144 CHAPTER 3. MATHEMATICAL FOUNDATIONS
with the metric
(3.3.43) (gμ ν) =
(
−1 0
0 G
)
.
HereM ̃is a closed Riemannian manifold, andG= (gij)is the Riemann metric ofM ̃.
In view of the Minkowski metric (3.3.43), we see that the operator divA·∇Ais a hyperbolic
differential operator expressed as
(3.3.44) divA·∇A=−
(
∂
∂t+A 0
) 2
+gijDAiDA j.Now a tensor fieldu∈L^2 (TrkM)has an orthogonal composition if the following hyper-
bolic equation
(3.3.45) divA·∇Aφ=divAu inM
has a weak solutionφ∈H^1 (Trk−^1 M)in the following sense:
(3.3.46)
∫M(DAφ,DAψ)√
−gdx=∫M(u,DAψ)√
−gdx ∀ψ∈H^1 (Trk−^1 M).Theorem 3.23.LetMbe a Minkowski manifold as defined by (3.3.42)-(3.3.43), and u∈
L^2 (TrlM) (k+r≥ 1 )be an(k,r)−tensor field. Then u can be orthogonally decomposed into
the following form
(3.3.47)
u=∇Aφ+v, divAv= 0 ,
∫M(∇Aφ,v)√
−gdx= 0 ,if and only if equation (3.3.45) has a weak solutionφ∈H^1 (Tk−^1 M)in the sense of (3.3.46).
3.4 Variations withdivA-Free Constraints
3.4.1 Classical variational principle
Variational approach originates from the minimization problem of a functional. LetXbe a
Banach space, andFbe a functional onX:
(3.4.1) F:X→R^1.
The minimization problem ofFis to find a pointu∈X, which is a minimal point ofF.
Namely, there is a neighborhoodU⊂Xofu, such thatFis minimal atuinU:
(3.4.2) F(u) =min
v∈U
F(v).In the classical variational principle we know that the minimal pointuofFin (3.4.2) is a
solution of the variational equation ofF:
(3.4.3) δF(u) = 0 ,