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A potential perturbation beyond the two-trip distance is invisible before time T

Example

Assume Countable Choice. Let $(M, g)$ be a compact, connected, nonempty smooth Riemannian manifold with nonempty smooth boundary, let $T > 0$ , and let $q_{1}, q_{2} \in C^{\infty} (M; R)$ be time independent. Put $Q_{0} = supp (q_{1} - q_{2})$ , the closure of the set on which $q_{1} - q_{2}$ is nonzero, and suppose $Q_{0} \neq \emptyset$ and $d_{g} (Q_{0}, \partial M) > T / 2$ , where $d_{g}$ is the intrinsic Riemannian distance. For every smooth Dirichlet input $h \in C_{c}^{\infty} ((0, T) \times \partial M)$ , the full time-dependent forward Dirichlet-to-Neumann maps satisfy $Λ_{q_{1}}^{+} h = Λ_{q_{2}}^{+} h in L^{2} ((0, T) \times \partial M, d S_{g} d t) .$ Thus a perturbation supported farther than half the observation time from the boundary cannot be detected by boundary measurements made before time $T$ .

Example references

Finite propagation for the wave equation on a Riemannian manifold

Smooth boundary tests separate locally integrable traces

Energy well-posedness for the variable-metric wave equation

Time-dependent Dirichlet-to-Neumann map for the wave equation

Alessandrini identity for the time-dependent wave Dirichlet-to-Neumann maps

Riemannian distance on a connected manifold

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: Countable Choice; the compact Riemannian manifold, static real potentials with $d_{g} (Q_{0}, \partial M) > T / 2$ , and smooth boundary input $h$ .

[A1]

Countable Choice is used through the DN-map and energy-solution interfaces and the uniqueness of the locally integrable representative of a regular distribution. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

For any forward solution for $q_{1}$ with boundary value $h$ and terminal solution for $q_{2}$ with boundary value $k$ , the interior product pairing is zero when $d_{g} (Q_{0}, \partial M) > T / 2$ . (Finite propagation for the wave equation on a Riemannian manifold)

[F2]

The forward DN map is the outward normal trace of the zero-initial solution; the terminal map is the outward normal trace of the zero-terminal solution. Both are in $L^{2}$ for the stated classical solution class. (Time-dependent Dirichlet-to-Neumann map for the wave equation)

[F3]

For these forward and terminal solutions, $\int_{Q} (q_{1} - q_{2}) u v d V_{g} d t = \int_{Σ} (k Λ_{q_{1}}^{+} h - h Λ_{q_{2}}^{-} k) d S_{g} d t .$ (Alessandrini identity for the time-dependent wave Dirichlet-to-Neumann maps)

[F4]

If a locally integrable function on the smooth boundary cylinder pairs to zero with every compactly supported smooth test, then it vanishes almost everywhere, by chart localization and regular-distribution uniqueness. (Smooth boundary tests separate locally integrable traces)

Proof

1.1A1F1F3given

Fix arbitrary $k \in C_{c}^{\infty} ((0, T) \times \partial M)$ . Let $u_{1}$ be the forward solution for $q_{1}$ with boundary input $h$ , and let $v_{2}$ be the terminal solution for $q_{2}$ with boundary input $k$ . By [F1], $\int_{Q} (q_{1} - q_{2}) u_{1} v_{2} d V_{g} d t = 0.$ Using [F3] therefore gives $\int_{Σ} (k Λ_{q_{1}}^{+} h - h Λ_{q_{2}}^{-} k) d S_{g} d t = 0.$

1.2F3given

Apply [F3] again with both potentials equal to $q_{2}$ . Its interior term is then zero, so $\int_{Σ} (k Λ_{q_{2}}^{+} h - h Λ_{q_{2}}^{-} k) d S_{g} d t = 0.$

2.1A1F2F4step 1.1step 1.2given∎

Subtract step 1.2 from step 1.1. The terminal terms cancel, yielding $\int_{Σ} k (Λ_{q_{1}}^{+} h - Λ_{q_{2}}^{+} h) d S_{g} d t = 0$ for every smooth compactly supported $k$ . The difference of the two traces belongs to $L^{2} (Σ)$ by [F2], hence is locally integrable. Its pairing with every $k$ is zero; [F4] implies that its $L^{2}$ class vanishes. This is the asserted equality of forward DN data.

Depends on

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Sources

Mikko Salo, On geometric inverse problems and microlocal analysis (full author lecture notes, arXiv:2312.08123v1) (standard reference, not scraped)
Medet Nursultanov and Lauri Oksanen, Introduction to inverse problems for hyperbolic PDEs (full lecture notes, arXiv:2311.06020v2) (standard reference, not scraped)