no. 1
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]
Daudé, Thierry1 ; Kamran, Niky2 ; Nicoleau, François3
1 Département de Mathématiques, UMR CNRS 8088 Université de Cergy-Pontoise 95302 Cergy-Pontoise (France)
2 Department of Mathematics and Statistics McGill University Montreal, QC, H3A 2K6 (Canada)
3 Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629 2 Rue de la Houssinière BP 92208 44322 Nantes Cedex 03 (France)
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 :
Publié le :
DOI : 10.5802/aif.3240
Classification : 81U40, 35P25, 58J50
Keywords: Anisotropic Calderón problem, Helmholtz equation on a Riemannian manifold, Sturm–Liouville problems, Weyl–Titchmarsh functions
Mot clés : Problème de Calderón anisotropique, équation d’Helmholtz sur une variété riemannienne, problèmes de Sturm–Liouville, fonctions de Weyl–Titchmarsh

Daudé, Thierry 1 ; Kamran, Niky 2 ; Nicoleau, François 3

1 Département de Mathématiques, UMR CNRS 8088 Université de Cergy-Pontoise 95302 Cergy-Pontoise (France)
2 Department of Mathematics and Statistics McGill University Montreal, QC, H3A 2K6 (Canada)
3 Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629 2 Rue de la Houssinière BP 92208 44322 Nantes Cedex 03 (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     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},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.3240/

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