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 films. The problem is considered part of the calculus of variations. The existence and regularity problems are part of geometric measure theory. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Verrill Minimal Surface In mathematics, a minimal surface is a surface with a mean curvature of zero. ... Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. ... Year 1760 (MDCCLX) was a leap year starting on Tuesday (link will display the full calendar) of the Gregorian calendar (or a leap year starting on Saturday of the 11-day slower Julian calendar). ... Plateaus phenakistiscope Joseph Antoine Ferdinand Plateau (October 14, 1801 - September 15, 1883) was a Belgian physicist. ... A soap film is a physical realization of a minimal surface. ... Calculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with functions. ... In mathematics, geometric measure theory (GMT) is the study of the geometric properties of the measures of sets (typically in Euclidean spaces), including such things as arc lengths and areas. ...

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. Jesse Douglas (July 3, 1897 - October 7, 1965) was an American mathematician. ... Tibor Rado (June 2, 1895 - December 29, 1965) 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. ... For other uses, see Garnier (disambiguation). ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ... The obverse of the Fields Medal The Fields Medal 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 years. ...

The extension of the problem to higher dimensions (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 . In the hypersurface case where k = n − 1, singularities occur only for . 2-dimensional renderings (ie. ... 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. ... In mathematics, a hypersurface is some kind of submanifold. ...

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. Ennio de Giorgi (1928 - 1996) was an Italian mathematician. ... Herbert Federer, an American mathematician, is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis. ...

See also

• Dirichlet principle
• Geometric measure theory

In mathematics, Dirichlets principle in potential theory states that the harmonic function on a domain with boundary condition on can be obtained as the minimizer of the Dirichlet integral amongst all functions such that on , provided only that there exists one such function making the Dirichlet integral finite. ... In mathematics, geometric measure theory (GMT) is the study of the geometric properties of the measures of sets (typically in Euclidean spaces), including such things as arc lengths and areas. ...

References

• Douglas, Jesse (1931). "Solution of the problem of Plateau". Trans. Amer. Math. Soc. 33 (1): 263–321. 
• Radó, Tibor (1930). "On Plateau's problem". Ann. of Math. (2) 31: 457–469. 
• T.C O'Neil (2001), “Geometric Measure Theory”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• R. Bonnett and A. T. Fomenko: The Plateau Problem (Studies in the Development of Modern Mathematics), ISBN 2-88124-702-4

This article incorporates material from Plateau's Problem on PlanetMath, which is licensed under the GFDL. Jesse Douglas (July 3, 1897 - October 7, 1965) was an American mathematician. ... Tibor Radó (June 2, 1895 - December 29, 1965) 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. ... The Encyclopaedia of Mathematics is a large reference work in mathematics. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...