Conformally Invariant Variational Problems. - SAM

small L 2 -norms (cf. theorem VIII.4 below). As a result, all coercive
conformally invariant Lagrangians with quadratic growth
will yield conservation laws written in divergence form. This
is quite an amazing fact. Indeed, while in cases of the CMC
and S n -valued harmonic map equations the existence of conservation
laws can be explained by Noether’s theorem 23 , one may
wonder which hidden symmetries yield the existence of
the general divergence form (VIII.13)? This profound
question shall unfortunately not be addressed here.
Prior to constructing A and B in the general case, we first
establish theorem VIII.3.
Proof of theorem VIII.3.
The first part of the theorem is the result of the elementary
calculation,
div(A∇u−B∇ ⊥ u) = A∆u+∇A·∇u−∇B ·∇ ⊥ u
= A ∆u+(∇A+∇ ⊥ B)·∇u
= A(∆u+Ω·∇u) = 0
Regularity matters are settled as follows. Just as in the previously
encountered problems, we seek to employ a Morrey-type
argument via the existence of some constant α > 0 such that
∫
sup ρ −α |∆u| < +∞ . (VIII.14)
p∈B 1/2 (0) , 0 2 we deduce
23 roughly speaking, symmetries give rise to conservation laws. In both the CMC and
S n -harmonic map equations, the said symmetries are tantamount to the corresponding
Lagrangians being invariant under the group of isometries of the target space R m .
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