Application of the second domain derivative in inverse electromagnetic scattering

A challenging class of inverse problems in scattering theory is the identification of scattering objects by the knowledge of far field patterns of scattered waves (see [3]). We have to distinguish the inverse scattering problems, if the response to any or at least to many incident fields is known, or if data are given only for one or a few incident fields. In this work we are going to consider the extreme situation of the reconstruction of the shape of a perfect conductor just from the knowledge of the far field pattern of one scattered time harmonic electromagnetic wave.

Derivative based iterative regularization schemes are known to be suitable numerical approaches for this class of problems (see the discussion in section 5.4 in [3] and references cited therein). Thus, we focus on linearization of the far field pattern with respect to variations of the shape of the scattering object. The derivative is given by the far field pattern of the so called domain derivative of the scattered wave. These domain derivatives are well established for most of the usually considered boundary value problems (see [14] and references cited therein). Furthermore, in case of acoustic scattering problems several numerical implementations are documented (see [6, 11] for three dimensional examples). Presumably due to the computational effort, there are only a few results for the full vector valued electromagnetic inverse scattering problem (see [9, 15, 21]). These approaches are based on boundary integral equations for the electromagnetic scattering problem and the first domain derivative. We will extend on these results, mainly by showing the existence and a characterization of the second domain derivative. This gives rise to an application of second degree regularization schemes (see [13, 16]).

After this introduction, in section 2, we collect some notations for the scattering problem and describe its weak formulation for later use. In section 3 the first domain derivative of the scattering problem is discussed. Although the derivative has already been established (see [19, 24]), we will present it in some detail in preparation of the following investigations for the second derivative. Moreover, the characterization of the domain derivative is used to show a condition, which ensures that a constant variation cannot lead to a vanishing domain derivative. This illuminates the challenging question on injectivity of the domain derivative operator. With these preparations we devote section 4 to the second domain derivative. It is shown that such a derivative exists and can be characterized again by an electromagnetic boundary value problem. Finally, based on these characterizations we explain and discuss in section 5 the regularized Halley method applied to the inverse problem and present its numerical performance by some examples.

Some results of this paper, e.g. theorem 4.5, are part of one of the authors' PhD thesis [8].

Let us assume a bounded scattering obstacle $D\subset {\mathbb{R}}^{3}$ with smooth boundary and simply connected complement ${\mathbb{R}}^{3}{\backslash}D$. The object is surrounded by a homogeneous, linear, isotropic and non-conductive medium with electric permittivity ɛ₀ > 0 and magnetic permeability μ₀ > 0, for instance vacuum. At frequency ω > 0, the time harmonic Maxwell system for the electric field E and the magnetic field H then reads as

$$\begin{equation}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E-\mathrm{i}kH=0,\quad \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace H+\mathrm{i}kE=0,\end{equation} \tag{ 2.1 }$$

with wave number $k=\omega \sqrt{{\varepsilon }_{0}{\mu }_{0}}$ (see [3]). Given an incident plane wave, Eⁱ(x) = p e^ikd⋅x , Hⁱ(x) = (d × p)e^ikd⋅x for $x\in {\mathbb{R}}^{3}\enspace$ with polarisation $p\in {\mathbb{C}}^{3}$ and direction $d\in {\mathbb{S}}^{2}$ satisfying p ⋅ d = 0 the scatterer gives rise to a radiating scattered field (E^s, H^s), a solution of the Maxwell system (2.1) in ${\mathbb{R}}^{3}{\backslash}\overline{D}$, which satisfies the Silver–Müller radiation condition

$$\begin{equation*}\underset{\vert x\vert \to \infty }{\mathrm{lim}}\enspace \left[{H}^{\mathrm{s}}\left(x\right){\times}x-\vert x\vert {E}^{\mathrm{s}}\left(x\right)\right]=0.\end{equation*}$$

The interaction of the perfect conductor D with the incident wave can be formulated as a boundary condition for the total field E = E^s + Eⁱ and is given by

$$\begin{equation*}\nu {\times}E=0\quad \text{on}\;\partial D,\end{equation*}$$

where ν denotes the outwards directed normal vector to ∂D.

The following investigations require a variational formulation of the scattering problem. Thus, we choose R > 0 large enough such that $\overline{D}\subset {B}_{R}\left(0\right)$, where B_R (0) denotes the open ball of radius R centered in the origin, and introduce the bounded domain ${\Omega}={B}_{R}\left(0\right){\backslash}\overline{D}$. In order to derive the weak formulation, let (E, H) be a pair of reasonable smooth solutions of the scattering problem and let V denote a test function with ν × V = 0 an ∂D. By partial integration and the Maxwell system (2.1) we arrive at

$$\begin{align}\hfill & {\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\cdot \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}-{k}^{2}\enspace E\cdot \overline{V}\right)\mathrm{d}x+\mathrm{i}k{\int }_{\partial {B}_{R}\left(0\right)}{\Lambda}\left(\nu {\times}E\right)\cdot \overline{V}\mathrm{d}s\hfill \\ \hfill & \quad ={\int }_{\partial {B}_{R}\left(0\right)}\left(\mathrm{i}k\enspace {\Lambda}\left(\nu {\times}{E}^{\mathrm{i}}\right)-\nu {\times}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\mathrm{i}}\right)\cdot \overline{V}\mathrm{d}s.\hfill \end{align} \tag{ 2.2 }$$

To ensure that a solution E^s = E − Eⁱ of (2.2) can be extended to a radiating solution of the Maxwell system in ${\mathbb{R}}^{3}{\backslash}\overline{D}$ we have introduced on the artificial boundary ∂B_R (0) the Calderon operator Λ, which maps ν × φ onto ν × H^s, where (E^s, H^s) denote the unique radiating solution of

$$\begin{align*}\hfill \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\mathrm{s}}-\mathrm{i}k{H}^{\mathrm{s}}=0,\quad \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {H}^{s}+\mathrm{i}k{E}^{s}=0& \quad \text{in}\;{\mathbb{R}}^{3}{\backslash}\overline{{B}_{R}\left(0\right)}\hfill \\ \hfill \nu {\times}{E}^{\mathrm{s}}=\nu {\times}\varphi & \quad \text{on}\;\partial {B}_{R}\left(0\right).\hfill \end{align*}$$

Equation (2.2) is considered in the Sobolev space $H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)=\left\{E\in {L}^{2}\left({\Omega},{\mathbb{C}}^{3}\right):\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\in {L}^{2}\left({\Omega},{\mathbb{C}}^{3}\right)\right\}$. The boundary integrals on ∂D and ∂B_R (0) exists in the sense of the dual pairing ${\langle \cdot ,\cdot \rangle }_{\partial {B}_{R}\left(0\right)}$ between the range spaces ${H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial {\Omega}\right)$ and ${H}^{-\frac{1}{2}}\left(\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l},\partial {\Omega}\right)$ of the traces γ_t φ = φ × ν and γ_T φ = ν × (φ × ν) for φ ∈ H(curl, Ω). Then, γ_t E = 0 in ${H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right)$ and we incorporate the boundary condition by the closed subspace

$$\begin{equation*}{H}_{\text{pc}}\left({\Omega}\right)=\left\{E\in H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right):\enspace {\gamma }_{t}E=0\right\}\end{equation*}$$

Since, the Calderon operator is extendable to a bounded operator ${\Lambda}:{H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial {B}_{R}\left(0\right)\right)\to {H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial {B}_{R}\left(0\right)\right)$ (see [22]), we finally can define the bounded sesquilinear form $\mathcal{A}:{H}_{\text{pc}}\left({\Omega}\right){\times}{H}_{\text{pc}}\left({\Omega}\right)\to \mathbb{C}$ and the antilinear map $\ell :{H}_{\text{pc}}\left({\Omega}\right)\to \mathbb{C}$ such that (2.2) reads as

$$\begin{equation}\mathcal{A}\left(E,V\right)=\ell \left(V\right).\end{equation} \tag{ 2.3 }$$

A weak solution of the scattering problem is then given by a function E ∈ H_pc(Ω) such that $\mathcal{A}\left(E,V\right)=\ell \left(V\right)$ holds for all V ∈ H_pc(Ω). Assuming D to be a Lipschitz domain it is known that for any ℓ ∈ H_pc(Ω)* there exists a unique solution E ∈ H_pc(Ω) of (2.3) for all V ∈ H_pc(Ω), and it exists c > 0 such that ${\Vert}E{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}{\leqslant}c\enspace {\Vert}\ell {{\Vert}}_{{H}_{\mathrm{p}\mathrm{c}}^{{\ast}}}$ (see [22, theorem 10.7]).

Due to the radiation condition, the scattered field E^s in ${\mathbb{R}}^{3}{\backslash}\overline{D}$ has the asymptotic behavior

$$\begin{equation*}{E}^{\mathrm{s}}\left(x\right)=\frac{{\mathrm{e}}^{\mathrm{i}k\vert x\vert }}{4\pi \vert x\vert }\left[{E}_{\infty }\left(\hat{x}\right)+\mathcal{O}\left(\frac{1}{\vert x\vert }\right)\right],\quad \vert x\vert \to \infty .\end{equation*}$$

E_∞ is called the (electric) far field pattern and is an analytic tangential vector field on the unit sphere ${\mathbb{S}}^{2}$. This motivates the definition of the non-linear boundary to far field operator F, which maps the boundary ∂D onto the far field pattern of E^s, i.e.,

$$\begin{equation}\mathbf{F}\left(\partial D\right)={E}_{\infty }.\end{equation} \tag{ 2.4 }$$

Of course, F depends also on the incident field (Eⁱ, Hⁱ) and the wave number k, which we assume to be fixed and known. The domain of F is given by a class of admissible boundaries, for which there is a unique solution of the scattering problem. Thus, the inverse obstacle problem under consideration is given by the inversion of equation (2.4).

It is known that the far field pattern uniquely determines the solution of the scattering problem, but, nevertheless, the whole inverse obstacle problem is severely ill-posed. For some more details on inverse electromagnetic scattering we refer to [3], where, for instance, uniqueness of the inverse problem is shown in the sense that if for a fixed wave number k the far field patterns of two objects for all incident plane waves coincide, the scattering objects must be identical. Such a result is not known in case of just one incident field.

In solving for the nonlinear equation (2.4), obviously a linearization is useful. Thus, a derivative of the far field pattern with respect to variations of the boundary of the scattering obstacle D is of specific interest. This leads to the concept of a domain derivative, which is well established in electromagnetic scattering (see [4, 12, 14, 19, 24]). For convenience to the reader and in preparation of the next section we present the variational approach in some detail, following very closely [12], where penetrable scattering objects are considered.

A perturbation of the scatterer is described by a vector field $h\in {C}^{1}\left({\mathbb{R}}^{3},{\mathbb{R}}^{3}\right)$ with compact support. Given a set $D\subset {\mathbb{R}}^{3}$, we denote by D_h the corresponding perturbed set D_h = {x + h(x) : x ∈ D}. If the C¹-norm of h is sufficiently small, the transformation x ↦ φ(x) = x + h(x) is a diffeomorphism. Throughout, we assume that a perturbation h does not change the artificial boundary ∂B_R (0). Thus, without loss of generality we have $h\in {C}_{0}^{1}\left({B}_{R}\left(0\right),{\mathbb{R}}^{3}\right)$. For functions $f:{D}_{h}\to {\mathbb{R}}^{d}$, d ∈ {1, 3}, on the perturbed set D_h we define the corresponding function $\tilde {f}:D\to {\mathbb{R}}^{d}$ on the unperturbed set by $\tilde {f}\left(x\right)=f\left(\varphi \left(x\right)\right)$.

Let E ∈ H_pc(Ω) be the weak solution of the scattering problem (2.2) and let E_h ∈ H_pc(Ω_h ) denote the solution of the scattering problem with respect to a perturbed scatterer D_h , i.e.

$$\begin{equation}{\int }_{{{\Omega}}_{h}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{h}\cdot \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {V}_{h}}-{k}^{2}{E}_{h}\cdot \overline{{V}_{h}}\right)\mathrm{d}x-\mathrm{i}k{\langle {\Lambda}{\gamma }_{t}{E}_{h},{\gamma }_{T}{V}_{h}\rangle }_{\partial {B}_{R}\left(0\right)}=\ell \left({V}_{h}\right)\end{equation} \tag{ 3.1 }$$

for all V_h ∈ H_pc(Ω_h ). Note the same right-hand sides of the weak formulations (2.2) and (3.1), since the boundary integral on the artificial boundary does not change. According to different sets of definition, E_h has to be transformed. We use the curl conserving transformation E_h ↦ Ê_h , given by

$$\begin{equation*}{\hat{E}}_{h}\left(x\right)={J}_{\varphi }^{\top }\left(x\right)\tilde {{E}_{h}}\left(x\right)=\left(I+{J}_{h}^{\top }\left(x\right)\right){E}_{h}\left(x+h\left(x\right)\right),\end{equation*}$$

where J_φ denotes the Jacobian matrix of φ (see [2, section 3.9]). Then E_h ∈ H_pc(Ω_h ) implies Ê_h ∈ H_pc(Ω), what can be seen from ${\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}_{\sim }\tilde {{E}_{h}}=\frac{1}{\mathrm{det}\enspace {J}_{\varphi }}{J}_{\varphi }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{E}}_{h},$ where curl_∼ denotes the curl operator with respect to the untransformed coordinates, and from ${\langle {\gamma }_{t}{\hat{E}}_{h},{\gamma }_{T}{\hat{V}}_{h}\rangle }_{\partial {\Omega}}={\langle {\gamma }_{t}{E}_{h},{\gamma }_{T}{V}_{h}\rangle }_{\partial {{\Omega}}_{h}}$, which follows by partial integration.

By the transformation we arrive for Ê_h ∈ H_pc(Ω) at

$$\begin{align}\hfill & {\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{E}}_{h}^{\top }\frac{{J}_{\varphi }^{\top }{J}_{\varphi }}{\mathrm{det}\enspace {J}_{\varphi }}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}-{k}^{2}\enspace {\hat{E}}_{h}^{\top }\mathrm{det}\left({J}_{\varphi }\right){J}_{\varphi }^{-1}{J}_{\varphi }^{-\top }\overline{V}\right)\mathrm{d}x\hfill \\ \hfill & \quad -\mathrm{i}k\enspace {\langle {\Lambda}{\gamma }_{t}{\hat{E}}_{h},{\gamma }_{T}V\rangle }_{\partial {B}_{R}\left(0\right)}=\ell \left(V\right)\hfill \end{align} \tag{ 3.2 }$$

for all V ∈ H_pc(Ω). We define the bounded sesquilinear form ${\mathcal{A}}_{h}:{H}_{\text{pc}}\left({\Omega}\right){\times}{H}_{\text{pc}}\left({\Omega}\right)\to \mathbb{C}$ such that (3.2) reads as ${\mathcal{A}}_{h}\left({\hat{E}}_{h},V\right)=\ell \left(V\right)$.

Straight forward calculations yield the asymptotic behavior

$$\begin{equation}\frac{{J}_{\varphi }^{\top }{J}_{\varphi }}{\mathrm{det}\enspace {J}_{\varphi }}=\left(1-\mathrm{d}\mathrm{i}\mathrm{v}\enspace h\right)I+{J}_{h}+{J}_{h}^{\top }+\mathcal{O}\left({\Vert}h{{\Vert}}_{{C}^{1}}^{2}\right),\end{equation} \tag{ 3.3 }$$

$$\begin{equation}\mathrm{det}\left({J}_{\varphi }\right)\enspace {J}_{\varphi }^{-1}{J}_{\varphi }^{-\top }=\left(1+\mathrm{d}\mathrm{i}\mathrm{v}\enspace h\right)I-{J}_{h}-{J}_{h}^{\top }+\mathcal{O}\left({\Vert}h{{\Vert}}_{{C}^{1}}^{2}\right).\end{equation} \tag{ 3.4 }$$

for ||h|| → 0. As a first step we show continuity with respect to the perturbation h.

Theorem 3.1. Let E ∈ H_pc(Ω) be the solution of (2.2) and Ê_h ∈ H_pc(Ω) of (3.2). Then,

$$\begin{equation*}\underset{{\Vert}h{{\Vert}}_{{C}^{1}}\to 0}{\mathrm{lim}}{\Vert}E-{\hat{E}}_{h}{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}=0.\end{equation*}$$

Proof. Let A, A_h : H_pc(Ω) → H_pc(Ω) denote the bounded linear operators defined by

$$\begin{equation*}\langle AE,{V\rangle }_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}=\mathcal{A}\left(E,V\right),\quad \langle {A}_{h}E,{V\rangle }_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}={\mathcal{A}}_{h}\left(E,V\right).\end{equation*}$$

We calculate

$$\begin{align*}\hfill {\Vert}\left({A}_{h}-A\right)V{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}^{2}& ={\mathcal{A}}_{h}\left(V,\left({A}_{h}-A\right)V\right)-\mathcal{A}\left(V,\left({A}_{h}-A\right)V\right)\hfill \\ \hfill & ={\int }_{{\Omega}}\left[\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {V}^{\top }\left(\frac{{J}_{\varphi }^{\top }{J}_{\varphi }}{\mathrm{det}\enspace {J}_{\varphi }}-I\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left({A}_{h}-A\right)V}\right.\hfill \\ \hfill & \quad -{k}^{2}\enspace {V}^{\top }\left(\mathrm{det}\enspace {J}_{\varphi }{J}_{\varphi }^{-1}{J}_{\varphi }^{-\top }-I\right)\overline{\left({A}_{h}-A\right)V}\left.\right]\mathrm{d}x,\hfill \end{align*}$$

and by (3.3) and (3.4) we obtain

$$\begin{equation*}{\Vert}\left({A}_{h}-A\right)V{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}^{2}{\leqslant}C{\Vert}h{{\Vert}}_{{C}^{1}}{\Vert}V{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}{\Vert}\left({A}_{h}-A\right)V{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}.\end{equation*}$$

Therefore, ||A_h − A|| → 0 as ${\Vert}h{{\Vert}}_{{C}^{1}}\to 0$. The operator A possesses a bounded inverse (see again [22, theorem 10.7]). Thus, a perturbation argument (see [20, theorem 10.1]) yields ||Ê_h − E||_H(curl,Ω) → 0 as ${\Vert}h{{\Vert}}_{{C}^{1}}\to 0$.□

Looking closely at the linearizations of the coefficients in the weak formulation (3.2), we can prove differentiability.

Theorem 3.2. Let E ∈ H_pc(Ω) be the solution of (2.2) and Ê_h ∈ H_pc(Ω) of (3.2). Then there exists a function W ∈ H_pc(Ω), depending linearly on $h\in {C}_{0}^{1}\left({B}_{R}\left(0\right),{\mathbb{R}}^{3}\right)$, such that

$$\begin{equation*}\underset{{\Vert}h{{\Vert}}_{{C}^{1}}\to 0}{\mathrm{lim}}\enspace \frac{1}{{\Vert}h{{\Vert}}_{{C}^{1}}}{\Vert}{\hat{E}}_{h}-E-W{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}=0.\end{equation*}$$

Proof. We define W ∈ H_pc(Ω) as the solution of

$$\begin{equation*}\mathcal{A}\left(W,V\right)={\int }_{{\Omega}}\left[\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left(\mathrm{d}\mathrm{i}\mathrm{v}\left(h\right)\enspace I-{J}_{h}-{J}_{h}^{\top }\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {E}^{\top }\left(\mathrm{d}\mathrm{i}\mathrm{v}\left(h\right)\enspace I-{J}_{h}-{J}_{h}^{\top }\right)\overline{V}, \right]\mathrm{d}x\end{equation*}$$

for all V ∈ H_pc(Ω). Then, from

$$\begin{align*}\hfill \mathcal{A}\left({\hat{E}}_{h}-E-W,V\right)& =\mathcal{A}\left({\hat{E}}_{h},V\right)-{\mathcal{A}}_{h}\left({\hat{E}}_{h},V\right)-\mathcal{A}\left(W,V\right)\hfill \\ \hfill & ={\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{E}}_{h}^{\top }\left(I-\frac{{J}_{\varphi }^{\top }{J}_{\varphi }}{\mathrm{det}\enspace {J}_{\varphi }}-\left(\mathrm{d}\mathrm{i}\mathrm{v}\left(h\right)\enspace I-{J}_{h}-{J}_{h}^{\top }\right)\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & \quad -{k}^{2}{\int }_{{\Omega}}{\hat{E}}_{h}^{\top }\left(I-{J}_{\varphi }^{-1}{J}_{\varphi }^{-\top }\enspace \mathrm{det}\enspace {J}_{\varphi }+\left(\mathrm{d}\mathrm{i}\mathrm{v}\left(h\right)\enspace I-{J}_{h}-{J}_{h}^{\top }\right)\right)\overline{V}\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left({\hat{E}}_{h}-E\right)}^{\top }\left(\mathrm{d}\mathrm{i}\mathrm{v}\left(h\right)\enspace I-{J}_{h}-{J}_{h}^{\top }\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\hfill \\ \hfill & \quad +\enspace {k}^{2}{\left({\hat{E}}_{h}-E\right)}^{\top }\left(\mathrm{d}\mathrm{i}\mathrm{v}\left(h\right)\enspace I-{J}_{h}-{J}_{h}^{\top }\right)\overline{V}\mathrm{d}x\hfill \end{align*}$$

for any V ∈ H(curl, Ω) we conclude with (3.3) and (3.4)

$$\begin{align*}\hfill & \frac{1}{{\Vert}h{{\Vert}}_{{C}^{1}}}\mathcal{A}\left({\hat{E}}_{h}-E-W,V\right)\hfill \\ \hfill & \quad {\leqslant}C\left({\Vert}{\hat{E}}_{h}{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}\mathcal{O}\left({\Vert}h{{\Vert}}_{{C}^{1}}\right)+{\Vert}{\hat{E}}_{h}-E{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}\right){\Vert}V{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}\to 0,\hfill \end{align*}$$

as h → 0 in C¹, which implies ${\mathrm{lim}}_{{\Vert}h{{\Vert}}_{{C}^{1}}\to 0}\enspace \frac{1}{{\Vert}h{{\Vert}}_{{C}^{1}}}{\Vert}{\hat{E}}_{h}-E-W{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}=0$ by a perturbation argument as in the proof of theorem 3.1.□

The function W ∈ H_pc(Ω) is called material derivative of E and it is no solution of the homogeneous Maxwell's equations. Note that W depends on values of h in the neighborhood of ∂D, which is not adequate in view of perturbations of the boundary ∂D. A formal Taylor expansion motivates to consider the domain derivative ${E}^{\prime }=W-{J}_{h}^{\top }E-{J}_{E}h$, which leads to the desired derivative of the operator F.

We introduce the notation V_ν = V ⋅ ν and V_τ = ν × (V × ν) for the decomposition V = V_ν ν + V_τ of a vector field on ∂D. Furthermore, we introduce the surface gradient ${\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}:{H}^{\frac{1}{2}}\left(\partial D\right)\to {H}^{-\frac{1}{2}}\left(\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l},\partial D\right)$, given for smooth functions u by ${\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace u=\nabla u-\frac{\partial u}{\partial \nu }\nu$, the vector surface rotation ${\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}:{H}^{\frac{1}{2}}\left(\partial D\right)\to {H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right)$, given by ${\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\enspace u={\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace u{\times}\nu$, and the surface divergence ${\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}:{H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right)\to {H}^{-\frac{1}{2}}\left(\partial D\right)$, which is defined for a smooth function V by Div_∂D V = div V − ν ⋅ J_V ν. Note that the surface divergence satisfies

$$\begin{equation}{\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left(V{\times}\nu \right)=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V\cdot \nu \end{equation} \tag{ 3.5 }$$

and is coupled by the duality

$$\begin{equation*}{\int }_{\partial D}u\enspace {\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left(V\right)\mathrm{d}s=-{\int }_{\partial D}V\cdot {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left(u\right)\enspace \mathrm{d}s.\end{equation*}$$

With these notations a representation of the domain derivative can be shown.

Theorem 3.3. Let ∂D be of class C². In the setting of theorem 3.2, define ${E}^{\prime }=W-{J}_{h}^{\top }E-{J}_{E}h\in H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)$. E' can be uniquely extended to the radiating weak solution of Maxwell's equations

$$\begin{equation*}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\prime }-\mathrm{i}k{H}^{\prime }=0,\quad \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {H}^{\prime }+\mathrm{i}k{E}^{\prime }=0\end{equation*}$$

in ${\mathbb{R}}^{3}{\backslash}\overline{D}$ with boundary condition

$$\begin{equation*}\nu {\times}{E}^{\prime }={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{\nu }{E}_{\nu }\right)-\mathrm{i}k\enspace {h}_{\nu }\enspace {H}_{\tau }\quad \text{on}\;\partial D.\end{equation*}$$

Proof. By the regularity of ∂D we have $E\in {H}^{1}\left({\Omega},{\mathbb{C}}^{3}\right)$ and therefore ${E}^{\prime }=W-{J}_{h}^{\top }E-{J}_{E}h\in {L}^{2}\left({\Omega},{\mathbb{C}}^{3}\right)$ (see [1]). Some basic vector calculus shows

$$\begin{align}\hfill \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left({J}_{h}^{\top }E+{J}_{E}h\right)& =\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\left({J}_{E}-{J}_{E}^{\top }\right)h+\nabla \left({h}^{\top }E\right)\right)=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}h\right)\hfill \\ \hfill & =\mathrm{d}\mathrm{i}\mathrm{v}\left(h\right)\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E+{J}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E}h-{J}_{h}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E=\mathrm{i}k\enspace \mathrm{d}\mathrm{i}\mathrm{v}\enspace hH+\mathrm{i}k{J}_{H}h-\mathrm{i}k{J}_{h}H,\hfill \end{align} \tag{ 3.6 }$$

which in particular implies $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\prime }\in {L}^{2}\left({\Omega},{\mathbb{C}}^{3}\right)$ and therefore E' ∈ H(curl, Ω). Additionally, by ν × W = 0 on ∂D, we find

$$\begin{equation*}\nu {\times}{E}^{\prime }=-\nu {\times}\left({J}_{E}h+{J}_{h}^{\top }E\right)=-\nu {\times}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}h\right)-\nu {\times}\nabla \left({h}^{\top }E\right)\end{equation*}$$

in ${H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right)$. From ν × E = 0 we obtain $-\nu {\times}\nabla \left({h}^{\top }E\right)={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{\nu }{E}_{\nu }\right)$. Furthermore, with ν × (curl E × h) = ik(h_ν H_τ + H_ν h_τ ) and H_ν = 0, which follows by Maxwell's equations and (3.5), we conclude the stated boundary condition for E'.

It remains to show, that E' is a radiating solution to Maxwell's equations, which will be achieved by showing $\mathcal{A}\left({E}^{\prime },V\right)=0$ for any V ∈ H_pc(Ω). We have

$$\begin{align*}\hfill \mathcal{A}\left({E}^{\prime },V\right)& =\mathcal{A}\left(W,V\right)-\mathcal{A}\left({J}_{h}^{\top }E+{J}_{E}h,V\right)\hfill \\ \hfill & ={\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left(\mathrm{d}\mathrm{i}\mathrm{v}\enspace hI-{J}_{h}-{J}_{h}^{\top }\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}{E}^{\top }\left(\mathrm{d}\mathrm{i}\mathrm{v}\enspace hI-{J}_{h}-{J}_{h}^{\top }\right)\overline{V}\right)\enspace \mathrm{d}x\hfill \\ \hfill & \quad -{\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left({J}_{h}^{\top }E+{J}_{E}h\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}-{k}^{2}{\left({J}_{h}^{\top }E+{J}_{E}h\right)}^{\top }\overline{V}\right)\enspace \mathrm{d}x.\hfill \end{align*}$$

Using again (3.6), we find

$$\begin{equation*}\mathcal{A}\left({E}^{\prime },V\right)={\int }_{{\Omega}}\left({k}^{2}{\left({J}_{E}h+\mathrm{d}\mathrm{i}\mathrm{v}\enspace h\enspace E-{J}_{h}E\right)}^{\top }\overline{V}-{\left({J}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E}h+{J}_{h}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\mathrm{d}x\end{equation*}$$

From div E = 0 in ${\mathbb{R}}^{3}{\backslash}\overline{D}$, we conclude curl(E × h) = div(h) E + J_E h − J_h E and Maxwell's equations yield

$$\begin{align*}\hfill {J}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E}h+{J}_{h}^{\top }E& =\left({J}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E}-{J}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E}^{\top }\right)+{J}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E}^{\top }h+{J}_{h}^{\top }E\hfill \\ \hfill & ={\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}^{2}\enspace E{\times}h+\nabla \left({h}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)={k}^{2}\left(E{\times}h\right)+\nabla \left({h}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right).\hfill \end{align*}$$

Together with $\mathrm{d}\mathrm{i}\mathrm{v}\left(\left(E{\times}h\right){\times}\overline{V}\right)=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left(E{\times}h\right)}^{\top }\overline{V}-{\left(E{\times}h\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}$, we finally arrive at

$$\begin{equation*}\mathcal{A}\left({E}^{\prime },V\right)={k}^{2}{\int }_{{\Omega}}\mathrm{d}\mathrm{i}\mathrm{v}\left(\left(E{\times}h\right){\times}\overline{V}\right)\enspace \mathrm{d}x-{\int }_{{\Omega}}\nabla {\left({h}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\enspace \mathrm{d}x.\end{equation*}$$

Since div curl = 0, we obtain by the divergence theorem

$$\begin{align*}\hfill \mathcal{A}\left({E}^{\prime },V\right)& ={\int }_{{\Omega}}\mathrm{d}\mathrm{i}\mathrm{v}\left[{k}^{2}\left(E{\times}h\right){\times}\overline{V}-\left({h}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right]\mathrm{d}x\hfill \\ \hfill & ={\int }_{\partial D}\left(\left({h}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\left({\nu }^{\top }\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)-{k}^{2}{\nu }^{\top }\left(\left(E{\times}h\right){\times}\overline{V}\right)\right)\mathrm{d}s.\hfill \end{align*}$$

Note that no boundary integrals on ∂B_R (0) occur, since h is compactly supported in B_R (0). The first term vanishes since ${\nu }^{\top }\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}={\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left(\overline{V}{\times}\nu \right)=0$ for V ∈ H_pc(Ω) and for the second term we compute

$$\begin{equation*}\left(\left(E{\times}h\right){\times}\overline{V}\right)\cdot \nu =\left(E\cdot \overline{V}\right)\left(h\cdot \nu \right)-\left(\overline{V}\cdot h\right)\left(E\cdot \nu \right)=\left(h{\times}E\right)\cdot \left(\nu {\times}\overline{V}\right)=0.\end{equation*}$$

Thus, we have $\mathcal{A}\left({E}^{\prime },V\right)=0$, which finishes the proof.□

Before establishing the second domain derivative in the next section let us consider the linearization of the operator F, which by the previous result is given by its Fréchet derivative F'[∂D]h = E'_∞, if we specify a linear space of admissible boundaries. Note, that the domain derivative E' of E in ${\mathbb{R}}^{3}{\backslash}\overline{D}$ also allows for a consideration of near field data instead of the far field pattern. In general solving an ill-posed nonlinear equation by iterative regularization schemes based on its derivative requires some additional conditions on the operator F. A quite general one is the tangential cone condition, which can be described by the existence of a constant c > 0 such that locally ||F(y) − F(x) − F'[x](y − x)|| ⩽ c||y − x|| ||F(x) − F(y)|| holds. It ensures to some extent the equivalence of local ill-posedness of a nonlinear equation and ill-posedness of its linearization (see [17]). To our knowledge the validity of such a condition is an essential open problem in any inverse obstacle scattering problem so far. Here, we just remark on injectivity of F', a necessary consequence from the cone condition, which is a severe problem in itself.

Corollary 3.4. Let the wave number −k² be no eigenvalue of the Laplace–Beltrami operator Δ_∂D on the boundary of D. Furthermore, let $h\in {C}^{1}\left(\partial D,{\mathbb{R}}^{3}\right)$ be a vector field with constant normal component ${h}_{\nu }=c\in \mathbb{R}$ on ∂D. Then F'[∂D]h = 0 on ${\mathbb{S}}^{2}$ implies h_ν = 0.

Proof. A vanishing far field pattern F'[∂D]h = 0 implies E' = 0 in ${\mathbb{R}}^{3}{\backslash}\overline{D}$ (see [3]). Since h_ν is constant we obtain from theorem 3.3

$$\begin{equation}{h}_{\nu }\left({\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({E}_{\nu }\right)-\mathrm{i}k\nu {\times}\left(H{\times}\nu \right)\right)=0\quad \text{on}\;\partial D.\end{equation} \tag{ 3.7 }$$

Assuming h_ν ≠ 0, a rotation and taking the surface divergence yields

$$\begin{align*}\hfill 0& ={\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left(\left({\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({E}_{\nu }\right)\right){\times}\nu -\mathrm{i}k\left(\nu {\times}\left(H{\times}\nu \right)\right){\times}\nu \right)\hfill \\ \hfill & =-{\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left({\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}{E}_{\nu }\right)-\mathrm{i}k\enspace {\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left(H{\times}\nu \right)=-{{\Delta}}_{\partial D}{E}_{\nu }-{k}^{2}{E}_{\nu },\hfill \end{align*}$$

where we have used ${\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({E}_{\nu }\right)={\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({E}_{\nu }\right){\times}\nu$ and Div_∂D (H × ν) = ν ⋅ curl H = −ikE_ν . Since −k² is no eigenvalue of the Laplace–Beltrami operator, we obtain E_ν = 0 on ∂D. Furthermore, by (3.7) we have ν × (H × ν) = 0 on ∂D. Applying the Stratton–Chu representation of Eⁱ in D and of the radiating solution E^s in ${\mathbb{R}}^{3}{\backslash}\overline{D}$ (see [18]) we obtain from vanishing boundary values ν × E = 0 and ν × (H × ν) = 0 on ∂D of the total field the contradiction

$$\begin{align*}\hfill {E}^{\mathrm{i}}\left(x\right)& =\frac{1}{\mathrm{i}k}\enspace {\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}^{2}{\int }_{\partial D}\left(\nu \left(y\right){\times}H\left(y\right)\right){\Phi}\left(x,y\right)\enspace \mathrm{d}s\left(y\right)\hfill \\ \hfill & \quad -\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\int }_{\partial D}\left(\nu \left(y\right){\times}E\left(y\right)\right){\Phi}\left(x,y\right)\enspace \mathrm{d}s\left(y\right)=0\hfill \end{align*}$$

for any x ∈ D. Thus we conclude h_ν = 0.□

Excluding eigenvalues of the Laplace–Beltrami operator seems to be necessary for injectivity of F'. This can be seen from scattering by a ball B_ρ (0) of radius ρ > 0, as it was already observed by H Haddar and R Kress in [7]. Since, if we consider an incident field

$$\begin{equation*}{E}^{\mathrm{i}}\left(x\right)=\sqrt{n\left(n+1\right)}\frac{{j}_{n}\left(kr\right)}{r}{Y}_{n}^{m}\left(\hat{x}\right)\hat{x}+\frac{{j}_{n}\left(kr\right)+kr{j}_{n}^{\prime }\left(kr\right)}{r}{U}_{n}^{m}\left(\hat{x}\right)\end{equation*}$$

in spherical coordinates, $x=r\hat{x}$, with spherical surface harmonics ${Y}_{n}^{m}$, ${U}_{n}^{m}=\frac{1}{\sqrt{n\left(n+1\right)}}\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{{\mathbb{S}}^{2}}{Y}_{n}^{m}$, ${V}_{n}^{m}=\hat{x}{\times}{U}_{n}^{m}$ and Bessel- and Hankel-functions j_n , ${h}_{n}^{\left(1\right)}$ for a positive integer $n\in \mathbb{N}$ and m ∈ {−n, ..., n} (see [18]), some calculations show for the total field E = Eⁱ + E^s on the boundary

$$\begin{align*}\hfill & {\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({E}_{\nu }\right)-\mathrm{i}k\left(\nu {\times}\left(H{\times}\nu \right)\right)\hfill \\ \hfill & \quad =\left({k}^{2}-\frac{n\left(n+1\right)}{{\rho }^{2}}\right)\left[\left(\right.{j}_{n}\left(k\rho \right)+\frac{{j}_{n}\left(k\rho \right)+k\rho {j}_{n}^{\prime }\left(k\rho \right)}{{h}_{n}^{\left(1\right)}\left(k\rho \right)+k\rho {h}_{n}^{\left(1\right)}\left(k\rho \right)}{h}_{n}^{\left(1\right)}\left(k\rho \right)\right]{V}_{n}^{m}\left(\hat{x}\right).\hfill \end{align*}$$

Thus, if ${k}^{2}=\frac{n\left(n+1\right)}{{\rho }^{2}}$ for $n\in \mathbb{N}$, i.e., if −k² is an eigenvalue of the Laplace–Beltrami operator on ∂B_ρ (0), there are nontrivial incident fields leading to a vanishing boundary condition (3.7) and therefore to a vanishing domain derivative.

Based on theorem 3.3 we can characterize the adjoint operator (F')*, which is of specific interest for iterative regularization schemes, and adds to comparable results for the exterior Dirichlet problem in acoustic scattering (see [10]). We specify the investigations to the case of star shaped domains, which also will be considered in the numerical tests below. Without changing notation we consider the operator F : r ↦ E_∞' for $r\in {C}^{2}\left({\mathbb{S}}^{2}\right)$ and a star shaped parameterized boundary $\partial D=\left\{y=r\left(\hat{y}\right)\hat{y}:\hat{y}\in {\mathbb{S}}^{2}\right\}$. Analogously, a variation is given by $h\left(y\right)=\tilde {h}\left(\hat{y}\right)\hat{y}$.

Corollary 3.5. The L²-adjoint operator of ${\mathbf{F}}^{\prime }\left[r\right]:{C}^{2}\left({\mathbb{S}}^{2}\right)\to {L}^{2}\left({\mathbb{S}}^{2},{\mathbb{C}}^{3}\right)$ is given by

$$\begin{equation*}{\left({\mathbf{F}}^{\prime }\left[r\right]\right)}^{{\ast}}A\left(\hat{y}\right)={k}^{2}{r}^{2}\left(\hat{y}\right)\left(\overline{{H}_{\tau }\left(y\right)\cdot {\left({\mathcal{H}}_{A}\left(y\right)\right)}_{\tau }\left(y\right)-{E}_{\nu }\left(y\right){\left({\mathcal{E}}_{A}\left(y\right)\right)}_{\nu }}\right),\end{equation*}$$

where ${\mathcal{E}}_{A}$, ${\mathcal{H}}_{A}$ denote the total fields of the scattering problem with incident field given by the Herglotz wave function ${\mathcal{E}}_{A}^{\mathrm{i}}\left(y\right){=\int }_{{\mathbb{S}}^{2}}\overline{A\left(\hat{x}\right)}{\mathrm{e}}^{-\mathrm{i}k\hat{x}\cdot y}\enspace \mathrm{d}{s}_{\hat{x}}$ for a tangential field $A\in {L}^{2}\left({\mathbb{S}}^{2},{\mathbb{C}}^{3}\right)$.

Proof. We introduce the notation ${E}^{\mathrm{s}}\left(y;a,\hat{x}\right)$ for a solution of the scattering problem with incident field ${E}^{\mathrm{i}}\left(y;a,\hat{x}\right)=a\enspace {\mathrm{e}}^{\mathrm{i}k\hat{x}\cdot y}$ and ${H}^{\mathrm{i}}\left(y;a,\hat{x}\right)=\hat{x}{\times}{E}^{\mathrm{i}}\left(y;a,\hat{x}\right)$ with direction $\hat{x}$ and polarization $a\perp \hat{x}$. From the integral representation of the far field pattern of the radiating solution E' (see [3]) we compute

$$\begin{align*}\hfill a\cdot {E}_{\infty }^{\prime }\left(\hat{x}\right)& =\mathrm{i}k\enspace a\cdot \left(\hat{x}{\times}{\int }_{\partial D}\left[\nu {\times}{E}_{\tau }^{\prime }+\left(\nu {\times}{H}_{\tau }^{\prime }\right){\times}\hat{x}\right]{\mathrm{e}}^{-\mathrm{i}k\hat{x}\cdot y}\enspace \mathrm{d}s\right)\hfill \\ \hfill & =\mathrm{i}k{\int }_{\partial D}\left(\nu {\times}{E}_{\tau }^{\prime }\right)\cdot {H}^{\mathrm{i}}\left(\cdot ;a,-\hat{x}\right)\enspace \mathrm{d}s+\mathrm{i}k{\int }_{\partial D}\left(\nu {\times}{H}_{\tau }^{\prime }\right)\cdot {E}^{\mathrm{i}}\left(\cdot ;a,-\hat{x}\right)\enspace \mathrm{d}s\hfill \end{align*}$$

for any $\hat{x}\in {\mathbb{S}}^{2}$ and $\hat{x}\perp a\in {\mathbb{C}}^{3}$. Applying Green's vector formula in the exterior of D together with the Silver–Müller radiation conditions implies

$$\begin{equation*}{\int }_{\partial D}\left(\nu {\times}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\prime }\right)\cdot {E}^{\mathrm{s}}\left(\cdot ;a,-\hat{x}\right)\enspace \mathrm{d}s={\int }_{\partial D}\left(\nu {\times}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{s}\left(\cdot ;a,-\hat{x}\right)\right)\cdot {E}^{\prime }\enspace \mathrm{d}s\end{equation*}$$

Thus, we conclude by $\nu {\times}{E}^{\mathrm{s}}\left(\cdot ;a,-\hat{x}\right)=-\nu {\times}{E}^{\mathrm{i}}\left(\cdot ;a,-\hat{x}\right)$ on ∂D that

$$\begin{align*}\hfill a\cdot {E}_{\infty }^{\prime }\left(\hat{x}\right)& =\mathrm{i}k{\int }_{\partial D}{E}_{\tau }^{\prime }\cdot \left({H}^{\mathrm{i}}\left(\cdot ;a,-\hat{x}\right){\times}\nu \right)\enspace \mathrm{d}s+\mathrm{i}k{\int }_{\partial D}{H}_{\tau }^{\prime }\cdot \left(\nu {\times}{E}^{\mathrm{s}}\left(\cdot ;a,-\hat{x}\right)\right)\enspace \mathrm{d}s\hfill \\ \hfill & =-\mathrm{i}k{\int }_{\partial D}{E}_{\tau }^{\prime }\cdot \left(\nu {\times}{H}^{\mathrm{i}}\left(\cdot ;a,-\hat{x}\right)\right)\enspace \mathrm{d}s-\mathrm{i}k{\int }_{\partial D}{E}_{\tau }^{\prime }\cdot \left(\nu {\times}{H}^{\mathrm{s}}\left(\cdot ;a,-\hat{x}\right)\right)\enspace \mathrm{d}s\hfill \\ \hfill & =-\mathrm{i}k{\int }_{\partial D}{E}_{\tau }^{\prime }\cdot \left(\nu {\times}H\left(\cdot ;a,-\hat{x}\right)\right)\enspace \mathrm{d}s.\hfill \end{align*}$$

Using this representation with polarizations given by a tangential field $A\in {L}_{t}^{2}\left({\mathbb{S}}^{2},{\mathbb{C}}^{3}\right)$ and substituting the boundary condition from theorem 3.3 yields

$$\begin{align*}\hfill {\langle {\mathbf{F}}^{\prime }\left[\partial D\right]h,A\rangle }_{{L}^{2}\left({\mathbb{S}}^{2}\right)}& =-\mathrm{i}k{\int }_{{\mathbb{S}}^{2}}{\int }_{\partial D}{E}_{\tau }^{\prime }\cdot \left(\right.\nu {\times}H\left(\cdot ;\overline{A\left(\hat{x}\right)},-\hat{x}\right)\enspace \mathrm{d}s\enspace \mathrm{d}s\left(\hat{x}\right)\hfill \\ \hfill & =\mathrm{i}k{\int }_{\partial D}\left(\nu {\times}{E}^{\prime }\right)\cdot \left({\int }_{{S}^{2}}H\left(\cdot ;\overline{A\left(\hat{x}\right)},-\hat{x}\right)\enspace \mathrm{d}s\left(\hat{x}\right)\right)\enspace \mathrm{d}s\hfill \\ \hfill & =\mathrm{i}k{\int }_{\partial D}\left({\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{\nu }{E}_{\nu }\right)-\mathrm{i}k\enspace {h}_{\nu }{H}_{\tau }\right)\cdot {\mathcal{H}}_{A}\enspace \mathrm{d}s\hfill \\ \hfill & =\mathrm{i}k{\int }_{\partial D}{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{\nu }{E}_{\nu }\right)\cdot \left(\nu {\times}{\mathcal{H}}_{A}\right)-\mathrm{i}k\enspace {h}_{\nu }{H}_{\tau }\cdot {\mathcal{H}}_{A}\enspace \mathrm{d}s\hfill \\ \hfill & ={\int }_{\partial D}\left[-\mathrm{i}k\enspace {h}_{\nu }{E}_{\nu }\enspace {\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left(\nu {\times}{\mathcal{H}}_{A}\right)+{k}^{2}{h}_{\nu }{H}_{\tau }\cdot {\mathcal{H}}_{A}\right]\enspace \mathrm{d}s\hfill \\ \hfill & ={\int }_{\partial D}{h}_{\nu }\left[{E}_{\nu }\left(\mathrm{i}k\nu \cdot \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\mathcal{H}}_{A}\right)+{k}^{2}{H}_{\tau }\cdot {\mathcal{H}}_{A}\right]\enspace \mathrm{d}s\hfill \\ \hfill & ={k}^{2}{\int }_{\partial D}{h}_{\nu }\left[-{E}_{\nu }{\left({\mathcal{E}}_{A}\right)}_{\nu }+{H}_{\tau }\cdot {\mathcal{H}}_{A}\right]\enspace \mathrm{d}s,\hfill \end{align*}$$

where we have denoted as before the solution of the actual scattering problem by E = E(⋅; p, d) and H = H(⋅; p, d). If ∂D is star shaped, parameterized by y = r(), and the variation is given by $h\left(y\right)=\tilde {h}\left(\hat{y}\right)\hat{y}$ we can calculate h_ν explicitly and arrive at

$$\begin{equation*}{\langle {\mathbf{F}}^{\prime }\left[r\right]\left.\right)h,A\rangle }_{{L}^{2}\left({\mathbb{S}}^{2}\right)}={k}^{2}{\int }_{{\mathbb{S}}^{2}}\tilde {h}\left(\hat{y}\right)\enspace {r}^{2}\left(\hat{y}\right)\left[-{E}_{\nu }\left(y\right){\left({\mathcal{E}}_{A}\left(y\right)\right)}_{\nu }+{H}_{\tau }\left(y\right)\cdot {\left({\mathcal{H}}_{A}\left(y\right)\right)}_{\tau }\right]\enspace \mathrm{d}s\left(\hat{y}\right),\end{equation*}$$

which shows the assertion.□

We continue in proving the scattered wave to be twice differentiable with respect to the boundary. If we use two small perturbations ${h}_{1},{h}_{2}\in {C}^{1}\left({\mathbb{R}}^{3},{\mathbb{R}}^{3}\right)$ with compact support in B_R (0) to perturb the boundary, we arrive at

$$\begin{equation*}{\left(\partial {D}_{{h}_{2}}\right)}_{{h}_{1}}=\left\{y={\varphi }_{1}\left({\varphi }_{2}\left(x\right)\right)=x+{h}_{2}\left(x\right)+{h}_{1}\left(x+{h}_{2}\left(x\right)\right):x\in \partial D\right\},\end{equation*}$$

which is not symmetric with respect to the variations h₁ and h₂. But we expect a second derivative to be symmetric, see [5, chapter VIII.12]. The perturbation becomes symmetric, if we replace h₁ by ${h}_{1}{\circ}{\varphi }_{2}^{-1}$. This motivates our goal: finding a radiating solution of Maxwell's equations E'', depending bilinearly on h₁ and h₂, being symmetric in h₁ and h₂, and satisfying

$$\begin{equation*}\underset{{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}\to 0}{\mathrm{lim}}\enspace \frac{1}{{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}}\enspace {\mathrm{sup}}_{{\Vert}{h}_{1}{{\Vert}}_{{C}^{1}}=1}{\Vert}{E}_{{h}_{1}{\circ}{\varphi }_{2}^{-1}}^{\prime }\left[\partial {D}_{{h}_{2}}\right]-{E}_{{h}_{1}}^{\prime }\left[\partial D\right]-{E}^{\prime \prime }{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}=0,\end{equation*}$$

where E'_h [∂D] denotes the domain derivative with respect to the variation h at the scatterer D. The Taylor expansion ${h}_{1}{\circ}{\varphi }_{2}^{-1}={h}_{1}-{J}_{{h}_{1}}{h}_{2}+\mathcal{O}\left({\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}^{2}\right)$ leads to

$$\begin{equation}{E}^{\prime \prime }={\left({E}_{1}^{\prime }\right)}_{2}^{\prime }-{E}_{h}^{\prime }\end{equation} \tag{ 4.1 }$$

with (E₁')'₂ being the domain derivative with respect to the variation h₂ of the domain derivative with respect to the variation h₁. The second term E_h ' is the domain derivative from theorem 3.3 with respect to the variation $h={J}_{{h}_{1}}{h}_{2}$.

We are going to prove that the second domain derivative is given by (4.1) and present a characterization of E'' as a radiating solution to Maxwell's equations. Similar to the first derivative, we start by showing existence of the material derivative of the material derivative.

Let W_i ∈ H(curl, Ω) denote the material derivative with respect to the perturbation h_i , i = 1, 2. W_i is the solution of

$$\begin{equation}\mathcal{A}\left({W}_{i},V\right)={\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }{A}_{i}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}{E}^{\top }{A}_{i}\overline{V}\right)\mathrm{d}x,\quad \text{for}\enspace \text{all}\;V\in H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right),\end{equation} \tag{ 4.2 }$$

where we introduced the abbreviation A_i for the symmetric matrix ${A}_{i}=\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{i}\enspace I-{J}_{{h}_{i}}-{J}_{{h}_{i}}^{\top }$. Let ${\tilde {W}}_{1}\in {H}_{\text{pc}}\left({{\Omega}}_{{h}_{2}}\right)$ denote the solution of (4.2) with Ω being replaced by ${{\Omega}}_{{h}_{2}}$. Again, we define ${\hat{W}}_{1,{h}_{2}}={J}_{{\varphi }_{2}}^{\top }{\tilde {W}}_{1}\in {H}_{\text{pc}}\left({\Omega}\right)$. Then, ${\hat{W}}_{1,{h}_{2}}$ solves

$$\begin{align}\hfill & {\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{W}}_{1,{h}_{2}}^{\top }\left(\frac{{J}_{{\varphi }_{2}}^{\top }{J}_{{\varphi }_{2}}}{\mathrm{det}\enspace {J}_{{\varphi }_{2}}}\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}-{k}^{2}\enspace {\hat{W}}_{1,{h}_{2}}^{\top }\left(\mathrm{det}\left({J}_{{\varphi }_{2}}\right){J}_{{\varphi }_{2}}^{-1}{J}_{{\varphi }_{2}}^{-\top }\right)\overline{V}\right)\mathrm{d}x\hfill \\ \hfill & \quad +\mathrm{i}k\enspace {\langle {\Lambda}\left(\nu {\times}{\hat{W}}_{1,{h}_{2}}\right),V\rangle }_{\partial {B}_{R}\left(0\right)}\hfill \\ \hfill & \quad ={\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{E}}_{{h}_{2}}\left(\frac{{J}_{{\varphi }_{2}}^{\top }\tilde {{A}_{1}}{J}_{{\varphi }_{2}}}{\mathrm{det}\enspace {J}_{{\varphi }_{2}}}\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {\hat{E}}_{{h}_{2}}\left(\mathrm{det}\enspace {J}_{{\varphi }_{2}}{J}_{{\varphi }_{2}}^{-1}\tilde {{A}_{1}}{J}_{{\varphi }_{2}}^{-\top }\right)\overline{V}\right)\mathrm{d}x\hfill \end{align} \tag{ 4.3 }$$

for all V ∈ H_pc(Ω). In the next lemma, we provide the linearization of the new matrices.

Lemma 4.1. Let $A\in {C}^{1}\left({\mathbb{R}}^{3},{\mathbb{R}}^{3{\times}3}\right)$ and φ(x) = x + h(x) with $h\in {C}^{1}\left({\mathbb{R}}^{3},{\mathbb{R}}^{3}\right)$ sufficiently small. Then we have

$$\begin{align*}\hfill \frac{\mathrm{det}\enspace {J}_{\varphi }^{\top }\tilde {A}{J}_{\varphi }}{\mathrm{det}\enspace {J}_{\varphi }}& =A+{J}_{h}^{\top }A+A{J}_{h}-\mathrm{d}\mathrm{i}\mathrm{v}\enspace h\enspace A+{A}^{\prime }\left(h\right)+\mathcal{O}\left({\Vert}h{{\Vert}}_{{C}^{1}}^{2}\right),\hfill \\ \hfill \mathrm{det}\enspace {J}_{\varphi }{J}_{\varphi }^{-1}\tilde {A}{J}_{\varphi }^{-\top }& =A-{J}_{h}A-A{J}_{h}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}\enspace h\enspace A+{A}^{\prime }\left(h\right)+\mathcal{O}\left({\Vert}h{{\Vert}}_{{C}^{1}}^{2}\right),\hfill \end{align*}$$

where the matrix ${A}^{\prime }\left(h\right)\in C\left({\mathbb{R}}^{3},{\mathbb{R}}^{3{\times}3}\right)$ is given by ${\left({A}^{\prime }\left(h\right)\right)}_{ij}={\left(\nabla {A}_{ij}\right)}^{\top }h$, i, j = 1, ..., 3.

Proof. The linearizations follow from (3.3), (3.4) and the Taylor expansion of the coefficients A_ij (x + h(x)).□

As a first step we prove that the material derivative W₁ depends continuously on perturbations h₂.

Theorem 4.2. Let W₁ ∈ H_pc(Ω) be the solution of (4.2) with i = 1 and ${\hat{W}}_{1,{h}_{2}}\in {H}_{\text{pc}}\left({\Omega}\right)$ a solution of (4.3). Then we have

$$\begin{equation*}\underset{{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}\to 0}{\mathrm{lim}}{\Vert}{W}_{1}-{\hat{W}}_{1,{h}_{2}}{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}=0.\end{equation*}$$

Proof. Let ${\ell }_{{h}_{2}}\left(V\right)$ denote the right-hand side of (4.2) with i = 1 and let ${\ell }_{{h}_{2},{h}_{1}}$ denote the right-hand side of (4.3). Recall the notation ${\mathcal{A}}_{{h}_{2}}$ for the sesquilinear form, such that the left-hand side of (4.3) is given by ${\mathcal{A}}_{{h}_{2}}\left({\hat{W}}_{1,{h}_{2}},V\right)$. Then we have

$$\begin{equation}\mathcal{A}\left({\hat{W}}_{1,{h}_{2}}-{W}_{1},V\right)=\mathcal{A}\left({\hat{W}}_{1,{h}_{2}},V\right)-{\mathcal{A}}_{{h}_{2}}\left({\hat{W}}_{1,{h}_{2}},V\right)+{\ell }_{{h}_{1},{h}_{2}}\left(V\right)-{\ell }_{{h}_{2}}\left(V\right).\end{equation} \tag{ 4.4 }$$

Adding and subtracting the integral

$$\begin{equation*}{\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{E}}_{{h}_{2}}^{\top }{A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {\hat{E}}_{{h}_{2}}^{\top }{A}_{1}\overline{V}\right)\enspace \mathrm{d}x\end{equation*}$$

leads to

$$\begin{align*}\hfill \mathcal{A}\left({\hat{W}}_{1,{h}_{2}}-{W}_{1},V\right)& ={\int }_{{\Omega}}\left[\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{W}}_{1,{h}_{2}}^{\top }\left(I-\frac{{J}_{{\varphi }_{2}}^{\top }{J}_{{\varphi }_{2}}}{\mathrm{det}\enspace {J}_{{\varphi }_{2}}}\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right.\hfill \\ \hfill & \quad -{k}^{2}\enspace {\hat{W}}_{1,{h}_{2}}^{\top }\left(I-\mathrm{det}\left({J}_{{\varphi }_{2}}\right){J}_{{\varphi }_{2}}^{-1}{J}_{{\varphi }_{2}}^{-\top }\right)\overline{V}\left.\right]\enspace \mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\left[\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{E}}_{{h}_{2}}^{\top }\left(\frac{{J}_{{\varphi }_{2}}^{\top }\tilde {{A}_{1}}{J}_{{\varphi }_{2}}}{\mathrm{det}\enspace {J}_{{\varphi }_{2}}}-{A}_{1}\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right.\hfill \\ \hfill & \quad +{k}^{2}\enspace {\hat{E}}_{{h}_{2}}^{\top }\left(\mathrm{det}\left({J}_{{\varphi }_{2}}\right){J}_{{\varphi }_{2}}^{-1}\tilde {{A}_{1}}{J}_{{\varphi }_{2}}^{-\top }-{A}_{1}\right)\overline{V}\left.\right]\enspace \mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left({\hat{E}}_{{h}_{2}}-E\right)}^{\top }{A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {\left({\hat{E}}_{{h}_{2}}-E\right)}^{\top }{A}_{1}\overline{V}\right)\enspace \mathrm{d}x.\hfill \end{align*}$$

With lemma 4.1 and theorem 3.1 we conclude $\mathcal{A}\left({\hat{W}}_{1,{h}_{2}}-{W}_{1},V\right)\to 0$ as h₂ → 0 in C¹, which finishes the proof.□

As before, we consider the linearizations and prove differentiability.

Theorem 4.3. Let W₁ ∈ H_pc(Ω) be the solution of (4.2) with i = 1 and ${\hat{W}}_{1,{h}_{2}}\in {H}_{\text{pc}}\left({\Omega}\right)$ of (4.3). Then there exists a function W₁' ∈ H_pc(Ω), depending linearly and continuously on h₂ ∈ C¹, such that

$$\begin{equation*}\underset{{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}\to 0}{\mathrm{lim}}\enspace \frac{1}{{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}}{\Vert}{\hat{W}}_{1,{h}_{2}}-{W}_{1}-{W}_{1}^{\prime }{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}=0.\end{equation*}$$

Proof. Motivated by (4.4) we define W₁' ∈ H_pc(Ω) as the solution of

$$\begin{align*}\hfill \mathcal{A}\left({W}_{1}^{\prime },V\right)=& {\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {W}_{1}^{\top }{A}_{2}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {W}_{1}^{\top }{A}_{2}\overline{V}+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {W}_{2}^{\top }{A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {W}_{2}^{\top }{A}_{1}\overline{V}\right)\mathrm{d}x\hfill \\ \hfill & +{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left({J}_{{h}_{2}}^{\top }{A}_{1}+{A}_{1}{J}_{{h}_{2}}-\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & +{k}^{2}{\int }_{{\Omega}}{E}^{\top }\left(-{J}_{{h}_{2}}{A}_{1}-{A}_{1}{J}_{{h}_{2}}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{V}\mathrm{d}x,\quad \text{for}\enspace \text{all}\;V\in {H}_{\text{pc}}\left({\Omega}\right).\hfill \end{align*}$$

As before, we consider the difference $\mathcal{A}\left({\hat{W}}_{1,{h}_{2}}-{W}_{1}-{W}_{1}^{\prime },V\right)$. We add and subtract the following integrals

$$\begin{align*}\hfill {I}_{1}& ={\int }_{{\Omega}}\left({\hat{W}}_{1,{h}_{2}}^{\top }{A}_{2}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}-{k}^{2}\enspace {\hat{W}}_{1,{h}_{2}}{A}_{2}\overline{V}\right)\mathrm{d}x,\hfill \\ \hfill {I}_{2}& ={\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{E}}_{{h}_{2}}^{\top }{A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {\hat{E}}_{{h}_{2}}^{\top }{A}_{1}\overline{V}\right)\mathrm{d}x,\hfill \\ \hfill {I}_{3}& ={\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{E}}_{{h}_{2}}^{\top }\left({J}_{{h}_{2}}^{\top }{A}_{1}+{A}_{1}{J}_{{h}_{2}}-\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x,\hfill \\ \hfill {I}_{4}& ={k}^{2}{\int }_{{\Omega}}{\hat{E}}_{{h}_{2}}^{\top }\left(-{J}_{{h}_{2}}{A}_{1}-{A}_{1}{J}_{{h}_{2}}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{V}\mathrm{d}x,\hfill \end{align*}$$

i.e. we consider

$$\begin{align*}\hfill & \mathcal{A}\left({\hat{W}}_{1,{h}_{2}}-{W}_{1}-{W}_{1}^{\prime },V\right)\hfill \\ \hfill & \quad =\mathcal{A}\left({\hat{W}}_{1,{h}_{2}}-{W}_{1}-{W}_{1}^{\prime },V\right)+\sum _{k=1}^{4}{I}_{k}-\sum _{l=1}^{4}{I}_{l}\hfill \\ \hfill & \quad ={\int }_{{\Omega}}\left[\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{W}}_{1,{h}_{2}}^{\top }\left(I-\frac{{J}_{{\varphi }_{2}}^{\top }{J}_{{\varphi }_{2}}}{\mathrm{det}\enspace {J}_{{\varphi }_{2}}}-{A}_{2}\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}-{k}^{2}\enspace {\hat{W}}_{1,{h}_{2}}^{\top }\left(I-\mathrm{det}\enspace {J}_{{\varphi }_{2}}{J}_{{\varphi }_{2}}^{-1}{J}_{{\varphi }_{2}}^{-\top }+{A}_{2}\right)\overline{V}\right]\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\left[\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left({\hat{W}}_{1,{h}_{2}}-{W}_{1}\right)}^{\top }{A}_{2}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}-{k}^{2}\enspace {\left({\hat{W}}_{1,{h}_{2}}-{W}_{1}\right)}^{\top }{A}_{2}\overline{V}\right)\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {\hat{E}}_{{h}_{2}}^{\top }\left(\frac{{J}_{{\varphi }_{2}}^{\top }\tilde {{A}_{1}}{J}_{{\varphi }_{2}}}{\mathrm{det}\enspace {J}_{{\varphi }_{2}}}-{A}_{1}-{J}_{{h}_{2}}^{\top }{A}_{1}-{A}_{1}{J}_{{h}_{2}}+\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace I-{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}{\hat{E}}_{{h}_{2}}^{\top }\left(\mathrm{det}\enspace {J}_{{\varphi }_{2}}{J}_{{\varphi }_{2}}^{-1}\tilde {{A}_{1}}{J}_{{\varphi }_{2}}^{-\top }-{A}_{1}+{J}_{{h}_{2}}{A}_{1}+{A}_{1}{J}_{{h}_{2}}^{\top }-\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace I-{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{V}\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\left[\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left({\hat{E}}_{{h}_{2}}-E-{W}_{2}\right)}^{\top }{A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {\left({\hat{E}}_{{h}_{2}}-E-{W}_{2}\right)}^{\top }{A}_{1}\overline{V}\right]\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left({\hat{E}}_{{h}_{2}}-E\right)}^{\top }\left({J}_{{h}_{2}}^{\top }{A}_{1}+{A}_{1}{J}_{{h}_{2}}-\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}{\left({\hat{E}}_{{h}_{2}}-E\right)}^{\top }\left(-{J}_{{h}_{2}}{A}_{1}-{A}_{1}{J}_{{h}_{2}}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{V}\mathrm{d}x.\hfill \end{align*}$$

This leads to the estimate

$$\begin{align*}\hfill & \mathcal{A}\left({\hat{W}}_{1,{h}_{2}}-{W}_{1}-{W}_{1}^{\prime },V\right)\hfill \\ \hfill & \quad {\leqslant}C{\Vert}V{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}\left({\Vert}{\hat{W}}_{1,{h}_{2}}{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}^{2}+{\Vert}{\hat{W}}_{1,{h}_{2}}-{W}_{1}{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}\right.\hfill \\ \hfill & \left.\quad +{\Vert}{\hat{E}}_{{h}_{2}}-E-{W}_{2}{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}+{\Vert}{\hat{E}}_{{h}_{2}}{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}^{2}+{\Vert}{\hat{E}}_{{h}_{2}}-E{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}^{2}\right)\hfill \end{align*}$$

for all V ∈ H(curl, Ω). Again by a perturbation argument, we conclude

$$\begin{equation*}\underset{{\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}\to 0}{\mathrm{lim}}{\Vert}{\hat{W}}_{1,{h}_{2}}-{W}_{1}-{W}_{1}^{\prime }{{\Vert}}_{H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)}=0.\end{equation*}$$

□

Since W₁' ∈ H_pc(Ω) is the material derivative with respect to h₂ of the material derivative with respect to h₁, it contains by linearity the domain derivative with respect to h₂ of the domain derivative with respect to h₁, which we denoted by ${\left({E}_{1}^{\prime }\right)}_{2}^{\prime }$ before. To calculate it, we consider the formal Taylor expansion

$$\begin{equation*}{\hat{W}}_{1,{h}_{2}}\left(x\right)=\left(I+{J}_{{h}_{2}}^{\top }\left(x\right)\right)\left({W}_{1}\left(x\right)+{J}_{{W}_{1}}\left(x\right){h}_{2}\left(x\right)+\frac{\mathrm{d}}{\mathrm{d}{h}_{2}}{W}_{1}\left(x\right)+\mathcal{O}\left({\Vert}{h}_{2}{{\Vert}}_{{C}^{1}}^{2}\right)\right).\end{equation*}$$

With the decomposition ${W}_{1}={E}_{1}^{\prime }+{J}_{{h}_{1}}^{\top }E+{J}_{E}{h}_{1}$ we formally conclude

$$\begin{equation*}\frac{\mathrm{d}}{\mathrm{d}{h}_{2}}{W}_{1}={\left({E}_{1}^{\prime }\right)}_{2}^{\prime }+{J}_{{h}_{1}}^{\top }{E}_{2}^{\prime }+{J}_{{E}_{2}^{\prime }}{h}_{1},\end{equation*}$$

and, furthermore, with ${W}_{1}^{\prime }=\frac{\mathrm{d}}{\mathrm{d}{h}_{2}}{\hat{W}}_{1,{h}_{2}}$ the Ansatz

$$\begin{equation*}{\left({E}_{1}^{\prime }\right)}_{2}^{\prime }={W}_{1}^{\prime }-{J}_{{h}_{2}}^{\top }{W}_{1}-{J}_{{W}_{1}}{h}_{2}-{J}_{{h}_{1}}^{\top }{E}_{2}^{\prime }-{J}_{{E}_{2}^{\prime }}{h}_{1}\end{equation*}$$

is motivated. Similarly to the first domain derivative, we need higher regularity of the solution and therefore higher regularity of the boundary, to ensure the Ansatz to be well defined.

Theorem 4.4. Let ∂D be of class C³. In the setting of the theorem 4.3, let

$$\begin{equation*}{\left({E}_{1}^{\prime }\right)}_{2}^{\prime }={W}_{1}^{\prime }-{J}_{{h}_{2}}^{\top }{W}_{1}-{J}_{{W}_{1}}{h}_{2}-{J}_{{h}_{1}}^{\top }{E}_{2}^{\prime }-{J}_{{E}_{2}^{\prime }}{h}_{1}.\end{equation*}$$

Then ${\left({E}_{1}^{\prime }\right)}_{2}^{\prime }\in H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)$. ${\left({E}_{1}^{\prime }\right)}_{2}^{\prime }$ can be uniquely extended to a radiating solution of Maxwell's equations.

Proof. See appendix A.□

In order to give a characterization of the second domain derivative ${E}^{{\prime\prime}}={\left({E}_{1}^{\prime }\right)}_{2}^{\prime }-{E}_{h}^{\prime }$ with $h={J}_{{h}_{1}}{h}_{2}$, we need to introduce the symmetric curvature operator $\mathcal{R}:\partial D\to {\mathbb{R}}^{3{\times}3}$, which acts on the tangential plane and is given by $\mathcal{R}\left(x\right)={J}_{\nu }\left(x\right)$, x ∈ ∂D. Furthermore we define the mean curvature $\kappa :\partial D\to \mathbb{R}$ by $\kappa =\frac{1}{2}\enspace \mathrm{d}\mathrm{i}\mathrm{v}\enspace \nu$. Note, that these definitions require differentiable extensions of the normal vector field ν in a neighborhood of ∂D which is constant in the direction of ν, see [23]. We state the main result of this paper.

Theorem 4.5. Let ∂D be of class C³. The second domain derivative E'' is a radiating solution of Maxwell's equations, satisfying the inhomogeneous boundary condition

$$\begin{align*}\hfill \nu {\times}{E}^{\prime \prime }=& \sum _{i\ne j=1}^{2}\left[{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{i,\nu }{E}_{j,\nu }^{\prime }-{E}_{\nu }{h}_{i,\tau }^{\top }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}{h}_{j,\nu }\right)\right.\hfill \\ \hfill & \left.-\enspace \mathrm{i}k\enspace {\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left({h}_{j,\nu }{H}_{\tau }\right){h}_{i,\tau }-\mathrm{i}k\enspace {h}_{i,\nu }{H}_{j,\tau }^{\prime }\right]\hfill \\ \hfill & -\mathrm{i}k\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right){H}_{\tau }+{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[\left(\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right)-2\kappa {h}_{1,\nu }{h}_{2,\nu }\right){E}_{\nu }\right]\hfill \\ \hfill & +2\mathrm{i}k\enspace {h}_{1,\nu }{h}_{2,\nu }\left(\mathcal{R}-\kappa \right){H}_{\tau }+\mathrm{i}k\sum _{i\ne j=1}^{2}{h}_{i,\tau }^{\top }\left(\nu {\times}H\right){\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}{h}_{j,\nu }.\hfill \end{align*}$$

Proof. Since ${E}^{{\prime\prime}}={\left({E}_{1}^{\prime }\right)}_{2}^{\prime }-{E}_{h}^{\prime }$ with $h={J}_{{h}_{1}}{h}_{2}$, the boundary values of E'' are given by $\nu {\times}{E}^{{\prime\prime}}=\nu {\times}{\left({E}_{1}^{\prime }\right)}_{2}^{\prime }-\nu {\times}{E}_{h}^{\prime }$. From theorem 3.3 we know $\nu {\times}{E}_{h}^{\prime }={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left(\left({\nu }^{\top }{J}_{{h}_{1}}{h}_{2}\right){E}_{\nu }\right)-\mathrm{i}k\left({\nu }^{\top }{J}_{{h}_{1}}{h}_{2}\right){H}_{\tau }.$ We calculate

$$\begin{align*}\hfill {\nu }^{\top }{J}_{{h}_{1}}{h}_{2}& =\left({\nu }^{\top }{J}_{{h}_{1}}\nu \right){h}_{2,\nu }+{\nu }^{\top }{J}_{{h}_{1}}{h}_{2,\tau }\hfill \\ \hfill & ={h}_{2,\nu }{\nu }^{\top }\left(\nabla {h}_{1,\nu }-{J}_{\nu }^{\top }{h}_{1}\right)+{h}_{2,\tau }^{\top }\left({J}_{{h}_{1}}^{\top }\nu +{J}_{\nu }{h}_{1}-{J}_{\nu }^{\top }{h}_{1}\right)\hfill \\ \hfill & ={h}_{2,\nu }\frac{\partial {h}_{1,\nu }}{\partial \nu }+{h}_{2,\tau }^{\top }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}{h}_{1,\nu }-{h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\hfill \end{align*}$$

since $\mathcal{R}$ is acting on the tangential plane. With W₁' ∈ H_pc(Ω) the boundary values of ${\left({E}_{1}^{\prime }\right)}_{2}^{\prime }$ are given by $\nu {\times}{\left({E}_{1}^{\prime }\right)}_{2}^{\prime }=\nu {\times}\left[-{J}_{{h}_{1}}^{\top }{E}_{2}^{\prime }-{J}_{{E}_{2}^{\prime }}{h}_{1}-{J}_{{h}_{2}}^{\top }{W}_{1}-{J}_{{W}_{1}}{h}_{2}\right].$ We use the decomposition of the material derivative ${W}_{i}={E}_{i}^{\prime }+{J}_{{h}_{i}}^{\prime }+{J}_{E}{h}_{i}$, for i = 1, 2 to find similarly as before

$$\begin{align*}\hfill \nu {\times}{\left({E}_{1}^{\prime }\right)}_{2}^{\prime }& =-\nu {\times}\left[{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1}^{\top }{E}_{2}^{\prime }+{h}_{2}^{\top }{E}_{1}^{\prime }\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{2}^{\prime }{\times}{h}_{1}+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{1}^{\prime }{\times}{h}_{2}\right]\hfill \\ \hfill & \quad -\nu {\times}\left[{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{2}^{\top }\left(\nabla \left({h}_{1}^{\top }E\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){\times}{h}_{2}\right].\hfill \end{align*}$$

As seen before, we have

$$\begin{equation*}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){\times}{h}_{2}=\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}+{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}.\end{equation*}$$

From the boundary condition ν × E = 0 on ∂D we conclude

$$\begin{align*}\hfill {h}_{2}^{\top }\nabla \left({h}_{1}^{\top }E\right)& ={h}_{2,\tau }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1}^{\top }E\right)+{h}_{2,\nu }\frac{\partial {h}_{1}^{\top }E}{\partial \nu }\hfill \\ \hfill & ={h}_{2,\tau }^{\top }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1,\nu }{E}_{\nu }\right)+{h}_{2,\nu }\frac{\partial {h}_{1,\nu }{E}_{\nu }}{\partial \nu }+{h}_{2,\nu }\frac{\partial {h}_{1,\tau }^{\top }{E}_{\tau }}{\partial \nu }.\hfill \end{align*}$$

We gather some identities, namely $\nu {\times}\left(\left(E{\times}{h}_{1}\right){\times}{h}_{2}\right)={E}_{\nu }{h}_{2,\nu }\left(\nu {\times}{h}_{1}\right)$ and

$$\begin{equation*}\nu {\times}\left({\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)=\mathrm{i}k{h}_{2,\nu }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1,\tau }^{\top }{H}_{\tau }\right)-\mathrm{i}k\frac{\partial {h}_{1,\tau }^{\top }{H}_{\tau }}{\partial \nu }{h}_{2,\tau },\end{equation*}$$

as well as $\nu {\times}\left(\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)=\mathrm{i}k{h}_{2,\nu }{A}_{1}{H}_{\tau }-\mathrm{i}k\left({\nu }^{\top }{A}_{1}{H}_{\tau }\right){h}_{2,\tau }$, and finally

$$\begin{equation*}\nu {\times}{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{2}^{\top }\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\right)={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{2,\nu }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left(\nu {\times}{h}_{1}\right)-{h}_{1,\nu }{\left(\nu {\times}{h}_{2}\right)}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right).\end{equation*}$$

Combining and substituting these into $\nu {\times}{E}^{\prime \prime }=\nu {\times}{\left({E}_{1}^{\prime }\right)}_{2}^{\prime }+\nu {\times}{E}_{h}^{\prime }$ yields the boundary condition

$$\begin{align*}\hfill \nu {\times}{E}^{\prime \prime }=& \;{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{1}^{\top }{E}_{2}^{\prime }+{h}_{2}^{\top }{E}_{1}^{\prime }\right)-\nu {\times}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{2}^{\prime }{\times}{h}_{1}+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{1}^{\prime }{\times}{h}_{2}\right)\hfill \\ \hfill & -\mathrm{i}k\left(\right.{h}_{2,\tau }\mathcal{R}{h}_{1,\tau }{H}_{\tau }+{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[\left(\right.{h}_{2,\tau }\mathcal{R}{h}_{1,\tau }{E}_{\nu }+{h}_{1,\nu }{h}_{2,\nu }\frac{\partial {E}_{\nu }}{\partial \nu }\right]\hfill \\ \hfill & +{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[{h}_{1,\nu }{h}_{2,\tau }^{\top }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}{E}_{\nu }+{h}_{2,\nu }{h}_{1,\tau }^{\top }\frac{\partial {E}_{\tau }}{\partial \nu }\right]\hfill \\ \hfill & -{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[{h}_{2,\nu }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left(\nu {\times}{h}_{1}\right)-{h}_{1,\nu }{\left(\nu {\times}{h}_{2}\right)}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right]\hfill \\ \hfill & -\mathrm{i}k\left({h}_{2,\nu }{A}_{1}{H}_{\tau }-\left({\nu }^{\top }{A}_{1}{H}_{\tau }\right){h}_{2,\tau }\right)-{k}^{2}{E}_{\nu }{h}_{2,\nu }\left(\nu {\times}{h}_{1}\right)\hfill \\ \hfill & -\mathrm{i}k{h}_{2,\nu }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1,\tau }^{\top }{H}_{\tau }\right)+\mathrm{i}k{h}_{2,\tau }\frac{\partial {h}_{1,\tau }^{\top }{H}_{\tau }}{\partial \nu }\hfill \\ \hfill & +\mathrm{i}k\left({h}_{2,\nu }\frac{\partial {h}_{1,\nu }}{\partial \nu }+{h}_{2,\tau }^{\top }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}{h}_{1,\nu }\right){H}_{\tau }.\hfill \end{align*}$$

For any vector field F, we have on the boundary ∂D

$$\begin{equation}\frac{\partial {F}_{\tau }}{\partial \nu }=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace F{\times}\nu +{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {F}_{\nu }-\mathcal{R}{F}_{\tau }\end{equation} \tag{ 4.5 }$$

(see (5.4.50) in [23]), and

$$\begin{equation}\frac{\partial {F}_{\nu }}{\partial \nu }-\mathrm{d}\mathrm{i}\mathrm{v}\enspace F=-{\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\enspace {F}_{\tau }-2\kappa {F}_{\nu }.\end{equation} \tag{ 4.6 }$$

For the tangential part of the curl we get

$$\begin{equation}{\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace F\right)}_{\tau }={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\enspace {F}_{\nu }+\left(\mathcal{R}-2\kappa -\frac{\partial }{\partial \nu }\right)\left(F{\times}\nu \right)\end{equation} \tag{ 4.7 }$$

(see theorem 2.5.20 in [23]). With equation (4.5) we conclude

$$\begin{equation*}\left({\nu }^{\top }{A}_{1}{H}_{\tau }\right)+\frac{\partial {h}_{1,\tau }^{\top }{H}_{\tau }}{\partial \nu }=-{H}_{\tau }^{\top \enspace }{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {h}_{1,\nu }.\end{equation*}$$

Furthermore, by div(H) = 0 and H_ν = 0, we have

$$\begin{equation*}{\left({A}_{1}H\right)}_{\tau }=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left(H{\times}{h}_{1}\right)}_{\tau }-{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1,\tau }^{\top }{H}_{\tau }\right)-\mathrm{i}k{E}_{\nu }\left(\nu {\times}{h}_{1}\right).\end{equation*}$$

With equation (4.7) we obtain

$$\begin{align*}\hfill \mathrm{i}k{h}_{2,\nu }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left(H{\times}{h}_{1}\right)}_{\tau }& ={h}_{2,\nu }{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left({h}_{1}{\times}\nu \right)\right)-\mathrm{i}k{h}_{1,\nu }{h}_{2,\nu }\left(\mathcal{R}-2\kappa \right){H}_{\tau }\hfill \\ \hfill & \quad +\mathrm{i}k{h}_{1,\nu }{h}_{2,\nu }\frac{\partial {H}_{\tau }}{\partial \nu }+\mathrm{i}k{h}_{2,\nu }\frac{\partial {h}_{1,\nu }}{\partial \nu }{H}_{\tau }.\hfill \end{align*}$$

Thus, we arrive at

$$\begin{align*}\hfill \nu {\times}{E}^{\prime \prime }& ={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{1}^{\top }{E}_{2}^{\prime }+{h}_{2}^{\top }{E}_{1}^{\prime }\right)-\nu {\times}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{2}^{\prime }{\times}{h}_{1}+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{1}^{\prime }{\times}{h}_{2}\right)\hfill \\ \hfill & \quad -\mathrm{i}k\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right){H}_{\tau }+{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right){E}_{\nu }+{h}_{1,\nu }\enspace {h}_{2,\nu }\frac{\partial {E}_{\nu }}{\partial \nu }\right]\hfill \\ \hfill & \quad +\mathrm{i}k{h}_{1,\nu }\enspace {h}_{2,\nu }\left(\mathcal{R}-2\kappa -\frac{\partial }{\partial \nu }\right){H}_{\tau }\hfill \\ \hfill & \quad +{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[{h}_{1,\nu }\enspace {h}_{2,\tau }^{\top }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {E}_{\nu }+{h}_{2,\nu }\enspace {h}_{1,\tau }^{\top }\frac{\partial {E}_{\tau }}{\partial \nu }\right]\hfill \\ \hfill & \quad -{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[{h}_{2,\nu }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left(\nu {\times}{h}_{1}\right)-{h}_{1,\nu }{\left(\nu {\times}{h}_{2}\right)}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right]\hfill \\ \hfill & \quad -{h}_{2,\nu }{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left({h}_{1}{\times}\nu \right)\right)+\mathrm{i}k\left({\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1,\nu }\right){\times}\left({H}_{\tau }{\times}{h}_{2,\tau }\right)\right).\hfill \end{align*}$$

By (4.6) we obtain

$$\begin{equation*}{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {E}_{\nu }=\frac{\partial {E}_{\tau }}{\partial \nu }-\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}\nu .\end{equation*}$$

Furthermore, it holds

$$\begin{align*}\hfill {\left[{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {h}_{1,\nu }{\times}\left({H}_{\tau }{\times}{h}_{2,\tau }\right)\right]}_{\tau }& ={\left[\left(\nu {\times}\left({\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1,\nu }\right){\times}\nu \right)\right){\times}\left({H}_{\tau }{\times}{h}_{2,\tau }\right)\right]}_{\tau }\hfill \\ \hfill & =\left[\left({\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1,\nu }\right){\times}\nu \right)\nu \cdot \left({H}_{\tau }{\times}{h}_{2,\tau }\right)\right.\hfill \\ \hfill & \quad {\left.-\nu {\left({\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{1,\nu }\right){\times}\nu \right)}^{\top }\left({H}_{\tau }{\times}{h}_{2,\tau }\right)\right]}_{\tau }\hfill \\ \hfill & =\frac{1}{\mathrm{i}k}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left(\nu {\times}{h}_{2}\right){\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{1,\nu }\right).\hfill \end{align*}$$

By the product rule we finally arrive at a symmetric characterization, i.e.,

$$\begin{align*}\hfill \nu {\times}{E}^{\prime \prime }& ={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{1}^{\top }{E}_{2}^{\prime }+{h}_{2}^{\top }{E}_{1}^{\prime }\right)-\nu {\times}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{2}^{\prime }{\times}{h}_{1}+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{1}^{\prime }{\times}{h}_{2}\right)\hfill \\ \hfill & \quad -\mathrm{i}k\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right){H}_{\tau }+{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right){E}_{\nu }+{h}_{1,\nu }\enspace {h}_{2,\nu }\frac{\partial {E}_{\nu }}{\partial \nu }\right]\hfill \\ \hfill & \quad +\mathrm{i}k{h}_{1,\nu }\enspace {h}_{2,\nu }\left(\mathcal{R}-2\kappa -\frac{\partial }{\partial \nu }\right){H}_{\tau }+{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[{h}_{1,\nu }{h}_{2,\tau }^{\top }\frac{\partial {E}_{\tau }}{\partial \nu }+{h}_{2,\nu }\enspace {h}_{1,\tau }^{\top }\frac{\partial {E}_{\tau }}{\partial \nu }\right]\hfill \\ \hfill & \quad -\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left(\nu {\times}{h}_{1}\right){\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{2,\nu }\right)-\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left(\nu {\times}{h}_{2}\right){\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{1,\nu }\right).\hfill \end{align*}$$

From (4.5) and from (4.6) we see $\frac{\partial {H}_{\tau }}{\partial \nu }=-\mathcal{R}{H}_{\tau }$, and $\frac{\partial {E}_{\nu }}{\partial \nu }=-2\kappa {E}_{\nu }$ and conclude

$$\begin{align*}\hfill \nu {\times}{E}^{\prime \prime }& ={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{1}^{\top }{E}_{2}^{\prime }+{h}_{2}^{\top }{E}_{1}^{\prime }\right)-\nu {\times}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{2}^{\prime }{\times}{h}_{1}+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{1}^{\prime }{\times}{h}_{2}\right)\hfill \\ \hfill & \quad -\mathrm{i}k\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right){H}_{\tau }+{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[\left(\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right)-2\kappa {h}_{1,\nu }\enspace {h}_{2,\nu }\right){E}_{\nu }\right]\hfill \\ \hfill & \quad +2\mathrm{i}k{h}_{1,\nu }\enspace {h}_{2,\nu }\left(\mathcal{R}-\kappa \right){H}_{\tau }\hfill \\ \hfill & \quad +{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[{h}_{1,\nu }{h}_{2,\tau }^{\top }\frac{\partial {E}_{\tau }}{\partial \nu }+{h}_{2,\nu }\enspace {h}_{1,\tau }^{\top }\frac{\partial {E}_{\tau }}{\partial \nu }\right]\hfill \\ \hfill & \quad +\mathrm{i}k\sum _{i\ne j=1}^{s}{h}_{i,\tau }^{\top }\left(\nu {\times}H\right){\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{j,\nu }\right).\hfill \end{align*}$$

We use the boundary condition $\nu {\times}{E}_{i}^{\prime }={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{i,\nu }{E}_{\nu }\right)-\mathrm{i}k{h}_{i,\nu }{H}_{\tau }$ of the domain derivative, i = 1, 2, which leads to E'_i,τ = −Grad_∂D (h_i,ν E_ν ) + ikh_i,ν (ν × H), and obtain

$$\begin{align*}\hfill {\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{i}^{\top }{E}_{j}^{\prime }\right)& ={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{i,\nu }{E}_{j,\nu }^{\prime }+{h}_{i,\tau }^{\top }{E}_{j,\tau }^{\top }\right)\hfill \\ \hfill & ={\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{i,\nu }{E}_{j,\nu }^{\prime }-{h}_{i,\tau }^{\top }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\left({h}_{j,\nu }{E}_{\nu }\right)+\mathrm{i}k{h}_{j,\nu }{h}_{i,\tau }^{\top }\left(\nu {\times}H\right)\right).\hfill \end{align*}$$

Furthermore, we have

$$\begin{align*}\hfill -\nu {\times}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{j}^{\prime }{\times}{h}_{i}\right)& =\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{j,\nu }^{\prime }{h}_{i,\tau }-{h}_{i,\nu }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{j,\tau }^{\prime }\hfill \\ \hfill & =-{\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left(\nu {\times}{E}_{j}^{\prime }\right){h}_{i,\tau }-\mathrm{i}k{h}_{i,\nu }{H}_{j,\tau }^{\prime }\hfill \\ \hfill & =-\mathrm{i}k{\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left({h}_{j,\nu }{H}_{\tau }\right){h}_{i,\tau }-\mathrm{i}k{h}_{i,\nu }{H}_{j,\tau }^{\prime }.\hfill \end{align*}$$

Together with $\frac{\partial {E}_{\tau }}{\partial \nu }=\mathrm{i}k\left(H{\times}\nu \right)+{\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {E}_{\nu }$ we arrive at

$$\begin{align*}\hfill \nu {\times}{E}^{\prime \prime }=& \sum _{i\ne j=1}^{2}\left[{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{i,\nu }{E}_{j,\nu }^{\prime }-{E}_{\nu }{h}_{i,\tau }^{\top }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {h}_{j,\nu }\right)\right.\hfill \\ \hfill & \left.-\mathrm{i}k\enspace {\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left({h}_{j,\nu }{H}_{\tau }\right){h}_{i,\tau }-\mathrm{i}k\enspace {h}_{i,\nu }{H}_{j,\tau }^{\prime }\right]\hfill \\ \hfill & -\mathrm{i}k\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right){H}_{\tau }+{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[\left(\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right)-2\kappa {h}_{1,\nu }{h}_{2,\nu }\right){E}_{\nu }\right]\hfill \\ \hfill & +2\mathrm{i}k\enspace {h}_{1,\nu }{h}_{2,\nu }\left(\mathcal{R}-\kappa \right){H}_{\tau }+\mathrm{i}k\sum _{i\ne j=1}^{2}{h}_{i,\tau }^{\top }\left(\nu {\times}H\right){\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}{h}_{j,\nu },\hfill \end{align*}$$

as stated in the theorem. □

We do not claim that the characterization is the most elegant or shortest way to describe the boundary condition of theorem 4.4. But it shows its symmetry with respect to h₁ and h₂. Note that the boundary condition of the second domain derivatives requires both the solution (E, H) as well as the first domain derivatives (E_i ', H_i ') to be sufficiently smooth in order to be well posed.

5.1. The regularized Halley method

Recall the boundary to far field operator defined by $\mathbf{F}\left(\partial D\right)={E}_{\infty }\in {L}_{t}^{2}\left({\mathbb{S}}^{2}\right)$. From the previous section, we know F to be twice differentiable where the derivatives are given by the far field patterns of the domain derivatives, i.e. F'[∂D]h = E_∞', ${F}^{{\prime\prime}}\left[\partial D\right]\left(\hat{h},h\right)={E}_{\infty }^{{\prime\prime}}$. To solve the equation

$$\begin{equation*}\mathbf{F}\left(\partial D\right)={E}_{\infty }\end{equation*}$$

for a given ${E}_{\infty }\in {L}_{t}^{2}\left({\mathbb{S}}^{2}\right)$, we apply Newton type methods. Choosing a starting guess ∂D⁰ a classical Newton step consists in solving the linear equation

$$\begin{equation}{\mathbf{F}}^{\prime }\left[\partial {D}^{i}\right]\hat{h}={E}_{\infty }-\mathbf{F}\left(\partial {D}^{i}\right).\end{equation} \tag{ 5.1 }$$

Due to ill-posedness the equation (5.1) has to be regularized in order to ensure solvability. Applying Tikhonov regularization, we consider the uniquely solvable equation

$$\begin{equation}\left({\mathbf{F}}^{\prime }{\left[\partial {D}^{i}\right]}^{{\ast}}{\mathbf{F}}^{\prime }\left[\partial {D}^{i}\right]+{\alpha }_{1}\mathbf{I}\right)\hat{h}={\mathbf{F}}^{\prime }{\left[\partial {D}^{i}\right]}^{{\ast}}\left({E}_{\infty }-\mathbf{F}\left(\partial {D}^{i}\right)\right)\end{equation} \tag{ 5.2 }$$

with some chosen regularization parameter α₁ > 0. Thus the regularized Newton scheme uses an update of the boundary by $\partial {D}^{i+1}=\partial {D}_{\hat{h}}^{i}$. For more details on iterative regularization methods we refer to [17].

For the second degree method, we modify this approach and use just as a predictor to linearize the quadratic approximation $\mathbf{F}\left(\partial {D}_{h}\right)\approx \mathbf{F}\left(\partial D\right)+{\mathbf{F}}^{\prime }\left[\partial D\right]h+\frac{1}{2}{\mathbf{F}}^{{\prime\prime}}\left[\partial D\right]\left(h,h\right)$. Thus, the corrector step becomes

$$\begin{equation*}{\mathbf{F}}^{\prime }\left[\partial {D}^{i}\right]h+\frac{1}{2}{\mathbf{F}}^{{\prime\prime}}\left[\partial {D}^{i}\right]\left(\hat{h},h\right)={E}_{\infty }-\mathbf{F}\left(\partial {D}^{i}\right).\end{equation*}$$

Again, we apply a Tikhonov regularization. Defining the linear operator T by $\mathbf{T}h={\mathbf{F}}^{\prime }\left[\partial {D}^{i}\right]h+\frac{1}{2}{\mathbf{F}}^{{\prime\prime}}\left[\partial {D}^{i}\right]\left(\hat{h},h\right)$ the corrector step solves for h by

$$\begin{equation}\left({\mathbf{T}}^{{\ast}}\mathbf{T}+{\alpha }_{2}\mathbf{I}\right)h={\mathbf{T}}^{{\ast}}\left(\right.{E}_{\infty }-\mathbf{F}\left(\partial {D}^{i}\right),\end{equation} \tag{ 5.3 }$$

again with some regularization parameter α₂ > 0. Then the so called Halley method is given by an update of the boundary by $\partial {D}^{i+1}=\partial {D}_{h}^{i}$.

To obtain a regularization scheme for the full non-linear problem it is known that we have to add a stopping condition. Therefore, we stop the iteration if the relative residual ${\Vert}\mathbf{F}\left(\partial {D}^{i}\right)-{E}_{\infty }{{\Vert}}_{{L}^{2}\left({\mathbb{S}}^{2}\right)}/{\Vert}{E}_{\infty }{{\Vert}}_{{L}^{2}\left({\mathbb{S}}^{2}\right)}$ falls below a chosen threshold, which depends on the noise level. This completes the regularized second degree method, which is sometimes called regularized Halley method. The method is introduced in [13, 16], where regularizing properties are shown providing assumptions similar to those used for the regularized Newton method. Mainly, the tangential cone condition, already mentioned in section 3, is required.

However, we consider a numerical implementation of the method, which requires the choice of a set $\mathcal{Y}$ of admissible boundaries as an open set of a normed space $\mathcal{X}$. Then, the domain derivatives become Frechet derivatives. We have chosen $\mathcal{Y}$ to be the set of star shaped domains with center in the origin and boundary of class C^∞, discretized in the same way as in [8, 9] by spherical harmonics ${Y}_{n}^{m}$ in the following way: first, we identify the boundary $\partial D\in \mathcal{Y}$ by the positive smooth function $r:{\mathbb{S}}^{2}\to \mathbb{R}$, such every x ∈ ∂D is given in spherical coordinates by x = r(d)d for some $d\in {\mathbb{S}}^{2}$, i.e. we choose the open set $\mathcal{Y}=\left\{r\in {C}^{\infty }\left({\mathbb{S}}^{2}\right):r{ >}0\right\}$ in the space $\mathcal{X}={C}^{\infty }\left({\mathbb{S}}^{2}\right)$ as the domain of F. To discretize $\mathcal{Y}$, we choose the finite dimensional subspace ${\mathcal{X}}_{N}\subset \mathcal{X}$, using the real and imaginary part of spherical harmonics ${Y}_{n}^{m}$ up to the degree $N\in \mathbb{N}$, i.e.,

$$\begin{equation*}{\mathcal{X}}_{N}=\left\{r\in {C}^{\infty }\left({\mathbb{S}}^{2}\right):r=\sum _{n=0}^{N}\sum _{m=0}^{n}{\alpha }_{n}^{m}\enspace \mathrm{R}\mathrm{e}\enspace {Y}_{n}^{m}+\sum _{n=1}^{N}\sum _{m=1}^{n}{\beta }_{n}^{m}\enspace \mathrm{I}\mathrm{m}\enspace {Y}_{n}^{m}\right\}\end{equation*}$$

with $\mathrm{dim}\enspace {\mathcal{X}}_{N}={\left(N+1\right)}^{2}$. The discretized set of admissible boundaries ${\mathcal{Y}}_{N}$ consists of the elements $r\in {\mathcal{X}}_{N}$ with r(d) > 0 for $d\in {\mathbb{S}}^{2}$. Fixing $M\in \mathbb{N}$ evaluation points ${\hat{x}}_{1},\dots {\hat{x}}_{M}\in {\mathbb{S}}^{2}$ for the far field patterns, we discretize F(∂D) = E_∞ by $\left({E}_{\infty }\left({\hat{x}}_{1}\right),\dots ,{E}_{\infty }\left({\hat{x}}_{M}\right)\right)\in {\mathbb{C}}^{3{\times}M}$. Using the linearity of the domain derivative F'[∂D] and the notation ${E}_{\infty }^{\prime }\left(\hat{x};h\right)$ for the domain derivative with respect to the perturbation h, evaluated at $\hat{x}\in {\mathbb{S}}^{2}$ and the ordered basis

$$\begin{equation*}\mathcal{B}=\left\{\mathrm{R}\mathrm{e}\enspace {Y}_{0}^{0},\enspace \mathrm{R}\mathrm{e}\enspace {Y}_{1}^{0},\enspace \mathrm{R}\mathrm{e}\enspace {Y}_{1}^{1},\dots ,\enspace \mathrm{R}\mathrm{e}\enspace {Y}_{N}^{N},\enspace \mathrm{I}\mathrm{m}\enspace {Y}_{1}^{1},\dots ,\enspace \mathrm{I}\mathrm{m}\enspace {Y}_{N}^{N}\right\},\end{equation*}$$

we arrive at the representation matrix for the discretized operator

$$\begin{equation*}{\mathbf{F}}^{\prime }\left[\partial D\right]:{\mathbb{R}}^{{\left(N+1\right)}^{2}}\to {\mathbb{C}}^{3{\times}M},\quad {\left({\mathbf{F}}^{\prime }\left[\partial D\right]\right)}_{ijk}={\left({E}_{\infty }^{\prime }\left({\hat{x}}_{j};{h}_{k}\right)\right)}_{i}\end{equation*}$$

with i = 1, 2, 3, j = 1, ..., M and k = 1, ..., (N + 1)², where h_k denotes the kth element of $\mathcal{B}$. The discretization of the identity I is given by the identity matrix ${I}_{{\left(N+1\right)}^{2}}$. We observed a better performance of our scheme by using a different penalty matrix J instead, which punishes the curvature of the boundary. Such a matrix J is for example given by the diagonal matrix with entries (J)_kk = 1 + λ(k), k = 1, ..., (N + 1)². Here, λ(k) is the corresponding eigenvalue of the spherical harmonic ${Y}_{n}^{m}$ with respect to the Laplace Beltrami operator ${{\Delta}}_{{\mathbb{S}}^{2}}={\mathrm{D}\mathrm{i}\mathrm{v}}_{{\mathbb{S}}^{2}}\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{{\mathbb{S}}^{2}}$, associated to the kth basis element of $\mathcal{B}$. For the predictor $\hat{h}=\left({\alpha }_{h},{\beta }_{h}\right)\in {\mathbb{R}}^{{\left(N+1\right)}^{2}}$ we solve the discretized version of equation (5.2) with I replaced by J. In general, a solution of this equation is complex-valued, so we discard the imaginary part.

Let ${E}_{\infty }^{{\prime\prime}}\left(\hat{x};h,\hat{h}\right)$ denote the far field pattern of the second domain derivative with respect to the perturbations h₁ = h and h₂ = , evaluated at $\hat{x}\in {\mathbb{S}}^{2}$. Then the representation matrix for the discretized operator $\mathbf{T}={\mathbf{F}}^{\prime }\left[\partial {D}^{i}\right]+\frac{1}{2}{\mathbf{F}}^{{\prime\prime}}\left[\partial {D}^{i}\right]\left(\hat{h},\cdot \right)$ is given by

$$\begin{equation*}\mathbf{T}:{\mathbb{R}}^{{\left(N+1\right)}^{2}}\to {\mathbb{C}}^{3{\times}M},\quad {\left(\mathbf{T}\right)}_{ijk}={\left({E}_{\infty }^{\prime }\left({\hat{x}}_{j};{h}_{k}\right)\right)}_{i}+\frac{1}{2}{\left({E}_{\infty }^{\prime \prime }\left({\hat{x}}_{j};{h}_{k},\hat{h}\right)\right)}_{i}\end{equation*}$$

with again i = 1, 2, 3, j = 1, ..., M and k = 1, ..., (N + 1)². The corrector h is then given by the real part of the solution of the discretized version of $\left({\mathbf{T}}^{{\ast}}\mathbf{T}+{\alpha }_{2}\mathbf{J}\right)h={\mathbf{T}}^{{\ast}}\left({E}_{\infty }-\mathbf{F}\left(\partial {D}^{i}\right)\right)$.

5.2. Solving the boundary value problems by BEMPP

Full discretization requires the numerical evaluation of F(∂D), F'[∂D] and F''[∂D]. Looking closely at the boundary conditions for the first and second domain derivative, we identify the traces of the solutions (E, H) and (E', H') and some terms involving surface differential operators to these traces. We therefore chose an integral equation approach for the full discretization. Our implementations were carried out in the open source Galerkin boundary element methods library BEMPP (https://bempp.com). For an overview of the library, see [26]. We will briefly present the tools needed to formulate the scattering from a perfect conductor as an integral equation. Let ${\Phi}\left(x,y\right)=\frac{1}{4\pi }\frac{{\mathrm{e}}^{\mathrm{i}k\vert x-y\vert }}{\vert x-y\vert },\enspace x\ne y$, denote the fundamental solution of the three-dimensional Helmholtz equation Δu + k² u = 0. We define the electric potential

$$\begin{equation*}\mathcal{E}\varphi \left(x\right)=\mathrm{i}k{\int }_{\partial D}\varphi \left(y\right){\Phi}\left(x,y\right)\mathrm{d}s\left(y\right)-\frac{1}{\mathrm{i}k}\nabla {\int }_{\partial D}{\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\enspace \varphi \left(y\right){\Phi}\left(x,y\right)\mathrm{d}s\left(y\right)\end{equation*}$$

and the magnetic potential

$$\begin{equation*}\mathcal{H}\varphi \left(x\right)=\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\int }_{\partial D}\varphi \left(y\right){\Phi}\left(x,y\right)\mathrm{d}s\left(y\right),\end{equation*}$$

which are bounded operators from ${H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right)$ to ${H}_{\text{loc}}\left({\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}^{2},{\mathbb{R}}^{3}{\backslash}\partial D\right)$. For any radiating solution $E\in {H}_{\text{loc}}\left({\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}}^{2},{\mathbb{R}}^{3}{\backslash}\overline{D}\right)$ of Maxwell's equations, we have the Stratton–Chu representation formula $E\left(x\right)=-\mathcal{H}{\gamma }_{T}E\left(x\right)-\mathcal{E}{\gamma }_{N}E\left(x\right)$, where we introduced the Neumann trace ${\gamma }_{N}\varphi =\frac{1}{\mathrm{i}k}{\gamma }_{t}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace \varphi$. The potentials satisfy the following jump conditions on the boundary ∂D,

$$\begin{equation*}{\left[{\gamma }_{t}\mathcal{E}\varphi \right]}_{{\pm}}={\left[{\gamma }_{N}\mathcal{H}\varphi \right]}_{{\pm}}=0\quad \text{and}\quad {\left[{\gamma }_{N}\mathcal{E}\varphi \right]}_{{\pm}}={\left[{\gamma }_{t}\mathcal{H}\varphi \right]}_{{\pm}}=-\varphi \quad \text{for}\enspace \varphi \in {H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right).\end{equation*}$$

By the mean of the interior and exterior traces of the potentials, we arrive at the electric boundary operator E and magnetic boundary operator H, both bounded linear operators from ${H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right)$ to ${H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right)$. These operators satisfy

$$\begin{equation*}{\gamma }_{t}\mathcal{E}=\mathbf{E},\quad {\gamma }_{N}\mathcal{E}=-\frac{1}{2}\mathbf{I}+\mathbf{H},\quad {\gamma }_{t}\mathcal{H}=-\frac{1}{2}\mathbf{I}+\mathbf{H},\quad \text{and}\quad {\gamma }_{N}\mathcal{H}=-\mathbf{E}.\end{equation*}$$

Let E be a radiating solution to Maxwell's equations, satisfying a Dirichlet boundary condition γ_t E = −F for some right-hand side F, in our case the scattered field E^s with F = γ_t Eⁱ or the domain derivatives E', E'' with the right hand sides presented in theorems 3.3 and 4.5. In each case, we make the Ansatz $E\left(x\right)=-\mathcal{E}\lambda \left(x\right),\quad x\in {\mathbb{R}}^{3}{\backslash}\overline{D}$ for some density $\lambda \in {H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right)$. Then, by the trace and ${\gamma }_{t}\mathcal{E}=\mathbf{E}$ we arrive at the indirect electric field equation (EFIE)

$$\begin{equation*}\mathbf{E}\lambda =F.\end{equation*}$$

Assuming k to be no interior eigenvalue of D, the EFIE is uniquely solvable for any right-hand side (see [2]). The major challenge arises from calculating the boundary conditions for the domain derivatives. Recall the boundary condition ν × E' = Grad_∂D (h_ν E_ν ) × ν − ikh_ν H_τ of E'. Numerically calculating the boundary condition requires access to the discrete version of the surface gradient Grad, the rotation operator R, defined by Rγ_T φ = γ_t φ, the magnetic trace H_τ = (ν × (H × ν)), and the normal component of the electric field E_ν . Since basic surface differential operators like surface gradient, surface divergence, and Laplace–Beltrami operator are available in BEMPP, we represent the boundary terms as follows.

We have to calculate discrete products of discretizations for the product h_ν E_ν and h_ν H_τ . From (3.5) we conclude ${E}_{\nu }=-\frac{1}{\mathrm{i}k}\enspace {\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left(H{\times}\nu \right)$, i.e. we can calculate the normal component of E by applying the surface divergence to H × ν. Considering

$$\begin{equation*}{\int }_{\partial D}{\gamma }_{T}\varphi \cdot {\gamma }_{t}\psi \enspace \mathrm{d}s=-{\int }_{\partial D}{\gamma }_{t}\varphi \cdot {\gamma }_{T}\psi \enspace \mathrm{d}s,\end{equation*}$$

we see, that the negative dual pairing $-{\langle {\gamma }_{t}\varphi ,{\gamma }_{T}\psi \rangle }_{\partial D}$ between ${H}^{-\frac{1}{2}}\left(\mathrm{D}\mathrm{i}\mathrm{v},\partial D\right)$ and its dual space ${H}^{-\frac{1}{2}}\left(\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l},\partial D\right)$ can be seen as the weak formulation for the rotation operator R.

Since we use the Ansatz ${E}^{\mathrm{s}}=-\mathcal{E}\lambda$, the tangential trace of the electric field is given by

$$\begin{equation*}{H}^{\mathrm{s}}{\times}\nu ={\gamma }_{N}{E}^{\mathrm{s}}=\left(\frac{1}{2}\mathbf{I}-\mathbf{H}\right)\lambda .\end{equation*}$$

For the discrete product f⋅_d g = ∑_i α_i ϕ_i of two functions f and g in a chosen basis of functions ϕ_i , we calculate the L² projection of the product onto the bases functions ϕ_i , i.e. we solve the linear system

$$\begin{equation*}{\int }_{\partial D}{\phi }_{j}\left(x\right)\cdot \left(f\left(x\right)g\left(x\right)\right)\mathrm{d}s\left(x\right)={\alpha }_{i}{\int }_{\partial D}{\phi }_{i}\left(x\right)\cdot {\phi }_{j}\left(x\right)\mathrm{d}s\left(x\right),\quad j=1,\dots .\end{equation*}$$

Note that we have to choose a basis of scalar functions for the product h_ν E_ν and a basis of vector valued functions for the product h_ν H_τ . For details on the above described implementations and the code of the actual implementations of the first domain derivative and its use to solve an inverse problem, we refer to [8, 9] and the tutorials on the homepage of BEMPP (https://bempp.com). For details on how to solve the EFIE with BEMPP, see also [25].

Lets consider now the boundary condition for the second domain derivative E'', given by

$$\begin{align*}\hfill \nu {\times}{E}^{{\prime\prime}}=& \sum _{i\ne j=1}^{2}\left[{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left({h}_{i,\nu }{E}_{j,\nu }^{\prime }-{E}_{\nu }{h}_{i,\tau }^{\top }\enspace {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {h}_{j,\nu }\right)\right.\hfill \\ \hfill & \left.-\mathrm{i}k\enspace {\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left({h}_{j,\nu }{H}_{\tau }\right){h}_{i,\tau }-\mathrm{i}k\enspace {h}_{i,\nu }{H}_{j,\tau }^{\prime }\right]\hfill \\ \hfill & -\mathrm{i}k\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right){H}_{\tau }+{\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\left[\left(\left({h}_{2,\tau }^{\top }\mathcal{R}{h}_{1,\tau }\right)-2\kappa {h}_{1,\nu }{h}_{2,\nu }\right){E}_{\nu }\right]\hfill \\ \hfill & +2\mathrm{i}k\enspace {h}_{1,\nu }{h}_{2,\nu }\left(\mathcal{R}-\kappa \right){H}_{\tau }+\mathrm{i}k\sum _{i\ne j=1}^{2}{h}_{i,\tau }^{\top }\left(\nu {\times}H\right){\vec {\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{l}}}_{\partial D}\enspace {h}_{j,\nu }.\hfill \end{align*}$$

Note, that we have formulated the boundary condition in a way, we can use the same tools as before. We only have to consider additionally a discretization of the curvature operator $\mathcal{R}$ and of the mean curvature κ. The discrete scalar product of two functions is another special case of the discrete product ⋅_d described above. Recall the definitions

$$\begin{equation*}\mathcal{R}={J}_{\nu }\quad \text{and}\quad \kappa =\frac{1}{2}\enspace \mathrm{d}\mathrm{i}\mathrm{v}\enspace \nu .\end{equation*}$$

We have $\frac{\partial \nu }{\partial \nu }=0$ and $\mathcal{R}$ acts only on the tangential plane. Since $\mathcal{R}={\mathcal{R}}^{\top }$, we arrive at

$$\begin{equation*}\mathcal{R}{H}_{\tau }=\left(\begin{matrix}\hfill {H}_{\tau }\cdot {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {\nu }_{1}\hfill \\ \hfill {H}_{\tau }\cdot {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {\nu }_{2}\hfill \\ \hfill {H}_{\tau }\cdot {\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}}_{\partial D}\enspace {\nu }_{3}\hfill \end{matrix}\right),\end{equation*}$$

which is in every component a discrete product of functions. Having calculated each component, we use again L² projections to calculate $\mathcal{R}{H}_{\tau }$. For the mean curvature κ, we use the relation −Δ_∂D x_i = 2κν_i , i ∈ {1, 2, 3}, of κ and the Laplace–Beltrami operator (see equation (2.5.212) in [23] with u = x_i ), to calculate

$$\begin{equation*}-\frac{1}{2}\sum _{i=1}^{3}{\nu }_{i}{{\Delta}}_{\partial D}{x}_{i}=\kappa \enspace \vert \nu {\vert }^{2}=\kappa .\end{equation*}$$

The left-hand side can again be implemented by using the discrete product of functions ⋅_d and applications of the surface gradient and the surface divergence.

5.3. Numerical examples

Before we consider some numerical examples let us summarize on one step of the iteration schemes using the above discretized operators. First for a Newton step:

• (a)  
  For given r_j and already computed far field pattern F(∂D^j ) compute the traces E_ν and H_τ and the right-hand side of the boundary values from theorem 3.3 for each of the (N + 1)² basis functions h_k . With these right-hand sides solve (N + 1)² times the integral equation (EFIE) and collect the computed corresponding far field pattern in the matrix F'[∂D^j ].
• (b)  
  Solve the discretized linear system (5.2) to compute the (N + 1)² expansion coefficients of and update r_j+1 = r_j + .
• (c)  
  Solve the integral equation (EFIE) for the new boundary ∂D^j+1 with right-hand side given through the incident plane wave, compute the current far field pattern F(∂D^j+1), and check on the stopping condition.

In case of the Halley method we have to modify the second part by:

• (b1)  
  Solve the discretized linear system (5.2) to compute the (N + 1)² expansion coefficients of the predictor .
• (b2)  
  Compute all required traces on the right-hand side of the boundary condition in theorem 4.5 with h₁ replaced by and h₂ replaced by each of the basis functions h_k . Solve the (N + 1)² integral equations (EFIE) with these right-hand sides and collect the corresponding far field pattern in the matrix F''[∂D^j ].
• (b3)  
  Solve the discrete version of (5.3) for the expansion coefficients of the corrector h and update r_j+1 = r_j + h.

In all presented experiments, we have chosen N = 4, i.e. we used 25 basis functions to describe our reconstructions. For the considered objects, choosing more basis functions did not seem to be worth the greatly increasing computational effort.

Now, since we know how to realize the boundary conditions, we present actual reconstructions using the second degree method. We ran reconstructions for exact and also noisy data. As in [3, 8, 9, 15] we consider the following shapes:

• A rounded cuboid, implicitly given by
  $$\begin{equation*}{\left(\frac{{x}_{1}}{{r}_{1}}\right)}^{n}+{\left(\frac{{x}_{2}}{{r}_{2}}\right)}^{n}+{\left(\frac{{x}_{3}}{{r}_{3}}\right)}^{n}={d}^{n}\end{equation*}$$
  with n = 6, a radius d = 1 and side lengths r₁ = 1, r₂ = 1.3, r₃ = 0.7.
• A cushion-shaped object with parametrization
  $$\begin{equation*}x=\sqrt{0.8+0.5\left(\mathrm{cos}\enspace 2\varphi -1\right)\left(\mathrm{cos}\enspace 4\theta -1\right)}\left(\begin{matrix}\hfill \mathrm{cos}\enspace \varphi \enspace \mathrm{sin}\enspace \theta \hfill \\ \hfill \mathrm{sin}\enspace \varphi \enspace \mathrm{sin}\enspace \theta \hfill \\ \hfill \mathrm{cos}\enspace \theta \hfill \end{matrix}\right),\quad \varphi \in \left[0,2\pi \right],\enspace \theta \in \left[0,\pi \right].\end{equation*}$$

Note, that the characterizations of the second domain derivative requires a smooth boundary. We therefore chose the rounded cuboid to challenge our reconstructions with an object close to the non-smooth cuboid. Additionally, we show the cushion as an example for a non-convex object with positive and negative curvature κ. In order to cancel any positive effects due to symmetry, we applied the translation $x{\mapsto}x+\frac{k}{8}{\left(1,1,1\right)}^{\top }$ such that the center of the rounded cuboid and the cushion-shaped object does not coincide with the center of the star shaped reconstructions.

Furthermore, we consider the plane wave, given by

$$\begin{equation*}\left(\begin{matrix}\hfill {E}^{i}\hfill \\ \hfill {H}^{i}\hfill \end{matrix}\right)\left(x\right)=\left(\begin{matrix}\hfill p\hfill \\ \hfill d{\times}p\hfill \end{matrix}\right){\mathrm{e}}^{\mathrm{i}kd\cdot x},\quad x\in {\mathbb{R}}^{3},\end{equation*}$$

with polarization $p={\left(i-1\enspace ,\enspace 2\enspace ,\enspace -1-\frac{1}{3}i\right)}^{\top }\in {\mathbb{C}}^{3}$ and direction $d=\frac{1}{\sqrt{14}}{\left(1\enspace ,\enspace 2\enspace ,\enspace 3\right)}^{\top }\in {\mathbb{S}}^{2}$ such that the direction of the plane wave does not coincide with the symmetries of the considered shapes. We also ran experiments where we have considered different star shaped objects, wave numbers, incident directions, or real or complex valued polarizations. All of these led to comparable results as in the presented examples. Additionally, similar to other iterative approaches in acoustic or electromagnetic inverse scattering the Halley method confirmed the known observation that there is some improvement if reconstructions from some far field patterns generated for instance by three or four incident directions are averaged in each iteration step (see [11, 15, 21]).

In the presented experiments, the wave number is k = 1.4. Note that the diameter of the rounded cuboid and the wavelength 2π/k are of the same magnitude. We calculate the exact data E_∞ = F(∂D) by picking M = 168 evaluation points on the unit sphere ${\mathbb{S}}^{2}$, i.e. ${E}_{\infty }\in {\mathbb{C}}^{3{\times}168}$. Note, that due to the offset of the exact data, we avoid an inverse crime, since the exact data is being calculated by using meshes unrelated to those used in the reconstructions. We additionally use finer meshes for the calculation of the exact data. For the presented examples of reconstructions, we have used grids with 800 elements and 1200 edges which results in 1200 degrees of freedom. To generate the exact data, we used a grid with 1152 elements and 1728 edges which results in 1728 degrees of freedom. On a machine with 32 CPU cores one iteration took on average about 15 min, while one iteration of the Halley method took on average about 90 min. Note that these computational times can almost surely be improved, since the code is not optimized with respect to a fast performance.

In the case of noisy data of level δ ⩾ 0, we multiply every element of the far field matrix ${E}_{\infty }\in {\mathbb{C}}^{3{\times}168}$ by a random complex number $1+\delta {\lambda }_{1}\enspace {\mathrm{e}}^{2\pi \mathrm{i}{\lambda }_{2}}$, where λ₁, λ₂ are uniformly distributed random numbers on (0, 1). In our experiments with noisy data, we have chosen a noise level δ = 0.2 which results in 10% relative error in comparison with the exact data.

In each experiment, besides the one presented in figure 5, we have chosen the unit ball as initial guess. To find an appropriate choice for the regularization parameters α₁, α₂, we tested a range of parameters α₁, α₂ and observed in the case of noise free data successful reconstructions for a wide range of parameters. In the case of noisy data, the choice of α₁ = 300 and α₂ = 150 led to reasonable reconstructions in all of our experiments. Considerably lower values led sometimes to deteriorated reconstructions.

To observe stability of the reconstructions with respect to random noise, we ran each experiment with noisy data ten times. In order to judge on the quality of the reconstructions, we considered the relative ${L}^{2}\left({\mathbb{S}}^{2}\right)$ error of the parametrizations of the exact object and the reconstruction. At the above mentioned noise level and with our chosen regularization parameters, one can hardly see which reconstruction is the worst and which one is the best. Considering the parametrizations of the best and the worst reconstructions we observed about 5% difference. For our figures, we picked neither the best nor the worst case to present.

The stopping rule applies, if the residual ${\Vert}\mathbf{F}\left(\partial {D}^{i}\right)-{E}_{\infty }{{\Vert}}_{{L}^{2}\left({\mathbb{S}}^{2}\right)}/{\Vert}{E}_{\infty }{{\Vert}}_{{L}^{2}\left({\mathbb{S}}^{2}\right)}$ falls below τ times the noise level, where τ satisfies 2 ⩾ τ > 1. Since there is no strategy for choosing an optimal τ, we have chosen τ = 1.5 for the final reconstruction in figure 2. With τ = 1.5, one can hardly observe any differences in the reconstructions of the regularized Newton and Halley method. However, by choosing smaller τ, we did observe qualitative differences in the reconstructions. This requires some fine tuning of the parameter τ. To present this behavior, we have chosen τ = 1.13 to pick the final reconstructions in figures 3 and 4.

In most of our experiments we observed a behavior of the residual as shown in figure 1 for the rounded cuboid and the cushion shaped object. It occurs a significant decrease in the first few iteration steps which then slows down rapidly. As seen for the rounded cuboid usually the second degree method shows a faster decrease in the first one to three iteration steps. But finally both methods, the iterative regularized Newton method and the second degree method, lead to similar residual errors in case of noise free data as well as in case of noisy data. Note that there was a case, where the second degree method shows only in the first iteration an improvement in comparison to the Newton type method.

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In comparing the reconstructions of both approaches some differences are remarkable. Both the Halley as well as the Newton scheme lead to reasonable, similar reconstructions, as shown in figure 2. The only difference are a few less iteration steps required in the Halley methods, which, of course, are payed for by more computational effort for the second derivative in each step. But, additionally, we observed a more stable performance with respect to the choice of the regularization parameter. The range of possible parameters α₁ within the Newton scheme leading to reasonable results is significantly smaller then for the second degree approach. To be more specific: using α₁/2 instead of α₁ within the Newton scheme led most of the time to deteriorated shapes after some iterations, whereas the using of α₂/2 instead of α₂ within the Halley method almost always still produced reasonable reconstructions.

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A more significant effect in the reconstructions occurs in case of noisy data. Due to regularization the iteration has to be stopped if the residual error is becoming to small. It is seen also in the acoustic case that the iterated shapes start to deteriorate if a receding cusp is developed. This effect is regularized slightly by choosing the matrix J instead of I in the Tikhonov regularization, but if it occurs, in general the iteration process can not compensate on it and the iteration must be stopped. Of course, by a larger regularization parameter we can avoid the effect but then reconstructions become worse, close to the initial guess. Here we observed the main advantage of considering the second degree method, since it turned out that the method reaches frequently the stopping level before such cusps occur. This can be seen from the figures 3 and 4 showing reconstructions from noisy data with the iterative regularized Newton method and with the Halley method.

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Finally, we consider the performance of the schemes in the case, where we know that the tangential cone condition fails. Thus we consider a ball, where the radius is chosen such that the wave number is an eigenvalue of the Laplace–Beltrami operator. From corollary 3.4 we have seen that injectivity of F' does not hold if we illuminate the ball by an incident field generated by a vector surface harmonics. Especially F'h = 0 if h_ν is constant. Thus the iteration schemes cannot just expand or shrink the size of the ball. Exactly this was observed. Using such a ball as an initial guess both the Newton as well as the second degree scheme slow down and do not reach as good reconstructions as in non critical cases (see figure 5).

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As a conclusion from these numerical investigations we can state a slightly more stable performance of the Halley method compared to an iterative regularized Newton approach as it was already observed for the acoustic case (see [13]), but by the prize of a higher computational effort in each iteration step. Additionally, the last observations in case of non injective domain derivatives confirm that further research is required in understanding the performance of iterative regularization schemes in inverse obstacle scattering.

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 258734477—CRC 1173.

Proof. Let ${\left({E}_{1}^{\prime }\right)}_{2}^{\prime }$ be defined as in the theorem. The regularity of the boundary implies $E,H\in {H}^{2}\left({\Omega},{\mathbb{C}}^{3}\right)$ and ${E}^{\prime },{H}^{\prime }\in {H}^{1}\left({\Omega},{\mathbb{C}}^{3}\right)$, see [1, corollary 2.15]. We conclude, similar as before ${\left({E}_{1}^{\prime }\right)}_{2}^{\prime }\in H\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l},{\Omega}\right)$. We will again show $\mathcal{A}\left({\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\right)=0$ for all V ∈ H_pc(Ω). For i = 1, 2 we have

$$\begin{equation}{W}_{i}={E}_{i}^{\prime }+{J}_{{h}_{i}}^{\top }E+{J}_{E}{h}_{i}={E}_{i}^{\prime }+\nabla \left({h}_{i}^{\top }E\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{i}.\end{equation} \tag{ 5.4 }$$

The curl of the material derivative is then given by

$$\begin{align*}\hfill \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {W}_{i}=& \;\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{i}^{\prime }+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{i}\right)\hfill \\ \hfill =& \;\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{i}^{\prime }+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\enspace \mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{i}+\left({J}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E}-{J}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E}^{\top }\right){h}_{i}\hfill \\ \hfill & +\left({J}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E}^{\top }{h}_{i}+{J}_{{h}_{i}}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)-\left({J}_{{h}_{i}}-{J}_{{h}_{i}}^{\top }\right)\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\hfill \\ \hfill =& \;\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}_{i}^{\prime }+{A}_{i}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E+{k}^{2}\left(E{\times}{h}_{i}\right)+\nabla \left({h}_{i}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right).\hfill \end{align*}$$

Note from the proof of theorem 3.3, that for any divergence free solution $F\in {H}^{1}\left({\Omega},{\mathbb{C}}^{3}\right)$ of Maxwell's equations and V ∈ H_pc(Ω) we have for i = 1, 2

$$\begin{equation*}{\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {F}^{\top }{A}_{i}\overline{V}+{k}^{2}\enspace {F}^{\top }{A}_{i}\overline{V}\right)\mathrm{d}x-\mathcal{A}\left({J}_{{h}_{i}}^{\top }F+{J}_{F}{h}_{i},V\right)=0.\end{equation*}$$

We apply the identity for F = E_i ', i = 1, 2. Let us continue by eliminating the terms involving the material derivative in (5.4). We obtain

$$\begin{equation*}{J}_{{h}_{2}}{W}_{1}+{J}_{{W}_{1}}{h}_{2}={J}_{{h}_{2}}^{\top }{E}_{1}^{\prime }+{J}_{{E}_{1}^{\prime }}{h}_{2}+{J}_{{h}_{2}}^{\top }{J}_{{h}_{1}}^{\top }E+{J}_{{h}_{2}}^{\top }{J}_{E}{h}_{1}+{J}_{{J}_{{h}_{1}}^{\top }E+{J}_{E}{h}_{1}}{h}_{2}.\end{equation*}$$

In order to deal with the Jacobian of the Jacobians, we write

$$\begin{align*}\hfill {J}_{{h}_{2}}^{\top }{J}_{E}{h}_{1}+{J}_{{J}_{{h}_{1}}^{\top }E+{J}_{E}{h}_{1}}{h}_{2}& =\nabla \left({h}_{2}^{\top }\left({h}_{1}^{\top }E\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right) \left.\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){\times}{h}_{2}\hfill \\ \hfill & =\nabla \left({h}_{2}^{\top }\left({h}_{1}^{\top }E\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right) \left.\right)+\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\hfill \\ \hfill & \quad +{k}^{2}\left(E{\times}{h}_{1}\right){h}_{2}+\nabla \left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}.\hfill \end{align*}$$

Let us consider now

$$\begin{align*}\hfill \mathcal{A}\left({\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\right)=& \;\mathcal{A}\left({W}_{1},V\right)-\mathcal{A}\left({J}_{{h}_{1}}^{\top }{E}_{2}^{\prime }+{J}_{{E}_{2}^{\prime }}{h}_{1}-{J}_{{h}_{2}}^{\top }{W}_{1}+{J}_{{W}_{1}}{h}_{2},V\right)\hfill \\ \hfill =& {\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {W}_{1}^{\top }{A}_{2}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {W}_{1}^{\top }{A}_{2}\overline{V}+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {W}_{2}^{\top }{A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {W}_{2}^{\top }{A}_{1}\overline{V}\right)\mathrm{d}x\hfill \\ \hfill & +{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left({J}_{{h}_{2}}^{\top }{A}_{1}+{A}_{1}{J}_{{h}_{2}}-\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & +{k}^{2}{\int }_{{\Omega}}{E}^{\top }\left(-{J}_{{h}_{2}}{A}_{1}-{A}_{1}{J}_{{h}_{2}}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\mathrm{d}x\hfill \\ \hfill & -\mathcal{A}\left({J}_{{h}_{1}}^{\top }{E}_{2}^{\prime }+{J}_{{E}_{2}^{\prime }}+{J}_{{h}_{2}}^{\top }{E}_{1}^{\prime }+{J}_{{E}_{1}^{\prime }}{h}_{2}+{J}_{{h}_{2}}^{\top }{J}_{{h}_{1}}^{\top }E+{J}_{{h}_{2}}^{\top }{J}_{E}{h}_{1}\right.\hfill \\ \hfill & \left.+{J}_{{J}_{{h}_{1}}^{\top }E+{J}_{E}{h}_{1}}{h}_{2},V\right).\hfill \end{align*}$$

With the previous calculations, we get

$$\begin{align*}\hfill \mathcal{A}\left({\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\right)& ={\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {{E}_{1}^{\prime }}^{\top }{A}_{2}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {{E}_{1}^{\prime }}^{\top }{A}_{2}\overline{V}\right)\mathrm{d}x-\mathcal{A}\left({J}_{{h}_{2}}^{\top }{E}_{1}^{\prime }+{J}_{{E}_{1}^{\prime }}{h}_{2},V\right)\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {{E}_{2}^{\prime }}^{\top }{A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+{k}^{2}\enspace {{E}_{2}^{\prime }}^{\top }{A}_{1}\overline{V}\right)\mathrm{d}x-\mathcal{A}\left({J}_{{h}_{1}}^{\top }{E}_{2}^{\prime }+{J}_{{E}_{2}^{\prime }}{h}_{1},V\right)\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\left({A}_{2}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E+{k}^{2}\left(E{\times}{h}_{2}\right)+\nabla \left({h}_{2}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\right){A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E+{k}^{2}\left(E{\times}{h}_{1}\right)+\nabla \left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\right){A}_{2}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}\left[\right.\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{2}+\nabla \left({h}_{2}^{\top }E\right)\right){A}_{1}\overline{V}+{\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}+\nabla \left({h}_{1}^{\top }E\right)\right)}^{\top }{A}_{2}\overline{V}\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left({J}_{{h}_{2}}^{\top }{A}_{1}+{A}_{1}{J}_{{h}_{2}}-\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}{E}^{\top }\left(-{J}_{{h}_{2}}{A}_{1}-{A}_{1}{J}_{{h}_{2}}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{V}\mathrm{d}x\hfill \\ \hfill & \quad -{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left[\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}+{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right]}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}V}\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}\left[\nabla \left({h}_{2}^{\top }\left(\nabla \left({h}_{1}^{\top }E\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\right)+\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right.\hfill \\ \hfill & {\left.\quad +{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right]}^{\top }\overline{V}\mathrm{d}x.\hfill \end{align*}$$

Recall the definition of the symmetric matrix ${A}_{i}=\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{i}\enspace I-{J}_{{h}_{i}}-{J}_{{h}_{i}}^{\top }$. We have

$$\begin{equation*}{A}_{2}{A}_{1}+{A}_{1}{A}_{2}=2\enspace \mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}-{J}_{{h}_{2}}{A}_{1}-{J}_{{h}_{2}}^{\top }{A}_{1}-{A}_{1}{J}_{{h}_{2}}-{A}_{1}{J}_{{h}_{2}}^{\top }\end{equation*}$$

and therefore

$$\begin{align}\hfill \mathcal{A}\left({\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\right)& ={k}^{2}{\int }_{{\Omega}}{E}^{\top }\left(-{J}_{{h}_{2}}{A}_{1}-{A}_{1}{J}_{{h}_{2}}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}{h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{V}\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{E}^{\top }\left(-{J}_{{h}_{2}}{A}_{1}-{A}_{1}{J}_{{h}_{2}}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}{h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}V}\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}{\left({k}^{2}\left(E{\times}{h}_{2}\right)+\nabla \left({h}_{2}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E\right)\right)}^{\top }{A}_{1}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}V}\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}{\left({k}^{2}\left(\right.E{\times}{h}_{1}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E\right)\right)}^{\top }{A}_{2}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}V}\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}\left[{\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E{\times}{h}_{2}+\nabla \left({h}_{2}^{\top }E\right)\right)}^{\top }{A}_{1}+{\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E{\times}{h}_{1}+\nabla \left({h}_{1}^{\top }E\right)\right)}^{\top }{A}_{2}\right]\overline{V}\mathrm{d}x\hfill \\ \hfill & \quad -{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left[\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E\right){\times}{h}_{2}+{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E\right){\times}{h}_{2}\right]}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}V}\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}\left[\nabla \left({h}_{2}^{\top }\left(\nabla \left({h}_{1}^{\top }E\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E{\times}{h}_{1}\right)\right)+\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E\right){\times}{h}_{2}\right.\hfill \\ \hfill & {\left.\quad +{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}E\right){\times}{h}_{2}\right]}^{\top }\overline{V}\mathrm{d}x.\hfill \end{align} \tag{ 5.5 }$$

Considering vector fields E, V, h and a symmetric matrix A, elementary calculus yields the following identities:

$$\begin{align*}\hfill \mathrm{d}\mathrm{i}\mathrm{v}\left(\left({h}^{\top }E\right)V\right)& =\left({h}^{\top }E\right)\mathrm{d}\mathrm{i}\mathrm{v}\enspace V+{V}^{\top }{J}_{h}^{\top }E+{V}^{\top }{J}_{E}^{\top }h,\hfill \\ \hfill \mathrm{d}\mathrm{i}\mathrm{v}\left(\left({h}^{\top }V\right)AE\right)& =\left({h}^{\top }V\right)\mathrm{d}\mathrm{i}\mathrm{v}\left(AE\right)+{E}^{\top }A{J}_{h}^{\top }V+{E}^{\top }A{J}_{V}^{\top }h,\hfill \\ \hfill \mathrm{d}\mathrm{i}\mathrm{v}\left(\left({E}^{\top }AV\right)h\right)& ={E}^{\top }AV\enspace \mathrm{d}\mathrm{i}\mathrm{v}\enspace h+{h}^{\top }{J}_{E}^{\top }AV+{E}^{\top }{A}^{\prime }\left(h\right)V+{E}^{\top }A{J}_{V}h.\hfill \end{align*}$$

With these identities, we conclude

$$\begin{align}\hfill & \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }\left(-{J}_{{h}_{2}}{A}_{1}-{A}_{1}{J}_{{h}_{2}}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\hfill \\ \hfill & \quad =-\mathrm{d}\mathrm{i}\mathrm{v}\left[\left({h}_{2}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){A}_{1}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}+\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right){A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E-\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }{A}_{1}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right){h}_{2}\right]\hfill \\ \hfill & \quad +\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}E\right)+\left({h}_{2}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\hfill \\ \hfill & \quad +{\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)}^{\top }\left(\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}{\times}{h}_{2}\right)-{k}^{2}{\left(E{\times}{h}_{2}\right)}^{\top }{A}_{1}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V},\hfill \end{align} \tag{ 5.6 }$$

since curl  curl  E = k² E. Similarly, we have

$$\begin{align}\hfill & {E}^{\top }\left(-{J}_{{h}_{2}}{A}_{1}-{A}_{1}{J}_{{h}_{2}}^{\top }+\mathrm{d}\mathrm{i}\mathrm{v}\enspace {h}_{2}\enspace {A}_{1}+{A}_{1}^{\prime }\left({h}_{2}\right)\right)\overline{V}\hfill \\ \hfill & \quad =\mathrm{d}\mathrm{i}\mathrm{v}\left[\left({h}_{2}^{\top }E\right){A}_{1}\overline{V}+\left({h}_{2}^{\top }\overline{V}\right){A}_{1}E+\left({E}^{\top }{A}_{1}\overline{V}\right){h}_{2}\right]+\left({h}_{2}^{\top }\overline{V}\right)\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}E\right)\hfill \\ \hfill & \quad +\left({h}_{2}^{\top }E\right)\mathrm{d}\mathrm{i}\mathrm{v}{\left({A}_{1}\overline{V}\right)}^{\top }\left(\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}{\times}{h}_{2}\right)-{\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{2}\right)}^{\top }{A}_{1}\overline{V}.\hfill \end{align} \tag{ 5.7 }$$

We combine (5.6), (5.7) and (5.5) to get

$$\begin{align*}\hfill \mathcal{A}\left({\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\right)& ={\int }_{{\Omega}}\mathrm{d}\mathrm{i}\mathrm{v}\left[\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }{A}_{1}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right){h}_{2}-\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right){A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right]\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}\left(\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)-{\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)}^{\top }\left(\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}{\times}{h}_{2}\right)\right)\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}\mathrm{d}\mathrm{i}\mathrm{v}\left[\left({E}^{\top }{A}_{1}\overline{V}\right){h}_{2}-\left({h}_{2}^{\top }\overline{V}\right){A}_{1}E\right]\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}\left({h}_{2}^{\top }\overline{V}\left.\right)\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}E\right){A}_{1}E\left(\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}{\times}{h}_{2}\right)\right)\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}{\left({k}^{2}\left(E{\times}{h}_{1}\right)+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\right)}^{\top }{A}_{2}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}{\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}+\nabla \left({h}_{1}^{\top }E\right)\right)}^{\top }{A}_{2}\overline{V}\mathrm{d}x\hfill \\ \hfill & \quad -{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left[\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}+{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right]}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}\left[\nabla \left({h}_{2}^{\top }\left(\nabla \left({h}_{1}^{\top }E\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\right)+\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right.\hfill \\ \hfill & {\left.\quad +{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right]}^{\top }\overline{V}\mathrm{d}x.\hfill \end{align*}$$

By the divergence theorem, we have

$$\begin{align*}\hfill & {\int }_{{\Omega}}\mathrm{d}\mathrm{i}\mathrm{v}\left[\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }{A}_{1}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right){h}_{2}-\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right){A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right]\mathrm{d}x\hfill \\ \hfill & \quad =-{\int }_{\partial D}\left({h}_{2,\nu }\left(\right.\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }{A}_{1}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}-\left({\nu }^{\top }{A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\right)\mathrm{d}s,\hfill \end{align*}$$

since h₁, h₂ are compactly supported in Ω. Application of the partial integration formula leads to

$$\begin{align*}\hfill & {\int }_{{\Omega}}\left(-{\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)}^{\top }\left(\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}{\times}{h}_{2}\right)-\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left(\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\mathrm{d}x\hfill \\ \hfill & \quad ={\int }_{{\Omega}}\left({\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{2}\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}-\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left(\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\mathrm{d}x\hfill \\ \hfill & \quad =-{\int }_{\partial D}{\left(\nu {\times}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)}^{\top }\left(\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)\mathrm{d}s.\hfill \end{align*}$$

In the last integral, we only need to consider the tangential component of (A₁ curl E) × h₂, which is given by

$$\begin{equation*}{\left[\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right]}_{\tau }=\left({\nu }^{\top }{A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\left(\nu {\times}{h}_{2}\right)-{h}_{2,\nu }\left(\nu {\times}\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\right).\end{equation*}$$

We use again (3.5) to conclude ${\nu }^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}=-{\mathrm{D}\mathrm{i}\mathrm{v}}_{\partial D}\left(\overline{\nu {\times}V}\right)=0$, since V ∈ H_pc(Ω), which yields

$$\begin{equation*}{\left(\nu {\times}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)}^{\top }\left(\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)=\left({h}_{2}^{\top }\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\left({\nu }^{\top }{A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)-{h}_{2,\nu }\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }{A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right).\end{equation*}$$

Finally, we have shown

$$\begin{align}\hfill & {\int }_{{\Omega}}\mathrm{d}\mathrm{i}\mathrm{v}\left[\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace {E}^{\top }{A}_{1}\enspace \overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right){h}_{2}-\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right){A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right]\mathrm{d}x\hfill \\ \hfill & \quad -{\int }_{{\Omega}}\left({\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)}^{\top }\left(\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}{\times}{h}_{2}\right)-\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left(\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\mathrm{d}x=0.\hfill \end{align} \tag{ 5.8 }$$

A second application of the divergence theorem, together with the boundary condition $\nu {\times}E=\nu {\times}\overline{V}=0$ on ∂D leads to

$$\begin{align}\hfill & {\int }_{{\Omega}}\mathrm{d}\mathrm{i}\mathrm{v}\left[\left({E}^{\top }{A}_{1}\overline{V}\right){h}_{2}-\left({h}_{2}^{\top }\overline{V}\right){A}_{1}E\right]\mathrm{d}x\hfill \\ \hfill & \quad =-{\int }_{\partial D}\left(\left({\nu }^{\top }{A}_{1}\nu \right){h}_{2,\nu }{E}_{\nu }\overline{{V}_{\nu }}-\left({\nu }^{\top }{A}_{1}\nu \right){h}_{2,\nu }{E}_{\nu }\overline{{V}_{\nu }}\right)\mathrm{d}s=0.\hfill \end{align} \tag{ 5.9 }$$

Applying (5.8) and (5.9), we have

$$\begin{align*}\hfill \mathcal{A}\left({\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\right)& ={\int }_{{\Omega}}\left(\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)+{k}^{2}\left({h}_{2}^{\top }\overline{V}\right)\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}E\right)\right)\mathrm{d}x\hfill \\ \hfill & \quad -{k}^{2}{\int }_{{\Omega}}{\left({A}_{1}E\right)}^{\top }\left(\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}{\times}{h}_{2}\right)\mathrm{d}x+{k}^{2}{\int }_{{\Omega}}{\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{2}+\nabla \left({h}_{1}^{\top }E\right)\right)}^{\top }{A}_{2}\overline{V}\mathrm{d}x\hfill \\ \hfill & \quad +{\int }_{{\Omega}}{\left({k}^{2}\left(E{\times}{h}_{1}\right)+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\right)}^{\top }{A}_{2}\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & \quad -{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left({k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & \quad +{k}^{2}{\int }_{{\Omega}}\left[\nabla \left({h}_{2}^{\top }\left(\nabla \left({h}_{1}^{\top }E\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\right)+\left({A}_{1}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right.\hfill \\ \hfill & {\left.\quad +{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{1}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right]}^{\top }\overline{V}\mathrm{d}x.\hfill \end{align*}$$

We consider the term $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\left(E{\times}{h}_{1}\right){\times}{h}_{2}\right)$. Elementary calculations yields

$$\begin{align*}\hfill \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\left(E{\times}{h}_{1}\right){\times}{h}_{2}\right)& ={A}_{2}\left(E{\times}{h}_{1}\right)-{h}_{2}\enspace \mathrm{d}\mathrm{i}\mathrm{v}\left(E{\times}{h}_{1}\right)+\nabla \left({h}_{2}^{\top }\left(E{\times}{h}_{1}\right)\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(E{\times}{h}_{1}\right){\times}{h}_{2}\hfill \\ \hfill & ={A}_{2}\left(E{\times}{h}_{1}\right)-{h}_{2}\enspace \mathrm{d}\mathrm{i}\mathrm{v}\left(E{\times}{h}_{1}\right)+\nabla \left({h}_{2}^{\top }\left(E{\times}{h}_{1}\right)\right)\hfill \\ \hfill & \quad +\left({A}_{1}E\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }E\right){\times}{h}_{2}+\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){\times}{h}_{2},\hfill \end{align*}$$

which leads to

$$\begin{align*}\hfill \mathcal{A}{\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\left.\right)=& {\int }_{{\Omega}}\left(\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)+{k}^{2}\left({h}_{2}^{\top }\overline{V}\right)\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}E\right)\right)\mathrm{d}x\hfill \\ \hfill & +{\int }_{{\Omega}}{\left(\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)+{k}^{2}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}+{k}^{2}\nabla \left({h}_{1}^{\top }E\right)\right)}^{\top }{A}_{2}\overline{V}\mathrm{d}x\hfill \\ \hfill & -{\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left(\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & +{k}^{2}{\int }_{{\Omega}}\left[\mathrm{d}\mathrm{i}\mathrm{v}\left(E{\times}{h}_{1}\right){h}_{2}-\nabla \left({h}_{2}^{\top }\left(E{\times}{h}_{1}\right)\right)\right.\hfill \\ \hfill & {\left.-\nabla \left({h}_{1}^{\top }\mathrm{E}\right){\times}{h}_{2}-\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){\times}{h}_{2}\right]}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x\hfill \\ \hfill & +{k}^{2}{\int }_{{\Omega}}\left[\nabla \left({h}_{2}^{\top }\left(\nabla \left({h}_{1}^{\top }E\right)+\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\right)+\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right.\hfill \\ \hfill & {\left.+{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right]}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x.\hfill \end{align*}$$

We apply again the divergence theorem to conclude

$$\begin{align*}\hfill {\int }_{{\Omega}}\nabla {\left({h}_{2}^{\top }\left(E{\times}{h}_{1}\right)\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x& =-{\int }_{\omega }\mathrm{d}\mathrm{i}\mathrm{v}\left[\nabla \left({h}_{2}^{\top }\left(E{\times}{h}_{1}\right)\right){\times}\overline{V}\right]\mathrm{d}x\hfill \\ \hfill & ={\int }_{\partial D}{\nu }^{\top }\left(\nabla \left({h}_{2}^{\top }\left(E{\times}{h}_{1}\right)\right){\times}\overline{V}\right)\mathrm{d}x\hfill \\ \hfill & ={\int }_{\partial D}\nabla {\left({h}_{2}^{\top }\left(E{\times}{h}_{1}\right)\right)}^{\top }\left(\nu {\times}\overline{V}\right)\mathrm{d}s=0,\hfill \end{align*}$$

since V ∈ H_pc(Ω). Similarly, we show

$$\begin{equation*}{\int }_{{\Omega}}\nabla {\left({h}_{2}^{\top }\nabla \left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x=0.\end{equation*}$$

The second integral occurs to

$$\begin{equation*}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\nabla \left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right)={A}_{2}\nabla \left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)-{h}_{2}{\Delta}\left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)+\nabla \left({h}_{2}^{\top }\nabla \left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\right).\end{equation*}$$

Combining all implies

$$\begin{align*}\hfill \mathcal{A}\left({\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\right)=& {\int }_{{\Omega}}\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\left(\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)+{\Delta}\left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)\right)\mathrm{d}x\hfill \\ \hfill & +{k}^{2}{\int }_{{\Omega}}\left({h}_{2}^{\top }\overline{V}\right)\left(\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}E\right)+{\Delta}\left({h}_{1}^{\top }E\right)\right)\mathrm{d}x+{k}^{2}{\int }_{{\Omega}}{\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)}^{\top }{A}_{2}\overline{V}\mathrm{d}x\hfill \\ \hfill & +{k}^{2}{\int }_{{\Omega}}\left[\nabla \left({h}_{2}^{\top }\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\right)+\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right.\hfill \\ \hfill & {\left.+{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\right]}^{\top }\overline{V}\mathrm{d}x\hfill \\ \hfill & +{k}^{2}{\int }_{{\Omega}}{\left(\mathrm{d}\mathrm{i}\mathrm{v}\left(E{\times}{h}_{1}\right){h}_{2}-\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){\times}{h}_{2}\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x.\hfill \end{align*}$$

Considering

$$\begin{align*}\hfill \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){\times}{h}_{2}\right)=& \;{A}_{2}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)-\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){h}_{2}+\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}\hfill \\ \hfill & +{k}^{2}\left(E{\times}{h}_{1}\right){\times}{h}_{2}+\nabla \left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right){\times}{h}_{2}+\nabla \left({h}_{2}^{\top }\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\right)\hfill \end{align*}$$

and the partial integration

$$\begin{equation*}{\int }_{{\Omega}}{\left(\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){\times}{h}_{2}\right)}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\mathrm{d}x={\int }_{{\Omega}}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}{\left(\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right){\times}{h}_{2}\right)}^{\top }\overline{V}\mathrm{d}x\end{equation*}$$

we arrive at

$$\begin{align*}\hfill \mathcal{A}\left({\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\right)=& \;{k}^{2}{\int }_{{\Omega}}\left({h}_{2}^{\top }\overline{V}\right)\left(\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}E\right)+{\Delta}\left({h}_{1}^{\top }E\right)+\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\right)\mathrm{d}x\hfill \\ \hfill & +{\int }_{{\Omega}}\left({h}_{2}^{\top }\overline{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace V}\right)\left(\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)+{\Delta}\left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)+{k}^{2}\enspace \mathrm{d}\mathrm{i}\mathrm{v}\left(E{\times}{h}_{1}\right)\right)\mathrm{d}x.\hfill \end{align*}$$

Since div E = 0 and curl  curl  E = k² E we have, again by elementary calculations, the identities

$$\begin{align*}\hfill 0& =\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(E{\times}{h}_{1}\right)\right)=\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}E\right)+{\Delta}\left({h}_{1}^{\top }E\right)\mathrm{d}\mathrm{i}\mathrm{v}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)\hfill \\ \hfill 0& =\mathrm{d}\mathrm{i}\mathrm{v}\left(\right.\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\left(\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E{\times}{h}_{1}\right)=\mathrm{d}\mathrm{i}\mathrm{v}\left({A}_{1}\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)+{\Delta}\left({h}_{1}^{\top }\enspace \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}\enspace E\right)+{k}^{2}\enspace \mathrm{d}\mathrm{i}\mathrm{v}\left(E{\times}{h}_{1}\right).\hfill \end{align*}$$

We finally conclude

$$\begin{equation*}\mathcal{A}\left({\left({E}_{1}^{\prime }\right)}_{2}^{\prime },V\right)=\mathcal{A}\left({W}_{1}^{\prime }-{J}_{{h}_{1}}^{\top }{E}_{2}^{\prime }-{J}_{{E}_{2}^{\prime }}{h}_{1}-{J}_{{h}_{2}}^{\top }{E}_{1}-{J}_{{W}_{1}}{h}_{2},V\right)=0\end{equation*}$$

for all V ∈ H_pc(Ω), i.e., ${\left({E}_{1}^{\prime }\right)}_{2}^{\prime }$ is a radiating solution to homogeneous Maxwell's equations. □