Global counterexamples to uniqueness for a Calderón problem with $C^k$ conductivities
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916403668320256 |
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| author | Daudé, Thierry Helffer, Bernard Kamran, Niky Nicoleau, François |
| author_facet | Daudé, Thierry Helffer, Bernard Kamran, Niky Nicoleau, François |
| contents | Let $Ω\subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $γ$ be a smooth conductivity in $\overlineΩ$. Consider a non-zero frequency $λ_0$ which does not belong to the Dirichlet spectrum of $L_γ= -{\rm div} (γ\nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of non-isometric $C^k$ conductivities $(γ_1, γ_2)$ on $\overlineΩ$, which are close to $γ$ such that the associated DN maps at frequency $λ_0$ satisfy \begin{equation*}
Λ_{γ_1,λ_0} = Λ_{γ_2,λ_0}. \end{equation*} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global counterexamples to uniqueness for a Calderón problem with $C^k$ conductivities Daudé, Thierry Helffer, Bernard Kamran, Niky Nicoleau, François Analysis of PDEs Mathematical Physics Spectral Theory 81U40, 35P25, 58J50 Let $Ω\subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $γ$ be a smooth conductivity in $\overlineΩ$. Consider a non-zero frequency $λ_0$ which does not belong to the Dirichlet spectrum of $L_γ= -{\rm div} (γ\nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of non-isometric $C^k$ conductivities $(γ_1, γ_2)$ on $\overlineΩ$, which are close to $γ$ such that the associated DN maps at frequency $λ_0$ satisfy \begin{equation*} Λ_{γ_1,λ_0} = Λ_{γ_2,λ_0}. \end{equation*} |
| title | Global counterexamples to uniqueness for a Calderón problem with $C^k$ conductivities |
| topic | Analysis of PDEs Mathematical Physics Spectral Theory 81U40, 35P25, 58J50 |
| url | https://arxiv.org/abs/2406.14063 |