Non-uniqueness results for the anisotropic Calderón problem with data measured on disjoint sets
[Non-unicité pour le problème de Calderón anisotropique avec données mesurées sur des ensembles disjoints]
Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 119-170.

Dans cet article, on montre qu’il y a non-unicité pour le problème de Calderón sur des variétés riemanniennes quand les données de Dirichlet et de Neumann sont mesurées sur des sous-ensembles disjoints du bord. On construit des contre-exemples à l’unicité en dimension 2 et 3 pour des variétés riemanniennes à bord topologiquement équivalentes à des cylindres dont les fibres sont des tores. La construction pourrait être aisément étendue à des variétés riemanniennes de dimensions supérieures.

In this paper, we show that there is non-uniqueness in the Calderón problem on Riemannian manifolds when the Dirichlet and Neumann data are measured on disjoint sets of the boundary. We provide counterexamples in the case of two and three dimensional Riemannian manifolds with boundary having the topology of circular cylinders in dimension two and toric cylinders in dimension three. The construction could be easily extended to higher dimensional Riemannian manifolds.

Reçu le :
Accepté le :
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DOI : https://doi.org/10.5802/aif.3240
Classification : 81U40,  35P25,  58J50
Mots clés : Problème de Calderón anisotropique, équation d’Helmholtz sur une variété riemannienne, problèmes de Sturm–Liouville, fonctions de Weyl–Titchmarsh
@article{AIF_2019__69_1_119_0,
     author = {Daud\'e, Thierry and Kamran, Niky and Nicoleau, Fran\c{c}ois},
     title = {Non-uniqueness results for the anisotropic Calder\'on problem with data measured on disjoint sets},
     journal = {Annales de l'Institut Fourier},
     pages = {119--170},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {1},
     year = {2019},
     doi = {10.5802/aif.3240},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3240/}
}
Daudé, Thierry; Kamran, Niky; Nicoleau, François. Non-uniqueness results for the anisotropic Calderón problem with data measured on disjoint sets. Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 119-170. doi : 10.5802/aif.3240. http://archive.numdam.org/articles/10.5802/aif.3240/

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