Lens rigidity with partial data in the presence of a magnetic field

Hanming Zhou

doi:10.3934/ipi.2018057

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2018, Volume 12, Issue 6: 1365-1387. Doi: 10.3934/ipi.2018057

This issue Previous Article Simultaneous reconstruction and segmentation with the Mumford-Shah functional for electron tomography Next Article Local block operators and TV regularization based image inpainting

Lens rigidity with partial data in the presence of a magnetic field

Hanming Zhou^,

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK

* Current address: Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106-3080, USA
* Current address: Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA 93106-3080, USA

Received: October 31, 2017

Revised: May 31, 2018

Published: November 2018

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In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension $≥ 3$ with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magnetic geodesics) boundary point where the lens data is accessible. We also prove a boundary rigidity result with partial data assuming the lengths of magnetic geodesics joining boundary points near a strictly convex boundary point are known. The local lens rigidity result also leads to a global rigidity result under some strictly convex foliation condition. A discussion of a weaker version of the lens rigidity problem with partial data for general smooth curves is given at the end of the paper.

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Mathematics Subject Classification: 53C24, 35R30, 53C65, 35S05.

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