Vol. 6, No. 8, 2013

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ISSN: 1948-206X (e-only)
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The Calderón problem with partial data on manifolds and applications

Carlos Kenig and Mikko Salo

Vol. 6 (2013), No. 8, 2003–2048

We consider Calderón’s inverse problem with partial data in dimensions n 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.

Calderón problem, partial data, inverse problem
Mathematical Subject Classification 2010
Primary: 35R30
Secondary: 35J10, 58J32
Received: 6 May 2013
Accepted: 13 November 2013
Published: 20 April 2014
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