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2.3 The Stokes Equations 67

boundary integral equations for the velocity u. We need, of course, the rep-
resentation formula for p implicitly when we deal with the stress operator.
In analogy to elasticity, we begin with the representation formula (2.3.13)
c
for the velocity u in Ω and (2.3.20) and (2.3.21) in Ω . Applying the trace
operator and the stress operator T to both sides of the representation for-
mula, we obtain the overdetermined system of boundary integral equations
(the Calder´on projection) for the interior problem
1
ϕ(x) = ( I − K)ϕ(x)+ V τ(x) , (2.3.26)
2
1

τ(x) = Dϕ +( I + K )τ(x)on Γ. (2.3.27)
2
Hence, the Calder´on projector for Ω can also be written in operator matrix
form as
1 I − K V

C Ω = 2 1 .
D I + K
2

Here V, K, K and D are the four corresponding basic boundary integral op-
erators of the Stokes flow. Hence, the Calder´on projector C Ω for the interior
domain has the same form as in (1.2.20) with the corresponding hydrody-
namic potential operators.
For the exterior problem, the Calder´on projector on solutions having the
decay properties (2.3.18) and (2.3.19) with Σ given by (2.3.22) is also given
by (1.4.11), i.e.,
1 I + K −V

C Ω c = I− C Ω = 2 1 . (2.3.28)
−D I − K
2
As always, the solutions of both Dirichlet problems as well as both Neumann
c
problems in Ω and Ω can be solved by using the boundary integral equations
of the first as well as of the second kind by employing the relations between
the Cauchy data given by the Calder´on projectors.
The four basic operators appearing in the Calder´on projectors for the
Stokes problem are defined in the same manner as in elasticity (see Lemmata
2.3.1 and 2.2.3) but with appropriate modifications involving the pressure
terms. More specifically, the double layer operator is defined as
1

Kϕ(x):= ϕ(x) + lim T E(z, y) ϕ(y)ds y (2.3.29)
y
2 Ω z→x∈Γ
Γ

= T E(x, y) ϕ(y)ds y
y
Γ \{x}
n

k ∂E ik (x, y) ∂E jk (x, y)
= Q (x, y)δ ij + µ + n j (y)ϕ i (y)ds y
∂y j ∂y i
i,j,k,=1
Γ \{x}

n (x − y) · n(y) (x − y) · ϕ(y) (x − y)
= ds y
2(n − 1)π |x − y| n+2
Γ \{x}