Plateau problem

In mathematics , the plateau problem is to find a minimal surface that has a given curve as an edge . It is named after Joseph Plateau , who experimentally determined the shapes of soap skins in wire frames. The problem was first formulated mathematically in 1760 by Joseph-Louis Lagrange . It belongs to the field of the calculus of variations .

In a more general sense, one understands by it a whole complex of problems, which are of the following form: one finds an element from a given set of "surfaces" which meet certain boundary conditions and which minimize a given "surface" function or a critical point this function are. In addition, the solutions should meet certain regularity conditions. Since its formulation in the 19th century, the plateau problem has given rise to too much research work and new developments in mathematics and, in its various generalizations, also poses open problems, for example with minimal areas . ${\displaystyle E}$ ${\displaystyle f\colon E\to \mathbb {R} }$

the solution of the problem

In the course of time various special forms of the problem were solved, for example by Schwarz in 1865. In 1928 René Garnier solved the plateau problem by solving a Riemann-Hilbert problem for polygonal edge curves. An approximation process then solves the plateau problem for continuous boundary curves. However, it was not until the early 1930s that Jesse Douglas and Tibor Radó , independently of one another, were able to prove the existence of a solution to the problem using the direct methods of the calculus of variations (cf. as an example the solution to the Dirichlet principle ). Douglas (who received the first Fields Medal for the solution ) originally only solved the problem for surfaces in three-dimensional Euclidean space (with a Jordan curve as the edge) that topologically correspond to a disk (genus 0). Douglas and Richard Courant generalized the solution to any topological gender and several disjoint curves as boundaries. While Douglas and Rado minimized a kind of energy functional, Herbert Federer and Wendell Fleming came up with a solution based on geometric dimension theory in 1960. In 1961 Ernst Robert Reifenberg gave a solution for any gender using novel methods.

Charles Morrey considered the generalized problem on surfaces in general Riemannian manifolds. Frederick Almgren investigated a variant of the problem in which the surfaces sought are better adapted to physical soap bubbles , followed up by Jean Taylor and Jenny Harrison , among others .

Regular solutions do not always exist in more than three dimensions and for hypersurfaces other than dimensions ( Ennio de Giorgi and others from 1961). In this case , however, singular solutions only appear in . ${\displaystyle k}$ ${\displaystyle k\geq n-1}$ ${\displaystyle k\geq n-1}$ ${\displaystyle n\geq 8}$

Parametric formulation of the problem

Let us assume a Jordan curve with three fixed points. We are looking for an image on the end of the open circular disk with the property with the edge of The following properties are required for the image : ${\displaystyle \Gamma \subset \mathbb {R} ^{3}}$ ${\displaystyle X_{1},X_{2},X_{3}\in \Gamma .}$ ${\displaystyle X\colon {\overline {B}}\to \mathbb {R} ^{3}}$ ${\displaystyle B=\{(u,v)\in B\,:\,u^{2}+v^{2}<1\}}$ ${\displaystyle X(\partial B)=\Gamma }$ ${\displaystyle \partial B=\{(u,v)\in \mathbb {R} ^{2}\,:\,u^{2}+v^{2}=1\}}$ ${\displaystyle B.}$ ${\displaystyle X}$

• Harmonicity: in ${\displaystyle \triangle X(u,v)=0}$ ${\displaystyle B}$
• Conformity: as well as in ${\displaystyle |X_{u}(u,v)|^{2}=|X_{v}(u,v)|^{2}}$ ${\displaystyle X_{u}\cdot X_{v}=0}$ ${\displaystyle B}$
• Topological boundary condition: homeomorphism on ${\displaystyle X\colon \partial B\to \Gamma }$ ${\displaystyle \Gamma }$
• 3-point condition: for ${\displaystyle X(e^{2\pi ik/3})=X_{k}}$ ${\displaystyle k=1,2,3.}$

Extended problem in higher dimensions

The expansion of the problem to higher dimensions, that is to say to k -dimensional surfaces in n -dimensional space, on the other hand, turns out to be much more difficult. In particular, solutions to the general problem are not necessarily regular, but can have singularities . This always applies to , but also to the case of a hypersurface , i.e. if . ${\displaystyle k\leq n-2}$ ${\displaystyle k=n-1}$ ${\displaystyle n\geq 8}$

literature

• Jenny Harrison , Harrison Pugh: Plateau's problem, in: John Forbes Nash jr., Michael Th. Rassias (eds.), Open problems in mathematics, Springer 2016, pp. 273-302

Original works:

• Jesse Douglas : Solution of the Problem of Plateau. In: Transactions of the American Mathematical Society. 33, 1, 1931, ISSN  0002-9947 , pp. 263-321, doi : 10.2307 / 1989472 .
• Tibor Radó : On Plateau's problem. In: The Annals of Mathematics. 31, 3, 1930, ISSN  0003-486X , pp. 457-469, doi : 10.2307 / 1968237 .

• AT Fomenko : The Plateau Problem. A Historical Survey , Gordon and Breach 1989
• Michael Struwe : Plateau's Problem and the Calculus of Variations , Princeton, NJ: Princeton University Press 1989

Web links

• Eric W. Weisstein : Plateau's Problem . In: MathWorld (English).
• Brian White's webpage
• Springer Online Reference Works

Individual evidence

1. ^ Douglas Solutions of the problem of Plateau , Transactions AMS, 33, 1941, 263-321
2. ^ Rado The problem of least area and the problem of Plateau , Mathematische Zeitschrift Vol. 32, 1930, p. 763, Rado On the problem of Plateau , Springer Verlag 1933
3. ↑ shown in Courant Dirichlet's principle conformal mapping and minimal surfaces , Interscience 1950
4. ↑ Federer, Fleming Normal and integral currents , Annals of Mathematics, 72, 1960, 458-520
5. ↑ Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces of varying topological type, Acta Mathematica, 80, 1960, No. 2, 1-14
6. ^ Morrey The problem of Plateau on a Riemannian manifold , Annals of Mathematics, Vol. 49, 1948, p. 807