Dirichlet problem in a cylindrical domain for a certain class of multidimensional hyperbolic-elliptic equations
https://doi.org/10.22405/2226-8383-2025-26-5-42-52
Abstract
Multidimensional Hyperbolic-Elliptic Equations describe important physical, astronomical,
and geometric processes. It is known that the oscillations of elastic membranes in space can
be modeled by multidimensional hyperbolic equations based on Hamilton’s principle. Assuming that the membrane is in equilibrium in the bending position, Hamilton’s principle also leads to multidimensional elliptic equations.
Consequently, the oscillations of elastic membranes in space can be described using
multidimensional hyperbolic-elliptic equations.
The problem of the well-posedness of the Dirichlet problem for mixed-type equations in
special domains has been the subject of research by many authors in both two-dimensional and multidimensional cases.
The author previously studied the Dirichlet problem for multidimensional hyperbolicparabolic equations, where the unique solvability of this problem was demonstrated, significantly depending on the height of the considered cylindrical domain.
In this work, a Dirichlet-type problem is studied in a cylindrical domain for a certain class
of multidimensional hyperbolic-elliptic equations, and an explicit form of its classical solution is obtained.
It is shown that the unique solvability depends only on the height of the hyperbolic part of
the cylindrical domain, and a uniqueness criterion for the solution is provided.
About the Author
Serik Aymurzaevich AldashevKazakhstan
doctor of physical and mathematical sciences
References
1. Shabat, B.V. 1957, “Examples of solving Dirichlet problems for mixed-type equations”, Doklady Akademii Nauk SSSR, vol. 112, no. 3, pp. 386–389.
2. Bitsadze, A.V. 1958, “Incorrectness of the Dirichlet problem for equations in mixed domains”, Doklady Akademii Nauk SSSR, vol. 112, no. 2, pp. 167–170.
3. Soldatov, A.P. 1993, “Dirichlet-type problems for the Lavrentiev-Bitsadze equation”, Doklady Rossiiskoi Akademii Nauk, vol. 332, no. 6, pp. 696–698; vol. 333, no. 1, pp. 16–18.
4. Sabitov, K.B. 2007, “Dirichlet problem for mixed-type equations in a rectangular domain”, Doklady Rossiiskoi Akademii Nauk, vol. 413, no. 1, pp. 23–26.
5. Nakhushev, A.M. 2006, Problems with shifts for partial differential equations, Nauka, Moscow, 287 p.
6. Khachev, M.M. 1998, The first boundary-value problem for linear mixed-type equations, Elbrus, Nalchik, 168 p.
7. Aldashev, S.A. 2010, “Uniqueness criterion for the solution of the Dirichlet problem in a cylindrical domain for the multidimensional Lavrentiev-Bitsadze equation”, in Proceedings of the International Russian-Bulgarian Symposium "Mixed-Type Equations and Related Problems of Analysis and Informatics", NII PMA KBNTS RAN, Nalchik, pp. 22–23.
8. Aldashev, S.A. 2014, “Correctness of the Dirichlet problem in a cylindrical domain for the multidimensional Lavrentiev-Bitsadze equation”, Izvestiya Natsional’noi Akademii Nauk Respubliki Kazakhstan. Seriya Fiziko-Matematicheskaya, no. 3, pp. 136–143.
9. Aldashev, S.A. 2013, “Correctness of the Dirichlet problem in a cylindrical domain for a class of multidimensional hyperbolic-elliptic equations”, Nonlinear Oscillations, vol. 16, no. 4, pp. 435–451.
10. Mikhlin, S.G. 1962, Multidimensional Singular Integrals and Integral Equations, Fizmatgiz, Moscow, 254 p.
11. Kamke, E. 1965, Handbook of Ordinary Differential Equations, Nauka, Moscow, 703 p.
12. Bitsadze, A.V. 1981, Some Classes of Partial Differential Equations, Nauka, Moscow, 448 p.
13. Bateman, G. & Erdelyi, A. 1974, Higher Transcendental Functions, Vol. 2, Nauka, Moscow, 295 p.
14. Kolmogorov, A.N. & Fomin, S.V. 1976, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 543 p.
15. Tikhonov, A.N. & Samarskii, A.A. 1966, Equations of Mathematical Physics, Nauka, Moscow, 724 p.
16. Aldashev, S.A. 2011, “Correctness of the Dirichlet problem in a cylindrical domain for multidimensional hyperbolic equations with a wave operator”, Doklady Adygeiskoi (Cherkesskoi) Mezhdunarodnoi Akademii Nauk, vol. 13, no. 1, pp. 21–29.
17. Smirnov, V.I. 1981, Course of Higher Mathematics, Vol. 4, Part 2, Nauka, Moscow, 550 p.
18. Bers, L., John, F. & Schechter, M. 1966, Partial Differential Equations, Mir, Moscow, 352 p.
19. Aldashev, S.A. 2018, “Correctness of the Dirichlet-type problem in a cylindrical domain for the multidimensional Lavrentiev-Bitsadze equation”, Vestnik Samarskogo Gosudarstvennogo Universiteta. Estestvennonauchnaya Seriya, vol. 24, no. 1, pp. 7–13.
Review
For citations:
Aldashev S.A. Dirichlet problem in a cylindrical domain for a certain class of multidimensional hyperbolic-elliptic equations. Chebyshevskii Sbornik. 2025;26(5):42-52. https://doi.org/10.22405/2226-8383-2025-26-5-42-52
JATS XML
Email this article 