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78 2. Boundary Integral Equations

where for n =3
1 1
L 2 ϕ(x)= M x 4µ 2 E st (x, y)− γ n (x, y)I M y ϕ(y)ds y ,
1+ c 2(n − 1)µπ
Γ \{x}
(2.3.49)
and for n = 2 the differential operators can be replaced as M x = d and
ds x
M y = d . Hence, (2.3.46) can be written as
ds y
1
D st u =( I − K )ψ − cL 2 u . (2.3.50)

2 e
In view of Theorem 2.3.2, one may decompose the solution u in the form
M

u(x)= u 0 (x)+ α j m j (x) (2.3.51)
j=1
where
1
u 0 · m j ds =0 for j =1,...,M with M := n(n +1) ,
2
Γ
and m j (x) are the traces of the rigid motions given in (2.2.55). These vector
valued functions form a basis of the kernel to D e as well as to D st which
implies also that

L 2 m j =0 for j =1,...,M and c ∈ IR . (2.3.52)
Substituting (2.3.51) into (2.3.50) yields the uniquely solvable system of equa-
tions
1

D st u 0 + cL 2 u 0 =( I − K )ψ ,
2 e
(2.3.53)
u 0 · m j ds =0 for j =1,...,M ;
Γ
or, in stabilized form
M
1

D st u 0 + u 0 · m j dsm j + cL 2 u 0 =( I − K )ψ . (2.3.54)
2 e
j=1 Γ
The right–hand side in (2.3.53) satisfies the orthogonality conditions
1
( I − K )ψ · m j ds =0 for j =1,...,M
2 e
Γ
since the given ψ satisfies the compatibility conditions

ψ · m j ds =0 for j =1,...,M
Γ
and the vector valued function m j satisfies
1
( I + K e )m j = 0 on Γ.
2