Nonsmooth optimal design problems for the Kirchhoff plate with unilateral conditions

The paper is concerned with the optimal design problems for the fourth order variational inequalities. Namely, the first order necessary optimality conditions are derived for the class of optimization problems under consideration. The differential stability of metric projection in the Sobolev space HQ{Q) onto the cone of nonnegative elements is considered by Mignot [9]. Mignot derived the form of the so-called conical differential of the metric projection. However, the technique of proof used by Mignot is based on potential theory in Dirichlet spaces, therefore, his argument cannot be directly applied in the Sobolev space H*(Q). The differential stability of metric projection in the Sobolev space HQ{Q) onto the cone of nonnegative elements is investigated by Rao and Sokolowski [14]. In particular, in [14] the sufficient conditions are obtained under which the set K is polyhedric at a given point / € A'. The question of polyhedricity is adressed in [14] since it implies directional differentiability of the metric projection onto K with an explicit form of the differential [5,9], i.e., the so-called conical differential of the metric projection onto the cone of nonnegative elements. It follows, we refer the reader to [19] for the details, that the conical differential is given as a metric projection onto the intersection of a tangent cone with a supporting hyperplane.The paper is organized as follows. In Section 2 the necessary optimality conditions for an optimal design problem for the Kirchhoff plate with an obstacle are derived. In Section 3 the optimal design of an obstacle is considered.