Skip to main content
Log in

Solvability of the Neumann Problem in a Disk for Fourth-Order Properly Elliptic Equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We establish and study sufficient conditions of solvability of the Neumann problem for fourth-order properly elliptic equations of the general form in a unit disk K in the space \( {C}^4(K)\cap {C}^{3,\upalpha}\left(\overline{K}\right) \).

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. A. O. Babayan, “On the Dirichlet problem for a fourth-order improperly elliptic equation,” in: Nonclassical Equations of Mathematical Physics [in Russian], Institute of Mathematics, Siberian Branch, Russ. Acad. Sci., Novosibirsk (2007), pp. 56–68.

  2. A. V. Bitsadze, “On the uniqueness of solution to the Dirichlet problem for partial elliptic equations,” Usp. Mat. Nauk, 3, No. 6(28), 211–212 (1948).

  3. V. P. Burskii, Methods for the Investigation of Boundary-Value Problems for General Differential Equations [in Russian], Naukova Dumka, Kiev (2002).

    Google Scholar 

  4. V. P. Burskii, “Breakdown of uniqueness of solutions of the Dirichlet problem for elliptic systems in a disc,” Mat. Zametki, 48, No. 3, 32–36 (1990); English translation: Math. Notes, 48, No. 3, 894–897 (1990).

  5. V. P. Burskii and K. A. Buryachenko, “On the breakdown of the uniqueness of a solution of the Dirichlet problem for typeless differential equations of arbitrary even order in a disk,” Ukr. Mat. Visnyk, 9, No. 4, 477–514 (2012); English translation: J. Math. Sci., 190, No. 4, 539–566 (2013).

  6. V. P. Burskii and E. V. Lesina, “Neumann problem for second-order improperly elliptic equations,” in: Trans. Inst. Appl. Math. Mech. [in Russian], Vol. 23 (2011), pp. 10–16.

  7. E. A. Buryachenko, “Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles,” Ukr. Mat. Zh., 62, No. 5, 591–603 (2010); English translation: Ukr. Math. J., 62, No. 5, 676–690 (2010).

  8. K. O. Buryachenko, “Solvability of inhomogeneous boundary-value problems for fourth-order differential equations,” Ukr. Mat. Zh., 63, No. 8, 1011–1020 (2011); English translation: Ukr. Math. J., 63, No. 8, 1165–1175 (2012).

  9. A. I. Markushevich, Recurrent Sequences [in Russian], Gostekhteorizdat, Moscow (1950).

    Google Scholar 

  10. B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).

    Google Scholar 

  11. A. P. Soldatov, “On the first and second boundary-value problems for elliptic systems on a plane,” Differents. Uravn., 39, No. 5, 674–686 (2003).

    MathSciNet  Google Scholar 

  12. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer, Berlin (1988).

    Book  MATH  Google Scholar 

  13. A. R. Aliev and A. L. Elbably, “Well-posedness of a boundary value problem for a class of third-order operator-differential equations,” Bound. Value Probl., 2013, 140 (2013).

    Article  MathSciNet  Google Scholar 

  14. S. Axler, P. Bourdon, and W. Ramey, Harmonic Function Theory, Springer, New York (2001).

    Book  MATH  Google Scholar 

  15. A. O. Babayan, “Dirichlet problem for properly elliptic equation in unit disk,” J. Contemp. Math. Anal. (=Izv. NAN Armen., Matematika), 38, No. 6, 39–48 (2003).

  16. A. O. Babayan, “On unique solvability of Dirichlet problem for fourth order properly elliptic equation,” Izv. NAN Armen., Matematika, 34, No. 5, 3–18 (1999).

    MathSciNet  Google Scholar 

  17. H. Begehr and A. Kumar, “Boundary-value problems for the inhomogeneous polyanalytic equation. I,” Analysis: Int. Math. J. Anal. & Its Appl., 25, No. 1, 55–71 (2005).

    MATH  MathSciNet  Google Scholar 

  18. A. L. Beklaryan, “On the existence of solutions of the first boundary-value problem for elliptic equations on unbounded domains,” Int. J. Pure Appl. Math., 88, No. 4, 499–522 (2013).

    Article  MATH  Google Scholar 

  19. G. Bonanno and P. F. Pizzimenti, “Neumann boundary-value problems with not coercive potential,” Mediter. J. Math., 9, No. 4, 601–609 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Brown, G. Grubb, and I. G. Wood, “M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems,” Math. Nachr., 282, No. 3, 314–347 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  21. B. Brown, M. Marletta, S. Naboko, and I. G. Wood, “Boundary triplets and M-functions for nonself-adjoint operators, with applications to elliptic PDEs and block operator matrices,” J. London Math. Soc., 77, No. 3, 700–718 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  22. G. Grubb, Distributions and Operators (Graduate Texts in Mathematics), Vol. 252, Springer, New York (2009).

    Google Scholar 

  23. L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, Springer, Berlin (1983).

    MATH  Google Scholar 

  24. N. B. Kerimov and U. Kaya, “Spectral properties of some regular boundary-value problems for fourth-order differential operators,” Cent. Eur. J. Math., 11, No. 1, 94–111 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Mokhtarzadeh, M. Pournaki, and A. Razani, “An existence-uniqueness theorem for a class of boundary-value problems,” Fixed Point Theory, 13, No. 2, 583–591 (2012).

    MATH  MathSciNet  Google Scholar 

  26. A. Posilicano, “Self-adjoint extensions of restrictions,” Operat. Matrices, 2, No. 4, 483–506 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  27. N. E. Tovmasyan, Non-Regular Differential Equations and Calculations of Electromagnetic Fields, World Scientific, Singapore (1998).

    Book  MATH  Google Scholar 

  28. N. E. Tovmasian and V. S. Zakarian, “Dirichlet problem for properly elliptic equations in multiply connected domains,” J. Contemp. Math. Anal. (=Izv. NAN Armen., Matematika), 37, No. 6, 2–34 (2002).

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 57, No. 1, pp. 7–17, January–March, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Buryachenko, K.O., Kharabara, A.I. Solvability of the Neumann Problem in a Disk for Fourth-Order Properly Elliptic Equations. J Math Sci 212, 1–15 (2016). https://doi.org/10.1007/s10958-015-2644-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-015-2644-6

Keywords

Navigation