Page 164 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Nonparametric minimal surfaces 151

We deduce from (5.22) and (5.24) (since (5.25) holds for every 1 ≤ j ≤ ν)that

ku j+1 − u j k ≤ K ku j − u j−1 k ≤ K j−1 ku 2 − u 1 k .

Returning to (5.27), we find
ν
X j−1 1
ku ν+1 k ≤ ku 1 k + ku 2 − u 1 k K ≤ ku 1 k + ku 2 − u 1 k .
1 − K
j=1
Appealing to (5.23) and then to (5.26), we obtain
à !
2 2 µ 4 2 ¶
C ku 1 k C
ku ν+1 k ≤ ku 1 k 1+ ≤ C 1+ ,
1 − K 1 − K
which is exactly (5.25). The theorem then follows.

5.5.1 Exercises

¡ ¢
2
Exercise 5.5.1 Let u ∈ C 2 R , u = u (x, y), be a solution of the minimal
surface equation
¡ 2 ¢ ¡ 2 ¢
Mu = 1+ u u xx − 2u x u y u xy + 1+ u u yy =0 .
y x
¡ ¢
2
Show that there exists a convex function ϕ ∈ C 2 R , so that
1+ u 2 x u x u y 1+ u 2 y
ϕ xx = q ,ϕ xy = q ,ϕ yy = q .
2
2
2
1+ u + u 2 1+ u + u 2 1+ u + u 2
x y x y x y
Deduce that
ϕ ϕ − ϕ 2 xy =1 .
xx yy