Reconstruction of singular and degenerate inclusions in Calderón's problem

Abstract

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 $\infty$ 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 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.

8 pages

References in corpus (2)

• Monotonicity Principle in Tomography of Nonlinear Conducting Materials
• Monotonicity-based reconstruction of extreme inclusions in electrical impedance tomography

Cited by in corpus (5)

• Linearised Calderón problem: Reconstruction and Lipschitz stability for infinite-dimensional spaces of unbounded perturbations
• Imaging of nonlinear materials via the Monotonicity Principle
• Piecewise nonlinear materials and Monotonicity Principle
• The inverse obstacle problem for nonlinear inclusions
• Reconstruction of cracks in Calderón's inverse conductivity problem using energy comparisons