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2013 The Calderón problem with partial data on manifolds and applications
Carlos Kenig, Mikko Salo
Anal. PDE 6(8): 2003-2048 (2013). DOI: 10.2140/apde.2013.6.2003

Abstract

We consider Calderón’s inverse problem with partial data in dimensions n3. 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.

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Carlos Kenig. Mikko Salo. "The Calderón problem with partial data on manifolds and applications." Anal. PDE 6 (8) 2003 - 2048, 2013. https://doi.org/10.2140/apde.2013.6.2003

Information

Received: 6 May 2013; Accepted: 13 November 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1335.35301
MathSciNet: MR3198591
Digital Object Identifier: 10.2140/apde.2013.6.2003

Subjects:
Primary: 35R30
Secondary: 35J10 , 58J32

Keywords: Calderón problem , inverse problem , partial data

Rights: Copyright © 2013 Mathematical Sciences Publishers

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Vol.6 • No. 8 • 2013
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