Lauri Oksanen: Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary from a restriction of the Dirichlet-to-Neumann operator for the wave equation on the manifold. The restriction corresponds to the case where the Dirichlet data is supported on a part of the boundary of the manifold and the Neumann data is measured on another part. We show that the restriction of the Dirichlet-to-Neumann operator determines the manifold and the metric tensor uniquely, assuming that the wave equation is exactly controllable from the set where the Dirichlet data is supported. The set where the Neumann data is measured can be an arbitrary open set. The exact controllability can be characterized geometrically by a billiard condition. Moreover, we show that the exact controllability assumption can be replaced by a weaker asymptotic condition on Neumann traces of Dirichlet eigenfunctions. This condition was first shown to hold by Hassell and Tao in the case where the Neumann traces are given on the whole boundary of a non-trapping manifold. The talk is based on a joint work with M. Lassas, see arXiv:1208.2105 for the related preprint.