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6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n 343

6.2.5 Strong Ellipticity and G˚arding’s Inequality

One of the advantages of considering integral operators as pseudodifferential
operators is that the mapping properties of the boundary integral operators
can be deduced by examining the symbols of the pseudodifferential operators.
On the other hand, G˚arding’s inequality for the integral operators plays a
fundamental role in the variational formulation of the integral equations. The
latter follows from the definition of uniform strong ellipticity of pseudodiffer-
ential operators.
Definition 6.2.4. (see Stephan et al [296]) We call a system of pseudodif-
ferential operators
s j +t k
A jk ∈L (Ω) uniformly strongly elliptic if to the principal part
c
matrix
0
∞
a (x; ξ)= a jk0 (x; ξ) there exist a C –matrix valued function
s j +t k p×p

Θ(x)= Θ jk (x) and a constant γ 0 > 0 such that
p×p
0 2
Re ζ Θ(x)a (x, ξ)ζ ≥ γ 0 |ζ| (6.2.62)
n
p
for all x ∈ Ω, ζ ∈ C and ξ ∈ IR with |ξ| =1.
α
Remark 6.2.1: In order to show G˚arding’s inequality let Λ be the Bessel
potential of order α ∈ IR which is the pseudodifferential operator (−∆+1) α/2
n
n
σ
2
with the symbol (|ξ| +1) α/2 defining isomorphisms H (IR ) → H σ−α (IR )
for every σ ∈ IR.
n
Theorem 6.2.7. Let Ω ⊂ IR be a bounded domain and let A be a strongly
elliptic system of pseudodifferential operators and let K Ω be a compact
subregion. Then there exist constants γ 1 > 0 and γ 2 ≥ 0 such that G˚arding’s
inequality holds in the form
σ
Re (w, Λ ΘΛ −σ Aw) % p H (t −s )/2 (Ω)
=1 (6.2.63)
2
≥ γ 1
w
2 % p H (Ω) − γ 2
w
% p H −1 (Ω)
t
t
=1 =1
% p σ
s j
for w ∈ H t (Ω) with supp w ⊂ K where Λ = Λ δ j (see
=1 comp p×p
(4.1.45))and where γ 1 > 0 and γ 2 ≥ 0 depend on Θ, A, Ω, the compact
K Ω and on the indices t j and s k .
The last term in (6.2.63) defines a linear compact operator
p % p
%
C : H t (Ω) → H −s (Ω) which is given by
=1 comp =1 comp
(v, Cw) % p H (t −s )/2 (Ω) = γ 2 (v, w) % p H −1 (Ω) .
t
=1 =1
With this operator C, the G˚arding inequality (6.2.63) becomes
σ −σ
Re w, Λ ΘΛ A + C w % p (t −s )/2 ≥ γ 1
w
2 % p H (Ω) . (6.2.64)
t
=1 H (Ω) =1