Page 156 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The Douglas-Courant-Tonelli method 143

0
Integrating this identity and changing the variables, letting (x ,y )= ϕ (x, y),
0
in the right hand side we find (recalling (5.11)) that
h i h i
RR 2 ¯ ¯ ¯ 2 RR 2 2
Ω |u 0 (x ,y )| + u 0 (x ,y ) dx dy = Ω |v x (x, y)| + |v y (x, y)| dxdy
0
0
0
0
0
0 ¯
x
y
h³ ´ i
RR 2 2 ¡ ¢
− |v x | − |v y | λ x − µ +2 hv x ; v y i (λ y + µ ) dxdy + o ( ) .
Ω y x
(5.12)
Step 3.2. We now use Riemann theorem to find a conformal mapping

α : Ω → Ω

which is also a homeomorphism from Ω onto Ω . We can also impose that the
mapping verifies

α (w i )= ϕ (w i )
where w i are the points that enter in the definition of S.
We finally let

v (x, y)= u ◦ α (x, y)= v ◦ ψ ◦ α (x, y)

where u is as in Step 3.1.

Since v ∈ S,we deduce that v ∈ S. Therefore using the conformal invariance
of the Dirichlet integral (see Exercise 5.3.1), we find that
ZZ
1 h 2 ¯ ¯ 2 i

¯
D (v )= |v (x, y)| + v (x, y) ¯ dxdy
x
y
2 Ω
ZZ h i
1 2 ¯ ¯ 2
¯
0
= |u 0 (x ,y )| + u 0 (x ,y ) dx dy 0
0
0
0
0 ¯
x
y
2
Ω
which combined with (5.12) leads to
ZZ
h³ ´ ¡ ¢ i
2 2
D (v )= D (v) − |v x | − |v y | λ x − µ y dxdy
2
Ω
ZZ
− [hv x ; v y i (λ y + µ )] dxdy + o ( ) .
x
Ω

Since v , v ∈ S and v is a minimizer of the Dirichlet integral, we find that
ZZ
h³ ´ i
2 2 ¡ ¢
|v x | − |v y | λ x − µ +2 hv x ; v y i (λ y + µ ) dxdy =0. (5.13)
y x
Ω
Step 3.3. We finally choose in an appropriate way the functions λ, µ ∈
¡ ¢
C ∞ Ω that appeared in the previous steps. We let σ, τ ∈ C ∞ (Ω) be arbitrary,
0