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 $Λ_{\Src, \Rec}$ 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 $Λ_{\Src, \Rec}$ 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 $Λ_{\Src, \Rec}$ 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, λ_j \le C \norm{\p_νϕ_j}_{L^2(\Src)}^2, \quad j =1, 2, ..., where ${\lambda }_j$ are the Dirichlet eigenvalues and $({\varphi }_j{)}_{j=1}^{\infty }$ is an orthonormal basis of the corresponding eigenfunctions.

References in corpus (1)

• An inverse problem for the wave equation with one measurement and the pseudorandom noise

Cited by in corpus (23)

• The Calderon problem with partial data on manifolds and applications
• Reconstruction of Lorentzian manifolds from boundary light observation sets
• Detection of Hermitian connections in wave equations with cubic non-linearity
• Uniqueness and stability of an inverse problem for a semi-linear wave equation
• On the Construction of Virtual Interior Point Source Travel Time Distances from the Hyperbolic Neumann-to-Dirichlet Map
• Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations
• Regularization strategy for inverse problem for 1+1 dimensional wave equation
• Inverse Scattering with Partial data on Asymptotically Hyperbolic Manifolds
• Global recovery of a time-dependent coefficient for the wave equation from a single measurement
• Inverse problems for non-linear hyperbolic equations with disjoint sources and receivers
• Unique determination of several coefficients in a fractional diffusion(-wave) equation by a single measurement
• Stable reconstruction of simple Riemannian manifolds from unknown interior sources
• Global uniqueness in an inverse problem for time fractional diffusion equations
• Determination of a compact Finsler manifold from its boundary distance map and an inverse problem in elasticity
• Hyperbolic inverse problem with data on disjoint sets
• A Non-Iterative Reconstruction Algorithm for the Acoustic Inverse Boundary Value Problem
• Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds
• Gel'fand's inverse problem for the graph Laplacian
• Inverse scattering on non-compact manifolds with general metric
• Reconstruction of a compact Riemannian manifold from the scattering data of internal sources
• Determination of the Spacetime from Local Time Measurements
• Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds
• An Exact Redatuming Procedure for the Inverse Boundary Value Problem for the Wave Equation