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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.
|Number of pages||24|
|Journal||SIAM Journal on Mathematical Analysis|
|Publication status||Published - 9 Jan 2020|
|MoE publication type||A1 Journal article-refereed|
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- 1 Finished
Centre of Excellence of Inverse Modelling and Imaging
Hyvönen, N., Ojalammi, A., Puska, J., Kuutela, T., Perkkiö, L. & Hirvi, P.
01/01/2018 → 31/12/2020
Project: Academy of Finland: Other research funding