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10
Thermoacoustic tomography with variable sound speed
 Inverse Problems
, 2009
"... Abstract. We study the mathematical model of thermoacoustic tomography in media with a variable speed for a fixed time interval [0, T] so that all signals issued from the domain leave it after time T. In case of measurements on the whole boundary, we give an explicit solution in terms of a Neumann s ..."
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Abstract. We study the mathematical model of thermoacoustic tomography in media with a variable speed for a fixed time interval [0, T] so that all signals issued from the domain leave it after time T. In case of measurements on the whole boundary, we give an explicit solution in terms of a Neumann series expansion. We give almost necessary and sufficient conditions for uniqueness and stability when the measurements are taken on a part of the boundary. 1.
Radiation fields, scattering and inverse scattering on asymptotically hyperbolic manifolds
, 2004
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Uniqueness results for ill posed characteristic problems in curved spacetimes
, 2007
"... Abstract. We prove two uniqueness theorems concerning linear wave equations; the first theorem is in Minkowski spacetimes, while the second is in the domain of outer communication of a Kerr black hole. Both theorems concern illposed Cauchy problems on bifurcate, characteristic hypersurfaces. In th ..."
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Abstract. We prove two uniqueness theorems concerning linear wave equations; the first theorem is in Minkowski spacetimes, while the second is in the domain of outer communication of a Kerr black hole. Both theorems concern illposed Cauchy problems on bifurcate, characteristic hypersurfaces. In the case of the Kerr spacetime, the hypersurface is precisely the event horizon of the black hole. The uniqueness theorem in this case, based on two Carleman estimates, is intimately connected to our strategy to prove uniqueness of the Kerr black holes among smooth, stationary solutions of the Einsteinvacuum equations, as formulated in [14]. Contents
FORWARD AND INVERSE SCATTERING ON MANIFOLDS WITH ASYMPTOTICALLY CYLINDRICAL ENDS
, 905
"... Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange perturbation of the metric of the form (dy) 2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codi ..."
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Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange perturbation of the metric of the form (dy) 2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to (dy) 2 + h(x, dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energy, we show that these two manifolds are isometric. 1.
On the Determination of Moving Boundaries for Hyperbolic Equations
, 902
"... Abstract. We consider wave equations in domains with timedependent boundaries (moving obstacles) contained in a fixed cylinder for all time. We give sufficient conditions for the determination of the moving boundary from the Cauchy data on part of the boundary of the cylinder. We also study the rel ..."
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Abstract. We consider wave equations in domains with timedependent boundaries (moving obstacles) contained in a fixed cylinder for all time. We give sufficient conditions for the determination of the moving boundary from the Cauchy data on part of the boundary of the cylinder. We also study the related problem of accessibility of the moving boundary by timelike curves from the boundary of the cylinder. In this article we study the possibility of determining a moving boundary from Cauchy data on a stationary boundary. The setting of this problem is as follows. Let Q be an exterior domain in Rn x ×Rt with smooth boundary ∂Q. We assume that the complement of Q is contained in the cylinder C = {(x, t) : x  < R, t ∈ R}, and
ASYMPTOTIC STABILITY OF THE WAVE EQUATION ON COMPACT SURFACES AND LOCALLY DISTRIBUTED DAMPING A SHARP
, 811
"... Abstract. This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by utt − ∆Mu + a(x)g(ut) = 0 on M ×]0, ∞ [, where M ⊂ R 3 is a smooth oriented embedded compact surface without boundary. Denoting by g the Riemannian metric induced ..."
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Abstract. This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by utt − ∆Mu + a(x)g(ut) = 0 on M ×]0, ∞ [, where M ⊂ R 3 is a smooth oriented embedded compact surface without boundary. Denoting by g the Riemannian metric induced on M by R 3, we prove that for each ǫ> 0, there exist an open subset V ⊂ M and a smooth function f: M → R such that meas(V) ≥ meas(M) − ǫ, Hessf ≈ g on V and inf x∈V ∇f(x) > 0. In addition, we prove that if a(x) ≥ a0> 0 on an open subset M ∗ ⊂ M which contains M\V and if g is a monotonic increasing function such that ks  ≤ g(s)  ≤ Ks for all s  ≥ 1, then uniform and optimal decay rates of the energy hold. 1.
Maxwell’s Equations with Scalar Impedance: Direct and Inverse Problems
, 2002
"... Abstract: The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell’s equations written for differential forms over a 3manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle ove ..."
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Abstract: The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell’s equations written for differential forms over a 3manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle over the manifold. The second part of the article deals with the dynamical inverse boundary value problem of determining the electromagnetic material parameters from boundary measurements. By using the boundary control method, it is proved that the dynamical boundary data determines the electromagnetic travel time metric as well as the scalar wave impedance on the manifold. This invariant result leads also to a complete characterization of the nonuniqueness of the corresponding inverse problem in bounded domains of R 3.
RADIATION FIELDS ON ASYMPTOTICALLY EUCLIDEAN MANIFOLDS
"... Abstract. F.G. Friedlander introduced the notion of radiation elds for asymptotically Euclidean manifolds. Here we answer some of the questions he proposed and apply the results to give a unitary translation representation of the wave group, and to obtain the scattering matrix for such manifolds. ..."
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Abstract. F.G. Friedlander introduced the notion of radiation elds for asymptotically Euclidean manifolds. Here we answer some of the questions he proposed and apply the results to give a unitary translation representation of the wave group, and to obtain the scattering matrix for such manifolds. We also obtain a support theorem for the radiation elds. 1. Introduction and Statement