The Calderon problem with partial data on manifolds and applications

TL;DR

This paper advances the Calderon inverse problem with partial boundary data on manifolds, establishing conditions for uniqueness and invertibility of related transforms, with implications for integral geometry and inverse problems.

Contribution

It unifies and extends previous approaches to partial data Calderon problems, introducing new techniques involving Carleman estimates and geodesic ray transform invertibility.

Findings

Local uniqueness in admissible geometries

Global uniqueness under concavity assumptions

Extension of previous methods to broader settings

Abstract

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.