Jessica Crosse | Search arXiv

arXiv2026

Stable boundary determination of a complex anisotropic admittivity and its derivatives from a local Neumann-to-Dirichlet map

We study the classical anisotropic Calderón problem associated to the elliptic equation $\text{div}(σ\nabla u)=0$, where the complex admittivity $σ$ is of the form $σ=A(\cdot,a(\cdot))$ in a domain $Ω\subset\mathbb{R}^n$, $n\ge3$. We establish boundary stability estimates for $σ$ and its derivatives of arbitrary order from a local Neumann-to-Dirichlet map. Our results extend those of Comm. Partial Differential Equations, 34 (2009) from the real-valued conductivity setting to complex anisotropic admittivities.

math.AP↗

arXiv2026

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

We address Calderón's problem of stably determining the anisotropic complex admittivity $σ$ in a domain $Ω\subset\mathbb{R}^n$, with $n\geq3$, representing a conducting medium, in terms of a Dirichlet-to-Neumann map locally prescribed on a non-empty portion $Σ$ of the boundary of $Ω$, $\partialΩ$. $σ$ is assumed to be of type $σ(\cdot)=A(\cdot,a(\cdot))$ in $Ω$, where the one-parameter family of complex-symmetric matrices $[λ^{-1},\:λ]\ni t\mapsto A(\cdot,\: t)$ 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 $σ$ and its derivatives of arbitrary order on $Σ$, respectively, in terms of the local map.

math.AP↗