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

TL;DR

This paper proves that the shape of a Riemannian manifold can be uniquely determined from wave measurements on disjoint boundary sets, even under less restrictive controllability conditions, advancing inverse boundary value problem theory.

Contribution

It establishes unique determination of a Riemannian manifold from boundary data on disjoint sets using controllability or eigenvalue conditions, extending previous inverse problem results.

Findings

Unique determination of manifold from disjoint boundary data

Controllability condition can be replaced by eigenvalue bounds

Results hold under Hassell-Tao eigenfunction condition

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.