Comptes Rendus
no. 9
Mathematical Problems in Mechanics
Uniqueness and continuous dependence on the initial data for a class of non-linear shallow shell problems
[Théorème d'unicité et de dépendence continue des solutions par rapport aux conditions initiales pour une classe de problèmes non linéaires de coques peu profondes]

Présenté par : Philippe G. Ciarlet

John Cagnol1 ; Irena Lasiecka2 ; Catherine Lebiedzik3 ; Richard Marchand4
1 Pôle universitaire Leonard-de-Vinci, ESILV, DER CS, 92916 Paris La Défense cedex, France
2 University of Virginia, Department of Mathematics, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 22904, USA
3 Wayne State University, Department of Mathematics, 656 W. Kirby, Room 1150, Detroit, MI 48202, USA
4 Slippery Rock University, Department of Mathematics, 229 Vincent Science Hall, Slippery Rock, PA 16057, USA
Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 711-716.

Dans cette Note, nous nous intéressons au modèle introduit en 1966 par W.T. Koiter, puis étudié par M. Bernadou et J.T. Oden. Nous démontrons l'unicité de la solution du modèle dynamique et que cette solution est continue par rapport aux conditions initiales.

This note is concerned with the non-linear shallow shell model introduced in 1966 by W.T. Koiter, and later studied by M. Bernadou and J.T. Oden. We show the uniqueness of the solution to the dynamical model and that this solution is continuous with respect to the initial data.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.02.034
John Cagnol 1 ; Irena Lasiecka 2 ; Catherine Lebiedzik 3 ; Richard Marchand 4

1 Pôle universitaire Leonard-de-Vinci, ESILV, DER CS, 92916 Paris La Défense cedex, France
2 University of Virginia, Department of Mathematics, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 22904, USA
3 Wayne State University, Department of Mathematics, 656 W. Kirby, Room 1150, Detroit, MI 48202, USA
4 Slippery Rock University, Department of Mathematics, 229 Vincent Science Hall, Slippery Rock, PA 16057, USA 
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John Cagnol; Irena Lasiecka; Catherine Lebiedzik; Richard Marchand. Uniqueness and continuous dependence on the initial data for a class of non-linear shallow shell problems. Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 711-716. doi : 10.1016/j.crma.2006.02.034. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.02.034/

[1] M. Bernadou Méthodes d'Eléments Finis pour les Problèmes de Coques Minces, Masson, Paris, 1994

[2] M. Bernadou; J.T. Oden An existence theorem for a class of nonlinear shallow shell problems, J. Math. Pures Appl. (9), Volume 60 (1981) no. 3, pp. 285-308

[3] J. Cagnol; I. Lasiecka; C. Lebiedzik; J.-P. Zolésio Uniform stability in structural acoustic models with flexible curved walls, J. Differential Equations, Volume 186 (2002) no. 1, pp. 88-121

[4] J. Cagnol, I. Lasiecka, C. Lebiedzik, R. Marchand, Hadamard wellposedness for a class of non-linear shallow shell problems, ESILV, DER-CS RR-28, Pôle Universitaire L. de Vinci, Paris La Défense, France, 2005

[5] P.G. Ciarlet Mathematical Elasticity, vol. III: Theory of Shells, Stud. Math. Appl., vol. 29, North-Holland Publishing Co., Amsterdam, 2000

[6] M.C. Delfour, J.-P. Zolésio, Intrinsic differential geometry and theory of thin shells, in press

[7] M.C. Delfour; J.-P. Zolésio Differential equations for linear shells: comparison between intrinsic and classical models, Adv. Math. Sci., CRM Proc. Lect. Notes, vol. 11, Amer. Math. Soc., 1997, pp. 41-124

[8] H. Koch; I. Lasiecka Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity—full von Karman systems, Progr. Nonlinear Differential Equations Appl., Volume 50 (2002), pp. 197-216

[9] W.T. Koiter On the nonlinear theory of thin elastic shells. III, Nederl. Akad. Wetensch. Proc. Ser. B, Volume 69 (1966), pp. 33-54

[10] J.-L. Lions; E. Magenes Problèmes aux limites non homogènes et applications, Dunod, 1968

[11] P.M. Naghdi Foundations of elastic shell theory, Progr. Solid Mech., vol. IV, North-Holland, Amsterdam, 1963, pp. 1-90

[12] Jr. Sanders; J. Lyell Nonlinear theories for thin shells, Quart. Appl. Math., Volume 21 (1963), pp. 21-36

[13] V.I. Sedenko On the uniqueness theorem for generalized solutions of initial-boundary problems for the Marguerre–Vlasov of shallow shells with clamped boundary conditions, Appl. Math. Optim., Volume 39 (1999) no. 3, pp. 309-326

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