The local Calderón problem and the determination at the boundary of a complex anisotropic admittivity

Abstract

We address Calderón's problem of stably determining the anisotropic complex admittivity $\sigma$ in a domain $\Omega \subset {\mathbb{R}}^n$, with $n\ge 3$, 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 Hölder stability estimates at the boundary for $\sigma$ and its derivatives of arbitrary order on $\Sigma$, respectively, in terms of the local map.

23 pages, planning a journal submission