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

Abstract

We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary $(M, g)$ from a restriction $\Lambda_{\Src, \Rec}$ of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here $\Src$ and $\Rec$ are open sets in $\p M$ and the restriction $\Lambda_{\Src, \Rec}$ corresponds to the case where the Dirichlet data is supported on $\R_+\times \Src$ and the Neumann data is measured on $\R_+\times \Rec$. In the novel case where $\bar \Src \cap \bar \Rec = \emptyset$, we show that $\Lambda_{\Src, \Rec}$ determines the manifold $(M,g)$ uniquely, assuming that the wave equation is exactly controllable from the set of sources $\Src$. Moreover, we show that the exact controllability can be replaced by the Hassell-Tao condition for eigenvalues and eigenfunctions, that is, \lambda_j \le C \norm{\p_\nu \phi_j}_{L^2(\Src)}^2, \quad j =1, 2, ..., where $\lambda_j$ are the Dirichlet eigenvalues and $(\phi_j)_{j=1}^\infty$ is an orthonormal basis of the corresponding eigenfunctions.