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Partial Data for the Neumann-to-Dirichlet Map

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Abstract

We show that measurements of a Neumann-to-Dirichlet map, with either inputs or outputs restricted to part of the boundary, can determine an electric potential on that domain. Given a convexity condition on the domain, either the set on which measurements are taken, or the set on which input functions are supported, can be made to be arbitrarily small. The result is analogous to the result by Kenig, Sjöstrand, and Uhlmann for the Dirichlet-to-Neumann map. The main new ingredient in the proof is an improved Carleman estimate for the Schrödinger operator with appropriate boundary conditions. This is proved by Fourier analysis of a conjugated operator along the boundary of the domain.

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Acknowledgments

The author would like to thank Mikko Salo for introducing him to this problem, for sharing the idea behind Proposition 3.1, for reading over the manuscript, and for several other helpful conversations. This research was partially supported by the Academy of Finland. Part of this work was also done at the University of Chicago, and here the author would also like to thank Carlos Kenig for his time and support. The author would also like to thank the referee for several helpful comments.

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  1. Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

    Francis J. Chung

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  1. Francis J. Chung
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Correspondence to Francis J. Chung.

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Communicated by Eric Todd Quinto.

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Chung, F.J. Partial Data for the Neumann-to-Dirichlet Map. J Fourier Anal Appl 21, 628–665 (2015). https://doi.org/10.1007/s00041-014-9379-5

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  • DOI: https://doi.org/10.1007/s00041-014-9379-5

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