Stability in the anisotropic Calderón problem for Painlevé-Liouville Riemannian manifolds
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| Format: | Preprint |
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2026
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| _version_ | 1866917274697334784 |
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| author | Daudé, Thierry Kamran, Niky Nicoleau, François |
| author_facet | Daudé, Thierry Kamran, Niky Nicoleau, François |
| contents | We study the question of stability of the global and partial anisotropic Calderón inverse problems for the class of Painlevé-Liouville Riemannian manifolds, that is compact $n$-dimensional manifolds with boundary $(M,g)$, where $M=[0,1]\times K\,$, $K$ is any smooth closed connected orientable manifold of dimension $n-1$ endowed with a Riemannian metric $g_K$, and $g=α^4 g_{0}$ is any conformal deformation of the product metric $g_{0}=dx^2+g_{K}$ on $M$ which is compatible with the Painlevé block-separability of the Laplace-Beltrami operator $Δ_{g_0}$. Given a pair of Painlevé-Liouville Riemannian manifolds $(M,g)$ and $(M,\tilde{g})$ satisfying some technical hypothesis, denoting the corresponding Dirichlet-to-Neumann maps by $Λ_{g}$ and $Λ_{\tilde{g}}$, and assuming that $\lVert Λ_{g}-Λ_{\tilde{g}}\rVert_{\mathcal{B}(H^{1/2}(\partial M), H^{-1/2}(\partial M))}\ = ε$, we show a logarithmic stability result for the global anisotropic Calderón problem which says that there exists constants $C$ and $0<θ<1$ such that $\| α- \tildeα \|_{C^{0,r}(M)} \leq C \left( \ln \frac{1}ε \right)^{-θ}$ for some $0<r<1$. Similar results are obtained for the partial anisotropic Calderón problem, corresponding to the case where the data are measured on only one connected component of the boundary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_13703 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability in the anisotropic Calderón problem for Painlevé-Liouville Riemannian manifolds Daudé, Thierry Kamran, Niky Nicoleau, François Analysis of PDEs Mathematical Physics Spectral Theory We study the question of stability of the global and partial anisotropic Calderón inverse problems for the class of Painlevé-Liouville Riemannian manifolds, that is compact $n$-dimensional manifolds with boundary $(M,g)$, where $M=[0,1]\times K\,$, $K$ is any smooth closed connected orientable manifold of dimension $n-1$ endowed with a Riemannian metric $g_K$, and $g=α^4 g_{0}$ is any conformal deformation of the product metric $g_{0}=dx^2+g_{K}$ on $M$ which is compatible with the Painlevé block-separability of the Laplace-Beltrami operator $Δ_{g_0}$. Given a pair of Painlevé-Liouville Riemannian manifolds $(M,g)$ and $(M,\tilde{g})$ satisfying some technical hypothesis, denoting the corresponding Dirichlet-to-Neumann maps by $Λ_{g}$ and $Λ_{\tilde{g}}$, and assuming that $\lVert Λ_{g}-Λ_{\tilde{g}}\rVert_{\mathcal{B}(H^{1/2}(\partial M), H^{-1/2}(\partial M))}\ = ε$, we show a logarithmic stability result for the global anisotropic Calderón problem which says that there exists constants $C$ and $0<θ<1$ such that $\| α- \tildeα \|_{C^{0,r}(M)} \leq C \left( \ln \frac{1}ε \right)^{-θ}$ for some $0<r<1$. Similar results are obtained for the partial anisotropic Calderón problem, corresponding to the case where the data are measured on only one connected component of the boundary. |
| title | Stability in the anisotropic Calderón problem for Painlevé-Liouville Riemannian manifolds |
| topic | Analysis of PDEs Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2602.13703 |