Publication: Non-Hookean Beams and Plates: Very Weak Solutions and their Numerical Analysis
Loading...
Files
Official URL
Full text at PDC
Publication Date
2014
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Institute for Scientific Computing and Information
Citations
Exportar
Abstract
We consider very weak solutions of a nonlinear version (non-Hookean materials) of the beam stationary Bernoulli-Euler equation, as well as the similar extension to plates, involving the bi-Laplacian operator, with Navier (hinged) boundary conditions. We are specially interested in the case in which the usual Sobolev space framework cannot be applied due to the singularity of the load density near the boundary. We present some properties of such solutions as well as some numerical experiences illustrating how the behaviour of the very weak solutions near the boundary is quite different to the one of more regular solutions corresponding to non-singular load functions.
Description
UCM subjects
Unesco subjects
Keywords
Citation
S. Antontsev, J. I. Díaz, S. Shmarev, Energy methods for free boundary problems. Applications to nonlinear PDEs and Fluid Mechanics, Series Progress in Nonlinear Differential Equations and Their Applications, No. 48, Birkäuser, Boston, 2002.
I. Arregui, J.I. Díaz and C. Vázquez, A nonlinear bilaplacian equation with hinged boundary conditions and very weak solutions: analysis and numerical solution. To appear
F. Bernis, “On some nonlinear singular boundary value problems of higher order,” Nonlinear. Analysis: Theory, Methods & Applications, vol. 26, no. 6, pp. 1061–1078, 1996.
H. Brezis and X. Cabr´e, Some simple nonlinear PDE’s without solutions, Bull UMI, 1(1998),223–262
H. Brezis, T. Cazenave, Y. Martel, A. Ramiandrisoa, Blow up for ut − ∆u = g(u) revisited, Advance in Diff. Eq. 1, 1996), 73-90.
E. Castillo, A. Iglesias and R. Ruiz-Cobo. Functional Equations in Applied Sciences. Elsevier, 2004
E. Castillo and R. Ruiz-Cobo. Ecuaciones Funcionales en la Ciencia, la Econom´ıa y la Ingenier´ıa. Editorial Reverté, Barcelona, 1993.
E. Castillo and J.I. Díaz, In preparation.
H. Castro, J. Dávila and H. Wang, A Hardy type inequality for W2,1 0 (Ω) functions, C. R. Acad. Sci. Paris, Ser. I 349 (2011) 765–767.
F. Chung and S.-T. Yau: Discrete Green’s functions, Journal of Combinatorial Theory (A), 91, (2000), 191-214.
P.G. Ciarlet, Discrete variational Green’s function. I. Aequationes Math. 4 (1970) 74–82.
P.G. Ciarlet, Introduction `a l’analyse numerique matricielle et l’optimisation, Masson, Paris, 1982.
P.G. Ciarlet and R.S.Varga, Discrete variational Green’s function. II. One dimensional problem. Numer. Math. 16 (1970) 115–128.
M.G. Crandall, P.H. Rabinowitz and L. Tartar. On a Dirichlet problem with singular nonlinearity.
Comm. P.D.E. 2(1977), 193-222.
M.G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving maps, Proc. AMS 78 (3) (1980) 385-390.
C. R. Deeter and G. J. Gray, The discrete Green’s function and the discrete kernel function, Discrete Math. 10 (1974), 29-42.
J.I. Díaz. Nonlinear Partial Differential equations and Free Boundaries. London, Pitman, 1985.
J.I. Díaz, On the very weak solvability of the beam equation. Rev. R. Acad. Cien. Serie A. Mat (RACSAM) 105 (2011), 167–173.
J.I.Díaz, J. Hernández and J.M.Rakotoson. On very weak positive solutions to some semilinear elliptic problems wtth simultaneous singular nonlinear and spatial dependence terms. Milan J.Math. 79 (2011), 233-245.
J.I.Díaz and J.M. Rakotoson. On the differentiability of very weak slutions with right-hand side integrable with respect to the distance to the boundary. J. Funct. Anal. 357(2009), 807-831.
J.I.Díaz and J.M. Rakotoson. On very weak solutions of semi-linear elliptic euqations in the framework of weighted spaces with respect to the distance to the boundary. Disc. Cont. Dyn. Syst. 27(2010), 1037-1058.
J.I.Díaz and J.M. Rakotoson, L1 Ω, dist(x, ∂Ω))-problems and their applications revisted. To appear in Electronic Journal Differential Equations.
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, EnglewwodCliffs, N.J. 1964.
M. Ghergu, A biharmonic equation with singular nonlinearity. Proc. Edinb. Math. Soc. (2)55 (2012), no. 1, 155–166.
Ph. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific, New Jersey, 2010.
D. H. Mugler, Green’s functions for the finite difference heat, Laplace and wave equations. In Anniversary volume on approximation theory and functional analysis (Oberwolfach, 1983), Internat. Schriftenreihe Numer. Math. 65, Birkhauser, Basel-Boston, Mass., 1984, 543-554.
J.M. Rakotoson, A few natural extension of the regularity of a very weak solution, Differential and Integral Equations, 24 11-12, (2011), 1125-1140.
G. Shi and J. Zhang: Positive solutions for higher order singular p-Laplacian boundary value problems, Proc. Indian Acad. Sci. (Math. Sci.), 118 (2008), 295-305
Ph. Souplet, A survey on L p δ spaces and their applications to nonlinear elliptic and parabolic problems. In Nonlinear partial differential equations and their applications, 464–479, GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakk¯otosho, Tokyo, 2004.
Ph. Souplet, Optimal regularity conditions for elliptic problems via L p δ-spaces. Duke Math. J. 127 (2005), no. 1, 175–192
I. Stakgold, Green’s Functions and Boundary Value Problems, Pure and Applied Mathematics Ser., Wiley and Sons, 1998.
T. Vejchodsk´y and P. Solín , Discrete Green’s function and Maximum Principles, In International Conference Programs and Algorithms of Numerical Mathematics 13 (in honor of Ivo Babuska’s 80th birthday), Edited by J. Chleboun, K. Segeth and T. Vejchodsky, Mathematical Institute Academy of Sciences of the Czech Republic, Prague 2006, 247-252.