EQUATIONS OF ELASTIC HYPERSURFACES

174 SHELLS
Lemma 5.31 Let S be µ–smooth and l ∈ N 0 , l ≤ µ. The operator A S (t, D) in (5.141) is
elliptic on the hypersurface S :
det A S ,pr (ξ) ≠ 0 , ξ ∈ R n ,
where , A S ,pr (ξ) is the principal symbol, defined on the cotangent manifold (see (5.145)).
Moreover, the operator
A S (t, D) : W 1 2(S ) → W −1
2 (S ) (5.148)
is self adjoint on the hypersurface
and positive definite modulo the set of killing vector fields:
(A S (t, D)) ∗ S = A S (t, D) (5.149)
u ∈ W 1 2(S ) , (A S (t, D)u, u) S > 0 iff u ∉ K (S ) . (5.150)
Proof: Since
∫
(A S (t, D)u, u) C =
∫
=
C
n∑
j,k,l,m=1
C
∥ ∥∥∥∥ n∑
Def ∗ c j,k,l,m D j,k u(t) ∥
j,k=1
∫
c j,k,l,m D j,k u(t)D l,m u(t)dS =
(see (5.141)) and E (u(t), u(t)) ≥ 0 (see (5.144)), we get
C
∥
n×n
(A S (t, D)u, u) S ≥ 0 ∀u ∈ W 1 2(S ) .
Moreover, due to (5.151), (5.143) and (5.144)
∫
(A S (t, D)u, u) S = 0 iff E (u(t), u(t)) S ◦ = 0
S
u(t) dS
E (u(t), u(t))dS (5.151)
⇐⇒ E (u(t), u(t)) ≡ 0 ⇐⇒ D j,k u(t) = 0 ∀j, k = 1, . . . , n . (5.152)
This accomplishes the proof.
Lemma 5.32 The set K (S ) of killing vector fields is finite dimensional.
If S is C ∞ smooth, then the killing vector fields are smooth as well K (S ) ⊂ C ∞ (S ).
Moreover, if C ⊂ S is an open subsurface, and K (C ) denotes the set of killing vector
fields restricted to the subsurface C , then
w ∈ K (C ) ⋂ ˜W1 2 (C ) =⇒ w ≡ 0 on C . (5.153)