Mathematical Principles of Theoretical Physics

138 CHAPTER 3. MATHEMATICAL FOUNDATIONS

Proof of Theorem3.17.We proceed in several steps as follows.

STEP1. PROOF OFASSERTION(1). Letu∈L^2 (E),E=TrkM(k+r≥ 1 ). Consider the
equation

(3.3.13) ∆φ=divAu inM,

where∆is the Laplace operator defined by

(3.3.14) ∆=divA·∇A.

Without loss of generality, we only consider the case where divAu∈E ̃=Trk−^1 M. It is
clear that if (3.3.13) has a solutionφ∈H^1 (E ̃), then by (3.3.14), the following vector field
must be divA-free

(3.3.15) v=u−∇Aφ∈L^2 (E).

Moreover, by (3.3.8), we have

(3.3.16) 〈v,∇Aψ〉L 2 =

∫

M

(v,∇Aψ)

√

−gdx= 0 ∀∇Aψ∈L^2 (TrkM).

Namelyvand∇Aφare orthogonal. Therefore, the orthogonal decompositionu=v+∇Aφ
follows from (3.3.15) and (3.3.16).
It suffices then to prove that (3.3.13) has a weak solutionφ∈H^1 (E ̃):

(3.3.17) 〈∇Aφ−u,∇Aψ〉L 2 = 0 ∀ψ∈H^1 (E ̃).

Obviously, ifφsatisfies

(3.3.18) ∆φ= 0 ,

where∆is as in (3.3.14), then, by (3.2.39),
∫

M

(∆φ,φ)

√

−gdx=−

∫

M

(∇Aφ,∇Aφ)

√

−gdx= 0.

Hence (3.3.18) is equivalent to

(3.3.19) ∇Aφ= 0.

Therefore, for allφsatisfying (3.3.18) we have
∫

M

(u,∇Aφ)

√

−gdx= 0.

By Theorem3.14, we derive that the equation (3.3.13) has a unique weak solutionφ∈H^1 (E ̃).
For Minkowski manifolds, by Theorem3.15, the equation (3.3.13) also has a solution.
Thus Assertion ( 1) is proved.

STEP2. PROOF OFASSERTION(2). Based on Assertion (1), we have

Hk(E) =HDk⊕Gk, L^2 (E) =L^2 D⊕G,