Operational approach for biharmonic equations in \({\varvec{L}}^{{\varvec{p}}}\)-spaces

Abstract

In this work, we study the existence, uniqueness and maximal \(L^p\)-regularity of the solution of different biharmonic problems. We rewrite these problems by a fourth-order operational equation and different boundary conditions, set in a cylindrical n-dimensional spatial region \(\Omega \) of \({\mathbb {R}}^n\). To this end, we give an explicit representation formula, using analytic semigroups, and invert explicitly a determinant operator in \(L^p\)-spaces thanks to \(\mathcal {E}_\infty \) functional calculus and operator sums theory.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    J. Bourgain, “Some remarks on Banach spaces in which martingale difference sequences are unconditional”, Ark. Mat., vol. 21, 1983, pp. 163–168.

    MathSciNet  Article  Google Scholar 

  2. 2.

    D.L. Burkholder, “A geometrical characterisation of Banach spaces in which martingale difference sequences are unconditional”, Ann. Probab., vol. 9, 1981, pp. 997–1011.

    MathSciNet  Article  Google Scholar 

  3. 3.

    F. Cakoni, G. C. Hsiao & W. L. Wendland, “On the boundary integral equation methodfor a mixed boundary value problem of thebiharmonic equation”, Complex variables, Vol. 50, No. 7–11, 2005, pp. 681–696.

    MATH  Google Scholar 

  4. 4.

    D.S. Cohen & J.D. Murray, “A generalized diffusion model for growth and dispersal in population”, Journal of Mathematical Biology, 12, Springer-Verlag, 1981, pp. 237–249.

  5. 5.

    M. Costabel, E. Stephan & W. L. Wengland, “On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain”, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, \(4^{\text{e}}\) serie, tome 10, no 2, 1983, pp. 197–241.

  6. 6.

    G. Da Prato & P. Grisvard, “Sommes d’opérateurs linéaires et équations différentielles opérationnelle”, J. Math. pures et appl., 54, 1975, pp. 305–387.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    G. Dore & A. Venni, “On the closedness of the sum of two closed operators”, Math. Z., 196, 1987, pp. 189–201.

    MathSciNet  Article  Google Scholar 

  8. 8.

    A. Favini, R. Labbas, S. Maingot, K. Lemrabet & H. Sidibé, “Resolution and Optimal Regularity for a Biharmonic Equation with Impedance Boundary Conditions and Some Generalizations”, Discrete and Continuous Dynamical Systems, Vol. 33, 11–12, 2013, pp. 4991–5014.

    MathSciNet  MATH  Google Scholar 

  9. 9.

    A. Favini, R. Labbas, S. Maingot, H. Tanabe & A. Yagi, “A simplified approach in the study of elliptic differential equations in UMD spaces and new applications”, Funkc. Ekv., 451, 2008, pp. 165–187.

    MathSciNet  Article  Google Scholar 

  10. 10.

    A. Favini, R. Labbas, S. Maingot, H. Tanabe & A. Yagi, “Complete abstract differential equations of elliptic type in UMD spaces”, Funkc. Ekv., 49, 2006, pp. 193–214.

    MathSciNet  Article  Google Scholar 

  11. 11.

    G. Geymonat & F. Krasucki, “Analyse asymptotique du comportement en flexion de deux plaques collées”, C. R. Acad. Sci. Paris - serie II b, Volume 325, Issue 6, 1997, pp. 307–314.

  12. 12.

    D. Gilbarg & N. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

  13. 13.

    P. Grisvard, “Spazi di tracce e applicazioni”, Rendiconti di Matematica, (4) Vol.5, Serie VI, 1972, pp. 657–729.

  14. 14.

    Z. Guo, B. Lai & D. Ye, “Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions”, Proc. Amer. Math. Soc., 142, 2014, pp. 2027–2034.

    MathSciNet  Article  Google Scholar 

  15. 15.

    M. Haase, The functional calculus for sectorial Operators, Birkhauser, Basel, 2006.

    Book  Google Scholar 

  16. 16.

    M. Haase, “Functional calculus for groups and applications to evolution equations”, Journal of Evolution Equations, Volume 7, Issue 3, 2007, pp. 529–554.

    MathSciNet  Article  Google Scholar 

  17. 17.

    M. A. Jaswon & G. T. Symm, “Integral equation methods in potential theory and elastostatics”, Academic Press, New York, San Francisco, London, 1977, pp. 1–10.

    MATH  Google Scholar 

  18. 18.

    P. C. Kunstmann & L. Weis, “New criteria for the \(H^\infty \)-calculus and the Stokes operator on bounded Lipschitz domains”, Journal of Evolution Equations, Volume 17, Issue 1, 2017, pp. 387–409.

    MathSciNet  Article  Google Scholar 

  19. 19.

    R. Labbas, K. Lemrabet, S. Maingot & A. Thorel, “Generalized linear models for population dynamics in two juxtaposed habitats”, Discrete and Continuous Dynamical Systems - A, Volume 39, Number 5, 2019, pp. 2933–2960.

    MathSciNet  Article  Google Scholar 

  20. 20.

    R. Labbas, S. Maingot, D. Manceau & A. Thorel, “On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain”, Journal of Mathematical Analysis and Applications, 450, 2017, pp. 351–376.

    MathSciNet  Article  Google Scholar 

  21. 21.

    C. Le Merdy, “A sharp equivalence between \(H^\infty \) functional calculus and square function estimates”, Journal of Evolution Equations, Volume 12, Issue 4, 2012, pp. 789–800.

    MathSciNet  Article  Google Scholar 

  22. 22.

    K. Limam, R. Labbas, K. Lemrabet, A. Medeghri & M. Meisner, “On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution”, Journal of Differential Equations, Volume 259, issue 7, 2015, p. 2695–2731.

    MathSciNet  Article  Google Scholar 

  23. 23.

    F. Lin & Y. Yang,“Nonlinear non-local elliptic equation modelling electrostatic actuation”, Proceedings of the Royal Society of London A, 463, 2007, pp. 1323–1337.

  24. 24.

    J.-L. Lions & J. Peetre, “Sur une classe d’espaces d’interpolation”, Publications mathématiques de l’I.H.É.S., tome 19, 1964, pp. 5–68.

  25. 25.

    A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Birkhauser, Basel, Boston, Berlin, 1995.

    Book  Google Scholar 

  26. 26.

    F.L. Ochoa, “A generalized reaction-diffusion model for spatial structures formed by motile cells”, BioSystems, 17, 1984, pp. 35–50.

    Article  Google Scholar 

  27. 27.

    A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.

    Book  Google Scholar 

  28. 28.

    J. Prüss & H. Sohr, “On operators with bounded imaginary powers in Banach spaces”, Mathematische Zeitschrift, Springer-Verlag, Math. Z., 203, 1990, pp. 429–452.

    MathSciNet  MATH  Google Scholar 

  29. 29.

    J. Prüss & H. Sohr, “Imaginary powers of elliptic second order differential operators in \(L^p\)-spaces”, Hiroshima Math. J., 23, no. 1, 1993, pp. 161–192.

    MathSciNet  Article  Google Scholar 

  30. 30.

    J.L. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, in: J. Bastero, M. San Miguel (Eds.), Probability and Banach Spaces, Zaragoza, 1985, in: Lecture Notes in Math., vol. 1221, Springer-Verlag, Berlin, 1986, pp. 195–222.

  31. 31.

    H. Saker & N. Bouselsal, “On the bilaplacian problem with nonlinear boundary conditions”, Indian J. Pure Appl. Math., Volume 47, Issue 3, 2016, pp. 425–435.

    MathSciNet  Article  Google Scholar 

  32. 32.

    I. Titeux & E. Sanchez-Palencia, “Conditions de transmission pour les jonctions de plaques minces”, C. R. Acad. Sci. Paris - serie II b, Volume 325, Issue 10, 1997, pp. 563–570.

  33. 33.

    H. Triebel, Interpolation theory, function Spaces, differential Operators, North-Holland publishing company Amsterdam New York Oxford, 1978.

    MATH  Google Scholar 

Download references

Acknowledgements

This research is supported by CIFRE Contract 2014/1307 with Qualiom Eco company. The author would like to thank the referee for the valuable and useful comments and corrections which help to improve this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alexandre Thorel.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Thorel, A. Operational approach for biharmonic equations in \({\varvec{L}}^{{\varvec{p}}}\)-spaces. J. Evol. Equ. 20, 631–657 (2020). https://doi.org/10.1007/s00028-019-00536-2

Download citation

Mathematics Subject Classification

  • 35B65
  • 35C15
  • 35J40
  • 35R20
  • 47A60
  • 47D06

Keywords

  • Operational differential equations
  • Functional calculus
  • Analytic semigroups
  • Interpolation spaces
  • Maximal regularity