Conformally Invariant Variational Problems. - SAM

egularity. Note that (VI.26) is equivalent to
⎧
⎨ (∆u,e 1 ) = 0
⎩
(∆u,e 2 ) = 0
(VII.33)
Using the fact that
∂ x u,∂ y u ∈ T u N n = vec{e 1 ,e 2 }
(∇e 1 ,e 1 ) = (∇e 2 ,e 2 ) = 0
(∇e 1 ,e 2 )+(e 1 ,∇e 2 ) = 0
we obtain that (VII.33) may be recast in the form
⎧
⎨ div((e 1 ,∇u)) = −(∇e 2 ,e 1 )·(e 2 ,∇u)
⎩
div((e 2 ,∇u)) = (∇e 2 ,e 1 )·(e 1 ,∇u)
On the other hand, there holds
⎧
⎨ rot((e 1 ,∇u)) = −(∇ ⊥ e 2 ,e 1 )·(e 2 ,∇u)
⎩
rot((e 2 ,∇u)) = (∇ ⊥ e 2 ,e 1 )·(e 1 ,∇u)
(VII.34)
(VII.35)
We next proceed by introducing the Hodge decompositions in
L 2 of the frames (e i ,∇u), for i ∈ {1,2}. In particular, there
exist four functions C i and D i in W 1,2 such that
(e i ,∇u) = ∇C i +∇ ⊥ D i .
SettingW := (C 1 ,C 2 ,D 1 ,D 2 ), theidentities(VII.34)et(VII.35)
become
−∆W = Ω·∇W , (VII.36)
where Ω is the vector field valued in the space of 4×4 matrices
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