Counterexamples to uniqueness in the inverse fractional conductivity problem with partial data

IF 1.2 4区 数学 Q2 MATHEMATICS, APPLIED
J. Railo, Philipp Zimmermann
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引用次数: 11

Abstract

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 \begin{document}$ \Omega \subset {\mathbb R}^n $\end{document} and any disjoint open sets \begin{document}$ W_1, W_2 \Subset {\mathbb R}^n \setminus \overline{\Omega} $\end{document} there always exist two positive, bounded, smooth, conductivities \begin{document}$ \gamma_1, \gamma_2 $\end{document}, \begin{document}$ \gamma_1 \neq \gamma_2 $\end{document}, with equal partial exterior Dirichlet-to-Neumann maps \begin{document}$ \Lambda_{\gamma_1}f|_{W_2} = \Lambda_{\gamma_2}f|_{W_2} $\end{document} for all \begin{document}$ f \in C_c^{\infty}(W_1) $\end{document}. The proof uses the characterization of equal exterior data from another work of the authors in combination with the maximum principle of fractional Laplacians. The main technical difficulty arises from the requirement that the conductivities should be strictly positive and have a special regularity property \begin{document}$ \gamma_i^{1/2}-1 \in H^{2s, \frac{n}{2s}}( {\mathbb R}^n) $\end{document} for \begin{document}$ i = 1, 2 $\end{document}. We also provide counterexamples on domains that are bounded in one direction when \begin{document}$ n \geq 4 $\end{document} or \begin{document}$ s \in (0, n/4] $\end{document} when \begin{document}$ n = 2, 3 $\end{document} using a modification of the argument on bounded domains.

部分数据下分数阶电导率逆问题唯一性的反例
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 \begin{document}$ \Omega \subset {\mathbb R}^n $\end{document} and any disjoint open sets \begin{document}$ W_1, W_2 \Subset {\mathbb R}^n \setminus \overline{\Omega} $\end{document} there always exist two positive, bounded, smooth, conductivities \begin{document}$ \gamma_1, \gamma_2 $\end{document}, \begin{document}$ \gamma_1 \neq \gamma_2 $\end{document}, with equal partial exterior Dirichlet-to-Neumann maps \begin{document}$ \Lambda_{\gamma_1}f|_{W_2} = \Lambda_{\gamma_2}f|_{W_2} $\end{document} for all \begin{document}$ f \in C_c^{\infty}(W_1) $\end{document}. The proof uses the characterization of equal exterior data from another work of the authors in combination with the maximum principle of fractional Laplacians. The main technical difficulty arises from the requirement that the conductivities should be strictly positive and have a special regularity property \begin{document}$ \gamma_i^{1/2}-1 \in H^{2s, \frac{n}{2s}}( {\mathbb R}^n) $\end{document} for \begin{document}$ i = 1, 2 $\end{document}. We also provide counterexamples on domains that are bounded in one direction when \begin{document}$ n \geq 4 $\end{document} or \begin{document}$ s \in (0, n/4] $\end{document} when \begin{document}$ n = 2, 3 $\end{document} using a modification of the argument on bounded domains.
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来源期刊
Inverse Problems and Imaging
Inverse Problems and Imaging 数学-物理:数学物理
CiteScore
2.50
自引率
0.00%
发文量
55
审稿时长
>12 weeks
期刊介绍: Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing. This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.
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