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Journal of Mathematical Sciences

, Volume 233, Issue 4, pp 541–554 | Cite as

Smoothness of Generalized Solutions of the Dirichlet Problem for Strongly Elliptic Functional Differential Equations with Orthotropic Contractions

  • A. L. TasevichEmail author
Article
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Abstract

In the disk, we consider the first boundary-value problem for a functional differential equation containing transformations of orthotropic contractions of independent variables of the unknown function. We study the smoothness of generalized solutions inside special-type subdomains and near their boundaries and pose strong ellipticity conditions.

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References

  1. 1.
    C. Auscher, A. McIntosh, and C. Tchamitchian, “The Kato square root problem for higher order elliptic operators and systems on ℝn ,J. Evol. Equ., 1, No. 4, 361–385 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Axelsson, S. Keith, and A. McIntosh, “The Kato square root problem for mixed boundary value problems,” J. London Math. Soc. (2), 74, No. 1, 113–130 (2006).Google Scholar
  3. 3.
    N. Danford and J. T. Schwartz, Linear Operators. Part 2 [Russian translation], Mir, Moscow (1966).Google Scholar
  4. 4.
    L. Gårding, “Dirichlet’s problem for linear elliptic partial differential equations,” Math. Scand., No. 1, 55–72 (1953).Google Scholar
  5. 5.
    O. V. Guseva, “On boundary problems for strongly elliptic systems,” Dokl. Akad. Nauk SSSR, 102, 1069–1072 (1955).MathSciNetGoogle Scholar
  6. 6.
    T. Kato, “Fractional powers of dissipative operators,” J. Math. Soc. Japan, 13, No. 3, 246–274 (1961).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).Google Scholar
  8. 8.
    J.-L. Lions, “Espaces d’interpolation et domaines de puissance fractionnaires d’opérateurs,” J. Math. Soc. Japan, 14, No. 2, 233–241 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. McIntosh, “On the comparability of A 1/2 and A*1/2 ,Proc. Amer. Math. Soc., 32, No. 2, 430–434 (1972).MathSciNetzbMATHGoogle Scholar
  10. 10.
    V. P. Mikhailov, Partial Differential Equations [in Russian], Nauka, Moscow (1976).Google Scholar
  11. 11.
    C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin–Heidelberg–New York (1966).zbMATHGoogle Scholar
  12. 12.
    D. A. Neverova and A. L. Skubachevskii, “On smoothness of generalized solutions to boundary value problems for strongly elliptic differential-difference equations on a boundary of neighboring domains,” Russ. J. Math. Phys., 22, No. 4, 504–517 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L. E. Rossovskii, “Coercitivity of functional differential equations,” Math. Notes, 59, No. 1-2, 75–82 (1996).MathSciNetCrossRefGoogle Scholar
  14. 14.
    L .E. Rossovskii, “On the coercivity of functional differential equations,” J. Math. Sci. (N. Y.), 201, No. 5, 663–672 (2014).Google Scholar
  15. 15.
    L. E. Rossovskii, “Elliptic functional differential equations with contractions and extensions of independent variables of the unknown function,” Sovrem. Mat. Fundam. Napravl., 54, 3–138 (2014).Google Scholar
  16. 16.
    L. E. Rossovskii and A. L. Tasevich, “The first boundary-value problem for strongly elliptic functional-differential equations with orthotropic contractions,” Math. Notes, 97, No. 5-6, 745–758 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    R. V. Shamin, “Spaces of initial data for differential equations in a Hilbert space,” Sb. Math., 194, No. 9-10, 1411–1426 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. L. Skubachevskii, “Smoothness of solutions of the first boundary value problem for an elliptic differential-difference equation,” Mat. Zametki, 34, No. 1, 105–112 (1983).MathSciNetGoogle Scholar
  19. 19.
    A. L. Skubachevskii, “The first boundary value problem for strongly elliptic differential-difference equations,” Differ. Equ., 63, No. 3, 332–361 (1986).MathSciNetCrossRefGoogle Scholar
  20. 20.
    A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel–Boston–Berlin (1997).zbMATHGoogle Scholar
  21. 21.
    A. L. Skubachevskii and E. L. Tsvetkov, “The second boundary value problem for elliptic differential-difference equations,” Differ. Equ., 25, No. 10, 1245–1254 (1990).MathSciNetGoogle Scholar
  22. 22.
    E. L. Tsvetkov, “On the smoothness of generalized solutions of the third boundary value problem for an elliptic differential-difference equation,” Ukrainian Math. J., 45, No. 8, 1272–1284 (1994).MathSciNetCrossRefGoogle Scholar
  23. 23.
    M. I. Vishik, “On strongly elliptic systems of differential equations,” Mat. Sb., 29, No. 3, 615–676 (2010).MathSciNetGoogle Scholar

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

Authors and Affiliations

  1. 1.RUDN UniversityMoscowRussia

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