Vol. 11, No. 6, 2018

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The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds

Plamen Stefanov and Yang Yang

Vol. 11 (2018), No. 6, 1381–1414
Abstract

We consider the Dirichlet-to-Neumann map $\Lambda$ on a cylinder-like Lorentzian manifold related to the wave equation related to the metric $g$, the magnetic field $A$ and the potential $q$. We show that we can recover the jet of $g,A,q$ on the boundary from $\Lambda$ up to a gauge transformation in a stable way. We also show that $\Lambda$ recovers the following three invariants in a stable way: the lens relation of $g$, and the light ray transforms of $A$ and $q$. Moreover, $\Lambda$ is an FIO away from the diagonal with a canonical relation given by the lens relation. We present applications for recovery of $A$ and $q$ in a logarithmically stable way in the Minkowski case, and uniqueness with partial data.

Keywords
Lorentz, DN map, inverse problem, light ray transform, microlocal
Mathematical Subject Classification 2010
Primary: 35R30
Secondary: 35A27, 53B30
Milestones
Received: 26 September 2016
Revised: 6 January 2018
Accepted: 14 February 2018
Published: 3 May 2018
Authors
 Plamen Stefanov Department of Mathematics Purdue University West Lafayette, IN United States Yang Yang Department of Computational Mathematics, Science and Engineering Michigan State University East Lansing, MI United States