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3.10 Remarks 157

Now, we apply γ and γ c to both sides of (3.9.41) and obtain after applying M:

2m−1 2m− −1

Mγu = MγF + MγM + Mγ K p P p+ +1 {N 1 [ϕ]+ N 2 [λ]}
=0 p=0
= MγF + MγM + MQ Ω P{N 1 [ϕ]+ N 2 [λ]} , (3.9.42)
2m−1 2m− −1

Mγ c u = Mγ c F + Mγ c M + M K p P p+ +1 {N 1 [ϕ]+ N 2 [λ]}
=0 p=0
= Mγ c F + Mγ c M + MQ P{N 1 [ϕ]+ N 2 [λ]} (3.9.43)
∼Ω c
where
[ϕ]= R(γ c u − γu) and [λ]= S(γ c u − γu) . (3.9.44)
Inserting (3.8.27) into (3.9.42) and (3.9.43), we arrive at the boundary inte-
gral equations

1
ϕ ∓ = RγF + RγM + RDP{N 1 [ϕ]+ N 2 [λ]}∓ [x] (3.9.45)
2
and
1
λ ∓ = SγF + SγM + SDP{N 1 [ϕ]+ N 2 [λ]}∓ [x] , (3.9.46)
2
where we use the properties γ c F = γF and γ c M = γM due to the C –
∞
continuity of F and M.
Since ϕ − and [ϕ] in (3.9.45) are given we may use the boundary integral
equation of the first kind for finding the unknown [λ]:

A[λ]:= −RDPN 2 [λ] (3.9.47)
1
−
= RγF + RγM(•; u) − ϕ − [ϕ]+ RDPN 1 [ϕ] .
2
Once [λ] is known, λ − may be obtained from the equation (3.9.46).
The solution u of the transmission problem is then given by the represen-
tation formula (3.9.41).
We notice that in all the exterior problems as well as in the transmission
problem, the corresponding boundary integral equations as (3.9.32), (3.9.33)
and (3.9.47) contain the term M(x; u) which describes the behavior of the so-
lution at infinity and may or may not be known a priori. In elasticity, M(x; u)
corresponds to the rigid motions. In order to guarantee the unique solvabil-
ity of these boundary integral equations, additional compatibility conditions
associated with M(x; u) need to be appended.

3.10 Remarks

In all of the derivations of Sections 3.4–3.9 we did not care for less regularity
∞
of Γ by assuming Γ ∈ C . It should be understood that with appropriate