Skip to main content
SpringerLink
Log in

Abstract elliptic equations with integral boundary conditons

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in L p spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed L p norm

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahmed, B., Alsaedi, A. and Alghamidi, B., Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions, Nonlinear Anal. Real World Appl., 9, 2008, 1727–1740.

    Article  MathSciNet  MATH  Google Scholar 

  2. Amann, H., Linear and Quasi-linear Equations, 1, Birkhauser, Basel, 1995.

    MATH  Google Scholar 

  3. Ashyralyev, A., On well-posedness of the nonlocal boundary value problem for elliptic equations, Numerical Functional Analysis & Optimization, 24(1–2), 2003, 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  4. Besov, O. V., Ilin, V. P. and Nikolskii, S. M., Integral Representations of Functions and Embedding Theorems, Nauka, Moscow, 1975.

    Google Scholar 

  5. Burkholder, D. L., A geometrical conditions that implies the existence certain singular integral of Banach space-valued functions, Proc. Conf. Harmonic analysis in honor of Antonu Zigmund, Chicago, 1981, Wads Worth, Belmont, 1983, 270–286.

    Google Scholar 

  6. Cannon, J. R., Perez Esteva S. and van Der Hoek J., A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal., 24, 1987, 499–515.

    Article  MathSciNet  MATH  Google Scholar 

  7. Choi, Y. S. and Chan, K. Y., A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear Anal., 18, 1992, 317–331.

    Article  MathSciNet  MATH  Google Scholar 

  8. Denk, R., Hieber, M. and Prüss, J., R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166, 2003, 788.

    MathSciNet  MATH  Google Scholar 

  9. Dore, C. and Yakubov, S., Semigroup estimates and non-coercive boundary value problems, Semigroup Forum., 60, 2000, 93–121.

    Article  MathSciNet  MATH  Google Scholar 

  10. Ewing, R. E. and Lin, T., A class of parameter estimation techniques for fluid flow in porous media, Adv. Water Resour., 14, 1991, 89–97.

    Article  MathSciNet  Google Scholar 

  11. Favini, A., Shakhmurov, V. and Yakubov, Y., Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces, Semigroup Forum., 79(1), 2009, 22–54.

    Article  MathSciNet  MATH  Google Scholar 

  12. Goldstain, J. A., Semigroups of Linear Operators and Applications, Oxfard University Press, Oxfard 1985.

    Google Scholar 

  13. Haller, R., Heck, H. and Noll, A., Mikhlin’s theorem for operator-valued Fourier multipliers in n variables, Math. Nachr., 244, 2002, 110–130.

    Article  MathSciNet  MATH  Google Scholar 

  14. Krein, S. G., Linear Differential Equations in Banach Space, American Mathematical Society, Providence, 1971.

    Google Scholar 

  15. Liang, J., Nagel, R. and Xiao, T. J., Approximation theorems for the propagators of higher order abstract Cauchy problems, Trans. Amer. Math. Soc., 360(4), 2008, 1723–1739.

    Article  MathSciNet  MATH  Google Scholar 

  16. Lions, J. L. and Magenes, E., Nonhomogenous Boundary Value Problems, Mir, Moscow, 1971.

    Google Scholar 

  17. Shahmurov, R., Solution of the Dirichlet and Neumann problems for a modified Helmholtz equation in Besov spaces on an annuals, Journal of Differential Equations, 249(3), 2010, 526–550.

    Article  MathSciNet  MATH  Google Scholar 

  18. Shakhmurov, V. B., Imbedding theorems and their applications to degenerate equations, Differential Equations, 24(4), 1988, 475–482.

    MathSciNet  MATH  Google Scholar 

  19. Shakhmurov, V. B., Coercive boundary value problems for regular degenerate differential-operator equations, J. Math. Anal. Appl., 292(2), 2004, 605–620.

    Article  MathSciNet  MATH  Google Scholar 

  20. Shakhmurov, V. B., Embedding theorems and maximal regular differential operator equations in Banachvalued function spaces, Journal of Inequalities and Applications, 2(4), 2005, 329–345.

    MathSciNet  MATH  Google Scholar 

  21. Shakhmurov, V. B., Embedding and maximal regular differential operators in Banach-valued weighted spaces, Acta Mathematica Sinica, 22(5), 2006, 1493–1508.

    Article  MathSciNet  MATH  Google Scholar 

  22. Shakhmurov, V. B., Linear and nonlinear abstract equations with parameters, Nonlinear Analysis, Method and Applications, 73, 2010, 2383–2397.

    MathSciNet  MATH  Google Scholar 

  23. Shi, P. and Shillor, M., Design of contact patterns in one-dimensional thermoelasticity, Theoretical Aspects of Industrial Design, SIAM, Philadelphia, PA, 1992.

    Google Scholar 

  24. Shklyar, A. Y., Complete Second Order Linear Differential Equations in Hilbert Spaces, Birkhauser Verlak, Basel, 1997.

    MATH  Google Scholar 

  25. Sobolevskii, P. E., Coerciveness inequalities for abstract parabolic equations, Doklady Akademii Nauk SSSR, 57(1), 1964, 27–40.

    MathSciNet  Google Scholar 

  26. Triebel, H., Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.

    MATH  Google Scholar 

  27. Weis, L., Operator-valued Fourier multiplier theorems and maximal Lp regularity, Math. Ann., 319, 2001, 735–758.

    Article  MathSciNet  MATH  Google Scholar 

  28. Xiao, T. J. and Liang, J., Second order differential operators with Feller-Wentzell type boundary conditions, J. Funct. Anal., 254, 2008, 1467–1486.

    Article  MathSciNet  MATH  Google Scholar 

  29. Xiao, T. J. and Liang, J., Nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions, J. Differential Equations, 252, 2012, 3953–3971.

    Article  MathSciNet  MATH  Google Scholar 

  30. Yakubov, S., Completeness of Root Functions of Regular Differential Operators, Longman, Scientific and Technical, New York, 1994.

    MATH  Google Scholar 

  31. Yakubov, S., A nonlocal boundary value problem for elliptic differential-operator equations and applications, Integr. Equ. Oper. Theory, 35, 1999, 485–506.

    Article  MathSciNet  MATH  Google Scholar 

  32. Yakubov, S. and Yakubov, Y., Differential-operator Equations, Ordinary and Partial Differential Equations, Chapman and Hall /CRC, Boca Raton, 2000.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Okan University, Akfirat, Tuzla Istanbul, 34959, Turkey

    Veli Shakhmurov

  2. Department of Mathematics, Azerbaycan Khazar University, Azerbaycan Khazar, Turkey

    Veli Shakhmurov

Authors
  1. Veli Shakhmurov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Veli Shakhmurov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shakhmurov, V. Abstract elliptic equations with integral boundary conditons. Chin. Ann. Math. Ser. B 37, 625–642 (2016). https://doi.org/10.1007/s11401-016-0948-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-016-0948-6

Keywords

2000 MR Subject Classification

Search

Navigation