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The goal of this article is the analyses of inverse source problem and null controllability for a heat equation with multipolar inverse-square potentials: $ \small {\sum _{j=1}^m}\frac {\mu _j}{|x-x^j|^2}, $ ∑ j = 1 m μ j | x − x j | 2 , where $ x^1, x^2,\ldots,x^m $ x 1 , x 2 , ... , x m are different points, defined in a bounded $ C^2 $ C 2 domain $ \Omega \subset \mathbb R^n $ Ω ⊂ R n , and $ \mu _1,\ldots,\mu _m $ μ 1 , ... , μ m are real constants. More precisely, we show that for any $ \mu _j\leq \frac {(n-2)^2}{4},\ j=1,\ldots,m, $ μ j ≤ (n − 2) 2 4 , j = 1 , ... , m , we can obtain a Lipschitz stability result in inverse source problems and we will show that the system is null controllable. Our proofs are mainly based on Carleman estimates, and due to multiple singularities, we give a new Carleman estimate by choosing an appropriate weight function. [ABSTRACT FROM AUTHOR]

For an inverse coefficient problem of determining a state‐varying factor in the corresponding Hamiltonian for a mean field game system, we prove the global Lipschitz stability by spatial data of one component and interior data in an arbitrarily chosen subdomain over a time interval. The proof is based on Carleman estimates with different norms. [ABSTRACT FROM AUTHOR]

In a bounded domain $ \Omega \subset \mathbb{R}^d $, $ d\geq 1 $, over a time interval $ (0,T) $, we consider mean field game equations whose principal coefficients depend on the time and the state variables with a general Hamiltonian. We attach a non-zero Robin boundary condition. We first prove the Lipschitz stability in $ \Omega \times (\varepsilon, T- \varepsilon) $ with given $ \varepsilon>0 $ for the determination of the solutions by the associated Dirichlet data on an arbitrarily chosen subboundary of $ \partial \Omega $. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at an intermediate time. [ABSTRACT FROM AUTHOR]

We study an inverse problem of determining a time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines the time-dependent damping coefficient and potential uniquely. [ABSTRACT FROM AUTHOR]

We study an inverse problem for variable coefficient fractional parabolic operators of the form (∂t - div(A(x)▽x))s + q(x, t) for s ∈ (0, 1) and show the unique recovery of q from exterior measured data. Similar to the fractional elliptic case, we use a Runge-type approximation argument, which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift. [ABSTRACT FROM AUTHOR]

The aim of this paper is to survey some recent works on inverse problems for stochastic partial differential equations (SPDEs for short), which were solved by means of Carleman estimate. For simplicity, we focus on two typical SPDEs, i.e, stochastic hyperbolic equations and stochastic parabolic equations. Our particular concerns are those inverse problems which are genuinely stochastic and therefore cannot be reduced to any deterministic ones. Also, some open problems are presented. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering problem involving the timedomain elastic wave equations in a bounded d-dimensional domain. First, an explicit formula for the density reconstruction is established by means of the Dirichlet-to-Neumann operator. The reconstruction is mainly based on the modified boundary control method and complex geometric optics solutions for the elastic wave. Next, the stable observability is obtained by a Carleman estimate. Finally, the stability for the density is presented by the connect operator. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying two multiobjective problems for stochastic degenerate parabolic equations. The first one is a hierarchical control problem, in which the controls are classified into a pair of leaders and a pair of followers. For each pair of leaders, a Nash equilibrium is searched for a noncooperative game problem. The aim of the pair of leaders is to achieve null controllability of the system. The other multiobjective problem is an inverse initial problem under a Nash equilibrium strategy. In contrast to the classical inverse initial problem, an optimization problem for stochastic degenerate parabolic equations is first investigated. Then, the conditional stability of determining initial information is derived through terminal observation. [ABSTRACT FROM AUTHOR]

The first globally convergent numerical method for a coefficient inverse problem for the Riemannian radiative transfer equation (RRTE) is constructed. This is a version of the so-called convexification method, which has been pursued by this research group for a number of years for some other CIPs for PDEs. Those PDEs are significantly different from the RRTE. The presence of the Carleman weight function in the numerical scheme is the key element which insures the global convergence. Convergence analysis is presented along with the results of numerical experiments, which confirm the theory. RRTE governs the propagation of photons in the diffuse medium in the case when they propagate along geodesic lines between their collisions. Geodesic lines are generated by the spatially variable dielectric constant of the medium. [ABSTRACT FROM AUTHOR]

For linearized Navier–Stokes equations, we consider an inverse source problem of determining a spatially varying divergence-free factor. We prove the global Lipschitz stability by interior data over a time interval and velocity field at t 0 > 0 over the spatial domain. The key machinery are Carleman estimates for the Navier–Stokes equations and the operator rot. [ABSTRACT FROM AUTHOR]

We consider stability in an inverse problem of determining the material coefficient matrix for a coupled system that describes acoustic interactions, by the Riemannian geometrical approach. The stability is proved by the Carleman estimates and observability inequalities. [ABSTRACT FROM AUTHOR]

A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problem with the initial condition being the delta function concentrated at a single plane (i.e. the plane wave). A certain associated operator is written in finite differences with respect to two out of three spatial variables, i.e. "partial finite differences". The grid step size is bounded from below by a fixed number. A Carleman estimate is applied to obtain, for the first time, a Hölder stability estimate for this problem. Another new result is an estimate from below the amplitude of the first term of the expansion of the solution of the forward problem near the characteristic wedge. [ABSTRACT FROM AUTHOR]

In this paper, we retrieve a stationary potential in the strongly coupled Schrödinger equations from either boundary or internal measurements. Lipschitz stabilities are obtained by means of new Carleman estimates for the system with different observations, respectively. [ABSTRACT FROM AUTHOR]

Motivated by the application to helioseismology, we demonstrate uniqueness and stability for a class of inverse problems of the weighted transport equation. Using 퐴-analytic functions, this inverse problem is expressed as a Cauchy problem. In this form, we show that, for a finite even trigonometric polynomial weight function, the resulting system is well-conditioned numerically and permits a Carleman-like estimate with boundary terms. [ABSTRACT FROM AUTHOR]

In this paper, we prove a stability estimate of the conductivity parameters identification problem in cardiac electrophysiology. The propagation of the electrical wave in the heart is described by the monodomain model coupled to an elliptic equation describing the diffusion of the electrical wave in the whole body. Our result concerns both heart and torso conductivity parameters. The main difficulty that we solve in this paper is related to the transmission conditions between the heart and the torso. We first, establish Carleman estimates for the coupled heart-torso system. Then, using these estimates and the Bukhgeim and Klibanov approach, we prove a Lipschitz stability estimate of cardiac and torso conductivity parameters. [ABSTRACT FROM AUTHOR]

This paper deals with the global uniqueness of an inverse problem for stochastic plate equations with structural damping. The key point is the Carleman estimate for the fourth‐order stochastic plate operators dyt−ρΔytdt+Δ2ydt$$ d{y}_t-\rho \Delta {y}_t dt+{\Delta}^2 ydt $$. For this aim, a weighted pointwise identity for a fourth‐order stochastic plate operator is established, via which we obtained the desired Carleman estimate for the corresponding stochastic plate equation with structural damping. [ABSTRACT FROM AUTHOR]

In this paper, we consider a transmission wave equation in N embedded domains with multiple interfaces of discontinuous coefficients in ℝ2. We study the global stability in determining the source term from a one-measurement data of wavefield velocity in a subboundary over a time interval. We prove the stability estimate for this inverse source problem by a combination of the local hyperbolic/elliptic Carleman estimates and the Fourier-Bros-Iagolniter transformation. Our method could be generalized to general dimensions since the weight functions and Carleman estimates are independent of the dimensions. [ABSTRACT FROM AUTHOR]

We develop an efficient and convergent numerical method for solving the inverse problem of determining the potential of nonlinear hyperbolic equations from lateral Cauchy data. In our numerical method we construct a sequence of linear Cauchy problems whose corresponding solutions converge to a function that can be used to efficiently compute an approximate solution to the inverse problem of interest. The convergence analysis is established by combining the contraction principle and Carleman estimates. We numerically solve the linear Cauchy problems using a quasi-reversibility method. Numerical examples are presented to illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

To compute the spatially distributed dielectric constant from the backscattering computationally simulated ane experimentally collected data, we study a coeffcient inverse problem for a 1D hyperbolic equation. To solve this inverse problem, we establish a new version of the Carleman estimate and then employ this estimate to construct a cost functional, which is strictly convex on a convex bounded set of an arbitrary diameter in a Hilbert space. The strict convexity property is rigorously proved. This result is called the convexification theorem and it is the central analytical result of this paper. Minimizing this cost functional by the gradient descent method, we obtain the desired numerical solution to the coeffcient inverse problems. We prove that the gradient descent method generates a sequence converging to the minimizer starting from an arbitrary point of that bounded set. We also establish a theorem confirming that the minimizer converges to the true solution as the noise in the measured data and the regularization parameter tend to zero. Unlike the methods, which are based on the optimization, our convexification method converges globally in the sense that it delivers a good approximation of the exact solution without requiring a good initial guess. Results of numerical studies of both computationally simulated and experimentally collected data are presented. [ABSTRACT FROM AUTHOR]

We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider Carleman estimates for the damped fourth-order plate operators ∂ t t − ρ ∂ t Δ + Δ 2 in a bounded smooth domain with Robin boundary conditions. Because the appearance of such kind of structural damping, the speed of propagation for solutions to the plate equation is infinite and the corresponding properties of the solution similar to heat equations, and is significantly different from that of the usual plate equations without damping. As applications, we consider two types of inverse problems in determining source terms for the plate equation with structural damping. The time-dependent measurements can be restricted to an arbitrary small sub-domain or arbitrary sub-boundary, respectively. Under some assumptions on the regularity of the solutions and coefficients, we prove the global stability results for these inverse problems. [ABSTRACT FROM AUTHOR]

We propose to combine the Carleman estimate and the Newton method to solve an inverse problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse problem for determination of initial condition is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed. [ABSTRACT FROM AUTHOR]

We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain. [ABSTRACT FROM AUTHOR]

In this paper, we study inverse source problems for the heat equation with an inverse-square potential localized on the boundary of a smooth bounded domain, and with a locally distributed observation. We first derive an improved Carleman estimate of that obtained by Cazacu [SIAM J Control Optim. 2014;52:2055–2089] and then use it to prove the Lipschitz stability result for the inverse source problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem for the Mindlin–Timoshenko plate system, which is a strongly coupled two-dimensional system consisting of a wave equation and a system of isotropic elasticity, that arises in modeling plate vibrations especially at high frequencies and thicker plates. More precisely, we prove the global uniqueness of recovering the plate density from a single boundary measurement under appropriate geometrical assumptions. Our approach relies on diagonalizing the principal part of the system and making it a system of wave-like equations. [ABSTRACT FROM AUTHOR]

The goal of this paper is to analyze control properties of the parabolic equation with variable coefficients in the principal part and with a singular inverse-square potential: ∂ t u (x , t) − d i v (p (x) ∇ u (x , t)) − (μ / | x | 2) u (x , t) = f (x , t). Here μ is a real constant. It was proved in the paper of Goldstein and Zhang (2003) that the equation is well-posedness when 0 ≤ μ ≤ p 1 (n − 2) 2 / 4 , and in this paper, we mainly consider the case 0 ≤ μ < (p 1 2 / p 2) (n − 2) 2 / 4 , where p 1 , p 2 are two positive constants which satisfy: 0 < p 1 ≤ p (x) ≤ p 2 , ∀ x ∈ Ω ¯ . We extend the specific Carleman estimates in the paper of Ervedoza [Control and stabilization properties for a singular heat equation with an inverse-square potential. Commun Partial Differ Equ. 2008;33:1996–2019] and Vancostenoble [Lipschitz stability in inverse source problems for singular parabolic equations. Commun Partial Differ Equ. 2011;36(8):1287–1317] to the system. We obtain that we can control the equation from any non-empty open subset as for the heat equation. Moreover, we will study the case μ > p 2 (n − 2) 2 / 4. We consider a sequence of regularized potentials μ / (| x | 2 + ϵ 2) , and prove that we cannot stabilize the corresponding systems uniformly with respect to ϵ > 0.. [ABSTRACT FROM AUTHOR]

In this paper, we address an inverse problem of reconstructing a space-dependent semilinear coefficient in the tumor growth model described by a system of semilinear partial differential equations (PDEs) with Dirichlet boundary condition using boundary-type measurement. We establish a new higher order weighted Carleman estimate for the given system with the help of Dirichlet boundary conditions. By deriving a suitable regularity of solutions for this nonlinear system of PDEs and the new Carleman estimate, we prove Lipschitz-type stability for the tumor growth model. [ABSTRACT FROM AUTHOR]

Consider an inverse problem of determining two stochastic source functions and the initial status simultaneously in a stochastic thermoelastic system, which is constituted of two stochastic equations of different types, namely a parabolic equation and a hyperbolic equation. To establish the conditional stability for such a coupling system in terms of some suitable norms revealing the stochastic property of the governed system, we first establish two Carleman estimates with regular weight function and two large parameters for stochastic parabolic equation and stochastic hyperbolic equation, respectively. By means of these two Carleman estimates, we finally prove the conditional stability for our inverse problem, provided the source in the elastic equation be known near the boundary and the solution be in an a priori bounded set. Due to the lack of information about the time derivative of wave field at the final time, the stability index with respect to the wave field at final time is found to be halved, which reveals the special characteristic of our inverse problem for the coupling system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a stability result regarding the inverse problem of retrieving the damping coefficient in Kawahara equation. We first establish an internal Carleman estimate for the linearized problem with the help of Dirichlet-Neumann type boundary conditions. Using the obtained Carleman estimate and the regularity of solutions for the Kawahara equation, we prove the Lipschitz type stability and uniqueness of the considered inverse problems.

The one-dimensional Saint-Venant equation describes unsteady water flow in channels and is derived from the one-dimensional Euler equation by imposing several physical assumptions. In this paper, we consider the linearized and simplified equation in the one-dimensional case featuring a mixed derivative term and prove the global Lipschitz stability of the inverse source problem via the global Carleman estimate. [ABSTRACT FROM AUTHOR]

In this paper, we consider a null controllability and an inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity. We construct two special weight functions to establish two Carleman estimates for the whole stochastic Grushin operator with singular potential by a weighted identity method. One is for the backward stochastic Grushin equation with singular weight function. We then apply it to prove the null controllability for stochastic Grushin equation for any T and any degeneracy γ > 0, when our control domain touches the degeneracy line {x = 0}. In order to study the inverse source problem of determining two kinds of sources simultaneously, we prove the other Carleman estimate, which is for the forward stochastic Grushin equation with regular weight function. Based on this Carleman estimate, we obtain the uniqueness of the inverse source problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a seventh-order generalized Korteweg–de Vries (GKdV) equation and study the boundary stability results concerning the inverse problem of recovering a space-dependent source term. We establish a new boundary Carleman estimate for the seventh-order linear operator with the Dirichlet–Neumann type boundary conditions. Using this crucial estimate along with regularity result of the nonlinear GKdV equation, we establish a Lipschitz stability estimate of GKdV equation. [ABSTRACT FROM AUTHOR]

In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems. [ABSTRACT FROM AUTHOR]

This paper concerns an inverse problem for a stochastic degenerate wave equation. This inverse problem aims to determine a source term and an initial displacement and an initial velocity from the boundary observation and the final time observation. More precisely, we will prove a global uniqueness for our inverse problem by a suitable Carleman estimate for the stochastic degenerate wave equation. [ABSTRACT FROM AUTHOR]

Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability estimate for an inverse problem of determining two spatially varying zeroth order non-diagonal elements of a coefficient matrix in a one-dimensional fractional diffusion system of half order in time. The proof relies on the conversion of the fractional diffusion system to a system of order 4 in the space variable and the Carleman estimate. [ABSTRACT FROM AUTHOR]

In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. Firstly, we state the stability results for our inverse coefficient problem. Secondly, we establish and prove two Carleman type inequalities both for the solutions and for the unknown coefficients. Finally, we complete the proofs of the stability results in terms of the above Carleman estimates. [ABSTRACT FROM AUTHOR]

We consider a fractional radiative transport equation, where the time derivative is of half order in the Caputo sense. By establishing Carleman estimates, we prove the global Lipschitz stability in determining the coefficients of the one-dimensional time-fractional radiative transport equation of half-order. [ABSTRACT FROM AUTHOR]

In this paper, we study two inverse problems for stochastic parabolic equations with additive noise. One is to determinate the history of a stochastic heat process and the random heat source simultaneously by the observation at the final time 𝑇. For this inverse problem, we obtain a conditional stability result. The other one is an inverse source problem to determine two kinds of sources simultaneously by the observation at the final time and on the lateral boundary. The main tool for solving the inverse problems is a new global Carleman estimate for the stochastic parabolic equation. [ABSTRACT FROM AUTHOR]

We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data. [ABSTRACT FROM AUTHOR]

A system of wave equations on a Lorentzian manifold, the coefficients of which depend on time relates to the Einstein equation in general relativity. We consider inverse source problem for the system in this paper. Having established the Carleman estimate with a second large parameter for the Laplace–Beltrami operator on a Lorentzian manifold under assumptions independent of a choice of local coordinates on a suitable weight function, we consider its application to the inverse source problem for the system and prove local Hölder stability. [ABSTRACT FROM AUTHOR]

We construct an explicit Green's function for the conjugated Laplacian e−ω⋅x/hΔ e−ω⋅x/h, which lets us control our solutions on roughly half of the boundary. We apply the Green's function to solve a partial data inverse problem for the Schrödinger equation with potential q ∈ Ln/2. Separately, we also use this Green's function to derive Lp Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329-347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate--rather, both results stem from the same explicit construction of the Green's function. This explicit Green's function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calderón problems which is presently only accessible with full data [Inverse Problems 27 (2011)]. [ABSTRACT FROM AUTHOR]

We study the inverse problem of determining a magnetic Schrodinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with general bounded and non-compactly supported electromagnetic potentials. By assuming that the electromagnetic potentials are known on the neighborhood of the boundary outside a compact set, we even prove the unique determination of the magnetic field and the electric potential from measurements restricted to a bounded subset of the infinite boundary. Finally, in the case of a waveguide taking the form of an infinite cylindrical domain, we prove the recovery of the magnetic field and the electric potential from partial data corresponding to restriction of Neumann boundary measurements to slightly more than half of the boundary. We establish all these results by mean of a new class of complex geometric optics solutions and of Carleman estimates suitably designed for our problem stated in an unbounded domain and with bounded electromagnetic potentials. [ABSTRACT FROM AUTHOR]

In this article we study an inverse problem for the space-time fractional parabolic operator (∂t-∆)s + Q with 0 < s < 1 in any space dimension. We uniquely determine the unknown bounded potential Q from infinitely many exterior Dirichlet-to-Neumann type measurements. This relies on Runge approximation and the dual global weak unique continuation properties of the equation under consideration. In discussing weak unique continuation of our operator, a main feature of our argument relies on a new Carleman estimate for the associated degenerate parabolic Caffarelli- Silvestre extension. Furthermore, we also discuss constructive single measurement results based on the approximation and unique continuation properties of the equation. [ABSTRACT FROM AUTHOR]

This paper is addressed to a semi-linear stochastic transport equation with three unknown parameters. It is proved that the initial displacement, the terminal state and the random term in diffusion are uniquely determined by the state on partial boundary and a Lipschitz stability of the inverse problem is established. The main tool we employ is a global Carleman estimate for stochastic transport equations. [ABSTRACT FROM AUTHOR]

We consider the reaction-diffusion equation with discontinuities in the diffusion coefficient and the potential term. We start by deriving the Carleman estimate for the discontinuous reaction-diffusion operator which is deployed in the inverse problems of finding the stability result of the two discontinuous coefficients from the internal observations of the given parabolic equation.

We consider inverse problems for the first- and half-order time-fractional equation. We establish the stability estimates of Lipschitz type in inverse source and inverse coefficient problems by means of the Carleman estimates. [ABSTRACT FROM AUTHOR]

We consider the linear system of viscoelasticity with the homogeneous Dirichlet boundary condition. First we prove a Carleman estimate with boundary values of solutions of the viscoelasticity system. Since a solution u under consideration is not assumed to have compact support, in the decoupling of the Lam'e operator by introducing the divergence and the n-dimensional rotation of u, we have no boundary condition for them, so that we have to carry out arguments by a pseudodifferential operator. Second we apply the Carleman estimate to an inverse source problem of determining a spatially varying factor of the external source in the linear viscoelastitiy by extra Neumann data on a suitable lateral subboundary over a sufficiently long time interval and establish a stability estimate. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. First, we derive quantitative uniqueness estimates in the form of three-ball inequalities for solutions of complex conductivity equations whose leading coefficients are Lipschitz. Second, we study the problem of estimating the size of an inclusion embedded inside a conductive body with anisotropic complex admittivity by one boundary measurement. Practical motivations for studying such inverse problems are also given. [ABSTRACT FROM AUTHOR]