Conformally Invariant Variational Problems. - SAM

holds
m∑
A j ik ∇uj = 0 ∀ i,k = 1···m . (VIII.9)
j=1
Inserting this identity into (VIII.8) produces
m∑
∆u i = − (Hjk(u)−H i j ik (u)) ∇⊥ u k ·∇u j
−
j,k=1
m∑
(A i jk (u)−Aj ik (u)) ∇uk ·∇u j .
j,k=1
(VIII.10)
The m×m matrix Ω := (Ω i j ) i,j=1···m defined via
m
Ω i j := ∑
(Hjk i (u)−Hj ik (u))∇⊥ u k +
k=1
m∑
(A i jk (u)−Aj ik (u))∇uk ,
isevidentlyantisymmetric,anditbelongstoL 2 . Withthisnotation,
(VIII.10) is recast in the form (VIII.6), and thus all of the
hypotheses of theorem VIII.1 are fulfilled, thereby concluding
the proof of theorem VIII.2.
✷
On the conservation laws for Schrödinger systems with
antisymmetric potentials.
Per the above discussion, there only remains to establish theorem
VIII.1 in order to reach our goal. To this end, we will
express the Schrödinger systems with antisymmetric potentials
in the form of conservation laws. More precisely, we have
Theorem VIII.3. [Riv1] Let Ω be a matrix-valued vector field
on D 2 in L 2 (∧ 1 D 2 ,so(m)). Suppose that A and B are two W 1,2
functions on D 2 taking their values in the same of square m×m
matrices which satisfy the equation
k=1
∇A−AΩ = −∇ ⊥ B .
(VIII.11)
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