The Calderón problem with partial data

By Carlos E. Kenig, J. Sjöstrand, Gunther Uhlmann. 2004

420 citations recorded by the source corpus.

In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n . 3, the knowledge of the Cauchy data for the Schr?Nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem. This implies a similar result for the problem of Electrical Impedance Tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5b62623290559ee358fbbe28430b0133358259a9

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Citations in the corpus

Electrical impedance tomography and Calderón's problem (2009, methodology): “The result [22] was substantially improved in [75].”
On Lp resolvent estimates for Laplace–Beltrami operators on compact manifolds (2011, methodology): “The motivation for introducing such weights in the context of general Riemannian manifolds lies in the application to inverse problems ([4,5,13]) where solutions to the Schrödinger equation with opposite exponential behaviours are constructed on manifolds using complex geometrical optics.”
On uniqueness in the inverse conductivity problem with local data (2007, methodology): “This assumption enables to reflect almost complex exponential solutions across 0 and to avoid use of a special fundamental solution (Green’s function) for the Schr¨odinger equation and of corresponding exponential weighted estimates with a large parameter in . Currently, such fundamental solution an”
Inverse problems: seeing the unseen (2014, methodology): “Examples of LCW are the linear phase φ(x) = x · ω,ω ∈ Sn−1, used previously, and the non-linear phase φ(x) = ln |x − x0|, where x0 ∈ Rn\ch ( ) which was used in [107].”
The Calderón problem with partial data in two dimensions (2010, result): “In [19], the result in [8] was generalized to show that by measuring all possible pairs of Dirichlet data on a possibly very small subsets of the boundary Γ+ and Neumann data on a slightly larger open domain than ∂Ω \ Γ+, one can uniquely determine the potential.”
Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets (2012, background): “[34] M. Lassas, M. Taylor, and G. Uhlmann.”
Determining an Unbounded Potential from Cauchy Data in Admissible Geometries (2011, background): “The concept of limiting Carleman weights was introduced in [11] as part of a general procedure for producing special complex geometrical optics solutions to elliptic equations, with applications to inverse problems.”
Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data (2013, background): “For the construction of various CGO solutions and their application, see [1, 2, 3, 4, 14, 17].”
REGULARIZED D-BAR METHOD FOR THE INVERSE CONDUCTIVITY PROBLEM (2009, background): “For recent development on new kinds of cgo solutions for inverse problems, see [16, 20, 47, 55].”