Volume 52, Number 3, May–June 2018
|Page(s)||1173 - 1193|
|Published online||14 September 2018|
Reconstruction of isotropic conductivities from non smooth electric fields
University of Rennes, INSA Rennes, CNRS, IRMAR – UMR 6625,
* Corresponding author: firstname.lastname@example.org
Accepted: 9 February 2018
In this paper we study the isotropic realizability of a given non smooth gradient field ∇u defined in ℝd, namely when one can reconstruct an isotropic conductivity σ > 0 such that σ∇u is divergence free in ℝd. On the one hand, in the case where ∇u is non-vanishing, uniformly continuous in ℝd and Δu is a bounded function in ℝd, we prove the isotropic realizability of ∇u using the associated gradient flow combined with the DiPerna, Lions approach for solving ordinary differential equations in suitable Sobolev spaces. On the other hand, in the case where ∇u is piecewise regular, we prove roughly speaking that the isotropic realizability holds if and only if the normal derivatives of u on each side of the gradient discontinuity interfaces have the same sign. Some examples of conductivity reconstruction are given.
Mathematics Subject Classification: 35B27 / 78A30 / 37C10
Key words: Isotropic conductivity / electric field / conductivity reconstruction / gradient flow / triangulation
© EDP Sciences, SMAI 2018
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