Results 1  10
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45
THE XRAY TRANSFORM FOR A GENERIC FAMILY OF CURVES AND WEIGHTS
, 2007
"... Abstract. We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T ∗ M. 1. ..."
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Cited by 15 (7 self)
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Abstract. We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T ∗ M. 1.
Inverse problems for the anisotropic Maxwell equations (preprint). 17 R. Leis, Initial boundary value problems in mathematical physics
, 1986
"... Abstract. We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the timeharmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and ..."
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Cited by 13 (12 self)
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Abstract. We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the timeharmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions. 1.
DETERMINING NONSMOOTH FIRST ORDER TERMS FROM PARTIAL BOUNDARY MEASUREMENTS
, 2006
"... Abstract. We extend results of Dos Santos FerreiraKenigSjöstrandUhlmann (arXiv:math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. ..."
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Cited by 12 (6 self)
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Abstract. We extend results of Dos Santos FerreiraKenigSjöstrandUhlmann (arXiv:math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the NakamuraUhlmann pseudodifferential conjugation method to logarithmic Carleman weights. 1.
Global uniqueness from partial Cauchy data in two dimensions. Arxiv preprint arXiv:0810.2286
, 2008
"... Abstract. We prove for a two dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an a ..."
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Cited by 12 (1 self)
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Abstract. We prove for a two dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results. 1.
INTEGRAL GEOMETRY OF TENSOR FIELDS ON A CLASS OF NONSIMPLE RIEMANNIAN MANIFOLDS
, 2006
"... Abstract. We study the geodesic Xray transform IΓ of tensor fields on a compact Riemannian manifold M with nonnecessarily convex boundary and with possible conjugate points. We assume that IΓ is known for geodesics belonging to an open set Γ with endpoints on the boundary. We prove generic sinjec ..."
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Cited by 12 (6 self)
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Abstract. We study the geodesic Xray transform IΓ of tensor fields on a compact Riemannian manifold M with nonnecessarily convex boundary and with possible conjugate points. We assume that IΓ is known for geodesics belonging to an open set Γ with endpoints on the boundary. We prove generic sinjectivity and a stability estimate under some topological assumptions and under the condition that for any (x,ξ) ∈ T ∗ M, there is a geodesic in Γ through x normal to ξ without conjugate points.
Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field
"... We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R n, n ≥ 3. The magnetic potential is assumed to be continuous with L ∞ divergence and zero boundary values. The me ..."
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Cited by 11 (5 self)
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We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R n, n ≥ 3. The magnetic potential is assumed to be continuous with L ∞ divergence and zero boundary values. The method is based on semiclassical pseudodifferential calculus and the construction of complex geometrical optics solutions in weighted Sobolev spaces. 1
Inverse problem for a Schrödinger operator in an unbounded strip
 J. Inverse IllPosed Problems V
, 2008
"... We consider the operator H: = i∂t + ∇ · (c∇) in an unbounded strip Ω in R 2, where c(x, y) ∈ C 3 (Ω). We prove adapted a global Carleman estimate and an energy estimate for this operator. Using these ..."
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Cited by 9 (4 self)
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We consider the operator H: = i∂t + ∇ · (c∇) in an unbounded strip Ω in R 2, where c(x, y) ∈ C 3 (Ω). We prove adapted a global Carleman estimate and an energy estimate for this operator. Using these
Stability estimates for coefficients of magnetic Schrödinger . . .
, 2008
"... In this paper we establish a log logtype estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrödinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly ..."
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Cited by 8 (0 self)
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In this paper we establish a log logtype estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrödinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly larger than half of the boundary ∂Ω. Furthermore, we prove that in the case when the measurement is taken on all of ∂Ω one can establish a better estimate that is of logtype. The proofs involve the use of the complex geometric optics (CGO) solutions of the magnetic Schrödinger equation constructed in [8] then follow a similar line of argument as in [1]. In the partial data estimate we follow the general strategy of [5] by using the Carleman estimate established in [4] and a continuous dependence result for analytic continuation developed in [14].
Calderón inverse problem with partial data on Riemann surfaces
 Duke Math. J
"... Abstract. On a fixed smooth compact Riemann surface with boundary (M0, g), we show that for the Schrödinger operator ∆ + V with potential V ∈ C 1,α (M0) for some α> 0, the DirichlettoNeumann map NΓ measured on an open set Γ ⊂ ∂M0 determines uniquely the potential V. We also discuss briefly the co ..."
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Cited by 7 (2 self)
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Abstract. On a fixed smooth compact Riemann surface with boundary (M0, g), we show that for the Schrödinger operator ∆ + V with potential V ∈ C 1,α (M0) for some α> 0, the DirichlettoNeumann map NΓ measured on an open set Γ ⊂ ∂M0 determines uniquely the potential V. We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends. 1.
INVISIBILITY AND INVERSE PROBLEMS
"... Abstract. We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that gove ..."
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Cited by 6 (4 self)
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Abstract. We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that govern a variety of wave phenomena, including electrostatics, electromagnetism, acoustics and quantum mechanics, have transformation laws under changes of variables which allow one to design material parameters that steer waves around a hidden region, returning them to their original path on the far side. Not only are observers unaware of the contents of the hidden region, they are not even aware that something is being hidden; the object, which casts no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other devices having striking effects on wave propagation, unseen in nature, are also possible. These designs are initially based on the transformation laws of the relevant PDEs, but due to the singular transformations needed for the desired effects, care needs to be taken in formulating and analyzing physically meaningful solutions. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved. 1.