Plateau's problem

Plateau's problem

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.

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 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 "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.

ee also

* Dirichlet principle

References

* cite journal
last = Douglas | first = Jesse
authorlink = Jesse Douglas
title = Solution of the problem of Plateau
journal = Trans. Amer. Math. Soc.
volume = 33
year = 1931
issue = 1
pages = 263–321
doi = 10.2307/1989472
* cite journal
first = Tibor | last = Radó
authorlink = Tibor Radó
title = On Plateau's problem
journal = Ann. of Math. (2)
volume = 31
year = 1930
pages = 457–469
doi = 10.2307/1968237
*springer|author=T.C O'Neil|id=G/g130040|title=Geometric Measure Theory
*R. Bonnett and A. T. Fomenko: "The Plateau Problem (Studies in the Development of Modern Mathematics)", ISBN 2-88124-702-4

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  • Plateau's problem — /pla tohz /, Math. the problem in the calculus of variations of finding the surface with the least area bounded by a given closed curve in space. [1910 15; named after J. A. F. Plateau (1801 83), Belgian physicist] * * * …   Universalium

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