The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

Abstract

We consider the semilinear wave equation on a Lorentzian manifold (M, g) with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric g and the coefficient a up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.