plateau

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.

or submarine plateau

Large submarine elevation rising sharply at least 660 ft (200 m) above the surrounding seafloor and having an extensive, relatively flat or gently tilted summit. Most plateaus are steplike interruptions of the continental slopes. Some, however, occur well beyond the continental margins. They stand alone, high above the surrounding seafloor, and are believed to be fragments of continents that were isolated during continental drift and seafloor spreading.

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Extensive area of flat upland, usually bounded by an escarpment on all sides but sometimes enclosed by mountains. Plateaus are extensive, and together with enclosed basins they cover about 45percnt of the Earth's land surface. The essential criteria for a plateau are low relative relief and some altitude. Low relief distinguishes plateaus from mountains, although their origin may be similar. Plateaus, being high, often create their own local climate; the topography of plateaus and their surroundings often produce arid and semiarid conditions.

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Tableland in the U.S. that forms the western section of the Appalachian Mountains and a part of the Allegheny Plateau. It extends southwest for 450 mi (725 km) from southern West Virginia to northeastern Alabama, averages 50 mi (80 km) in width, and is 2,000–4,145 ft (600–1,263 m) high. The roughest and highest portion is a narrow ridge about 140 mi (225 km) long that forms its eastern margin in eastern Kentucky and northeastern Tennessee; the name Cumberland Mountains is generally applied to this area, which includes the Cumberland Gap National Historical Park. The plateau has large deposits of coal, limestone, and sandstone.

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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

• Dirichlet principle

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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