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2013-07-23T16:06:09Z

130.164.77.95: /* Harmonic distortion */ Changed sentence from "distortion" to "harmonic distortion"

{{for|Bernstein's problem in mathematical genetics|Genetic algebra}}
{{for|Bernstein's Degrees-of-Freedom problem in motor control|Degrees of Freedom Problem (Motor Control)}}
In [[differential geometry]], '''Bernstein's problem''' is as follows: if the graph of a function on '''R'''''n''−1 is a [[minimal surface]] in '''R'''''n'', does this imply that the function is linear?
This is true in dimensions ''n'' at most 8, but false in dimensions ''n'' at least 9. The problem is named for [[Sergei Natanovich Bernstein]] who solved the case ''n'' = 3 in 1914.

==Statement==

Suppose that ''f'' is a function of ''n'' − 1 real variables. The graph of ''f'' is a surface in '''R'''''n'', and the condition that this is a minimal surface is that ''f'' satisfies the minimal surface equation

:<math>\sum_{i=1}^{n-1} \frac{\partial}{\partial x_i}\frac{\frac{\partial f}{\partial x_i}}{\sqrt{1+\sum_{j=1}^{n-1}(\frac{\partial f}{\partial x_j})^2}} = 0</math>

Bernstein's problem asks whether an ''entire'' function (a function defined throughout '''R'''''n''−1 ) that solves this equation is necessarily a degree-1 polynomial.

==History==

{{harvtxt|Bernstein|1915–1917}} proved Bernstein's theorem that a graph of a real function on '''R'''2 that is also a minimal surface in '''R'''3 must be a plane.

{{harvtxt|Fleming|1962}} gave a new proof of Bernstein's theorem by deducing it from the fact that there is no non-planar area-minimizing cone in '''R'''3.

{{harvtxt|De Giorgi|1965}} showed that if there is no non-planar area-minimizing cone in '''R'''''n''−1 then the analogue of Bernstein's theorem is true in '''R'''''n'', which in particular implies that it is true in '''R'''4.

{{harvtxt|Almgren|1966}} showed there are no non-planar minimizing cones in '''R'''4, thus extending Bernstein's theorem to '''R'''5.

{{harvtxt|Simons|1968}} showed there are no non-planar minimizing cones in '''R'''7, thus extending Bernstein's theorem to '''R'''8. He also gave examples of locally stable cones in '''R'''8 and asked if they were globally area-minimizing.

{{harvtxt|Bombieri|De Giorgi|Giusti|1969}} showed that Simon's cones are indeed globally minimizing, and showed that in '''R'''''n'' for ''n''≥9 there are graphs that are minimal but not hyperplanes. Combined with the result of Simons, this shows that the analogue of Bernstein's theorem is true in dimensions up to 8, and false in higher dimensions.

==References==
*{{Citation | last1=Almgren | first1=F. J. | title=Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem | jstor=1970520 | mr=0200816 | year=1966 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=84 | pages=277–292}}
*{{Citation | last1=Bernstein | first1=S.N. | title=Sur une théorème de géometrie et ses applications aux équations dérivées partielles du type elliptique | year=1915–1917 | journal=Comm. Soc. Math. Kharkov | volume=15 | pages=38–45}} German translation in {{Citation | last1=Bernstein | first1=Serge | title=Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus | doi=10.1007/BF01475472 | publisher=Springer Berlin / Heidelberg | language=German | year=1927 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=26 | pages=551–558}}
*{{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | last2=De Giorgi | first2=Ennio | last3=Giusti | first3=E. | title=Minimal cones and the Bernstein problem | doi=10.1007/BF01404309 | mr=0250205 | year=1969 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=7 | pages=243–268}}
*{{Citation | last1=De Giorgi | first1=Ennio | title=Una estensione del teorema di Bernstein | url=http://www.numdam.org/item?id=ASNSP_1965_3_19_1_79_0 | mr=0178385 | year=1965 | journal=Ann. Scuola Norm. Sup. Pisa (3) | volume=19 | pages=79–85}}
*{{Citation | last1=Fleming | first1=Wendell H. | title=On the oriented Plateau problem | doi=10.1007/BF02849427 | mr=0157263 | year=1962 | journal=[[Rendiconti del Circolo Matematico di Palermo]]. Serie II | issn=0009-725X | volume=11 | pages=69–90}}
*{{eom|id=b/b015750|title=Bernstein theorem|first=I.Kh. |last=Sabitov}}
*{{Citation | last1=Simons | first1=James | author1-link=James Harris Simons | title=Minimal varieties in riemannian manifolds | jstor=1970556 | mr=0233295 | year=1968 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=88 | pages=62–105}}
*{{eom|id=b/b110360|title=Bernstein problem in differential geometry|first=E. |last=Straume}}

==External links==
* [http://www.encyclopediaofmath.org/index.php/Bernstein_theorem Encyclopaedia of Mathematics article on the Bernstein theorem]

[[Category:Differential geometry]]

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