Spectral Properties and Boundary Conditions for Second Order Elliptic Operators
DOI:
https://doi.org/10.22105/kmisj.v2i1.80Keywords:
Operator-differential equation, Boundary value problem, Hilbert space, Self-adjoint operator, A regular solutionAbstract
This article investigates the regular solvability of boundary value problems for second-order elliptic operator-differential equations with discontinuous coefficients and non-standard boundary conditions in a Hilbert space setting. The authors focus on operator equations defined over a finite interval and aim to identify the structural and spectral conditions under which these problems admit unique, stable solutions. They construct the problem using self-adjoint and positively defined operators, incorporating piecewise-constant coefficients and boundary operators. The study introduces a specialized function space to accommodate the differential and boundary conditions and establishes conditions ensuring the existence, uniqueness, and continuity of solutions with respect to input data. By analyzing the properties of associated linear operators and leveraging classical results from functional analysis, such as the Banach inverse operator theorem, the authors demonstrate that the solution operator is bounded and invertible under specific spectral constraints. This work contributes to the theoretical understanding of elliptic operator-differential equations and provides valuable tools for further analysis in mathematical physics, engineering, and applied mathematics contexts where such boundary value problems frequently arise.
References
Agayeva, G.A.( 2017). On a boundary value problem for operator-differential equations of the second order. Proceedings of the
Pedagogical University, section of the natural sciences. 9–17,
Agayeva, G.A.(2014). On the existence and uniqueness of the generalized solution of a boundary value problem for second order
operator-differential equations, Trans-actions of NAS of Azerbaijan, 4, 3–8.
Agayeva, G.A. (2015). On the solvability of a boundary value problem for elliptic opera-tor-differential equations with an
operator coefficient in the boundary condition. Bulletin of Baku University, 1, 37–42.
Aliyev, A.R. (2003). Boundary Value Problems for a Class of High-Order Operator-Differential Equations with Variable
Coefficients. Matem. Zametki, 74(6), 803–814.
Aliyev, A.R. (2007). Solvability of a class of boundary value problems for second-order operator-differential equations with a
discontinuous coefficient in a weighted space. Differential Equations, 43(10), 1459–1463.
Dezin, A.A.(1980). General issues of theory of boundary value problem M: Nauka, 207 p.
Dubinsky, Yu.A. (1972). On some arbitrary order differential operator equations. Matemat. Sbornik, 90(1), 3–22.
Gasımov, M.G. (1972). On solvability of boundary value problem for a class operator-differential equation. DAN SSSR, 235(3),
–508.
Gasimova, G.M. (2014). On solvability conditions of a boundary value problem with an operator in the boundary condition for
a second order elliptic operator-differential equations. Proceeding of the Institute of Mathematics and Mechanics. National Academy of Sciences of Azerbaijan, 40, 172–177.
Gasimova, G.M. (2015). On well-defined solvability of a boundary value problem for an elliptic differential equation in Hilbert space. Transactions of National Academy of Sciences of Azerbajan, Series of Physical-Technical and Mathematical Sciences,
Issuse Mathematics, 35(1), 31–36.
Gorbacuk, V.I., & Gorbacuk, M.L.,(1984). Boundary Value Problem for Differential-Operator Equations. Kiev, Naumova
Dumka, 284 p.
Krein, S.G. (1967). Linear Differential Equation in Banach Space. Nauka, 464.
Lions-J.L., & Magenes, E. (1977). Inhomogeneous Boundary Value Problems and Their Applications. Moscow: Mir, 371 p. [14] Mirzoyev, S.S., Aghayeva, G.A. (2014). On the solvability conditions of solvability of one boundary value problems for the
second order differential equations with operator coefficients. Inter. Journal of Math. Analysis, 8(4), 149–156.
Mirzoyev, S.S., Agayeva, G.A. (2013). On correct solvability of one boundary value problems for the differential equations of
the second order on Hilbert space. Applied Mathematical Sciences, 7(79), 3935–3945.
Mirzoev, S.S., Aliyev, A.R., Gasimova, G.M. (2016). Solvability conditions for a boundary value problem with operator
coefficients and related estimations of the norms of intermediate derivatives. Dokl. RAN, 2, 511–513.
Mirzoyev, S.S., Aliyev, A.R., Rustamova, L.A. (2013). On the boundary value prob-lem with operator in boundary conditions
for the operator-differential equations of second order with discontinuous coefficients. Journal of Mathem. Physics, Analysis, Geometry, 9(2), 207–206.
Mirzoyev S.S., Aliyev A.R., Rustamova L.A. (2012). Solvability conditions for boundary value problem for elliptic operator-
differential equations with discontinuers coefficient. Math. Zametki. 92(5). 789–793.
Mirzoyev, S.S.(1983). Conditions for well-defined solvability of boundary value problems for operator differential equations.
DAN, SSSR, 2, 291–295.
Skalikov, A.A. (1990). Elliptic equations in hilbert space and associated spectral problems. Journal of Soviet Mathematics , 51,
–2467. https://doi.org/10.1007/BF01097162
Yakubov, S.Ya. (1985). Linear-Differential Equations and Their Applications. Baku, Elm, 220 p.
Aghayeva, G.A., Ibrahimov, V.R., Juraev, D.A. (2024). On some comparision of the numerical methods applied to solve ODEs,
Volterra integral and integro differential equations. Karshi Multidisciplinary International Scientific Journal. 1(1). 39–46. [23] Shafiyeva, G.Kh., Ibrahimov, V.R., Juraev, D.A. (2024). On some comparison of Adam’s methods with multistep methods
and application them to solve initial-value problem for the ODEs first order. Karshi Multidisciplinary International Scientific Journal. 1(2). 181–188.
Liu, T., Ding, B., Saray, B.N., Juraev, D.A., Elsayed, E.E. (2025). On the pseudospectral method for solving the fractional Klein–Gordon equation using Legendre cardinal functions. Fractal and Fractional. 9(3). 1–19. https://doi.org/10.3390/fractalfract9030177
Niyozov, I.E., Juraev, D.A., Efendiev, R.F., Abdalla, M. (2025). Cauchy problem for the biharmonic equation. Journal of Umm
Al-Qura University for Applied Sciences. 2025. 1–11. https://doi.org/10.1007/s43994-025-00244-3
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