论文标题
反对偏分的电导率问题的唯一性与部分数据的唯一性
Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data
论文作者
论文摘要
我们在一般有限的开放集中的所有维度中为部分数据构造的部分数据逆问题构建反向问题。特别是，我们表明，对于任何有限的域$ω\ subset \ mathbb {r}^n $以及任何不连接的打开集$ w_1，w_2 \ subset \ mathbb {r}^n \ setMinus \barΩ部分外部dirichlet to-neumann映射$λ_{γ_1} f | _ {w_2} =λ_{γ_2} f | _ {w_2} $ for c_c^\ infty（w_1）$中的所有$ f \ for c_c^\ in证明使用了作者的另一项作品的同等外部数据以及分数拉普拉奇人的最大原理的表征。主要的技术难度是由于要求的电导率应严格为正，并且具有特殊的规律性属性$γ_i^{1/2} -1 \ in H^{2S，\ frac {n} {n} {2s}}}}}}}（\ Mathbb {r}^n）$ i = 1,2,2 $。当$ n \ geq 4 $或$ s \ in（0，n/4] $时，当$ n = 2,3 $使用有限域上的参数修改时，我们还会在一个方向上提供反示例。
We construct counterexamples for the partial data inverse problem for the fractional conductivity equation in all dimensions on general bounded open sets. In particular, we show that for any bounded domain $Ω\subset \mathbb{R}^n$ and any disjoint open sets $W_1,W_2 \Subset \mathbb{R}^n \setminus \barΩ$ there always exist two positive, bounded, smooth, conductivities $γ_1,γ_2$, $γ_1 \neq γ_2$, with equal partial exterior Dirichlet-to-Neumann maps $Λ_{γ_1}f|_{W_2} = Λ_{γ_2}f|_{W_2}$ for all $f \in C_c^\infty(W_1)$. The proof uses the characterization of equal exterior data from another work of the authors in combination with the maximum principle of fractional Laplacians. The main technical difficulty arises from the requirement that the conductivities should be strictly positive and have a special regularity property $γ_i^{1/2}-1 \in H^{2s,\frac{n}{2s}}(\mathbb{R}^n)$ for $i=1,2$. We also provide counterexamples on domains that are bounded in one direction when $n \geq 4$ or $s \in (0,n/4]$ when $n=2,3$ using a modification of the argument on bounded domains.
