Lagrange multipliers for evolution problems with constraints on the derivatives

We prove the existence of generalized Lagrange multipliers for a class of evolution problems for linear differential operators of different types subject to constraints on the derivatives. Those Lagrange multipliers and the respective solutions are stable for the vanishing of the coercive parameter and are naturally associated with evolution variational inequalities with time-dependent convex sets of gradient type. We apply these results to the sandpile problem, to superconductivity problems, to flows of thick fluids, to problems with the biharmonic operator, and to first order vector fields of subelliptic type.