Reconstruction of singular and degenerate inclusions in Calderón's problem

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \infty $\end{document}</tex-math></inline-formula> in some parts of the domain and values bounded away from <inline-formula><tex-math id="M3">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \infty $\end{document}</tex-math></inline-formula> elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an <inline-formula><tex-math id="M5">\begin{document}$ A_2 $\end{document}</tex-math></inline-formula>-Muckenhoupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In particular, the coefficient may tend to <inline-formula><tex-math id="M6">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \infty $\end{document}</tex-math></inline-formula> 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.</p>