由非光滑电场重构各向同性电导率
Reconstruction of isotropic conductivities from non smooth electric fields
摘要：
本文研究了定义在$\mathbb{R}^d$中的给定非光滑梯度场$\nabla u$的各向同性可实现性，即当可以重构各向同性电导率$σ>0$使得$σ\nabla u$在$\mathbb{R}^d$中无发散时.一方面，当$\nabla u$在$\mathbb{R}^d$中是非零的、一致连续的，且$\triangle u$在$\mathbb{R}^d$中是有界函数的情况下，我们在合适的索伯列夫空间中，利用相关的梯度流结合求解常微分方程的DiPerna，Lions方法，证明了$\nabla u$的各向同性可实现性.另一方面，在$\nabla u$是分段正则的情况下，我们粗略地证明了各向同性可实现性成立的充要条件是$u$在梯度不连续界面两侧的法向导数具有相同的符号。给出了电导率重建的一些实例。
英文摘要：
In this paper we study the isotropic realizability of a given non smooth gradient field $\nabla u$ defined in $\mathbb{R}^d$, namely when one can reconstruct an isotropic conductivity $σ>0$ such that $σ\nabla u$ is divergence free in $\mathbb{R}^d$. On the one hand, in the case where $\nabla u$ is non-vanishing, uniformly continuous in $\mathbb{R}^d$ and $\triangle u$ is a bounded function in $\mathbb{R}^d$, we prove the isotropic realizability of $\nabla u$ 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 $\nabla u$ 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.
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