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On regularity of the logarithmic forward map of electrical impedance tomography

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  • Henrik Garde
  • Nuutti Hyvönen, Department of Mathematics and Systems Analysis, Aalto University, Finland
  • Topi Kuutela, Department of Mathematics and Systems Analysis, Aalto University, Finland

This work considers properties of the logarithm of the Neumann-to-Dirichlet boundary map for the conductivity equation in a Lipschitz domain. It is shown that the mapping from the (logarithm of) the conductivity, i.e., the (logarithm of) the coefficient in the divergence term of the studied elliptic partial differential equation, to the logarithm of the Neumann-to-Dirichlet map is continuously Frechet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Frechet derivative defines a bounded linear operator on the space of square integrable functions living on the domain boundary, although the logarithm of the Neumann-to-Dirichlet map itself is unbounded in that topology. In particular, it follows from the fundamental theorem of calculus that the difference between the logarithms of any two Neumannto- Dirichlet maps is always bounded on the space of square integrable functions. All aforementioned results also hold if the Neumann-to-Dirichlet boundary map is replaced by its inverse, i.e., the Dirichlet-to-Neumann map.

OriginalsprogEngelsk
TidsskriftSIAM Journal on Mathematical Analysis
Vol/bind52
Nummer1
Sider (fra-til)197-220
Antal sider24
ISSN0036-1410
DOI
StatusUdgivet - 2020
Eksternt udgivetJa

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