Conformally Invariant Variational Problems. - SAM

maps, namely
∑n+1
(
−∆u i = ν(u) i ∇(ν(u)) j −ν(u) j ∇(ν(u)) i)·∇u j .
j=1
(VIII.4)
On the contrary, the second step of the process can not a priori
be extended. Indeed, one cannot justify that the vector field
ν(u) i ∇(ν(u)) j −ν(u) j ∇(ν(u)) i
is divergence-free. This was true so long as N n was the sphere
S n , butitfailssosoonasthemetriciseversoslightlyperturbed.
What remainshoweverrobustisthe antisymmetryofthe matrix
Ω := ( ν(u) i ∇(ν(u)) j −ν(u) j ∇(ν(u)) i) i,j=1···n+1
. (VIII.5)
It turns out that the antisymmetry of Ω lies in the heart of
the problem we have been tackling in these lecture notes. The
following result sheds some light onto this claim.
Theorem VIII.1. [Riv1] Let Ω be a vector field in L 2 (∧ 1 D 2 ⊗
so(m)), thus takings values in the space antisymmetric m × m
matrices so(m). Suppose that u is a map in W 1,2 (D 2 ,R m ) satisfying
the equation 21
−∆u = Ω·∇u in D ′ (D 2 ) . (VIII.6)
Then there exists some p > 2 such that u ∈ W 1,p
loc (D2 ,R m ). In
particular, u is Hölder continuous.
✷
Prior to delving into the proof of this theorem, let us first examine
some of its implications towards answering the questions
we aim to solve.
21 In local coordinates, (VIII.6) reads
−∆u i =
m∑
Ω i j ·∇u j ∀ i = 1···m .
j=1
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