Conformally Invariant Variational Problems. - SAM

We note that Γ i (∇u,∇u) := ∑ m
k,l=1 Γi kl ∇uk·∇u l , so that the
harmonic map equation can be recast as
∆u+Γ(∇u,∇u) = 0 .
(VI.6)
This equation raises several analytical questions:
(i) Weak limits : Let u n be a sequence of solutions of (VI.6)
with uniformly bounded energy E g . Can one extract a subsequence
converging weakly in W 1,2 to a harmonic map ?
(ii) Palais-Smale sequences : Let u n be a sequence of solutions
of (VI.6) in W 1,2 (D 2 ,R m ) with uniformly bounded
energy E g , and such that
∆u n +Γ(∇u n ,∇u n ) = δ n → 0 strongly in H −1 .
Can one extract a subsequence converging weakly in W 1,2
to a harmonic map ?
(iii) Regularity of weak solutions : Let u be a map in
W 1,2 (D 2 ,R m ) which satisfies (VI.4) distributionally. How
regular is u ? Continuous, smooth, analytic, etc...
The answer to (iii) is strongly tied to that of (i) and (ii).
We shall thus restrict our attention in these notes on regularity
matters.
Prior to bringing into light further examples of conformally
invariant Lagrangians, we feel worthwhile to investigate deeper
the difficulties associated with the study of the regularity of
harmonic maps in two dimensions.
The harmonic map equation (VI.6) belongs to the class of
elliptic systems with quadratic growth, also known as natural
growth, of the form
∆u = f(u,∇u) ,
(VI.7)
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