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Construction of Exact Solutions of Irregularly Degenerate Elliptic Equations with Analytic Coefficients

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Abstract

We solve a boundary value problem (problem E in the sense of M.V. Keldysh) for an irregularly degenerate elliptic operator in a rectangle. The exact solution of the problem is constructed as a series in the eigenfunctions of the limit operator. The method of spectral isolation of singularities, which generalizes the method of regularization of singular perturbations to the case of degenerate elliptic equations, is developed.

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References

  1. Keldysh, M.V., On some cases of degenerate elliptic-type equations on the boundary of a domain, Dokl. Akad. Nauk SSSR, 1951, vol. 77, no. 2, pp. 181–183.

    Google Scholar 

  2. Keldysh, M.V., Izbrannye trudy. Matematika (Selected Works. Mathematics), Moscow: Nauka, 1985.

    MATH  Google Scholar 

  3. Lomov, I.S., Small denominators in analytic theory of degenerate differential equations, Differ. Equations, 1993, vol. 29, no. 12, pp. 1811–1820.

    MathSciNet  MATH  Google Scholar 

  4. Lomov, I.S., The method of spectral separation of variables for irregularly degenerate elliptic differential operators, Dokl. Math., 2001, vol. 63, no. 1, pp. 88–91.

    MATH  Google Scholar 

  5. Lomov, S.A. and Lomov, I.S., Osnovy matematicheskoi teorii pogranichnogo sloya (Foundations of the Mathematical Theory of Boundary Layers), Moscow: Mosk. Gos. Univ., 2011.

    Google Scholar 

  6. Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, New York: McGraw-Hill, 1955. Translated under the title Teoriya obyknovennykh differentsial’nykh uravnenii, Moscow: Inostrannaya Literatura, 1958.

    MATH  Google Scholar 

  7. Yanushauskas, A.I., Analiticheskaya teoriya ellipticheskikh uravnenii (Analytic Theory of Elliptic Equations), Novosibirsk: Nauka, 1979.

    MATH  Google Scholar 

  8. Fel’dman, N.I., Sed’maya problema Gil’berta (Hilbert’s Seventh Problem), Moscow: Mosk. Gos. Univ., 1982.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    D. P. Emel’yanov & I. S. Lomov

Authors
  1. D. P. Emel’yanov
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  2. I. S. Lomov
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Correspondence to D. P. Emel’yanov.

Additional information

Russian Text © D.P. Emel’yanov, I.S. Lomov, 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 1, pp. 45–58.

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Emel’yanov, D.P., Lomov, I.S. Construction of Exact Solutions of Irregularly Degenerate Elliptic Equations with Analytic Coefficients. Diff Equat 55, 46–59 (2019). https://doi.org/10.1134/S0012266119010051

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  • DOI: https://doi.org/10.1134/S0012266119010051

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