Reconstruction of isotropic conductivities from non smooth electric\n fields

2018/02/12 by Marc Briane, Briane, Marc
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Physics (physics.class-ph) #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1802.04857

openalex publication_date 2018/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the isotropic realizability of a given non smooth\ngradient field \∇ u defined in \ℝ^d, namely when one can\nreconstruct an isotropic conductivity \σ>0 such that \σ\∇ u is\ndivergence free in \ℝ^d. On the one hand, in the case where \∇ν is non-vanishing, uniformly continuous in \ℝ^d and triangle u\nis a bounded function in \ℝ^d, we prove the isotropic realizability\nof \∇ u using the associated gradient flow combined with the DiPerna,\nLions approach for solving ordinary differential equations in suitable Sobolev\nspaces. On the other hand, in the case where \∇ u is piecewise regular,\nwe prove roughly speaking that the isotropic realizability holds if and only if\nthe normal derivatives of u on each side of the gradient discontinuity\ninterfaces have the same sign. Some examples of conductivity reconstruction are\ngiven.\n

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