Calculus of variations and optimal control; optimization Geometric measure theory and optimization on manifolds Theorem

Douglas–Radó solution of Plateau's problem

Builds on Plateau's problem Harmonic function Conformal map Minimal surface Sobolev space Rectifiable curve

Statement

Let $n \ge 2$ and let $\Gamma \subseteq \R^n$ be a rectifiable Jordan curve, that is, a curve of finite length homeomorphic to a circle. Let $D$ be the open unit disc in $\R^2$ , with coordinates $(u, v)$ , and let $\overline{D}$ be its closure. Let $\mathcal{C}(\Gamma)$ be the set of continuous maps $Y \colon \overline{D} \to \R^n$ in the Sobolev space $W^{1,2}(D; \R^n)$ . The restriction of each such $Y$ to $\partial D$ must be a monotone parametrization of $\Gamma$ . Then there is a map $X \in \mathcal{C}(\Gamma)$ such that:

$X$ restricted to $\partial D$ is a homeomorphism onto $\Gamma$ ;
$X$ is harmonic in $D$ ;
$X$ is weakly conformal in $D$ : $\abs{X_u}^2 = \abs{X_v}^2$ and $X_u \cdot X_v = 0$ ;
the area $\int_D \abs{X_u \wedge X_v} \, du \, dv$ of $X$ is the least area of all maps in $\mathcal{C}(\Gamma)$ .

Remarks

Jesse Douglas and Tibor Radó found the theorem independently, around 1930. The idea is to minimize the Dirichlet energy $\frac12 \int_D (\abs{Y_u}^2 + \abs{Y_v}^2)$ instead of the area. The energy is easier to control, and it is at least the area, with equality for conformal maps. The energy is invariant under the conformal maps of the disc, so one fixes the images of three boundary points. Then the Courant–Lebesgue lemma makes a minimizing sequence equicontinuous on the boundary, and a minimizer exists. A minimizer of the energy is harmonic, and varying the parametrization of the disc shows that it is weakly conformal, so it also minimizes area. Points where $X_u = 0$ are isolated branch points, and away from them the image is a minimal surface. For example, if $\Gamma$ is a planar circle, the solution parametrizes the flat disc that it bounds. The theorem controls only disc-type surfaces: a curve may bound a surface of another topological type with smaller area.

Going deeper

For a student

The theorem answers Plateau's problem for disc-type surfaces, and it is a model of the direct method of the calculus of variations. The trick of replacing area by the Dirichlet energy reappears in the theory of harmonic maps and of minimal surfaces in Riemannian manifolds. Later work studied the branch points: for area-minimizing solutions in $\R^3$ there are no interior branch points. Geometric measure theory treats the problem in other dimensions and without fixing the topology, with currents and varifolds.

Reading path

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Conformal map concept
Euclidean space object
Function concept
Minimal surface concept
Sobolev space concept
Total derivative concept
Function of several variables concept
Vector-valued function concept
Differentiable function of several variables concept
Partial derivative concept
Parametrised curve concept
Vector field concept
Gradient concept
Divergence concept
Rectifiable curve concept
Laplacian concept
Harmonic function concept
Plateau's problem problem
Douglas–Radó solution of Plateau's problem statement

Tag	`KM8Q5HZ48G`
URI	`m2://KM8Q5HZ48G`
Kind	statement, theorem
Writing	W0 stub
MSC	49Q05, 53A10, 58E12
Status	This copy of the site has no build data. Run `m2 status KM8Q5HZ48G`.
About	Plateau's problem
Uses	Plateau's problem, Harmonic function, Conformal map, Minimal surface, Sobolev space, Rectifiable curve