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Vestnik SamU. Estestvenno-Nauchnaya Ser., 2019, Volume 25, Issue 1, Pages 7–20 (Mi vsgu584)  

Mathematics

The correctness of a Dirichlet type problem for the degenerate multidimensional hyperbolic-elliptic equations

S. A. Aldashev

Institute of Mathematics and Mathematical Modeling, 125, Pushkin street, Almaty, 050010, Republic of Kazakhstan

Abstract: Multidimensional hyperbolic-elliptic equations describe important physical, astronomical, and geometric processes. It is known that the oscillations of elastic membranes in space according to Hamilton’s prism can be modeled by multidimensional degenerate hyperbolic equations. Assuming that the membrane is in equilibrium in half the bend, from Hamilton’s principle we also obtain degenerate elliptic equations. Consequently, vibrations of elastic membranes in space can be modeled as multidimensional degenerate hyperbolic-elliptic equations. When studying these applications, it is necessary to obtain an explicit representation of the investigated boundary value problems. The author has previously studied the Dirichlet problem for multidimensional hyperbolic-elliptic equations, where the unique solvability of this problem is shown, which essentially depends on the height of the cylindrical domain under consideration. However, the Dirichlet problem in a cylindrical domain for multidimensional degenerate hyperbolic-elliptic equations has not been studied previously. In this paper, the Dirichlet problem is studied for a class of degenerate multidimensional hyperbolic-elliptic equations. Moreover, the existence and uniqueness of the solution depends on the height of the considered cylindrical domain and on the degeneration of the equation. A uniqueness criterion for a regular solution is also obtained.

Keywords: correctness, Dirichlet problem, cylindrical domain, degeneration of Bessel function, criteria.

DOI: https://doi.org/10.18287/2541-7525-2019-25-1-7-20

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UDC: 517.956
Received: 15.01.2019
Accepted:20.02.2019

Citation: S. A. Aldashev, “The correctness of a Dirichlet type problem for the degenerate multidimensional hyperbolic-elliptic equations”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 25:1 (2019), 7–20

Citation in format AMSBIB
\Bibitem{Ald19}
\by S.~A.~Aldashev
\paper The correctness of a Dirichlet type problem for the degenerate multidimensional hyperbolic-elliptic equations
\jour Vestnik SamU. Estestvenno-Nauchnaya Ser.
\yr 2019
\vol 25
\issue 1
\pages 7--20
\mathnet{http://mi.mathnet.ru/vsgu584}
\crossref{https://doi.org/10.18287/2541-7525-2019-25-1-7-20}
\elib{https://elibrary.ru/item.asp?id=39454580}


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