Multi-Valued Parabolic Variational Inequalities on Convex Sets

This chapter is devoted to multi-valued evolutionary variational inequalities of the abstract form and related systems under constraints given by closed convex sets K ⊂ L ^p (0, τ;V ), where u′=du/dt denotes the generalized derivative of u: (0, τ) → V in the sense of vector space-valued distributions (see Chapter 2 ). It should be noted that unlike in the stationary case, in the treatment of its evolutionary counterpart (5.1) an additional difficulty arises. This difficulty is due to the appearance of the indicator function I _K representing the constraint, so that no growth condition can be assumed on ∂I _K , and therefore, in general, no estimate of the time derivative du/dt in the dual space L ^p′ (0, τ; V ^∗ ) is available, which would be needed for proving existence of solutions.