On uniqueness in the inverse conductivity problem with local data

By Victor Isakov. 2007

160 citations recorded by the source corpus.

We show that the Dirichlet-to-Neumann map given on an arbitrary part of the boundary of a three-dimensional domain with zero Dirichlet (or Neumann) data on the remaining (spherical or plane) part of the boundary uniquely determines conductivity or potential coefficients. This is the first uniqueness result for the Calderon problem with zero data on unaccessible part of the boundary. Proofs use some modification of the method of complex geometrical solutions due to Calderon-Sylvester-Uhlmann.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/a206bc531a299a39dd084d27649b009a2d9f8151

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In Computational Theory and Mathematics

In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

The Calderón problem with partial data on manifolds and applications (2012, methodology): “We extend the method of [Isakov 2007], which dealt with inaccessible parts that are part of a hyperplane, to the case of inaccessible parts that are part of the graph of a function independent of one of the variables.”
Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data (2013, methodology): “Constructing complex geometric optics solutions using a reflection argument was initiated in [8].”
Inverse problems: seeing the unseen (2014, background): “Isakov [95] proved a uniqueness result in dimension three or higher when the DN map is given on an arbitrary part of the boundary assuming that the remaining part is an open subset of a plane or a sphere and the DN map”
Monotonicity and local uniqueness for the Helmholtz equation (2017, result): “We refer to [Kenig et al. 2007; Isakov 2007; Kenig and Salo 2013; Krupchyk and Uhlmann 2016] and the overview article [Kenig and Salo 2014] for known results, which either impose strong geometric restrictions on the inaccessible part of the boundary or require measurements of Dirichlet and Neumann…”
Electrical impedance tomography and Calderón's problem (2009, background): “Isakov [63] proved a uniqueness result in dimension three or higher when the DN map is given on an arbitrary part of the boundary assuming that the remaining part is an open subset of a plane or a sphere and the DN map is measured on the plane or sphere.”
The Calderón problem with partial data in two dimensions (2010, background): “In dimension n ≥ 3 Isakov [17] proved global uniqueness assuming that Γ0 is a subset of a plane or a sphere.”