Journal of Mathematical Sciences

, Volume 235, Issue 4, pp 473–483 | Cite as

On the Dirichlet Problem for Differential-Difference Elliptic Equations in a Half-Plane

  • A. B. MuravnikEmail author


The Dirichlet problem is considered in a half-plane (with continuous and bounded boundaryvalue function) for the model elliptic differential-difference equation
$$ {u}_{xx}+a{u}_{xx}\left(x+h,y\right)+{u}_{yy}=0,\mid a\mid <1. $$

Its solvability is proved in the sense of generalized functions, the integral representation of the solution is constructed, and it is proved that everywhere but the boundary hyperplane this solution satisfies the equation in the classic sense as well.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. N. Denisov and A. B. Muravnik, “On asymptotic behavior of solutions of the Dirichlet problem in half-space for linear and quasi-linear elliptic equations,” Electron. Res. Announc. Am. Math. Soc., 9, 88–93 (2003).MathSciNetCrossRefGoogle Scholar
  2. 2.
    V. N. Denisov and A. B. Muravnik, “On asymptotic form of solution of the Dirichlet problem for an elliptic equation in a halfspace,” In: Nonlinear Analysis and Nonlinear Differential Equations [in Russian], 397–417, Fizmatlit, Moscow (2003).zbMATHGoogle Scholar
  3. 3.
    N. Dunford and J. T. Schwartz, Linear Operators. Spectral Theory [in Russian], Mir, Moscow (1966).Google Scholar
  4. 4.
    I. M. Gel’fand and G. E. Shilov, “Fourier transformations of rapidly growing functions and questions of unique solvability of the Cauchy problem,” Usp. Mat. Nauk, 8, No. 6, 3–54 (1953).Google Scholar
  5. 5.
    I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol. 3: Some Questions of the Theory of Differential Equations [in Russian], Fizmatgiz, Moscow (1958).zbMATHGoogle Scholar
  6. 6.
    D. Gilbarg and N. Trudinger, Second-Order Elliptic Equations [in Russian], Mir, Moscow (1989).zbMATHGoogle Scholar
  7. 7.
    P. L. Gurevich, “Elliptic problems with nonlocal boundary-value conditions and Feller semigroups,” Sovrem. Mat. Fundam. Napravl., 38, 3–173 (2010).Google Scholar
  8. 8.
    V. A. Kondrat’ev and E. M. Landis, “Qualitative theory of second-order linear partial differential equations,” Itogi Nauki Tekhn. Sovrem. Probl. Mat., 32, 99–218 (1988).MathSciNetzbMATHGoogle Scholar
  9. 9.
    G. E. Shilov, Analysis. The Second Special Course [in Russian], MGU, Moscow (1984).Google Scholar
  10. 10.
    A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel–Boston–Berlin (1997).zbMATHGoogle Scholar
  11. 11.
    A. L. Skubachevskii, “Nonclassic boundary-value problems. I,” Sovrem. Mat. Fundam. Napravl., 26, 3–132 (2007).Google Scholar
  12. 12.
    A. L. Skubachevskii, “Nonclassic boundary-value problems. II,” Sovrem. Mat. Fundam. Napravl., 33, 3–179 (2009).Google Scholar
  13. 13.
    V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1976).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.JSC Concern “Sozvezdie”VoronezhRussia
  2. 2.RUDN UniversityMoscowRussia

Personalised recommendations