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6.2 Elliptic Pseudodifferential Operators on Ω ⊂ IR n 329

and its principal part is now defined by

jk
a jk (x; ξ):= a (x)i |β| β (6.2.12)
ξ
s j +t k β
|β|=s j +t k
jk jk
where a (x; ξ) is set equal to zero if the order of a (x; ξ) is less than
s j +t k
s j + t k . Then
ξ
jk0 s j +t k jk
a (x; ξ)= |ξ| a x; (6.2.13)
s j +t k s j +t k
|ξ|
is a representative of the corresponding homogeneous principal symbol.
For the more general system of operators A jk ∈L s j +t k (Ω) we can asso-
ciate in the same manner as for the differential operators in (6.2.10) represen-
tatives (6.2.11) by first using the principal symbol class (6.1.31) of A jk and
then taking the homogeneous representatives (6.2.13) and neglecting those
jk
a from (6.1.31) of order less than s j + t k .
s j +t k
Definition 6.2.3. Let the characteristic determinant H(x, ξ) for the system
((A jk )) be defined by
jk0
H(x, ξ) := det a (x; ξ) . (6.2.14)
s j +t k p×p
2
Then the system is elliptic in the sense of Agmon–Douglis–Nirenberg if
n
H(x, ξ) =0 for all x ∈ Ω and ξ ∈ IR \{0} . (6.2.15)
With this definition of ellipticity, Definition 6.2.2 of a parametrix Q 0 =((Q jk ))
s j +t k
for the operator A =((A jk )) p×p with A jk ∈L (Ω) as well as Theorem
s j +t k
6.2.2 with Q jk ∈L (Ω) remain valid.
The Stokes system (2.3.1) is a simple example of an elliptic system in
the sense of Agmon–Douglis–Nirenberg. If we identify the pressure in (2.3.1)
with u n+1 for n = 2 or 3 then the Stokes system takes the form (6.2.10) with
p = n +1, for n =3:
−µ∆ 0 0 u 1
⎛ ∂ ⎞ ⎛ ⎞
4 s j +t j ∂x 1
∂
jk β ⎜ 0 −µ∆ 0 ⎟ ⎜ u 2 ⎟
a D u k = ⎜ ∂x 2 ⎟ ⎜ ⎟ = f
β ∂ ⎠ ⎝ u 3
⎝ 0 0 −µ∆ ⎠
k=1 |β|=0 ∂x 3
∂ ∂ ∂
0 p
∂x 1 ∂x 2 ∂x 3
where s j =0 for j = 1,n, s n+1 = −1and t k =2 for k =1,n, t n+1 =1. The
jk jk
coefficients are given by a = −µδ for |β| =2 and j, k = 1,n;
β
a 14 = a 24 = a 34 = a 41 = a 42 = a 43 =1 for n =3 ;
(1,0,0) (0,1,0) (0,0,1) (1,0,0) (0,1,0) (0,0,1)
a 13 = a 23 = a 31 = a 32 =1 for n =2
(1,0) (0.1) (1,0) (0,1)
2
Originally called Douglis–Nirenberg elliptic.