Sensitivity of certain constrained systems and application to shell theory

Abstract We consider constrained variational problems in mixed formulation. Denoting by V and M the energy space and the space of the Lagrange multipliers, the resolvent is continuous from V′×M′ into V×M . In applications to PDE, it is said that the problem is Lagrange multiplier sensitive when M′ does not contain the space D of test functions of distributions. This amounts to some kind of instability as very small and smooth perturbations of the data imply that the solution goes out of V×M . We give a criterion of sensitivity and relations with penalty perturbation problems. We apply the abstract theory to thin elastic shells (in the case when the middle surface S is not geometrically rigid) and give several examples of sensitive and non-sensitive shells.