Mediterranean Journal of Mathematics

, Volume 13, Issue 4, pp 1669–1683 | Cite as

Nonlocal General Boundary Value Problems of Elliptic Type in Lp Cases

  • Houari Hammou
  • Rabah Labbas
  • Stéphane Maingot
  • Ahmed Medeghri
Article
  • 52 Downloads

Abstract

This paper is devoted to the study of operational second-order differential equations of elliptic type with nonregular coefficient-operator boundary conditions. In the framework of UMD spaces, we give some new results by using semi-groups and interpolation theory. We define two types of solutions: semi-strict and strict solutions. We then characterizes the existence and uniqueness of such solutions.

Keywords

Nonlocal boundary conditions fractional powers of operators interpolation spaces semigroup theory UMD spaces 

Mathematics Subject Classification

34G10 35J05 35J15 35J25 35J99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balakrishnan A.V.: Fractional powers of closed operators and the semigroups generated by them. Pac. J. Math. 10, 419–437 (1960)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cheggag M., Favini A., Labbas R., Maingot S., Medeghri A.: Sturm–Liouville problems for an abstract differential equation of elliptic type in UMD spaces. Differ. Integr. Equ. 21(9–10), 981–1000 (2008)MathSciNetMATHGoogle Scholar
  3. 3.
    Da Prato, G., Grisvard, P.: Sommes d’Opérateurs Linéaires et Equations Différentielles Opérationnelles. J. Math. Pures Appl. IX Ser. 54, 305–387 (1975)Google Scholar
  4. 4.
    Dore, G., Venni, A.: On the closedness of the sum of two closed operators. Math. Z. 196, 124–136 (1987)Google Scholar
  5. 5.
    Dore G., Yakubov S.: Semigroup Estimates And Noncoercive Boundary Value Problems. Semigroup Forum 60, 93–121 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Eltaief, A., Maingot, S.: Second order abstract differential equations of elliptic type set in \({ \mathbb{R} _{+}}\). Demonstr. Math., vol. XLVI, No. 4 (2013), 709–727Google Scholar
  7. 7.
    Favini, A., Labbas, R., Maingot, S., Meisner, M.: Boundary Value Problem for Elliptic Differential Equations in Noncommutative Cases, DCDS, vol. 33 (volume in honor of Jerry’s 70th birthday), Number 11&12, November & December, pp. 4967–4990Google Scholar
  8. 8.
    Gurevich, PL.: Elliptic Problems with Nonlocal Boundary Conditions and Feller Semigroups. J. Math. Sci. 182(3), 255–440 (2012)Google Scholar
  9. 9.
    Haase M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169. Birkhäuser Verlag, Basel-Boston-Berlin (2006)Google Scholar
  10. 10.
    Hammou, H, Labbas, R, Maingot, S, Medeghri, A.: On some elliptic problems with nonlocal boundary coefficient-operator conditions inthe framework of Hölderian spaces. Electron. J. Qual. Theory Differ. Equ. 36, 1–32 (2013)Google Scholar
  11. 11.
    Holmes E.E., Lewis M.A., Banks J.E., Veit R.R.: Partial differential equation in ecology: spatial interactions and population dynamics. Ecology 75(1), 17–29 (1994)CrossRefGoogle Scholar
  12. 12.
    Krein, SG.: Linear Differential Equations in Banach Spaces, Moscou (1967)Google Scholar
  13. 13.
    Martin-Vaquero J., Vigo-Aguiar J.: A note on efficient techniques for the second-order parabolic equation subject to non-local conditions. Appl. Numer. Math. 59, 1258–1264 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Prüss J., Sohr H.: On operators with bounded imaginary powers in banach spaces. Math. Z. 203, 429–452 (1990)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Prüss, H., Sohr, J.: Boundedness of imaginary powers of second-order elliptic differential operators in Lp . Hiroshima Math. J. 23, 161–192 (1993)Google Scholar
  16. 16.
    Skubachevskii, A.L.: Nonclassical boundary-value problems. I, J. Math. Sci. 155(2), 199–334 (2008)Google Scholar
  17. 17.
    Triebel, H.: (1978) Interpolation theory, Function spaces, differential operators. North Holland, AmsterdamGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Houari Hammou
    • 1
  • Rabah Labbas
    • 2
  • Stéphane Maingot
    • 2
  • Ahmed Medeghri
    • 1
  1. 1.Laboratoire de Mathématiques Pures et AppliquéesUniversité de MostaganemMostaganemAlgérie
  2. 2.Laboratoire de Mathématiques AppliquéesUniversité du HavreLe HavreFrance

Personalised recommendations