An inverse anisotropic conductivity problem induced bytwisting a homogeneous cylindrical domain

TL;DR

This paper investigates the inverse problem of determining a twisting function in a cylindrical waveguide from boundary measurements, proving Lipschitz stability when the function is constant or close to a fixed constant.

Contribution

It establishes Lipschitz stability for the inverse anisotropic conductivity problem in a twisted waveguide when the twisting function is constant or near a fixed constant.

Findings

Lipschitz stability for constant twisting functions

Stability results extend to approximate DN maps

Applicable to functions close to a fixed constant

Abstract

We consider the inverse problem of determining the unknown function $\alpha: \mathbb{R} \rightarrow \mathbb{R}$ from the DN map associated to the operator $\mbox{div}(A(x',\alpha (x\_3))\nabla \cdot)$ acting in the infinite straight cylindrical waveguide $\Omega =\omega \times \mathbb{R}$, where $\omega$ is a bounded domain of $\mathbb{R}^2$. Here $A=(A\_{ij}(x))$, $x=(x',x\_3) \in \Omega$, is a matrix-valued metric on $\Omega$ obtained by straightening a twisted waveguide. This inverse anisotropic conductivity problem remains generally open, unless the unknown function $\alpha$ is assumed to be constant. In this case we prove Lipschitz stability in the determination of $\alpha$ from the corresponding DN map. The same result remains valid upon substituting a suitable approximation of the DN map, provided the function $\alpha$ is sufficiently close to some {\it a priori} fixed constant.