The Calderón problem with partial data on manifolds and applications

By Carlos E. Kenig, Mikko Salo. 2012

128 citations recorded by the source corpus.

We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flatness condition include parts of cylindrical sets, conical sets, and surfaces of revolution. We prove local uniqueness in the Calderon problem with partial data in admissible geometries, and global uniqueness under an additional concavity assumption. This work unifies two earlier approaches to this problem (\cite{KSU} and \cite{I}) and extends both. The proofs are based on improved Carleman estimates with boundary terms, complex geometrical optics solutions involving reflected Gaussian beam quasimodes, and invertibility of (broken) geodesic ray transforms. This last topic raises questions of independent interest in integral geometry.

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

View Paper: https://www.semanticscholar.org/paper/42c1bc21c2bec7745262a217d5d246dae7c3412d

How researchers see this paper

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

In Computational Theory and Mathematics

In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

An inverse problem for a semi-linear elliptic equation in Riemannian geometries (2019, unknown): “The presentation here follows [9, 23] with some modifications.”
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…”
The Calderón problem in transversally anisotropic geometries (2013, background): “Calderón problem in a cylinder: continuous spectrum 38 References 46”
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].”