3.3. ORTHOGONAL DECOMPOSITION FOR TENSOR FIELDS 139
where
HDk={u∈Hk(E)|divAu= 0 },
Gk={u∈Hk(E)|u=∇Aψ}.Define an operator∆ ̃:HD^2 (E)→L^2 D(E)by
(3.3.20) ∆ ̃u=P∆u,
whereP:L^2 (E)→L^2 D(E)is the canonical orthogonal projection.
We known that the Laplace operator∆can be expressed as
(3.3.21) ∆=divA·∇A=gkl
∂^2
∂xk∂xl+B,
whereBis a lower-order differential operator. SinceMis compact, the Sobolev embeddings
H^2 (E)֒→H^1 (E)֒→L^2 (E)are compact. Hence the lower-order differential operator
B:H^2 (M,RN)→L^2 (M,RN)is a linear compact operator. Therefore the operator in (3.3.21) is a linear completely contin-
uous field
∆:H^2 (E)→L^2 (E),
which implies that the operator of (3.3.20) is also a linear completely continuous field
̃∆=P∆:HD^2 (E)→L^2 D(E).By the spectrum theorem of completely continuous fields (Ma and Wang, 2005 ), the space
H ̃={u∈HD^2 (E)| ̃∆u= 0 }is finite dimensional, and is the eigenspace of the eigenvalueλ=0. By (3.2.39), foru∈H ̃
we have
∫
M( ̃∆u,u)√
−gdx=∫M(∆u,u)√
−gdx (by divAu= 0 )=−∫M(∇Au,∇Au)√
−gdx= 0 (by ̃∆u= 0 ).It follows that
u∈H ̃ ⇔ ∇Au= 0 ,
which implies thatH ̃is the same as the harmonic spaceHof (3.3.11), i.e.H ̃=H. Thus we
have
L^2 D(E) =H⊕L^2 N(E),
L^2 N(E) ={u∈L^2 D(E)|∇Au 6 = 0 }.The proof of Theorem3.17is complete.