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

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.