Skip to main content
SpringerLink
Log in

H2 Convergence of Solutions of a Biharmonic Problem on a Truncated Convex Sector Near the Angle π

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

We consider a biharmonic problem Δ2uω = fω with Navier type boundary conditions uω = Δuω = 0, on a family of truncated sectors Ωω in ℝ2 of radius r, 0 < r < 1 and opening angle ω, ω ∈ (2π/3, π] when ω is close to π. The family of right-hand sides (fω)ω∈(2π/3, π] is assumed to depend smoothly on ω in L2(Ωω). The main result is that uω converges to uπ when ω → π with respect to the H2-norm. We can also show that the H2-topology is optimal for such a convergence result.

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. H. Blum, R. Rannacher: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980), 556–581.

    Article  MathSciNet  Google Scholar 

  2. M. Costabel, M. Dauge: General edge asymptotics of solutions of second-order elliptic boundary value problems I. Proc. R. Soc. Edinb., Sect. A 123 (1993), 109–155.

    Article  MathSciNet  Google Scholar 

  3. M. Costabel, M. Dauge: General edge asymptotics of solutions of second-order elliptic boundary value problems II. Proc. R. Soc. Edinb., Sect. A 123 (1993), 157–184.

    Article  MathSciNet  Google Scholar 

  4. M. Dauge: Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics 1341. Springer, Berlin, 1988.

    Book  Google Scholar 

  5. M. Dauge, S. Nicaise, M. Bourlard, J. M.-S. Lubuma: Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques I.: Résultats généraux pour le problème de Dirichlet. RAIRO, Modélisation Math. Anal. Numér. 24 (1990), 27–52. (In French.)

    Article  MathSciNet  Google Scholar 

  6. F. Gazzola, H.-C. Grunau, G. Sweers: Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Lecture Notes in Mathematics 1991. Springer, Berlin, 2010.

    MATH  Google Scholar 

  7. P. Grisvard: Alternative de Fredholm rélative au problème de Dirichlet dans un polygone ou un polyedre. Boll. Unione Mat. Ital., IV. Ser. 5 (1972), 132–164. (In French.)

    MATH  Google Scholar 

  8. P. Grisvard: Elliptic Problems in Nonsmooth Domains. Monograhs and Studies in Mathematics 24. Pitman, Boston, 1985.

    MATH  Google Scholar 

  9. V. A. Kondrat’ev: Boundary problems for elliptic equation in domains with conical or angular points. Trans. Mosc. Math. Soc. 16 (1967), 227–313; translation from Tr. Mosk. Mat. O.-va 16 (1967), 209–292.

    MathSciNet  MATH  Google Scholar 

  10. V. G. Maz’ya, B. A. Plamenevskij: Estimates in Lp and in Holder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Transl., Ser. 2, Am. Math. Soc. 123 (1984), 1–56; translation from Math. Nachr. 81 (1978), 25–82.

    MATH  Google Scholar 

  11. V. G. Maz’ya, B. A. Plamenevskij: Lp-estimates of solutions of elliptic boundary value problems in domains with edges. Trans. Mosc. Math. Soc. 1 (1980), 49–97; translation from Tr. Mosk. Mat. O.-va 37 (1978), 49–93.

    Google Scholar 

  12. V. G. Maz’ya, J. Rossmann: On a problem of Babuska. (Stable asymptotics of the solution to the Dirichlet problem for elliptic equations of second order in domains with angular points). Math. Nachr. 155 (1992), 199–220.

    Article  MathSciNet  Google Scholar 

  13. S. Nicaise: Polygonal interface problems for the biharmonic operator. Maths. Methods Appl. Sci. 17 (1994), 21–39.

    Article  MathSciNet  Google Scholar 

  14. S. Nicaise, A.-M. Sändig: General interface problems I. Math. Methods Appl. Sci. 17 (1994), 395–429.

    Article  MathSciNet  Google Scholar 

  15. S. Nicaise, A.-M. Sändig: General interface problems II. Math. Methods Appl. Sci. 17 (1994), 431–450.

    Article  MathSciNet  Google Scholar 

  16. A. Stylianou: Comparison and Sign Preserving Properties of Bilaplace Boundary Value Problems in Domains with Corners. PhD Thesis. Universität Köln, München, 2010.

    MATH  Google Scholar 

  17. A. Tami: Etude d’un problème pour le bilaplacien dans une famille d’ouverts du plan. PhD Thesis. Aix-Marseille University, Marseille, 2016. Available at https://www.theses.fr/2016AIXM4362. (In French.)

    Google Scholar 

  18. A. Tami: The elliptic problems in a family of planar open sets. Appl. Math., Praha 64 (2019), 485–499.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgment

Gratefully, the authors would like to express their thanks and appreciation for the valuable efforts made by the reviewer in order to substantially improve the latest version of the draft.

This work was initiated differently during the PhD studies of the first author. He wishes to express his gratitude to everyone who contributed to this. In particular, his supervisors: Philippe Tchamitchian, Professor at Aix-Marseille University, Marseille, France, for his valuable and helpful guidance to achieve some results in this context, and Boubakeur Merouani, Professor at the Ferhat Abbas University Setif 1 of Algeria, whose remarks and comments were very beneficial for him to understand some interesting aspects of the biharmonic problems.

Author information

Authors and Affiliations

  1. Département de Mathématiques, Université des Sciences et de la Technologie d’Oran Mohamed-Boudiaf USTOMB, El Mnaouar, BP 1505, Bir El Djir, 31000, Oran, Algeria

    Abdelkader Tami & Mounir Tlemcani

Authors
  1. Abdelkader Tami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mounir Tlemcani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdelkader Tami.

Additional information

The research has been supported by the Ministry of Higher Education and Scientific Research within the framework of PRFU university training projects.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tami, A., Tlemcani, M. H2 Convergence of Solutions of a Biharmonic Problem on a Truncated Convex Sector Near the Angle π. Appl Math 66, 383–395 (2021). https://doi.org/10.21136/AM.2021.0284-19

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/AM.2021.0284-19

Keywords

MSC 2020

Search

Navigation