Sharp estimate of electric field from a conductive rod and application in EIT

Initially focusing on the mathematics, but not the physics, we consider the following elliptic PDE system in $\mathbb{R}^{2}$:

where $u(\mathbf{x})\in H_{loc}^{1}(\mathbb{R}^{2})$ is a potential field, and $\sigma(\mathbf{x})$ is of the following form:

with $D$ being a bounded domain with a connected complement $\mathbb{R}^{2}\backslash\overline{D}$. $H(\mathbf{x})$ in (1.1) is a (nontrivial) harmonic function in $\mathbb{R}^{2}$, which stands for a background potential. We are mainly concerned with the quantitative properties of the solution $u(\mathbf{x})$ in (1.1) and in particular, its dependence on the geometry of $D$. To that end, we next introduce the rod-geometry of $D$ for our subsequent study. Let $\Gamma_{0}$ be a straight line of length $L\in\mathbb{R}_{+}$ with $\Gamma_{0}=(x_{1},0)$, $x_{1}\in(-L/2,L/2)$. Let $\mathbf{n}:=(0,1)$ and define the two points $P:=(-L/2,0)$ and $Q:=(L/2,0)$. Then the rod $D$ is introduced as $\overline{D}=\overline{D^{a}}\cup\overline{D^{f}}\cup\overline{D^{b}}$, where $D^{f}$ is defined by

The two end-caps $D^{a}$ and $D^{b}$ are two half disks with radius $\delta$ and centering at $P$ and $Q$, respectively. More precisely,

It can be verified that $D$ is of class $C^{1,\alpha}$ for a certain $\alpha\in\mathbb{R}_{+}$. In what follows, we define $S^{c}:=\partial D^{c}=\partial(D^{a}\cup D^{b})$, and $S^{f}:=\partial D^{f}$. Specially, $S^{f}=\Gamma_{1}\cup\Gamma_{2}$, where $\Gamma_{j}$, $j=1,2$ are defined by

Moreover, we shall always use $\mathbf{z}_{x}$ and $\mathbf{z}_{y}$ to signify the projections of $\mathbf{x}\in S^{f}$ and $\mathbf{y}\in S^{f}$ on $\Gamma_{0}$, respectively.

The elliptic PDE system (1.1) describes the perturbation of an electric field $H(\mathbf{x})$ due to the presence of a conductive body $D$. $u(\mathbf{x})$ signifies the electric potential and $\sigma(\mathbf{x})$ signifies the conductivity of the space. The homogeneous background space possesses a conductivity being normalized to be 1, whereas the conductive rod $D$ possesses an inhomogeneous conductivity being $\sigma_{0}$. The perturbed electric potential is $u-H$ and the gradient field $\nabla(u-H)$ is the corresponding electric current.

The conductivity equation (1.1) is a fundamental problem in many existing studies. It is the master equation for the electrical impedance tomography (EIT) which is an important medical imaging modality and it can also find important applications in materials science; see e.g. [5, 6, 15, 43] and the references cited therein. There are rich results in the literature devoted to the quantitative properties of the solution to (1.1) and its geometric relationship to the conductive inclusion $D$. In this work, we shall derive an accurate characterisation of the perturbed electric field $u-H$ and its dependence on the geometry of $D$. There are mainly two motivations for our study as described in what follows.

First, in [4, 9, 14, 34], the authors studied the electric field perturbation from thin/slender structures, which are originated from the study of imaging crack defects in EIT [4, 9, 41]. This is closely related to the current study. Indeed, the geometric setup in the aforementioned works are more general than the one considered in the present article. However, with the specific rod-geometry, we can derive an accurate characterisation of the perturbed electric field and its geometric dependence on $D$. In fact, in our asymptotic formula of the electric field $u-H$ (with respect to $\delta\ll 1$), the leading-order term is exact and can be used to fully decode $D$. This is in a sharp contrast to the existing studies mentioned above, which inevitably involve some qualitative estimates due to the more general geometries. The rod-geometry, though special, also possesses several local features that are worth our investigation, which is the second motivation of our study as described below.

Clearly, the studies mentioned above are mainly concerned with extracting the global geometry of $\partial D$ from the perturbed electric field $u-H$. On the other hand, there are studies on the relationships between the local geometry of $\partial D$ and the perturbed field $u-H$. In fact, there are classical results concerning the singularities of the solution near a boundary corner point [28]. Roughly speaking, if $\partial D$ possesses a corner, then the solution $u$ locally around the corner point can be decomposed into the sum of a regular part and a singular part. Such a qualitative property has been used to establish novel uniqueness and stability results for the Calderón inverse inclusion problem in EIT by a single partial boundary measurement [17, 37, 38]. The Calderón inverse inclusion problem is a longstanding problem in the literature and we shall present more background discussion in Section 4. In [18], the corner singularities of a conductive inclusion have been characterized in terms of the generalised polarisation tensors associated with the electric potential $u$, and the results are directly applied to EIT. Recently, in [39], the authors consider the case that $\partial D$ is smooth, but possesses high-curvature points. In two dimensions, a high-curvature point means that the extrinsic curvature of the boundary curve $\partial D$ at that point is sufficiently large. It is shown in [39] that the quantitative property of $\nabla u$ around the high-curvature point enables one to recover the local part of $\partial D$ around that high-curvature point. However, the sharp curvature-dependence of $\nabla u$ in [39] is established through numerically refining the upper-bound estimate in [24]. As mentioned before, the rod-geometry possesses a few interesting local features that consolidate the numerical study in [39]. First, it is geometrically anisotropic where the two dimensions are of different scales. In fact, the curvature at the two end-caps (i.e. $S^{c}=\partial D^{a}\cup\partial D^{b}$) is $\delta^{-1}\geq 1$, whereas the curvature at the facade part (i.e. $S^{f}$) is $0$. Hence, the rod-geometry, though special, provides rich insights on the curvature dependence of the electric field with respect to the shape of the conductive inclusion. In fact, we shall see that the perturbed electric energy is localized at the two end-caps of the rod. Similar to [39], the result enables us to rigorously justify that one can uniquely determine the conductive rod by a single measurement in EIT. It is emphasized that in three dimensions or in the case that the rod is curved, the situation would become much more complicated. Hence, we mainly consider the case with a straight rod in the two dimensions. Nevertheless, even in such a case, the mathematical analysis is technically involved and highly nontrivial. We shall develop layer potential techniques to tackle the problem. It turns out that the so-called Neumann-Poincaré (NP) operator shall play a critical role in our analysis. We would like to mention that the NP operator and its spectral properties have received considerable attentions recently in the literature due to its important applications in several intriguing fields of mathematical physics, including plasmon resonances and invisibility cloaking [7, 8, 10, 11, 19, 20, 23, 26, 29, 32, 31, 35]. Finally, we would also like to mention in passing that more general rod-geometries were also considered in the literature in different contexts of physical importance [21, 22, 36].

The rest of the paper is organized as follows. In Section 2, we derive several auxiliary results and in Section 3, we present the main results on the quantitative analysis of the solution $u$ to (1.1) with respect to the geometry of the inclusion $D$. Finally, we consider in Section 4 the application of the quantitative result derived in Section 3 to Calderón’s inverse inclusion problem.

Let $G$ be the radiating fundamental solution to the Laplacian $\Delta$ in $\mathbb{R}^{2}$, which is given by

For any bounded Lipschitz domain $B\subset\mathbb{R}^{2}$, we denote by $\mathcal{S}_{B}:L^{2}(\partial B)\rightarrow H^{1}(\mathbb{R}^{2}\setminus\partial B)$ the single-layer potential operator given by

and $\mathcal{K}_{B}^{*}:H^{-1/2}(\partial B)\rightarrow H^{-1/2}(\partial B)$ the Neumann-Poincaré (NP) operator:

where p.v. stands for the Cauchy principle value. In (2.3) and also in what follows, unless otherwise specified, $\nu$ signifies the exterior unit normal vector to the boundary of the domain concerned. It is known that the single-layer potential operator $\mathcal{S}_{B}$ is continuous across $\partial B$ and satisfies the following trace formula

where $\frac{\partial}{\partial\nu}$ stands for the normal derivative and the subscripts $\pm$ indicate the limits from outside and inside of a given inclusion $B$, respectively.

By using the layer-potential techniques, one can readily find the integral solution to (1.1) by

where the density function $\varphi\in H^{-1/2}(\partial D)$ is determined by

Here, the constant $\lambda$ is defined by

In what follows, we always suppose that $\delta\ll 1$. We shall present some asymptotic expansions of the Neumann-Poincaré operator with respect to $\delta$. Recalling that $\partial D=S^{a}\cup\overline{S^{f}}\cup S^{b}$, we decompose the Neumann-Poincaré operator into several parts accordingly. To that end, we introduce the following boundary integral operator:

It is obvious that $\mathcal{K}_{\mathcal{S},\mathcal{S}^{\prime}}$ is a bounded operator from $L^{2}(\mathcal{S})$ to $L^{2}(\mathcal{S}^{\prime})$. For the case $\mathcal{S}=S^{c}$, we mean $S^{c}=S^{a}\cup S^{b}$. In what follows, we define $S_{1}^{a}$ and $S_{1}^{b}$ by

For the subsequent use, we also introduce the following regions:

Define $\tilde{\phi}(\tilde{\mathbf{x}}):=\phi(\mathbf{x})$, where $\mathbf{x}\in S^{a},S^{b}$ and $\tilde{\mathbf{x}}\in S_{1}^{a},S_{1}^{b}$.

We can prove the following result on the asymptotic expansion of the Neumann-Poincaré operator.

Recall that $u$ is represented by (2.5). We first derive some asymptotic properties of the density function $\varphi$ in (2.6). Let $\mathbf{z}_{x}\in\Gamma_{0}$ be defined by

One has the following asymptotic expansion for $H$ around $\Gamma_{0}$:

for $\mathbf{x}\in S^{f}$ and $\tilde{\mathbf{x}}\in S_{1}^{f}$. Similarly, one has

for $\mathbf{x}\in S^{a}$ and $\tilde{\mathbf{x}}\in S_{1}^{a}$. Moreover,

for $\mathbf{x}\in S^{b}$ and $\tilde{\mathbf{x}}\in S_{1}^{b}$.

We now can show the following asymptotic result.