EQUATIONS OF ELASTIC HYPERSURFACES

120 SHELLS
Proof: We quote [Hr1, Shu1, Ta2] for the proof.
Theorem 4.25 Let C be a l-smooth manifold and X θ p(C ) denote either the space H θ p(C ) or
W θ p(C ).
Let an operator A : X θ+r
p (C ) → X θ p(C ) be Fredholm for all θ, θ + r ∈ [−l, l] and
all p ∈]1, ∞[. Moreover, there exists a regularizer R (a parametrix) such that RA − I and
AR − I a re compact in the appropriate spaces and R independent of θ. The kernel Ker A,
the cokernel, the kernel of the adjoint operator Ker A ∗ and the index Ind A are independent
of the admissible values of the parameters θ and p.
If, in particular, a(x, D) is a differential (or a pseudodifferential) operator with elliptic
symbol of order r (of the Hörmander’s class S r 1,0(C , T ∗ C )) then a(x, D) : X θ+r
p (C ) →
X θ p(C ) is elliptic and has parametrix for all θ, θ + r ∈ [−l, l] and all p ∈]1, ∞[. Therefore
the above conclusion holds and the Kernels Ker a(x, D), Ker a ∗ (x, D) are infinitely smooth
provided l = ∞.
Proof: We quote [DNS1] for the proof of more general assertion. Also see [Ag2, Du5, Ka1]
for similar results and [DNS2] for a most general one.
Theorem 4.26 Let S be a smooth, closed and compact manifold, θ ∈ R, 1 < p < ∞ and
a(x, D) : X θ p(S ) −→ X θ−r
p (S ) (4.86)
be a pseudodifferential operator of order r with a N × N matrix symbol a(x, ξ) in the
Hörmanrder’s class a ∈ S r (TS ). If a(x,D) is positive definite
(a(x, D)ϕ, ϕ) S ≥ C‖ϕ ∣ ∣ H
r/2
2 (S )| 2 (4.87)
(cf. (2.263)), the symbol is elliptic and, moreover, is positive definite
for some C > 0.
1 < p < ∞.
Proof: The operator
〈a(x, ξ)η, η〉 ≥ C|ξ| r |η| 2 ∀ (x, ξ) ∈ TS , η ∈ C N (4.88)
The operator a(x, D) in (4.86) is then invertible for all θ ∈ R and
a(x, D) : H r/2
p (S ) −→ H −r/2
p (S ) (4.89)
has a trivial kernel: if a(x, D)ϕ = 0, then, due to the inequality (4.87) ϕ = 0. Moreover, the
operator a(x, D) in (4.89) has a closed range. In fact, from (4.87) we derive
‖a(x, D)ϕ ∣ ∣ H
−r/2
2 (S )‖ ‖ϕ ∣ ∣ H
r/2
2 (S )‖ ≥ (a(x, D)ϕ, ϕ) S ≥ C‖ϕ ∣ ∣ H
r/2
2 (S )‖ 2
and, consequently,
‖a(x, D)ϕ ∣ −r/2 H 2 (S )‖ ≥ C‖ϕ ∣ r/2 H 2 (S )‖ . (4.90)
The latter inequality (4.90) implies that for every convergent sequence in the range ψ j =
a(x, D)ϕ j ∈ H −r/2
2 (S ), j = 1, 2, . . ., the sequence ψ 1 , ψ 2 , . . . converges as well
‖a(x, D)(ϕ j − ϕ k ) ∣ ∣ H
r/2
2 (S )‖ ≥ C‖(ϕ j − ϕ k ) ∣ ∣ H
r/2
2 (S )‖