The Calderón problem with partial data in two dimensions

By Oleg Yu. Imanuvilov, Gunther Uhlmann, Masahiro Yamamoto. 2010

174 citations recorded by the source corpus.

We prove for a two dimensional bounded domain that the Cauchy data for the Schrodinger 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.

Subfields: Biomedical Engineering, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e498ab1df3eb06d8a5752b5a21e4ead882dd8c46

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biomedical Engineering

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Partial Cauchy data for general second order elliptic operators in two dimensions (2010, background): “The case of the conductivity equation has been considered in [15].”
An analysis of electrical impedance tomography with applications to Tikhonov regularization (2012, background): “dimension two, the uniqueness in the class of L∞(Ω) functions was shown, and [31] for partial data.”
Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities (2019, background): “In dimension n = 2 in the linear setting, the global identifiability in the partial data inverse problem is established in [14] when Γ1 = Γ2 is an arbitrary open non-empty portion of ∂Ω, and in [15] when Γ1 ∩Γ2 = ∅, provided that some additional geometric assumptions are satisfied.”
Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets (2012, background): “Also, the local data problem for anisotropic elliptic equations in Ω ⊂ R2 was recently solved by Imanuvilov, Uhlmann and Yamamoto [23, 24].”
The Calderón problem with partial data on manifolds and applications (2012, methodology): “In two dimensions, the main partial data result is that of [Imanuvilov et al. 2010], which has been extended in [Imanuvilov et al. 2011a; 2011b; Imanuvilov and Yamamoto 2012a; 2012b] to, respectively,”
The Calderón problem for variable coefficients nonlocal elliptic operators (2017, unknown): “Analogues resembles can be made with the study of the partial data Calderón’s type problem, the richness of such works can be found in [23, 24, 25, 26].”
A remark on partial data inverse problems for semilinear elliptic equations (2019, methodology): “We refer to [6] and [7] for the study of the partial data inverse problem in the linear and non-linear settings, respectively, in dimension n = 2.”
Electrical impedance tomography and Calderón's problem (2009, methodology): “It is shown in [59] that for a two-dimensional bounded domain the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determine uniquely the potential.”
Inverse problems: seeing the unseen (2014, methodology): “Finally we consider the work of Imanuvilov, Uhlmann andYamamoto on the partial data problem in two dimensions [87,88].”