Conformally Invariant Variational Problems. - SAM

where u still denotes the normal unit-vector to the submanifold
N n ⊂ R n+1 . In particular, if N n is the sphere S n , there holds
ν(u) = u, and the equation reads
−∆u = u |∇u| 2 .
(VII.21)
Another characterization of (VII.21) states that the function
u ∈ W 1,2 (D 2 ,S n ) satisfies (VII.21) if and only if
u∧∆u = 0 in D ′ (D 2 ) . (VII.22)
Indeed, any S n -valued map u obeys
0 = ∆ |u|2
2 = div(u∇u) = |∇u|2 +u∆u
sothat∆uisparalleltouasin(VII.22)ifandonlyiftheproportionality
is −|∇u| 2 . This is equivalent to (VII.21). Interestingly
enough, J. Shatah [Sha] observed that (VII.22) is tantamount
to
∀i,j = 1···n+1
div(u i ∇u j −u j ∇u i ) = 0 . (VII.23)
This formulation of the equation for S n -valued harmonic maps
enables one to pass to the weak limit, just as we previously did
in the CMC equation.
The regularity of S n -valued harmonic maps was obtained
by F.Hélein, [He]. It is established as follows.
For each pair of indices (i,j) in {1···n + 1} 2 , the equation
(VII.23) reveals that the vector field u i ∇u j − u j ∇u i forms a
curl term, and hence there exists B i j ∈ W1,2 with
∇ ⊥ B i j = u i ∇u j −u j ∇u i .
In local coordinates, (VII.21) may be written
−∆u i =
∑n+1
u i ∇u j ·∇u j . (VII.24)
j=1
65