We derive bounds on the volume of an inclusion in a body in two or three dimensions when the conductivities of the inclusion and the surrounding body are complex and assumed to be known. The bounds are derived in terms of average values of the electric field, current, and certain products of the electric field and current. All of these average values are computed from a single electrical impedance tomography measurement of the voltage and current on the boundary of the body. Additionally, the bounds are tight in the sense that at least one of the bounds gives the exact volume of the inclusion for certain geometries and boundary conditions.
This paper concerns the inverse problem of determining a planar conductivity inclusion. Our aim is to analytically recover from the generalized polarization tensors (GPTs), which can be obtained from exterior measurements, a homogeneous inclusion with arbitrary constant conductivity. The primary outcome of recovering a homogeneous inclusion is an inversion formula in terms of the GPTs for conformal mapping coefficients associated with the inclusion. To prove the formula, we establish matrix factorizations for the GPTs.
Using a distributed representation formula of the Gateaux derivative of the Dirichlet to Neumann map with respect to movements of a polygonal conductivity inclusion, [11], we extend the results obtained in [8] proving global Lipschitz stability for the determination of a polygonal conductivity inclusion embedded in a layered medium from knowledge of the Dirichlet to Neumann map.
We consider the problem of determining a polyhedral conductivity inclusion embedded in a homogeneous isotropic medium from boundary measurements. We prove global Lipschitz stability for the polyhedral inclusion from the local Dirichlet-to-Neumann map extending in a highly nontrivial way the results obtained in [18] and [19] in the two-dimensional case to the three-dimensional setting.
A conductivity inclusion, inserted in a homogeneous background, induces a perturbation in the background potential. This perturbation admits a multipole expansion whose coefficients are the so-called generalized polarization tensors (GPTs). GPTs can be obtained from multistatic measurements. As a modification of GPTs, the Faber polynomial polarization tensors (FPTs) were recently introduced in two dimensions. In this study, we design two novel analytical non-iterative methods for recovering the shape of a simply connected inclusion from GPTs by employing the concept of FPTs. First, we derive a…
We consider an insulated conductivity model with two neighboring inclusions of $m$-convex shapes in $\mathbb{R}^{d}$ when $m\geq2$ and $d\geq3$. We establish the pointwise gradient estimates for the insulated conductivity problem and capture the gradient blow-up rate of order $\varepsilon^{-1/m+β}$ with $β=[-(d+m-3)+\sqrt{(d+m-3)^{2}+4(d-2)}]/(2m)\in(0,1/m)$, as the distance $\varepsilon$ between these two insulators tends to zero. In particular, the optimality of the blow-up rate is also demonstrated for a class of axisymmetric $m$-convex inclusions.
We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium. The governing equation may be degenerate of $p-$Laplace type, with $1<p \leq N$. We prove optimal $L^\infty$ estimates for the blow-up of the gradient of the solution as the distance between the inclusions tends to zero.
We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear $p$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $\operatorname{div}(σ\lvert\nabla u\rvert^{p-2}\nabla u)=0$ where the measurable conductivity $σ\colonΩ\to[0,\infty]$ is zero or infinity in large sets and $1<p<\infty$.
In this paper, the insulated conductivity model with two touching or close-to-touching inclusions is considered in $\mathbb{R}^{d}$ with $d\geq3$. We establish the pointwise upper bounds on the gradient of the solution for the generalized $m$-convex inclusions under these two cases with $m\geq2$, which show that the singular behavior of the gradient in the thin gap between two inclusions is described by the first non-zero eigenvalue of an elliptic operator of divergence form on $\mathbb{S}^{d-2}$. Finally, the sharpness of the estimates is also proved for two touching axisymmetric insulators, …
In the perfect conductivity problem, it is interesting to study whether the electric field can become arbitrarily large or not, in a narrow region between two adjacent perfectly conducting inclusions. In this paper, we show that the relative convexity of two adjacent inclusions plays a key role in the blowup analysis of the electric field and find some new phenomena. By energy method, we prove the boundedness of the gradient of the solution if two adjacent inclusions fail to be locally relatively strictly convex, namely, if the top and bottom boundaries of the narrow region are partially "flat…
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb R^n$, for $n \ge 3$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. We established in a recent paper optimal gradient estimates for a class of inclusions including balls. In this paper, we prove such gradient estimates for general strictly convex inclusions. Unlike the perfect conductivity problem, the estimates depend on the principal curvatures of the inclusions, and we show that these estimates are characterized by the first non-ze…
Ellipsoidal characterization of neutral inclusions for imperfect bonding of high-conductivity type
Gunma University（群马大学）；Korea Institute for Advanced Study（韩国高等研究院）；Inha University（仁荷大学）；Zhejiang University of Technology（浙江工业大学）；Tohoku University（东北大学）
When a convex perfectly conducting inclusion is closely spaced to the boundary of the matrix domain, a bigger convex domain containing the inclusion, the electric field can be arbitrary large. We establish both the pointwise upper bound and the lower bound of the gradient estimate for this perfect conductivity problem by using the energy method. These results give the optimal blow-up rates of electric field for conductors with arbitrary shape and in all dimensions. A particular case when a circular inclusion is close to the boundary of a circular matrix domain in dimension two is studied earli…
The multifrequency electrical impedance tomography is considered in order to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous stability estimates. Our results in this paper are quite surprising in inverse conductivity problem si…
We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension which characterize the singular behavior of the electric field as the distance between the inclusions goes to zero.
In composite materials, the inclusions are frequently spaced very closely. The electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large. In this paper, we establish an asymptotic formula of the electric field in the zone between two spherical inclusions with different radii in three dimensions. An explicit blowup factor relying on radii is obtained, which also involves the digamma function and Euler-Mascheroni constant, and so the role of inclusions' radii played in such blowup analysis is identified.
In the perfect conductivity problem of composite material, the electric field concentrates in a narrow region in between two inclusions and always becomes arbitrarily large when the distance between inclusions tends to zero. To characterize such singular behavior, we capture the leading term of the gradient and reveal that the blow-up rates are determined by their relative convexity of the two adjacent inclusions. On the other hand, a blow-up factor, which is a linear functional of boundary data, is found to determine the blow-up will occur or not.
In high-contrast composite materials, the electric field concentration is a common phenomenon when two inclusions are close to touch. It is important from an engineering point of view to study the dependence of the electric field on the distance between two adjacent inclusions. In this paper, we derive the upper and lower bounds of the gradient of solutions to the conductivity problem where two perfectly conducting inclusions are located very close to each other. To be specific, we extend the known results of Bao-Li-Yin (ARMA 2009) in two folds: First, we weaken the smoothness of the inclusion…
We consider a large number of randomly dispersed spherical, identical, perfectly conducting inclusions (of infinite conductivity) in a bounded domain. The host medium's conductivity is finite and can be inhomogeneous. In the dilute limit, with some boundedness assumption on a large number (proportional to the global volume fraction raised to the power of -1/2) of marginal probability densities, we prove convergence in H^1 norm of the expectation of the solution of the steady state heat equation, to the solution of an effective medium problem, where the conductivity is given by the Clausius-Mos…
We investigate the homogenization of inclusions of infinite conductivity, randomly stationary distributed inside a homogeneous conducting medium. A now classical result by Zhikov shows that, under a logarithmic moment bound on the inter-particle distance, an effective model with finite homogeneous conductivity exists. Relying on ideas from network approximation, we provide a relaxed criterion ensuring homogenization. Several examples not covered by the previous theory are discussed.
In the perfect conductivity problem of composite material, the gradient of solutions can be arbitrarily large when two inclusions are located very close. To characterize the singular behavior of the gradient in the narrow region between two inclusions, we capture the leading term of the gradient and give a fairly sharp description of such asymptotics.
In this paper we derive rigorously the derivative of the Dirichlet to Neumann map and of the Neumann to Dirichlet map of the conductivity equation with respect to movements of vertices of triangular conductivity inclusions. We apply this result to formulate an optimization problem based on a shape derivative approach.
This work considers the inclusion detection problem of electrical impedance tomography with stochastic conductivities. It is shown that a conductivity anomaly with a random conductivity can be identified by applying the Factorization Method or the Monotonicity Method to the mean value of the corresponding Neumann-to-Dirichlet map provided that the anomaly has high enough contrast in the sense of expectation. The theoretical results are complemented by numerical examples in two spatial dimensions.
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. It was known that the optimal blow up rate in dimension $n = 2$ is of order $\varepsilon^{-1/2}$. It has recently been proved that in dimensions $n \ge 3$, an upper bound of the gradient is of order $\varepsilon^{-1/2 + β}$ for some $β> 0$. On the other hand, optimal values of $β$ have not been identified. In this paper, we prove that when the inclusions are balls, the optimal valu…
We investigate a generalization of Calderón's problem of recovering the conductivity coefficient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation with p strictly between one and infinity, which reduces to the standard conductivity equation when p equals two, and to the p-Laplace equation when the conductivity is constant. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity may be zero or infinity in la…
We consider well-posedness of the boundary value problem in presence of an inclusion with complex conductivity $k$. We first consider the transmission problem in $\mathbb{R}^d$ and characterize solvability of the problem in terms of the spectrum of the Neumann-Poincaré operator. We then deal with the boundary value problem and show that the solution is bounded in its $H^1$-norm uniformly in $k$ as long as $k$ is at some distance from a closed interval in the negative real axis. We then show with an estimate that the solution depends on $k$ in its $H^1$-norm Lipschitz continuously. We finally s…
We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as $\varepsilon$, the distance between inclusions, approaches to $0$. An upper bound for the blow up rate was proved to be of order $\varepsilon^{-1/2}$. The upper bound was known to be sharp in dimension $n = 2$. However, whether this upper bound is sharp in dimension $n \ge 3$ has remained open. In this paper, we improve the upper bound in dimension $n \ge 3$ to be of order $\varepsilon^{-1/2 + β}$, for some $β> 0$.
We consider a gradient estimate for a conductivity problem whose inclusions are two neighboring insulators in three dimensions. When inclusions with an extreme conductivity (insulators or perfect conductors) are closely located, the gradient can be concentrated in between inclusions and then becomes arbitrarily large as the distance between inclusions approaches zero. The gradient estimate in between insulators in three dimensions has been regarded as a challenging problem, while the optimal blow-up rates in terms of the distance were successfully obtained for the other extreme conductivity pr…
We consider an inverse boundary value problem for the heat equation with a nonsmooth coefficient of conductivity which models the displacement of a moving body inside a nonhomogeneous background. We prove the uniqueness of the moving inclusion from the knowledge of the Dirichlet-to-Neumann operator by using a dynamical probe method.
We consider the problem of reconstructing of the boundary of an unknown inclusion together with its conductivity from the localized Dirichlet-to-Neumann map. We give an exact reconstruction procedure and apply the method to an inverse boundary value problem for the system of the equations in the theory of elasticity.
Imposing either Dirichlet or Neumann boundary conditions on the boundary of a smooth bounded domain $Ω$, we study the perturbation incurred by the voltage potential when the conductivity is modified in a set of small measure. We consider $\left(γ_{n}\right)_{n\in\mathbb{N}}$, a sequence of perturbed conductivity matrices differing from a smooth $γ_{0}$ background conductivity matrix on a measurable set well within the domain, and we assume $\left(γ_{n}-γ_{0}\right)γ_{n}^{-1}\left(γ_{n}-γ_{0}\right)\to0$ in $L^{1}(Ω)$. Adapting the limit measure, we show that the general representation formula …
The Neumann-Poincaré operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the Neumann-Poincaré operator was developed in two dimensions based on geometric function theory. In this paper, we investigate geometric properties of composite materials by using this series expansion. In particular, we obtain explicit formulas for the polarization tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary…
We deal with the problem of determining the shape of an inclusion embedded in a homogenous background medium. The multifre-quency electrical impedance tomography is used to image the inclusion. For different frequencies, a current is injected at the boundary and the resulting potential is measured. It turns out that the potential solves an elliptic equation in divergence form with discontinuous leading coefficient. For this inverse problem we aim to establish a logarithmic type stability estimate. The key point in our analysis consists in reducing the original problem to that of determining an…
We consider a boundary value problem for the conductivity equation in a bounded domain containing an inclusion which is nearly touching to the domain's boundary. We assume that the domain and the inclusion are disks with conductivity jump on the boundary of the inclusion. By using the layer potential technique and adopting the bipolar coordinates, we derive the asymptotic formulas which explicitly describe the gradient blow-up of the solution as the distance between the inclusion and the domain's boundary tends to zero. It turns out that the gradient blow-up term can be identified with the ele…
