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

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We study linear and nonlinear bilaplacian problems with hinged boundary conditions and right hand side in L1( : δ), with δ = dist(x, ∂). More precisely, the existence and uniqueness of the very weak solution is obtained and some numerical techniques are proposed for its approximation.
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) Brezis, H.: Une équation Semi-linéaire Avec Conditions Aux Limites Dans L1. Personal communication to J.I. Díaz (unpublished) Brezis, H., Cabré, X.: Some simple nonlinear PDE’s without solutions. Bull. UMI 1, 223–262 (1998) Brezis, H., Cazenave, T., Martel, Y., Ramiandrisoa, A.: Blow up for ut − �u = g(u) revisited. Adv.Differ. Equ. 1, 73–90 (1996) 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 L1. Numer.Math. 105, 337–374 (2007) Crandall, M.G., Tartar, L.: Some relations between nonexpansive and order preserving maps. Proc. AMS 78(3), 385–390 (1980) Díaz, J.I.: On the very weak solvability of the beam equation. Rev. R. Acad. Cien. Ser. A (RACSAM) 105, 167–172 (2011) Díaz, J.I.: Non Hookean Beams and Plates: Very Weak Solutions and Their Numerical Analysis (2013).(submitted) 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) 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) 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) 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) Friedman, A.: Generalized Functions and Partial Differential Equations. Prentice-Hall, Englewood Cliffs (1963) Ghergu, M.: A biharmonic equation with singular nonlinearity. Proc. Edinb. Math. Soc. 55, 155–166(2012) Souplet, Ph.: A survey on L p δ 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) Stakgold, I.: Green’s functions and boundary value problems. In: Pure and Applied Mathematics Series. Wiley, New York (1998)