The local complex Calder\'on problem. Stability in a layered medium for a special type of anisotropic admittivity

TL;DR

This paper establishes stability estimates for the Calderón problem in layered anisotropic media with complex admittivity, providing quantitative bounds based on boundary measurements and known layer structures.

Contribution

It introduces new Hölder and Lipschitz stability estimates for the inverse problem in layered anisotropic media with complex admittivity, assuming known layers and affine unknown functions.

Findings

Hölder stability estimates for admittivity reconstruction

Lipschitz stability bounds based on boundary data

Quantitative relation between measurements and admittivity

Abstract

We deal with Calder\'on's problem in a layered anisotropic medium $\Omega\subset\mathbb{R}^n$, $n\geq 3$, with complex anisotropic admittivity $\sigma=\gamma A$, where $A$ is a known Lipschitz matrix-valued function. We assume that the layers of $\Omega$ are fixed and known and that $\gamma$ is an unknown affine complex-valued function on each layer. We provide H\"{o}lder and Lipschitz stability estimates of $\sigma$ in terms of an ad hoc misfit functional as well as the more classical Dirichlet to Neumann map localised on some open portion $\Sigma$ of $\partial\Omega$, respectively.