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Uniqueness for Inverse Boundary Value Problems by Dirichlet-to-Neumann Map on Subboundaries

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Abstract

We consider inverse boundary value problems for elliptic equations of second order of determining coefficients by Dirichlet-to-Neumann map on subboundaries, that is, the mapping from Dirichlet data supported on subboundary \({\partial \Omega \setminus \Gamma_{-}}\) to Neumann data on subboundary \({\partial \Omega \setminus \Gamma_{+}}\). First we prove uniqueness results in three dimensions under some conditions such as \({\overline{\Gamma_{+}\cup\Gamma_{-}}= \partial\Omega}\) Next we survey uniqueness results in two dimensions for various elliptic systems for arbitrarily given \({\Gamma_{-} = \Gamma_{+}}\) Our proof is based on complex geometric optics solutions which are constructed by a Carleman estimate.

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Authors and Affiliations

  1. Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, CO, 80523-1874, U.S.A.

    Oleg Y. Imanuvilov

  2. Department of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan

    Masahiro Yamamoto

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  1. Oleg Y. Imanuvilov
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  2. Masahiro Yamamoto
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Correspondence to Masahiro Yamamoto.

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Oleg. Y. Imanuvilov: The author was partially supported by NSF DMS 1312900.

Lecture held by M. Yamamoto in the Seminario Matematico e Fisico on March 21, 2012.

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Imanuvilov, O.Y., Yamamoto, M. Uniqueness for Inverse Boundary Value Problems by Dirichlet-to-Neumann Map on Subboundaries. Milan J. Math. 81, 187–258 (2013). https://doi.org/10.1007/s00032-013-0205-3

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