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Abstract
This work considers properties of the logarithm of the NeumanntoDirichlet 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 NeumanntoDirichlet map is continuously Fréchet differentiable between natural topologies. Moreover, for any essentially bounded perturbation of the conductivity, the Fréchet derivative defines a bounded linear operator on the space of square integrable functions living on the domain boundary, although the logarithm of the NeumanntoDirichlet 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 NeumanntoDirichlet maps is always bounded on the space of square integrable functions. All aforementioned results also hold if the NeumanntoDirichlet boundary map is replaced by its inverse, i.e. the DirichlettoNeumann map.
Original language  English 

Journal  SIAM Journal on Mathematical Analysis 
Volume  52 
Issue number  1 
Pages (fromto)  197220 
Number of pages  24 
ISSN  00361410 
DOIs  
Publication status  Published  2020 
Keywords
 NeumanntoDirichlet map
 Fréchet derivative
 logarithm
 functional calculus
 electrical impedance tomography
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Activities
 1 Visiting another research institution

Aalto University
Henrik Garde (Visiting researcher)
1 Feb 2019 → 31 Dec 2019Activity: Visiting another research institution