Conformally Invariant Variational Problems. - SAM

yields the existence of the solution (Ã,B) of the system
⎧
∆à = ∇÷∇⊥ ξ +∇ ⊥ B ·∇P
⎪⎨
∆B = −div(Ã∇ξ P−1 )+∇ ⊥ ÷∇P −1
∂Ã
= 0 and B = 0 on ∂D2
∂ν
∫
⎪⎩ Ã = π 2 Id m
D 2
(VIII.41)
By construction, this solution satisfies the estimate (VIII.26)
with A = ÃP−1 .
The proof of theorem VIII.4 will then be finished once it is
established that (A,B) is a solution of (VIII.27).
To do so, we introduce the following linear Hodge decomposition
in L 2 :
∇Ã−Ã∇⊥ ξ +∇ ⊥ B P = ∇C +∇ ⊥ D
where C = 0 on ∂D 2 . The first equation in (VIII.41) states
that ∆C = 0, so that C ≡ 0 sur D 2 . The second equation
in (VIII.41) along with the boundary conditions imply that D
satisfies
⎧⎨
⎩
div(∇D P −1 ) = 0 in ∂D 2
D = 0 on ∂D 2 .
(VIII.42)
Thus, there exists E ∈ W 1,2 (D 2 ,M n (R)) such that
⎧
⎪⎨ −∆E = ∇ ⊥ D·∇P −1 in D 2
(VIII.43)
⎪⎩
∂E
∂ν = 0 on ∂D2
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