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SUFFICIENT CONDITIONS FOR THE EXISTENCE OF LIMITING CARLEMAN WEIGHTS

Part of: Miscellaneous topics - Partial differential equations Classical differential geometry

Published online by Cambridge University Press:  02 March 2017

PABLO ANGULO-ARDOY ,
DANIEL FARACO  and
LUIS GUIJARRO
PABLO ANGULO-ARDOY
Affiliation:
E.T.S de Ingenieros Navales, Universidad Politécnica de Madrid, Madrid, Spain; [email protected]
DANIEL FARACO
Affiliation:
Department of Mathematics, Universidad Autónoma de Madrid, Madrid, Spain ICMAT CSIC-UAM-UCM-UC3M, Madrid, Spain; [email protected], [email protected]
LUIS GUIJARRO
Affiliation:
Department of Mathematics, Universidad Autónoma de Madrid, Madrid, Spain ICMAT CSIC-UAM-UCM-UC3M, Madrid, Spain; [email protected], [email protected]

Abstract

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In Angulo-Ardoy et al. [Anal. PDE, 9(3) (2016), 575–596], we found some necessary conditions for a Riemannian manifold to admit a local limiting Carleman weight (LCW), based on the Cotton–York tensor in dimension 3 and the Weyl tensor in dimension 4. In this paper, we find further necessary conditions for the existence of local LCWs that are often sufficient. For a manifold of dimension 3 or 4, we classify the possible Cotton–York, or Weyl tensors, and provide a mechanism to find out whether the manifold admits local LCW for each type of tensor. In particular, we show that a product of two surfaces admits an LCW if and only if at least one of the two surfaces is of revolution. This provides an example of a manifold satisfying the eigenflag condition of Angulo-Ardoy et al. [Anal. PDE, 9(3) (2016), 575–596] but not admitting LCW.

MSC classification

Primary: 35R30: Inverse problems
Secondary: 53A30: Conformal differential geometry
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e7
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

References

Angulo-Ardoy, P., ‘On the set of metrics without local limiting Carleman weights’, Inv. Prob. Imag. 11(1) (2017), 47–64.Google Scholar
Angulo-Ardoy, P., Faraco, D., Guijarro, L. and Ruiz, A., ‘Obstructions to the existence of limiting Carleman weights’, Anal. PDE 9(3) (2016), 575–596.CrossRefGoogle Scholar
Besse, A. L., Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 10 (Springer, Berlin, 1987).CrossRefGoogle Scholar
Caro, P. and Rogers, K. M., ‘Global uniqueness for the Calderón problem with Lipschitz conductivities’, Forum Math. Pi 4 (2016), e2.CrossRefGoogle Scholar
Caro, P. and Salo, M., ‘Stability of the Calderón problem in admissible geometries’, Inv. Prob. Imag. 8 (2014), 939–957.Google Scholar
Do Carmo, M. P., Riemannian Geometry Mathematics: Theory and Applications (Birkhaüser Boston, Inc., Boston, MA, 1992).CrossRefGoogle Scholar
Dos Santos Ferreira, D., Kenig, C. E. and Salo, M., ‘Determining an unbounded potential from Cauchy data in admissible geometries’, Comm. Partial Differential Equations 38(1) (2013), 50–68.CrossRefGoogle Scholar
Dos Santos Ferreira, D., Kenig, C. E., Salo, M. and Uhlmann, G., ‘Limiting Carleman weights and anisotropic inverse problems’, Invent. Math. 178(1) (2009), 119–171.CrossRefGoogle Scholar
Dos Santos Ferreira, D., Yaroslav, K., Lassa, M. and Salo, M., ‘The Calderon problem in transversally anisotropic geometries’, J. Euro. Math. Soc. 18(11) (2016), 2579–2626.CrossRefGoogle Scholar
Haberman, B., ‘Tataru, Daniel Uniqueness in Calderón’s problem with Lipschitz conductivities’, Duke Math. J. 162(3) (2013), 496–516.CrossRefGoogle Scholar
Haberman, B., ‘Uniqueness in Calderón’s problem for conductivities with unbounded gradient’, Comm. Math. Phys. 340(2) (2015), 639–659.CrossRefGoogle Scholar
Kenig, C. E., Salo, M. and Uhlmann, G., ‘Reconstructions from boundary measurements on admissible manifolds’, Inverse Probl. Imaging 5(4) (2011), 859–877.CrossRefGoogle Scholar
Kenig, C. E., Sjöstrand, J. and Uhlmann, G., ‘The Caldern problem with partial data’, Ann. of Math. (2) 165(2) (2007), 567–591.CrossRefGoogle Scholar
Lax, P. D., Linear Algebra, Pure and Applied Mathematics (Wiley and Sons, New York, 1997).Google Scholar
Liimatainen, T. and Salo, M., ‘Nowhere conformally homogeneous manifolds and limiting Carleman weights’, Inverse Probl. Imaging 6(3) (2012), 523–530.CrossRefGoogle Scholar
Salo, M., ‘The Calderón problem on Riemannian manifolds’, inInverse Problems and Applications: Inside Out II (ed. Uhlmann, G.) (MSRI Publications, Cambridge University Press, Cambridge, 2012).Google Scholar
Salo, M., ‘The Calderón problem on Riemannian manifolds’, inInverse Problems and Applications: Inside Out. II, Mathematical Sciences Research Institute Publications, 60 (Cambridge University Press, Cambridge, 2013), 167–247.Google Scholar
Sylvester, J. and Uhlmann, G., ‘A global uniqueness theorem for an inverse boundary value problem’, Ann. of Math. (2) 125 (1987), 153–169.CrossRefGoogle Scholar
Tojeiro, R., ‘Conformal de Rham decomposition of Riemannian manifolds’, Houston J. Math. 32(3) (2006), 725–743.Google Scholar