Time Asymptotics for the Polyharmonic Wave Equation in Waveguides

P. H. Lesky

Mathematisches Institut A der Universität Stuttgart, Pfaffenwaldring 57, D 70569 Stuttgart

Abstract

Let Ω denote an unbounded domain in ℝ^{n} having the form Ω = ℝ^{l} × D with bounded cross-section D ⊂ ℝ^{n−l}, and let m ∈ ℕ be fixed. This article considers solutions u to the scalar wave equation ∂_{t}^{2}u(t, x) + (−Δ)^{m}u(t, x) = f(x) e^{−iωt} satisfying homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t → ∞ is investigated. Depending on the choice of f, ω and Ω, two cases occur: Either u shows resonance, which means that |u(t, x)| → ∞ as t → ∞ for almost every x ∈ Ω, or u satisfies the principle of limiting amplitude. Furthermore the spatial operators resolvent and the validity of the principle of limiting absorption is studied.

1 Introduction

Let Ω denote a domain in ℝ^{n} (n ≥ 2) having the form

Ω = ℝ^{l} × D, 1 ≤ l ≤ n − 1, D ⊂ ℝ^{n−l} bounded, ∂D ∈ C^{∞}.  (1.1)

If n = 3, l = 1 and D is connected, then Ω is an infinite tube with cross-section D. In the case l = n − 1, D = (a, b) with −∞ < a < b < ∞, Ω is the domain between two parallel hyperplanes.

Consider the Dirichlet initial-boundary-value problem

∂_{t}^{2}u(t, x) + (−Δ)^{m}u(t, x) = f(x) e^{−iωt} for t ≥ 0, x ∈ Ω,
∂_{x_{1}}^{α_{1}} ⋯ ∂_{x_{n}}^{α_{n}} u(t, x) = 0  for t ≥ 0, x ∈ ∂Ω, α ∈ ℕ_{0}^{n} with
|α| = α_{1} + … + α_{n} ≤ m − 1,
u(x, 0) = u_{0}(x), ∂_{t}u(x, 0) = u_{1}(x)  for x ∈ Ω,

(1.2)

Datei: /arbeiten/2001/polyharmonische-Wellen/Poly-Wellen

where Δ = ∂_{x_{1}}^{2} +…+∂_{x_{n}}^{2}, m ∈ ℕ, ω ≥ 0. This equation is used in different physical problems. In the case m = 1, n = 2, the solution u(t, x) describes the perpendicular deviation of a membran stimulated by a periodic perpendicular force, where the membran is fixed at the boundary. In the theory of linear acoustic waves, equation (1.2) with m = 1, n = 3 is used. The case m = 2, n = 2 appears in the study of the perpendicular displacement of a thin plate which is clamped at the boundary.

We are interested in the asymptotic behaviour of the solution u(t, x) as t → ∞. It turns out, that the occurence of resonances depends on the number l of unbounded spatial directions and not on the spatial operator’s order 2m: In the cases l = 1 or l = 2, resonances occur at countable many frequencies of incitation. The amplitude of the resonant solution grows like respectively \sqrt{t} (if l = 1) or ln t (if l = 2). These resonances are not connected with eigenvalues of the spatial operator in (1.2). In order to be more precise, suppose that f, u_{0}, u_{1} ∈ C_{0}^{∞}(Ω) := {φ ∈ C^{∞}(Ω) : supp φ bounded, \overline{supp φ} ⊂ Ω\}. Then (1.2) has a solution u ∈ C^{∞}([0, ∞) × \overline{Ω}), which is uniquely determined by an integrability condition (see (5.1) and the remarks there). Set x = (x′, x′′) with x′ := (x_{1}, …, x_{l}) ∈ ℝ^{l}, x′′ := (x_{l+1}, …, x_{n}) ∈ D and let 0 < μ_{1} < μ_{2} < … denote the eigenvalues of the boundary value problem in cross-direction

\begin{matrix}(−∂_{x_{l+1}}^{2} − … − ∂_{x_{n}}^{2})^{m} v_{jk}(x′′) & = & μ_{j} v_{jk}(x′′) & for x′′ ∈ D, \\ ∂_{x_{l+1}}^{α_{1}} ⋯ ∂_{x_{n}}^{α_{n−l}} v_{jk}(x′′) & = & 0 & for x′′ ∈ ∂D, α ∈ ℕ_{0}^{n−l} \\ & & & with |α| ≤ m − 1, \end{matrix} \} (1.3)

where {v_{jk}} (j ∈ ℕ, 1 ≤ k ≤ K(j) := multiplicity of μ_{j}) is supposed to be a suitable chosen orthonormal system of eigenfunctions being complete in L_{2}(D). We define the resonance frequencies by ω_{j} := \sqrt{μ_{j}} (j ∈ ℕ).

Theorem 1.1 Let Ω ⊂ ℝ^{n} be given by (1.1) and let f, u_{0}, u_{1} ∈ C_{0}^{∞}(Ω). Denote by u the unique solution u of (1.2) and (5.1).

1) Suppose that l ≥ 3 or that ω ∈ [0, ∞) \ {ω_{1}, ω_{2}, … }. Then u satisfies the principle of limiting amplitude:

u(t, x) = e^{−iωt} u_{ω}(x) + o_{M}(1)  as t → ∞,  (1.4)

where the notation o_{M}(1) means that the remainder turns to 0 uniformly with respect to x ∈ M, where M is an arbitrary bounded subset of \overline{Ω}. Furthermore u_{ω} is solution of

\begin{matrix}((−Δ)^{m} − ω^{2}) u_{ω} & = & f & in Ω, \\ ∂_{x_{1}}^{α_{1}} ⋯ ∂_{x_{n}}^{α_{n}} u_{ω} & = & 0 & on ∂Ω for |α| ≤ m − 1. \end{matrix} \}  (1.5)

2) If l = 1 and ω = ω_{j} (j ∈ ℕ fixed), then u shows resonance of order t^{1/2}:

u(t, x′, x′′) = \frac{1 + i}{2\sqrt{πωm}} \sqrt{t} e^{−iωt} ∑_{k=1}^{K(j)} \frac{v_{jk}(x′′)}{∥∇′′^{m−1}v_{jk}∥_{D}} ∫_{Ω} f(y′, y′′)v_{jk}(y′′) d(y′, y′′)
+ e^{−iωt} u_{ω}(x′, x′′) + o_{M}(1)  as t → ∞,  (1.6)

u_{ω} being a solution of (1.5); here the eigenfunctions {v_{jk}} of (1.3) have to be chosen according to Definition 3.3 and ∇′′^{k} := Δ′′^{k/2} for even k ∈ ℕ and ∇′′^{k} := ∇′′Δ′′^{(k−1)/2} for odd k ∈ ℕ. Note that the unbounded term depends only on x′′ ∈ D and is constant with respect to x′ ∈ ℝ^{l}.

3) If l = 2 and ω = ω_{j} (j ∈ ℕ fixed), then resonance of order ln t occurs:

u(t, x′, x′′) = \frac{ln t}{4πm} e^{−iωt} ∑_{k=1}^{K(j)} \frac{v_{jk}(x′′)}{∥∇′′^{m−1}v_{jk}∥_{D}^{2}} ∫_{Ω} f(y′, y′′)v_{jk}(y′′) d(y′, y′′)
+ e^{−iωt} u_{ω}(x′, x′′) + o_{M}(1)  as t → ∞,  (1.7)

where u_{ω} solves (1.5).

Remarks: 1) The principle of limiting amplitude holds even in the case l ≤ 2 for ω = ω_{j}, if f satisfies the orthogonality conditions

∫_{Ω} f(y′, y′′)v_{jk}(y′′) d(y′, y′′) = 0 for 1 ≤ k ≤ K(j).  (1.8)

2) Denote by R_{z} (z ∈ ℂ \ ℝ) the resolvent of the spatial operator. The principle of limiting absorption holds at ρ ∈ ℝ, if v_{ρ} := lim_{τ↓0} R_{ρ+iτ}f exists. It turns out, that the principle of limiting absorption holds at ρ ∈ ℝ if and only if the solution of (1.2), (5.1) with ω := \sqrt{ρ} satisfies the principle of limiting amplitude.

3) The resonance effects are connected to the so-called “admissible standing waves” (compare [9]) given by

S_{jk}(x′, x′′) := \frac{1}{∥∇′′^{m−1}v_{jk}∥_{D}^{l/2}} v_{jk}(x′) for (x′, x′′) ∈ Ω.

They are solution of the boundary value problem

((−Δ)^{m} − μ_{j}) S_{jk} = 0 in Ω,
∂_{x_{1}}^{α_{1}} ⋯ ∂_{x_{n}}^{α_{n}} u_{ω} = 0 on ∂Ω for |α| ≤ m − 1

Note that S_{jk} ∉ L_{2}(Ω). The resonance term in (1.6) is like a projection of f on the linear space spanned by {S_{j1}, …, S_{jK(j)}}.

4) The assumptions ∂D ∈ C^{∞} and f ∈ C_{0}^{∞}(Ω) can be weakend (cp. Satz 1.22 in [8]), but we skip these details for sake of simplicity.

In the case m = 1 these results are already known. They were proved by P. Werner in [12] (for n = 3, l = 2), in [13] (n ≥ 2, l = 1) and, together with A. G. Ramm, in [10] (n ≥ 2, l = n − 1). Furthermore the case m = 1 with n ≥ 2 and arbitrary l is contained in the article [3] of B. A. Iskenderov, Eh. Kh. Ehjvazov and A. N. Ehfendieva (with partly wrong results) and in [7] of the author. The present paper generalizes the results to the case of arbitrary m ∈ ℕ.

The proofs in these papers are based on the fact, that the spatial operator −Δ can be decomposed in the sum of a longitudinal part −Δ′ := −∂_{1}^{2} − … − ∂_{l}^{2} and a transversal part −Δ′′ := −∂_{l+1}^{2} − … − ∂_{n}^{2}. The resolvent of each part can be studied separately to obtain resolvent and spectral family of the full operator. Such a decomposition in longitudinal- and cross-derivatives is not possible for the operator (−Δ)^{m} in the case m ≥ 2. The present article uses another method, which is based on Fourier-transform with respect to the longitudinal variable (x_{1}, …, x_{l}). The same method was used by M. D. Groves and the author in [2]. Application to the equations of linear elasticity in an infinite tube will be given in a further paper (see [6]).

This method can roughly be described as follows: Let A be a self-adjoint extension of (−Δ)^{m} in L_{2}(Ω) corresponding to Dirichlet boundary condition. Fouriertransform of A with respect to x′ = (x_{1}, …, x_{l}) leads to the operator family

B_{|ξ|} := ( |ξ|^{2} − Δ′′)^{m} , D(B_{|ξ|}) := ∘ H^{m}(D) ∩ H^{2m}(D)

depending on |ξ| = \sqrt{ξ_{1}^{2} + … + ξ_{l}^{2}} =: r ≥ 0 and being self-adjoint in L_{2}(D). Note that D is bounded. In a first step, the (suitable enumerated) eigenvalues μ_{1}^{(r)}, μ_{2}^{(r)}, … of B_{r} are studied. For each fixed j ∈ ℕ it is proved that μ_{j}^{(.)} depends analytically on r and

\frac{dμ_{j}^{(r)}}{dr} > 0 for r > 0, \frac{dμ_{j}^{(r)}}{dr} = 0 for r = 0, μ_{j}^{(r)} → ∞ as r → ∞.

Hence the mapping r ↦ μ_{j}^{(r)} has an inverse r_{j} : [μ_{j}^{(0)}, ∞) → [0, ∞) being differentiable on (μ_{j}^{(0)}, ∞). The derivative r′_{j} has a singularity at μ_{j}^{(0)}.

The second step contains the heart of the method. Denote by {P_{λ}}_{λ∈ℝ} the spectral family of A. The local behaviour of \frac{dP_{λ}}{dλ} is studied, which proves to be closely connected to the behaviour of those of the derivatives r′_{j}, which are defined at λ. The derivative of the spectral family is shown to be Hölder-continuous (in a certain sense) on ℝ \ {μ_{1}^{(0)}, μ_{2}^{(0)}, … } and to have singularities at λ = μ_{1}^{(0)}, μ_{2}^{(0)}, . . ..

In the last step, the resolvent of A and the solution u of (1.2) are given by spectral integrals, which can be estimated by the properties of \frac{dP_{λ}}{dλ} given in the second step.

2 The operator family B_{r}

Set

\begin{matrix}D(B_{r}) & := & \{v ∈ ∘ H^{m}(D) : (r^{2} − Δ′′)^{m} v ∈ L_{2}(D)\} , \\ B_{r}v & := & (r^{2} − Δ′′)^{m} v = ∑_{k=0}^{m} \binom{m}{k}r^{2k} (−Δ′′)^{m−k} v \end{matrix} \}  (2.1)

( ∘ H^{m}(D) := closure of C_{0}^{∞}(D) in H^{m}(D)). For every fixed r ∈ ℝ, B_{r} is coercitive, which means that

⟨B_{r}v, v⟩_{D} ≥ c_{1}^{(r)} ∥v∥_{H^{m}(D)}^{2} − c_{2}^{(r)} ∥v∥_{D}^{2}  for v ∈ D(B_{r})  (2.2)

with suitable coefficients c_{1}^{(r)}, c_{2}^{(r)} > 0 depending on r but not on v; here ⟨. , .⟩_{D} and ∥.∥_{D} denote the inner product and associated norm in L_{2}(D), respectively. Standard theory shows that B_{r} is self-adjoint in L_{2}(D) and positive (see e.g. [5]).

The aim of this section is to prove the following theorem.

Theorem 2.1 Let D ⊂ ℝ^{n−l} be bounded. Suppose that m ≤ 2 and D has the segment property or that m ≥ 3 and ∂D ∈ C^{2m}. Then:

1) For every fixed r ∈ ℝ, B_{r} has an orthonormal system of eigenfunctions being complete in L_{2}(D). The eigenfunctions v_{1}^{(r)}, v_{2}^{(r)}, … and associated eigenvalues μ_{1}^{(r)}, μ_{2}^{(r)}, … can be chosen in a way, such that the mappings r ↦ v_{j}^{(r)} and r ↦ μ_{1}^{(r)} are analytic and

0 < μ_{1}^{(0)} ≤ μ_{2}^{(0)} ≤ …,  μ_{j}^{(0)} → ∞ as j → ∞.  (2.3)

2) For every j ∈ ℕ,

\frac{dμ_{j}^{(r)}}{dr} (0) = 0  and  \frac{dμ_{j}^{(r)}}{dr} (r) ≥ 2m r^{2m−1} if r ≥ 0.  (2.4)

3) For every j ∈ ℕ, the mapping μ_{j}^{(.)} : [0, ∞) → [μ_{j}^{(0)}, ∞) : r ↦ μ_{j}^{(r)} has an invers r_{j} ∈ C([μ_{j}^{(0)}, ∞)) ∩ C^{∞}((μ_{j}^{(0)}, ∞)), and

\begin{matrix}r_{j}(μ) & = & \frac{1}{\sqrt{m} ∥∇′′^{m−1}v_{j}^{(0)}∥_{D}} \sqrt{μ − μ_{j}^{(0)}} + O(|μ − μ_{j}^{(0)}|^{3/2}) \\ \frac{d}{dμ} r_{j}(μ) & = & \frac{1}{2\sqrt{m} ∥∇′′^{m−1}v_{j}^{(0)}∥_{D} \sqrt{μ − μ_{j}^{(0)}}} + O(|μ − μ_{j}^{(0)}|^{1/2}) \end{matrix} \} (2.5)

as μ ↓ μ_{j}^{(0)}, where ∇′′^{k} := Δ′′^{k/2} for even k ∈ ℕ and ∇′′^{k} := ∇′′Δ′′^{(k−1)/2} for odd k ∈ ℕ.

Proof: First let r ∈ ℝ be fixed. From (2.2) and Rellich’s selection theorem one

obtains that (B_{r} + c_{2}^{(r)}I)^{−1} : L_{2}(D) → L_{2}(D) is a compact operator. Furthermore it is symmetric and postive. Every eigenfunction of it is an eigenfunction of B_{r} and vice versa. Application of the theory for compact symmetric operators shows: B_{r} has a countable set of eigenvalues {μ_{1}^{(r)}, μ_{2}^{(r)}, … }, where μ_{j}^{(r)} → ∞ as j → ∞, and the eigenfunctions can be chosen such that they form a complete orthonormal basis of L_{2}(D). Choose the enumeration of eigenvalues, such that that every eigenvalue appears in the enumeration so many times, as his finite multiplicity counts and that 0 < μ_{1}^{(r)} ≤ μ_{2}^{(r)} ≤ . . ..

Now let r vary. If m = 1 or m = 2, then

D(B_{r}) = D(B) := {u ∈ ∘ H^{m}(D) : Δ′′^{m}u ∈ L_{2}(D)} for r ∈ ℝ  (2.6)

by definition of D(B_{r}). This equation holds also in the cases m ≥ 3, if ∂D ∈ C^{2m} is supposed, since elliptic regularity theory then shows that

D(B_{r}) = ∘ H^{m}(D) ∩ H^{2m}(D) = D(B) for r ∈ ℝ

(see e.g. [1]). In the notation of [4], B_{r} is a self-adjoint holomorphic family of type (A) with compact resolvent, defined for r ∈ ℂ. According to Theorem 3.9 of Chapter VII in [4], eigenvalues and eigenfunctions can be chosen in a way, such that, for every j ∈ ℕ, the eigenvalue μ_{j}^{(r)} and associated eigenfunction v_{j}^{(r)} depend analytically on r ∈ ℝ. Since different graphs G_{j} := {(r, μ_{j}^{(r)}) : r ∈ ℝ} ⊂ ℝ^{2} (j ∈ ℕ) may intersect, the succesion of enumeration has to be changed eventually for r > 0. Note that μ_{1}^{(0)} > 0, since B_{0} is positive and B_{0}u = 0 would imply u = 0. This proves Part 1 of Theorem 2.1.

For every fixed j ∈ ℕ, we have

\frac{d}{dr} μ_{j}^{(r)} = \frac{d}{dr} ⟨B_{r}v_{j}^{(r)}, v_{j}^{(r)}⟩_{D}

= lim_{h→0} \frac{1}{h} (⟨(B_{r+h} − B_{r}) v_{j}^{(r+h)}, v_{j}^{(r+h)}⟩_{D} + ⟨B_{r} (v_{j}^{(r+h)} − v_{j}^{(r)}) , v_{j}^{(r+h)}⟩_{D}
+ ⟨B_{r}v_{j}^{(r)}, v_{j}^{(r+h)} − v_{j}^{(r)}⟩_{D}) .

According to (2.6), D(B_{r}) is independent of r. Using the symmetry of B_{r}, we obtain

lim_{h→0} \frac{1}{h} (⟨B_{r} (v_{j}^{(r+h)} − v_{j}^{(r)}) , v_{j}^{(r+h)}⟩_{D} + ⟨B_{r}v_{j}^{(r)}, v_{j}^{(r+h)} − v_{j}^{(r)}⟩_{D}) =

= 2 Re ⟨\frac{d}{dr}v_{j}^{(r)}, B_{r}v_{j}^{(r)}⟩_{D}
= μ_{j}^{(r)} \frac{d}{dr} ∥v_{j}^{(r)}∥_{D}^{2}
= 0

(since ∥v_{j}^{(r)}∥_{D} = 1), and hence

\frac{d}{dr} μ_{j}^{(r)} = lim_{h→0} \frac{1}{h} ⟨(B_{r+h} − B_{r}) v_{j}^{(r+h)}, v_{j}^{(r+h)}⟩_{D}
= ∑_{k=1}^{m} \binom{m}{k}2k r^{2k−1} ⟨(−Δ′′)^{m−k}v_{j}^{(r)}, v_{j}^{(r)}⟩_{D}

= ∑_{k=1}^{m} \binom{m}{k}2k r^{2k−1} ∥∇′′^{m−k}v_{j}^{(r)}∥_{D}^{2} ,  (2.7)

This implies that (2.4) holds.

Assertion 1 of Theorem 2.1 and (2.4) imply that the inverse function r_{j} exists and that r_{j} ∈ C([μ_{j}^{(0)}, ∞)) ∩ C^{∞}((μ_{j}^{(0)}, ∞)). It remains to prove (2.5).

According to (2.1), B_{r} and μ_{j}^{(r)} depend analytically on r^{2}, since the perturbation argument used in the first step of this proof also holds for the mapping r ↦ B\sqrt{r}. Hence

μ_{j}^{(r)} = ∑_{j=0}^{∞} a_{j} r^{2j}  for r ∈ ℝ,  (2.8)

with suitable constants a_{j} ∈ ℝ. Obviously a_{0} = μ_{j}^{(0)}, and a_{1} = m∥∇′′^{m−1}v_{j}^{(0)}∥_{D}^{2} by

(2.7). Note that a_{1} > 0, since otherwise ∇′′^{m−1}v_{j}^{(0)} = 0 and v_{j}^{(0)} ∈ D(B) ⊂ ∘ H^{m}(D) would imply v_{j}^{(0)} = 0. By the theory of analytic functions,

r_{j}(μ) = ∑_{k=0}^{∞} b_{k}(μ − μ_{j}^{(0)})^{k/2} for μ_{j}^{(0)} ≤ μ ≤ μ_{j}^{(0)} + δ

with suitable δ > 0 and b_{k} ∈ ℝ. Inserting this in the expansion for μ_{j}^{(r)}, we obtain that b_{0} = 0 and b_{1} = _{\frac{1}{\sqrt{m}∥∇′′^{m−1}v_{j}^{(0)}∥_{D}}}. This proves (2.5). □

3 Spectral properties of (−Δ)^{m}

Consider the operator A given by

\begin{matrix}D(A) & := & \{u ∈ ∘ H^{m}(Ω) : (−Δ)^{m} ∈ L_{2}(Ω)\} , \\ A u & := & (−Δ)^{m} u. \end{matrix} \}  (3.1)

Standard theory shows coercivity (cp. (2.2)) and self-adjointness of A in L_{2}(Ω). This section contains representations of the operators resolvent R_{z} (see (3.5)) and spectral family {P_{λ}}_{λ∈ℝ} (see (3.8)), and gives certain properties of {P_{λ}} (see Theorem 3.4) needed later.

Define the space S of functions having rapid decrease by

S := \{f ∈ C^{∞}(\overline{Ω}) | ∀α, β ∈ ℕ_{0}^{n} : sup_{x∈\overline{Ω}} |x^{α} ∂_{1}^{β_{1}} ⋯ ∂_{n}^{β_{n}} f(x)| < ∞\}  (3.2)

and the partial Fourier-transform F : S → S by

(Fφ)(ξ, x′′) := \frac{1}{(2π)^{l/2}} ∫_{ℝ^{l}} e^{−iξ·x′}φ(x′, x′′) dx′ (ξ ∈ ℝ^{l}, x′′ ∈ D).  (3.3)

Theorem 3.1 Suppose that f ∈ C_{0}^{∞}(Ω), z ∈ ℂ \ ℝ and set

u_{z}(ξ, .) := (B_{|ξ|} − z Id)^{−1} (Ff)(ξ, .)  for ξ ∈ ℝ^{l}.  (3.4)

Then u_{z} ∈ S and the resolvent R_{z} := (A − z Id)^{−1} can be represented by

(R_{z}f)(x′, x′′) = (F^{−1}u_{z})(x′, x′′) = \frac{1}{(2π)^{l/2}} ∫_{ℝ^{l}} e^{i ξ·x′} u_{z}(ξ, x′′) dξ  (3.5)

for (x′, x′′) ∈ \overline{Ω}.

Proof: From ⟨B_{|ξ|}u_{z}(ξ, .), u_{z}(ξ, .)⟩_{D} ∈ ℝ and (B_{|ξ|} − z)u_{z} = Ff one obtains that

|−Im z ∥u_{z}(ξ, .)∥_{D}^{2}| = |Im ⟨(B_{|ξ|} − z)u_{z}(ξ, .), u_{z}(ξ, .)⟩_{D}|
= |⟨(Ff)(ξ, .), u_{z}(ξ, .)⟩_{D}|
≤ ∥(Ff)(ξ, .)∥_{D} ∥u_{z}(ξ, .)∥_{D}

and hence

∥u_{z}(ξ, .)∥_{D} ≤ \frac{1}{|Im z|} ∥(Ff)(ξ, .)∥_{D}.  (3.6)

Elliptic estimates applied to

B_{0}u_{z}(ξ, .) = (B_{0} − B_{|ξ|} + z)u_{z}(ξ, .) + (Ff)(ξ, .)

yield

∥u_{z}(ξ, .)∥_{H^{2m+j}(D)} ≤ c_{j} (∥(B_{0} − B_{|ξ|} + z)u_{z}(ξ, .) + (Ff)(ξ, .)∥_{H^{j}(D)} + ∥u_{z}(ξ, .)∥_{D})

≤ c_{j} |ξ|^{2m} (∥u_{z}(ξ, .)∥_{H^{2m+j−2}(D)} + (Ff)(ξ, .)∥_{H^{j}(D)})
≤ c_{j} |ξ|^{2m} (ε∥u_{z}(ξ, .)∥_{H^{2m+j−2}(D)} + \frac{1}{ε} ∥u_{z}(ξ, .)∥_{D} + (Ff)(ξ, .)∥_{H^{j}(D)}) ,

c_{j} ≥ denoting various constants depending at most on j ∈ ℕ; for the last estimate compare Theorem 3.4 in [1], which can be extended to unbounded domains of form (1.1). By ε := \frac{1}{2c_{j}|ξ|^{2m}}, this together with (3.6) proves

∥u_{z}(ξ, .)∥_{H^{2m+j}(D)} ≤ \frac{c_{j} |ξ|^{4m}}{|Im z|} ∥(Ff)(ξ, .)∥_{H^{j}(D)} for each ξ ∈ ℝ^{l}, j ∈ ℕ.

Sobolev’s Lemma shows that u(ξ, .) ∈ C^{∞}(D) and

|∂_{x_{l+1}}^{α_{l+1}} ⋯ ∂_{x_{n}}^{α_{n}}u_{z}(ξ, x′′)| ≤ \frac{c_{j} |ξ|^{4m}}{|Im z|} ∥(Ff)(ξ, .)∥_{H^{j}(D)} for (ξ, x′′) ∈ \overline{Ω}

if j ≥ α_{l+1} + … + α_{n} + \frac{n−l+1}{2}.

Set ũ(ξ, .) := (B_{|ξ|} − z Id)^{−1} ∂_{ξ_{1}}(Ff)(ξ, .). From

(B_{|ξ|} − z)u (ũ_{z}(ξ, .) − \frac{u_{z}(ξ + h e_{1}, .) − u_{z}(ξ, .)}{h}) =

= ∂_{ξ_{1}}(Ff)(ξ, .) − \frac{(Ff)(ξ + h e_{1}, .) − (Ff)(ξ, .)}{h}

one can conlude that ũ_{z} = ∂_{ξ_{1}}u_{z}(ξ, .). The same argument used above proves that

|∂_{ξ_{1}}^{α_{1}} ⋯ ∂_{ξ_{l}}^{α_{l}} ∂_{x_{l+1}}^{α_{l+1}} ⋯ ∂_{n}^{α_{n}}u_{z}(ξ, x′′)| ≤ \frac{c_{j} |ξ|^{4m}}{|Im z|} ∥∂_{ξ_{1}}^{α_{1}} ⋯ ∂_{ξ_{l}}^{α_{l}}(Ff)(ξ, .)∥_{H^{j}(D)}

for (ξ, x′′) ∈ \overline{Ω} if j ≥ α_{l+1} + … + α_{n} + \frac{n−l+1}{2}. Since Ff ∈ S, this proves u_{z} ∈ S and, interchanging differentiation and integration, that

((−Δ)^{m} − z) F^{−1}u_{z} = f.

Furthermore u_{z}(ξ, .) ∈ ∘ H^{m}(D) for every ξ ∈ ℝ^{n} implies that F^{−1}u_{z} ∈ ∘ H^{m}(Ω). Since F^{−1}u_{z} ∈ S ⊂ H^{2m}(Ω), (3.5) is proved. □

Theorem 3.2 Let Ω ⊂ ℝ^{n} and the operator A be given by (1.1) and (3.1), respectively. Denote by {P_{λ}} the left-hand continuous spectral family of A. Then

P_{λ} = 0  for λ ≤ μ_{1}^{(0)},  (3.7)

μ_{1}^{(0)} > 0 denoting the smallest eigenvalue of B_{0} (see Theorem 2.1). If f ∈ C_{0}^{∞}(Ω) and λ > μ_{1}^{(0)}, then

(P_{λ}f)(x) = \frac{1}{(2π)^{l/2}} ∫_{μ_{1}^{(0)}}^{λ} ∑_{j=1}^{J(ρ)} \frac{d r_{j}(ρ)}{dρ} r_{j}(ρ)^{l−1}I_{j}(r_{j}(ρ), x) dρ,  (3.8)

with r_{j}(ρ) being defined in Theorem 2.1,

J(ρ) := min{j ∈ ℕ : μ_{j}^{(0)} ≥ ρ} − 1  for ρ ∈ ℝ  (3.9)

and

I_{j}(r, x′, x′′) := ∫_{|ξ|=r} e^{iξ·x′}⟨(Ff)(ξ, .), v_{j}^{(r)}⟩_{D} dS_{ξ} v_{j}^{(r)}(x′′)  (3.10)

for j ∈ ℕ, r ≥ 0, (x′, x′′) ∈ \overline{Ω}; here the integral on the right-hand side has to be read as

∫_{|ξ|=r} φ(ξ) dS_{ξ} := φ(r) + φ(−r)  if l = 1.  (3.11)

Proof: We use Stone’s formula

\frac{1}{2} ⟨P_{β+0}f + P_{β}f − P_{α+0}f − P_{α}f, g⟩ =

= \frac{1}{2πi} lim_{τ↓0} ∫_{α}^{β} ⟨R_{ρ+iτ}f − R_{ρ−iτ}f, g⟩ dρ for f, g ∈ C_{0}^{∞}(Ω).  (3.12)

Note that the Fourier-transform u_{z} of R_{z} (cp. (3.5)) can be represented as Fourierseries according to Theorem 2.1:

u_{z}(ξ, x′′) = ∑_{j=1}^{∞} \frac{1}{μ_{j}^{(|ξ|)} − z} ⟨(Ff)(ξ, .), v_{j}^{(|ξ|)}⟩_{D}v_{j}^{(|ξ|)}(x′′).

Using elliptic estimates, pointwise convergence for ξ ∈ ℝ^{l}, x′′ ∈ \overline{D} can be shown. This implies that

(R_{ρ+iτ}f)(x′, x′′) − (R_{ρ−iτ}f)(x′, x′′) =

= \frac{1}{(2π)^{l/2}} ∫_{ℝ^{l}} e^{iξ·x′} (u_{ρ+iτ}(ξ, x′′) − u_{ρ−iτ}(ξ, x′′)) dξ

= \frac{1}{(2π)^{l/2}} ∫_{ℝ^{l}} e^{iξ·x′} (∑_{j=1}^{∞} \frac{2iτ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} ⟨(Ff)(ξ, .), v_{j}^{(|ξ|)}⟩_{D}v_{j}^{(|ξ|)}(x′′)) dξ

(3.13)

for (ξ, x′′) ∈ \overline{Ω}, z ∈ ℂ \ ℝ. Set L(β) := min\{j ∈ ℕ : μ_{j}^{(0)} > β\} and

ψ_{ρ}(ξ, x′′) := ∑_{j=L(β)}^{∞} \frac{2iτ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} ⟨(Ff)(ξ, .), v_{j}^{(|ξ|)}⟩_{D}v_{j}^{(|ξ|)}(x′′)

for (ξ, x′′) ∈ Ω and ρ ≤ β. With μ_{j}^{(|ξ|)} ≥ μ_{j}^{(0)} ≥ μ_{L(β)}^{(0)} for j ≥ L(β) one obtains

∥ψ_{ρ}(ξ, .)∥_{D}^{2} = ∑_{j=L(β)}^{∞} (\frac{2τ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} )^{2} |⟨(Ff)(ξ, .), v_{j}^{(|ξ|)}⟩_{D}|^{2}

≤ \frac{4τ^{2}}{((μ_{k}^{(0)} − ρ)^{2} + τ^{2})^{2}} ∑_{j=L(β)}^{∞} |⟨(Ff)(ξ, .), v_{j}^{(|ξ|)}⟩_{D}|^{2}

≤ \frac{4τ^{2}}{((μ_{k}^{(0)} − β)^{2} + τ^{2})^{2}} ∥(Ff)(ξ, .)∥_{D}^{2}

and

∥F^{−1}ψ_{ρ}∥_{Ω}^{2} = ∥ψ_{ρ}|∥_{Ω}^{2} = ∫_{ℝ^{l}} ∥ψ_{ρ}(ξ, .)∥_{D}^{2} dξ
≤ \frac{4τ^{2}}{((μ_{k}^{(0)} − β)^{2} + τ^{2})^{2}} ∫_{ℝ^{l}} ∥(Ff)(ξ, .)∥_{D}^{2} dξ

= \frac{4τ^{2}}{((μ_{k}^{(0)} − β)^{2} + τ^{2})^{2}} ∥f∥_{Ω}^{2}

for every ξ ∈ ℝ^{l}. This implies that

lim_{τ↓0} ∫_{ℝ^{l}} e^{iξ·x′} ∑_{j=L(β)}^{∞} \frac{2iτ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} ⟨(Ff)(ξ, .), v_{j}^{(|ξ|)}⟩_{D}v_{j}^{(|ξ|)}(x′′) dξ = 0 (3.14)

in sense of L_{2}(Ω)-norm uniformly with respect to ρ ≤ β. In case of β < μ_{1}^{(0)}, this proves that the right-hand side of (3.12) vanishes and hence that P_{λ}f = 0 for λ < μ_{1}^{(0)}, f ∈ C_{0}^{∞}(Ω). By the left-hand continuity of P_{λ}, (3.7) follows.

In case of β ≥ μ_{1}^{(0)}, (3.12), (3.13) and (3.14) imply that

\frac{1}{2} ⟨P_{β+0}f + P_{β}f − P_{α+0}f − P_{α}f, g⟩ =

= \frac{1}{i (2π)^{1+l/2}} lim_{τ↓0} ∫_{α}^{β} (∫_{ℝ^{l}} ( ∑_{j=1}^{L(β)−1} \frac{2iτ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} φ_{j}(ξ)) dξ) dρ (3.15)

with

φ_{j}(ξ) := ∫_{Ω} e^{iξ·x′′}⟨(Ff)(ξ, .), v_{j}^{(|ξ|)}⟩_{D} v_{j}^{(|ξ|)}(x′′) \overline{g(x′, x′′)} d(x′, x′′)  (3.16)

for (x′, x′′) ∈ \overline{Ω}. Let j ∈ ℕ be fixed and suppose that ρ > μ_{j}^{(0)}. We prove that

∫_{ℝ^{l}} \frac{2iτ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} φ_{j}(ξ) dξ = 2πi \frac{dr_{j}(ρ)}{dρ} ∫_{|ξ|=r_{j}(ρ)} φ_{j}(ξ) dS_{ξ} + o(1) (3.17)

as τ ↓ 0 uniformly with respect to ρ in every compact subset of (μ_{j}^{(0)}, ∞). Note that r_{j} is a strictly increasing function. Fix δ ∈ (0, r_{j}(ρ)) and set

Φ_{j}(ξ, x′′) := \{ \begin{matrix}0 & if |ξ − r_{j}(ρ)| < δ, \\ \frac{2iτ}{(λ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} ⟨(Ff)(ξ, .), v_{j}^{(|ξ|)}⟩_{D}v_{j}^{(|ξ|)}(x′′) & otherwise. \end{matrix}

Since |λ_{j}^{(|ξ|)} − ρ| ≥ c > 0 for |ξ − r_{j}(ρ)| ≥ δ, one obtains

∥Φ_{j}(ξ, .)∥_{D} ≤ \frac{2τ}{c^{2}} |⟨(Ff)(ξ, .), v_{j}^{(|ξ|)}⟩_{D}| ≤ \frac{2τ}{c^{2}} ∥(Ff)(ξ, .)∥_{D}

for ξ ∈ ℝ^{l} and

∥F^{−1}Φ_{j}∥_{Ω}^{2} = ∥Φ_{j}∥_{Ω}^{2} ≤ \frac{4 τ^{2}}{c^{4}} ∥Ff∥_{Ω}^{2} = \frac{4 τ^{2}}{c^{4}} ∥f∥_{Ω}^{2} (1 ≤ j ≤ L(ρ)).

This shows that

∫_{|ξ−r_{j}(ρ)|≥δ} \frac{2iτ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} φ_{j}(ξ) dξ = O(τ)

as τ ↓ 0. The substitution μ = μ_{j}^{(r)} (⇔ r = r_{j}(μ)) and δ_{1} := ρ − μ_{j}^{(r_{j}(ρ)−δ_{0})} > 0, δ_{2} := μ_{j}^{(r_{j}(ρ)+δ_{0})} − ρ > 0 yields

∫_{|ξ−r_{j}(ρ)|≤δ} \frac{2iτ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} φ_{j}(ξ) dξ =

= ∫_{r_{j}(ρ)−δ_{0}}^{r_{j}(ρ)+δ_{0}} \frac{2iτ}{(μ_{j}^{(r)} − ρ)^{2} + τ^{2}} (∫_{|ξ|=r} φ(ξ) dS_{ξ}) dr

= ∫_{ρ−δ_{1}}^{ρ+δ_{2}} \frac{2iτ}{(μ − ρ)^{2} + τ^{2}} \frac{dr_{j}(μ)}{dμ} (∫_{|ξ|=r_{j}(μ)} φ(ξ) dS_{ξ}) dμ
= 2πi \frac{dr_{j}(ρ)}{dρ} ∫_{|ξ|=r_{j}(ρ)} φ_{j}(ξ) dS_{ξ} + o(1) as τ ↓ 0.

The same argument shows that

∫_{ℝ^{l}} \frac{2iτ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} φ_{j}(ξ) dξ = +o(1)  (3.18)

as τ ↓ 0 uniformly with respect to ρ in every compact subset of (−∞, μ_{j}^{(0)}) and that

|∫_{ℝ^{l}} \frac{2iτ}{(μ_{j}^{(|ξ|)} − ρ)^{2} + τ^{2}} φ_{j}(ξ) dξ| ≤ \frac{c_{j}}{ρ − μ_{j}^{(0)}}  (3.19)

for 0 ≤ ρ ≤ μ_{j}^{(1)}. Insert (3.17), (3.18) and (3.19) to obtain

\frac{1}{2} ⟨P_{β+0}f + P_{β}f − P_{α+0}f − P_{α}f, g⟩ =

= \frac{1}{(2π)^{l/2}} ∫_{α}^{β} (∑_{j=1}^{J(ρ)} \frac{dr_{j}(ρ)}{dρ} ∫_{|ξ|=r_{j}(ρ)} φ_{j}(ξ) dS_{ξ}) dρ.

Since the integrand on the right-hand side depends continuously on ρ ∈ ℝ \ {μ_{1}^{(0)}, μ_{2}^{(0)}, … } and is integrabel at ρ = μ_{1}^{(0)}, μ_{2}^{(0)}, …, this proves

⟨P_{λ}f, g⟩ = ∫_{μ_{1}^{(0)}}^{λ} (∑_{j=1}^{J(ρ)} \frac{dr_{j}(ρ)}{dρ} ∫_{|ξ|=r_{j}(ρ)} φ_{j}(ξ) dS_{ξ}) dρ

for λ > μ_{1}^{(0)}. Note that φ_{j} is defined by (3.16). Since g ∈ C_{0}^{∞}(Ω) is arbitrary, (3.8) follows . □

For the next result we have to define a new enumeration of eigenvalues μ_{j}^{(0)} and eigenfunctions v_{j}^{(0)} of B_{0} (the old one was defined in Theorem 2.1).

Definition 3.3 Let 0 < μ_{1} < μ_{2} < … denote all eigenvalues of B_{0}. Here every eigenvalue appears only one time in contrary to (2.3). Each eigenvalue μ_{j} has finite multiplicity, denoted by K(j). Let {v_{j1}, …, v_{j K(j)}} be an enumeration of the eigenfunctions {v_{i}^{(0)} : μ_{i}^{(0)} = μ_{j}}.

Theorem 3.4 If f ∈ C_{0}^{∞}(Ω), then, for every x ∈ Ω, (P_{λ}f)(x) is differentiable with respect to λ for λ ∈ ℝ \ {μ_{1}, μ_{2}, … } and

\frac{dP_{λ}f}{dλ} (x) − \frac{dP_{μ}f}{dμ} (x) = O_{M}(|λ − μ|)  as λ → μ  (3.20)

if μ ∈ ℝ \ {μ_{1}, μ_{2}, … }; here the notation O_{M}(|λ − μ|) means that the remainder is of order O(|λ − μ|) uniformly with respect to x in every compact subset of \overline{Ω}. Furthermore, for every j ∈ ℕ and every x ∈ Ω, lim_{λ↑μ_{j}} \frac{dP_{λ}f}{dλ} (x) exists and

\frac{dP_{λ}f}{dλ} (x) − lim_{λ↑μ_{j}} \frac{dP_{λ}f}{dλ} (x) =

= \{ \begin{matrix}O_{M}(|λ − μ_{j}|) & as λ ↑ μ_{j} (l ∈ ℕ), \\ \frac{1}{\sqrt{λ − μ_{j}}} (P^{(j)}f)(x) + O_{M}(\sqrt{λ − μ_{j}}) & as λ ↓ μ_{j} (l = 1), \\ (P^{(j)}f)(x) + O_{M}((λ − μ_{j})^{3/4}) & as λ ↓ μ_{j} (l = 2), \\ O_{M}(λ − μ_{j}) & as λ ↓ μ_{j} (l ≥ 3), \end{matrix}  (3.21)

where

(P^{(j)}f)(x′, x′′) := \frac{1}{π(4m)^{l/2}} ∑_{k=1}^{K(j)} \frac{1}{∥∇′′^{m−1}v_{jk}∥_{D}^{l}} ∫_{Ω} f(y′, y′′)v_{jk}(y′′) d(y′, y′′) v_{jk}(x′′)

(3.22)

for (x′, x′′) ∈ \overline{Ω}.

Proof: According to (3.8),

\frac{dP_{λ}f}{dλ} (x) = \frac{1}{(2π)^{l/2}} ∑_{\{k∈ℕ:μ_{k}^{(0)}<λ\}} \frac{d r_{k}(λ)}{dλ} r_{k}(λ)^{l−1}I_{k}(r_{k}(λ), x) dλ  (3.23)

holds for λ ∈ [μ_{1}, ∞)\{μ_{1}, μ_{2}, … }. From (3.10), Ff ∈ S and analycity of r ↦ v_{j}^{(r)}, one obtains that

I_{j}(r, x) − I_{j}(s, x) = O_{M}(|r − s|) as r → s > 0.

This together with Theorem 2.1 proves (3.20). Furthermore the same argument shows that

lim_{λ↑μ_{j}} \frac{dP_{λ}f}{dλ} (x) = \frac{1}{(2π)^{l/2}} ∑_{\{k∈ℕ:μ_{k}^{(0)}<μ_{j}\}} \frac{d r_{k}(λ)}{dλ} r_{k}(λ)^{l−1}I_{k}(r_{k}(λ), x) dλ

and

\frac{dP_{λ}f}{dλ} (x) − lim_{λ↑μ_{j}} \frac{dP_{λ}f}{dλ} (x) = O_{M}(|λ − μ_{j}|) as λ ↑ μ_{j}

for each k ∈ ℕ. Hence the first estimate in (3.21) is proved.

Note that v_{j}^{(r)} = v_{j}^{(0)} + O(r^{2}) (cp. (2.8)). If l = 1, this implies that

I_{j}(r, x) = (e^{irx′}⟨(Ff)(r, .), v_{j}^{(r)}⟩_{D} + e^{−irx′}⟨(Ff)(−r, .), v_{j}^{(−r)}⟩_{D}) v_{j}^{(r)}(x′′)
= 2 ∫_{Ω} f(x′, x′′) v_{j}^{(0)}(x′′) d(x′, x′′) + O_{M}(r^{2})  as r ↓ 0.

If l = 2, then

I_{j}(r, x) = 2π\sqrt{r} ∫_{Ω} f(x′, x′′) v_{j}^{(0)}(x′′) d(x′, x′′) + O_{M}(r^{3/2}) as r ↓ 0.

Finally, if l ≥ 3, then

I_{j}(r, x) = O_{M}(r^{l−1}) as r ↓ 0.

Inserting these estimates and (2.5) in (3.23) proves the remaining three parts of (3.21). □

4 Limiting absorption principle

Theorem 4.1 Suppose that Ω and A are given by (1.1) and (3.1), respectively, and that f ∈ C_{0}^{∞}(Ω). Let R_{z} := (A − z Id)^{−1} denote the resolvent of A. Then:

1) If l ≤ 2 and ρ ∈ ℝ \ {μ_{1}, μ_{2}, … }, or if l ≥ 3 and ρ ∈ ℝ,

v_{ρ}(x) := lim_{ε↓0} ∫_{|λ−ρ|≥ε} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x) dλ + iπ \frac{dP_{ρ}f}{dρ} (x) for x ∈ \overline{Ω} (4.1)

exists with v_{ρ} ∈ C^{∞}(\overline{Ω}), and the principle of limiting absorption

(R_{ρ+iτ}f)(x) = v_{ρ}(x) + O_{M}(τ^{1/4})  as τ ↓ 0  (4.2)

holds.

2) If l = 1 and j ∈ ℕ, set

v_{μ_{j}}(x) := lim_{ε↓0} (∫_{|λ−μ_{j}|≥ε} \frac{1}{λ − μ_{j}} \frac{dP_{λ}f}{dλ} (x) dλ − \frac{2}{\sqrt{ε}} (P^{(j)}f)(x))
+ iπ lim_{λ↑μ_{j}} \frac{dP_{λ}f}{dλ} (x)  (4.3)

for x ∈ \overline{Ω} with P^{(j)}f being defined by (3.22). Then v_{μ_{j}} ∈ C^{∞}(\overline{Ω}) and

(R_{μ_{j}+iτ}f)(x) = \frac{1}{\sqrt{τ}} \frac{(1 + i)π}{\sqrt{2}} (P^{(j)}f)(x) + v_{μ_{j}}(x) + O_{M}(\sqrt{τ}) as τ ↓ 0. (4.4)

3) If l = 2 und j ∈ ℕ, set

v_{μ_{j}}(x) := lim_{ε↓0} (∫_{|λ−μ_{j}|≥ε} \frac{1}{λ − μ_{j}} \frac{dP_{λ}f}{dλ} (x) dλ + ln ε · (P^{(j)}f)(x))
+ iπ lim_{λ↑μ_{j}} \frac{dP_{λ}f}{dλ} (x) + \frac{iπ}{2} (P^{(j)}f)(x)  (4.5)

for x ∈ \overline{Ω}. Then v_{μ_{j}} ∈ C^{∞}(\overline{Ω}) and

(R_{μ_{j}+iτ}f)(x) = ln(\frac{1}{τ}) (P^{(j)}f)(x) + v_{μ_{j}}(x) + O_{M}(τ^{1/4})  as τ ↓ 0. (4.6)

Remarks: 1) Let j ∈ ℕ be fixed. If f satisfies the orthogonality condition P^{(j)}f = 0 (cp. (3.22)), then the principle of limiting absorption holds also in the cases l = 1, 2 at ρ = μ_{j}.

2) It is easy to show that v_{ρ} given by (4.1) is solution of

\begin{matrix}((−Δ)^{m} − ρ) v_{ρ} & = & f & in Ω, \\ ∂_{x_{1}}^{α_{1}} ⋯ ∂_{x_{n}}^{α_{n}} v_{ρ} & = & 0 & on ∂Ω for |α| ≤ m − 1. \end{matrix} \}  (4.7)

In the case m = 1, if ρ ∈ ℝ \ {μ_{1}, μ_{2}, … }, v_{ρ} can be uniquely characterized by this equation and some radiation conditions (see e.g. [7]). The radiation conditions show that u_{ρ}(x)e^{−iωt} is an outgoing wave. In the case m ≥ 2, the question is open, if such a characterization is possible.

Proof: Suppose that ρ ∈ ℝ, τ ≠ 0 and k ∈ ℕ. Then

A^{k} R_{ρ+iτ}f = ∫_{0}^{∞} \frac{λ^{k}}{λ − ρ − iτ} d(P_{λ}f).

Since f ∈ C_{0}^{∞}(Ω) ⊂ D(A^{k}) for every k ∈ ℕ, elliptic regularity theory shows that R_{ρ+iτ}f ∈ C^{∞}(\overline{Ω}) and

(R_{ρ+iτ}f)(x) = ∫_{0}^{∞} \frac{1}{λ − ρ − iτ} \frac{dP_{λ}f}{dλ} (x) for x ∈ \overline{Ω}.

If δ > 0 is fixed, then

∥A^{k} (∫_{|λ−ρ|≥δ} \frac{1}{λ − ρ − iτ} d(P_{λ}f) − ∫_{|λ−ρ|≥δ} \frac{1}{λ − ρ} d(P_{λ}f))∥^{2} =

= ∫_{|λ−ρ|≥δ} \frac{τ^{2} λ^{2k}}{|λ − ρ − iτ|^{2} |λ − ρ|^{2}} d(∥P_{λ}f∥^{2})
≤ \frac{τ^{2}}{δ^{4}} ∥A^{k} f∥^{2}.

Again using elliptic regularity theory, one obtains that

∫_{|λ−ρ|≥δ} \frac{1}{λ − ρ − iτ} \frac{dP_{λ}f}{dλ} (x) = ∫_{|λ−ρ|≥δ} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x) + O(τ) as τ ↓ 0 (4.8)

uniformly with respect to x ∈ \overline{Ω}. Hence it is sufficient to study

I_{δ}(τ, x) := ∫_{ρ−δ}^{ρ+δ} \frac{1}{λ − ρ − iτ} \frac{dP_{λ}f}{dλ} (x) for x ∈ \overline{Ω}  (4.9)

with fixed δ > 0. If ρ < μ_{1} and δ ≤ μ_{1} − λ, this integral vanishes, since P_{λ} = 0 for λ ≤ μ_{1} = μ_{1}^{(0)}.

First suppose that ρ ∈ [μ_{1}, ∞) \ {μ_{1}, μ_{2}, … } and choose δ > 0 and j ∈ ℕ such that [ρ − δ, ρ + δ] ⊂ (μ_{j}, μ_{j+1}). From

∫_{ρ−δ}^{ρ+δ} \frac{1}{λ − ρ − iτ} dλ = iπ + O(τ) as τ ↓ 0,
∫_{ρ−δ}^{ρ+δ} |λ − ρ| |\frac{1}{λ − ρ − iτ} − \frac{1}{λ − ρ}| dλ = O(τ ln \frac{1}{τ})  as τ ↓ 0

one obtains by (3.20) that

I_{δ}(τ, x) = iπ [\frac{dP_{λ}f}{dλ} (x)]_{λ=ρ} + ∫_{ρ−δ}^{ρ+δ} \frac{1}{λ − ρ} (\frac{dP_{λ}f}{dλ} (x) − [\frac{dP_{λ}f}{dλ} (x)]_{λ=ρ}) dλ

+ O_{M}(τ ln \frac{1}{τ}) as τ ↓ 0.

Using

lim_{ε↓0} (∫_{ρ−δ}^{ρ−ε} \frac{1}{λ − ρ} dλ + ∫_{ρ+ε}^{ρ+δ} \frac{1}{λ − ρ} dλ) = 0,

one can rewrite this as

I_{δ}(τ, x) = iπ \frac{dP_{ρ}f}{dρ} (x) + lim_{ε↓0} (∫_{ρ−δ}^{ρ−ε} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x) + ∫_{ρ+ε}^{ρ+δ} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x))

+ O_{M}(τ ln \frac{1}{τ})  as τ ↓ 0.  (4.10)

This together with (4.8) proves (4.2) in the case of arbitrary l ∈ ℕ, if ρ ∈ ℝ \ {μ_{1}, μ_{2}, … }.

Second suppose that ρ = μ_{j} for a fixed j ∈ ℕ and choose δ > 0 such that [ρ − δ, ρ + δ] ⊂ (μ_{j−1}, μ_{j+1}). Note that

∫_{ρ−δ}^{ρ+δ} |λ − ρ|^{α} |\frac{1}{λ − ρ − iτ} − \frac{1}{λ − ρ}| dλ = O(τ^{α}) as τ ↓ 0  (4.11)

(0 < α < 1). If l ≥ 3, then (4.2) is proved in the same way as above using the last equation with α = \frac{1}{4}.

In the case l = 1, set

Ψ(λ, x) := \{ \begin{matrix}\frac{dP_{λ}f}{dλ} (x) & for λ ∈ [ρ − δ, ρ), \\ lim_{λ↑ρ} \frac{dP_{λ}f}{dλ} (x) & for λ = ρ, \\ \frac{dP_{λ}f}{dλ} (x) − \frac{1}{\sqrt{λ − ρ}} (P^{(j)}f)(x) & for λ ∈ (ρ, ρ + δ] \end{matrix}  (4.12)

for x ∈ \overline{Ω}. Using

Ψ(λ, x) − Ψ(ρ, x) = O_{M}(\sqrt{|λ − ρ|}) as λ → ρ

(see (3.21) with ρ = μ_{j}) and (4.11) with α = \frac{1}{2}, one obtains as above that

∫_{ρ−δ}^{ρ+δ} \frac{1}{λ − ρ − iτ} Ψ(λ, x) dλ =

= iπΨ(ρ, x) + lim_{ε↓0} (∫_{ρ−δ}^{ρ−ε} \frac{1}{λ − ρ} Ψ(λ, x) dλ + ∫_{ρ+ε}^{ρ+δ} \frac{1}{λ − ρ} Ψ(λ, x) dλ)

+ O_{M}(\sqrt{τ})

= lim_{ε↓0} (∫_{ρ−δ}^{ρ−ε} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x) dλ + ∫_{ρ+ε}^{ρ+δ} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x) dλ − \frac{2}{\sqrt{ε}} (P^{(j)}f)(x))

+ iπ lim_{λ↑ρ} \frac{dP_{λ}f}{dλ} (x) + \frac{2}{\sqrt{δ}} (P^{(j)}f)(x) + O_{M}(\sqrt{τ})

as τ ↓ 0. By

∫_{ρ}^{ρ+δ} \frac{1}{λ − ρ − iτ} \frac{1}{\sqrt{λ − ρ}} dλ = \frac{π(1 + i)}{\sqrt{2}} \frac{1}{\sqrt{τ}} − \frac{2}{\sqrt{δ}} + O(τ) as τ ↓ 0,

(see e.g. (6.16), (6.17) in [7]), this implies that

I_{δ}(τ, x) = ∫_{ρ−δ}^{ρ+δ} \frac{1}{λ − ρ − iτ} Ψ(λ, x) dλ + ∫_{ρ}^{ρ+δ} \frac{1}{λ − ρ − iτ} \frac{1}{\sqrt{λ − ρ}} dλ (P^{(j)}f)(x)

= \frac{π(1 + i)}{\sqrt{2}} \frac{1}{\sqrt{τ}} (P^{(j)}f)(x) + iπ lim_{λ↑ρ} \frac{dP_{λ}f}{dλ} (x)

+ lim_{ε↓0} (∫_{ρ−δ}^{ρ−ε} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x) dλ + ∫_{ρ+ε}^{ρ+δ} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x) dλ − \frac{2}{\sqrt{ε}}(P^{(j)}f)(x))

+ O_{M}(\sqrt{τ})  as τ ↓ 0,  (4.13)

which together with (4.8) proves (4.4).

In the case l = 2, similar calculations and

∫_{ρ}^{ρ+δ} \frac{1}{λ − ρ − iτ} dλ = ln \frac{1}{τ} + \frac{iπ}{2} + ln δ + O(τ) as τ ↓ 0  (4.14)

show that

I_{δ}(τ, x) = ln(\frac{1}{τ}) (P^{(j)}f)(x) + iπ lim_{λ↑ρ} \frac{dP_{λ}f}{dλ} (x)

+ lim_{ε↓0} (∫_{ρ−δ}^{ρ−ε} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x) ∫_{ρ+ε}^{ρ+δ} \frac{1}{λ − ρ} \frac{dP_{λ}f}{dλ} (x) (\frac{iπ}{2} + ln ε) (P^{(j)}f)(x))

+ O_{M}(τ^{1/4})  as τ ↓ 0.  (4.15)

This proves (4.6). □

5 Long-time asymptotics

Consider the solution u of

u ∈ ⋂_{j=0}^{2} C^{j}([0, ∞), H^{2m−jm}(Ω)) ,  (5.1)
u(t) ∈ D(A)  for t ≥ 0,  (5.2)
u′′(t) + A u(t) = f e^{−iωt}  for t ≥ 0,  (5.3)
u(0) = u_{0},  u′(0) = u_{1},  (5.4)

given by

u(t) = ∫_{0}^{∞} \frac{1}{λ − ω^{2}} (e^{−iωt} − cos \sqrt{λ}t + \frac{iω}{\sqrt{λ}} sin \sqrt{λ}t) d(P_{λ}f)
+ ∫_{0}^{∞} cos \sqrt{λ}t d(P_{λ}u_{0}) + ∫_{0}^{∞} \frac{sin \sqrt{λ}t}{\sqrt{λ}} d(P_{λ}u_{1}).  (5.5)

Standard theory shows, that this solution is unique. If u_{0}, u_{1}, f ∈ C_{0}^{∞}(Ω) is supposed, then u ∈ C^{∞}([0, ∞) × \overline{Ω}) by elliptic regularity theory, and u solves (1.2). On the other hand, every solution of (1.2) satisfying (5.1) is solution of (5.2) – (5.4). Hence (1.2) has a unique solution satisfying (5.1).

By Theorems 3.2, 3.4 and 4.1, we can estimate the spectral integrals in (5.5) by the same arguments used in [13] and [7] to prove Theorem 1.1.

References

[1] S. Agmon. Lectures on Elliptic Boundary Value Problems. Van Nostrand, New York, 1965.

[2] M. D. Groves and P. L. Lesky. Resonance phenomena for water waves in channels of arbitrary cross-section. Math. Meth. in the Appl. Sci., 22:837 – 865, 1999.

[3] B. A. Iskenderov, Eh. Kh. Ehjvazov, and A. N. Ehfendieva. Radiation principles for an elliptic equation of high order in a cylindrical domain. Differ. Uravn., 23(10):1804–1807, 1987.

[4] T. Kato. Perturbation theory for linear operators. Springer-Verlag, Berlin, Heidelberg, New York, 1966.

[5] R. Leis. Initial Boundary Value Problems in Mathematical Physics. Teubner/Wiley, Stuttgart/Chichester et al., 1986.

[6] P. Lesky. Time asymptotics for the linear elastic wave equation in waveguides. to appear.

[7] P. Lesky. Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides. Math. Meth. in the Appl. Sci., 11:697– 723, 1989.

[8] P. H. Lesky. Zur Langzeitasymptotik periodisch angeregter polyharmonischer und elastischer Wellen. Shaker Verlag, Aachen, 1999.

[9] K. Morgenröther and P. Werner. Resonances and standing waves. Math. Meth. in the Appl. Sci., 9:105–126, 1987.

[10] A.G. Ramm and P. Werner. On the limit amplitude principle for a layer. J. Reine Angew. Math., 360:19 – 46, 1985.

[11] M. Reed and B. Simon. Methods of Modern Mathematical Physics, volume I. Academic Press Inc., New York, 1995 (8. ed).

[12] P. Werner. Ein Resonanzphänomen in der Theorie akustischer und elektromagnetischer Wellen. Math. Meth. in the Appl. Sci., 6:104–128, 1984.

[13] P. Werner. Resonance phenomena in cylindrical waveguides. J. Math. Anal. Appl., 121:173–214, 1987.