Conformally Invariant Variational Problems. - SAM

In other words u is either constant or is the translationof the
composition of an isometry with a dilation or the composition
of an inversion, an isometry and a dilation. Regarding the assumptionontheregularity,thereisaweakformulationforbeing
conformal (see [Iwaniez Martin]) which permits to consider the
conformal condition for W 1,1 maps from R n into R n .
Iwaniez and Martin proved that the rigidity result given by
Liouville Theorem still holds for W 1,p (U,R n ) maps when p n 2
and n is even and they provided counterexamples to Liouville
Theorem in W 1,p (U,R n ) for any p < n 2
. Whether the threshold
holds also for n odd is still open. We shall give a
n
2
Proof of Theorem II.1 for u being a C 4 diffeomorphism. In
order to do so we shall first prove the following Lemma.
Lemma II.3. Let UCR n be open and u be a C 4 conformal
diffeomorphism from U into u(U)CR n then there exists A and
B in R such that
∀x ∈ U ∀X,Y ∈ R n
< du x·X,du x·Y >=
1
(A | x−x 0 | 2 +B) 2 < X,Y >
where < ·,· > denotes the canonical scalar product in R n .
✷
Proof of Lemma II.3
Denote e λ =| ∂u
∂x i
| for any i = 1...n
−λ ∂u
By assumption e
∂x i
forms an orthonormal basis of R n . Thus
there exist coefficients X j ki
∈ C 2 (U) such that
∂ 2 u
∂x k ∂x i
=
n∑
j=1
X j −λ ∂u
ki
e
∂x j
(II.11)
We have with those notations for any choice of i,j,k ∈ {1...n}
8