Conformally Invariant Variational Problems. - SAM

following equation
⎧ [ ]
∂ ⎪⎨
ij ∂λ
a = ∂a ∂b
− ∂a ∂b
on D 2
∂x i ∂x j ∂x 1 ∂x 2 ∂x 2 ∂x 1
⎪⎩
ϕ = 0 on ∂D 2 .
(X.161)
Then ϕ ∈ L ∞ ∩W 1,2 (D 2 ,R) and there exists C > 0 independent
of a and b such that
‖ϕ‖ L
∞ +‖∇ϕ‖ L
2 ≤ C ‖∇a‖ L
2 ‖∇b‖ L
2 . (X.162)
Using theorem X.11 we obtain that λ ∈ L ∞ (D 2 ,R). Let e i
be the frame on D 2 given by
✷
d ⃗ Φ·e i =⃗e i
and denote e i = e 1 i ∂ x 1
+e 2 i ∂ x 2
. We have
2∑
e k i g kj = .
k=1
from which we deduce
2∑
e k i = g kj
j=1
Because of (X.145) the maps g kj are in L ∞ . Thus
2∑ [
e ∗ i = g 1j dx1
j=1
+g 2j dx2
]
∈ L ∞ (D 2 ) .
(X.163)
Denote by (f 1 ,f 2 ) the frame on D 2 such that d ⃗ Φ · f i = ⃗ f i , we
have f a st = f ∗ 1 + if∗ 2 = e−iθ (e ∗ 1 + ie∗ 2 ) which is in L∞ (D 2 ) due
to (X.163). Hence the map φ = (φ 1 ,φ 2 ) which is given by
dφ i := e −λ f ∗ i .
172