Limiting Carleman weights and anisotropic inverse problems
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In this article, the authors considered the anisotropic Calderon problem and related inverse problems, and characterized those Riemannian manifolds which admit limiting Carleman weights, and gave a complex geometrical optics construction for a class of such manifolds.Abstract:
In this article we consider the anisotropic Calderon problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.read more
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Inverse problems with partial data for a Dirac system: a Carleman estimate approach
Mikko Salo,Leo Tzou +1 more
TL;DR: In this paper, it was shown that the material parameters in a Dirac system with magnetic and electric potentials are uniquely determined by measurements made on a possibly small subset of the boundary.
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On L-p resolvent estimates for Laplace-Beltrami operators on compact manifolds
TL;DR: In this article, it was shown that the Laplace-Beltrami resolvents of compact Riemannian manifolds can be approximated with Carleman weights.
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Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities
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TL;DR: In this article, the stability of the inverse conductivity problem for a conformal class of anisotropic conductivities in terms of the local Dirichlet-Neumann map was studied.
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TL;DR: In this paper, the spectral theory and associated forward and inverse scattering problems for the Laplace-Beltrami operators on asymptotically hyperbolic manifolds are studied.
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The X-ray transform for connections in negative curvature
Colin Guillarmou,Gabriel P. Paternain,Mikko Salo,Gunther Uhlmann,Gunther Uhlmann,Gunther Uhlmann +5 more
TL;DR: In this article, integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature were considered and the results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics.
References
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The Analysis of Linear Partial Differential Operators I
TL;DR: In this article, the analysis of linear partial differential operators i distribution theory and fourier rep are a good way to achieve details about operating certain products using instruction manuals, which are clearlybuilt to give step-by-step information about how you ought to go ahead in operating certain equipments.
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TL;DR: A very readable introduction to Riemannian geometry and geometric analysis can be found in this paper, where the author focuses on using analytic methods in the study of some fundamental theorems in Riemmannian geometry, e.g., the Hodge theorem, the Rauch comparison theorem, Lyusternik and Fet theorem and the existence of harmonic mappings.
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A global uniqueness theorem for an inverse boundary value problem
John Sylvester,Gunther Uhlmann +1 more
TL;DR: In this paper, the single smooth coefficient of the elliptic operator LY = v yv can be determined from knowledge of its Dirichlet integrals for arbitrary boundary values on a fixed region 2 C R', n? 3.
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