Lagrange multipliers for evolution problems with constraints on the derivatives

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