论文标题
全球反向电导率问题
The global inverse fractional conductivity problem
论文作者
论文摘要
我们证明了\ emph {global}唯一性对于沿一个方向界限的域上的分数电导率方程的逆问题。假定电导率在域的外部是各向同性和非平凡的，而数据的形式是在外部的Nondischoint开放子集中测得的部分dirichlet到Neumann（DN）映射的形式。这可以看作是经典逆电导率问题的分数。该证明基于DN地图的唯一延续属性（UCP）和从部分外部DN地图中的外部确定方法。这与Kohn和Vogelius的经典边界确定方法类似。最重要的技术新颖性是构造分数电导率方程式的特殊溶液序列，其在极限中的差异能量可以集中在外部的任何给定点。这是独立于UCP实现的，尽管方程式非局部性。由于最近两位作者最近的反例，我们的结果几乎完全表征了与各向同性全球电导率的部分数据的反向电导率问题的唯一性。
We prove \emph{global} uniqueness for an inverse problem for the fractional conductivity equation on domains that are bounded in one direction. The conductivities are assumed to be isotropic and nontrivial in the exterior of the domain, while the data is given in the form of partial Dirichlet-to-Neumann (DN) maps measured in nondisjoint open subsets of the exterior. This can be seen as the fractional counterpart of the classical inverse conductivity problem. The proof is based on a unique continuation property (UCP) for the DN maps and an exterior determination method from the partial exterior DN maps. This is analogous to the classical boundary determination method by Kohn and Vogelius. The most important technical novelty is the construction of sequences of special solutions to the fractional conductivity equation whose Dirichlet energies in the limit can be concentrated at any given point in the exterior. This is achieved independently of the UCP and despite the nonlocality of the equation. Due to the recent counterexamples by the last two authors, our results almost completely characterize uniqueness for the inverse fractional conductivity problem with partial data for isotropic global conductivities.
