Reconstruction of non-self-adjoint anisotropic and complex inclusions in the Calder\'on problem

TL;DR

This paper extends the monotonicity method for inclusion detection in the anisotropic Calderón problem to handle very general non-self-adjoint perturbations in various dimensions, broadening applicability.

Contribution

It generalizes existing methods to include non-self-adjoint perturbations and accounts for anisotropic real conductivity and permittivity in the forward model.

Findings

Results hold in any spatial dimension $d \,\geq\, 2$.

Method requires only unique continuation based on the self-adjoint part of the background.

Applicable to inclusions reachable from the boundary under certain conditions.

Abstract

We generalize recent results on the monotonicity method, for inclusion detection in the partial data anisotropic Calder\'on problem, to very general non-self-adjoint perturbations. This involves a forward model that accounts for both the anisotropic real conductivity and the anisotropic permittivity, and the results hold in any spatial dimension $d \geq 2$. We assume that the inclusion boundaries can be reached from the domain boundary via a set on which the background conductivity is self-adjoint, and that a definiteness condition holds near the inclusion boundaries. Away from the inclusion boundaries we allow general $L^\infty$ non-self-adjoint perturbations. We only require unique continuation based on the self-adjoint part of the background conductivity, thus making the methods compatible with generic unique continuation results.