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 ${\Lambda }_{\mathcal{S},\mathcal{R}}$ of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here $\mathcal{S}$ and $\mathcal{R}$ are open sets in $\partial M$ and the restriction ${\Lambda }_{\mathcal{S},\mathcal{R}}$ corresponds to the case where the Dirichlet data is supported on ${\mathbb{R}}_{+}\times \mathcal{S}$ and the Neumann data is measured on ${\mathbb{R}}_{+}\times \mathcal{R}$. In the novel case where $\bar{\mathcal{S}}\cap \bar{\mathcal{R}}=\varnothing$, we show that ${\Lambda }_{\mathcal{S},\mathcal{R}}$ determines the manifold $(M,g)$ uniquely, assuming that the wave equation is exactly controllable from the set of sources $\mathcal{S}$. Moreover, we show that the exact controllability can be replaced by the Hassell–Tao condition for eigenvalues and eigenfunctions, that is,

$${\lambda }_j\le C\Vert {\partial }_{\nu }{\varphi }_j{\Vert }_{L^2\left(\mathcal{S}\right)}^2,j=1,2,\dots ,$$

where ${\lambda }_j$ are the Dirichlet eigenvalues and where $({\varphi }_j{)}_{j=1}^{\infty }$ is an orthonormal basis of the corresponding eigenfunctions.