3.3. ORTHOGONAL DECOMPOSITION FOR TENSOR FIELDS 143
3.3.4 Orthogonal decomposition on manifolds with boundary
In the above subsections, we mainly consider the orthogonaldecomposition of tensor fields
on the closed Riemannian and Minkowski manifolds. In this section we discuss the problem
3.3.4 Orthogonal decomposition on manifolds with boundary.
1.Orthogonal decomposition on Riemannian manifolds with boundaries. The Leray
decomposition (3.3.5) is for the vector fields on a domainΩ⊂Rnwith∂Ω 6 =/0. This result
can be also generalized to general(k,r)-tensor fields defined on manifolds with boundaries.
Theorem 3.22.LetMbe a Riemannian manifold with boundary∂M 6 =/0, and
(3.3.36) u:M→TrkM
be a(k,r)-tensor field. Then we have the following orthogonal decomposition:
(3.3.37) u=∇Aφ+v,
divAv= 0 , v·n|∂M= 0 ,∫M(∇Aφ,v)√
−gdx= 0 ,where∂v/∂n=∇Av·n is the derivative of v in the direction of outward normal vector n on
∂Ω.
Proof.For the tensor fielduin (3.3.36), consider
(3.3.38)
divA·∇Aφ=divAu ∀x∈M,
∂ φ
∂n
=u·n ∀x∈∂M.This Neumann boundary problem possesses a solution provided the following condition holds
true:
(3.3.39)
∫∂M∂ φ
∂nds=∫∂Mu·nds,which is ensured by the boundary condition in (3.3.38). Hence by (3.3.38) the field
(3.3.40) v=u−∇Aφ
is divA-free, and satisfies the boundary condition
(3.3.41) v·n|∂M= 0.
Then it follows from (3.3.40) and (3.3.41) that the tensor fielduin (3.3.36) can be orthogo-
nally decomposed into the form of (3.3.37). The proof is complete.
2.Orthogonal decomposition on Minkowski manifolds.LetMbe a Minkowski manifold
in the form
(3.3.42) M=M ̃×( 0 ,T),