The asymptotic behaviour of weak solutions to the forward problem of electrical impedance tomography on unbounded three‐dimensional domains

The forward problem of electrical impedance tomography on unbounded domains can be studied by introducing appropriate function spaces for this setting. In this paper we derive the point‐wise asymptotic behaviour of weak solutions to this problem in the three‐dimensional case. Copyright © 2008 John Wiley & Sons, Ltd.     

The impedance boundary-value problem of diffraction by a strip

We consider the impedance boundary-value problem for the Helmholtz equation originated by the problem of wave diffraction by an infinite strip with imperfect conductivity. The two possible different situations of real and complex wave numbers are considered. Bessel potential spaces are used to deal with the problem, and the identification of corresponding operators of single and double layer potentials allow a reformulation of the problem into a system of integral equations. The well-posedness of the problem is obtained for a set of impedance parameters (and wave numbers), after the incorporation of some compatibility conditions on the data. At the end, an improvement of the regularity of the solution is derived for the same set of parameters previously considered.     

The geometrical problem of electrical impedance tomography in the disk

The geometrical problem of electrical impedance tomography consists of recovering a Riemannian metric on a compact manifold with boundary from the Dirichlet-to-Neumann operator (DNoperator) given on the boundary. We present a new elementary proof of the uniqueness theorem: A Riemannian metric on the two-dimensional disk is determined by its DN-operator uniquely up to a conformal equivalence. We also prove an existence theorem that describes all operators on the circle that are DN-operators of Riemannian metrics on the disk.     

An inverse spectral problem for the Laplacian

We propose an algorithm for the recovery of a potential from the knowledge of the eigenvalues of the Laplacian operator and the traces of its eigenfunctions. This inverse spectral problem is solved by recasting the operator as an infinite matrix and using transition matrices together with spectral projections on the boundary.     

Stepwise solution to an inverse problem for the radiative transfer equation as applied to tomography

An X-ray tomography problem is formulated and analyzed within the framework of a mathematical model based on the polychromatic stationary radiative transfer equation with no collision integral. It is assumed that the outgoing radiation density is only given, and the task is to find the surface of an internal inclusion on whose boundary the coefficients of the equation may have jump discontinuities. The uniqueness of the solution is proved, and the corresponding solution algorithm is outlined. A feature of this work is that the research technique is local in character. This makes it possible to use only some of the available data, and the procedure can be stopped at an intermediate stage of the reconstruction, which can be useful in applications.     

Identification of an inclusion in multifrequency electric impedance tomography

The multifrequency electrical impedance tomography is considered 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 since in general infinitely many input currents are needed to obtain the uniqueness in the determination of the conductivity.     

The two-dimensional Prouhet-Tarry-Escott problem

In this paper we generalize the Prouhet-Tarry-Escott problem (PTE) to any dimension. The one-dimensional PTE problem is the classical PTE problem. We concentrate on the two-dimensional version which asks, given parameters n,k∈N, for two different multi-sets {(x₁,y₁),…,(xn,yn)}, of points from Z² such that for all d,j∈{0,…,k} with j?d. We present parametric solutions for n∈{2,3,4,6} with optimal size, i.e., with k=n−1. We show that these solutions come from convex 2n-gons with all vertices in Z² such that every line parallel to a side contains an even number of vertices and prove that such convex 2n-gons do not exist for other values of n. Furthermore we show that solutions to the two-dimensional PTE problem yield solutions to the one-dimensional PTE problem. Finally, we address the PTE problem over the Gaussian integers.     

Reconstruction and uniqueness of an inverse scattering problem with impedance boundary

Consider an inverse scattering problem for a scatterer D ⊂ R³ with impedance type boundary condition. By defining the scatterer shape in a completely new way, we give a constructive method to recover the scatterer shape with unknown impedance coefficient. The uniqueness for this inverse problem is also obtained.     

Boundary feedback stabilization of the transmission problem of Naghdi's model

We consider the stabilization of the transmission problem of Naghdi's model by boundary feedbacks where the model has a middle surface of any shape. The exponential decay rate for the problem is established under some checkable geometric conditions on the middle surface.     

On a boundary value problem for integro-differential equations on the halfline

The following boundary value problem

     

A boundary value problem for Bitsadze equation in the unit disc

A boundary value problem for the Bitsadze equation

$\frac{{\partial ^2 }}{{\partial \bar z^2 }}u(x,y) \equiv \frac{1}{4}\left( {\frac{\partial }{{\partial x}} + i\frac{\partial }{{\partial y}}} \right)^2 u(x,y) = 0$

in the interior of the unit disc is considered. It is proved that the problem is Noetherian and its index is calculated, and solvability conditions for the non-homogeneous problem are proposed. Some solutions of the homogeneous problem are explicitely found.

     

The statement and numerical solution of an optimization problem in X-ray tomography

A nonclassical problem is considered for the transport equation with coefficients depending on the energy of radiation. The task is to find the discontinuity surfaces for the coefficients of the equation from measurements of the radiation flux leaving the medium. For this tomography problem, an optimization problem is stated and numerically analyzed. The latter consists in determining the radiation energy that ensures the best reconstruction of the unknown medium. A simplified optimization problem is solved analytically.     

The indicator of contact boundaries for an integral geometry problem

We pose and study a rather particular integral geometry problem. In the two-dimensional space we consider all possible straight lines that cross some domain. The known data consist of the integrals over every line of this kind of an unknown piecewise smooth function that depends on both points of the domain and the variables characterizing the lines. The object we seek is the discontinuity curve of the integrand. This problem arose in the author’s previous research in X-ray tomography. In essence, it is a generalization of one mathematical aspect of flaw detection theory, but seems of interest in its own right. The main result of this article is the construction of a special function that can be unbounded only near the required curve. Precisely for this reason we call the function the indicator of contact boundaries. A uniqueness theorem for the solution follows rather easily from the property of indicators.     

Multiple symmetric positive solutions for a second order boundary value problem

For the second order boundary value problem, , , , where growth conditions are imposed on which yield the existence of at least three symmetric positive solutions.

     

Existence of a positive solution of a fourth-order boundary value problem

In this paper, the existence of a positive solution of the boundary value problem of the following fourth-order nonlinear differential equation:

is discussed.     

Limit behaviour of solutions to boundary value problem with equivalued surface

In this paper, we discuss the limit behaviour of solutions to boundary value problem with equivalued surface with m inner holes and give a different proof from that of Li Ta-tsien et al. (1998).     

On the numerical solution of Plateau's problem

The present paper is dedicated to the numerical computation of minimal surfaces by the boundary element method. Having a parametrization γ of the boundary curve over the unit circle at hand, the problem is reduced to seeking a reparametrization κ of the unit circle. The Dirichlet energy of the harmonic extension of γ○κ has to be minimized among all reparametrizations. The energy functional is calculated as boundary integral that involves the Dirichlet-to-Neumann map. First and second order necessary optimality conditions of the underlying minimization problem are formulated. Existence and convergence of approximate solutions is proven. An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented.