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 experimented with soap films. The problem is considered part of the calculus of variations. The existence and regularity problems are part of geometric measure theory.

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 and Tibor Radó. 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 in 1936 for his efforts.

In higher dimensions

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 in special cases, the theory of perimeters (De Giorgi) for codimension 1 and the theory of rectifiable currents (Federer and Fleming) for higher codimension have been developed. Multidimensional Plateau problem in the class of spectral surfaces (parametrized by the spectra of the manifolds with a fixed boundary) was solved in 1969 by Anatoly Fomenko.

Physical applications

Physical soap films are more accurately modeled by the (M,0,delta)-minimal sets of Frederick Almgren, 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 solved such a "universal Plateau's problem" for boundaries which are homeomorphic to single embedded spheres. In his book Almgren claimed to use varifolds to solve the problem for more than one sphere, as well as more general boundaries, but Allard's compactness theorem for integral varifolds produces a minimal surface, not necessarily an area minimizer. In 2015 Jenny Harrison and Harrison Pugh used cohomology to define spanning sets and Hausdorff measure weighted by a bounded Lipschitz function to define area, and solved the problem in this setting. Their paper is currently under review.

See also

References

  • Douglas, Jesse (1931). "Solution of the problem of Plateau". Trans. Amer. Math. Soc. Transactions of the American Mathematical Society, Vol. 33, No. 1. 33 (1): 263–321. doi:10.2307/1989472. JSTOR 1989472.
  • Reifenberg, Ernst Robert (1960). "Solution of the {Plateau} problem for m-dimensional surfaces of varying topological type". Acta Mathematica. 104 (2): 1–92. doi:10.1007/bf02547186.
  • Fomenko, A.T. (1989). The Plateau Problem: Historical Survey. Williston, VT: Gordon & Breach. ISBN 978-2-88124-700-2.
  • Morgan, Frank (2009). Geometric Measure Theory: a Beginner's Guide. Academic Press. ISBN 978-0-12-374444-9.
  • O'Neil, T.C. (2001) [1994], "Geometric Measure Theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Radó, Tibor (1930). "On Plateau's problem". Ann. of Math. 2. The Annals of Mathematics, Vol. 31, No. 3. 31 (3): 457–469. doi:10.2307/1968237. JSTOR 1968237.
  • Struwe, Michael (1989). Plateau's Problem and the Calculus of Variations. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08510-4.
  • Almgren, Frederick (1966). Plateau's problem, an invitation to varifold geometry. New York-Amsterdam: Benjamin. ISBN 978-0-821-82747-5.
  • Harrison, Jenny (2012). "Soap Film Solutions to Plateau's Problem". Journal of Geometric Analysis. Springer Online First. 24: 271–297. arXiv:1106.5839. doi:10.1007/s12220-012-9337-x.

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