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

Stefanov, Plamen; Yang, Yang

doi:10.2140/apde.2018.11.1381

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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

DOI: 10.2140/apde.2018.11.1381

Abstract

We consider the Dirichlet-to-Neumann map $Λ$ 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 $Λ$ up to a gauge transformation in a stable way. We also show that $Λ$ recovers the following three invariants in a stable way: the lens relation of $g$ , and the light ray transforms of $A$ and $q$ . Moreover, $Λ$ 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

Vol. 11, No. 6, 2018