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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]

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]

We consider the inverse problem for the general transport equation with external field, source term, and absorption coefficient. We show that the source and the absorption coefficients can be uniquely reconstructed from the boundary measurement in a Lipschitz stable manner. Specifically, the uniqueness and stability are obtained by using the Carleman estimate, in which a special weight function is designed to pick up information on the desired parameter. [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]

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]

We consider a second-order hyperbolic equation on an open bounded domain Ω in ℝn for n ≥ 2, with C²-boundary Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. subject to nonhomogeneous Neumann boundary conditions on the entire boundary Γ. We then study the inverse problem of determining the interior damping coefficient of the equation by means of an additional measurement of the Dirichlet boundary trace of the solution, in a suitable, explicit subportion Γ1 of the boundary Γ, and over a computable time interval T > 0. Under sharp conditions on the complementary part Γ0 = Γ\Γ1, and T > 0, and under weak regularity requirements on the data, we establish the two canonical results in inverse problems: (i) global uniqueness and (ii) Lipschitz stability (at the L²-level). The latter is the main result of this paper. Our proof relies on three main ingredients: (a) sharp Carleman estimates at the H¹ x L2-level for second-order hyperbolic equations [I. Lasiecka, R Triggiani, and X. Zhang, Contemp. Math.. 268 (2000), pp. 227-325]; (b) a correspondingly implied continuous observability inequality at the same energy level [I. Lasiecka, R. Triggiani, and X. Zhang, Contemp. Math., 268 (2000), pp. 227-325]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Neumann boundary data [I. Lasiecka and R. Triggiani, Ann. Mat. Pura. Appl. (4), 157 (1990), pp. 285-367], [I. Lasiecka and R. Triggiani, J. Differential Equations, 94 (1991), pp. 112-164], [I. Lasiecka and R. Triggiani, Recent advances in regularity of second-order hyperbolic mixed problems, and applications, in Dynamics Reported: Expositions in Dynamical Systems, Vol. 3, Springer-Verlag, Berlin, 1994, pp. 104-158], [D. Tataru, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), pp. 185-206]. The proof of the linear uniqueness result (section 4, step 5) also takes advantage of a convenient tactical route "post-Carleman estimates" suggested by Isakov in [V. Isakov, Inverse Problems for Partial Differential Equations, 2nd ed., Springer, New York, 2006, Thm. 8.2.2, p. 231]. [ABSTRACT FROM AUTHOR]

In this paper we study an inverse problem of determining spatially varying density and two Lamé coefficients by a single measurement of solution in a subboundary over a time interval. By assuming that, in a neighborhood of the boundary of the spatial domain, the density and the Lamé coefficients are known, we prove a logarithmic stability estimate for the inverse problem with a single measurement of data on an arbitrarily given subboundary. [ABSTRACT FROM AUTHOR]

In this paper, we consider Maxwell's equations in a bi-isotropic and inhomogeneous medium. We discuss an inverse problem of determining the coefficients $\varepsilon$, $\zeta$, $\mu$ in the constitutive relations from a finite number of interior measurements. We prove a Lipschitz stability estimate for the inverse problem by applying the argument on the basis of Carleman estimate. [ABSTRACT FROM AUTHOR]

We consider an inverse scattering problem (ISP) for the acoustic equation utt = c2(x)Δu, u¦ t = 0, u t u¦ t=0 = ∂(x), x ∈ R³. The ISP consists of the determination of the speed of sound c(x) inside a bounded domain Ω ≡ R³ given c(x) outside Ω and measurements of the amplitude u(x; t) of the sound at the boundary ∂Ω u¦∂Ω = φ(x, t). This problem is nonoverdetermined since only a single source location at {0} is counted. Assuming regularity of the rays generated by c(x) and using the Carleman's weight functions, we construct a cost functional Jλ. The main result is Theorem 3.1, which claims global strict convexity of Jλ on "reasonable" compact sets of solutions. Therefore, global convergence on such a set of a number of standard minimization algorithms to the unique global minimum of Jλ (i.e., solution of the ISP) is guaranteed. This in turn shows a possibility of constructions of numerical methods for this ISP which would not be affected by the problem of local minima. [ABSTRACT FROM AUTHOR]