Conformally Invariant Variational Problems. - SAM

well known variation of the constant method : a solution is
constructed to some auxiliary equation - (IX.46) or (IX.51) -
which has been carefully chosen in order to absorb the ”worst”
part of the r.h.s. of the original equation while comparing our
given solution to the constructed solution of the auxiliary equation.
We shall then in the sequel forget the geometrical interpretation
of the auxiliary equation (VIII.33) in terms of gauge theory
that we see as being specific to the kind of equation (IX.45) we
were looking at and keep the general philosophy of the classical
variation of the constant method for Ordinary Differential
Equations that we are now extending to other classes of Partial
Differential Equations different from (IX.45).
We establish the following result.
Theorem IX.6. [Riv2] Let n > 2 and m ≥ 2 there exists ε 0 > 0
and C > 0 such that for any Ω ∈ L n/2 (B n ,so(m)) there exists
A ∈ L ∞ ∩W 2,n/2 (B n ,Gl m (R) satisfying
i)
ii)
‖A‖ W
2,n/2
(B n ) ≤ C ‖Ω‖ L
n/2
(B n ) ,
∆A+AΩ = 0 .
(IX.53)
(IX.54)
Moreover for any map v in L n/(n−2) (B n ,R m )
−∆v = Ωv ⇐⇒ div(A∇v−∇Av) = 0 (IX.55)
and we deduce that v ∈ L ∞ loc (Bn ).
✷
Remark IX.1. Again the assumptions v ∈ L n/(n−2) (B n ,R m )
and Ω ∈ L n/2 (B n ,so(m)) make equation (IX.55) critical in dimension
n : Inserting this information in the r.h.s. of (IX.55)
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