Conformally Invariant Variational Problems. - SAM

Using Rellich Kondrachov theorem 70 we deduce that
⃗Φ k −→ ⃗ Φ ∞ strongly in W 1,p
loc (D2 ,R m ) ∀p < +∞ .
Because of the strong convergence of the gradient of ⃗ Φ k in L p
for any p < +∞,
∇ ⃗ Φ k
converges almost everywhere towards ∇ ⃗ Φ ∞
and then we can pass to the limit in the conformalityconditions
⎧
⎨ |∂ x1Φk ⃗ | = |∂ x2Φk ⃗ | = e λ k
,
⎩
in order to deduce
⎧
⎨
⎩
< ∂ x1
⃗ Φk ,∂ x2
⃗ Φk >= 0
|∂ x1
⃗ Φ∞ | = |∂ x2
⃗ Φ∞ | = e λ ∞
,
< ∂ x1
⃗ Φ∞ ,∂ x2
⃗ Φ∞ >= 0
The passage to the limit in the condition iii) gives
λ ∞ = log|∂ x1
⃗ Φ∞ | = log|∂ x1
⃗ Φ∞ | ∈ L ∞ loc (D2 ) .
(X.182)
Hence ⃗ Φ ∞ realizes a conformal lipschitz immersion of the disc
D 2 .
Because of the pointwise convergence of ∇ ⃗ Φ k towards ∇ ⃗ Φ ∞
we have that
∂ x1
⃗ Φk ∧∂ x2
⃗ Φk −→ ∂ x1
⃗ Φ∞ ∧∂ x2
⃗ Φ∞
almost everywhere,
and, because of iii)|∂ x1
⃗ Φk ∧∂ x2
⃗ Φk | = e 2λ k
is bounded from below
by a positive constant on each compact set included in the open
disc D 2 . Therefore
⃗n ⃗Φk = e −2λ k
∂ x1
⃗ Φk ∧∂ x2
⃗ Φk −→ e −2λ ∞
∂ x1
⃗ Φ∞ ∧∂ x2
⃗ Φ∞ = ⃗n ⃗Φ∞ a. e.
70 See Chapter 6 of [AdFo].
182