Page 152 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The Douglas-Courant-Tonelli method 139

(ii) Consider the problem (where α> 0)

(P α ) inf {I (w): w (0) = w (1) = α} .
2
Prove that any C ([0, 1]) minimizer is necessarily of the form
2x − 1 1
w (x)= a cosh with a cosh = α.
2a 2a
Discuss the existence of such solutions, as function of α.
Exercise 5.2.4 Prove the first part of Theorem 5.12.

5.3 The Douglas-Courant-Tonelli method

We now present the main ideas of the method of Douglas, as modified by Courant
3
and Tonelli, for solving Plateau problem in R . For a complete proof, we refer
to Courant [24], Dierkes-Hildebrandt-Küster-Wohlrab [39], Nitsche [78] or for a
slightly different approach to a recent article of Hildebrandt-Von der Mosel [59].
© 2 2 2 ª 3
Let Ω = (x, y) ∈ R : x + y < 1 and Γ ⊂ R be a rectifiable (i.e., of finite
length) Jordan curve. Let w i ∈ ∂Ω (w i 6= w j ) and p i ∈ Γ (p i 6= p j ) i =1, 2, 3 be
fixed. The set of admissible surfaces will then be
⎧ ¡ ¢ 3 ⎫
⎪ Σ = v Ω where v : Ω → Σ ⊂ R so that ⎪
⎪ ¡ ¢ ¡ ¢ ¡ ¢ ⎪
⎨ (S1) v ∈ M Ω = C 0 Ω; R 3 ∩ W 1,2 Ω; R 3 ⎬
S = .
⎪ (S2) v : ∂Ω → Γ is weakly monotonic and onto ⎪
⎪ ⎪
(S3) v (w i )= p i ,i =1, 2, 3
⎩ ⎭
Remark 5.18 (i) The set of admissible surfaces is then the set of parametric
¡ ¢
surfaces of the type of the disk with parametrization in M Ω . The condition
weakly monotonic in (S2) means that we allow the map v to be constant on some
parts of ∂Ω;thus v is not necessarily a homeomorphism of ∂Ω onto Γ. However
the minimizer of the theorem will have the property to map the boundary ∂Ω
topologically onto the Jordan curve Γ. The condition (S3) may appear a little
strange, it will help us to get compactness (see the proof below).
(ii) A first natural question is to ask if S is non empty. If the Jordan curve
Γ is rectifiable then S 6= ∅ (see for more details Dierkes-Hildebrandt-Küster-
Wohlrab [39] pages 232-234 and Nitsche [78], pages 253-257).
(iii) Recall from the preceding section that for Σ ∈ S we have
ZZ
Area (Σ)= J (v)= |v x × v y | dxdy .
Ω
The main result of this chapter is then