Mathematical Principles of Theoretical Physics

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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.

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