#### Vol. 6, No. 8, 2013

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
The Calderón problem with partial data on manifolds and applications

### Carlos Kenig and Mikko Salo

Vol. 6 (2013), No. 8, 2003–2048
##### Abstract

We consider Calderón’s inverse problem with partial data in dimensions $n\ge 3$. 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 Calderón problem with partial data in admissible geometries, and global uniqueness under an additional concavity assumption. This work unifies two earlier approaches to this problem—one by Kenig, Sjöstrand, and Uhlmann, the other by Isakov—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.

##### Keywords
Calderón problem, partial data, inverse problem
##### Mathematical Subject Classification 2010
Primary: 35R30
Secondary: 35J10, 58J32
##### Milestones
Received: 6 May 2013
Accepted: 13 November 2013
Published: 20 April 2014
##### Authors
 Carlos Kenig Department of Mathematics University of Chicago 5734 S. University Avenue Chicago, Illinois 60637 United States Mikko Salo Department of Mathematics and Statistics University of Jyväskylä FI-40014 Jyväskylä Finland