Page 162 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Nonparametric minimal surfaces 149

From
n
X
¡ ¢
N (u) − N (v)= u x i x j u x i x j − v x i x j
u
i,j=1
n
X ¡ ¢
v
u
+ v x i x j u x i x j − v x i x j
i,j=1
³ ´
2 2 2
+ |∇u| (∆v − ∆u)+ ∆v |∇v| − |∇u|
we deduce that (5.17) holds from Proposition 1.10 and its proof (cf. Exercise
1.2.1).
Step 3. We are now in a position to show the theorem. We define a sequence
¡ ¢
∞ 2,α
{u ν } of C Ω functions in the following way
ν=1
⎧
⎨ ∆u 1 =0 in Ω
(5.18)
⎩
u 1 = u 0 on ∂Ω
and by induction
⎧
⎨ ∆u ν+1 = N (u ν ) in Ω
(5.19)
⎩
u ν+1 = u 0 on ∂Ω .
The previous estimates will allow us to deduce that for ku 0 k C 2,α ≤ , to be
determined, we have

ku ν+1 − u ν k C 2,α ≤ K ku ν − u ν−1 k C 2,α (5.20)
for some K< 1. Banach fixedpoint theoremwillthenimply that u ν → u in
C 2,α and hence ⎧
⎨ ∆u = N (u) in Ω

⎩
u = u 0 on ∂Ω
which is the claimed result.
We now establish (5.20), which amounts to find the appropriate > 0.We
start by choosing 0 <K < 1 andwethenchoose > 0 sufficiently small so that
µ 4 2 ¶
C √
2
2C 1+ ≤ K (5.21)
1 − K
where C is the constant appearing in Step 1 and Step 2 (we can consider, without
loss of generality, that they are the same).