
Harrison grew up in Tuscaloosa, Alabama. On graduating from the University of Alabama, she won a Marshall Scholarship which she used to ...
and
Harrison Pugh
Harrison may refer to:
People
* Harrison (name)
* Harrison family of Virginia, United States
Places
In Australia:
* Harrison, Australian Capital Territory, suburb in the Canberra district of Gungahlin
In Canada:
* Inukjuak, Quebec, or " ...
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
Camillo De Lellis (born 11 June 1976) is an Italian mathematician who is active in the fields of calculus of variations, hyperbolic systems of conservation laws, geometric measure theory and fluid dynamics. He is a permanent faculty member in ...
, Francesco Ghiraldin and
Francesco Maggi
Francesco, the Italian language, Italian (and original) version of the personal name "Francis (given name), Francis", is the List of most popular given names, most common given name among males in Italy. Notable persons with that name include:
Pe ...
.
Physical applications
Physical soap films are more accurately modeled by the
-minimal sets of
Frederick Almgren
Frederick Justin Almgren Jr. (July 3, 1933 – February 5, 1997) was an American mathematician working in geometric measure theory. He was born in Birmingham, Alabama.
Almgren received a Guggenheim Fellowship in 1974. Between 1963 and 1992 he wa ...
, 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
Ernst is both a surname and a given name, the German, Dutch, and Scandinavian form of Ernest. Notable people with the name include:
Surname
* Adolf Ernst (1832–1899) German botanist known by the author abbreviation "Ernst"
* Anton Ernst (1975-) ...
solved such a "universal Plateau's problem" for boundaries which are homeomorphic to single embedded spheres.
See also
*
Double Bubble conjecture
In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a ''standard double bubble'': three spherical surfaces meet ...
*
Dirichlet principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Formal statement
Dirichlet's principle states that, if the functio ...
*
Plateau's laws
Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws.
Laws f ...
*
Stretched grid method
The stretched grid method (SGM) is a numerical technique for finding approximate solutions of various mathematical and engineering problems that can be related to an elastic grid behavior.
In particular, meteorologists use the stretched grid meth ...
*
Bernstein's problem
In differential geometry, Bernstein's problem is as follows: if the graph of a function on R''n''−1 is a minimal surface in R''n'', does this imply that the function is linear?
This is true in dimensions ''n'' at most 8, but false in dimens ...
References
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{{PlanetMath attribution, id=4286, title=Plateau's Problem
Calculus of variations
Minimal surfaces
Mathematical problems