Page 90 -

74 2. Boundary Integral Equations

Since each of the integral equations in Table 2.3.3 has a nonempty kernel,
we now modify these equations in the same manner as in elasticity by incor-
porating eigenspaces to obtain uniquely solvable boundary integral equations.
Again, in order not to be repetitious, we summarize the modified equations
in Table 2.3.4.
A few comments are in order.
In the two–dimensional case, it should be understood that V should be

replaced by V and that kerD = span {v j, } with v j, a basis of {a+b x 2 }| Γ

−x 1
2
with a ∈ IR ,b ∈ IR. Moreover, as in elasticity in Section 2.2, one has to

incorporate σds appropriately, in order to take into account the decay
Γ
conditions (2.3.18).
For exterior problems, special attention has to be paid to the behavior at
infinity. In particular, u has the representation (2.3.20), i.e.,
c
u = Wϕ − V τ + ω in Ω .
Then the Dirichlet condition leads on Γ to the system

1
V τ − ω = −( I − K)ϕ and τds = Σ , (2.3.34)
2
Γ
where in the last equation Σ is a given constant vector determining the log-
arithmic behavior of u at infinity (see (2.3.18)). For uniqueness, this system
is modified by adding the additional conditions

τ · n ds =0 = 1,L.
Γ
Then the system (2.3.34) is equivalent to the uniquely solvable system
L
1
V τ − ω + ω 3 n = −( I − K)ϕ ,
2
=1

τ · n ds =0 , σds = Σ , =1,L, or (2.3.35)
Γ Γ
L
1
V τ − ω + τ · n dsn = −( I − K)ϕ , σds = Σ . (2.3.36)
2
=1 Γ Γ
These last two versions (2.3.35) and (2.3.36) correspond to mixed for-
mulations and have been analyzed in detail in Fischer et al [80] and
[134, 135, 137, 139].
In the same manner, appropriate modifications are to be considered for
other boundary co nditions and the time harmonic unsteady problems and
corresponding boundary integral equations as well [137, 139], Kohr et al [163,
162] and Varnhorn [310]. There, as in this section, the boundary integral
equations are considered for charges in H¨older spaces. We shall come back to
these problems in a more general setting in Chapter 5.