Stability in the anisotropic Calder\'{o}n problem for Painlev\'e-Liouville Riemannian manifolds

TL;DR

This paper establishes a logarithmic stability estimate for the anisotropic Calderón inverse problem on Painlevé-Liouville Riemannian manifolds, showing how small boundary measurement differences imply close conformal factors.

Contribution

It provides the first stability results for the anisotropic Calderón problem on this class of manifolds with partial data, extending inverse problem theory to new geometric settings.

Findings

Logarithmic stability estimate for global inverse problem

Similar stability results for partial boundary data

Quantitative bounds relating boundary data differences to metric conformal factors

Abstract

We study the question of stability of the global and partial anisotropic Calder\'on inverse problems for the class of Painlev\'e-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 $g=\alpha^4 g_{0}$ is any conformal deformation of the product metric $g_{0}=dx^2+g_{K}$ on $M$ which is compatible with the Painlev\'e block-separability of the Laplace-Beltrami operator $\Delta_{g_0}$. Given a pair of Painlev\'e-Liouville Riemannian manifolds $(M,g)$ and $(M,\tilde{g})$ satisfying some technical hypothesis, denoting the corresponding Dirichlet-to-Neumann maps by $\Lambda_{g}$ and $\Lambda_{\tilde{g}}$, and assuming that $\lVert \Lambda_{g}-\Lambda_{\tilde{g}}\rVert_{\mathcal{B}(H^{1/2}(\partial M), H^{-1/2}(\partial M))}\ = \epsilon $, we show a logarithmic stability result for the global anisotropic Calder\'on problem which says that there exists constants $C$ and $0<\theta<1$ such that $\| \alpha - \tilde{\alpha} \|_{C^{0,r}(M)} \leq C \left( \ln \frac{1}{\epsilon} \right)^{-\theta}$ for some $0<r<1$. Similar results are obtained for the partial anisotropic Calder\'on problem, corresponding to the case where the data are measured on only one connected component of the boundary.