Conformally Invariant Variational Problems. - SAM

the whole regularity result stated in the theorem by using the
following lemma.
Lemma VIII.1. Let m ∈ N \ {0} and u ∈ W 1,p
loc (D2 ,R m ) for
some p > 2 satisfying
−∆u = Ω·∇u
where 24 Ω ∈ L 2 (D 2 ,M m (R) ⊗ R 2 ) then u ∈ W 1,q
loc (D2 ,R m ) for
any q < +∞.
✷
Let ε 0 > 0 be some constant whose value will be adjusted in
due time to fit our needs. There exists a radius ρ 0 such that for
every r < ρ 0 and every point p dans B 1/2 (0), there holds
∫
|∇A| 2 +|∇B| 2 < ε 0 . (VIII.15)
B r (p)
Henceforth, we consider only radii r < ρ 0 .
Note that A∇u satisfies the elliptic system
⎧
⎨ div(A∇u) = ∇B ·∇ ⊥ u = ∂ y B∂ x u−∂ x B∂ y u
⎩
rot(A∇u) = −∇A·∇ ⊥ u = ∂ x A∂ y u−∂ y A∂ x u
We proceed by introducing on B r (p) the linear Hodge decomposition
in L 2 of A∇u. Namely, there exist two functions C and
D, unique up toadditiveconstants, elementsof W 1,2
0 (B r (p))and
W 1,2 (B r (p)) respectively, and such that
A∇u = ∇C +∇ ⊥ D .
(VIII.16)
To see why such C and D do indeed exist, consider first the
equation
⎧
⎨ ∆C = div(A∇u) = ∂ y B∂ x u−∂ x B∂ y u
(VIII.17)
⎩
C = 0 .
24 Observe that in this lemma no antisymmetry assumption is made for Ω which is an
arbitrary m×m−matrix valued L 2 −vectorfield on D 2 .
83