We begin with simple models of contact problem in elasticity. Difference for the solution
of second order and fourth order elasticity problems are commented. To this end we
recall some results on optimization with inequality constraints.
 Starting with the three-dimensional unilateral contact problem (the so-called Signorini
problem) we show how to get its two-dimensional limit (the obstacle problem).We illustrate
that procedure with the case of an elastic shell as considered in [1]. The asymptotics
follow [3].
 When subjected to small changes of applied forces, the contact zone of an elastic body in
contact with a plane changes, the free boundary of this contact zone moves.We introduce
the problem of the stability with Schaeffer’s work in the scalar case for the Laplacian
operator [2]. Some ideas on the extension are given in the framework of a simplified set
of equations which describes the equilibrium equations of a shallow membrane.
References
[1] L´eger A. and Miara B., The obstacle problem for shallow shells: a curvilinear approach,
Int. J. Numerical Analysis and Modeling, Series B, Vol. 2, (1), 1-26, 2010.
[2] Schaeffer D.G., A stability theorem for the obstacle problem, Advances in mathematics
16 , 34-47, 1975.
[3] Ciarlet P.-G., Miara B., A justification of the two-dimensional equations of a linearly
elastic shallow shell, Comm. Pure Appl. Math., 45, 327-360, 1993.