Stability in the anisotropic Calderón problem for Painlevé-Liouville Riemannian manifolds

Abstract

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 $g={\alpha }^4{g}_0$ is any conformal deformation of the product metric $g_0=d{x}^2+g_K$ on $M$ which is compatible with the Painlevé block-separability of the Laplace-Beltrami operator ${\Delta }_{g_0}$. 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 ${\Lambda }_g$ and ${\Lambda }_{\tilde{g}}$, and assuming that $\Vert {\Lambda }_g-{\Lambda }_{\tilde{g}}{\Vert }_{\mathcal{B}(H^{1/2}(\partial M),H^{-1/2}(\partial M))}=\epsilon$, we show a logarithmic stability result for the global anisotropic Calderón problem which says that there exists constants $C$ and $0<\theta <1$ such that $\Vert \alpha -\tilde{\alpha }{\Vert }_{C^{0,r}(M)}\le C{\left(\ln \frac{1}{\epsilon }\right)}^{-\theta }$ 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.