Skip to main content
SpringerLink
Log in

On a class of elliptic variational inequalities with a spectral parameter and discontinuous nonlinearity

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We consider coercive second order elliptic variational inequalities with a spectral parameter and discontinuous nonlinearity. Using the variational method, we establish solvability of these problems and apply the results to the Goldshtik problem.

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. Pavlenko V. N., “Existence theorems for elliptic variational inequalities with quasipotential operators,” Differential Equations, 24, No. 8, 913–916 (1988).

    MathSciNet  MATH  Google Scholar 

  2. Pavlenko V. N., “Semiregular solutions of elliptic variational inequalities with discontinuous nonlinearities,” Ukrainian Math. J., 43, No. 2, 201–205 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  3. Pavlenko V. N., “On solvability of variational inequalities with discontinuous semimonotone operators,” Ukrainian Math. J., 45, No. 3, 475–480 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  4. Pavlenko V. N. and Chizh E. A., “Strongly resonance elliptic variational inequalities with discontinuous nonlinearities,” Russian Math. (Izv. VUZ. Mat.), 49, No. 7, 47–54 (2005).

    MathSciNet  MATH  Google Scholar 

  5. Pavlenko V. N. and Pribyl’ M. A., “Resonance elliptic variational inequalities with discontinuous nonlinearities,” Differential Equations, 42, No. 1, 132–138 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  6. Chang K. C., “Free boundary problems and the set-valued mappings,” J. Differential Equations, 49, No. 1, 1–28 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  7. Pavlenko V. N., Equations and Variational Inequalities with Discontinuous Nonlinearities [in Russian], Dis. Dokt. Fiz.-Mat. Nauk, Chelyabinsk (1995).

    Google Scholar 

  8. Pavlenko V. N. and Potapov D. K., “Existence of a ray of eigenvalues for equations with discontinuous operators,” Siberian Math. J., 42, No. 4, 766–773 (2001).

    Article  MathSciNet  Google Scholar 

  9. Potapov D. K., Problems with a Spectral Parameter and Discontinuous Nonlinearity [in Russian], Izdat. IBP, St. Petersburg (2008).

    Google Scholar 

  10. Potapov D. K., “On an upper bound for the value of the bifurcation parameter in eigenvalue problems for elliptic equations with discontinuous nonlinearities,” Differential Equations, 44, No. 5, 737–739 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  11. Vainberg M. M., Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, John Wiley and Sons, New York and Toronto (1973).

    MATH  Google Scholar 

  12. Gilbarg D. and Trudinger N. S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin etc. (1983).

    MATH  Google Scholar 

  13. Pavlenko V. N., “Existence of solutions to nonlinear equations with discontinuous monotone operators,” Vestnik Moskov. Univ. Mat. Mekh., No. 6, 21–29 (1973).

  14. Dunford N. and Schwartz J. T., Linear Operators. Vol. 2: Spectral Theory. Selfadjoint Operators in Hilbert Space, John Wiley and Sons, New York (1988).

    Google Scholar 

  15. Goldshtik M. A., “A mathematical model of separated flows in an incompressible liquid,” Dokl. Akad. Nauk SSSR, 147, No. 6, 1310–1313 (1962).

    Google Scholar 

  16. Potapov D. K., “A mathematical model of separated flows in an incompressible fluid,” Izv. RAEN Ser. MMMIU, 8, No. 3–4, 163–170 (2004).

    Google Scholar 

  17. Potapov D. K., “Continuous approximations of Goldshtik’s model,” Math. Notes, 87, No. 2, 244–247 (2010).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    D. K. Potapov

Authors
  1. D. K. Potapov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. K. Potapov.

Additional information

Original Russian Text Copyright © 2012 Potapov D. K.

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 1, pp. 205–212, January–February, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Potapov, D.K. On a class of elliptic variational inequalities with a spectral parameter and discontinuous nonlinearity. Sib Math J 53, 168–173 (2012). https://doi.org/10.1134/S0037446612010144

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446612010144

Keywords

Search

Navigation