Trapped Modes in Piezoelectric and Elastic Waveguides

Abstract

We derive a sufficient condition for the existence of trapped modes in a cylindrical piezoelectric waveguide $\varOmega$ with a compact void $\varTheta$. An infinite part $\varGamma_{D}$ of the exterior boundary $\partial\varOmega$ is clamped along an electric conductor and the remaining part $\varGamma_{N}=(\partial\varOmega\setminus \varGamma_{D})\cup\partial\varTheta$ is traction-free and is in contact with an insulator, e.g., a vacuum. The condition permits the limit passage either to an elastic or a compound waveguide but it crucially differs from the pure elastic case due to the involved electric enthalpy instead of the energy functional. Examples of concrete waveguides and voids supporting trapped modes are given, and open questions are formulated. In particular, in contrast to a pure elastic waveguide where “almost” any crack traps a wave, no example of a crack trapping a wave in a piezoelectric waveguide is known yet.

Appendices

Appendix A: Korn’s Inequality and Singular Weyl Sequence

1.1 A.1 Korn’s Inequality

Here we formulate the well known Korn’s inequality in elasticity and by an evident argument show that it holds in piezoelectricity, too.

Lemma 4

For every $u\in H^{1}_{0}(\varOmega_{\varTheta};\varGamma_{D})^{4}$ there holds the inequality

$$\begin{aligned} \Vert u; H^{1}(\varOmega_{\varTheta})\Vert\leq C_{\omega,\varTheta}\Vert D(\nabla)u; L^{2}(\varOmega_{\varTheta})\Vert, \end{aligned}$$

(A.1)

where the constant $C_{\omega,\varTheta}$ depends only on the cross-section $\omega$ of the cylindrical waveguide and on the set $\varTheta$.

Proof

We define the truncated waveguide $\varOmega_{\varTheta}^{N}$ containing the defect $\varTheta$ by setting $\varOmega_{\varTheta}^{N}=\{ x\in\varOmega_{\varTheta}: \vert z\vert < N\}$ and the finite cylinders $\xi^{n}_{\pm}=\{x=(y,z):y\in\omega, \pm z\in(n,n+1)\}$, where $n = n^{\varTheta}, n^{\varTheta}+1, \ldots$ and $n^{\varTheta}$ is a natural number such that $\varTheta\subset\omega\times [-n^{\varTheta},n^{\varTheta}]$. On the bounded Lipschitz domains $\varOmega_{\varTheta}^{N}$ and $\xi^{n}_{\pm}$ we have the conventional Korn’s and Friedrich’s inequalities for the mechanical and electrical components

$$\begin{aligned} \big\Vert u^{\mathsf{M}}; H^{1}\bigl( \varOmega _{\varTheta}^{n_{\varTheta}}\bigr)\big\Vert ^{2} \leq& C^{2}_{\varTheta}\big\Vert D^{\mathsf{M}}(\nabla)u^{\mathsf{M}}; L^{2}\bigl(\varOmega_{\varTheta}^{n_{\varTheta}} \bigr)\big\Vert ^{2}, \end{aligned}$$

(A.2)

$$\begin{aligned} \big\Vert u^{\mathsf{M}}; H^{1}\bigl( \xi^{n}_{\pm }\bigr)\big\Vert ^{2} \leq& C^{2}_{\omega}\big\Vert D^{\mathsf{M}}(\nabla)u^{\mathsf{M}}; L^{2}\bigl(\xi^{n}_{\pm}\bigr)\big\Vert ^{2}, \end{aligned}$$

(A.3)

$$\begin{aligned} \big\Vert u^{\mathsf{E}}; H^{1}\bigl( \varOmega _{\varTheta}^{n_{\varTheta}}\bigr)\big\Vert ^{2} \leq& C^{2}_{\varTheta}\big\Vert \nabla u^{\mathsf{E}}; L^{2} \bigl(\varOmega _{\varTheta}^{n_{\varTheta}}\bigr)\big\Vert ^{2}, \end{aligned}$$

(A.4)

$$\begin{aligned} \big\Vert u^{\mathsf{E}}; H^{1}\bigl( \xi^{n}_{\pm }\bigr)\big\Vert ^{2} \leq& C^{2}_{\omega}\big\Vert \nabla u^{\mathsf{E}}; L^{2} \bigl(\xi ^{n}_{\pm}\bigr)\big\Vert ^{2}. \end{aligned}$$

(A.5)

These inequalities are valid because the Dirichlet conditions (3) are imposed on the non-empty parts with a positive area of the lateral surfaces of the sets $\varOmega_{\varTheta}^{n_{\varTheta}}$ and $\xi^{n}_{\pm}$. Furthermore, the constants $C_{\omega}$ in inequalities (A.3) and (A.5) do not depend on the indexes $n$ and ± due to the similarity of the sets. By taking the smallest possible $n_{\varTheta}$, we can conclude that the constants $C_{\varTheta}$ in inequalities (A.2) and (A.4) depend only on $\varTheta$ and $\omega$. Summing up all the inequalities (A.2)–(A.5) we obtain (A.1). □

1.2 A.2 The Singular Weyl Sequence

Lemma 5

For $\lambda\geq\lambda_{\dagger}$ the operator $\mathcal{A}(\lambda):H^{1}_{0}(\varOmega_{\varTheta};\varGamma_{D})\to H^{1}_{0}(\varOmega_{\varTheta};\varGamma_{D})^{\ast}$ is not a Fredholm operator.

To prove the statement, we make use of the wave (16) multiplied with the plateau function $X_{N}(z)\in C^{\infty}_{c}(\mathbb{R})$ defined as follows:

$$ X_{N}(z)=\left\{ \textstyle\begin{array}{l@{\quad}l} 1 & \mbox{for } z\in(2^{N}+1,2^{N+1}-1),\\ 0 & \mbox{for } z\notin(2^{N},2^{N+1}). \end{array}\displaystyle \right. $$

The graph of this function is depicted in Fig. 7. Clearly, we have

$$\begin{aligned} \Vert X_{N}w;H^{1}( \varOmega_{\varTheta})\Vert^{2}&\geq\Vert X_{N}w;L^{2}( \varOmega_{\varTheta})\Vert^{2} =\Vert X_{N}w;L^{2}( \varOmega)\Vert^{2} \\ &\geq \int^{2^{N+1}-1}_{2^{N}+1}dz \Vert U;L^{2}( \omega)\Vert^{2} =c_{w}\bigl(2^{N+1}-2^{N}-2 \bigr). \end{aligned}$$

(A.6)

Fig. 7

The plateau function

On the other hand, by the definition of the wave (16), the expressions

$$L(x,\nabla) \bigl(X_{N}(z)e^{i\zeta z}U(y)\bigr)-\lambda\rho(y)E X_{N}(z)e^{i\zeta z}U(y) $$

and

$$B(x,\nabla) \bigl(X_{N}(z)e^{i\zeta z}U(y)\bigr) $$

differ from zero only for $z\in(2^{N},2^{N+1})$ and $z\in (2^{N+1}-1,2^{N+1})$ so that

$$\begin{aligned} \Vert \mathcal{A}(\lambda)X_{N}w; \bigl(H^{1}_{0}(\varOmega _{\varTheta}; \varGamma_{N})^{4}\bigr)^{*}\Vert^{2} \leq& \Vert LX_{N} w-\lambda\rho EX_{N}w;L^{2}( \varOmega_{\varTheta})\Vert^{2} \\ &{}+\Vert BX_{N}w;L^{2}(\varGamma_{N})\Vert^{2} \leq2C_{w}, \end{aligned}$$

(A.7)

where the right-hand side of the inequality is a constant independent on $N$. Thus the estimate

$$\begin{aligned} \Vert u;H^{1}(\varOmega_{\varTheta})\Vert\leq K\big\Vert \mathcal{A}( \lambda)u;\bigl(H^{1}_{0}(\varOmega_{\varTheta}; \varGamma _{D})^{4}\bigr)^{*}\big\Vert \end{aligned}$$

is not valid for the infinite family $\{X_{N}w\}_{N\geq N(K)}$ of linearly independent vector functions (due to mutually disjoint supports) and estimate (42) cannot hold true for any $u\in H^{1}_{0}(\varOmega_{\varTheta};\varGamma_{D})^{4}\setminus\operatorname{ker} \mathcal{A}(\lambda)$. Here $N(K)\in\mathbb{N}$ is fixed such that, in accordance with inequalities (A.6) and (A.7),

$$\begin{aligned} c_{w}\bigl(2^{N(K)+1}-2^{N(K)}-2\bigr)\geq 2KC_{w}. \end{aligned}$$

With the help of the same family of trial functions we can construct directly the singular Weyl sequence for the operator ℳ at the point $\mu=\lambda ^{-1}$ and apply the Weyl criterion, see [2, Thm. 9.1.2] and [34, Thm. VII.12]. However, in view of the reduction scheme this needs long routine calculations which are omitted here for brevity. It should be mentioned the circuitous route based on the Dirichlet condition (3): the assumed Fredholm property of $\mathcal{M}-\mu$ and the isomorphism of the operator related to the scalar problem (9) pass the first one to the operator (41), for which it has been disproved.

Remark 4

The expressions in the previous considerations have sense in the case of the smooth boundary $\partial\omega$. If the boundary is Lipschitz, we have to consider the integral identity with a test function $v\in H^{1}_{0}(\varOmega _{\varTheta},\varGamma_{D})^{4}$:

$$\begin{aligned} & \big(AD\bigl(\nabla(X_{N}w)\bigr),D(\nabla)v\bigr)_{\varOmega_{\varTheta}}-\lambda\bigl( \rho X_{N} w^{\mathsf{M}},v^{M}\bigr) \\ &\quad=\bigl(AD(\nabla)w,D(\nabla) (X_{n}v)\bigr)_{\varOmega}-\lambda \bigl(\rho w^{\mathsf{M}},X_{N}v^{M}\bigr)_{\varOmega}+F_{N}(v), \end{aligned}$$

where

$$F_{N}(v)=\bigl(A\bigl[D(\nabla),X_{N}\bigr]w,D(\nabla)v \bigr)_{\varOmega}- \bigl(AD(\nabla)w,\bigl[D(\nabla),X_{N}\bigr]v \bigr)_{\varOmega}. $$

Here $[D(\nabla),X_{N}]$ is the commutator of $D(\nabla)$ and $X_{N}$, i.e. just a matrix function $z\mapsto D(0,0,\partial_{z})X_{N}(z)$.

Using the formula (16) for $w$, we have

$$\begin{aligned} &\bigl(AD(\nabla)w,D(\nabla) (X_{N} v)\bigr)_{\varOmega}- \lambda \bigl(\rho w^{\mathsf{M}},X_{N}v^{M}\bigr)_{\varOmega} \\ &\quad= \int_{\mathbb{R}} \bigl\{ \bigl(AD(\nabla_{y},i\zeta)U, D( \nabla_{y},i\zeta) \bigl(e^{-i\zeta z}X_{N}(z)v(\cdot,z) \bigr)\bigr)_{\omega}\\ &\quad\quad{}-\lambda\bigl(\rho U^{\mathsf{M}},e^{-i\zeta z}X_{N}(z)v( \cdot,z)\bigr)_{\omega}\bigr\} dz \\ &\quad\quad{}+ \int_{\mathbb{R}}\bigl(AD(\nabla_{y},i\zeta)U, D(0, \partial_{z}) \bigl(e^{-i\zeta z}X_{N}(z)v(\cdot,z)\bigr) \bigr)_{\omega}dz. \end{aligned}$$

The first integral is zero because $U,\zeta$ and $\lambda$ satisfy (20). The second integral vanishes, because after computing the scalar product in $L^{2}(\omega)$, it becomes

$$\int_{\mathbb{R}}\partial_{z}\bigl(X_{N}(z) \rho(z)\bigr)dz=0. $$

Finally,

$$ |F_{N}(v)|\leq c \big\| v;H^{1}\bigl(\omega\times \mathrm{supp}(\partial_{z}X_{N})\bigr)\big\| , $$

(A.8)

where $c$ does not depend on $N$ and support of $\partial_{z}X_{N}$. The estimate (A.8) is nothing but the interpretation of (42) in the case when $\omega$ has a Lipschitz boundary.

Appendix B: Calculation of the Rayleigh Quotient

2.1 B.1 Calculation of the Numerator in the Rayleigh Quotient

In this section we evaluate the norm of the operator ℳ

$$\begin{aligned} \Vert \mathcal{M}\Vert=\sup_{u^{\mathsf{M}}\in\mathcal{H}\setminus {0}}\frac{\langle\mathcal{M}u^{\mathsf{M}}, u^{\mathsf{M}}\rangle_{\varOmega_{\varTheta}}}{\langle u^{\mathsf{M}},u^{\mathsf{M}}\rangle_{\varOmega_{\varTheta}}}, \end{aligned}$$

(B.1)

which provides the sufficient condition for the existence of the trapped wave.

Let us define

$$\begin{aligned} w^{\mathsf{M}}_{\delta}(y,z)=e^{-\delta|z|}w^{\mathsf{M}}_{\dagger}(y,z) \end{aligned}$$

(B.2)

with small $\delta>0$ and $w^{\mathsf{M}}_{\dagger}$ taken from (43). Note that, in view of the exponential factor $e^{-\delta|z|}$, this vector function falls into $H^{1}_{0}(\varOmega_{\varTheta};\varGamma_{D})^{3}$. To calculate the numerator in the Rayleigh quotient, we apply (14) and (43) to obtain

$$\begin{aligned} \bigl\langle \mathcal{M}w^{\mathsf{M}}_{\delta},w^{\mathsf {M}}_{\delta} \bigr\rangle &=\bigl(\rho w^{\mathsf{M}}_{\delta},w^{\mathsf {M}}_{\delta} \bigr)_{\varOmega_{\varTheta}}= \bigl(\rho w^{\mathsf{M}}_{\delta },w^{\mathsf{M}}_{\delta} \bigr)_{\varOmega}-\bigl(\rho w^{\mathsf{M}}_{\delta},w^{\mathsf {M}}_{\delta} \bigr)_{\varTheta} \\ &= \int_{\mathbb{R}}e^{-2\delta|z|}dz \bigl(\rho U^{\mathsf {M}}_{\dagger},U^{\mathsf{M}}_{\dagger} \bigr)_{\omega}- \int_{\varTheta}\rho(y)e^{-2\delta|z|}|U^{\mathsf{M}}_{\dagger }(y)|^{2}dydz \\ &=\delta^{-1}\bigl(\rho U^{\mathsf{M}}_{\dagger},U^{\mathsf {M}}_{\dagger} \bigr)_{\omega}-\bigl(\rho U^{\mathsf{M}}_{\dagger}, U^{\mathsf{M}}_{\dagger}\bigr)_{\varTheta}+O(\delta). \end{aligned}$$

(B.3)

We have used the Taylor expansion $e^{-2\delta|z|}=1+O(\delta)$ and the simple formula

$$ \int_{\mathbb{R}}e^{-2\delta|z|}dz=\frac{1}{\delta}. $$

(B.4)

2.2 B.2 Calculation of the Denominator

To evaluate the denominator we shall use the following auxiliary lemmata.

Lemma 6

The problem

$$\begin{aligned} &\bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}w^{\mathsf {E}}_{\bullet},D^{\mathsf{E}}v^{\mathsf{E}} \bigr)_{\varOmega_{\varTheta}}= \bigl(A^{\mathsf{E}\mathsf {M}}D^{\mathsf{M}}w^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}}w^{\mathsf{E}}_{\dagger},D^{\mathsf{E}}v^{E} \bigr)_{\varOmega_{\varTheta}} \\ &\quad\forall v^{E}\in H^{1}_{0}( \varOmega_{\varTheta},\varGamma_{D}) \end{aligned}$$

(B.5)

has a unique solution $w^{\mathsf{E}}_{\bullet}\in H^{1}_{0}(\varOmega _{\varTheta},\varGamma_{D})$. Moreover, the solution $w^{\mathsf{E}}_{\bullet}$ decays exponentially at infinity, i.e. the inclusion $e^{\beta|z|}w^{\mathsf{E}}_{\bullet}\in H^{1}_{0}(\varOmega_{\theta},\varGamma_{D})$ holds for some $\beta>0$.

Proof

If the coefficients are smooth, the differential formulation of the problem (B.5) takes the form

$$\begin{aligned} &-D^{\mathsf{E}}(\nabla)^{\top}A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}}( \nabla)w^{\mathsf{E}}_{\bullet}(x) \\&\quad=f^{\mathsf{E}}_{\bullet}(x) :=D^{\mathsf{E}}(\nabla)^{\top}\bigl(A^{\mathsf{E}\mathsf {E}}(y)D^{\mathsf{E}}( \nabla)w^{\mathsf{E}}_{\dagger}(x)- A^{\mathsf{E}\mathsf {M}}(y)D^{\mathsf{M}}( \nabla)w^{\mathsf{M}}_{\dagger}(x) \bigr),\quad x\in\varOmega_{\varTheta}, \\ &-D^{\mathsf{E}}\bigl(\nu(x)\bigr)^{\top}A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}} \bigl(\nu(x)\bigr)w^{\mathsf{E}}_{\bullet}(x)\\&\quad=g^{\mathsf {E}}_{\bullet}(x) :=D^{\mathsf{E}}\bigl(\nu(x)\bigr)^{\top}\bigl(A^{\mathsf{E}\mathsf{E}}(y)D^{\mathsf{E}}(\nabla)w^{\mathsf {E}}_{\dagger}(x)- A^{\mathsf{E}\mathsf{M}}(y)D^{\mathsf{M}}(\nabla)w^{\mathsf {M}}_{\dagger}(x) \bigr),\quad x\in\varGamma_{N}\cup\partial\varTheta, \\ & w^{\mathsf{E}}_{\bullet}=0,\quad x\in\varGamma_{D}. \end{aligned}$$

(B.6)

Since $w_{\dagger}=(w^{\mathsf{M}}_{\dagger},w^{\mathsf{E}}_{\dagger})$ satisfies the problem (1)–(3) in the unperturbed cylinder $\varOmega$, we see that $f^{\mathsf{E}}_{\bullet}=0$ in $\varOmega_{\varTheta}$, $g^{\mathsf {E}}_{\bullet}=0$ on $\varGamma_{N}$. Therefore, the right-hand sides in (B.6) have compact supports. To conclude a similar property in the variational formulation of the problem (B.5), we will verify that the right-hand side of (B.5) depends only on the restriction of a test function $v^{E}$ onto the compact set $\varOmega_{\varTheta}^{n}=\{x\in\varOmega_{\varTheta}: |z|< n_{\varTheta}\}$, cf. the proof of Lemma 2.1. In other words, the functional $F^{\mathsf{E}}_{\bullet}(v^{\mathsf {E}})$ on the right-hand side of (B.5) is compactly supported. Then the assertion follows from the theory of elliptic problems in domains with cylindrical outlets to infinity (see in particular [31, §2.2, §5.1 ], [20, Example 1.12 and Thm. 3.4] and [24, §2]). The latter is a consequence of the cylindrical structure of the nonempty Dirichlet zone $\varGamma_{D}$. We, however, will provide a simple argument to this conclusion in Remark 5.

Next we proceed by evaluating the integral

$$\begin{aligned} & \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf {M}}w^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf{E}}D^{\mathsf {E}}w^{\mathsf{E}}_{\dagger},D^{\mathsf{E}}v^{E} \bigr)_{ \omega\times(n_{\varTheta}, \infty)} \\ &\quad= \int_{n_{\varTheta}}^{\infty} \bigl(A^{\mathsf{E}\mathsf {M}}D^{\mathsf{M}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}-A^{\mathsf {E}\mathsf{E}}D^{\mathsf{E}}( \nabla_{y}, i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}(\nabla_{y},i\zeta_{\dagger}) \bigl(e^{-i\zeta_{\dagger}|z|}v^{E}(\cdot,z) \bigr) \bigr)_{\omega}dz \\ &\quad\quad{}+ \int_{n_{\varTheta}}^{\infty} \bigl(A^{\mathsf{E}\mathsf {M}}D^{\mathsf{M}}( \nabla,i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}} (\nabla_{y}, i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}(0,0,\partial_{z}) \bigl(e^{-i\zeta_{\dagger}|z|}v(\cdot,z) \bigr) \bigr)_{\omega}dz. \end{aligned}$$

(B.7)

The first integral on the right in (B.7) vanishes due to the integral identity (20) where we set $\zeta=\zeta_{\dagger}, \varLambda (\zeta)=\lambda_{\dagger}$ and $V(y)=(0,e^{-i\zeta_{\dagger}|z|}v(y,z))$. Since the last integrand takes the form $a(y)\partial_{z} b(y,z)$, the Newton-Leibnitz formula shows that the second integral in (B.7) becomes

$$- \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}(\nabla_{y},i \zeta_{\dagger})U^{\mathsf{M}}_{\dagger}- A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger},D^{\mathsf {E}}(0,0,1)e^{-i\zeta_{\dagger}|n_{\varTheta}|} v \bigr)_{\omega}$$

and depends clearly only on the restriction $v |_{\varOmega ^{n_{\varTheta}}_{\varTheta}}$. The opposite end $\omega\times(-\infty,-n_{\varTheta})$ is treated similarly and hence the lemma is proved. □

Remark 5

Let us reproduce a simple argument presented in [25] to ensure the exponential decay of the solution $w^{\mathsf{E}}_{\bullet}\in H^{1}_{0}(\varOmega_{\varTheta};\varGamma_{D})$ of the problem (B.5). We introduce the continuous weight function

$$ \mathcal{R}(z)= \textstyle\begin{cases} \mathrm{e}^{\beta|z|},& |z|< T,\\ \mathrm{e}^{\beta T},& |z|\geq T, \end{cases} $$

(B.8)

depending on two positive parameters $\beta$ and $T$. Since the weight function is bounded, we may insert the product $v^{\mathsf{E}}=\mathcal{R}^{2} w^{\mathsf{E}}_{\bullet}=:\mathcal{R}W^{\mathsf{E}}_{\bullet}$ into (B.5) as a test function, and after transporting ℛ from $W^{\mathsf{E}}_{\bullet}$ onto $w^{\mathsf{E}}_{\bullet}$ we obtain

$$\begin{aligned} F^{\mathsf{E}}_{\bullet}\bigl(\mathcal{R}W^{\mathsf{E}}_{\bullet}\bigr) =&\bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}w^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}\bigl(\mathcal{R}W^{\mathsf{E}}_{\bullet}\bigr) \bigr)_{\varOmega_{\varTheta}} \\ =&\bigl(A^{\mathsf{E}\mathsf{E}}\mathcal{R}D^{\mathsf{E}} w^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}W^{\mathsf{E}}_{\bullet}\bigr)_{\varOmega_{\varTheta}}+ \bigl(A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}}w^{\mathsf{E}}_{\bullet}, \bigl[D^{\mathsf{E}}, \mathcal{R}\bigr]W^{\mathsf{E}}_{\bullet}\bigr)_{\varOmega_{\varTheta}} \\ =&\bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}W^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}W^{\mathsf{E}}_{\bullet}\bigr)_{\varOmega_{\varTheta}}- \bigl(A^{\mathsf{E}\mathsf{E}}\mathcal{R}^{-1} \bigl[D^{\mathsf{E}}, \mathcal{R}\bigr] W^{\mathsf{E}}_{\bullet}, \mathcal{R}^{-1} \bigl[D^{\mathsf{E}}, \mathcal{R}\bigr]W^{\mathsf{E}}_{\bullet}\bigr)_{\varOmega_{\varTheta}} \\ &{}-\bigl(A^{\mathsf{E}\mathsf{E}}\mathcal{R}^{-1} \bigl[D^{\mathsf{E}},\mathcal{R}\bigr] W^{\mathsf{E}}_{\bullet},D^{\mathsf{E}} W^{\mathsf{E}}_{\bullet}\bigr)_{\varOmega_{\varTheta}}+ \bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}} W^{\mathsf{E}}_{\bullet}, \mathcal{R}^{-1} \bigl[D^{\mathsf{E}},\mathcal{R}\bigr]W^{\mathsf{E}}_{\bullet}\bigr)_{\varOmega_{\varTheta}}. \end{aligned}$$

(B.9)

Recalling the symmetry and positivity of the matrix $A^{\mathsf {E}\mathsf{E}}$, we observe that the last two terms in (B.9) compose a purely imaginary number. Secondly, according to (A.4) and (A.5),

$$ \bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}W^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}W^{\mathsf{E}}_{\bullet}\bigr)_{\varOmega_{\varTheta}}\geq c_{\varTheta}\|W^{\mathsf{E}}_{\bullet}; H^{1}( \varOmega_{\varTheta})\|^{2}. $$

(B.10)

Moreover, by (B.8) the vector function $[D^{\mathsf{E}},\mathcal{R}]=D^{\mathsf{E}}(0,0,\partial_{z})\mathcal{R}$ satisfies

$$ \big|\bigl[D^{\mathsf{E}},\mathcal{R}\bigr](z)\big|\leq c\beta \mathcal{R}(z),\ |z|< T\quad\mbox{and}\quad\bigl[D^{\mathsf{E}},\mathcal{R} \bigr](z)= 0,\ |z|>T. $$

(B.11)

Since the functional $F^{\mathsf{E}}_{\bullet}$ is compactly supported, the left-hand side does not depend on $T>n_{\varTheta}$. Hence, for a sufficiently small $\beta>0$ we take the real part of (B.9) and derive from (B.10), (B.11) the relation

$$\|W^{\mathsf{E}}_{\bullet};H (\varOmega_{\varTheta})\|^{2} \leq c_{\beta}\big|\operatorname{Re} F^{\mathsf{E}}_{\bullet}\bigl(w^{\mathsf{E}}_{\bullet}\bigr)\big|. $$

Letting $T$ tend to infinity, we finally observe that $e^{\beta |z|}w^{\mathsf{E}}_{\bullet}\in H^{1}_{0}(\varOmega_{\varTheta};\varGamma_{D})$.

Lemma 7

For

$$\begin{aligned} w^{\mathsf{M}}_{\delta}(y,z)=e^{-\delta|z|}w^{\mathsf{M}}_{\dagger}(y,z) \end{aligned}$$

(B.12)

and

$$\begin{aligned} w^{\mathsf{E}}_{\delta}(y,z)=e^{-\delta|z|}e^{i\zeta _{\dagger}z}U^{\mathsf{E}}_{\dagger}(y), \widehat{w}^{\mathsf{E}}_{\delta}=Rw^{\mathsf{M}}_{\delta}-w^{\mathsf{E}}_{\delta}, \end{aligned}$$

(B.13)

the problem

$$\begin{aligned} &\bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}\widehat {w}^{\mathsf{E}}_{\delta},D^{\mathsf{E}}v^{\mathsf{E}} \bigr)_{\varOmega_{\varTheta}}= \bigl(A^{\mathsf{E}\mathsf {M}}D^{\mathsf{M}}w^{\mathsf{M}}_{\delta}-A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}}w^{\mathsf{E}}_{\delta},D^{\mathsf{E}}v^{\mathsf{E}} \bigr)_{\varOmega_{\varTheta}} \\ &\quad\forall v^{\mathsf{E}}\in H^{1}_{0}( \varOmega_{\varTheta};\varGamma_{D}). \end{aligned}$$

(B.14)

has a unique solution $\widehat{w}^{\mathsf{E}}_{\delta}\in H^{1}_{0}(\varOmega_{\varTheta},\varGamma_{D})$ and it takes the form

$$ \widehat{w}^{\mathsf{E}}_{\delta}(y,z)=e^{-\delta|z|} \biggl(w^{\mathsf{E}}_{\bullet}(y,z)+ \delta\sum _{\pm}\pm\chi_{\pm}(z)e^{i\zeta_{\dagger}z}U^{\mathsf{E}}_{\bullet}(y) \biggr) +\widetilde{w}^{\mathsf{E}}_{\delta}, $$

(B.15)

where $w^{\mathsf{E}}_{\bullet}$ is the exponentially decaying function given in Lemma 6, $U^{\mathsf{E}}_{\bullet}$ is a function in $H^{1}_{0}(\omega,\gamma_{D})$, $\chi_{\pm}$ are smooth cut-off functions such that $0\leq\chi_{\pm}\leq1$ and

$$\chi_{\pm}(z)= \textstyle\begin{cases} 1, &\textit{for }\pm z\geq2n_{\varTheta},\\ 0,&\textit{for }\pm z\leq n_{\varTheta}. \end{cases} $$

The remainder $\widetilde{w}^{\mathsf{E}}_{\delta}$ satisfies the estimate

$$ \|\widetilde{w}^{E}_{\delta};H^{1}_{0}( \varOmega_{\varTheta};\varGamma_{D})\|\leq c\delta. $$

(B.16)

Proof

The remainder $\widetilde{w}^{E}_{\delta}$ in (B.15) must be a solution of the problem

$$ \bigl(A^{EE}D^{\mathsf{E}}\widetilde{w}^{\mathsf{E}}_{\delta},D^{\mathsf{E}}v^{\mathsf{E}} \bigr)=\widetilde{f}_{\delta}\bigl(v^{\mathsf{E}}\bigr)\quad\forall v^{\mathsf{E}}\in H^{1}_{0}(\varOmega_{\varTheta}, \varGamma_{D}) $$

with the right-hand side

$$\begin{aligned} \widetilde{f}_{\delta}\bigl(v^{\mathsf{E}}\bigr) =& \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}\mathrm{e}^{-\delta |z|}w^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}} \mathrm{e}^{-\delta|z|}\bigl(w^{\mathsf{E}}_{\dagger}-w^{\mathsf{E}}_{\bullet}\bigr),D^{\mathsf{E}}v^{\mathsf{E}}\bigr)_{\varOmega_{\varTheta}} \\ &{}-\delta\sum_{\pm}\pm\bigl( A^{\mathsf{E}\mathsf{E}}D^{\mathsf {E}}e^{-\delta|z|}w^{\mathsf{E}}_{\bullet},D^{\mathsf{E}}v^{\mathsf{E}} \bigr)_{\varOmega_{\varTheta}} \\ &{}-\delta\sum_{\pm} \pm\bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}} \chi_{\pm}e^{-\delta|z|} e^{i\zeta_{\dagger}z}U^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}v^{E} \bigr)_{\varOmega_{\varTheta}} \\ =:& \mathcal{F}^{\mathsf{E}}_{\delta}\bigl(v^{\mathsf{E}}\bigr)- \delta\mathcal{F}^{\mathsf{E}}_{\delta_{\dagger}}\bigl(v^{\mathsf{E}}\bigr). \end{aligned}$$

The first term $\mathcal{F}^{\mathsf{E}}_{\delta}(v^{\mathsf{E}})$ on the right can be written as follows:

$$\begin{aligned} \mathcal{F}^{\mathsf{E}}_{\delta}\bigl(v^{\mathsf{E}} \bigr) =& \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}w^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}\bigl(w^{\mathsf{E}}_{\dagger}-w^{\mathsf{E}}_{\bullet}\bigr), D^{\mathsf{E}}v^{\mathsf{E}}_{\delta}\bigr)_{\varOmega _{\varTheta}} \\ &{}+\delta\mathcal{F}^{\mathsf{E}}_{\dagger}\bigl(v^{\mathsf{E}}_{\delta}\bigr)-\delta\mathcal{F}^{\mathsf{E}}_{\delta\bullet} \bigl(v^{\mathsf{E}}_{\delta}\bigr)- \delta^{2}\widetilde{ \mathcal{F}}^{\mathsf{E}}_{\delta}\bigl(v^{\mathsf{E}}_{\delta}\bigr), \end{aligned}$$

(B.17)

where $v^{\mathsf{E}}_{\delta}(y,z)=\mathrm{e}^{-\delta|z|}v^{\mathsf{E}}(y,z)$ is a modified test function and

$$\begin{aligned} \mathcal{F}^{\mathsf{E}}_{\dagger}\bigl(v^{\mathsf{E}}_{\delta}\bigr) =& \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}\bigl(0,0,\mathrm{sign}(z) \bigr)w^{\mathsf{M}}_{\dagger}- A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}} \bigl(0,0,\mathrm{sign}(z)\bigr)w^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}} v^{\mathsf{E}}_{\delta}\bigr)_{\varOmega_{\varTheta}} \\ &{}- \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}w^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf{E}} D^{\mathsf{E}}w^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}\bigl(0,0,\mathrm{sign}(z)\bigr) v^{\mathsf{E}}_{\delta}\bigr)_{\varOmega_{\varTheta}}, \\ \mathcal{F}^{\mathsf{E}}_{\delta\bullet}\bigl(v^{\mathsf{E}}_{\delta}\bigr) =& \bigl( A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}\bigl(0,0,\mathrm{sign}(z) \bigr)w^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}v^{\mathsf{E}}_{\delta}\bigr)_{\varOmega_{\varTheta}} - \bigl( A^{\mathsf{E}\mathsf{E}} D^{\mathsf{E}} w^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}\bigl(0,0,\mathrm{sign}(z)\bigr) v^{\mathsf{E}}_{\delta}\bigr)_{\varOmega_{\varTheta}}, \\ \widetilde{\mathcal{F}}^{\mathsf{E}}_{\delta}\bigl(v^{\mathsf{E}}_{\delta}\bigr) =& \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}\bigl(0,0,\mathrm{sign}(z) \bigr)w^{\mathsf{M}}_{\dagger}\\ &{}- A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}} \bigl(0,0,\mathrm{sign}(z)\bigr) \bigl(w^{\mathsf{E}}_{\dagger}-w^{\mathsf{E}}_{\bullet}\bigr), D^{\mathsf{E}}\bigl(0,0,\mathrm{sign}(z)\bigr) v^{\mathsf{E}}_{\delta}\bigr)_{\varOmega_{\varTheta}}. \end{aligned}$$

The first scalar product in (B.17) vanishes according to the definition of $w^{\mathsf{E}}_{\bullet}$, see the integral identity (B.5) with the change $v^{\mathsf{E}}\mapsto v^{\mathsf{E}}_{\delta}$. We subtract from $\mathcal{F}^{\mathsf{E}}_{\dagger}(v^{\mathsf{E}}_{\delta})$ the sum

$$\begin{aligned} \mathcal{F}^{\mathsf{E}}_{\pm}\bigl(v^{\mathsf{E}}_{\delta}\bigr) =&\sum_{\pm}\pm\bigl( A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}(0,0,1)w^{\mathsf{M}}_{\dagger}- A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}(0,0,1) w^{\mathsf{E}}_{\dagger}, \chi_{\pm}D^{\mathsf{E}}(\nabla)v^{\mathsf{E}}_{\delta}\bigr)_{\varOmega_{\varTheta}} \\ &{}- \bigl( A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}(\nabla)w^{\mathsf{M}}_{\dagger}- A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}(\nabla) w^{\mathsf{E}}_{\dagger}, \chi_{\pm}D^{\mathsf{E}}(0,0,1)v^{\mathsf{E}}_{\delta}\bigr)_{\varOmega_{\varTheta}} \end{aligned}$$

(B.18)

and observe that the remaining parts on the first position in the scalar products get compact supports. So the estimate

$$ \delta\big|\mathcal{F}^{\mathsf{E}}_{\dagger}\bigl(v^{\mathsf{E}}_{\delta}\bigr)- \mathcal{F}^{\mathsf{E}}_{\pm}\bigl(v^{\mathsf{E}}_{\delta}\bigr)\big| \leq c\delta\|v^{\mathsf{E}};H^{1}( \varOmega_{\varTheta})\| $$

(B.19)

becomes evident. Since $w^{\mathsf{E}}_{\bullet}$ decays exponentially, see Lemma 6, we also have

$$\delta|\mathcal{F}^{\mathsf{E}}_{\delta\bullet}|\leq c \delta \|v^{\mathsf{E}};H^{1}(\varOmega_{\varTheta})\|. $$

Moreover, the equality (B.4) (the functions $w^{\mathsf{M}}_{\dagger}$ and $w^{\mathsf{E}}_{\bullet}$ are bounded) yields

$$\begin{aligned} \delta^{2}\big|\widetilde{\mathcal{F}}^{\mathsf{E}}_{\delta}\bigl(v^{\mathsf{E}}_{\delta}\bigr)\big| \leq& c\delta^{2} \bigl( \|w^{\mathsf{E}}_{\bullet};L^{2}(\varOmega_{\varTheta})\| +\| \mathrm{e}^{-\delta|z|}w^{\mathsf{M}}_{\dagger};L^{2}( \varOmega_{\varTheta})\| \\ &{}+\|\mathrm{e}^{-\delta|z|}w^{\mathsf{M}}_{\dagger};L^{2}( \varOmega_{\varTheta})\| \bigr) \|v^{\mathsf{E}};H^{1}( \varOmega_{\varTheta})\|\leq c\delta^{3/2} \|v^{\mathsf{E}};H^{1}( \varOmega_{\varTheta})\|. \end{aligned}$$

(B.20)

Finally, recalling the definition (43), we choose $U^{\mathsf{E}}_{\bullet}$ as a solution of the following problem on the cross-section $\omega$:

$$\begin{aligned} & \big( A^{\mathsf{E}\mathsf{E}} D^{\mathsf{E}}(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}(\nabla_{y},i\zeta_{\dagger}) V^{\mathsf{E}} \big)_{\omega}\\ &\quad= \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}(0,0,1)U^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}(0,0,1)U^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}(\nabla_{y},i\zeta_{\dagger})V^{\mathsf{E}} \bigr)_{\omega}\\ &\quad\quad{}+ \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}(\nabla_{y},i \zeta_{\dagger})U^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}(0,0,1)V^{\mathsf{E}}\bigr)_{\omega} \end{aligned}$$

for all $V^{\mathsf{E}}\in H^{1}_{0}(\omega;\gamma_{D})$, which is uniquely solvable due to the Dirichlet condition on $\gamma _{D}$. The justification of this is exactly the same as for the problem (26). Our choice of $U^{\mathsf{E}}_{\bullet}$ means that the principal terms in the expressions $\delta\mathcal {F}^{\mathsf{E}}_{\delta\dagger}$ in (B.17) and $\delta\mathcal{F}^{\mathsf{E}}_{\delta\pm}$ (B.18) cancel each other out leaving just two sorts of scalar products. First, scalar products containing the factor $\delta$ and compactly supported functions at the first position which appear due to the commuting of the cut-off functions with the differential operators. Secondly, we have scalar products with the factor $\delta^{2}$ and bounded functions multiplied by $\mathrm {e}^{-\delta|z|}$ at the first position. Both cases have been examined in (B.19) and (B.20). Thus, the inequality

$$\delta\big|\mathcal{F}^{\mathsf{E}}_{\delta\dagger}\bigl(v^{\mathsf{E}}\bigr)- \mathcal{F}^{\mathsf{E}}_{\pm}\bigl(v^{\mathsf{E}}\bigr)\big|\leq c \bigl(\delta+\delta^{3/2}\bigr) \|v^{\mathsf{E}};H^{1}( \varOmega_{\varTheta})\| $$

completes the proof of the lemma. □

Now we are in position to evaluate the denominator of the Rayleigh quotient. First of all, we use the definitions of the bi-linear form (12) and the scalar product $\langle\cdot,\cdot\rangle$ in ℋ to obtain

$$\begin{aligned} \bigl\langle w^{\mathsf{M}}_{\delta},w^{\mathsf{M}}_{\delta} \bigr\rangle =&\bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}(\nabla )w^{\mathsf{M}}_{\delta},D^{\mathsf{M}}( \nabla)w^{\mathsf{M}}_{\delta}\bigr)_{\varOmega_{\varTheta}} \\ &{}+\bigl(A^{\mathsf{M}\mathsf{E}}\nabla Rw^{\mathsf{M}}_{\delta },D^{\mathsf{M}}( \nabla)w^{\mathsf{M}}_{\delta}\bigr)_{\varOmega_{\varTheta}}, \end{aligned}$$

(B.21)

where $R\omega^{\mathsf{M}}_{\delta}\in H^{1}_{0}(\varOmega _{\varTheta};\varGamma_{D})$ is a unique solution of the problem, cf. (9),

$$\begin{aligned} \bigl(A^{\mathsf{E}\mathsf{E}}\nabla Rw^{\mathsf{M}}_{\delta}, \nabla v^{\mathsf{E}}\bigr)_{\varOmega_{\varTheta}}= \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}( \nabla)w^{\mathsf{M}} _{\delta},\nabla v^{\mathsf{E}} \bigr)_{\varOmega_{\varTheta}} \quad\forall v^{\mathsf{E}}\in H^{1}_{0}( \varOmega_{\varTheta};\varGamma_{D}). \end{aligned}$$

(B.22)

We set, as in Lemma 7,

$$\begin{aligned} w^{\mathsf{E}}_{\delta}(y,z)=e^{-\delta|z|}e^{i\zeta_{\dagger }z}U^{\mathsf{E}}_{\dagger}(y), \widehat{w}^{\mathsf{E}}_{\delta}=Rw^{\mathsf{M}}_{\delta}-w^{\mathsf{E}}_{\delta}. \end{aligned}$$

(B.23)

Using (B.22), the function $\widehat{w}^{\mathsf{E}}_{\delta}$ becomes a solution to the problem (B.14) in Lemma 7. Moreover, the denominator of the Rayleigh quotient turns into

$$\begin{aligned} \bigl\langle w^{\mathsf{M}}_{\delta},w^{\mathsf{M}}_{\delta}\bigr\rangle =& \bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}w^{\mathsf{M}}_{\delta}+A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}} w^{\mathsf{E}}_{\delta},D^{\mathsf{M}}w^{\mathsf{M}}_{\delta}\bigr)_{\varOmega_{\varTheta}} \\ &{}+\bigl(A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}\widehat{w}^{\mathsf{E}}_{\delta}, D^{\mathsf{M}}w^{\mathsf{M}}_{\delta}\bigr)_{\varOmega_{\varTheta}}. \end{aligned}$$

(B.24)

For the computation of the first scalar product, we proceed in the same way as in (B.3):

$$\begin{aligned} &\bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}w^{\mathsf{M}}_{\delta}+A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}w^{\mathsf {E}}_{\delta},D^{\mathsf{M}}w^{\mathsf{M}}_{\delta}\bigr)_{\varOmega_{\varTheta}} \\ &\quad= \int_{\mathbb{R}}e^{-2\delta|z|} \bigl(\bigl(A^{\mathsf{M}\mathsf {M}}D^{\mathsf{M}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}(y), D^{\mathsf{M}}(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}(y) \bigr)_{\omega}\bigr)dz \\ &\quad\quad{}+ \int_{\mathbb{R}}e^{-2\delta|z|} \bigl(\bigl(A^{\mathsf{M}\mathsf {E}}D^{\mathsf{E}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}(y), D^{\mathsf{M}}(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}(y) \bigr)_{\omega}\bigr)dz \\ &\quad\quad{}-\bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}(\nabla)w^{\mathsf {M}}_{\dagger}+A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}( \nabla)w^{\mathsf{E}}_{\dagger}, D^{\mathsf{M}}(\nabla)w^{\mathsf {M}}_{\dagger}, \bigr)_{\varTheta}+O(\delta) \\ &\quad=\frac{1}{\delta}\lambda_{\dagger}\bigl(\rho U^{\mathsf {M}}_{\dagger},U^{\mathsf{M}}_{\dagger}\bigr)_{\omega} \\ &\quad\quad{}-\bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}(\nabla_{y},i \zeta_{\dagger})U^{\mathsf{M}}_{\dagger}+A^{ME }D^{\mathsf{E}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}, D^{\mathsf{M}}(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}\bigr)_{\varTheta}+O(\delta). \end{aligned}$$

(B.25)

The calculation of (B.25) needs an explanation. We made use of the transformation

$$\begin{aligned} &\bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}(\nabla )e^{-\delta|z|}w^{\mathsf{M}}_{\dagger}+ A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}(\nabla)e^{-\delta |z|}w^{\mathsf{E}}_{\dagger},D^{\mathsf{M}}( \nabla)e^{-\delta|z|} w^{\mathsf{M}}_{\dagger}\bigr)_{\varOmega} \\ &\quad= \bigl(e^{-2\delta|z|}\bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}( \nabla)w^{\mathsf{M}}_{\dagger}+A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}( \nabla)w^{\mathsf{E}} _{\dagger}\bigr), D^{\mathsf{M}}( \nabla)w^{\mathsf{M}}_{\dagger}\bigr)_{\varOmega} \\ &\quad\quad{}-\delta \big(e^{-2\delta|z|} A^{\mathsf{M}\mathsf{M}}D^{\mathsf {M}}\bigl(0,0, \mathrm{sign}(z)\bigr)U^{\mathsf{M}}_{\dagger},D^{\mathsf{M}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}\bigr)_{\varOmega} \\ &\quad\quad{}-\delta\bigl(e^{-2\delta|z|}A^{\mathsf{M}\mathsf{E}}D^{\mathsf {E}}\bigl(0,0, \mathrm{sign}(z)\bigr)U^{\mathsf{E}}_{\dagger},D^{\mathsf{M}}( \nabla_{y}, i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}\bigr)_{\varOmega} \\ &\quad\quad{}-\delta\big(e^{-2\delta|z|} A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}, D^{\mathsf{M}}\bigl(0,0,\mathrm{sign}(z)\bigr)U^{\mathsf {M}}_{\dagger}\bigr)_{\varOmega} \\ &\quad\quad{}-\delta\bigl(e^{-2\delta|z|}A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}, D^{\mathsf{M}}\bigl(0,0,\mathrm{sign}(z)\bigr)U^{\mathsf {M}}_{\dagger}\bigr)_{\varOmega} \\ &\quad\quad{}+\delta^{2} \bigl(e^{-2\delta|z|} A^{\mathsf{M}\mathsf {M}}D^{\mathsf{M}} \bigl(0,0,\mathrm{sign}(z)\bigr)U^{\mathsf{M}}_{\dagger}, D^{\mathsf{M}} \bigl(0,0,\mathrm{sign}(z)\bigr)U^{\mathsf{M}}_{\dagger}\bigr)_{\varOmega} \\ &\quad\quad{}+\delta^{2} \bigl(A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}\bigl(0,0, \mathrm{sign}(z)\bigr)U^{\mathsf{E}}_{\dagger}, D^{\mathsf{M}}\bigl(0,0, \mathrm{sign}(z)\bigr)U^{\mathsf{M}}_{\dagger}\bigr)_{\varOmega}. \end{aligned}$$

(B.26)

The last term on the right-hand side of (B.26) is of order $O(\delta)$ whereas, in view of (B.4), the first term equals

$$\begin{aligned} &\delta^{-1} \bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}+ A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger},D^{\mathsf{M}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}\bigr)_{\omega}\\ &\quad=\delta^{-1} \bigl(\rho U^{\mathsf{M}}_{\dagger},U^{\mathsf{M}}_{\dagger}\bigr)_{\omega}, \end{aligned}$$

where we have employed the integral identity (20) with $\zeta=\zeta_{\dagger}, U=U_{\dagger}$, $\varLambda(\zeta)=\lambda _{\dagger}$ and $V=(U^{\mathsf{M}}_{\dagger},0)$. The integrands in the second to fifth terms on the right in (B.26) are odd functions in the $z$-variable. Therefore, the sum of these integrals vanishes.

Finally, in dealing with the scalar product

$$\bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}(\nabla)w^{\mathsf {M}}_{\delta}+A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}( \nabla)w^{\mathsf{E}}_{\delta}, D^{\mathsf{M}}(\nabla)w^{\mathsf {M}}_{\delta}\bigr)_{\varTheta}, $$

we have replaced $e^{-\delta|z|}$ by 1 in the compact set $\varTheta$ providing an error of order $O(\delta)$.

Our evaluation so far has not taken into account the non-local perturbation of the elasticity problem which is contributed in the last scalar product in (B.24), which depends on the solution $\widehat{w}^{\mathsf {E}}_{\delta}$ of the problem (B.14). Indeed, by Lemma 7, we recall (46) and write

$$\begin{aligned} \bigl(A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}} \widehat {w}^{\mathsf{E}}_{\delta}, D^{\mathsf{M}}w^{\mathsf{M}}_{\delta}\bigr)_{\varOmega_{\varTheta}} =& \bigl(A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}} \mathrm{e}^{-\delta |z|}w^{\mathsf{E}}_{\bullet}, D^{\mathsf{M}}\mathrm{e}^{-\delta|z|}w^{\mathsf{M}}_{\dagger}\bigr)_{\varOmega_{\varTheta}} \\ &{}+\delta\biggl(A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}\sum _{\pm} \pm\chi_{\pm}\mathrm{e}^{-\delta|z|} \mathrm{e}^{i\zeta_{\dagger}z}U^{\mathsf{E}}_{\bullet}, D^{\mathsf{M}} \mathrm{e}^{-\delta|z|} w^{\mathsf{M}}_{\dagger}\biggr)_{\varOmega _{\varTheta}} \\ &{}+ \bigl(A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}\widetilde{w}^{\mathsf{E}}_{\delta}, D^{\mathsf{M}}\mathrm{e}^{-\delta|z|}w^{\mathsf{M}}_{\dagger}\bigr)_{\varOmega_{\varTheta}}. \end{aligned}$$

(B.27)

Since $\|\mathrm{e}^{-\delta|z|}w^{\mathsf{M}}_{\dagger};L^{2}(\varOmega _{\varTheta})\|\leq c\delta^{-1/2}$, the modulo of the last term in (B.27) is less than $c\delta^{1/2}$, see (B.16). A similar bound occurs for second term, because its modulo can be estimated from above by the expression

$$c\delta\biggl( \int _{-n_{\varTheta}}^{n_{\varTheta}}e^{-2\delta|z|}dz+ \bigg|\sum _{\pm}\pm \int _{-n_{\varTheta}}^{\infty}e^{-2\delta|z|}dz \bigg|+\delta \int _{-\infty}^{\infty}e^{-2\delta|z|}dz \biggr)\leq C\delta. $$

By virtue of the exponential decay of $w^{\mathsf{E}}_{\bullet}$ and the integral identity (B.5) with $v^{\mathsf{E}}=w^{\mathsf{E}}_{\bullet}$, the first term on the right-hand side of (B.27) equals

$$\begin{aligned} \bigl(A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}w^{\mathsf{E}}_{\bullet}, D^{\mathsf{M}}w^{\mathsf{M}}_{\dagger}\bigr)_{\varOmega_{\varTheta}}+O( \delta) =& \bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}w^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}w^{\mathsf{E}}_{\bullet}\bigr)_{\varOmega_{\varTheta}} \\ &{}+ \bigl(A^{\mathsf{M}\mathsf{E}}D^{\mathsf{E}}w^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}w^{\mathsf{E}}_{\dagger}\bigr)_{\varOmega_{\varTheta}}+O( \delta). \end{aligned}$$

By the same arguments, but with $v^{\mathsf{E}}=\mathrm{e}^{-\delta |z|}w^{\mathsf{E}}_{\dagger}$ as the test function, we show that

$$\begin{aligned} &\bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}w^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}w^{\mathsf{E}}_{\dagger}\bigr)_{\varOmega_{\varTheta}} \\&\quad= \lim _{\delta\to+0} \bigl( A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}} w^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}\mathrm{e}^{-\delta|z|} w^{\mathsf{E}}_{\dagger}\bigr)_{\varOmega_{\varTheta}} \\ &\quad=\lim_{\delta\to+0} \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}} w^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}} w^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}\mathrm{e}^{-\delta |z|}w^{\mathsf{E}}_{\dagger}\bigr)_{\varOmega_{\varTheta}} \\ &\quad=- \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}w^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}w^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}w^{\mathsf{E}}_{\dagger}\bigr)_{\varTheta} \\ &\quad\quad{}+\lim_{\delta\to+0} \int_{\mathbb{R}} \mathrm{e}^{-\delta|z|} \bigl( A^{\mathsf {E}\mathsf{M}}D^{\mathsf{M}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}\bigr)_{\omega}dz \\ &\quad\quad{}-\lim_{\delta\to+0} \int_{\mathbb{R}} \mathrm{e}^{-\delta|z|} \bigl( A^{\mathsf {E}\mathsf{M}}D^{\mathsf{M}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}, \\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad D^{\mathsf{E}}\bigl(0,0,\delta\mathrm{sign}(z)\bigr)U^{\mathsf{E}}_{\dagger}\bigr)_{\omega}dz. \end{aligned}$$

(B.28)

The limits vanish due to the following reasons. The scalar product in $L^{2}(\omega)$ with $D(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}$ is zero according to the integral identity (26), where $\zeta=\zeta_{\dagger}$ and $U^{\mathsf{E}}= V^{\mathsf{E}}=U^{\mathsf{E}}_{\dagger}$. The integral containing the term $D(0,0,\delta\mathrm{sign}(z))U^{\mathsf{E}}_{\dagger}$ becomes null, since it can be written in the form

$$\begin{aligned} \int_{-\infty}^{0}\delta\mathrm{e}^{-\delta|z|} K(\omega) dz - \int_{0}^{\infty}\delta\mathrm{e}^{-\delta|z|} K(\omega) dz, \end{aligned}$$

where

$$K(\omega)= \bigl( A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}(\nabla_{y},i \zeta_{\dagger})U^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}(0,0,1)U^{\mathsf{E}}_{\dagger}\bigr)_{\omega}$$

is independent on $z$.

Collecting the calculations above, the denominator can be estimated as

$$\begin{aligned} \bigl\langle w^{\mathsf{M}}_{\delta},w^{\mathsf{M}}_{\delta}\bigr\rangle &= \frac{\lambda_{\dagger}}{\delta} \bigl(\rho U^{\mathsf{M}}_{\dagger}, U^{\mathsf{M}}_{\dagger}\bigr)_{\omega} \\ &\quad{}- \bigl(A^{\mathsf{M}\mathsf{M}}D^{\mathsf{M}}(\nabla_{y},i \zeta_{\dagger})U^{\mathsf{M}}_{\dagger}+ A^{\mathsf{M}\mathsf {E}}D^{\mathsf{E}}( \nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}, D^{\mathsf{M}}(\nabla_{y},i\zeta_{\dagger}) U^{\mathsf{M}}_{\dagger}\bigr)_{\varTheta}\\ &\quad{}- \bigl(A^{\mathsf{E}\mathsf{M}}D^{\mathsf{M}}(\nabla_{y},i \zeta_{\dagger})U^{\mathsf{M}}_{\dagger}-A^{\mathsf{E}\mathsf {E}}D^{\mathsf{E}}( \nabla_{y},i\zeta_{\dagger}) U^{\mathsf{E}}_{\dagger}, D^{\mathsf{E}}(\nabla_{y},i\zeta_{\dagger})U^{\mathsf{E}}_{\dagger}\bigr)_{\varTheta} \\ &\quad{}+ \bigl(A^{\mathsf{E}\mathsf{E}}D^{\mathsf{E}}(\nabla)w^{\mathsf{E}}_{\bullet}, D^{\mathsf{E}}(\nabla)w^{\mathsf{E}}_{\bullet}\bigr)_{\varOmega _{\varTheta}}. \end{aligned}$$

(B.29)