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On an Ill-Posed Problem for the Laplace Operator with Data on the Whole Boundary

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Functional Analysis in Interdisciplinary Applications (FAIA 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 216))

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Abstract

In this paper a nonlocal problem for the Poisson equation in a rectangular domain is considered. It is shown that this problem is ill-posed as well as the Cauchy problem for the Laplace equation. The method of spectral expansion in eigenfunctions of the nonlocal problem for equations with deviating argument establishes a criterion of the strong solvability of the considered nonlocal problem. It is shown that the ill-posedness of the nonlocal problem is equivalent to the existence of an isolated point of the continuous spectrum for a nonself-adjoint operator with the deviating argument.

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Acknowledgements

This research is financially supported by a grant No. 0820/GF4 and by the target program 0085/PTSF-14 from the Ministry of Science and Education of the Republic of Kazakhstan.

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Authors and Affiliations

  1. Institute of Mathematics and Mathematical Modeling, 125 Pushkin st., Almaty, 050010, Kazakhstan

    Berikbol T. Torebek

Authors
  1. Berikbol T. Torebek
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Correspondence to Berikbol T. Torebek .

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Editors and Affiliations

  1. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Tynysbek Sh. Kalmenov

  2. Department of Mechanics and Mathematics, Kazakhstan Branch of Lomonosov Moscow State University, Astana, Kazakhstan

    Erlan D. Nursultanov

  3. Department of Mathematics, Imperial College London, London, United Kingdom

    Michael V. Ruzhansky

  4. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Makhmud A. Sadybekov

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Torebek, B.T. (2017). On an Ill-Posed Problem for the Laplace Operator with Data on the Whole Boundary. In: Kalmenov, T., Nursultanov, E., Ruzhansky, M., Sadybekov, M. (eds) Functional Analysis in Interdisciplinary Applications. FAIA 2017. Springer Proceedings in Mathematics & Statistics, vol 216. Springer, Cham. https://doi.org/10.1007/978-3-319-67053-9_16

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