Conformally Invariant Variational Problems. - SAM

Proposition IX.1. There exists ε 0 > 0 and C > 0 such that
for any Ω ∈ L 2 (B 4 ,so(m)) satisfying ‖Ω‖ L
2 < ε 0 there exists
P ∈ W 2,2 (B 4 ,SO(m)) such that
ASym(∆P P −1 )+P ΩP −1 = 0 ,
and
‖∇P‖ W
1,2
(B 4 ) ≤ C ‖Ω‖ L2 (B 4 ) .
(IX.59)
✷
Taking the gauge P given by the previous proposition and
denoting w := P v the system (IX.56) becomes
L P w := −∆w−(∇P P −1 ) 2 w+2 div(∇P P −1 w) = 0 ,
(IX.60)
where we have used that
Symm(∆P P −1 ) = 2 −1( ∆P P −1 +P ∆P −1)
= 2 −1 div ( ∇P P −1 +P ∇P −1) +∇P ·∇P −1
= −(∇P P −1 ) 2 .
Denote forany Q ∈ W 2,2 (B 4 ,M m (R)) L ∗ PQ the ”formaladjoint”
to L P acting on Q
L ∗ P Q := −∆Q−2 ∇Q·∇P P−1 −Q (∇P P −1 ) 2 .
In order to factorize the divergence operator in (IX.60) it is
natural to look for Q satisfying L ∗ PQ = 0. One has indeed
0 = QL P w−wL ∗ PQ
= div(Q∇w−[∇Q+2∇P P −1 ] w) .
(IX.61)
It is not so difficult to construct Q solving L ∗ P Q = 0 for a
Q ∈ W 2,p (B 4 ,M m (R) (for p < 2). However, in order to give
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