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The existence of generalized Lagrange multipliers is proved for a class of evolution problems for linear differential operators of various 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. These results are applied 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.

Lagrange multipliers for evolution problems with constraints on the derivatives