Plateau's problem
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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 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
In higher dimensions
The extension of the problem to higher dimensions (that is, for -dimensional surfaces in -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
The axiomatic approach of Jenny Harrison and Harrison Pugh[4] treats a wide variety of special cases. In particular, they solve the anisotropic Plateau problem in arbitrary dimension and codimension for any collection of rectifiable sets satisfying a combination of general homological, cohomological or homotopical spanning conditions. A different proof of Harrison-Pugh's results were obtained by Camillo De Lellis, Francesco Ghiraldin and Francesco Maggi.[5]
Physical applications
Physical soap films are more accurately modeled by the -minimal sets of
See also
- Double Bubble conjecture
- Dirichlet principle
- Plateau's laws
- Stretched grid method
- Bernstein's problem
References
- ^
Bombieri, Enrico; De Giorgi, Ennio; Giusti, Enrico (1969), "Minimal cones and the Bernstein problem", Inventiones Mathematicae, 7 (3): 243–268, S2CID 59816096
- ^
Chang, Sheldon Xu-Dong (1988), "Two-dimensional area minimizing integral currents are classical minimal surfaces", Journal of the American Mathematical Society, 1 (4): 699–778, JSTOR 1990991
- MR 3470822
- S2CID 119704344
- S2CID 29820759
- JSTOR 1989472.
- .
- 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", Encyclopedia of Mathematics, EMS Press
- JSTOR 1968237.
- Struwe, Michael (1989). Plateau's Problem and the Calculus of Variations. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08510-4.
- ISBN 978-0-821-82747-5.
- Harrison, Jenny; Pugh, Harrison (2016). Open Problems in Mathematics (Plateau's Problem). Springer. ISBN 978-3-319-32160-8.
This article incorporates material from Plateau's Problem on