Page 159 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

146 Minimal surfaces

Theorem 5.24 Let Γ be an analytical Jordan curve with total curvature not
exceeding 4π, then Plateau problem has a unique solution.
We now give a non uniqueness result (for more details we refer to Dierkes-
Hildebrandt-Küster-Wohlrab [39] and to Nitsche [78]).
¡ √ ¢
Example 5.25 (Enneper surface, see Example 5.10). Let r ∈ 1, 3 and
½µ ¶ ¾
1 3 1 3 2
Γ r = r cos θ − r cos 3θ, −r sin θ − r sin 3θ, r cos 2θ : θ ∈ [0, 2π) .
3 3
We have seen (Example 5.10) that

½µ 3 3 ¶ ¾
r 3 3 2 r 3 2 ¡ 2 2 ¢ 2 2
2
3
Σ r = rx + r xy − x , −ry − r x y + y ,r x − y : x + y ≤ 1
3 3
is a minimal surface and that ∂Σ r = Γ r . It is possible to show (cf. Nitsche [78],
¡ √ ¢
page 338) that Σ r is not of minimal area if r ∈ 1, 3 ; therefore it is distinct
from the one found in Theorem 5.19.
5.5 Nonparametric minimal surfaces

n
We now discuss the case of nonparametric surfaces. Let Ω ⊂ R be a bounded
domain (in the present section we do not need to limit ourselves to the case
n =2). The surfaces that we will consider will be of the form
© ª
Σ = (x, u (x)) = (x 1 , ..., x n ,u (x 1 , ..., x n )) : x ∈ Ω .
The area of such a surface is given by
Z q
2
I (u)= 1+ |∇u (x)| dx .
Ω
¡ ¢
As already seen in Theorem 5.12, we have that any C 2 Ω minimizer of
(P) inf {I (u): u = u 0 on ∂Ω}

satisfies the minimal surface equation
n
³ ´ X
2
(E) Mu ≡ 1+ |∇u| ∆u − u x i x j =0
u u x ix j
i,j=1
⎡ ⎤
n
X ∂ u x i
⇔ ⎣ q ⎦ =0
∂x i 2
i=1 1+ |∇u|