Conformally Invariant Variational Problems. - SAM

can then deduce from the previous facts that
⃗H k ⇀ ⃗ H ⃗Φ∞ weakly in L 2 loc(D 2 ) . (X.184)
At this preliminary stage of our analysis of the passage to the
limit inside the Willmore equation in conservative form (X.86)
it is not possible to identify the limits of the bilinearities such
as
∇ ⊥ ⃗n k ∧ ⃗ H k −→ ?
Indeed both ∇ ⊥ ⃗n k and H ⃗ k converge weakly in L 2 loc but, because
oftheseweakconvergences,onecannota-prioriidentifythelimit
of the product ∇ ⊥ ⃗n k ∧ H ⃗ k as being the product of the limit
∇ ⊥ ⃗n ⃗Φ∞ ∧H ⃗ ⃗Φ∞ .
Before any more advanced study of the passage to the limit
the best one can deduce at this stage is the existence of a locally
Radon measure vector fields taking values in R m : µ = µ 1 ∂ x1 +
µ 2 ∂ x2 on D 2 such that 71
[
div ∇H ⃗ ∞ −3π ⃗n∞ (∇H ⃗ ∞ )+⋆(∇ ⊥ ⃗n ∞ ∧H ⃗ ]
∞ ) = divµ (X.185)
where ⃗ H ∞ and ⃗n ∞ stand for ⃗ H ⃗Φ∞ and ⃗n ⃗Φ∞ . In order to understand
the possible limits of Willmore discs with small L 2 −norm
ofthesecondfundamentalform,itremainstoidentifythenature
of µ. This is the purpose of the following 2 subsections.
X.7.2 Conservation laws for Willmore surfaces.
As we saw in the first part of the course critical non-linear elliptic
systems and equations do not always pass to the limit.
For the systems issued from conformally invariant Lagrangians
we discovered conservation laws which were the key objects for
passing in the limit in these systems. We shall here also, for
Willmore surfaces, find divergence free quantities that will help
us to describe the passage to the limit in Willmore equation.
71 Each µ 1 and µ 2 are R m -valued Radon measures on D 2 .
184