Well-Posed Solvability of the Neumann Problem for a Generalized Mangeron Equation with Nonsmooth Coefficients


For a fourth-order generalized Mangeron equation with nonsmooth coefficients defined on a rectangular domain, we consider the Neumann problem with nonclassical conditions that do not require matching conditions. We justify the equivalence of these conditions to classical boundary conditions for the case in which the solution to the problem is sought in an isotropic Sobolev space. The problem is solved by reduction to a system of integral equations whose well-posed solvability is established based on the method of integral representations. The well-posed solvability of the Neumann problem for the generalized Mangeron equation is proved by the method of operator equations.

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  1. 1.

    Bitsadze, A.V., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1976.

    Google Scholar 

  2. 2.

    Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1976.

    Google Scholar 

  3. 3.

    Ladyzhenskaya, O.A., Kraevye zadachi matematicheskoi fiziki (Boundary Value Problems of Mathematical Physics), Moscow: Nauka, 1973.

    Google Scholar 

  4. 4.

    Mangeron, D., New methods for determining solution of mathematical models governing polyvibrating phenomena, Bul. Inst. Politehn. Jasi. Sec. 1, 1968, vol. 14, no. 1, pp. 433–436.

    MATH  Google Scholar 

  5. 5.

    Colton, D., Pseudoparabolic equations in one space variable, J. Differ. Equations, 1972, vol. 12, no. 3, pp. 559–565.

    MathSciNet  Article  Google Scholar 

  6. 6.

    Soldatov, A.P. and Shkhanukov, M.Kh., Boundary value problems with A.A. Samarskii’s general nonlocal condition for higher-order pseudoparabolic equations, Dokl. Akad. Nauk SSSR, 1987, vol. 297, no. 3, pp. 547–552.

    MathSciNet  Google Scholar 

  7. 7.

    Berezanskii, Yu.M., On a Dirichlet type problem for the string vibration equation, Ukr. Mat. Zh., 1960, vol. 12, no. 4, pp. 363–372.

    MathSciNet  Article  Google Scholar 

  8. 8.

    Nakhushev, A.M., Uravneniya matematicheskoi biologii (Equations of Mathematical Biology), Moscow: Vysshaya Shkola, 1995.

    Google Scholar 

  9. 9.

    Utkina, E.A., The Neumann problem for one fourth-order equation, Vestn. Samar. Gos. Univ. Ser. Fiz.-Mat. Nauki, 2009, vol. 2(19), pp. 29–37.

    Article  Google Scholar 

  10. 10.

    Vakhania, N.N., On a boundary value problem with imposing the condition on the entire boundary for a hyperbolic system equivalent to the string vibration equation, Dokl. Akad. Nauk SSSR, 1957, vol. 116, no. 6, pp. 906–909.

    MathSciNet  Google Scholar 

  11. 11.

    Abdul Latif, A.I., Dirichlet, Neumann, and mixed Dirichlet-Neumann value problems for u xy = 0in rectangles, Proc. Roy. Soc. Edinburgh. Ser. A, 1978, no. 82, pp. 107–110.

  12. 12.

    Fokin, M.V., On the Dirichlet problem for the string equation, in Nekorrektnye kraevye zadachi dlya neklassicheskikh uravnenii matematicheskoi fiziki (Ill-Posed Boundary Value Problems for Nonclassical Equations of Mathematical Physics), Novosibirsk: Sib. Branch Akad. Nauk SSSR, 1981, pp. 178–182.

    Google Scholar 

  13. 13.

    Utkina, E.A., The Dirichlet problem for one three-dimensional equation, Vestn. Samar. Gos. Univ. Estestvennonauchn. Ser., 2010, no. 2(76), pp. 84–95.

  14. 14.

    Utkina, E.A., Dirichlet problem for a fourth-order equation, Differ. Equations, 2011, vol. 47, no. 4, pp. 599–603.

    MathSciNet  Article  Google Scholar 

  15. 15.

    Mamedov, I.G., On the well-posed solvability of the Dirichlet problem for a generalized Mangeron equation with nonsmooth coefficients, Differ. Equations, 2015, vol. 51, no. 6, pp. 745–754.

    MathSciNet  Article  Google Scholar 

  16. 16.

    Amanov, T.I., Investigation of the properties of classes of functions with dominant mixed derivatives. Representation, embedding, continuation, and interpolation theorems, Extended Abstract of Doctoral (Phys.-Math.) Dissertation, Novosibirsk, 1967.

  17. 17.

    Nikol’skii, S.M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya (Approximation of Functions of Many Variables and Embedding Theorems), Moscow: Nauka, 1969.

    Google Scholar 

  18. 18.

    Lizorkin, P.I. and Nikol’skii, S.M., Classification of differentiable functions based on spaces with a dominant derivative, Tr. Mat. Inst. im. V.A. Steklova Akad. Nauk SSSR, 1965, vol. 77, pp. 143–167.

    MATH  Google Scholar 

  19. 19.

    Besov, O.V., Il’in, V.P., and Nikol’skii, S.M., Integral’nye predstavleniya funktsii i teoremy vlozheniya (Integral Representations of Functions and Embedding Theorems), Moscow: Nauka, 1975.

    Google Scholar 

  20. 20.

    Dzhabrailov, A.D., Investigation of differential-difference properties of a function defined on an n-dimensional domain, Extended Abstract of Doctoral (Phys.-Math.) Dissertation, Moscow, 1971.

  21. 21.

    Akhiev, S.S., On the general form of linear bounded functionals in a function space of the S.L. Sobolev type, Dokl. Akad. Nauk AzSSR, 1979, vol. 35, no. 6, pp. 3–7.

    MATH  Google Scholar 

  22. 22.

    Nadzhafov, A.M., On integral representations of functions from spaces with a dominant mixed derivative, Vestn. Bakinskogo Gos. Univ. Ser. Fiz.-Mat. Nauk, 2005, no. 3, pp. 31–39.

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This work was supported by a grant from the Presidium of Azerbaijan National Academy of Sciences in the framework of funding scientific research for the period of 2018–2019.

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Correspondence to I. G. Mamedov or M. Dzh. Mardanov or T. K. Melikov or R. A. Bandaliev.

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Russian Text © The Author(s), 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 10, pp. 1405–1415.

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Mamedov, I.G., Mardanov, M.D., Melikov, T.K. et al. Well-Posed Solvability of the Neumann Problem for a Generalized Mangeron Equation with Nonsmooth Coefficients. Diff Equat 55, 1362–1372 (2019). https://doi.org/10.1134/S0012266119100112

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