Reconstruction of singular and degenerate inclusions in Calderón's problem

Henrik Garde; Nuutti Hyvönen

doi:10.3934/ipi.2022021

Article Contents

2022, Volume 16, Issue 5: 1219-1227. Doi: 10.3934/ipi.2022021

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Reconstruction of singular and degenerate inclusions in Calderón's problem

Henrik Garde^1, , and
Nuutti Hyvönen^2,

1.
Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark
2.
Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076 Helsinki, Finland

^*Corresponding author
^*Corresponding author

Received: October 31, 2021

Revised: February 28, 2022

Early access: April 2022

Published: October 2022

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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.

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

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References

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