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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The Calderón problem with partial data in two dimensions


Authors: Oleg Yu. Imanuvilov, Gunther Uhlmann and Masahiro Yamamoto
Journal: J. Amer. Math. Soc. 23 (2010), 655-691
MSC (2010): Primary 35R30; Secondary 35Q60
Published electronically: February 16, 2010
MathSciNet review: 2629983
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Abstract: We prove for a two-dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary uniquely determines the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can uniquely determine the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.


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Additional Information

Oleg Yu. Imanuvilov
Affiliation: Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, Colorado 80523
Email: oleg@math.colostate.edu

Gunther Uhlmann
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: gunther@math.washington.edu

Masahiro Yamamoto
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153, Japan
Email: myama@ms.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0894-0347-10-00656-9
PII: S 0894-0347(10)00656-9
Keywords: Calder\'on's problem, partial data, complex geometrical optics solutions
Received by editor(s): November 25, 2008
Published electronically: February 16, 2010
Additional Notes: The first author was partly supported by NSF grant DMS 0808130.
The second author was partly supported by the NSF and a Walker Family Endowed Professorship.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.