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On the Hidden Mechanism Behind Non-uniqueness for the Anisotropic Calderón Problem with Data on Disjoint Sets

  • Thierry Daudé
  • Niky Kamran
  • François NicoleauEmail author
Article
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Abstract

We show that there is generically non-uniqueness for the anisotropic Calderón problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary (Mg) of dimension \(n\ge 3\), there exist in the conformal class of g an infinite number of Riemannian metrics \(\tilde{g}\) such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet data \(\Gamma _D\) and Neumann data \(\Gamma _N\) are measured on disjoint sets and satisfy \(\overline{\Gamma _D \cup \Gamma _N} \ne \partial M\). The conformal factors that lead to these non-uniqueness results for the anisotropic Calderón problem satisfy a nonlinear elliptic PDE of Yamabe type on the original manifold (Mg) and are associated with a natural but subtle gauge invariance of the anisotropic Calderón problem with data on disjoint sets. We then construct a large class of counterexamples to uniqueness in dimension \(n\ge 3\) to the anisotropic Calderón problem at fixed frequency with data on disjoint sets and modulo this gauge invariance. This class consists in cylindrical Riemannian manifolds with boundary having two ends (meaning that the boundary has two connected components), equipped with a suitably chosen warped product metric.

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Notes

Acknowledgements

The authors would like to warmly thank Yves Dermenjian for suggesting the crucial role of the transformation law of the Laplacian under conformal scaling in the gauge invariance for the Calderon problem with disjoint data and also Gilles Carron for his help in solving the nonlinear PDE of Yamabe type encountered in Sects. 2 and 4.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Thierry Daudé
    • 1
  • Niky Kamran
    • 2
  • François Nicoleau
    • 3
    Email author
  1. 1.Département de Mathématiques, UMR CNRS 8088Université de Cergy-PontoiseCergy-PontoiseFrance
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629Nantes Cedex 03France

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