Scattering by the local perturbation of an open periodic waveguide in the half plane

TL;DR

This paper addresses the scattering problem caused by local perturbations in open periodic waveguides in the half plane, establishing well-posedness under certain conditions using a recently introduced radiation condition.

Contribution

It extends the application of a new radiation condition to the perturbed case, proving well-posedness for the scattering problem with local perturbations.

Findings

Well-posedness of the perturbed scattering problem established

Utilizes a recently introduced radiation condition for analysis

Provides theoretical foundation for waveguide perturbation analysis

Abstract

We consider the scattering problem of the local perturbation of an open periodic waveguide in the half plane. Recently by Kirsch and Lechleiter, a new radiation condition was introduced in order to solve the unperturbed case. In this paper, with the same radiation condition and an additional assumption we show the well-posedness of the perturbed scattering problem.

Equations

$$\Delta u^s+k^2(1+q)n{u}^s=-k^{2}qn{u}^i(\cdot ,y)\mathrm{in}{\mathbb{R}}_{+}^2,$$

$$u^s=0\mathrm{on}{\Gamma }_0,$$

$$G_n(x,y):=G(x,y)+{\tilde{u}}^s(x,y),$$

$${\Phi }_k(x,y):=\frac{i}{4}{H}_0^{(1)}(k|x-y|),x\ne y.$$

$$\Delta {\tilde{u}}^s+k^{2}n{\tilde{u}}^s=k^2(1-n)G(\cdot ,y)\mathrm{in}{\mathbb{R}}_{+}^2.$$

$$\Delta v+k^2(1+q)nv=0\mathrm{in}{\mathbb{R}}_{+}^2,$$

$$v=0\mathrm{on}{\Gamma }_0,$$

$$\Delta u+k^2(1+q)nu=f\mathrm{in}{\mathbb{R}}_{+}^2,$$

$$u=0\mathrm{on}{\Gamma }_0,$$

$$\Delta u+k^{2}nu=f\mathrm{in}{\mathbb{R}}_{+}^2,$$

$$u=0\mathrm{on}{\Gamma }_0.$$

$$\int_{\mathbb{R}^{2}_{+}}\bigl{[}\nabla u\cdot\nabla\overline{\varphi}-k^{2}nu\overline{\varphi}\bigr{]}dx=-\int_{W}f\overline{\varphi}dx,$$

$$u(x)=2{\int }_{{\Gamma }_h}u(y)\frac{\partial {\Phi }_k(x,y)}{\partial y_2}ds(y)=0,x_2>h.$$

$$T_{per}f(t,\alpha )={\tilde{f}}_{\alpha }(t):=\sum _{m\in \mathbb{Z}}f(t+2\pi m)e^{-i\alpha (t+2\pi m)},$$

$$T_{per}^{-1}g(t)={\int }_{-1/2}^{1/2}g(t,\alpha )e^{i\alpha t}d\alpha ,t\in \mathbb{R}.$$

$$\Delta {\tilde{u}}_{\alpha }+2i\alpha \frac{\partial {\tilde{u}}_{\alpha }}{\partial x_1}+(k^{2}n-{\alpha }^2){\tilde{u}}_{\alpha }={\tilde{f}}_{\alpha }\mathrm{in}(0,2\pi )\times (0,\infty ).$$

$${\tilde{u}}_{\alpha }=0\mathrm{on}(0,2\pi )\times \{0\}.$$

$${\tilde{u}}_{\alpha }(x)=\sum _{n\in \mathbb{Z}}{u}_n(\alpha )e^{in{x}_1+i\sqrt{k^2-(n+\alpha {)}^2}(x_2-h)},x_2>h,$$

$$⟨u,v{⟩}_{\ast }={\int }_{C_h}\nabla u\cdot \nabla \overline{v}dx+2\pi \sum _{n\in \mathbb{Z}}\sqrt{n^2+1}{u}_n\overline{v_n},$$

$${\tilde{u}}_{\alpha }-K_{\alpha }{\tilde{u}}_{\alpha }={\tilde{f}}_{\alpha }\mathrm{in}{H}_{0,per}^1(C_h),$$

$$⟨K_{\alpha }u,v{⟩}_{\ast }$$

$$X_{j}:=\left\{\phi\in H^{1}_{loc}(\mathbb{R}^{2}_{+}):\begin{array}[]{cc}\Delta\phi+2i\alpha_{j}\frac{\partial\phi}{\partial x_{1}}+(k^{2}n-\alpha^{2})\phi=0\ \mathrm{in}\ \mathbb{R}^{2}_{+},\\ \phi=0\ \mathrm{for}\ x_{2}=0,\ \ \ \phi\ \mathrm{is}\ 2\pi\mathrm{-periodic}\ \mathrm{for}\ x_{1},\\ \phi\ \mathrm{satisfies\ the\ Rayleigh\ expansion}\ (\ref{2.9})\end{array}\right\}$$

$${\int }_{C_{\infty }}\left[-i\frac{\partial \varphi }{\partial x_1}+{\alpha }_j\varphi \right]\overline{\psi }dx=dk{\int }_{C_{\infty }}n\varphi \overline{\psi }dx,$$

$${\int }_{C_{\infty }}\left[-i\frac{\partial {\varphi }_{l,j}}{\partial x_1}+{\alpha }_j{\varphi }_{l,j}\right]\overline{\psi }dx=d_{l,j}k{\int }_{C_{\infty }}n{\varphi }_{l,j}\overline{\psi }dx,$$

$$k{\int }_{C_{\infty }}n{\varphi }_{l,j}\overline{{\varphi }_{l^{\prime },j}}dx={\delta }_{l,l^{\prime }},$$

$${\psi }^{\pm }(x_1):=\frac{1}{2}\left[1\pm \frac{2}{\pi }{\int }_0^{x_1/2}\frac{sint}{t}dt\right],x_1\in \mathbb{R}.$$

$$u^{(2)}(x)={\psi }^{+}(x_1)\sum _{j\in J}\sum _{d_{l,j}>0}{a}_{l,j}{\varphi }_{l,j}(x)+{\psi }^{-}(x_1)\sum _{j\in J}\sum _{d_{l,j}<0}{a}_{l,j}{\varphi }_{l,j}(x)$$

$$u(x)=k^2{\int }_W(n(y)-1)u(y)G(x,y)dy-{\int }_{W}f(y)G(x,y)dy,x\in {\mathbb{R}}_{+}^2$$

$$\Delta u_{ϵ}+(k+iϵ{)}^{2}n{u}_{ϵ}=f\mathrm{in}{\mathbb{R}}_{+}^2,$$

$$u_{ϵ}=0\mathrm{on}{\Gamma }_0.$$

Full text

1 Introduction

Let $k>0$ be the wave number, and let $\mathbb{R}^{2}_{+}:=\mathbb{R}\times(0,\infty)$ be the upper half plane, and let $W:=\mathbb{R}\times(0,h)$ be the waveguide in $\mathbb{R}^{2}_{+}$. We denote by $\Gamma_{a}:=\mathbb{R}\times\{a\}$ for $a>0$. Let $n\in L^{\infty}(\mathbb{R}^{2}_{+})$ be real value, $2\pi$-periodic with respect to $x_{1}$ (that is, $n(x_{1}+2\pi,x_{2})=n(x_{1},x_{2})$ for all $x=(x_{1},x_{2})\in\mathbb{R}^{2}_{+}$), and equal to one for $x_{2}>h$. We assume that there exists a constant $n_{0}>0$ such that $n\geq n_{0}$ in $\mathbb{R}^{2}_{+}$. Let $q\in L^{\infty}(\mathbb{R}^{2}_{+})$ be real value with the compact support in $W$. We denote by $Q:=\mathrm{supp}q$. In this paper, we consider the following scattering problem: For fixed $y\in\mathbb{R}^{2}_{+}\setminus\overline{W}$, determine the scattered field $u^{s}\in H^{1}_{loc}(\mathbb{R}^{2}_{+})$ such that

[TABLE]

[TABLE]

Here, the incident field $u^{i}$ is given by $u^{i}(x,y)=G_{n}(x,y)$, where $G_{n}$ is the Dirichlet Green’s function in the upper half plane $\mathbb{R}^{2}_{+}$ for $\Delta+k^{2}n$, that is,

[TABLE]

where $G(x,y):=\Phi_{k}(x,y)-\Phi_{k}(x,y^{*})$ is the Dirichlet Green’s function in $\mathbb{R}^{2}_{+}$ for $\Delta+k^{2}$, and $y^{*}=(y_{1},-y_{2})$ is the reflected point of $y$ at $\mathbb{R}\times\{0\}$. Here, $\Phi_{k}(x,y)$ is the fundamental solution to Helmholtz equation in $\mathbb{R}^{2}$, that is,

[TABLE]

$\tilde{u}^{s}$ is the scattered field of the unperturbed problem by the incident field $G(x,y)$, that is, $\tilde{u}^{s}$ vanishes for $x_{2}=0$ and solves

[TABLE]

If we impose a suitable radiation condition introduced by Kirsch and Lechleiter [6], the unperturbed solution $\tilde{u}^{s}$ is uniquely determined. Later, we will explain the exact definition of this radiation condition (see Definition 2.4).

In order to show the well-posedness of the perturbed scattering problem (1.1)–(1.2), we make the following assumption.

Assumption 1.1**.**

We assume that $k^{2}$ is not the point spectrum of $\frac{1}{(1+q)n}\Delta$ in $H^{1}_{0}(\mathbb{R}^{2}_{+})$, that is, evey $v\in H^{1}(\mathbb{R}^{2}_{+})$ which satisfies

[TABLE]

[TABLE]

has to vanish for $x_{2}>0$.

If we assume that $q$ and $n$ satisfy in addition that \partial_{2}\bigl{(}(1+q)n\bigr{)}\geq 0 in $W$, then $v$ which satisfies (1.6)–(1.7) vanishes, that is, under this assumption all of $k^{2}$ is not the point spectrum of $\frac{1}{(1+q)n}\Delta$. We will prove it in Section 6. Our aim in this paper is to show the following theorem.

Theorem 1.2**.**

Let Assumptions 1.1 and 2.1 hold and let $k>0$ be regular in the sense of Definition 2.3 and let $f\in L^{2}(\mathbb{R}^{2}_{+})$ such that $\mathrm{supp}f=Q$. Then, there exists a unique solution $u\in H^{1}_{loc}(\mathbb{R}^{2}_{+})$ such that

[TABLE]

[TABLE]

and $u$ satisfies the radiation condition in the sense of Definition 2.4.

Roughly speaking, the radiation condition of Definition 2.4 requires that we have a decomposition of the solution $u$ into $u^{(1)}$ which decays in the direction of $x_{1}$, and a finite combination $u^{(2)}$ of propagative modes which does not decay, but it exponentially decays in the direction of $x_{2}$.

This paper is organized as follows. In Section 2, we briefly recall a radiation condition introduced in [6], and show that the solution of (2.1)–(2.2) has an integral representation (2.18). Under the radiation condition in the sense of Definition 2.4, we show the uniqueness of $u^{(2)}$ and $u^{(1)}$ in Section 3 and 4, respectively. In Section 5, we show the existence of $u$. In Section 6, we will give an example of $n$ and $q$ with respect to Assumption 1.1.

2 A radiation condition

In Section 2, we briefly recall a radiation condition introduced in [6]. Let $f\in L^{2}(\mathbb{R}^{2}_{+})$ have the compact support in $W$. First, we consider the following problem: Find $u\in H^{1}_{loc}(\mathbb{R}^{2}_{+})$ such that

[TABLE]

[TABLE]

(2.1) is understood in the variational sense, that is,

[TABLE]

for all $\varphi\in H^{1}(\mathbb{R}^{2}_{+})$, with compact support. In such a problem, it is natural to impose the upward propagating radiation condition, that is, $u(\cdot,h)\in L^{\infty}(\mathbb{R})$ and

[TABLE]

However, even with this condition we can not expect the uniqueness of this problem. (see Example 2.3 of [6].) In order to introduce a suitable radiation condition, Kirsch and Lechleiter discussed limiting absorption solution of this problem, that is, the limit of the solution $u_{\epsilon}$ of $\Delta u_{\epsilon}+(k+i\epsilon)^{2}nu_{\epsilon}=f$ as $\epsilon\to 0$. For the details, we refer to [5, 6].

Let us prepare for the exact definition of the radiation condition. First we recall that the Floquet Bloch transform T_{per}:L^{2}(\mathbb{R})\to L^{2}\bigl{(}(0,2\pi)\times(-1/2,1/2)\bigr{)} is defined by

[TABLE]

for $(t,\alpha)\in(0,2\pi)\times(-1/2,1/2)$. The inverse transform is given by

[TABLE]

By taking the Floquet Bloch transform with respect to $x_{1}$ in (2.1)–(2.2), we have for $\alpha\in[-1/2,1/2]$

[TABLE]

[TABLE]

By taking the Floquet Bloch transform with respect to $x_{1}$ in (2.4), $\tilde{u}_{\alpha}$ satisfies the Rayleigh expansion of the form

[TABLE]

where $u_{n}(\alpha):=(2\pi)^{-1}\int_{0}^{2\pi}u_{\alpha}(x_{1},h)e^{-inx_{1}}dx_{1}$ are the Fourier coefficients of $u_{\alpha}(\cdot,h)$, and $\sqrt{k^{2}-(n+\alpha)^{2}}=i\sqrt{(n+\alpha)^{2}-k^{2}}$ if $n+\alpha>k$.

We denote by $C_{R}:=(0,2\pi)\times(0,R)$ for $R\in(0,\infty]$, and $H^{1}_{per}(C_{R})$ the subspace of the $2\pi$-periodic function in $H^{1}(C_{R})$. We also denote by $H^{1}_{0,per}(C_{R}):=\{u\in H^{1}_{per}(C_{R}):u=0\ \mathrm{on}\ (0,2\pi)\times\{0\}\}$ that is equipped with $H^{1}(C_{R})$ norm. The space $H^{1}_{0,per}(C_{R})$ has the inner product of the form

[TABLE]

where $u_{n}=(2\pi)^{-1}\int_{0}^{2\pi}u(x_{1},R)e^{-inx_{1}}dx_{1}$. The problem (2.7)–(2.9) is equivalent to the following operator equation (see section 3 in [6]),

[TABLE]

where the operator $K_{\alpha}:H^{1}_{0,per}(C_{h})\to H^{1}_{0,per}(C_{h})$ is defined by

[TABLE]

For several $\alpha\in(-1/2,1/2]$, the uniqueness of this problem fails. We call $\alpha$ exceptional values if the operator $I-K_{\alpha}$ fails to be injective. For the difficulty of treatment of $\alpha$ such that $|\alpha+l|=k$ for some $l\in\mathbb{Z}$ in periodic scattering problem, we set $A_{k}:=\{\alpha\in(-1/2,1/2]:\exists l\in\mathbb{Z}\ s.t.\ |\alpha+l|=k\}$, and make the following assumption:

Assumption 2.1**.**

For every $\alpha\in A_{k}$, $I-K_{\alpha}$ has to be injective.

The following properties of exceptional values was shown in [6].

Lemma 2.2**.**

Let Assumption 2.1 hold. Then, there exists only finitely many exceptional values $\alpha\in(-1/2,1/2]$. Furthermore, if $\alpha$ is an exceptional value, then so is $-\alpha$. Therefore, the set of exceptional values can be described by $\{\alpha_{j}:j\in J\}$ where some $J\subset\mathbb{Z}$ is finite and $\alpha_{-j}=-\alpha_{j}$ for $j\in J$. For each exceptional value $\alpha_{j}$ we define

[TABLE]

Then, $X_{j}$ are finite dimensional. We set $m_{j}=\mathrm{dim}X_{j}$. Furthermore, $\phi\in X_{j}$ is evanescent, that is, there exists $c>0$ and $\delta>0$ such that $|\phi(x)|,\ |\nabla\phi(x)|\leq ce^{-\delta|x_{2}|}$ for all $x\in\mathbb{R}^{2}_{+}$.

Next, we consider the following eigenvalue problem in $X_{j}$: Determine $d\in\mathbb{R}$ and $\phi\in X_{j}$ such that

[TABLE]

for all $\psi\in X_{j}$. We denote by the eigenvalues $d_{l,j}$ and eigenfunction $\phi_{l,j}$ of this problem, that is,

[TABLE]

for every $l=1,...,m_{j}$ and $j\in J$. We normalize the eigenfunction $\{\phi_{l,j}:l=1,...,m_{j}\}$ such that

[TABLE]

for all $l,l^{\prime}$. We will assume that the wave number $k>0$ is regular in the following sense.

Definition 2.3**.**

$k>0$ is regular if $d_{l,j}\neq 0$ for all $l=1,...m_{j}$ and $j\in J$.

Now we are ready to define the radiation condition.

Definition 2.4**.**

Let Assumptions 2.1 hold, and let $k>0$ be regular in the sense of Definition 2.3. We set

[TABLE]

Then, $u\in H^{1}_{loc}(\mathbb{R}^{2}_{+})$ satisfies the radiation condition if $u$ satisfies the upward propagating radiation condition (2.4), and has a decomposition in the form $u=u^{(1)}+u^{(2)}$ where u^{(1)}\bigl{|}_{\mathbb{R}\times(0,R)}\in H^{1}(\mathbb{R}\times(0,R)) for all $R>0$, and $u^{(2)}\in L^{\infty}(\mathbb{R}^{2}_{+})$ has the following form

[TABLE]

where some $a_{l,j}\in\mathbb{C}$, and $\{d_{l,j},\phi_{l,j}:l=1,...,m_{j}\}$ are normalized eigenvalues and eigenfunctions of the problem (2.8).

Remark 2.5**.**

It is obvious that we can replace $\psi^{+}$ by any smooth functions $\tilde{\psi}^{\pm}$ with $\tilde{\psi}^{+}(x_{1})=1+\mathcal{O}(1/x_{1})$ as $x_{1}\to\infty$ and $\tilde{\psi}^{+}(x_{1})=\mathcal{O}(1/x_{1})$ as $x_{1}\to-\infty$ and $\frac{d}{dx_{1}}\tilde{\psi}^{+}(x_{1})\to 0$ as $|x_{1}|\to\infty$ (and analogously for $\psi^{-}$).

The following was shown in Theorems 2.2, 6.6, and 6.8 of [6].

Theorem 2.6**.**

Let Assumptions 2.1 hold and let $k>0$ be regular in the sense of Definition 2.3. For every $f\in L^{2}(\mathbb{R}^{2}_{+})$ with the compact support in $W$, there exists a unique solution $u_{k+i\epsilon}\in H^{1}(\mathbb{R}^{2}_{+})$ of the problem (2.1)–(2.2) replacing $k$ by $k+i\epsilon$. Furthermore, $u_{k+i\epsilon}$ converge as $\epsilon\to+0$ in $H^{1}_{loc}(\mathbb{R}^{2}_{+})$ to some $u\in H^{1}_{loc}(\mathbb{R}^{2}_{+})$ which satisfy (2.1)–(2.2) and the radiation condition in the sense of Definition 2.4. Furthermore, the solution $u$ of this problem is uniquely determined.

We have recalled the radiation condition and its properties. Finally in this section, we will show the following integral representation.

Lemma 2.7**.**

Let $f\in L^{2}(\mathbb{R}^{2}_{+})$ have a compact support in $W$, and let $u$ be a solution of (2.1)–(2.2) which satisfying the radiation condition in the sense of Definition 2.4. Then, $u$ has an integral representation of the form

[TABLE]

Proof of Lemma 2.7.

Let $\epsilon>0$ be small enough and let $u_{\epsilon}\in H^{1}(\mathbb{R}^{2}_{+})$ be a solution of the problem (2.1)–(2.2) replacing $k$ by $k+i\epsilon$, that is, $u_{\epsilon}$ satisfies

[TABLE]

[TABLE]

Let $G_{\epsilon}(x,y)$ be the Dirichlet Green’s function in the upper half plane $\mathbb{R}^{2}_{+}$ for $\Delta+(k+i\epsilon)^{2}$. Let $x\in\mathbb{R}^{2}_{+}$ be always fixed such that $x_{2}>R$. Let $r>0$ be large enough such that $x\in B_{r}(0)$ where $B_{r}(0)\subset\mathbb{R}^{2}$ be a open ball with center [math] and radius $r>0$. By Green’s representation theorem in $B_{r}(0)\cap\mathbb{R}^{2}_{+}$ we have

[TABLE]

Since $u_{\epsilon}\in H^{1}(\mathbb{R}^{2}_{+})$, the first term of the right hand side converges to zero as $r\to\infty$. Therefore, as $r\to\infty$ we have for $x\in\mathbb{R}^{2}_{+}$

[TABLE]

We will show that (2.22) converges as $\epsilon\to 0$ to

[TABLE]

Indeed, by the argument in (3.8) and (3.9) of [2], $G_{\epsilon}(x,y)$ is of the estimation

[TABLE]

where above $C$ is independent of $\epsilon>0$. Then, by Lebesgue dominated convergence theorem we have the second integral in (2.22) converges as $\epsilon\to 0$ to one in (2.23). So, we will consider the convergence of the first integral in (2.22).

By the beginning of the proof of Theorem 6.6 in [6], $u_{\epsilon}$ can be of the form $u_{\epsilon}=u^{(1)}_{\epsilon}+u^{(2)}_{\epsilon}$ where $u^{(1)}_{\epsilon}$ converges to $u^{(1)}$ in $H^{1}(W)$, and $u^{(2)}_{\epsilon}$ is of the form for $x\in W$

[TABLE]

which converges pointwise to $u^{(2)}(x)$. Here, $y_{l,j}\in\mathbb{C}$ is some constant. From the convergence of $u^{(1)}_{\epsilon}$ in $H^{1}(W)$ we obtain that $\int_{W}(n(y)-1)u^{(1)}_{\epsilon}(y)G_{\epsilon}(x,y)dy$ converges $\int_{W}(n(y)-1)u^{(1)}(y)G(x,y)dy$ as $\epsilon\to 0$.

By the argument of (b) in Lemma 6.1 of [6] we have

[TABLE]

which implies that for all $x_{1}\in\mathbb{R}$

[TABLE]

where above $C$ is independent of $\epsilon>0$. Then, we have that for $y\in W$

[TABLE]

where above $C$ is independent of $y$ and $\epsilon$. Then, right hand side of (2.28) is an integrable function in $W$ with respect to $y$. Then, by Lebesgue dominated convergence theorem $\int_{W}(n(y)-1)u^{(2)}_{\epsilon}(y)G_{\epsilon}(x,y)dy$ converges to $\int_{W}(n(y)-1)u^{(2)}(y)G(x,y)dy$ as $\epsilon\to 0$. Therefore, (2.23) has been shown. ∎

3 Uniqueness of $u^{(2)}$

In Section 3, we will show the uniqueness of $u^{(2)}$ in Theorem 1.2.

Lemma 3.1**.**

Let Assumptions 2.1 hold and let $k>0$ be regular in the sense of Definition 2.3. If $u\in H^{1}_{loc}(\mathbb{R}^{2}_{+})$ such that

[TABLE]

[TABLE]

and $u$ satisfies the radiation condition in the sense of Definition 2.4, then $u^{(2)}=0$ in $\mathbb{R}^{2}_{+}$.

Proof of Lemma 3.1.

By the definition of the radiation condition, $u$ is of the form $u=u^{(1)}+u^{(2)}$ where u^{(1)}\bigl{|}_{\mathbb{R}\times(0,R)}\in H^{1}(\mathbb{R}\times(0,R)) for all $R>0$, and $u^{(2)}\in L^{\infty}(\mathbb{R}^{2}_{+})$ has the form

[TABLE]

where some $a_{l,j}\in\mathbb{C}$, and $\{d_{l,j},\phi_{l,j}:l=1,...,m_{j}\}$ are normalized eigenvalues and eigenfunctions of the problem (2.13). Here, by Remark 2.5 the function $\psi^{+}$ is chosen as a smooth function such that $\psi^{+}(x_{1})=1$ for $x_{1}\geq\eta$ and $\psi^{+}(x_{1})=0$ for $x_{1}\leq-\eta$, and $\psi^{-}:=1-\psi^{+}$ where $\eta>0$ is some positive number.

Step1 (Green’s theorem in $\Omega_{N}$): We set $\Omega_{N}:=(-N,N)\times(0,\phi(N))$ where $\psi(N):=N^{s}$. Later we will choose a appropriate $s\in(0,1)$. Let $R>h$ be large and always fixed, and let $N$ be large enough such that $\phi(N)>R$. We denote by $I_{\pm N}^{R}:=\{\pm N\}\times(0,R)$, $I_{\pm N}^{\phi(N)}:=\{\pm N\}\times(R,\phi(N))$, and $\Gamma_{\phi(N),N}:=(-N,N)\times\{\phi(N)\}$. (see the figure below.) We set $I_{\pm N}:=I_{\pm N}^{R}\cup I_{\pm N}^{\phi(N)}$.

x_{1}$$x_{2}$$O$$N$$-N$$\phi(N)$$R$$\Gamma_{\phi(N),N}

\Biggl{\{}

\Biggl{\{}$$\Biggl{\{}$$\Biggr{\}}$$\Biggr{\}}$$I_{-N}^{R}$$I_{-N}^{\phi(N)}$$I_{N}^{R}$$I_{N}^{\phi(N)}

By Green’s first theorem in $\Omega_{N}$ and $u=0$ on $(-N,N)\times\{0\}$, we have

[TABLE]

[TABLE]

By the same argument in Theorem 4.6 of [5] and Lemma 6.3 of [6], we can show that

[TABLE]

and the first and second term in the right hand side converge as $N\to\infty$ to $\frac{ik}{2\pi}\sum_{j\in J}\sum_{d_{l,j}>0}|a_{l,j}|^{2}d_{l,j}$ and $-\frac{ik}{2\pi}\sum_{j\in J}\sum_{d_{l,j}<0}|a_{l,j}|^{2}d_{l,j}$ respectively. Therefore, taking an imaginary part in (3.4) yields that

[TABLE]

We set

[TABLE]

and we will show that $\mathrm{limsup_{N\to\infty}}J_{\pm}(N)\geq 0$.

Step2 ($\mathrm{limsup_{N\to\infty}}J_{\pm}(N)\geq 0$): By Cauchy Schwarz inequality we have

[TABLE]

In order to estimate $u^{(1)}$, we will show the following lemma.

Lemma 3.2**.**

$u^{(1)}$* has an integral representation of the form*

[TABLE]

where $\sigma:=\Delta u^{(2)}+k^{2}nu^{(2)}$.

Proof of Lemma 3.2.

First, we will consider an integral representation of $u^{(2)}$. Let $N>0$ be large enough. By Green’s representation theorem in $(-N,N)\times(0,N^{1/4})$, we have

[TABLE]

[TABLE]

By Lemma 3.1 of [2], the Dirichlet Green’s function $G(x,y)$ is of the estimation

[TABLE]

By Lemma 2.2 we have that |u^{(2)}(x)|,\ \bigl{|}\frac{\partial u^{(2)}(x)}{\partial x_{2}}\bigr{|}\leq ce^{-\delta|x_{2}|} for all $x\in\mathbb{R}^{2}_{+}$, and some $c,\delta>0$. Then, we obtain

[TABLE]

Furthermore,

[TABLE]

Therefore, as $N\to\infty$ in (3.10) we get

[TABLE]

By Lemma 2.7, we have (substitute $-k^{2}qnu$ for $f$ in (2.18))

[TABLE]

Combining (3.14) with (3.15) we have

[TABLE]

[TABLE]

Therefore, Lemma 3.2 has been shown. ∎

We set $u^{\pm}(x):=\sum_{j\in J}\sum_{d_{l,j}\lessgtr 0}a_{l,j}\phi_{l,j}(x)$. Then, by simple calculation we can show

[TABLE]

which implies that $\mathrm{supp}\sigma\subset(-\eta,\eta)\times(0,\infty)$. By Lemma 3.2 we have for $R<x_{2}<\phi(N)$

[TABLE]

We have to estimate the second term in right hand side. The following lemma was shown in Lemma 4.12 of [1].

Lemma 3.3**.**

Assume that $\varphi\in L^{2}_{loc}(\mathbb{R})$ such that

[TABLE]

for some $\epsilon>0$. Then, for every $\alpha\in[0,\frac{1}{2}-\epsilon)$ there exists a constant $C>0$ and a sequence $\{A_{m}\}_{m\in\mathbb{N}}$ such that $A_{m}\to\infty$ as $m\to\infty$ and

[TABLE]

where $K_{A}:=K^{+}_{A}\cup K^{-}_{A}$, $K^{+}_{A}:=(-A^{+},A^{+})\setminus(-A,A)$, $K^{-}_{A}:=(-A,A)\setminus(-A^{-},A^{-})$, and $A^{\pm}:=A\pm A^{1/2}$ for $A\in[1,\infty)$.

Applying Lemma 3.3 to \varphi=\bigl{(}\int^{h}_{0}\bigl{|}u^{(1)}(\cdot,y_{2})\bigr{|}^{2}dy_{2}\bigr{)}^{1/2}\in L^{2}(\mathbb{R}), there exists a sequence $\{N_{m}\}_{m\in\mathbb{N}}$ such that $N_{m}\to\infty$ as $m\to\infty$ and

[TABLE]

Then, by Cauchy Schwarz inequality we have

[TABLE]

With (3.18) we have for $m\in\mathbb{N}$,

[TABLE]

Therefore, by (3.8) we have

[TABLE]

Since $\phi(N)=N^{s}$, if we choose $s\in(0,1)$ such that $3s<\frac{1}{8}$, that is, $0<s<\frac{1}{24}$ the right hand side in (3.24) converges to zero as $m\to\infty$. Therefore, $\mathrm{limsup_{N\to\infty}}J_{+}(N)\geq 0$. By the same argument of $J_{+}$, we can show that $\mathrm{limsup_{N\to\infty}}J_{-}(N)\geq 0$, which yields Step 2.

Next, we discuss the last term in (3.6). By the same argument in Lemma 3.2 that we apply Green’s representation theorem in $x_{2}>h$ and use the Dirichlet Green’s function $G_{h}$ of $\mathbb{R}^{2}_{x_{2}>h}(:=\mathbb{R}\times(h,\infty))$ insted of $G$, $u^{(1)}$ can also be of another integral representation for $x_{2}>h$

[TABLE]

where $G_{h}$ is defined by $G_{h}(x,y):=\Phi_{k}(x,y)-\Phi_{k}(x,y^{*}_{h})$ where $y^{*}_{h}=(y_{1},2h-y_{2})$. We define approximation $u^{(1)}_{N}$ of $u^{(1)}$ by

[TABLE]

where $\chi_{a}$ is defined by for $a>0$,

[TABLE]

By Lemma 3.4 of [4] and Lemma 2.1 of [3] we can show that $v^{1}_{N}$ and $v^{2}_{N}$ satisfy the upward propagating radiation condition, which implies that so does $u^{(1)}_{N}$. Furthermore, by the definition of $u^{(1)}_{N}$ we can show that $u^{(1)}_{N}(\cdot,\phi(N)-1)\in L^{2}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$. Then, by Lemma 6.1 of [4] we have that

[TABLE]

Combining (3.6) with (3.28) we have

[TABLE]

We observe the last term

[TABLE]

where

[TABLE]

[TABLE]

By Lemma 3.2 we can show $|u^{(1)}(x_{1},\phi(N))|$, $|\frac{\partial u^{(1)}}{\partial x_{2}}(x_{1},\phi(N))|\leq C\phi(N)$ for $x_{1}\in\mathbb{R}$, and by Lemma 2.2 we have $|u^{(2)}(x_{1},\phi(N))|$, $|\frac{\partial u^{(2)}}{\partial x_{2}}(x_{1},\phi(N))|\leq Ce^{-\delta\phi(N)}$ for $x_{1}\in\mathbb{R}$. Then, we have

[TABLE]

which implies that $M(N)=o(1)$ as $N\to\infty$. Hence, we will show that $\mathrm{limsup_{N\to\infty}}L(N)\geq 0$.

Step3 ($\mathrm{limsup_{N\to\infty}}L(N)\geq 0$): First, we observe that

[TABLE]

[TABLE]

By Lemma 2.2 $\sigma$ has a exponential decay in $y_{2}$. Then, we have for $x_{1}\in\mathbb{R}$,

[TABLE]

and

[TABLE]

Since the fundamental solution to Helmholtz equation $\Phi(x,y)$ is of the following estimation (see e.g., [2]) for $|x-y|\geq 1$

[TABLE]

we can show that for $x_{1}\in\mathbb{R}$

[TABLE]

and

[TABLE]

and

[TABLE]

and

[TABLE]

where $W_{N}$ is defined by for $N\in(0,\infty]$

[TABLE]

Using (3.35)–(3.41), we continue to estimate (3.34). By Cauchy Schwarz inequality we have

[TABLE]

[TABLE]

Finally, we will estimate \bigl{(}W_{\infty}(x_{1})-W_{N}(x_{1})\bigr{)} and $W_{N}(x_{1})$. Since $u^{(1)}(\cdot,h)\in L^{2}(\mathbb{R})$, by Lemma 3.3 there exists a sequence $\{N_{m}\}_{m\in\mathbb{N}}$ such that $N_{m}\to\infty$ as $m\to\infty$ and

[TABLE]

where $K_{A}:=K^{+}_{A}\cup K^{-}_{A}$, $K^{+}_{A}:=(-A^{+},A^{+})\setminus(-A,A)$, $K^{-}_{A}:=(-A,A)\setminus(-A^{-},A^{-})$, and $A^{\pm}:=A\pm A^{1/2}$ for $A\in[1,\infty)$.

By Cauchy Schwarz inequality we have for $|x_{1}|>N_{m}$,

[TABLE]

and

[TABLE]

Therefore, we obtain

[TABLE]

By Cauchy Schwarz inequality we have for $|x_{1}|<N_{m}$,

[TABLE]

and

[TABLE]

Therefore, we obtain

[TABLE]

Therefore, Collecting (3.43), (3.47), and (3.50) we conclude that $|L(N_{m})|\leq C\frac{\phi(N_{m})^{3}}{N_{m}^{1/8}}$. Since $\phi(N)=N^{s}$, if we choose $s\in(0,1)$ such that $3s<\frac{1}{8}$, that is, $0<s<\frac{1}{24}$, the term $\frac{\phi(N_{m})^{3}}{N_{m}^{1/8}}$ converges to zero as $m\to\infty$. Therefore, $\mathrm{limsup_{N\to\infty}}L(N)\geq 0$, which yields Step 3.

By taking $\mathrm{limsup_{N\to\infty}}$ in (3.29) we have that

[TABLE]

By Steps 2 and 3 and choosing $0<s<\frac{1}{24}$ the right hand side is non-negative. Therefore, $a_{l,j}=0$ for all $l,j$, which yields $u^{(2)}=0$. Theorem 3.1 has been shown, and in next section we will show the uniqueness of $u^{(1)}$. ∎

4 Uniqueness of $u^{(1)}$

In Section 4, we will show the following lemma.

Lemma 4.1**.**

If $u\in H^{1}_{loc}(\mathbb{R}^{2}_{+})$ satisfies

(i)

$u\in H^{1}(\mathbb{R}\times(0,R))$* for all $R>0$,*

(ii)

$\Delta u+k^{2}(1+q)nu=0\ \mathrm{in}\ \mathbb{R}^{2}_{+}$,

(iii)

$u$* vanishes for $x_{2}=0$,*

(iv)

There exists $\phi\in L^{\infty}(\Gamma_{h})\cap H^{1/2}(\Gamma_{h})$ with $u(x)=2\int_{\Gamma_{h}}\phi(y)\frac{\partial\Phi_{k}(x,y)}{\partial y_{2}}ds(y)$ for $x_{2}>h$,

then, $u\in H^{1}_{0}(\mathbb{R}^{2}_{+})$.

By using Lemma 4.1, we have the uniqueness of solution in Theorem 1.2.

Theorem 4.2**.**

Let Assumptions 1.1 and 2.1 hold and let $k>0$ be regular in the sense of Definition 2.3. If $u\in H^{1}_{loc}(\mathbb{R}^{2}_{+})$ satisfies (3.1), (3.2), and the radiation condition in the sense of Definition 2.4, then $u$ vanishes for $x_{2}>0$.

Proof of Theorem 4.2.

Let $u\in H^{1}_{loc}(\mathbb{R}^{2}_{+})$ satisfy (3.1), (3.2), and the radiation condition in the sense of Definition 2.4. By Lemma 3.1, $u^{(2)}=0$ for $x_{2}>0$. Then, $u^{(1)}$ satisfies the assumptions (i)–(iv) of Lemma 4.1, which implies that $u^{(1)}\in H^{1}_{0}(\mathbb{R}^{2}_{+})$. By Assumption 1.1, $u^{(1)}$ vanishes for $x_{2}>0$, which yields the uniqueness. ∎

Proof of Lemma 4.1.

Let $R>h$ be fixed. We set $\Omega_{N,R}:=(-N,N)\times(0,R)$ where $N>0$ is large enough. We denote by $I^{R}_{\pm N}:=\{\pm N\}\times(0,R)$, $\Gamma_{R,N}:=(-N,N)\times\{R\}$, and $\Gamma_{R}:=(-\infty,\infty)\times\{R\}$. By Green’s first theorem in $\Omega_{N,R}$ and assumptions (ii), (iii) we have

[TABLE]

By the assumption (i), the first and second term in the right hands side of (4.1) go to zero as $N\to\infty$. Then, by taking an imaginary part and as $N\to\infty$ in (4.1) we have

[TABLE]

By considering the Floquet Bloch transform with respect to $x_{1}$ (see the notation of (2.5)), we can show that

[TABLE]

Since the upward propagating radiation condition is equivalent to the Rayleigh expansion by the Floquet Bloch transform (see the proof of Theorem 6.8 in [6]), we can show that

[TABLE]

where $u_{n}(\alpha):=(2\pi)^{-1}\int_{0}^{2\pi}u_{\alpha}(x_{1},h)e^{-inx_{1}}dx_{1}$. From (4.2)–(4.4) we obtain that

[TABLE]

Here, we denote by $k=n_{0}+r$ where $n_{0}\in\mathbb{N}_{0}$ and $r\in[-1/2,1/2)$. Then by (4.5) we have

[TABLE]

[TABLE]

[TABLE]

By (4.6) we have

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

By the same argument in (4.8) we have

[TABLE]

[TABLE]

It is well known that the Floquet Bloch Transform is an isomorphism between $H^{1}(\mathbb{R}^{2}_{+})$ and L^{2}\bigl{(}(-1/2,1/2)_{\alpha};H^{1}((0,2\pi)\times\mathbb{R})_{x}\bigr{)} (e.g., see Theorem 4 in [7]). Therefore, we obtain from (4.7)–(4.9)

[TABLE]

If we can show that

[TABLE]

then the right hands side of (4.10) is finite, which yield Lemma 4.1.

Finally, we will show (4.11). By the same argument in section 3 of [6] we have

[TABLE]

where the operator $K_{\alpha}$ is defined by (2.12) and $f_{\alpha}:=-(T_{per}k^{2}nqu)(\cdot,\alpha)$. Since the function $k^{2}nqu$ has a compact support, $\left\|f_{\alpha}\right\|^{2}_{H^{1}(C_{h})}$ is bounded with respect to $\alpha$. By Assumption 2.1 and the operator $K_{\alpha}$ is compact, $(I-K_{\alpha})$ is invertible if $\alpha\in A_{k}$. Since $\pm r\in A_{k}$, $(I-K_{\pm})$ is invertible. Since the exceptional values are finitely many (see Lemma 2.2), $(I-K_{\alpha})$ is also invertible if $\alpha$ is close to $\pm r$. Therefore, there exists $\delta>0$ such that $(I-K_{\alpha})$ is invertible for all $\alpha\in(-\delta+r,\delta+r)\cup(-\delta-r,\delta-r)$.

The operator $(I-K_{\alpha})$ is of the form

[TABLE]

where $M_{\alpha}:=(I-K_{\pm r})^{-1}(K_{\alpha}-K_{\pm r})$. Next, we will estimate $(K_{\alpha}-K_{\pm r})$. By the definition of $K_{\alpha}$ we have for all $v,w\in H^{1}_{0,per}(C_{h})$,

[TABLE]

Since

[TABLE]

[TABLE]

we have for all $\alpha\in(-\delta+r,\delta+r)\cup(-\delta-r,\delta-r)$

[TABLE]

(we retake very small $\delta>0$ if needed.) This implies that there is a constant number $C>0$ which is independent of $\alpha$ such that $\left\|K_{\alpha}-K_{\pm r}\right\|\leq C\sqrt{|\alpha\mp r|}$. Therefore, by the property of Neumann series, there is a small $\delta>0$ such that for all $\alpha\in(-\delta+r,\delta+r)\cup(-\delta-r,\delta-r)$

[TABLE]

By Cauchy-Schwarz, the boundedness of trace operator, and (4.19) we have

[TABLE]

where constant number $C>0$ is independent of $\alpha$. Therefore, we have shown (4.11). ∎

5 Existence

In previous sections we discussed the uniqueness of Theorem 1.2. In Section 5, we will show the existence. Let Assumptions 1.1 and 2.1 hold and let $k>0$ be regular in the sense of Definition 2.3. Let $f\in L^{2}(\mathbb{R}^{2}_{+})$ such that $\mathrm{supp}f=Q$. We define the solution operator $S:L^{2}(Q)\to L^{2}(Q)$ by Sg:=v\bigl{|}_{Q} where $v$ satisfies the radiation condition and

[TABLE]

[TABLE]

Remark that by Theorem 2.6 we can define such a operator $S$, and $S$ is a compact operator since the restriction to $Q$ of the solution $v$ is in $H^{1}(Q)$. We define the multiplication operator $M:L^{2}(Q)\to L^{2}(Q)$ by $Mh:=k^{2}nqh$. We will show the following lemma.

Lemma 5.1**.**

$I_{L^{2}(Q)}+SM$* is invertible.*

Proof of Lemma 5.1.

By the definition of operators $S$ and $M$ we have SMg=v\bigl{|}_{Q} where $v$ is a radiating solution of (5.1)–(5.2) replacing $g$ by $k^{2}nqg$. If we assume that $(I_{L^{2}(Q)}+SM)g=0$, then g=-v\bigl{|}_{Q}, which implies that $v$ satisfies $\Delta v+k^{2}n(1+q)v=0$ in $\mathbb{R}^{2}_{+}$. By the uniqueness we have $v=0$ in $\mathbb{R}^{2}_{+}$, which implies that $I_{L^{2}(Q)}+SM$ is injective. Since the operator $SM$ is compact, by Fredholm theory we conclude that $I_{L^{2}(Q)}+SM$ is invertible. ∎

We define $u$ as the solution of

[TABLE]

satisfying the radiation condition and $u=0$ on $\Gamma_{0}$. Since

[TABLE]

we have that

[TABLE]

and $u$ is a radiating solution of (1.8)–(1.9). Therefore, Theorem 1.2 has been shown.

6 Example of Assumption 1.1

In Section 6, we will show the following lemma in order to give one of the example of Assumption 1.1.

Lemma 6.1**.**

Let $q$ and $n$ satisfy that \partial_{2}\bigl{(}(1+q)n\bigr{)}\geq 0 in $W$, and let $v\in H^{1}(\mathbb{R}^{2}_{+})$ satisfy (1.6)–(1.7). Then, $v$ vanishes for $x_{2}>0$.

Proof of Lemma 6.1.

Let $R>h$ be fixed. For $N>0$ we set $\Omega_{N,R}:=(-N,N)\times(0,R)$ and $I_{\pm N}^{R}:=\{\pm N\}\times(0,R)$ and $\Gamma_{R,N}:=(-N,N)\times\{R\}$. By Green’s first theorem in $\Omega_{N,R}$ we have

[TABLE]

Since $v\in H^{1}(\mathbb{R}^{2}_{+})$ the first and second term in the right hand side of (1.6) go to zero as $N\to\infty$. Then, by taking an imaginary part in (6.1) and as $N\to\infty$ we have

[TABLE]

By the simple calculation, we have

[TABLE]

which implies that

[TABLE]

Since \partial_{2}\bigl{(}(1+q)n\bigr{)}\geq 0 in $W$, we have

[TABLE]

By taking limit as $N\to\infty$ we have

[TABLE]

By Lemma 6.1 of [4] we have

[TABLE]

From (6.2), (6.6), and (6.7) we obtain that $\partial_{2}v=0$ on $\Gamma_{0}$. We also have $v=0$ on $\Gamma_{0}$, which implies that by Holmgren’s theorem and unique continuation principle we conclude that $v=0$ in $\mathbb{R}^{2}_{+}$. ∎

Acknowledgments

The author thanks to Professor Andreas Kirsch, who supports him in the study of this waveguide problem, and gives him many comments to improve this paper.