Conformally Invariant Variational Problems. - SAM

quantity
div
[
2∇H ⃗ −3H∇⃗n−∇ ⊥ ⃗n× H ⃗ ]
is an ”honest” distribution in D ′ (D 2 ) whereas, under such minimal
assumption
∆ g H +2H (H 2 −K)
has no distributional meaning at all. This is why the conservative
form of the Willmore surface equation is more suitable
to solve the analysisquestions i)···vi) we are asking. The same
happens in higher codimension as well. Using (X.88), one sees
that, under the assumption that ∇⃗n ∈ L 2 one has
∇ ⃗ H −3π ⃗n (∇ ⃗ H)+⋆(∇ ⊥ ⃗n∧ ⃗ H) ∈ H −1 +L 1
which is again an honest distribution.
The second equation in (X.89) is in conservative-elliptic
form which is critical in 2 dimension under the assumption
that⃗n ∈ W 1,2 . For a sake of claritywe present it in codimension
1 though this holds identically in arbitrary codimension.
In codimension 1 we write the Willmore surface equation as
follows
∆H ⃗ 3
= div[
2 H ∇⃗n+ 1 ]
2 ∇⊥ ⃗n× H ⃗ (X.90)
the right-hand-side of (X.90) is the flat Laplacian of H ⃗ and the
left-hand-side is the divergence of a R m vector-field wich is a
bilinear map of the second fundamental form. Assuming hence
⃗n ∈ W 1,2 wededuce 3/2H∇⃗n+1/2∇ ⊥ ⃗n× H ⃗ ∈ L 1 (D 2 ). Adams
result on Riesz potentials [Ad] implies that
[ 3
∆ −1
0 div 2 H∇⃗n+ 1 ]
2 ∇⊥ ⃗n× H ⃗ ∈ L 2,∞
where ∆ −1
0 is the Poisson Kernel on the disc D 2 . Inserting this
information back in (X.90) we obtain H ⃗ ∈ L 2,∞
loc (D2 ) which is
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