Conformally Invariant Variational Problems. - SAM

The assumption i) implies that, modulo extraction of a subsequence,
⃗n ⃗Φk converges weakly in W 1,2 (D 2 ,∧ m−2 R m ) to a limit
⃗n ∞ ∈ W 1,2 (D 2 ,∧ m−2 R m ). Using againRellich KondrachovcompactnessresultweknowthatthisconvergenceisstronginL
p (D 2 )
for any p < +∞. The almost everywhere convergence of ⃗n ⃗Φk towards
⃗n ⃗Φ∞ implies that
⃗n ∞ = ⃗n ⃗Φ∞
hence the limit is unique and the whole sequence ⃗n ⃗Φk converges
weakly in W 1,2 (D 2 ,∧ m−2 R m ) to ⃗n ⃗Φ∞ . From the lower semicontinuity
of the W 1,2 norm, we deduce in particular that
∫
|∇⃗n ⃗Φ∞ | 2 ≤ 8π/3 . (X.183)
D 2
Hence ⃗ Φ ∞ isaconformalLipschitzimmersionwithL 2 −bounded
second fundamentalform. It is naturalto ask whether the equation
iv) passes to the limit or in other words we are asking the
following question :
Does the weak limit ⃗ Φ ∞ define a weak Willmore immersion in
the sense of definition X.4 ?
Since ∇ ⃗ Φ k converges strongly in L p loc (D2 ) to ∇ ⃗ Φ ∞ for any
p < +∞ and since inf p∈B
2 ρ (0)|∇ ⃗ Φ k |(p) is bounded away from
zero uniformly in k, we have that
2 e −2λ k
= |∇ ⃗ Φ k | −2 −→ |∇ ⃗ Φ ∞ | −2 = 2 e −2λ ∞
stronlgy in L p loc (D2 ) ∀p < +∞
Since ∆ ⃗ Φ k ⇀ ∆ ⃗ Φ ∞ weakly in L 2 loc (D2 ) we deduce that
⃗H k = e−2λ k
2 ∆⃗ Φ k −→ e−2λ ∞
2
∆ ⃗ Φ ∞ in D ′ (D 2 )
Since | H ⃗ k | 2 e 2λ k
≤ 2 −1 |∇⃗n k | 2 , because of the assumption i) H ⃗ k
is uniformly bounded w.r.t. k in L 2 (Bρ 2 (0)) for any ρ < 1. We
183