Conformally Invariant Variational Problems. - SAM

In the simpler case when Ω is divergence-free, we can write Ω
in the form
Ω = ∇ ⊥ ξ ,
for some ξ ∈ W 1,2 (D 2 ,so(m)). In particular, the statement of
theorem VIII.4 is settled by choosing
A ij = δ ij and B ij = ξ ij . (VIII.28)
Accordingly, it seems reasonable in the general case to seek
a solution pair (A,B) which comes as “close” as can be to
(VIII.28). A first approach consists in performing a linear
Hodge decomposition in L 2 of Ω. Hence, for some ξ and
P in W 1,2 , we write
Ω = ∇ ⊥ ξ −∇P .
(VIII.29)
In this case, we see that if A exists, then it must satisfy the
equation
∆A = ∇A·∇ ⊥ ξ −div(A∇P) .
(VIII.30)
This equation is critical in W 1,2 . The first summand ∇A·∇ ⊥ ξ
on the right-hand side of (VIII.30) is a Jacobian. This is a desirable
feature with many very good analytical properties, as
we have previously seen. In particular, using integration by
compensation (Wente’s theorem VII.1), we can devise a bootstrapping
argument beginning in W 1,2 . On the other hand, the
second summand div(A∇P) on the right-hand side of (VIII.30)
displays no particular structure. All which we know about it,
is that A should a-priori belong to W 1,2 . But this space is not
embedded in L ∞ , and so we cannot a priori conclude that A∇P
lies in L 2 , thereby obstructing a successful analysis...
However, not all hope is lost for the antisymmetric structure
of Ω still remains to be used. The idea is to perform
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