EQUATIONS OF ELASTIC HYPERSURFACES

5. BOUNDARY VALUE PROBLEMS: EXAMPLES 175
Proof: Let us consider an auxiliary operator
B S (t, D) := Def ∗ Def : W 1 (S ) −→; W −1 (S ) . (5.154)
We get the following:
B S (t, D) := 1 2 Def∗ [D j u k (t) + D k u j (t))] n×n
+ B 1 S (t, D)u
[ n∑
]
= − 1 D j [D j u k (t) + D k u j (t)] + B 2 S (t, D)u
2
j=1
= −∆ S u − 1 2 [D jD k ] n×n
u + B 2 S (t, D)u , t ∈ S , (5.155)
where B 1 S (t, D) and B 2 S (t, D) have order 1. The principal symbol B pr (t, ξ) of B S (t, D)
then equals
n∑
B pr (t, ξ) = B pr (ξ) = (δ j,k |ξ| 2 + ξ j ξ k )η k η j = . (5.156)
and is positive definite:
j,k=1
n×n
〈B pr (ξ)η, η〉 = 1 2
+ 1 2
n∑
ξ j η j
j=1
n∑
[δ j,k |ξ| 2 + ξ j ξ k ]η k η j = |ξ| 2 1 2
j,k=1
n∑
j=1
ξ k η k = 1 2 |η|2 |ξ| 2 + 1 |〈ξ, η〉|2
2
n∑
|η k | 2
k=1
≥ 1 2 |η|2 |ξ| 2 ∀ξ ∈ R 3 , ∀η ∈ C 3 . (5.157)
The positive definiteness property (5.157) implies the following Gårding’s (Korn’s) inequality
‖Def u ∣ ( ) 1
∣L 2 (S )‖ = (B S (t, D)u, u) S ≥
2 − ε ‖u|W 1 2(C )‖ 2 + (R(t, D)u, u)
( ) 1 − 2ε
≥ ‖u|W 1
2
2(C )‖ 2 − C s ‖u|W s 2(C )‖ 2 (5.158)
with some smoothing ΨDO R(t, D) : W m (S ) → C ∞ S of order −∞ (see [Ta1] and
[Du4, Theorem 4.21]).
From (5.158) immediately follows:
w ∈ K (C ) ⋂ W 1 2(S ) =⇒
√
1 − 2ε
‖u|W s 2(C )‖ ≤ ‖u|W 1
2C
2(C )‖ .
s
(5.159)
By taking s < 1 we derive from (5.159) that the set K (S ) of killing vector fields is finite
dimensional. In fact, from arbitrary bounded sequence {u j } ∞ j=1 ⊂ K (S ) we can choose a