The local Calder\'on problem and the determination at the boundary of a complex anisotropic admittivity

TL;DR

This paper investigates the local Calderón problem for anisotropic complex admittivity, establishing stability estimates at the boundary based on partial boundary measurements.

Contribution

It introduces a method to determine the unknown scalar function in a known matrix family from local boundary data, with stability estimates for the boundary values.

Findings

Established Lipschitz stability at the boundary.

Proved Hölder stability for derivatives of any order on the boundary.

Achieved boundary determination with partial data.

Abstract

We address Calder\'on's problem of stably determining the anisotropic complex admittivity $\sigma$ in a domain $\Omega\subset\mathbb{R}^n$, with $n\geq3$, representing a conducting medium, in terms of a Dirichlet-to-Neumann map locally prescribed on a non-empty portion $\Sigma$ of the boundary of $\Omega$, $\partial\Omega$. $\sigma$ is assumed to be of type $\sigma(\cdot)=A(\cdot,a(\cdot))$ in $\Omega$, where the one-parameter family of complex-symmetric matrices $[\lambda^{-1},\:\lambda]\ni t\mapsto A(\cdot,\: t)$ is assumed to be a-priori known and the scalar function $a$ is unknown. We establish Lipschitz and H\"older stability estimates at the boundary for $\sigma$ and its derivatives of arbitrary order on $\Sigma$, respectively, in terms of the local map.