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88 2. Boundary Integral Equations

◦ 2
V σ(x)= u 0 (x)= p(x)for x ∈ IR .
Then
◦ ◦
σ 1 =[Mu 0 ]| Γ =0 and σ 2 =[Nu 0 ]| Γ =0 .

We now conclude this section by summarizing the boundary integral equa-
tions associated with the two boundary value problems of the biharmonic
equation considered here in the following Tables 2.4.5 and 2.4.6. However,
the missing details will not be pursued here. We shall return to these equa-
tions in later chapters.

We remark that in Table 2.4.5 we did not include orthogonality conditions
for the right–hand sides in the equations INP (1) and (2), EDP (2) and ENP
(1) since due to the direct approach it is known that the right–hand sides
always lie in the range of the operators. Hence, we know that the solutions
exist due to the basic results in Theorem 2.4.1, and, moreover, the classical
Fredholm alternative holds for the systems in Table 2.4.5. From this table we
now consider the modified systems so that the latter will always be uniquely
solvable. The main idea here is to incorporate additional side conditions as
well as eigensolutions. These modifications are collected in Table 2.4.6. In
particular, we have augmented the systems by including additional unknowns
3
ω ∈ IR in the same manner as in Section 2.2 for the Lam´e system. Note that
in Table 2.4.6 the matrix valued function S is defined by
◦ 1 ◦ 2 ◦ 3
S(x):= σ (x), σ (x), σ (x)
where the columns of S are the three linearly independent eigensolutions of
the operator on the left–hand side of EDP (2) in Table 2.4.5. If we solve the
exterior Neumann problem with the system ENP (1) in Table 2.4.6, then we
c
obtain a particular solution with p(x)=0 in Ω , and for given p(x) =0, the
latter is to be added to the representation formula (2.4.14). For the interior
Neumann problem, the modified boundary integral equation INP (1) and (2)
provide a particular solution which presents the general solution only up to
linear polynomials.
Note that here we needed Γ ∈ C 2,α and even jumps of the curvature
are excluded. For piecewise Γ ∈ C 2,α –boundary, Green’s formula, the rep-
resentation formula as well as the boundary integral equations need to be
modified appropriately by including certain functionals at the discontinuity
points (Kn¨opke [160]).