# Plateau's Problem and the Calculus of Variations. (MN-35):

Michael Struwe
Pages: 158
https://www.jstor.org/stable/j.ctt7zv371

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-2)
Michael Struwe
4. ### A. The “classical” Plateau problem for disctype minimal surfaces.

• I. Existence of a solution.
(pp. 5-32)

Let Г be a Jordan curve inIRn. The “classical” problem of Plateau asks for a disc-type surfaceXof least area spanning Г. Necessarily, such a surface must have mean curvature 0. If we introduce isothermal coordinates onX(assuming that such a surface exists) we may parametrizeXby a function X(w) = (X¹(w),..., Xn(w)) over the disc

$B = \left\{ {w = (u,v) \in I{R^2}|{u^2} + {v^2} < 1} \right\}$

satisfying the following system of nonlinear differential equations

(1.1)$\Delta X = 0$inB,

(1.2)$|{X_u}{|^2} - |{X_v}{|^2} = 0 = {X_u} \cdot {X_v}$inB,

(1.3)$X{|_{\partial B}}:\partial B \to \Gamma$is an (oriented) parametrization of Г.

Here and in the following${X_u} = \frac{\partial }{{\partial u}}X$, etc., and$\cdot$denotes the scalar product in Euclidian...

• II. Unstable minimal surfaces
(pp. 33-88)

The method of gradient line deformations and the minimax-principle are the most general avaible tools for obtaining unstable critical points in the calculus of variations. Historically, the use of these methods can be traced back to the beginning of this century, cf. Birkhoff’s [1] theorem on the existence of closed geodesies on surfaces of genus 0 . Through their famous improvement of BirkhofF’s result the names of Ljusternik and Schnirelman [1] became intimately attached to these methods. In 1964 a major extension of these techniques was proposed by Palais [1], [2], Smale [1] and Palais - Smale [1]. Their fundamental...

5. ### B. Surfaces of prescribed constant mean curvature.

• III. The existence of surfaces of prescribed constant mean curvature spanning a Jordan curve in IR³.
(pp. 91-110)

Let Г be a Jordan curve inIR³. In part A we studied minimal surfaces spanned by Г, find we observed that any solutionXto the parametric Plateau problem (1.1.1) - (1.1.3) parametrizes a surface of vanishing mean curvature (away from branch points where$\nabla X(w) = 0$

A natural generalization of the classical Plateau problem therefore is the following question: Given$\Gamma \subset I{R^3}$,$H \in IR$, is there a surfaceXwith mean curvatureH(for short “H-surface”) spanning Г?

We restrict ourselves to surfaces of the type of the discB. Introducing isothermal coordinates overBon such aXwe derive the...

• IV. Unstable H – surfaces
(pp. 111-140)

In the analysis of unstable minimal surfaces we relied on the existence of harmonic extensions of admissible parametrizations of Г in order to reformulate the Plateau problem in terms of a variational problem on a convex set. To imitate this procedure forH—surfaces we now considerDirichlet’s problem for the H—surface system:

Given${X_o} \in {H^{1,2}} \cap {L^\infty }(B;I{R^3})$,$H \in IR$find$X \in {H^{1,2}} \cap {C^2}(B;I{R^3})$such that

(1.1)$\Delta X = 2H{X_u} \wedge {X_v}$inB,

(1.2)$X = {X_o}$on$\partial B$

or, equivalently, find$X \in {X_o} + H_0^{1,2}(B;I{R^3})$such that

$d{D_H}(X) = 0 \in (H_0^{1,2}(B;I{R^3})) *$

Recall that forH= 0 the harmonic extension$\underline X$of${X_o} \in {H^{1,2}} \cap {L^\infty }(B;I{R^3})$is uniquely characterized by the relations

(1.3)$\underline X \in {X_o} + H_0^{1,2}(B;I{R^3})$,

(1.4)$D(\underline X ) = \inf \{ D(X)|X \in {X_o} + H_0^{1,2}(B;I{R^3})\}$

Moreover,X...

6. References:
(pp. 141-148)