Advertisement

Ukrainian Mathematical Journal

, Volume 62, Issue 1, pp 1–14 | Cite as

Solvability of the boundary-value problem for the second-order elliptic differential-operator equation with spectral parameter in the equation and boundary conditions

  • B. A. Aliev
Article

We study the solvability of a boundary-value problem for the second-order elliptic differential-operator equation with spectral parameter both in the equation and in boundary conditions. We also analyze the asymptotic behavior of the eigenvalues corresponding to the uniform boundary-value problem.

Keywords

Banach Space Spectral Parameter Separable Hilbert Space Order Linear Differential Equation Nonlocal Elliptic Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. G. Krein, Linear Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1967).Google Scholar
  2. 2.
    A. A. Dezin, General Problems of the Theory of Boundary-Value Problems [in Russian], Nauka, Moscow (1980).zbMATHGoogle Scholar
  3. 3.
    S. Ya. Yakubov, Linear Differential-Operator Equations and Their Applications [in Russian], Élm, Baku (1985).Google Scholar
  4. 4.
    S. Yakubov, Completeness of Root Functions of Regular Differential Operators, Longman, New York (1994).zbMATHGoogle Scholar
  5. 5.
    S. Yakubov and Ya. Yakubov, Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC, Boca Raton (2000).zbMATHGoogle Scholar
  6. 6.
    V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Differential-Operator Equations [in Russian], Naukova Dumka, Kiev (1984).Google Scholar
  7. 7.
    A. Ya. Shklyar, Complete Second Order Linear Differential Equations in Hilbert Spaces, Birkhäuser, Basel (1997).zbMATHGoogle Scholar
  8. 8.
    G. I. Laptev, “Strongly elliptic equations in Hilbert spaces,” Lit. Mat. Sb., 8, No. 1, 87–99 (1968).zbMATHMathSciNetGoogle Scholar
  9. 9.
    P. E. Sobolevskii, “Elliptic equations in Banach spaces,” Differents. Uravn., 4, No. 7, 1346–1348 (1968).MathSciNetGoogle Scholar
  10. 10.
    M. G. Gasymov, “On the solvability of boundary-value problems for one class of operator-differential equations,” Dokl. Akad. Nauk SSSR, 235, No. 3, 505–508 (1977).MathSciNetGoogle Scholar
  11. 11.
    V. I. Gorbachuk and M. L. Gorbachuk, “Some problems of the spectral theory of elliptic differential equations in spaces of vector functions,” Ukr. Mat. Zh., 28, No. 3, 313–324 (1976).zbMATHGoogle Scholar
  12. 12.
    V. A. Il’in and V. S. Filippov, “On the character of the spectrum of self-adjoint extension of the Laplace operator in a bounded domain,” Dokl. Akad. Nauk SSSR, 191, No. 2, 167–169 (1970).MathSciNetGoogle Scholar
  13. 13.
    H. Amann, “Dual semigroups and second order linear elliptic boundary value problems,” Isr. J. Math., 45, 225–254 (1983).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Aibeche, “Coerciveness estimates for a class of nonlocal elliptic problems,” Different. Equat. Dynam. Syst., 4, No. 1, 341–351 (1993).MathSciNetGoogle Scholar
  15. 15.
    S. Yakubov, “Problems for elliptic equations with operator-boundary conditions,” Integr. Equat. Oper. Theory, 43, 215–236 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    G. Dore and S. Yakubov, “Semigroup estimates and noncoercive boundary value problems,” Semigroup Forum, 60, 93–121 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    S. Ya. Yakubov and B. A. Aliev, “Boundary-value problem with operator in the boundary conditions for a second-order elliptic differential-operator equation,” Sib. Mat. Zh., 26, No. 4, 176–188 (1985).MathSciNetGoogle Scholar
  18. 18.
    V. M. Bruk, “On a class of boundary-value problems with spectral parameter in the boundary condition,” Mat. Sb., 100(142), No. 2(6), 210–216 (1976).MathSciNetGoogle Scholar
  19. 19.
    V. I. Gorbachuk and M. A. Rybak, “Boundary-value problems for the operator Sturm–Liouville equation with spectral parameter both in the equation and in the boundary condition,” in: Direct and Inverse Problems of Scattering Theory [in Russian], Kiev (1981), pp. 3–16.Google Scholar
  20. 20.
    M. A. Rybak, “On the asymptotic distribution of eigenvalues of some boundary-value problems for the operator Sturm–Liouville equation,” Ukr. Mat. Zh., 32, No. 2, 248–252 (1980).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    B. A. Aliev, “Asymptotic behavior of eigenvalues of one boundary-value problem for the second-order elliptic differential-operator equation,” Ukr. Mat. Zh., 58, No. 8, 1146–1152 (2006).zbMATHCrossRefGoogle Scholar
  22. 22.
    B. A. Aliev, “Asymptotic behavior of eigenvalues of a boundary value problem with spectral parameter in the boundary conditions for the second order elliptic differential-operator equation,” Trans. NAS Azer. Ser. Phys-Tech. Math. Sci., 25, No. 7, 3–8 (2005).Google Scholar
  23. 23.
    L. A. Oleinik, “Inhomogeneous boundary-value problems for differential-operator equations with spectral parameter in the boundary conditions,” in: Spectral Theory of Differential-Operator Equations [in Russian], Kiev (1986), pp. 25–28.Google Scholar
  24. 24.
    B. A. Aliev and Ya. Yakubov, “Elliptic differential-operator problems with a spectral parameter in both the equation and boundary-operator conditions,” Adv. Different. Equat., 11, No. 10, 1081–1110 (2006).zbMATHMathSciNetGoogle Scholar
  25. 25.
    L. A. Kotko and S. G. Krein, “Completeness of the system of eigenfunctions and associated functions for boundary-value problems containing a parameter in the boundary conditions,” Dokl. Akad. Nauk SSSR, 227, No. 2, 288–300 (1976).MathSciNetGoogle Scholar
  26. 26.
    A. N. Kozhevnikov, “Separate asymptotics of two series of eigenvalues of one elliptic boundary-value problem,” Mat. Zametki, 22, No. 5, 699–711 (1977).zbMATHMathSciNetGoogle Scholar
  27. 27.
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB, Berlin (1978).Google Scholar
  28. 28.
    K. Maurin, Metody Przestrzeni Hilberta, Państwowe Wydawnictwo Naukowe, Warsawa (1959).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  • B. A. Aliev
    • 1
  1. 1.Institute of Mathematics and MechanicsAzerbaijan National Academy of SciencesBakuAzerbaijan

Personalised recommendations