The anisotropic Calderón problem for singular metrics of warped product type: the borderline between uniqueness and invisibility
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Bibliographic record
Abstract
In this paper, we investigate the anisotropic Calderón problem on cylindrical Riemannian manifolds with boundary having two ends and equipped with singular metrics ofwarped product type, that iswhose coefficients only depend on the horizontal direction of the cylinder. By singular, we mean that these coefficients are positive almost everywhere and belong to some L^p, 1 \leq p \leq \infty spaces only. Using the recent developments on Weyl–Titchmarsh’s theory for singular Sturm–Liouville operators, we prove that the local Dirichlet to Neumann maps at each end are well defined and determine the metric uniquely if Eventually, we show (in the warped product case and for zero frequency) that these uniqueness results are sharp by giving simple counterexamples for a class of singular metrics whose coefficients do not belong to the critical L^p space. All these counterexamples lead in fact to a region of space that is invisible to boundary measurements.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.010 |
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| Open science | 0.000 | 0.000 |
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| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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