Conformally Invariant Variational Problems. - SAM

order to hope to construct energy controlled liftings. However,
if one removes the requirement to control the energy, every map
⃗n ∈ W 1,2 (D 2 ,Gr m−2 (R m )) admits a W 1,2 lifting. The following
theorem is proved in [He] chapter 5.2.
Theorem X.9. Let⃗n in W 1,2 (D 2 ,Gr m−2 (R m )), then thereexists
⃗e 1 ,⃗e 2 ∈ W 1,2 (D 2 ,S m−1 ) such that
⃗e 1 ∧⃗e 2 = ⋆⃗n .
Wecanthenmakethefollowingobservation. Let⃗n ∈ W 1,2 (D 2 ,S m−1 )
and ⃗e 1 ,⃗e 2 ∈ W 1,2 (D 2 ,S m−1 ) given by the previous theorem.
Consider θ ∈ W 1,2 to be the unique solution of
⎧
⎪⎨
⎪⎩
∆θ = −div in D 2
〈
∂θ
∂ν = − ⃗e 1 , ∂⃗e 〉
2
∂ν
then the lifting ⃗ f = ⃗ f 1 +i ⃗ f 2 given by
⃗f 1 +i ⃗ f 2 = e iθ (⃗e 1 +i⃗e 2 ) ,
∫
is Coulomb. Let λ such that −∇ ⊥ λ =< f ⃗ 1 ,∇f ⃗ 2 > and such that
∂D
λ = 0, one easily sees that λ satisfies
2
⎧
⎨ −∆λ =< ∇ ⊥ f1 ⃗,∇f ⃗ 2 > in D 2
(X.153)
⎩
λ = 0 on ∂D 2
Observe that since ∇⃗e 1 is perpendicular to ⃗e 1 and since ∇⃗e 2 is
perpendicular to ⃗e 2 one has
< ∇ ⊥ ⃗ f1 ,∇ ⃗ f 2 >=< π ⃗n (∇ ⊥ ⃗ f1 ),π ⃗n (∇ ⃗ f 2 ) > (X.154)
✷
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