Conformally Invariant Variational Problems. - SAM

vi) Is there a weak notion of Willmore immersions and, if yes,
is any such a weak solution smooth ?
We will devote the rest of the course to these questions which
are very much related to another - as an experienced non-linear
analysts could anticipate ! -. To this aim we have to find a
suitableframeworkfordevelopingcalculusofvariationquestions
for Willmore functional.
The first step consists naturally in trying to confront the
Euler Lagrange Willmore equation (X.64) we obtained to the
questions i)···vi).
In codimension 3 the Schadow’s-Thomsen equation of Willmore
surfaces is particularly attractive because of it’s apparent
simplicity :
i) The term ∆ g H is the application of a somehow classical
linear elliptic operator - the Laplace Beltrami Operator
- on the mean-curvature H.
ii) Thenonlinearterms2H (H 2 −K)isanalgebraic function
of the principal curvatures.
Despite it’s elegance, Schadow-Thomsen’s equation is however
verycomplexforananalysisapproachandforthepreviousquestions
i)···vi) we posed. Indeed
i) The term ∆ g H is in fact the application of the Laplace
Beltrami Operator to the mean curvature but this operator
it depends on the metric g which itself is varying
dependingoftheimmersion ⃗ Φ. Hence∆ g could, intheminimization
procedures we aim to follow, strongly degenerate
as the immersion ⃗ Φ degenerates.
ii) Already in codimension 1 where the problem admits a simpler
formulation,the non-linear term 2H (H 2 −K) is somehow
supercritical with respect to the Lagrangian: indeed
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