Conformally Invariant Variational Problems. - SAM

a meaning to (IX.61) we need at least Q ∈ W 2,2 and, moreover
the invertibility of the matrix Q almost everywhere is also
needed for the conservation law. In the aim of producing a
Q ∈ W 2,2 (B 4 ,Gl m (R)) solving L ∗ PQ = 0 it is important to observe
first that −(∇P P −1 ) 2 is a non-negative symmetric matrix
29 but that it is in a smaller space than L 2 : the space
L 2,1 . Combining the improved Sobolev embeddings 30 space
W 1,2 (B 4 ) ֒→ L 4,2 (B 4 ) and the fact that the product of two functions
in L 4,2 is in L 2,1 (see [Tar2]) we deduce from (IX.59) that
‖(∇P P −1 ) 2 ‖ L
2,1
(B 4 ) ≤ C ‖Ω‖ 2 L 2 (B 4 ) .
(IX.62)
Grantingthesethreeimportantpropertiesfor−(∇P P −1 ) 2 (symmetry,
positiveness and improved integrability) one can prove
the following result (see [Riv3]).
Theorem IX.7. There exists ε > 0 such that
∀P ∈ W 2,2 (B 4 ,SO(m)) satisfying ‖∇P‖ W
1,2
(B 4 ) ≤ ε
there exists a unique Q ∈ W 2,2 ∩L ∞ (B 4 ,Gl m (R)) satisfying
⎧
⎨ −∆Q−2 ∇Q·∇P P −1 −Q (∇P P −1 ) 2 = 0
(IX.63)
⎩
Q = id m
and
‖Q−id m ‖ L∞ ∩W 2,2 ≤ C ‖∇P‖2 W 1,2 (B 4 ) .
(IX.64)
✷
Taking A := P Q we have constructed a solution to
∆A+AΩ = 0 .
(IX.65)
29 Indeed it is asum ofnon-negativesymmetric matrices: eachof the matrices∂ xj P P −1
is antisymetric and hence its square (∂ xj P P −1 ) 2 is symmetric non-positive.
30 L 4,2 (B 4 ) is the Lorentz space of measurable functions f such that the decreasing
rearangement f ∗ of f satisfies ∫ ∞
0
t −1/2 (f ∗ ) 2 (t) dt < +∞.
99