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 S,R of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here S and R are open sets in ∂M and the restriction Λ S,R corresponds to the case where the Dirichlet data is supported on R + × S and the Neumann data is measured on R + × R. In the novel case where S ∩ R = ∅, we show that Λ S,R determines the manifold (M, g) uniquely, assuming that the wave equation is exactly controllable from the set of sources S. Moreover, we show that the exact controllability can be replaced by the Hassell-Tao condition for eigenvalues and eigenfunctions, that is,

Inverse problems for the connection Laplacian

“…For the convenience of the reader, we give a proof of the following lemma. An analogous lemma is stated in [28] without a proof.…”

“…As described in the scalar valued case in Section 4.4 of [28], in order to determine the cut distance σ Γ from the restricted Dirichlet-to-Neumann map, we need to use a perturbation argument that is based on a refined version of approximate controllability and modified domains of influence. Let Γ ⊂ ∂M and h : Γ → R, and define…”

Global uniqueness in an inverse problem for time fractional diffusion equations

“…Following [39] we assume that µ k = 1 and q k = 0 identically, and that both (M k , g k ), k = 1, 2, satisfy the spectral inequality λ k,n ≤ C ψ k,n,p 2 L 2 (S in ) , (5.5) where the constant C > 0 is independent of n and p. Hassell and Tao [27] showed that all non-trapping Riemannian manifolds (M k , g k ) satisfy (5.5) when S in is replaced by ∂M k . Moreover, (5.5) follows from (and is strictly weaker than) the geometric control condition by Bardos, Lebeau and Rauch [2], see [39]. We will now give a reduction to the result in [39].…”

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“…Below C > 0 denotes a constant that may grow between inequalities, and that depends only on m, C 0 , C 1 , L. Lemma 1 gives us f δ for which u f δ (x, T ) = w δ (x). Thus (38) implies…”