Kurylev, Y; Lassas, M; Uhlmann, G; (2018) Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations. Inventiones mathematicae , 212 pp. 781-857. 10.1007/s00222-017-0780-y.

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Abstract

We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are: / 1. Passive observations in spacetime: consider observations in an open set V⊂M . The light observation set corresponding to a point source at q∈M is the intersection of V and the light-cone emanating from the point q. Let W⊂M be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points q∈W determine uniquely the conformal type of W. / 2. Active measurements in spacetime: we develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood V of the time-like path μ and the source-to-solution operator that maps the source supported on V to the restriction of the solution of the wave equation to V. When M is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from μ and return back to μ .

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