Investigation of the correctness of non-local boundary value problems for elliptic type differential equations with discontinuous coefficient
https://doi.org/10.25587/2411-9326-2025-1-32-45
Abstract
The paper investigates the solvability of nonlocal boundary value problems with the generalized Samarsky–Ionkin condition for elliptic second order differential equations with a discontinuous coefficient in the higher part. The existence and uniqueness theorems for regular solutions to the studied problems are proved, i.e. solutions having all required generalized derivatives.
About the Authors
A. I. Kozhanov
Sobolev Institute of Mathematics
Russian Federation
Russian Federation
Alexander I. Kozhanov
4 Koptyug Avenue, 630090 Novosibirsk
N. N. Shadrina
Department of Fundamental Sciences at Bauman Moscow State Technical University, Department of Higher Mathematics at RTU MIREA
Russian Federation
Russian Federation
Natalia N. Shadrina
Moscow
References
1. Ladyzhenskaya O. A., “On the solution of the general problem of diffraction [in Russian],” Dokl. Akad. Nauk SSSR, 96, No. 3, 433–436 (1954).
2. Oleinik O. A., “Solution of the main boundary value problems for the second order equation with discontinuous coefficients [in Russian],” Dokl. Akad. Nauk SSSR, 124, No. 6, 1219–1222 (1959).
3. Schechter M. “A generalization of the problem of transmission,” Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e serie, 207–236 3, No. 3 (1960).
4. Oleinik O. A., “On a method for solving the general problem of diffraction,” Dokl. Akad. Nauk SSSR, 135, No. 5, 1054–1057 (1960).
5. Il’in V. A., “Solvability of the Dirichlet and Neumann problems for a linear elliptic operator with discontinuous coefficients [in Russian],” Dokl. Akad. Nauk SSSR, 137, No. 1, 28–30 (1961).
6. Il’in V. A. and Shishmarev I. A., “The problem on eigenfunctions for the operator Lu = div[p(x) grad u] − q(x)u with discontinuous coefficients [in Russian],” Sib. Math. J., 2, No. 4, 520–536 (1961).
7. Il’in V. A. and Shishmarev I. A., “Potential method for Dirichlet and Neumann problems in the case of equations with discontinuous coefficients [in Russian],” Sib. Math. J., 2, No. 1, 46–58 (1961).
8. Ladyzhenskaya O. A., Rivkind V. Ya., and Ural’tseva N. N., “On the classic solvability of diffraction problems [in Russian],” Tr. MIAN SSSR, 92, 116–146 (1966).
9. Ladyzhenskaya O. A., Linear and Quasilinear Elliptic Equations, Acad. Press, New York (1968). xviii
10. Shadrina N. N., “On the solvability of some conjugation problems for elliptic equations [in Russian],” Mat. Zamet. SVFU, 21, No. 1, 75–89, (2014).
11. Shadrina N. N., “On the influence of parameters on the solvability of some conjugate problems for elliptical equations [in Russian],” Sib. Elektron. Mat. Izv., 13, 411–425 (2016).
12. Kozhanov A. I. and Shadrina N. N., “Boundary value problems with conjugation conditions for quasi-parabolic equations of the third order with a discontinuous sign-variable coefficient [in Russian],” Sib. Elektron. Mat. Izv., 18, No. 1, 599–616 (2021).
13. Kozhanov A. I. and Shadrina N. N., “Study of the influence of parameters on the correctness of the conjugation problem for the Boussinesq–Love differential equation [in Russian],” Chelyab. Fiz.-Mat. Zhurn., 7, No. 1, 30–42 (2022).
14. Ionkin N. I., “The stability of a problem in the theory of heat conduction with nonclassical boundary conditions [in Russian],” Differ. Uravn., 15, No. 7, 1279–1283 (1979).
15. Ionkin N. I., “The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition [in Russian],” Differ. Uravn., 13, No. 2, 294–304 (1977).
16. Ashyraliev A. and Akay N., “A note on the well-posedness of the nonlocal boundary value problem for elliptic difference equations,” Appl. Math. Comput, 175, No. 1, 49–60 (2006).
17. Skubachevskii A. L., “Nonclassical boundary value problems, I,” J. Math. Sci., 155, No. 2, 199–334 (2008).
18. Ashyraliev A. and Akay N., “A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space,” Math. Anal. Appl., 344, 557–563 (2008).
19. Kozhanov A. I. and Dyuzheva A. V., “Well-posedness of the generalized Samarskii–Ionkin problem for elliptic equations in a cylindrical domain [in Russian],” Differ. Uravn., 59, No. 2, 223–235 (2023).
20. Samarskii A. A., “Some problems of the theory of differential equations [in Russian],” Differ. Uravn., 16, No. 11, 1925–1935 (1980).
21. Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics [in Russian], Nauka, Moscow (1988).
22. Triebel H., Interpolation Theory, Functional Spaces, Differential Operators, VEB Deutscher Verl. Wiss., Berlin (1978).
23. Kozhanov A. I., “Initial-boundary value problems with generalized Samarskii–Ionkin condition for parabolic equations with arbitrary evolution direction,” J. Math. Sci., 274, No. 2, 228–240 (2023).
Review
For citations:
Kozhanov A.I., Shadrina N.N. Investigation of the correctness of non-local boundary value problems for elliptic type differential equations with discontinuous coefficient. Mathematical notes of NEFU. 2025;32(1):32-45. (In Russ.) https://doi.org/10.25587/2411-9326-2025-1-32-45
Views:
2
JATS XML
Email this article 