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3.9 Direct Boundary Integral Equations 151

Integral equations of the “second kind”:

B Ω λ := λ + SQ Ω PN 2 λ = −SQ Ω PN 1 ϕ + SγF
(3.9.24)
1
= λ + SDPN 2 λ = −SDN 1 ϕ + SγF on Γ.
2
Note that in the above equations the tangential differential operators R and
S are applied to finite part integral operators. These operators can be shown
to commute with the finite part integration (see Corollary 8.3.3).
We now have shown how the boundary value problem could be reduced
to each of these boundary integral equations. Consequently, the solution λ of
one of the boundary integral equations generates the solution of the boundary
value problem with the help of the representation formula (3.9.18). More
precisely, we now establish the following equivalence theorem.
This theorem assures that every solution of the boundary integral equa-
tion of the first kind generates solutions of the interior as well as of the
exterior boundary value problem and that every solution of the boundary
value problems can be obtained by solving the first kind integral equations.
This equivalence, however, does not guarantee that the numbers of solutions
of boundary value problems and corresponding boundary integral equations
coincide.

Theorem 3.9.1. Assume the fundamental assumption holds. Let f ∈
∞
∞
C (Ω) and ϕ ∈ C (Γ). Then every solution u ∈ C (Ω) of the bound-
∞
ary value problem (3.9.1), (3.9.2) defines by (3.9.14) asolution λ of both
∞
integral equations (3.9.23) and (3.9.24). Conversely, let λ ∈ C (Γ) be a
solution of the integral equation of the first kind (3.9.23).Then λ together
with ϕ defines via (3.9.18) asolution u ∈ C (Ω) of the boundary value
∞
problem (3.9.1), (3.9.2) and, in addition, λ solves also the the second kind
equation (3.9.24).
∞
If λ ∈ C (Γ) is a solution of the integral equation of the second kind
(3.9.24) and if, in addition, the operator SDPN 1 is injective, then the poten-
tial u defined by λ and ϕ via (3.9.18) is a C –solution of the boundary value
∞
problem (3.9.1), (3.9.2) and, in addition, λ solves also the first kind equation
(3.9.23).
∞
Proof: i.) If u ∈ C (Ω) is a solution to (3.9.1) and (3.9.2) then the previous
∞
derivation shows that λ ∈ C (Γ) and that λ satisfies both boundary integral
equations.
∞
ii.) If λ ∈ C (Γ) is a solution of the integral equation of the first kind
∞
(3.9.23) then the potentials in (3.9.18) define a C –solution u of the partial
differential equation (3.9.1) due to Lemma 3.8.1 and (3.8.3), (3.8.4). Applying
to both sides of (3.9.18) the trace operator yields

ϕ
−1
γu = γF + C Ω M on Γ. (3.9.25)
λ