Plateau's problem

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To find the minimal surface with a given boundary

title: "Plateau's problem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["calculus-of-variations", "minimal-surfaces", "mathematical-problems"] description: "To find the minimal surface with a given boundary" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Plateau's_problem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary To find the minimal surface with a given boundary ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/f2/Bulle_caténoïde.png" caption="A soap bubble in the shape of a [[catenoid"] ::

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 \leq n - 2. In the hypersurface case where k = n - 1, singularities occur only for n \geq 8. An example of such singular solution of the Plateau problem is the Simons cone, a cone over S^3 \times S^3 in \mathbb{R}^8 that was first described by Jim Simons and was shown to be an area minimizer by Bombieri, De Giorgi and Giusti. | 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 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. | 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 | 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}}

The axiomatic approach of Jenny Harrison and Harrison Pugh{{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 | 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

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.

References

  • {{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}}

  • {{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

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calculus-of-variationsminimal-surfacesmathematical-problems