{"title":"Counterexamples to inverse problems for the wave equation","authors":"Tony Liimatainen, L. Oksanen","doi":"10.3934/ipi.2021058","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>We construct counterexamples to inverse problems for the wave operator on domains in <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\mathbb{R}^{n+1} $\\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">\\begin{document}$ n \\ge 2 $\\end{document}</tex-math></inline-formula>, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\mathbb{R}^{n+1} $\\end{document}</tex-math></inline-formula> the metrics are conformal to the Minkowski metric.</p>","PeriodicalId":50274,"journal":{"name":"Inverse Problems and Imaging","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inverse Problems and Imaging","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/ipi.2021058","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 2
Abstract
We construct counterexamples to inverse problems for the wave operator on domains in \begin{document}$ \mathbb{R}^{n+1} $\end{document}, \begin{document}$ n \ge 2 $\end{document}, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On \begin{document}$ \mathbb{R}^{n+1} $\end{document} the metrics are conformal to the Minkowski metric.
We construct counterexamples to inverse problems for the wave operator on domains in \begin{document}$ \mathbb{R}^{n+1} $\end{document}, \begin{document}$ n \ge 2 $\end{document}, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On \begin{document}$ \mathbb{R}^{n+1} $\end{document} the metrics are conformal to the Minkowski metric.
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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