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Reconstruction of singular and degenerate inclusions in Calderón's problem

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  • 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.

    Mathematics Subject Classification: 35R30, 35R05, 47H05.


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