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76 2. Boundary Integral Equations

A necessary and sufficient condition for the solvability of this system is
the orthogonality condition

1 2 2
(1 + c)( I + K e )ϕ − c V ∆ σ 0 − αc V ∆ n · nds =0 .
2
µ µ
Γ
Now we combine (2.2.68) with (2.3.29) and obtain the relation
(1 + c)K e ϕ = K st ϕ + c(K ∆ ϕ + L 1 ϕ) (2.3.42)

between the double layer potential operators of the Lam´e and the Stokes sys-
tems where K ∆ is the double layer potential operator (1.2.8) of the Laplacian
and L is the linear Cauchy singular integral operator defined by

1 n(y) · ϕ(y)(x − y) − (x − y) · ϕ(y)n(y)
L 1 ϕ = p.v. ds y .
2π(n − 1) |x − y| n
Γ \{x}
(2.3.43)
Therefore the orthogonality condition becomes
#
1 1
{( I + K st )ϕ}· nds + c {( I + K ∆ + L 1 )ϕ}· nds
2 2
Γ Γ
2 $
2
− (V ∆ σ 0 ) · nds − α β ∆ =0
µ µ
Γ

where β ∆ := (V ∆ n) · nds. Since nds = 0, it can be shown that β ∆ > 0
Γ Γ
(see [138], [141, Theorem 3.7]). In the first integral, however, we interchange
orders of integration and obtain
1 1
( I + K st )ϕ · nds = ϕ · ( I + K )n ds = ϕ · nds

2 2 st
Γ Γ Γ
from Theorem 2.3.2. Hence, the orthogonality condition implies that α must
be chosen as
1 µ µ 1
1
α = ϕ · nds + {( I + K ∆ + L 1 )ϕ}· nds − (V ∆ σ 0 ) · nds.
2
c 2β ∆ Γ 2β ∆ Γ β ∆ Γ
(2.3.44)
Replacing α from (2.3.44) in (2.3.41) we finally obtain the corresponding
stabilized equation,

V st σ 0 + σ 0 · ndsn + cBσ 0 = f (2.3.45)
Γ
where the linear operator B is defined by
2 1
Bσ 0 := V ∆ σ 0 − (V ∆ σ 0 ) · ndsV ∆ n
µ β ∆ Γ