Page 148 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Generalities about surfaces 135

2
Theorem 5.12 Let Ω ⊂ R be a bounded Lipschitz domain.
¡ ¢ 2 ¡ 3 ¢
Part 1. Let Σ 0 = v Ω where v ∈ C Ω; R , v = v (x, y),with v x × v y 6=0
in Ω.If
Area (Σ 0 ) ≤ Area (Σ)

2
among all regular surfaces Σ of class C with ∂Σ = ∂Σ 0 ,then Σ 0 is a minimal
surface.

Part 2. Let S Ω be the set of nonparametric surfaces of the form Σ u =
© ª 2 ¡ ¢
(x, y, u (x, y)) : (x, y) ∈ Ω with u ∈ C Ω and let Σ u ∈ S Ω . The two fol-
lowing assertions are then equivalent.

(i) Σ u is a minimal surface, which means
¡ 2 ¢ ¡ 2 ¢
Mu = 1+ u y u xx − 2u x u y u xy + 1+ u x u yy =0 .
(ii) For every Σ u ∈ S Ω with u = u on ∂Ω

ZZ
q
2
Area (Σ u ) ≤ Area (Σ u )= I (u)= 1+ u + u dxdy .
2
x y
Ω
Moreover, Σ u is, among all surfaces of S Ω with u = u on ∂Ω, the only one to
have this property.
Remark 5.13 (i) The converse of Part 1, namely that if Σ 0 is a minimal sur-
face then it is of minimal area, is, in general, false. The claim of Part 2 is that
the converse is true when we restrict our attention to nonparametric surfaces.
(ii) This theorem is easily extended to R n+1 , n ≥ 2.

Proof. We will only prove Part 2 of the theorem and we refer to Exercise
5.2.4 for Part 1. Let

v (x, y)= (x, y, u (x, y)) , (x, y) ∈ Ω

we then have
ZZ ZZ
q
2
2
J (v)= |v x × v y | dxdy = 1+ u + u dxdy ≡ I (u) .
x y
Ω Ω
(ii) ⇒ (i). We write the associated Euler-Lagrange equation. Since u is a
minimizer we have
I (u) ≤ I (u + ϕ) , ∀ϕ ∈ C ∞ (Ω) , ∀ ∈ R
0