Conformally Invariant Variational Problems. - SAM

gives ∆v ∈ L 1 which implies in return v ∈ L n/(n−2),∞
loc
, which
corresponds to our definition of being critical for an elliptic system.
Remark IX.2. We have then been able to write critical systems
of the kind −∆v = Ωv in conservative form whenever Ω is
antisymmetric. This ”factorization of the divergence” operator
is obtain through the constructionof a solutionA to the auxiliary
equation ∆A+AΩ = 0 exactly like in the constant variation
method in 1-D, a solution to the auxiliary equation A ′′ +AΩ =
0 permits to factorize the derivative in the ODE given by −z ′′ =
Ωz which becomes, after multiplication by A : (Az ′ −A ′ z) ′ = 0.
Proof of theorem IX.6 for n = 4.
The goal again here, like in the previous sections, is to establish
a Morrey type estimatefor v that could be re-injected in the
equationand convertedintoan L q loc
due toAdamsresultin [Ad].
We shall look for some map P from B 4 into the space SO(m)
solving some ad-hoc auxiliary equation. Formal computation -
we still don’t kow which regularity for P we should assume at
this stage - gives
−∆(P v) = −∆P v −P∆v −2∇P ·∇v
= ( ∆P P −1 +P ΩP −1) Pv
(IX.56)
−2div(∇P P −1 Pv) .
In view of the first term in the r.h.s. of equation (IX.56) it is
natural to look for ∆P having the same regularity as Ω that is
L 2 . HencewearelookingforP ∈ W 2,2 (B 4 ,SO(m)). Undersuch
an assumption the second term in the r.h.s. is not problematic
while working in the function space L 2 for v indeed, standard
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