An inverse anisotropic conductivity problem induced by twisting a homogeneous cylindrical domain

Abstract

We consider the inverse problem of determining the unknown function $\alpha :\mathbb{R}\to \mathbb{R}$ from the DN map associated with the operator div $(A(x^{\prime },\alpha (x_3))\nabla \cdot )$ acting in the infinite straight cylindrical waveguide $\Omega =\omega \times \mathbb{R}$, where $\omega$ is a bounded domain of ${\mathbb{R}}^2$. Here $A=(A_{ij}(x))$, $x=(x^{\prime },x_3)\in \Omega$, 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 a priori fixed constant.