Conformally Invariant Variational Problems. - SAM

We are now in positionto start the Chern movingframemethod
in order to produce a conformal parametrization of ⃗ Φ on this
disc. This however has to be done with the additional difficulty
of keeping track of the regularity of the different actors at each
step of the construction.
First we introduce the function λ ∈ W 1,2 satisfying (X.133).
The second equation of (X.159) implies that the restriction to
the boundary of D 2 of the one form dλ is equal to zero. Hence
this last fact combined with the second equation of (X.133) implies
that λ is identically equal to zero on ∂D 2 .
We have then
⎧
⎨ d∗ g dλ = − < d⃗e 1 ,d⃗e 2 > on D 2
(X.160)
⎩
λ = 0 on ∂D 2
which reads in the canonical coordinates of D 2
⎧ [ ]
∂ g
ij
∂λ ⎪⎨ √ =< ∂⃗e 1
, ∂⃗e 2
> − < ∂⃗e 1
, ∂⃗e 2
> on D 2
∂x i detg ∂x j ∂x 1 ∂x 2 ∂x 2 ∂x 1
⎪⎩
λ = 0 on ∂D 2
where we are using an implicit summation in i and j and where
g ij arethecoefficienttotheinversematrixtog ij :=< ∂ xi
⃗ Φ,∂xj ⃗ Φ >.
We are now in position to make use of the following generalization
of Wente’s theorem due to S.Chanillo and Y.Y. Li [ChLi].
Theorem X.11. Let a and b be two functions in W 1,2 (D 2 ,R).
Let (a ij ) 1≤i,j≤2 be a2×2symmetricmatrixvaluedmapinL ∞ (D 2 )
such that there exists C > 0 for which
∀ξ = (ξ 1 ,ξ 2 ) ∈ R 2 ∀x ∈ D 2 C −1 |ξ| 2 ≤ a ij (x)ξ i ξ j ≤ C |ξ| 2
Let ϕ be the solution in W 1,p (D 2 ,R) for any 1 ≤ p < 2 of the
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