We finish this chapter with results on the continuous dependence of the potential operators on variations of the boundary and the wavenumber.
We start by proving that the single- and double-layer potential operators depend continuously on variations in the boundaryΓf :={(y, f(y)) :y∈R2}of the domain Df as defined in (3). In the statement of the following theorem, B =B(C1, C2) is the set defined inRemark 2.5, for some constants C2 > C1 >0.
To make the dependence of the operators on the scattering surface more explicit, we write Bf for either S or K defined on a surface Γf given by some f ∈ B. As before we prove the results for the associated operators B˜f =IfBfIf−1 ∈L2(R2).
Denoting the kernel of B˜f by˜bf, we see that, wherex= (x, f(x)), y= (y, f(y)), and b(x, y) :=G(x, y) or b(x, y) := ∂G(x, y)/∂ν(y), in the respective cases Bf = S and Bf =K, it holds that
˜bf(x,y) =b(x, y)Jf(y), Jf(y) :=p
1 +|∇f(y)|2.
Theorem2.9. The single- and double-layer potential operators depend continuously on the boundary Γf of the unbounded domain Df in the sense that
sup
f,g∈B kf−gkBC1,α(R2)≤
kB˜f −B˜gkL2(R2)→L2(R2)→0, →0. (2.21)
Proof. Similarly to how we proceeded when proving Theorem 2.6, we decompose the operator B˜f −B˜g into a global and a local part, i.e. B˜f −B˜g = ˜Bglobal+ ˜Blocal with B˜global,B˜local defined similarly to (2.3) and (2.4). We now carry out the proof for the case of the single-layer operator. The necessary changes for the double-layer operator are straightforward.
2.2 Continuity properties of the potential operators
The global operator. The kernel of the global operator B˜global is given by
˜bglobal(x,y) := 1−χ(|x−y|)
[˜bf(x,y)−˜bg(x,y)]. (2.22) We use the expansion (2.7) and equation (2.9), denoting l by lf and lg to indicate its dependence on f and g. We obtain
˜bglobal(x,y) =−iκ which we rewrite in the more convenient way
˜bglobal(x,y) =−iκ
The integral operators whose kernel are the first three terms of (2.23) can be bounded using Lemma 1.33 and Lemma 1.46 and the estimate (2.8), noting that Remark 2.5 guarantees the uniformity of (2.8) for f ∈ B. To bound the integral operator whose kernel is the last term of (2.23), we construct, for every η ∈(0,1), a function `η ∈L1(R2)such that
|lf(x,y)−lg(x,y)| ≤`η(x−y), x,y ∈R2, (2.24) whenever f, g ∈B and kf −gkBC1,α(R2) is sufficiently small, and such that
k`ηkL1(R2)→0, η→0,
and then we use the estimate (1.54). Together, the bounds on the four parts of B˜global show (2.21) for the global part of the operator.
The construction of `η is as follows. We choose (possible by Remark 2.5) a constant C >0 so that (2.8) holds for allf ∈B. Then, where`˜∈L1(R2)is defined as in Lemma 2.3, we set
`η(y) :=
(η3, η <|y|< η−1,
2C`(y),˜ otherwise. (2.25)
Clearly this satisfies that k`ηkL1(R2) →0as η→0. Since, for everyη ∈(0,1),
|lf(x,y)−lg(x,y)| →0, kf −gkBC1,α(R2)→0,
uniformly inf and g for f, g∈B, and uniformly inx and y for η≤ |x−y| ≤η−1, the bound (2.24) holds for all f, g∈B with kf −gkBC1,α(R2) sufficiently small.
The local operator. For the local operator we argue in a similar way as for the global operator, in particular in a similar way as for the integral operator cor-responding to the last term in (2.23). In particular, where˜blocal is the kernel of the local operator, it holds for everyη >0that|˜blocal(x,y)| →0askf−gkBC1,α(R2) →0, uniformly in f and g for f, g ∈ B, and uniformly in x and y for |x−y| ≥η, and (2.5) takes the role of (2.8).
To show later that the limiting absorption condition (8) is satisfied in the case κ >0 we need the following theorem.
Theorem 2.10. Denote S and K temporarily by Sκi and Kκi to indicate their de-pendence on κi. Then, where Bκi denotes either Sκi or Kκi, it holds that
kBκi −B0kL2(Γ)→L2(Γ) →0 (2.26)
as κi →0.
Proof. As we did when proving Theorem 2.6 we split Bκi into global and local parts, as Bκi =B1+B2, withB1, B2 defined by (2.3) and (2.4). As in the proofs of Lemma 2.1 and Lemma 2.3 we denote the kernel of Bj bybj.
To show (2.26) for the local partB2 we note thatb2(x, y)depends continuously on κi, uniformly in xand yfor|x−y| ≥η and every η >0, and that, by Remark2.5, the bound (2.5) holds uniformly in κi for κi ∈[0,1]. We then argue as for the local part in the proof of Theorem 2.9, showing that the kernel of the local part of Bκi −B0 is bounded by an L1 convolution kernel `(x−y) with k`kL1(R2) → 0 as κi →0. Finally we apply (1.54).
To show (2.26) for the global part B2 we use the representation (2.7) for b1(x, y), which splitsb1 into a weakly singular partl(x,y), bounded by (2.8), and a strongly singular part l∗(x,y), given explicitly by (2.9) or (2.10). To show (2.26) for the weakly singular part ofB2 we argue exactly as we did in the proof of Theorem2.9, noting that, by Remark 2.5, (2.8) holds uniformly in κi for κi ∈ [0,1], and that l(x,y) depends continuously onκi, uniformly inx and yfor η≤ |x−y| ≤η−1, for every η∈(0,1). That (2.26) holds for the strongly singular part ofB2 follows from Lemma 1.34 and (1.51).
Chapter 3
Existence and Uniqueness
In this chapter we prove results on the equivalence of the boundary integral equation and the boundary value problem. We show that the boundary value problem has at most one solution and prove a result on the solvability of the integral equation, in a first step in the case of a flat surface and then in a second step for the general case of an arbitrary rough surface.
3.1 Uniqueness
Our first objective is to prove the following theorem on the equivalence of the integral equation and boundary value problem, using various results already shown above.
Theorem 3.1 (Equivalence). Suppose that v is defined by (15)–(17) with ϕ∈X∞. Then, in the case κi > 0, v satisfies the boundary value problem if and only if ϕ satisfies the BIE (18). In the case κi = 0 (i.e., κ > 0), if v satisfies the boundary value problem, then ϕsatisfies (18). Conversely, if κ >0, ϕ(κ+i) ∈X∞ satisfies the integral equation (18) with κ replaced by κ+i, for all sufficiently small >0, and kϕ−ϕ(κ+i)kL2(Γ) →0 as →0, then v satisfies the boundary value problem.
Proof. Let v be the combined single- and double-layer potential v, defined in (15), with density ϕ ∈ X∞. By Theorem 2.8, v ∈ C2(D)∩ C( ¯D) and satisfies the Helmholtz equation in D. Further, due to the jump relations (2.13) and (2.14), v = g ∈ X∞ on Γ if and only if the density ϕ satisfies the boundary integral equation (18). ApplyingTheorem 2.8 again, we see thatv satisfies the bound (7).
This yields the equivalence statement for κi >0.
For realκ, in addition, we need to show the limiting absorption principle (8). Let a(x, y) = ∂G(x, y)/∂ν(y)−iηG(x, y), so that
v(x) = Z
Γ
a(x, y)ϕ(y)ds(y), x∈D. (3.1) Suppose, as stated in the theorem, that ϕ(κ+i)∈X∞ satisfies the integral equation (18) with κ replaced by κ +i, for all sufficiently small > 0, and that kϕ−
ϕ(κ+i)kL2(Γ) → 0 as → 0. Let a(κ+i) denote a with κ replaced with κ+i and define v(κ+i) by (3.1) with a, ϕ, replaced by a(κ+i), ϕ(κ+i), respectively. We have shown in the previous chapter that v(κ+i) satisfies Problem 2 (with κ replaced by κ +i). To show the limiting absorption principle (8) we need to show that v(κ+i)(x)→v(x)as →0. We have
v(κ+i)(x)−v(x) = Z
Γ
a(κ+i)(x, y)−a(x, y)
ϕ(κ+i)(y)ds(y) +
Z
Γ
a(x, y) ϕ(κ+i)(y)−ϕ(y)
ds(y).
We see that the second term tends to zero as →0 by the bound (2.16). Clearly, a(κ+i)(x, y)−a(x, y)→0as→0, for everyy∈Γ. Thus, applying Cauchy-Schwarz and then the dominated convergence theorem, noting that the bound (2.19) holds uniformly inκ, we see that the first term tends to 0 as →0.
We now prove the following uniqueness result.
Theorem3.2 (Uniqueness). The boundary value problem has at most one solution.
Proof. Due to [15, Theorem 1], see also [50, Theorem 3.1], a solution u∈ C2(G)∩ C(G) to the Helmholtz equation (4) with =(κ) >0 on an open set G⊂ Rn which satisfies the growth condition |u(x)| ≤ Ceθ|x|, with some constant θ < =(κ), and the boundary condition u(x) = 0 for x ∈ ∂G will vanish identically on G. This result directly implies uniqueness for the scattering problem and the boundary value problem forκi >0. Forκi = 0uniqueness is a consequence of the limiting absorption principle we require, i.e. of the convergence (8).
Before we turn to establishing existence of solutions, we show that to establish unique solvability of the integral equation in the spaceX(Γ)orX∞(Γ), it is enough to study the solvability in L2(Γ).
Theorem3.3. If the integral operatorA, given through (19), is invertible inBL(L2(Γ)), the Banach algebra of bounded linear operators on L2(Γ), then it is also invertible in the subalgebras BL(X(Γ)) and BL(X∞(Γ)).
Proof. We first note that the second statement can be proven in complete analogy to the first.
Assume that A is invertible in BL(L2(Γ)), i.e. the integral equation (18) has exactly one solution ψ ∈ L2(Γ) for every g ∈ X(Γ) ⊂ L2(Γ). Then, defining B =K−iηS, it holds that ψ =Bψ+ 2g and, by induction, that, for every n∈N,
ψ =Bnψ+ 2(Bn−1+· · ·+B0)g.
3.2 Invertibility of I+K−iηS