Skip to main content
SpringerLink
Log in

Elliptic Differential-Difference Equations in the Half-Space

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The Dirichlet problem in the half-space for elliptic differential-difference equations with operators that are compositions of differential and difference operators is considered. For this problem, classical solvability or solvability almost everywhere (depending on the constraints imposed on the boundary data) is proved, an integral representation of the found solution in terms of a Poisson-type formula is constructed, and its convergence to zero as the time-like independent variable tends to infinity is proved.

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. V. A. Kondrat’ev and E. M. Landis, “Qualitative theory of second-order linear partial differential equations,” in Partial Differential Equations–3, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. (VINITI, Moscow, 1988), Vol. 32, pp. 99–215 [in Russian].

    MathSciNet  MATH  Google Scholar 

  2. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer- Verlag, Berlin–New York, 1983).

    Book  Google Scholar 

  3. M. A. Vorontsov, N. G. Iroshnikov, and R. I. Abernathy, “Diffractive patterns in a nonlinear optical the two-dimensional feedback system with field rotation,” Chaos Solitons Fractals 4 (8-9), 1701–1716 (1994).

    Article  Google Scholar 

  4. A. L. Skubachevskii, “On the Hopf bifurcation for a quasilinear parabolic functional- differential equation,” Differ. Equ. 34 (10), 1395–1402 (1998).

    MathSciNet  Google Scholar 

  5. A. L. Skubachevskii, “Bifurcation periodic solutions for nonlinear parabolic functional- differential equations arising in optoelectronics,” Nonlinear Anal. 32 (2), 261–278 (1998).

    Article  MathSciNet  Google Scholar 

  6. A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications (Birkhäuser Verlag, Basel, 1997).

    MATH  Google Scholar 

  7. A. L. Skubachevskii, “Boundary-value problems for elliptic functional-differential equations and their applications,” Russian Math. Surveys 71 (5), 801–906 (2016).

    Article  MathSciNet  Google Scholar 

  8. A. L. Skubachevskii, “Nonclassical boundary-value problems. I,” J. Math. Sci. (N. Y.) 155 (2), 199–334 (2008).

    Article  MathSciNet  Google Scholar 

  9. A. L. Skubachevskii, “Nonclassical boundary-value problems. II,” J. Math. Sci. (N. Y.) 166 (4), 377–561 (2010).

    Article  MathSciNet  Google Scholar 

  10. P. L. Gurevich, “Elliptic problems with nonlocal boundary conditions and Feller semigroups,” J. Math. Sci. (N. Y.) 182 (3), 255-440 (2012).

    Article  MathSciNet  Google Scholar 

  11. A. B. Muravnik, “On the Dirichlet problem for differential-difference elliptic equations in a half-plane,” in Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 3 (RUDN, Moscow, 2016), Vol. 60, pp. 102–113.

    Google Scholar 

  12. A. B. Muravnik, “Asymptotic properties of solutions of the Dirichlet problem in the half-plane for differential-difference elliptic equations,” Math. Notes 100 (4), 579–588 (2016).

    Article  MathSciNet  Google Scholar 

  13. A. B. Muravnik, “On the half-plane Dirichlet problem for differential-difference elliptic equations with several nonlocal terms,” Math. Model. Nat. Phenom. 12 (6), 130–143 (2017).

    Article  MathSciNet  Google Scholar 

  14. A. B. Muravnik, “Asymptotic properties of solutions of two-dimensional differential- difference elliptic problems,” in Differential and Functional-Differential Equations (RUDN, Moscow, 2017), Vol. 63, pp. 678–688 [in Russian].

    MathSciNet  Google Scholar 

  15. A. B. Muravnik, “On stabilization of solutions of singular elliptic equations,” J. Math. Sci. (N. Y.) 150 (5), 2408–2421 (2008).

    Article  MathSciNet  Google Scholar 

  16. I. M. Gel’fand and G. E. Shilov, “Fourier transforms of rapidly increasing functions and questions of uniqueness of the solution of Cauchy’s problem,” Uspekhi Mat. Nauk 8 (6 (58)), 3–54 (1953).

    MathSciNet  Google Scholar 

  17. I. M. Gel’fand and G. E. Shilov, Generalized Functions (Distributions), in Some Questions of the Theory of Differential Equations (Fizmatgiz, Moscow, 1958), Vol. 3 [in Russian].

    MATH  Google Scholar 

  18. G. E. Shilov, Mathematical Analysis, Second Special Course (Izd. Moskov. Univ., Moscow, 1984) [in Russian].

    MATH  Google Scholar 

Download references

Acknowledgments

The author wishes to express deep gratitude to A. L. Skubachevskii for his permanent attention to this work.

Funding

This work was supported by the RUDN Program “5-100” (Secs. 1, 2) and by the Russian Foundation for Basic Research under grant 20-01-00288 A (Secs. 3, 4).

Author information

Authors and Affiliations

  1. JSC Concern “Sozvezdie”, 394018, Voronezh, Russia

    A. B. Muravnik

  2. Peoples’ Friendship University of Russia (RUDN University), 117198, Moscow, Russia

    A. B. Muravnik

Authors
  1. A. B. Muravnik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. B. Muravnik.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Muravnik, A.B. Elliptic Differential-Difference Equations in the Half-Space. Math Notes 108, 727–732 (2020). https://doi.org/10.1134/S0001434620110115

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434620110115

Keywords

Search

Navigation