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326 6. Introduction to Pseudodifferential Operators

6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n

m
Elliptic pseudodifferential operators form a special class of L (Ω) which are
essentially invertible. This section we devote to some basic properties of this
class of operators on Ω. In particular, we will discuss the connections between
the parametrix and the Green’s operator for elliptic boundary value prob-
lems. The development here will also be useful for the treatment of boundary
integral operators later on.
n
m
Definition 6.2.1. Asymbol a(x, ξ) in S S S (Ω ×IR ) is elliptic of order m
n
if there exists a symbol b(x, ξ) in S S S −m (Ω × IR ) such that
n
ab − 1 ∈ S S S −1 (Ω × IR ) . (6.2.1)
This definition leads at once to the following criterion.
n
m
Lemma 6.2.1. Asymbol a ∈ S S S (Ω × IR ) is elliptic if and only if for any
compact set K Ω there are constants c(K) > 0 and R(K) > 0 such that
m
|a(x, ξ)|≥ c(K) ξ for all x ∈ K and |ξ|≥ R(K) . (6.2.2)
We leave the proof to the reader.
m
For a pseudodifferential operator A ∈L (Ω) we say that A is elliptic of
order m, if one of the representatives of the complete symbol σ A is elliptic
of order m.
m
For classical pseudodifferential operators A ∈L (Ω), the ellipticity of
c
order m can easily be characterized by the homogeneous principal symbol
a 0 and the ellipticity condition
m
0
a (x, ξ) = 0 for all x ∈ Ω and |ξ| =1 . (6.2.3)
m
As an example we consider the second order linear differential operator P as
in (5.1.1) for the scalar case p = 1 which has the complete symbol
n n
n

∂a jk

a(x, ξ)= a jk (x)ξ j ξ k − i b k (x) − (x) ξ k + c(x) . (6.2.4)
∂x j
j,k=1 k=1 j=1
In particular, Condition (6.2.3) is fulfilled for a strongly elliptic P satisfying
(5.4.1); then
n

0
a (x, ξ):= a jk (x)ξ j ξ k =0 for |ξ| =0 and x ∈ Ω, (6.2.5)
2
j,k=1
and P is elliptic of order m =2.