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

We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderón’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 ∞ in some parts of the domain and values bounded away from 0 and ∞ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an A2-Muckenhaupt 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 ∞ in a controlled manner, which goes beyond the standard setting of Calderón’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.