论文标题
低规律性理论的逆电导率问题
Low regularity theory for the inverse fractional conductivity problem
论文作者
论文摘要
我们用$ h^{s，n/s} $规则性假设在所有维度上的逆电导率问题表征了部分数据唯一性。这扩展了$ h^{2s，\ frac {n} {2s}} \ cap h^s $电导率的较早结果。每当在与域具有正距离的不相交的打开集中进行测量时，我们将在一个方向界定的域上构建反示例。特别是，我们在特殊情况下提供反例，在（n/4,1）$，$ n = 2,3 $中，由于较早的规律性条件而缺少。我们还提供了不基于Runge近似属性的独特结果的新证明。当$ n = 3,4 $时，我们的作品可以看作是Haberman独特定理的分数。这项工作的一个动机是布朗的猜想是，对于经典的calderón问题的独特性，$ w^{1，n} $电导率也在维度上，$ n \ geq 5 $。
We characterize partial data uniqueness for the inverse fractional conductivity problem with $H^{s,n/s}$ regularity assumptions in all dimensions. This extends the earlier results for $H^{2s,\frac{n}{2s}}\cap H^s$ conductivities by Covi and the authors. We construct counterexamples to uniqueness on domains bounded in one direction whenever measurements are performed in disjoint open sets having positive distance to the domain. In particular, we provide counterexamples in the special cases $s \in (n/4,1)$, $n=2,3$, missing in the literature due to the earlier regularity conditions. We also give a new proof of the uniqueness result which is not based on the Runge approximation property. Our work can be seen as a fractional counterpart of Haberman's uniqueness theorem for the classical Calderón problem with $W^{1,n}$ conductivities when $n=3,4$. One motivation of this work is Brown's conjecture that uniqueness for the classical Calderón problem holds for $W^{1,n}$ conductivities also in dimensions $n \geq 5$.
