Reconstruction of isotropic conductivities from non smooth electric fields
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 1173-1193.

In this paper we study the isotropic realizability of a given non smooth gradient field u defined in R d , namely when one can reconstruct an isotropic conductivity σ > 0 such that σ u is divergence free in R d . On the one hand, in the case where u is non-vanishing, uniformly continuous in R d and Δ u is a bounded function in R d , 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.

DOI : 10.1051/m2an/2018013
Classification : 35B27, 78A30, 37C10
Mots clés : Isotropic conductivity, electric field, conductivity reconstruction, gradient flow, triangulation
Briane, Marc 1

1
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     title = {Reconstruction of isotropic conductivities from non smooth electric fields},
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     pages = {1173--1193},
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Briane, Marc. Reconstruction of isotropic conductivities from non smooth electric fields. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 1173-1193. doi : 10.1051/m2an/2018013. http://www.numdam.org/articles/10.1051/m2an/2018013/

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