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A nonlinear bilaplacian equation with hinged boundary conditions and very weak solutions: analysis and numerical solution

  • Iñigo Arregui
  • Jesús Ildefonso Díaz
  • Carlos VázquezEmail author
Original Paper
  • 111 Downloads

Abstract

We study linear and nonlinear bilaplacian problems with hinged boundary conditions and right hand side in \(L^{1}(\Omega :\delta )\), with \(\delta =\text{ dist }\,(x,\partial \Omega )\). More precisely, the existence and uniqueness of the very weak solution is obtained and some numerical techniques are proposed for its approximation.

Keywords

Very weak solutions Distance to the boundary Nonlinear bilaplacian operator Hinged boundary conditions Numerical methods Finite elements 

Mathematics Subject Classification

35G50 35G60 74G25 74G15 

References

  1. 1.
    Bayada, G., Durany, J., Vázquez, C.: Existence of a solution for a lubrication problem in elastic journal-bearing devices with thin bearing. Math. Methods Appl. Sci. 18, 255–266 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Brezis, H.: Une équation Semi-linéaire Avec Conditions Aux Limites Dans \(L^1\). Personal communication to J.I. Díaz (unpublished)Google Scholar
  3. 3.
    Brezis, H., Cabré, X.: Some simple nonlinear PDE’s without solutions. Bull. UMI 1, 223–262 (1998)zbMATHGoogle Scholar
  4. 4.
    Brezis, H., Cazenave, T., Martel, Y., Ramiandrisoa, A.: Blow up for \(u_t-\Delta u = g(u)\) revisited. Adv. Differ. Equ. 1, 73–90 (1996)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Casado-Díaz, J., Chacón-Rebollo, T., Girault, V., Gómez-Mármol, M., Murat, F.: Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in \(L^1\). Numer. Math. 105, 337–374 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Crandall, M.G., Tartar, L.: Some relations between nonexpansive and order preserving maps. Proc. AMS 78(3), 385–390 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Díaz, J.I.: On the very weak solvability of the beam equation. Rev. R. Acad. Cien. Ser. A (RACSAM) 105, 167–172 (2011)zbMATHCrossRefGoogle Scholar
  8. 8.
    Díaz, J.I.: Non Hookean Beams and Plates: Very Weak Solutions and Their Numerical Analysis (2013). (submitted)Google Scholar
  9. 9.
    Díaz, J.I., Hernández, J., Rakotoson, J.M.: On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms. Milan J. Math. 79, 233–245 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Díaz, J.I., Rakotoson, J.M.: On the differentiability of very weak solutions with right hand side data integrable with respect to the distance to the boundary. J. Funct. Anal. 257, 807–831 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Díaz, J.I., Rakotoson, J.M.: On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discret. Contin. Dyn. Syst. 27, 1037–1058 (2010)zbMATHCrossRefGoogle Scholar
  12. 12.
    Durany, J., García, G., Vázquez, C.: An elastohydrodynamic coupled problem between a piezoviscous Reynolds equation and a hinged plate model. RAIRO Modél. Math. Anal. Numér. 31, 495–516 (1997)zbMATHGoogle Scholar
  13. 13.
    Friedman, A.: Generalized Functions and Partial Differential Equations. Prentice-Hall, Englewood Cliffs (1963)Google Scholar
  14. 14.
    Ghergu, M.: A biharmonic equation with singular nonlinearity. Proc. Edinb. Math. Soc. 55, 155–166 (2012)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Souplet, Ph.: A survey on \(L_{\delta }^{p}\) spaces and their applications to nonlinear elliptic and parabolic problems. Nonlinear partial differential equations and their applications. GAKUTO Int. Ser. Math. Sci. Appl. 20, 464–479 (2004)Google Scholar
  16. 16.
    Stakgold, I.: Green’s functions and boundary value problems. In: Pure and Applied Mathematics Series. Wiley, New York (1998)Google Scholar

Copyright information

© Springer-Verlag Italia 2013

Authors and Affiliations

  • Iñigo Arregui
    • 1
  • Jesús Ildefonso Díaz
    • 2
  • Carlos Vázquez
    • 1
    Email author
  1. 1.Department of Mathematics, Faculty of InformaticsUniversity of A Coruña CoruñaSpain
  2. 2.Department of Applied Mathematics, Instituto de Matemática InterdisciplinarComplutense University of Madrid MadridSpain

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