On sensitivity and related phenomena in thin shells which are not geometrically rigid

We consider the asymptotic behavior as the thickness 2 epsilon tends to zer o of thin elastic shells which are not geometrically rigid for the kinemati c boundary conditions (noninhibited shells). It is known that the limit dis placement belongs to the subspace G of inextensional displacements. We writ e the corresponding mixed formulation with a Lagrange multiplier. It is the n proved that the corresponding problem (equations and boundary conditions) is not elliptic, whatever the type of the surface. Examples are given wher e the interior smoothness of the data does not imply interior smoothness of the solutions. The topology of the space M of the multipliers is weaker th an the L-2 topology. In some cases it is even weaker than that of distribut ions (sensitivity phenomenon). As a consequence, the convergence of the pro blem in mixed formulation for thickness 2 epsilon as a tends to zero only h olds in very poor topologies, implying non-uniformity with respect to epsil on of the finite element mixed formulations.