Mathematical Principles of Theoretical Physics

134 CHAPTER 3. MATHEMATICAL FOUNDATIONS

Theorem 3.14.Let the metric gijbe W^2 ,∞, and f,g be L^2. If

(3.2.41)

∫

M

[(f,∇φ)−(g,φ)]

√

−gdx= 0

holds for allφsatisfying

(3.2.42) gijDiDjφ= 0 ,

then the equation (3.2.40) possesses a weak solution u∈H^1 (TrkM). In particular, if gij,f,g
are C∞, then u∈C∞(TrkM).

Two remarks are now in order. First, the solutionsφ 6 =0 of (3.2.42) are the eigenfunctions
of the Laplace operator div·∇=DiDicorresponding to the zero eigenvalueλ=0, and the
condition (3.2.41) represents that divf+gis orthogonal to all eigenfunctions forλ=0 of
div·∇.
Second, by theLp-estimate theorem, ifgij∈Wm+^2 ,p,f∈Wm+^1 ,p, andg∈Wm,p, then
u∈Wm,p(TrkM).Therefore, by the Sobolev Embedding Theorem3.7, ifgij,f,gareC∞,
then the solutionuis alsoC∞.

Linear hyperbolic equations

Let{M,gμ ν}be a Minkowski manifold, i.e., its metricgμ νcan be written in some coor-
dinate system as

(3.2.43) (gμ ν) =

(

−1 0

0 G

)

, G= (gij)

whereGis an(n− 1 )×(n− 1 )positive definite symmetric matrix. Then, the Laplace operator

DμDμ=gμ νDμDν=−

∂^2

∂t^2

+gijDiDj

is a hyperbolic operator.

LetM=S^1 ×M ̃with the metric (3.2.43), andM ̃is a Riemannian manifold with metric
{gij}as in (3.2.43), and∂M ̃=/0. Then, for the Minkowski manifoldS^1 ×M ̃, the equation

(3.2.44) gμ νDμDνu=divf,

is a hyperbolic equation, written as

(3.2.45) −

∂^2 u

∂t^2

+gijDiDju=divf,

with the periodic condition

(3.2.46) u(t+T) =u(t) ∀t∈R^1.

The following existence theorem is classical.

Theorem 3.15.LetM=S^1 ×M ̃is a Minkowski manifold with metric (3.2.43), and∂M ̃=
/0. Assume that gμ νand f are C∞, then the problem (3.2.45)-(3.2.46) has a C∞solution u.