Editing Plateau's problem (section)

==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>