Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 6 (2018), 1381-1414.

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

Plamen Stefanov and Yang Yang

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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.

Article information

Source
Anal. PDE, Volume 11, Number 6 (2018), 1381-1414.

Dates
Received: 26 September 2016
Revised: 6 January 2018
Accepted: 14 February 2018
First available in Project Euclid: 23 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.apde/1527040815

Digital Object Identifier
doi:10.2140/apde.2018.11.1381

Mathematical Reviews number (MathSciNet)
MR3803714

Zentralblatt MATH identifier
06881246

Subjects
Primary: 35R30: Inverse problems
Secondary: 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE [See also 32C38, 58J15] 53B30: Lorentz metrics, indefinite metrics

Keywords
Lorentz DN map inverse problem light ray transform microlocal

Citation

Stefanov, Plamen; Yang, Yang. The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds. Anal. PDE 11 (2018), no. 6, 1381--1414. doi:10.2140/apde.2018.11.1381. https://projecteuclid.org/euclid.apde/1527040815


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