Conformally Invariant Variational Problems. - SAM

such that
‖dist(A,SO(m))‖ ∞ +‖A−Id‖ W
2,2 ≤ C ‖Ω‖ L
2
(IX.66)
and then for any map v in L 2 (B 4 ,R m ) the following equivalence
holds
−∆v = Ωv ⇐⇒ div(A∇v−∇Av) = 0 (IX.67)
Having now the equation −∆v = Ωv in the form
div ( ∇w−2∇AA −1 w ) = 0
where w := Av permits to obtain easily the following Morrey
estimate : ∀ρ < 1
∫
∀ρ < 1 sup r −ν |w| 2 < +∞ , (IX.68)
x 0 ∈B ρ (0), r 0. As in the previous sections one deduces using
Adams embeddings that w ∈ L q loc
for some q > 2. Bootstarping
this information in the equation gives v ∈ L ∞ loc (see [Riv3] for a
complete description of these arguments).
IX.1 Concluding remarks.
Morejacobianstructuresoranti-symmetricstructureshavebeen
discovered in other conformallyinvariantproblemssuch as Willmore
surfaces [Riv1], bi-harmonic maps into manifolds [LaRi],
1/2-harmonic maps into manifolds [DR1] and [DR2]...etc. Applying
then integrabilityby compensation results in the spirit of
what has been presented above, analysis questions such as the
regularity of weak solutions, the behavior of sequences of solutions
or the compactness of Palais-Smale sequences....have been
solved in these works.
Moreover,beyondtheconformaldimension,whileconsidering
the same problems, but in dimension larger than the conformal
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