Editing Plateau's problem

{{Short description|To find the minimal surface with a given boundary}}
[[File:Bulle caténoïde.png|thumb|A soap bubble in the shape of a [[catenoid]]]]
{{calculus|expanded=specialized}}
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 film]]s.  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 curve|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 [[dimension]]s (that is, for <math>k</math>-dimensional surfaces in <math>n</math>-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 [[mathematical singularity|singularities]] if <math>k \leq n - 2</math>. In the [[hypersurface]] case where <math>k = n - 1</math>, singularities occur only for <math>n \geq 8</math>. An example of such singular solution of the Plateau problem is the [[Simons cone]], a cone over <math> S^3 \times S^3 </math> in <math>\mathbb{R}^8</math> that was first described by [[Jim Simons]] and was shown to be an area minimizer by [[Enrico Bombieri|Bombieri]], [[Ennio De Giorgi|De Giorgi]] and [[Enrico Giusti|Giusti]].<ref>
{{citation
 | last1 = Bombieri | first1 = Enrico
 | last2 = De Giorgi | first2 = Ennio
 | last3 = Giusti | first3 = Enrico
 | title = Minimal cones and the Bernstein problem
 | journal = Inventiones Mathematicae
 | pages = 243–268
 | volume = 7
 | year = 1969
 | issue = 3
 | doi=10.1007/BF01404309| bibcode = 1969InMat...7..243B
 | s2cid = 59816096
 }}</ref> To solve the extended problem in certain special cases, the [[Caccioppoli set#De Giorgi definition|theory of perimeters]] ([[Ennio De Giorgi|De Giorgi]]) for codimension 1 and the theory of [[rectifiable current]]s ([[Herbert Federer|Federer]] and Fleming) for higher codimension have been developed. The theory guarantees existence of codimension 1 solutions that are smooth away from a closed set of [[Hausdorff dimension]] <math>n-8</math>. In the case of higher codimension [[Frederick J. Almgren, Jr.|Almgren]] proved existence of solutions with [[singular set]] of dimension at most <math>k-2</math> in his [[Almgren regularity theorem|regularity theorem]]. S. X. Chang, a
student of Almgren, built upon Almgren’s work to show that the singularities of 2-dimensional area
minimizing integral currents (in arbitrary codimension) form a finite discrete set.<ref>
{{citation
 | last1 = Chang | first1 = Sheldon Xu-Dong
 | title = Two-dimensional area minimizing integral currents are classical minimal surfaces
 | journal = Journal of the American Mathematical Society
 | pages = 699–778
 | volume = 1
 | issue = 4
 | year = 1988
 | doi=10.2307/1990991| jstor = 1990991
 }}</ref><ref>{{citation
 | last = De Lellis | first = Camillo
 | doi = 10.1007/s40574-016-0057-1
 | issue = 1
 | journal = Bollettino dell'Unione Matematica Italiana
 | mr = 3470822
 | pages = 3–67
 | title = Two-dimensional almost area minimizing currents
 | url = https://www.math.stonybrook.edu/~bishop/classes/math638.F20/deLellis_survey_BUMI_24.pdf
 | volume = 9
 | year = 2016}}</ref>

The axiomatic approach of [[Jenny Harrison]] and [[Harrison Pugh]]<ref>{{citation
| last1 = Harrison | first1 = Jenny
| last2 = Pugh | first2 = Harrison
| title = General Methods of Elliptic Minimization
| journal = [[Calculus of Variations and Partial Differential Equations]]
| volume = 56
| year = 2017
| issue = 1
| doi = 10.1007/s00526-017-1217-6
| arxiv = 1603.04492
| s2cid = 119704344
}}</ref> 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]].<ref>{{citation
 | last1 = De Lellis | first1 = Camillo
 | last2 = Ghiraldin | first2 = Francesco
 | last3 = Maggi | first3 = Francesco
 | title = A direct approach to Plateau's problem
 | journal = [[Journal of the European Mathematical Society]]
 | pages = 2219–2240
 | volume = 19
 | issue = 8
 | year = 2017
 | doi=10.4171/JEMS/716| s2cid = 29820759
 | url = https://www.zora.uzh.ch/id/eprint/141580/1/DeLDeRGhi_15apr17.pdf
 }}</ref>

==Physical applications==
Physical soap films are more accurately modeled by the <math>(M, 0, \Delta)</math>-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.

==See also==
{{Portal|Mathematics|Physics}}
* [[Double Bubble conjecture]]
* [[Dirichlet principle]]
* [[Plateau's laws]]
* [[Stretched grid method]]
* [[Bernstein's problem]]

==References==
{{Reflist}}
* {{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
 | jstor = 1989472
 | doi-access = free
 }}
* {{cite journal
 | last = Reifenberg | first = Ernst Robert
 | authorlink = Ernst Robert Reifenberg
 | title = Solution of the {Plateau} problem for m-dimensional surfaces of varying topological type
 | journal = Acta Mathematica
 | volume = 104
 | year = 1960
 | issue = 2
 | pages = 1–92
 | doi = 10.1007/bf02547186
 | doi-access = free
 }}
* {{cite book
  | last = Fomenko | first = A.T. | title = The Plateau Problem: Historical Survey
  | url = https://archive.org/details/plateauproblem0000fome | url-access = registration | publisher = Gordon & Breach | year = 1989 | location = Williston, VT
  | isbn = 978-2-88124-700-2}}
* {{cite book
  | last = Morgan | first = Frank | title = Geometric Measure Theory: a Beginner's Guide
  | publisher = Academic Press | year = 2009
  | isbn = 978-0-12-374444-9}}
*{{springer|author=O'Neil, T.C.|id=G/g130040|title=Geometric Measure Theory}}
* {{cite journal
 | first = Tibor | last = Radó
 | authorlink = Tibor Radó
 | title = On Plateau's problem
 | journal = Ann. of Math. |series =  2
 | volume = 31
 | year = 1930
 | pages = 457–469
 | doi = 10.2307/1968237
 | jstor = 1968237
 | issue = 3
 | bibcode = 1930AnMat..31..457R
 }}
* {{cite book
  | last = Struwe | first = Michael | title = Plateau's Problem and the Calculus of Variations
  | publisher = Princeton University Press | year = 1989 | location = Princeton, NJ
  | isbn = 978-0-691-08510-4
}}
* {{cite book
  | last = Almgren | first = Frederick 
  | authorlink = Frederick Almgren
  | title = Plateau's problem, an invitation to varifold geometry 
  | url = https://archive.org/details/plateausproblemi0000almg | url-access = registration | publisher = Benjamin | year = 1966 | location = New York-Amsterdam
  | isbn =  978-0-821-82747-5 
}}

* {{cite book
| last1 = Harrison | first1 = Jenny
| last2 = Pugh | first2 = Harrison
| title = Open Problems in Mathematics (Plateau's Problem)
| publisher = Springer
| year = 2016
| doi = 10.1007/978-3-319-32162-2
| isbn = 978-3-319-32160-8
| arxiv = 1506.05408
}}

{{PlanetMath attribution|id=4286|title=Plateau's Problem}}

[[Category:Calculus of variations]]
[[Category:Minimal surfaces]]
[[Category:Mathematical problems]]