Reconstruction of singular and degenerate inclusions in Calder\'on's problem

TL;DR

This paper extends Calderón's inverse conductivity problem to include singular and degenerate conductivities, providing a constructive method to recover the support of such perturbations using boundary measurements.

Contribution

It generalizes previous monotonicity-based reconstruction results to include A_2-Muckenhoupt weights, allowing for singular and degenerate conductivities in Calderón's problem.

Findings

Constructive characterization of the support of singular and degenerate perturbations.

Extension of monotonicity methods to A_2-weighted conductivities.

Recovery of the shape of perturbations from boundary measurements.

Abstract

We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calder\'on's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $0$ and $\infty$ in some parts of the domain and values bounded away from $0$ and $\infty$ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an $A_2$-Muckenhoupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In particular, the coefficient may tend to $0$ and $\infty$ in a controlled manner, which goes beyond the standard setting of Calder\'on's problem. Our main result constructively characterises the outer shape of the support of such a general perturbation, based on a local Neumann-to-Dirichlet map defined on an open subset of the domain boundary.