plateau

plateau

[pla-toh or, especially Brit., plat-oh]
plateau, elevated, level or nearly level portion of the earth's surface, larger in summit area than a mountain and bounded on at least one side by steep slopes, occurring on land or in oceans. Some plateaus, such as the Deccan of India and the Columbia Plateau of the NW United States, are basaltic and were formed as the result of a succession of lava flows covering hundreds of thousands of square miles that built up the land surface. Others are the result of upward folding; still others have been left elevated by the erosion of adjacent lands. Plateaus, like all elevated regions, are subject to dissection by erosion, which removes greater amounts of the upland surface. Low plateaus are often agricultural regions, while high plateaus are usually fit chiefly for stock grazing. Many of the world's high plateaus are deserts. Other notable plateaus are the Colorado Plateau of the W United States, the Bolivian plateau in South America, and the plateaus of Anatolia, Arabia, Iran, and the Tibet region of China.
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.

See also

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.
  • 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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