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

TL;DR

This work addresses the stability of the anisotropic Calderón problem on Painlevé-Liouville manifolds, where the metric is conformally related to a product and the potential separates as $q(x,\omega)=\phi_1(x)+\phi_2(\omega)$. By exploiting block separability, the authors reduce the problem to 1D radial Schrödinger equations, analyze the associated Weyl-Titchmarsh data, and formulate a moment problem to recover the radial part $\phi_1$ from the Dirichlet-to-Neumann map. They obtain a logarithmic stability rate for the global problem and a corresponding rate for the partial data problem, with Hölder control at the boundary and localized logarithmic stability in the interior. The results hinge on a combination of spectral analysis, Carleman estimates, and elliptic regularity, and they extend classical log-type stability to this geometric setting, even when CGO constructions are unavailable due to the transversal geometry.

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 $g=α^4 g_{0}$ 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.

Theorems & Definitions (39)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Corollary 3.1
• proof
• Corollary 3.2
• proof
• Lemma 4.1
• proof