Page 155 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

142 Minimal surfaces

¡ ¢
Step 3.1. Let λ, µ ∈ C ∞ Ω ,to be chosen later,and let ∈ R be sufficiently
small so that the map
µ ¶ µ ¶ µ ¶
x 0 ϕ (x, y) x + λ (x, y)
1
= ϕ (x, y)= =
y 0 ϕ (x, y) y + µ (x, y)
2
¡ ¢
is a diffeomorphism from Ω onto a simply connected domain Ω = ϕ Ω .We

denote its inverse by ψ and we find that
µ ¶ µ ¶ µ ¶
x ψ (x ,y ) x − λ (x ,y )+ o ( )
0
0
0
0
0
1
= ψ (x ,y )= =
0
0
0
0
0
0
y ψ (x ,y ) y − µ (x ,y )+ o ( )
0
2
where o (t) stands for a function f = f (t) so that f (t) /t tends to 0 as t tends
to 0. We therefore have

0
0
ϕ (ψ (x ,y )) = (x ,y ) and ψ (ϕ (x, y)) = (x, y)
0
0
moreover the Jacobian is given by
¢
¡

det ∇ϕ (x, y)= 1 + λ x (x, y)+ µ (x, y) + o ( ) . (5.11)
y
We now change the independent variables and write

0
0
0
0
u (x ,y )= v (ψ (x ,y )) .
We find that
∂ ∂

0
0
0
0
0
0
0
0
u 0 (x ,y )= v x (ψ (x ,y )) ψ (x ,y )+ v y (ψ (x ,y )) ψ (x ,y )
0
0
2
1
x
∂x 0 ∂x 0

= v x (ψ ) − [v x (ψ ) λ x (ψ )+ v y (ψ ) µ (ψ )] + o ( )
x
and similarly
¤
£

0
0
u 0 (x ,y )= v y (ψ ) − v x (ψ ) λ y (ψ )+ v y (ψ ) µ (ψ ) + o ( ) .
y
y
This leads to
2
¯
¯

0
|u 0 (x ,y )| + u 0 (x ,y ) ¯ 2
0
0
0 ¯
x
y
2 2
= |v x (ψ )| + |v y (ψ )|
h i

2

2

−2 |v x (ψ )| λ x (ψ )+ |v y (ψ )| µ (ψ )
y

−2 [hv x (ψ ); v y (ψ )i (λ y (ψ )+ µ (ψ ))] + o ( ) .
x