Page 226 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 5: Minimal surfaces 213

7.5.2 The Douglas-Courant-Tonelli method
Exercise 5.3.1. We have

w x = v λ λ x + v µ µ ,w y = v λ λ y + v µ µ y
x
and thus
2 2 2 2 2
|w x | = |v λ | λ +2λ x µ hv λ ; v µ i + µ |v µ |
x
x
x
2 2 2 2 2
|w y | = |v λ | λ +2λ y µ hv λ ; v µ i + µ |v µ | .
y y y
Since λ x = µ and λ y = −µ , we deduce that
y
x
h i
2 2 2 2 £ 2 2 ¤
|w x | + |w y | = |v λ | + |v µ | λ + λ y
x
and thus
ZZ ZZ
h i h i
2 2 2 2 £ 2 2 ¤
|w x | + |w y | dxdy = |v λ | + |v µ | λ + λ y dxdy .
x
Ω Ω
Changing variables in the second integral, bearing in mind that
2
2
λ x µ − λ y µ = λ + λ ,
x
x
y
y
we get the result, namely
ZZ ZZ
h i h i
2 2 2 2
|w x | + |w y | dxdy = |v λ | + |v µ | dλdµ .
Ω B
7.5.3 Nonparametric minimal surfaces
Exercise 5.5.1. Set
1+ u 2 x u x u y 1+ u 2 y
f = q ,g = q ,h = q .
2
2
2
1+ u + u 2 y 1+ u + u 2 y 1+ u + u 2 y
x
x
x
A direct computation shows that
f y = g x and g y = h x ,
since
¡ 2 ¢ ¡ 2 ¢
Mu = 1+ u u xx − 2u x u y u xy + 1+ u u yy =0 .
y x
Setting
Z x Z y Z x Z t Z y Z t
ϕ (x, y)= g (s, t) dtds + f (s, 0) dsdt + h (0,s) dsdt
0 0 0 0 0 0